[
    {
        "video_name": "SIU_9SMd5q0",
        "Q": "At 5:15, what is the difference btw the voltage of the battery and the voltage across the capacitor? I just can't seem to get the difference. An in-depth explanation with examples will be great. Thanks.",
        "A": "When they are hooked together, there is no difference."
    },
    {
        "video_name": "AxaWXWd2pw4",
        "Q": "4:43, will the bacteria still survive once all the oxygen is depleted? Since they require oxygen to perform cellular respiration, won't they also die off, meaning EVERYTHING is dead now in a dead zone?",
        "A": "Yes, the bacteria will die too. But I guess the big deal is, The oxygen is depleted by respiration of the bacteria, which causes aquatic plants and fish to die, which causes even more oxygen to be consumed, and so on. So there is this bad positive feedback going on. Yes. Everything dies. It is literally, dead zone"
    },
    {
        "video_name": "x34OTtDE5q8",
        "Q": "how does 1/k become k? @7:08",
        "A": "If k is just an arbitrary constant, then it makes no difference whether you call it k or 1/k."
    },
    {
        "video_name": "x34OTtDE5q8",
        "Q": "At 3:37, why does he divide both sides by 10 and not 20?",
        "A": "he divides succesively by 10 and 2"
    },
    {
        "video_name": "x34OTtDE5q8",
        "Q": "After viewing what you explained at 10:31 Can you work a problem or explain a case where the temp increases as the pressure decreases and the volume is constant?",
        "A": "That is given by him .you can have your own certain volume as well . If willing ,you can change it , and then the problem becomes more difficult."
    },
    {
        "video_name": "x34OTtDE5q8",
        "Q": "At 2:20, Sal said P1V1=P2V2 assuming no work was done, but doesn't squeezing the box count as work?",
        "A": "Practically squeezing the box counts as a work but we have to assume that no work has been done. Or it can be taken as the particles were shifted from a larger box to a smaller box without changing the kinetic energy."
    },
    {
        "video_name": "x34OTtDE5q8",
        "Q": "@6:20. Why is it KE=nKT instead of KE=nT/K.\nHere's my reasoning T=K*KE/n, then\nnT=K*KE\nnT/K=KE, right\nWhat am I doing wrong?",
        "A": "K is just a constant. Do your equation. make a new constant, call it M, and definite it as the reciprocal of your K, so M =1/K. Now you have KE = nMT, right? But M is still just some constant, we can call it whatever we want. So let s rename it to K. KE = nKT. Ultimately if you have a constant, you can call it whatever you want. If you want to call it K, you can. If you want to call it 1/K, you can, because if K is a constant then 1/K is also a constant."
    },
    {
        "video_name": "x34OTtDE5q8",
        "Q": "At 2:25, Sal said that the relationship between pressure and volume is constant unless work is done by the system. How does the amount of work done by the system affect this relationship, exactly?",
        "A": "PV = nrT. If PV is going to change, that means T has to change. To change T requires energy. Energy requires work."
    },
    {
        "video_name": "x34OTtDE5q8",
        "Q": "At 3:41 to about 4:01 Mr. Khan divided both sides of the equation by 10 but he didn't divide 500 by 10, but 20 and 50, how come?",
        "A": "I wondered the same the math works however."
    },
    {
        "video_name": "x34OTtDE5q8",
        "Q": "At 3:30, Sal said meter cube is the same as L.\nI am not sure, but isn't L the same as decimeter cube not meter cube?",
        "A": "You are correct in saying that 1 dm^3 = 1 L. I think what Sal was getting at was that the units of volume on both sides of the equation must be the same. He probably should have added the words after converting m^3 to L or something similar."
    },
    {
        "video_name": "x34OTtDE5q8",
        "Q": "in 6:57, how could he just switch 1/k to k like that?",
        "A": "If it is just an unknown constant, it does not matter whether you call it k or 1/k."
    },
    {
        "video_name": "x34OTtDE5q8",
        "Q": "At 1:07, Sal writes Pascal as N/m3, shouldn't it be N/m2?",
        "A": "Yes it should be N/m^2"
    },
    {
        "video_name": "x34OTtDE5q8",
        "Q": "i cant figure it out..why volume is increased, if you want more pressure...?? according to Boyle's law volume is inversely proportional to pressure... I Didn't GET IT FROM 9:34..?",
        "A": "haha...no need to answer,, i myself got it ... :D :D :) silly me!!"
    },
    {
        "video_name": "gSzxeL64Cn0",
        "Q": "In an intermediate step of forming an oxime: why doesn't the OH on the hydroxylamine form water and make a leaving group (as it already does with the double bonded O on the reactant)? approx @3:00",
        "A": "I think the N and the O are very similar in electronegativity, so the O can t pull out the electrons from the N. Hence, they stay stuck together."
    },
    {
        "video_name": "gSzxeL64Cn0",
        "Q": "at 7:10 why doesn't the other nitrogen of the hydrozine react with another molecule of cyclohexanone resulting in R=N-N=R (where R=cyclohexane)?",
        "A": "The lone pair on the NH\u00e2\u0082\u0082 group in the hydrazone R\u00e2\u0082\u0082C=N-NH\u00e2\u0082\u0082 is delocalized by resonance with the C=N bond, so it isn t as nucleophilic as the lone pair in H\u00e2\u0082\u0082N-NH\u00e2\u0082\u0082 itself."
    },
    {
        "video_name": "gSzxeL64Cn0",
        "Q": "At 6:00 he says that these are stereoisomers of oximes.Are they enantiomers or other kind of stereoisomers?",
        "A": "No, they ain t enantiomers because there is not a chiral Carbon."
    },
    {
        "video_name": "gSzxeL64Cn0",
        "Q": "At 3:37 why does oxygen all of a sudden give away its 2 electrons to nitrogen?",
        "A": "He is displaying the resonance stabilization, the electrons are actually delocalized throughout the molecule, but it is displayed as if the oxygen made a double bond with the nitrogen."
    },
    {
        "video_name": "hHteUIS0OFY",
        "Q": "8:45 Does that mean that convection only happens in the lithosphere and the asthenosphere?",
        "A": "Convection happens in the upper mantle (asthenosphere). When these currents are strong enough, the move the lithospheric plates above causing plate tectonics."
    },
    {
        "video_name": "hHteUIS0OFY",
        "Q": "At 2:31, Sal said that you could drill the oceanic crust and reach the mantle. If we ever did that, would the lava meet the water and cover the opening by solidifying or will any other type of change in the earth's surface or a natural disaster occur?",
        "A": "If we ever do that, we would die of the heat and pressure (magma can melt iron). And if we ever did that, the magma will mix with the water and from sulfur dioxide, which is poisonous and can form acid rain clouds. So basically, even if it sounds stupid, it will be natural disaster."
    },
    {
        "video_name": "CJzuu_k9Nv0",
        "Q": "At 0:20, if we start naming the compound from benzene ring, moving halfway and then towards right in the bottom carbon chain(which you actuay used) there would have been nine carbon i.e. nonane, will it be wrong?",
        "A": "It will be wrong. You first choose the longest continuous chain (nonane). Then you pick the highest priority group (OH) and number the chain from the end closest to that group."
    },
    {
        "video_name": "CJzuu_k9Nv0",
        "Q": "At 2:47 why isn't isn't it 3-octanol, like In the alcohol nomenclature video, instead of octan-3-ol?",
        "A": "3-Octanol is acceptable, but the IUPAC preference is to put the locant (the number) as close as possible to the group it locates."
    },
    {
        "video_name": "CJzuu_k9Nv0",
        "Q": "At 5:37, why is it not called 4-bromo-3-nitroaniline?",
        "A": "Because counting to keep numbers low."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "At 4:30 he says \"The cell doesn\u00c2\u00b4t want viruses.\" Isn\u00c2\u00b4t that partially false? There is a reason why those viruses exist, isn\u00c2\u00b4t there? Or do most scientists think nowadays that it\u00c2\u00b4s just a random joke of mother nature? There may be many situations in which viruses give a useful genetic code to the cell so it makes it better in some way, what do you think?",
        "A": "No, there is no reason why a virus would have to be useful to a cell in order for the virus to exist. THe virus will exist if it has a way to propagate. Whether that propagation is favorable or unfavorable to the host cell is not relevant."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "At 6:44, Sal says something about tRNA. What is tRNA?",
        "A": "tRNA stands for transcribed RNA."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "Wait , at 4:45, sal draws a cell that he later(5-10 sec) mentions that the othe two sides will merge, and the virus is consumed within the cell. What kind of cell did he draw, and what kind of cell does that",
        "A": "I think it might have been a white blood cell, which is what attacks problems it is alerted to within the body."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "in 19:05, what is that red thing on the white blood cell?",
        "A": "It s a red blood cell"
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "1:54 what s an icosehedran ?",
        "A": "It is a solid with twenty congruent faces which are all equilateral triangles."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "at 1:00 what is the taxonomy system",
        "A": "It s a system scientists use to classify and identify organisms. It s made up of different ranks from that go from least to most specific - domain, kingdom, phylum (division), class, order, family, genus, and species."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "so is virus life or not 23:17",
        "A": "no, a virus is not alive cause it cannot reproduce without a host cell to do everything for it."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "at 16:06 it states the virus can take D.N.A. from the host sell to another organism, does this speed up evelution",
        "A": "When a virus takes DNA from a host cell and then brings it to another organism, all it s really doing is spreading the DNA of the original organism it was in. It may also spread the disease or whatever it is that the virus originally had. Its basically like when someone in your family gets a cold, and then the virus spreads and you get it too."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "At 19:18, I saw this big red thing on the side of the white blood cell. What was it?",
        "A": "I m not positive, but I think that it is a platelet."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "Can I say that viruses are a very useful tool of nature because they make DNA \"jump\"? Sal talks about it at 16:00. So this process is only possible because of viruses and make communication between DNA of different species possible, is that right?",
        "A": "A virus can not naturally transfer DNA from one organism to another. When the cell is destroyed and the viruses come out, they carry no DNA from the cell, just viral DNA."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "What is SARS? At 21:31, he mentions \"SARS\". When I looked up what SARS is, Google simply told me that SARS is a respiratory illness caused by a coronavirus.... what even is a coronavirus?",
        "A": "A corona virus is any virus that can cause a variety of illnesses in an animal or human, in this case SARS (severe acute respiratory syndrome) is a virus that affects the respiratory system, just in case you were wondering the symptoms for this illness are flu like and can cause pneumonia."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "At 3:15 ,Sal said that virus seems like life when in comes in contact with something living.\nSo....we must be having , like a million \"dormant\" viruses around us , also maybe on our window sill , which have not yet come into contact with a living thing , right?",
        "A": "Our existence to some extent is caused by Viruses. Viruses are constantly swarming into our bodies. Sometimes they make us sick; sometimes our immune systems vanquish them; and sometimes they become a part of ourselves. A type of virus called a retrovirus makes copies of itself by inserting its genes into the DNA of a cell. The cell then uses those instructions to make the parts for new viruses. HIV makes a living this way, as do a number of viruses that can trigger cancer."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "Sal said that 5-8% of us is virus so what part of our body is that 5-8%?\nand if you don't get what i'm saying see for yourself -> 14:13",
        "A": "no particular part, just all over"
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "I thought HIV was a virus that attacked cells, mainly white blood cells. Is that true? I wonder because you said otherwise at about 10:00 ish, and from what i know, aids is what is after HIV, could you help me understand? thanks.",
        "A": "You are correct, although at 10:00 he doesn t really go into it. Later, more at the 20:00 mark, he does show a picture of HIV infecting a white blood cell. Keep in mind that AIDS is the syndrome caused by HIV after you have lost too many white blood cells to stay healthy."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "At around 20:30, in the first picture (the colored one), what is the mass of pink stuff?",
        "A": "The picture is HIV infecting a cell, so the other cell is a T cell. It s hard to say what it is specifically because we don t know where in the body the picture is taken, but my guess is the pink mass could be any cell in the blood."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "good video. From 19:00 to about 20:00, can you clarify what is the red stringy mass on the left side in the white blood cell picture?",
        "A": "I don t believe that the red mass is a virus - HIV viruses are fairly large for viruses, and if they are the green particles, there is not a known virus the size of the red mass. To me, it resembles a platelet - but I m not positive."
    },
    {
        "video_name": "0h5Jd7sgQWY",
        "Q": "8:30 What happens to all the other matter in the cell when the membrane dissolves?",
        "A": "Usually other cells take it in as food, while some of it is just expelled from the body as waste."
    },
    {
        "video_name": "0w9R_foNLrg",
        "Q": "at about 3:51 can you explain how it is possible to have a 4 dimensional sphere with a 3 dimensional surface? Is it like the space / time continuum from Star Trek. and if you go one direction and reach the end you go back where you started? like a doughnut or Torus?",
        "A": "A 4-d sphere would have to have a 3-d surface Just like a 3d sphere has a 2-d surface We can t picture a 4-d sphere but we know that if you were on one then if you traveled for a long enough time in the direction you thought was a straight line, you would end up back where you started."
    },
    {
        "video_name": "dclP0R2wf7g",
        "Q": "At 9:09, david says force of kinetic energy is equal to coefficient of kinetic energy multiplied by the normal force. But how? Can you please post a derivation or a link where he explains it?",
        "A": "This is a good question. I have been using this formula for quite some time and have never seen a derivation either. My best educated guess would be that it was experimentally determined, meaning physicists investigated the kinetic friction force and found that it was proportional to the normal force by some factor, and they named the factor the coefficient of kinetic friction."
    },
    {
        "video_name": "dclP0R2wf7g",
        "Q": "Why does David use the coefficient of kinetic friction at 9:04?And why is it miu?",
        "A": "watch sal s vids about friction"
    },
    {
        "video_name": "dclP0R2wf7g",
        "Q": "At 10:00 you say that the peguins of the same mass will have the same speed, but what about momentum? Isn't momentum dependent on mass?",
        "A": "Momentum is mass * velocity. So it depends on mass as well as velocity. So if they have the same mass and velocity, they have the same momentum. Coming to the question, he says that penguins with same speed will travel the same distance no matter what the mass of the penguin is."
    },
    {
        "video_name": "kCM2mSb4qIU",
        "Q": "At 4:24 How was jay certain that zn will lose 2 electrons?",
        "A": "+2 is the commonest oxidation state for Zn although other oxidation states are known. You have compounds such as ZnO, ZnCl2, Zn(NO3)2, Zn(OH)2, etc, where it has this oxidation state of +2. To get to an oxidation state of +2 requires a Zn atom to lose two electrons."
    },
    {
        "video_name": "kCM2mSb4qIU",
        "Q": "At 3:25, he starts to write down the configuration for Zn [Ar]. Why does he ignore the p-block section of the periodic table. Shouldn't Zn [Ar] be: 1s2, 2s2, 2p6, 3s2, 3p6, 3d12 , not Zn [Ar]: 4s2, 3d10 ?",
        "A": "There are only five d orbitals per energy level, which can hold a maximum of 10 electrons. So [Ar] 4s2 3d10 is the correct electron configuration for zinc. The [Ar] here just represents the electron configuration of argon which is 1s2 2s2 2p6 3s2 3p6, so the full electron configuration of zinc is 1s2 2s2 2p6 3s2 3p6 4s2 3d10."
    },
    {
        "video_name": "kCM2mSb4qIU",
        "Q": "When are you able to write the 4s shell before the 3d shell? Is this only the case with transition metals, or are their exceptions to the rule as seen in 2:34?",
        "A": "4s fills before 3d in most atoms. It s the norm, but there are exceptions."
    },
    {
        "video_name": "kCM2mSb4qIU",
        "Q": "in 3:19-3:29 they say zn is next to ar,? can someone explain",
        "A": "It s the short hand notation for writing electronic configuration where we reference to the nearest noble gas (in this case, Ar) and continue writing the rest of the electronic configuration from there. e.g. Mg is [Ne] 3s2 just search short hand notation electronic configuration on google and you ll find it"
    },
    {
        "video_name": "kCM2mSb4qIU",
        "Q": "At 3:42,I noticed you used a technique to calculate electrons in d-sub shell. Could you please tell me more about it?so i can use it onward.",
        "A": "All he was doing was counting along from Sc to Zn. Zn was the 10th in the row which means 10 d electrons. Note that these electrons go into the 3rd shell (3d) - the 4s subshell fills before the 3d. Likewise, 5s fills before 4d, etc."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 6:30 Sal said molecules bump into each other.on solids it's known as friction. In air drag. What's it called in water",
        "A": "It would also be drag in water, or fluid resistance . The term drag applies to fluid flow, which includes both air and water. The study of air flow would be aerodynamics, while the study of water flow would be hydrodynamics."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 1:30 Sal says that the rock will stay at rest but if a force is put upon that rock from all directions and equal force it will stay at rest like if it is mid air and equal force is put upon it from all directions will it stay in the air?",
        "A": "Yes, we call that picking up the rock."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 7:23 he said that in a world without gravity and air you can't stop anything but why in the space the things don't move in a line?",
        "A": "Gravity affects them, moving the thing into orbit around other objects, such as planets, stars, asteroids, etc.. Or these objects have a gravitational force that doesn t send the thing into orbit, but slightly alters their course, so they don t move in a straight line."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 23:00, according to newton's 1st law of motion object will be in the state of rest or in uniform motion in a straight line unless and until acted upon by any external unbalanced force. but motion is a relative term which depends on the observer or reference point so how does it became a law when it keeps changing depending on observer? It should be same in every aspect.",
        "A": "Any observer will see the object maintain its state of motion as that observer sees it."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "at 6:41 Salman says that the block of ice will bump into particles of air. Would aerodynamics elliminate the force of air?",
        "A": "Eliminate? No. Reduce? Yes."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "I thought that if something was thrown in a void, that that object would have a constant ACCELERATION,not velocity (0:38). My science teacher told me that if an asteroid was accelerating at 1 meters / seconds, it would continue to accelerate until some force stopped it. Is this true?",
        "A": "Acceleration is measured in meters / seconds*seconds, this is probably just a confusion between words. An asteroid can fly in constant acceleration only given strict condition: no gravity field and constant gas erruption from it, or a constant gravity field, which doesn t exist in that scales, only as aproximation."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "7:56\nThese laws then cannot be applied on earth where there is friction and gravity is acting upon every object. So then why do we still consider these laws?",
        "A": "They do apply. Friction and gravity are just additional forces to throw into the mix."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 7:18 Sal said the air particles will exert a pressure on the tennis ball as a reason it stops; then why doesn't it happen with light?",
        "A": "For something to be acted on by air resistance, it has to have mass. Photons, which light consists of, have no mass. They are simply little balls of energy, so they re not affected by air."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 5:12, Sal says everything comes to a stop. What about planets, or an asteroid?",
        "A": "Everything on earth comes to a stop, because of friction. If there s no friction, a moving object won t stop. Planets experience almost no friction, so they keep going."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 7:52 he says planets have a constant speed, is that the same for stars\nI know Halleys comet moves much faster when its next to the sun and much slower when at its aphelion",
        "A": "interesting quesiton. As you know, the planets do not really have constant speed. But if we assume circular motion (which is an OK assumption) then the motion will be circular. As to whether stars have constant speed, it depend very much what the system is like where they belong. Does out Sun have a constant speed? What is your opinion?"
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "at 2:50 he says that if two people push with the same force the rock won't move, but where does that energy that those two people go to because it is not getting converted into friction since there is no movement.... so where does that energy go to",
        "A": "It doesn t really go anywhere, the forces cancel each other out and the people just get worn out."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 1:32, what about the gravitation forces applied on the rock by other objects present in the system like trees, other rocks, etc. ?",
        "A": "Good question! As is the case for many physics problem, we try to simplify our explanation by neglecting the effects that you are listing. Indeed, the gravitational pull from other objects is so incredibly small that we can forget about it."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "at 7:54 he says that the speed never changes on planets is that true because I heard somewhere that the earth moves a bit faster every year?",
        "A": "Earth s rotation is slowing down ever so slightly. So slightly that the change is immeasurable over the time that human civilization has been paying close attention to the length of a day. Earth s rate of revolution around the sun does not change in any measurable way."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 3:00 Sal mentions that the forces on the rock cancel each other out. What if there are slight differences in the forces?",
        "A": "Then the rock would move"
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 8:45 when we were talking about the plane example......we said that if the plane was moving or at rest and we were inside it, we wont realize that whether the plane is movinr or is at rest.......However even if the plane was accelerating with we being inside it......wont we still feel that we are at rest? Isn't it that both our velocity and the acceleration w.r.t. plane zero?",
        "A": "If the plane is accelerating, you will feel yourself pushed backward into your seat. This is because the plane has to exert a force on you to get you to accelerate along with it. Put a ball in a wagon and give the wagon a pull. See the ball appear to roll toward the back of the wagon? That s how the ball knows the wagon is accelerating."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "6:47 Does this mean that objects will not stop in space? Because there is no resistance in the void and no gravity either?",
        "A": "yes, that s why the earth keeps going around the sun and the moon keeps going around the earth."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "Also, at 3:40 in the video, isn't the ice at equilibrium?? Because all of its forces are at balance?",
        "A": "I guess you could say that. That s one of the several definitions of equilibrium =)"
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "9:20 Can't you tell if the plane turns?",
        "A": "This certainly is a great question,but according to me if you won t even feel the plane going so how can you tell the plane would turn,but in second case if we think that the plane takes a very very sharp turn we may feel as we also may move slightly on our seats Hope this helps."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "With the Airplane example 8:39 wouldn't you feel that the airplane was moving if it was falling?",
        "A": "You would only feel that you have become weightless inside the airplane (assuming the airplane is in free fall). NASA simulates zero-g by putting astronauts on a plane that flies a parabolic path so that it is descending at a free fall rate for short periods of time. They call it the Vomit Comet."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "6:56\nLike you said there is going to be some amount of friction and the body in motion will eventually come to rest. So my question is why hasn't this law been changed or rephrased?",
        "A": "Friction is just another force. The law holds if you take the forces into account. There is nothing in the law that indicates you can ignore frictional forces."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "From 8:15 to 8:25, he says the ball would keep going on forever, so does that mean that if I were in space or vacuum and threw a ball it would go on forever?\nThe reference plane is also a little confusing to me. Because I'm in the plane and it is stationary from my point of view hence its difficult to tell whether the plane is moving, is that the concept?",
        "A": "yes that is what he meant because there is no gravity and so no force would be applied on the ball from the bottom.Thus it would go on forever ball stops because of frictional gravity but in vaccum where there is no gravity there is no force being applied on the ball.So it would never come down nad the second doubt of your s is not clear could you please explain it to me a bit more clearly? :)"
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "The ice will melt at 3:25 ... If you press on an ice cube on opposite sides, not only will it stay at rest, but it will melt. Where is the science behind that? Chemistry?",
        "A": "Water is one of the few liquids which expands when it freezes. When you apply pressure to ice, the melting point is lowered because the ice wants to revert back into a liquid in order to decrease its volume from the pressure."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 4:25, he mentions that an object will continue to move at a constant velocity unless it is acted upon by an external force. Won't an object only 'continue' to move like that if the surface it is moving on is frictionless? Otherwise the object would stop moving, wouldn't it?",
        "A": "Friction is caused by an external force pushing one object against another. I don t think a surface can really be frictionless, because friction is not really a property of an object. If an object really has no external force acting upon it, it will not be effected by friction and will continue at a constant velocity"
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 23:00, Sal said about the lattice structure of the water molecules,I didn't understood what it meant? Please explain",
        "A": "He s describing the bonds that hold the water in a solid shape and structure. As these lattice structures rub together, there will be friction between the pieces of ice. We perceive ice to be very slippery. That s because we usually experience it as a solid with a thin coat of liquid water. Liquid water experiences much less friction than solid water. If the temperature is cold enough to prevent our body heat and the pressure of our weight from melting the surface, ice is much less slippery."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 0:27, what does it mean by force impressed?",
        "A": "Being acted on by a force"
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 8:45, when Sal says that he'd have no idea if he was on a moving plane or a stationary one, is he still talking about Newton's first law or Einstein equivalence principle? Is Newton's first law analogous to it? Is it a direct consequence? Help!",
        "A": "newton I he is saying if there is no acceleration, i dont feel a force"
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 7:57 the comment corrects Sal saying he meant the objects never come to a stop rather than what he actually said which was they never slow down. Why is it incorrect to say they never slow down? (if there are no unbalanced forces to act on them in space?)",
        "A": "Because objects do slow down if they are not in a perfectly symmetrical orbit. (speeding up the closer you are, slowing down the further away). The correction is to distinct from slowing down vs stopping completely."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 6:27, when you gave the example of the ice cube moving on ice, you said that the molecules would bump and grind into each other, producing a little bit of heat. But then, wouldn't the heat cause the ice to melt, further lubricating the surface? The friction would have decreased then and it would continue moving, right?",
        "A": "Even if the ice melted enough to reduce the coefficient of friction, there is still friction."
    },
    {
        "video_name": "CQYELiTtUs8",
        "Q": "At 6:50-7:00, you said the ball would fall to the ground because of gravity but it would eventually stop because of friction. But, what if it was falling down an infinite hill wouldn\u00e2\u0080\u0099t it gain speed no matter the amount of friction because it would be going downhill",
        "A": "Yes, in the absence of air resistance, a ball falling down an infinite hill would gain speed forever, according to Newton s laws. There is of course no such thing as an infinite hill but we can instead imagine a continuous accelerating force, which gives the same effect as an infinite hill). (We know now that the ball cannot exceed the speed of light, according to relativity, but we don t worry about that when we are working with Newtonian physics)"
    },
    {
        "video_name": "rC165FcI4Yg",
        "Q": "@0:43, can someone explain to me why the N is positively charged? I mean if we count the electrons, we notice that the N has completed its octet, so why the positive charge? thank you.",
        "A": "Yes, the N atom has an octet. But the rules for calculating formal charge say that the N atom owns only half of the shared electrons. Since it owns only 4 of the 8 shared electrons, it has officially lost one electron compared to what it has as an isolated atom. Its formal charge is therefore +1."
    },
    {
        "video_name": "rC165FcI4Yg",
        "Q": "@6:10 Jay is using the H2O molecule as a base instead of the HSO4-. Although the pKa of H3O+ is higher than that of H2SO4, would or wouldn't you recommend using HSO4-, as the catalyst is then showed to be \"restored\"? Or perhaps would you recommend to draw another reaction with a equilibrium to the left using H30+ and HSO4- to restore H2SO4? Is it generally considered to be ok (at exams) if you leave the recovery of the catalyst out?",
        "A": "H\u00e2\u0082\u0082O is a stronger base than HSO\u00e2\u0082\u0084\u00e2\u0081\u00bb, but either one can be used as the base. Many prof/teachers don t care if you don t show the regeneration of the catalyst, but many do care. Check with your own teacher to make sure of the policy. In this particular reaction, the H\u00e2\u0082\u0082SO\u00e2\u0082\u0084 is not regenerated, because it is a strong acid and reacts completely with the water to form H\u00e2\u0082\u0083O\u00e2\u0081\u00ba and HSO\u00e2\u0082\u0084\u00e2\u0081\u00bb."
    },
    {
        "video_name": "rC165FcI4Yg",
        "Q": "At 0:50, why doesn't the proton from H2SO4 get picked up by the oxygen with a -1 formal charge? Won't that happen instead, knowing that H+ will be more attracted to a greater negative charge?",
        "A": "It does. It happens much more frequently than attack on the other oxygen. But all that does is regenerate the starting nitric acid starting material. It gets us nowhere. It is the few that get picked up by the other oxygen that lead eventually to the generation of the nitronium ion and nitrobenzene."
    },
    {
        "video_name": "rC165FcI4Yg",
        "Q": "at 0:50, why does the oxygen in the hydroxyl group share electrons with the hydrogen, and not the oxygen on the left that has a lone pair of electrons?",
        "A": "Both protonated intermediates have a +1 formal charge. The \u00e2\u0080\u0093O- would indeed be more likely to pick up the proton, but then what? By which I mean that that is reversible, and either water or the conjugate base HSO4- would then deprotonate the H2NO3+, regenerating nitric acid. On the other hand, if the \u00e2\u0080\u0093OH is protonated, and it leaves as H2O, that reaction is less easily reversible, and the resulting nitronium ion has a chance to be attacked by benzene s \u00cf\u0080-bond."
    },
    {
        "video_name": "rC165FcI4Yg",
        "Q": "at 0:53, Why does the oxygen with no formal charge take the proton rather than the oxygen with a -1 formal charge? Is this just a trick to remember or can someone please further explain why this happens?",
        "A": "The oxygen with the -1 formal charge is more strongly attracted to the + Nitrogen than the OH oxygen. Forming a diol is not favorable and the formation of a great leaving group in H2O leads to the protonation of the OH group instead of the -1 Oxygen. Protonation of the -1 Oxygen does happen but to keep neutrality the Hydrogen in the OH group is ejected and you end up with the starting product again."
    },
    {
        "video_name": "rC165FcI4Yg",
        "Q": "Why there is only one electron in the lone pair of O when Jay drew the structure @1:17?\nThere should be 2 electrons nah??",
        "A": "If you mean the top oxygen double bonded to the nitrogen, then yes. There s a missing electron."
    },
    {
        "video_name": "rC165FcI4Yg",
        "Q": "At 5:00, Jay draws in the formal charge on the nitrogen of the first resonance structure. What happens to this +1 formal charge? He doesn't show what happens and just names it as a nitro group.",
        "A": "The + charge on N shows that N is the atom that will be attacked by the \u00cf\u0080 electrons of the benzene ring."
    },
    {
        "video_name": "TX7-Kdn6lJQ",
        "Q": "8:35, cleavages form during meiosis in animal cells. Plant cells will create cell plates and not create cleavage, right?",
        "A": "A cleavage furrow forms when an animal cell begins cytokenesis, while plant cells grow a cell plate that will eventually become the cell wall in the places where the cell split."
    },
    {
        "video_name": "TX7-Kdn6lJQ",
        "Q": "At 2:30 in the chromosome picture the 23rd chromosomes are the same size, so does that mean that the person whose chromosomes these are is a girl?",
        "A": "Yup youre right Erin"
    },
    {
        "video_name": "uVWquWFjnCw",
        "Q": "should it be Li2+ (2:21)",
        "A": "No, it should be Li+, because Lithium is a Alkali metal and Alkali metals are in Group 1. Metals in group 1 give up 1 electron making its charge +. I hope that answers your question."
    },
    {
        "video_name": "uVWquWFjnCw",
        "Q": "At 5:50, you wrote that there was an increase in ionization energy due to the electron shielding that was occurring; however, at 5:13 you said that there was not much electron shielding occurring within the first shell between the two electrons. I need a little bit of clarification on whether the increase of ionization energy was really suppose to be due to the nuclear charge on that one magenta electron within the first shell or whether I am just mis-understanding electron shielding?",
        "A": "This is not so. In the case of F and every atom. Every time with every successive electron taken away the Ionization energy goes up!"
    },
    {
        "video_name": "uVWquWFjnCw",
        "Q": "At 4:48 he mentions that you get a nuclear charge of 1.3 when you do the \"more complicated calculation\". I realized he hasn't been showing us how to do this, anyone want to explain how he is getting a nuclear charge of 1.3?",
        "A": "For that, we actually need to know the force and the angle at which it is exerted by the electrons in last shell and then the resultant of all the vectors of repulsion. We need to subtract this resultant from the charge of nucleus(atomic number)."
    },
    {
        "video_name": "XMkUDyl1ZRo",
        "Q": "at 5:11, Sal says \"it switches the direction of the current\".\ni know that change in magnetic poles occurs when the coil flips more than 90 degree which leads to reverse direction of current.\nBUT WHY ?",
        "A": "Because charges (eg electrons in a wire) experience a force when they move through a magnetic, and the direction of that force depends on the direction they are moving."
    },
    {
        "video_name": "XMkUDyl1ZRo",
        "Q": "Why Right Hand Rule @ 4:30 ??",
        "A": "yeah..but i thought the right hand rule was for induced current cases"
    },
    {
        "video_name": "6GB_kcdVMQo",
        "Q": "At 2:30 he's talking about sound waves. Does he mean that when the air molecules create areas of high and low pressure it creates sound waves? Or is there something I missed?",
        "A": "Yes, sound waves propagate by oscillating air molecules, which create areas of high and low pressure as a result of the oscillations."
    },
    {
        "video_name": "COZqJb-HT-k",
        "Q": "At about 6:30, when referring to the carbon that is drawn in navy blue, why is this charge considered localized and why are there no resonance structures for it? Thanks in advance!",
        "A": "You can t draw resonance structures for the structure with the carbon in blue, because the adjacent carbon atoms already have four bonds and a complete octet, so the pair of electrons are localized. That pair of electrons cannot be moved to form a double bond with the carbon in blue and one of the adjacent carbons, because any of these carbons would get five bonds, which cannot be."
    },
    {
        "video_name": "COZqJb-HT-k",
        "Q": "At 4:20, why does a carbon have -1 formal charge ? It is bonded to 2 hydrogen atoms and 2 carbon atoms so it's sharing 4 electrons, and 4-4=0. I don't get it, please help. :)",
        "A": "It has 5 electrons: 1 from the H, 1 from each of the C atoms on either side of it, and the two lone pair electrons. 4 - 5 = -1"
    },
    {
        "video_name": "COZqJb-HT-k",
        "Q": "Regarding the Cyclopentanone around 6:00, how are the chances of it going through Keto-Enol tautomery?",
        "A": "Keto-enlol tautomerization is actually a static process. In most cases, you will not see the enol-structure, cuz it is unstable. The product, in most cases, favors the keto-structure. So to me it is strange to ask about the chances , instead you make ask about the portion."
    },
    {
        "video_name": "mlrOJgyPySw",
        "Q": "At 4:02, Carole was mentioning a way of remembering things - She was saying A Banana with an orange hat juggling blueberries. This is a great example, but are there any more?",
        "A": "That is a Mnemonic Device-Imagery, the best examples are ones that connect you to things that are familiar for you."
    },
    {
        "video_name": "sPz5RrFus1Q",
        "Q": "At 7:00, from C to D work was done TO the system. Why did Sal say Work was done BY the system.\nWork was done by the system from A to B, not from C to D.\nIs this an error?\nAlso why is Q2 negative.",
        "A": "From C to D, work is done TO the system. That s the same as negative work being done ON the system."
    },
    {
        "video_name": "sPz5RrFus1Q",
        "Q": "At 7:06, when Sal defines Q2 as the integral of P*deltaV, why isn't the P different from the P of Q1? Because it is very clear from the graph that the pressure at C and B are very different.",
        "A": "Good eye - P does vary throughout every process. It varies with V, so P is a function of V. This is why Sal s integral does NOT call P a constant and bring it out of the integral. Instead, he uses the ideal gas law to show that P varies according to 1/V (nRT/V) during the isothermal sections when T is constant, and plugs that into the integral, and gets the natural log of V as a result. Make sense? Then he plugs in the V values at A,B,C,D which as you pointed out are very different."
    },
    {
        "video_name": "sPz5RrFus1Q",
        "Q": "Around 1:05 Sal says that in the second isothermic process (from C to D) you take away less heat than you needed to add in the first isothermic process (from A to B). Can anyone please help clarify why this is the case and why the heat added and taken are not the same?",
        "A": "The second isotherm is at a lower temperature and pressure, so it takes less work to compress it, and therefore less heat has to leave the system after the compression to keep the system at the same temperature (on that same lower isotherm)."
    },
    {
        "video_name": "sPz5RrFus1Q",
        "Q": "@4:55 Sal says \"we'll see that Qsub2/Tsub2 is a negative value\" which makes sense, but we never \"see\" that. We get to delta S =zero by ln1, not by demonstrating that the above represents a negative value which it must in order to get to zero. Please explain.\nthx",
        "A": "Note that 0 = \u00ce\u0094s = Q1 / T1 + Q2 / T2 \u00e2\u0087\u0092 Q1 / T1 = \u00e2\u0088\u0092Q2 / T2. Does this make sense?"
    },
    {
        "video_name": "sPz5RrFus1Q",
        "Q": "At 13:17 Sal multiplies the numerator and denominator by Vsub-b/Vsub-d.\nIs this a valid operation and why?",
        "A": "Yes, it s valid. Note that you can multiply anything by 1 without changing it (1x = x), and also that dividing something by itself gives 1 (y/y = 1). Combining these, we have that multiplying anything by something over itself does not change the original value (x * (y/y) = x). This is the same as multiplying the numerator and denominator by any nonzero value, because x * (y/y) = (x * y) / y."
    },
    {
        "video_name": "sPz5RrFus1Q",
        "Q": "why does sal says at 1:19 that the process from C to D, took out less heat than the added in the process from A to B????",
        "A": "This follows from the previous video, but the argument goes like this. In the complete cycle the net change of the internal energy is zero. But delU = Q - W, so 0 = Q - W, or Q = W, where W, the work done in the cycle and represented by the area of the PV diagram inside the cycle, is positive. So Q = W > 0. Now Q = Q(AB) + Q(CD) where Q(AB) > 0 and Q(CD) < 0. Therefore, Q(AB) + Q(CD) = W > 0, or Q(AB) > - Q(CD), which is the questioned claim."
    },
    {
        "video_name": "9q-T8o1HUcw",
        "Q": "At 8:44, I do not understand how you find the direction the current flows. IF you do the right-hand rule for both of the possibilities, don't you get the same result. With each current, at one point, it creates magnetic field in the direction of magnetic flux and opposite direction of magnetic flux. If someone could explain this, it would be great. Thanks!",
        "A": "This may be a point of confusion for others too! With each current, try the right hand grip rule, only with your fist open and your palm in the direction of the field. You will find that you can trace the loop with your thumb, all the while your palm in the same direction. This means the current will also move in the same direction throughout. Can t understand? Search Right Hand Palm Rule ."
    },
    {
        "video_name": "8bCL8TQZFKo",
        "Q": "At 11:36, Sal states that the enthalpy for the reaction is -74.8. When I calculated it, it came out to be -74.5. Please explain",
        "A": "Rounding errors most likely. Some people like to use intermediatary values for further calculations others prefer to use exact value."
    },
    {
        "video_name": "8bCL8TQZFKo",
        "Q": "At 2:23 when Sal flips the arrow so that CH4 is now a product, is there a specific rule that allows this?",
        "A": "No there is not a specific rule to do this. this is the requirement of the solution to flip the arrow so that we can get the CH4 on the product side where we need it."
    },
    {
        "video_name": "aOSlXuDO4UU",
        "Q": "At 9:54 Sal says \" If the system does work it looses energy\" , but shouldnt it gain energy according to the equation , we get that 10J of heat is ADDED to the system",
        "A": "If the system does work on something else, it gave energy to something else, and the principle of conservation of energy tells us the system must have less energy than it had before. If you add thermal energy to the system you can replace the energy it lost by doing work."
    },
    {
        "video_name": "aOSlXuDO4UU",
        "Q": "At 2:48, what is the meaning of \"inherent macrostate of the system\" ?",
        "A": "Inherent macrostate of the system just means The state of the system at the time of measurement in this case."
    },
    {
        "video_name": "aOSlXuDO4UU",
        "Q": "07:40 - Technically , If a balloon is expanding isn't work BEING DONE ON IT rather than it doing work ?? i mean gas molecules cant just automatically expand without an external force right ?? (e.g. I blow the balloon ) unless ur considering the gas molecules as a separate entity/system from the balloon wherein the gas molecules do work ... are you ??",
        "A": "The balloon expands because the gas molecules hit the inside of the balloon and push on it. That work is being done BY the gas molecules ON the balloon. Your intuition is that this can t happen all by itself, and you are right that a balloon in equilibrium wont just suddenly start expanding. But if heat is added to the balloon, the gas molecules would become more energetic and do work that expands the balloon."
    },
    {
        "video_name": "aOSlXuDO4UU",
        "Q": "At 5:55 why sal is comfortable with \u00e2\u0088\u0086E=Q+W and avoids using \u00e2\u0088\u0086E=\u00e2\u0088\u0086Q+\u00e2\u0088\u0086W?\ncan you explain it ?",
        "A": "That is because heat and work actually alredy represent a certain change! You might know that work is defined as an integral, thus represents a change (you can easily look this up). Same goes for heat, which has as unit the Joule as well. So when you say Delta Q, you re referring to the change of something that designates change, and that isn t what we re trying to say here."
    },
    {
        "video_name": "qMaVLzWnS80",
        "Q": "10:27 shouldn't the blood pressure increase, due to the increase of resistance, caused by the the decraese of blood vessel diameter?",
        "A": "The pressure should rise at the site of narrowing but past the vasculitis the blockage reduces blood pressure in the rest of the vessel past that site."
    },
    {
        "video_name": "dNp7vErqlaA",
        "Q": "7:56 (In video).\nSo does the pair of Homologous Chromosomes only code for one thing such as hair color or can they code for multiple things? If so what?",
        "A": "A pair of homologous chromosomes consists of one maternal chromosome and one paternal chromosome. Each homologous pair has copies of the same genes at the same loci. Since genes are the functional unit responsible for coding a characteristic like hair colour, chromosomes can have many genes that code for different things (any phenotypic characteristic you can think of). Some traits or characteristics are coded by multiple genes."
    },
    {
        "video_name": "dNp7vErqlaA",
        "Q": "At 4:28\nIs it necessary that m0ther wiII have X?\nIf there is an XY c0mbinati0n what w0uId be the gender?\nCant it be femaIe 0r is the X recessive f0r femaIe ?",
        "A": "Females are XX, so yes, it is necessary for a normal, ferrite woman to have an X (two in fact!). XY combo is male. X chromosome from the mother, and Y from the father."
    },
    {
        "video_name": "dNp7vErqlaA",
        "Q": "On 3:00 Sal calls the ovum a sperm cell. Isn't the sperm cell the \"tadpole\"? Also when is a cell called a gamete. I know the ovum is a gamete but I'm not exactly sure about the sperm cell. For male gamete is the \"tadpole\" a gamete or the unification of gamete and ovum?",
        "A": "The unification of sperm and ovum is not a gamete, it is just a zygote, but the sperm or the ovum individually is a gamete."
    },
    {
        "video_name": "dNp7vErqlaA",
        "Q": "In the beginning, around 1:17, you say that the sperm that reaches the ovum is a \"winner\" because he won the race, but isn't it true that many other sperm got to the egg as well but had to deteriorate the zona pellucida, and the sperm that was lucky enough to find a spot in the zona pellucida that had already been deteriorated \"wins\"?",
        "A": "yes, I believe it s the first sperm to get through the zona pellucida that wins"
    },
    {
        "video_name": "dNp7vErqlaA",
        "Q": "In the video at about 7:11 Sal mentions the 23 homologous chromosome pairs that are present in the zygote's nucleus. He gives the example that on one chromosome pair each chromosome codes for the same things (proteins) and he uses hair color as an example. Do both traits get expressed in the offspring or does the sex of the offspring determine which genes are expressed?",
        "A": "If you re asking if, for example, the offspring is male, all traits from the father are dominant and all traits are from the mother are recessive, then no. Offspring are very unlikely to get every trait from one parent and no traits from the other."
    },
    {
        "video_name": "dNp7vErqlaA",
        "Q": "at 4:30 he talks about the pair of sex defining chromosomes, so if it's XX you are female, and if it's XY you are male. but can it be YY?",
        "A": "No. While there are disorders that cause unusual numbers of sex chromosomes, the X chromosome has numerous genes on it that are not found on the Y chromosome but which are essential to forming a viable body during gestation. Thus in the extremely unusual situation that a zygote had a YY, then it could not form a functional body and the embryo would die during gestation, probably very early on."
    },
    {
        "video_name": "NnlAI4ZiUrQ",
        "Q": "At 5:22 what did Sal meant when he said \" The cross product cares about the vectors that are perpendicular to each other\" ? Please help!",
        "A": "Sal has videos about cross product, which is a way to multiply vectors. It s result is maximized when the vectors you are multiplying are perpendicular to one another. It is zero when they are parallel."
    },
    {
        "video_name": "NnlAI4ZiUrQ",
        "Q": "At 3:04 I wondered what is the precise equation for the field lines? And are they discrete, as in quantized?",
        "A": "I can t remember the equation by head, but you can find an approximate value by considering a magnet as two monopoles at a small distance apart... Whenever you work with fields, you assume a continuity to it, but they can be considered as the manifestation of virtual particles (thus quantum-like) that carry the information hey, there s a field there!"
    },
    {
        "video_name": "NnlAI4ZiUrQ",
        "Q": "I don't understand what Sal means when he says \"monopole\" at 0:45; anyone care to explain simply?",
        "A": "Think of an electron or a proton, these are single points of electric charge which would make them electric monopoles. With magnets we have never found a single point charge of magnitism so you alwasy have a North and South on every magnet which is a dipole."
    },
    {
        "video_name": "NnlAI4ZiUrQ",
        "Q": "At 5:20, Sal says that the cross product only cares about vectors that are perpendicular to each other. How is that so? In the video where he introduced cross products earlier, he used vectors that had an angle that had less than 90 degrees between them, so they weren't perpendicular, so cross products do work between 2 vectors that aren't perpendicular. I'm confused! (Of course I understand that no result is obtained if they are parallel as sin(0) would just be 0).",
        "A": "Thanks :) Makes logical sense to me but just hoping you re right, it didn t seem that way when Sal explained it in earlier videos but maybe he felt he didn t need to, thanks again all the same!"
    },
    {
        "video_name": "NnlAI4ZiUrQ",
        "Q": "at 1:00 Sal atarts to draw some lines , what are these lines called ?\nWhat is the distance between each of these lines ?",
        "A": "These are called field lines. They always point from North to South. There is no specified distance between the lines, but the closer together they are (the more dense the lines are), the stronger the field at that point - and vice versa."
    },
    {
        "video_name": "NnlAI4ZiUrQ",
        "Q": "At 7:45, Is Webber the SI unit or Tesla?",
        "A": "SI Unit of magnetic field is Tesla which is also equal to Weber/m^2"
    },
    {
        "video_name": "NnlAI4ZiUrQ",
        "Q": "5:51 why does a magnetic field have to be perpendicular to exert a force? why wouldn't it accelerate a charge parallel to it?",
        "A": "This was determined by experiment and how the math worked. There is no consistent way of coming up with a mathematical definition of the magnetic field without having the force on a moving particle to be perpendicular to the direction of motion and the field."
    },
    {
        "video_name": "NnlAI4ZiUrQ",
        "Q": "At about 5:31 he says if the velocity is perpendicular to the magnetic field then we will get a number.So my question is that if its not perpendicular to the magnetic field then what? And can anyone help me what he says at 5:50 im really confused\nThank You",
        "A": "If the velocity is not perpendicular, then you have to use some trigonometry to figure out what component of the velocity is perpendicular. If the velocity is parallel, then the answer is 0."
    },
    {
        "video_name": "4SQDybFjhRE",
        "Q": "I'm having trouble with an initial concept Sal makes around 3:05 - Why does AC = 9.81m/s^2? The 9.81 is from the force to due gravity, not centripetal acceleration. The 9.81 is in effect on the car at all points around the circle, not just at the top.",
        "A": "Only at the top of the loop is the pull of gravity maximally unopposed by the normal force that the track applies to the car."
    },
    {
        "video_name": "4SQDybFjhRE",
        "Q": "At 5:26, Sal says something about the traction of the road. What does he mean by traction?",
        "A": "Traction - maximum frictional force that can be produced by two surfaces without slipping"
    },
    {
        "video_name": "4SQDybFjhRE",
        "Q": "At 1:51 how did we get acceleration as 9.81 m/s^2?",
        "A": "That is the acceleration due to gravity, it is usually a given in any problem involving gravity on earth. A = G * m/r^2 = ((5.9736 * 10^24 kg) * ((6.67398 * 10^-11 (m^3)) / (kg (s^2)))) / ((6.371 * 10^6 m)^2) = 9.82213342 m / s2"
    },
    {
        "video_name": "yg0XJWHPqOA",
        "Q": "at 4:19 what is formal charge",
        "A": "Charge present on the individual atom in a molecule"
    },
    {
        "video_name": "yg0XJWHPqOA",
        "Q": "At 3:00, there are 3 atomic orbitals, which means there are 3 atomic orbitals.\nIf this is the case, why don't you fill the bonding molecular orbitals (with 2 electrons) AND assign 1 electron in the anti bonding orbitals in the Frost diagram? (which would give you a non positive integer)",
        "A": "The cyclopropenyl carbocation has only two \u00cf\u0080 electrons. There is no third electron to put in an antibonding orbital."
    },
    {
        "video_name": "yg0XJWHPqOA",
        "Q": "At 6:17 it says that it is just 4n, not 4n+2 Why? How do we know when is it 4n+2 or just simply 4n?",
        "A": "To be aromatic it has to have 4n+2 pi electrons. Cyclobutadiene does not have 4n+2 pi electrons and so it does not meet the criteria for aromaticity. This is explained in the video."
    },
    {
        "video_name": "yg0XJWHPqOA",
        "Q": "How come \"n\" is equal to zero in a Huckel's rule ? at 3:05, he says that we have 0 pi electrons in the anti bonding MO and that's why 4n = 4*0 =0, but in the previous video on stability of benzene ring we also have 0 pi electrons in the anti bonding MO but instead of putting n as a zero he put n as one. Why is that? What does n represent in the Huckel's rule ? Thank you.",
        "A": "I think he should have paused in between these bits. Huckel s rule gives us the number of pi electrons that mean a ring could be aromatic, the solutions to Huckel s rule are 2, 6, 10, 14, etc. It has nothing to do with the number of electrons in only the antibonding MOs, just the total number of pi electrons. n is simply an integer value starting from 0, eg 0,1,2,3"
    },
    {
        "video_name": "yg0XJWHPqOA",
        "Q": "@1:56 why does he draw the triangle like that and not turned? I mean why not 2 mo in the bonding part and 1 in the anti-bong part?",
        "A": "because while drawing the polygon in the frost circle u always start from a point at the bottom of the circle.Similarly he drew the square of cyclobutadiene in a tilted manner so that one of its vertices touches the circle at its bottom."
    },
    {
        "video_name": "yg0XJWHPqOA",
        "Q": "At 2:22 why did he draw the triangle inverted in the frost circle? pls answer...",
        "A": "According to the rule ,one of its vertices should point downwards . Therefore its drawn inverted such that one of its vertices is at the base of the circle."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 6:23, what is a torroid?",
        "A": "Basically the shape of a donut or bagel is a torrid."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 2:09 is space expanding all the time?",
        "A": "Space will always continue to expand because the initial explosion from the Big Bang (theoretically) gave it momentum ."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 6:14 he says that we know there is a slight curvature to space. I thought this topic was still up for debate and there have been tests indicating it is actually flat.",
        "A": "You are correct. Sal s video is a bit out of date. Our most recent observations indicate that the universe is almost certainly flat, to within about +/- 0.5%."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 0:16, if all the 'space' and 'matter' we know today was in that speck, what was this speck sitting in prior to the Big Bang? Nothing?",
        "A": "We don t know anything about the universe before the big bang. It s not even clear that there was a before the big bang. Time itself may not have existed until the big bang."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "4:49 a sphere is a 3D object if it is 2D it would also be 4D and the 4D is time 5D is a tesser; tesser is the act of a tesseract which is going to one different point in an infinite small time",
        "A": "infinite 3d\u00e2\u0080\u00a6 look at my answer to cameron.norton under Sweep N Clear s question"
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 5:03, Sal mentioned having a 'profound' brain might enable you to see in more than 3 dimensions. Is this actually possible for a human being? Might evolution ever get us there?\nThanks!",
        "A": "It s possible, but improbable. More likely scientists will develop technology for seeing more dimensions."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "Wait, I thought at around 4:17, that the universe is constantly expanding. If it is constantly expanding, and assuming it always will, how is the area still finite?",
        "A": "When you expand a finite area by a finite amount it never becomes infinite."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 6:20 sal mentions a toroide i am sure of misspelling it but what is it?",
        "A": "Think of a donut."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 2:22, Sal talks about a two-dimensional space. Isn't a sphere three-dimensional?",
        "A": "The surface of a sphere is 2d. The sphere itself is 3d."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 5:50, Sal said there is a four dimensional sphere , that means there is four dimensions, what is the fourth dimension?",
        "A": "It is simply a fourth spatial dimension. As we only intuit three spatial dimensions in our everyday life, it is difficult to describe a fourth spatial dimension aside from pointing to analogy. So, as a two dimensional surface of a sphere is to the three dimensional sphere, our current three dimensional universe can be thought of as the surface of a four dimensional hypersphere."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "at 1:40 sal starts talking about what the matter is expanding into and the edge of the universe. how is it possible that we're not expanding into something else? doesn't everything need to be within something larger?",
        "A": "No, it doesn t. If the universe is inside something larger, then why isn t that thing part of the universe? THe universe is everything. What s happening is that space itself is expanding, everywhere. It s not that there s some outer boundary that s moving farther and farther into some new territory."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "I'm slightly confused at 2:18 . He says he has a 2-dimensional space with a finite area and no edges, and he draws a sphere. If it's a sphere, you can keep walking in one direction and come back to where you started, hence having no edges, but it's not a 2D space if it's a sphere. Thanks for your help.",
        "A": "For something to be N dimensional means that any point that is part of that object requires N coordinates identify its location. Even through a sphere occupies 3 dimensions any point on its surface can be specified by 2 coordinates therefore it is 2 dimensional."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 00:46 Sal said that all the matter exploded and started to condense? Why after explosion matter started condense?",
        "A": "As the universe expanded, it got cooler. As it got cooler, matter could condense. Sort of like steam becomes water as it cools. Not exactly the same, but that s the idea"
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "When a 2-dimensional space is mentioned in 2:20, I'm confused. Isn't a 2-dimensional space impossible?",
        "A": "I understand some of what you re saying, but those things are 3-dimensional, so it would make sense that they form very flat 3-dimensional disks."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 10:50, aren't some galaxies travelling towards other galaxies?",
        "A": "yes but at the same time some galaxies are actually spreading apart."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "at 0:15 Sal said that the universe was infinitely small but if Einstein made the equation E=mc^2 then if practically had no mass how did it get energy to explode?",
        "A": "You are mistaken in assuming that because the universe was small it no mass/energy. It had enormous mass/energy."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 10:41 it says the universe is expanding everywhere. HOW IS THE POSSIBLE?",
        "A": "Why do you think it is not possible? The expansion and contraction of space is part of what Einsteins theory of Relativity describes."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "0:09 Why is the Big Bang called the Big Bang? The name seems a little odd to me.",
        "A": "Georges Lemaitre called the theory big bang as a joke when it was first proposed because he disliked it."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 9:47 sal said more dimensions. Can their be any more dimensions than the 3 we know?",
        "A": "Basically in String Theory, M Theory and even supersymmetric quantum theories have dimensions in addition to the usual 3 spacial and 1 time dimension. The extra dimensions are either to small to detect or not accessible."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 1:33 if space itself was expanding, is it nothingness its expanding into (not even space, just the absence of matter) or is it just growing and there is nothing else around it, or nothing else had ever been there before? (I know this is a confusing question, sorry.)",
        "A": "At 1:33 if space was expanding that means that there is more space. Space is not nothing. It s space as in a continuous area or expanse that is free, available, or unoccupied. .That is why space is called space.Sort of like continuous free area."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 6:24, Sal mentions something called a toroid, what is that? Also when speaking of the 4th dimension, what does it look like? Theoretically, I mean.",
        "A": "A toroid is a geometrical shape, similar to a doughnut, that is generated if you revolve, for instance, a circle around an axis that lies outside the circle itself. An o-ring has a toroidal shape."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 00:16, Sal talks about how the universe started from an infinitely small point. How did this point exist, according to the Big Bang Theory?",
        "A": "Well, if you think of black hole theory, then it shows that these infinity small objects can exist. Since the universe is expanding, then if you turn the clock back, it should begin with everything coming from an infinitely small point."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 0:42, Sal says that the point exploded outward. That made me wonder, could the universe implode?",
        "A": "It s one possible eventual fate of the universe, but we think we have ruled it out."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "Hi, I'm not sure if Khan's statemant at 6:15 that \"we know it has a slight curvature\" is correct. I read that the universe is flat with only a very small margin of error and although it might mean that the curvature is so small that the universe appears to be flat, we can't be sure because there is no proof for this statement, or is there?",
        "A": "The measured curvature is so slight that it is within the margin of error of the measurements. So, it could very well be flat, or it could be curved and very big."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 7:30 Sal says that light might not be able to ever 'go around' the universe, and that it is continually expanding. My question is, does that mean that the universe is expanding faster than the speed of light? If so, isn't the big bang and all that contradictory?",
        "A": "Expansion allows parts that are far enough away to recede from each other faster than the speed of light. This in no way contradicts the big bang theory."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 7:26, Sal says that even if light were to travel around the universe, it would take an unbelievably large amount of time - so does that mean that light actually bends with a really small curvature?",
        "A": "If the universe is curved, then light would follow a curved path through it. Our best current evidence indicates that the universe is not curved, or if it is, the curvature has to be very, very small."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 9:11 is there there space beyond the small sphere if not, how does it keep expanding?",
        "A": "There is currently no reason to think otherwise."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "at about the 3:00 minute mark in the video you talk about a curved 2D finite space as a shpere... well how can a 3D object be 2D or vise versa?",
        "A": "the only possible way i can see this happening is if a 3D shape was sucked into a black hole the technically speaking it is a 3d shape the just looks like a 2D shape"
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 1:25, I am confused. Can someone explain expanding space to me? I don't get it at all...",
        "A": "Space IS everything. It HAS everything. But even the best scientists don t know it completely. So, don t worry."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 5:17, why can't you move diagonally?",
        "A": "that s just a combination of the other two perpendicular directions"
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "at 3:03 sal said about 2 dimensional space but he gave the example of a sphere which is 3 dimensional",
        "A": "He is referring the the surface of a sphere. You can uniquely specify any point on the spheres surface with only 2 coordinates making it 2 dimensional."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At 0:23 seconds Sol as I see the speaker being called that in various comments, anyway, what is a singularity?",
        "A": "Of course, we re not even sure about that yet; a number of experts have doubts on the matter."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "At around 6:23. he mentions something like a sphere. Can someone tell me how to spell that name, because I want to see a visual of that shape.",
        "A": "I believe Sal uses a sphere for a 3-D surface on a 4-D sphere to illustrate how we know space can curve. However, this is simply analogous with Earth s finite area yet no boundary idea. There is no picture that can describe 4-D because we live in a 3-D reality. Its just as abstract as asking for a picture of time. Luckily we can create mathematical models to draw these ideas and abstractions."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "As Sal mentioned at 8:20 that everything in the universe goes in the same direction and follows it. So if fire a gun in the space and we are also in space, there are no limitations and blockade, will the bullet hit us back?",
        "A": "If the universe is curved in that way, yes. WE don t know if it is."
    },
    {
        "video_name": "eUF59jCFcyQ",
        "Q": "4:10 isn't a sphere a 3 dimensional space and a circle a 2 dimensional space? right...",
        "A": "Yes you are correct! Sphere 3D Circle 2D"
    },
    {
        "video_name": "NM0WycKCCDU",
        "Q": "In 3:36 , why is he finding the mole?",
        "A": "Because that will let us find the molar ratio between the atoms which will give us the empirical formula (which is the entire point of the video)"
    },
    {
        "video_name": "NM0WycKCCDU",
        "Q": "At 6:04 shouldn't it be Hg2Cl? For every one mole of Hg we have 2 moles of Cl. Why is it HgCl2?",
        "A": "HgCl2 means 1 Hg atom and 2 Cl atoms Hg2Cl means 2 Hg atoms and 1 Cl atom."
    },
    {
        "video_name": "NM0WycKCCDU",
        "Q": "This may have been answered in another video, but if you got a ratio of let's say exactly 1:1.5, would you round up or round down in the empirical formula?",
        "A": "Good question. Multiply them both by 2 so you get a ratio of 2:3."
    },
    {
        "video_name": "NM0WycKCCDU",
        "Q": "At 5:18, where does the number .3639 come from?",
        "A": "It is the number of moles of Hg. It is written on the right of the calculator as a rounded number (0.364)"
    },
    {
        "video_name": "NM0WycKCCDU",
        "Q": "Why do we assume that the percent compositions are in given in mass rather than in volume or numerically? Why can't the percents be saying that we have a mole ratio just over 3:1?",
        "A": "Because atoms tend to differ widely in terms of mass. If all atoms weighed the same then we could indeed use weight percentages to determine empirical formulas (formulae?), but, as Sal showed us in this video, there are two Cl atoms for each Hg atom, instead of the one Cl atom to each three Hg atoms that the percentages seemed to indicate. In other words: There are six times fewer Cl atoms than it seemed. This is because Cl atoms are about six times LIGHTER than Hg atoms."
    },
    {
        "video_name": "NM0WycKCCDU",
        "Q": "If the ratios aren't exact and he just rounded to 2:1, how exactly do these non-whole ratios work out in the real world? Do chemists just round the way he did to find the formula since you can't have a fractional molecule?",
        "A": "If the ratios are reasonably close to integers (say, within a few percent), they round off as in the video. If the ratios aren t close to integers, they repeat the analysis one or more times."
    },
    {
        "video_name": "NM0WycKCCDU",
        "Q": "At around 5:14 why did he divide it by .3636 where did he get that number from?",
        "A": ".3639 came from the finding the number of moles in 73 grams of Mercury."
    },
    {
        "video_name": "NM0WycKCCDU",
        "Q": "In aluminium chloride (AlCl3) the ratio of chlorine to aluminum (2.25mol:0.749mol) is about 1:3. Can someone help explain why the empirical formula is different from sal's example where in this senario the chlorine is greater in mass yet three Cl atoms are used for each Al atom in the compound.",
        "A": "The ratio of chlorine to aluminium is 3:1, not 1:3. So the empirical formula of aluminium chloride is AlCl\u00e2\u0082\u0083."
    },
    {
        "video_name": "NM0WycKCCDU",
        "Q": "Why was it necesarry to change into mole form? 27% to 73% is roughly a 2:1 ratio, is it just luck that the percentage ratio and the mole ratio were similar or will this always be so?",
        "A": "73%27% is closer to a 3:1 than 2:1 ratio. You convert it into moles so that you can get a closer estimate."
    },
    {
        "video_name": "45qlm6cfHgg",
        "Q": "At 2:15, what is episodic memory?",
        "A": "It is the collection of past personal experiences that occurred at a particular time and place."
    },
    {
        "video_name": "45qlm6cfHgg",
        "Q": "Does IQ mean intelligent at 1:35?",
        "A": "IQ itself stands for intelligence quotient. It is based off a person s score on a standardized test. Often IQ is used to measure a person intelligence. So for example a person with an IQ of 100 would have an average intelligence. I m sure Wikipedia has a much more comprehensive explanation, if mine was not helpful."
    },
    {
        "video_name": "zTUnjPALX_U",
        "Q": "If an ion is the loss/gain of electron(s) and isotope involves the collective numerical value of protons and (sometimes changed) neutrons, is there ever an atom that can lose a proton??\n\nHow come when you have a cation, like at 6:21 with Pt+4, it doesn't add 4 of those to protons? Is it because they're electrons and it wouldn't make sense to add electrons to become protons?",
        "A": "Atoms only gain or lose protons during radioactive processes, not during chemical reactions. If a proton is gained or lost, then the atom changes into a different element, because an element is defined by the number of protons that it has. Pt4+ is a platinum atom that has lost 4 electrons to give it a 4+ positive charge. There is no change in proton number, otherwise it wouldn t be platinum any longer."
    },
    {
        "video_name": "zTUnjPALX_U",
        "Q": "At 3:00, a practice question is given. The problem was solved easily as the periodic table was right there, and Sal could find out the atomic number of platinum by referring to the periodic table. But, what if the same question is asked in an exam, and a periodic table is not given? How do we find out the number of protons then?",
        "A": "You should be given a periodic table in any exam, check with your teacher. It is a silly idea to memorise the periodic table but you may have to..."
    },
    {
        "video_name": "zTUnjPALX_U",
        "Q": "at 1:12, how did you get 5 electrons instead of 6 electrons?",
        "A": "If it lost an electron, as in some other element stole an electron from carbon. Don t overthink it too much."
    },
    {
        "video_name": "GFiizJ-jGVw",
        "Q": "At 1:12 why don't you also include the ethyl and propyl group when naming the alkene? Should you have to include the methyl, ethyl, and propyl?",
        "A": "Those aren t groups, they are already accounted for by saying the main carbon chain is heptene"
    },
    {
        "video_name": "GFiizJ-jGVw",
        "Q": "At 1:02 how do you know if it's a methyl group or not? Why doesn't the lone hydrogen count",
        "A": "CH3 by itself is a methyl group"
    },
    {
        "video_name": "GFiizJ-jGVw",
        "Q": "at 3:23 Sal only named the 4-methyl constituent ... you don't have to name the two big priority groups on either side of the double bond? Just saying \"Z\" is sufficient?",
        "A": "The name of the compound is 4-methylhept-3-ene. The Z specifies the configuration of the double bond. No more information is needed to completely identify the compound. Just saying Z is sufficient."
    },
    {
        "video_name": "GFiizJ-jGVw",
        "Q": "At 5:31, where the second example of an Alkene is named using the E-Z naming Scheme, would it be the same thing as 3-Bromo-Trans-Oct-3-ene?",
        "A": "Strictly speaking, you can use the cis/trans nomenclature only for disubstituted alkenes. Without the Br group, this would be cis-oct-3-ene. With the Br group, you have to use the E/Z notation."
    },
    {
        "video_name": "f_Z1zsR9lFM",
        "Q": "At 7:20, what does LH do specifically in the male and female body? I don't think they went over that every well",
        "A": "The luteinizing hormone in females stimulates ovulation. In males it stimulates testosterone production."
    },
    {
        "video_name": "Z4F88tTx9-8",
        "Q": "At 2:58,in the example of (N) ,oxygen is having a partial charge and it turns to (N) then why hydrogen with a positive charge didn't turn to a (E) and what happens when both charges come together",
        "A": "Sorry I m having trouble understanding your question. Could you please try rewording it?"
    },
    {
        "video_name": "jUUJSOM1ihU",
        "Q": "At 0:19, what is a prime carbon?",
        "A": "The carbons in the sugars are given the little dashes so that they can be distinguished from any numbers given to atoms in the other rings :)"
    },
    {
        "video_name": "jUUJSOM1ihU",
        "Q": "At 0:15 why does Sal say that we can look at the 5' carbon to tell if the molecule is DNA or RNA? Isn't the 5' carbon connected to a phosphate group in both RNA and DNA? The carbon he refers to later at 0:26, that distinguishes RNA from DNA, is the 2' carbon that lacks an -OH group, therefore, making the sugar deoxyribose instead of ribose.",
        "A": "I have not seen the video, but the ribose-deoxyribose differentiation is based on the -OH group on the second carbon only. Hence, the deoxyribose sugar, is actually named 2-deoxyribose."
    },
    {
        "video_name": "qbCZbP6_j48",
        "Q": "At 1:20 what makes or defines that it is higher concentrated?",
        "A": "Hi Jasmine, Concentration is defined as the amount of the constituent per unit volume of solution. Therefore, if the volume is held constant and the amount of the constituent is raised, the concentration becomes higher. Similarly, if the amount of the constituent is held constant and the volume is decreased, the concentration also becomes higher. Hope that helps :)"
    },
    {
        "video_name": "qbCZbP6_j48",
        "Q": "At 1:28-1:30, Sal says 'shining at a wavelength of light that is specifically sensitive to the solute'. What does that mean ?",
        "A": "Depending on what is in a solution, it will absorb light at only some of the wavelengths of light. We usually choose a wavelength where the component we are studying absorbs most strongly and where other components of the solution do not absorb."
    },
    {
        "video_name": "qbCZbP6_j48",
        "Q": "At 9:25, what is that symbol next to the \"A\" and what does that mean?",
        "A": "It means is proportional to"
    },
    {
        "video_name": "qbCZbP6_j48",
        "Q": "At 10:20, he seems to be indicating that if you know the concentration and absorption of the solute, you can figure out the chemical composition? Or am I just totally off?",
        "A": "Perhaps. If every solvent has a unique constant variable (epsilon) in the beer-lambert formula, then yes. However, this is usually not the way spectrophotometry is used; there are better ways to find composition."
    },
    {
        "video_name": "qbCZbP6_j48",
        "Q": "So molecules can absorb light? @ 2:00",
        "A": "Yes they can. If they do, an electron gets promoted to a higher energy level. Some (or all) of this extra energy can be transformed into vibrational energy of the molecule. And the remaining part can be radiated away by giving off light (of a lesser frequency than the one that was absorbed)."
    },
    {
        "video_name": "qbCZbP6_j48",
        "Q": "Sal, at 2:02, I heard you say \"...some of this light at certain frequencies is going to be absorbed...\" Did you mean wavelength instead? I know that in the analysis of DNA or RNA concentration, DNA and RNA have a maximum absorption at 260 nanometers, which explains why this technique is called UV spectrophotometry since 260 nm is within the UV range.",
        "A": "Frequency and wavelength are sometimes used interchangeably because they can both describe the energy of a light source. Energy = (constant) * frequency = (constant) * 1/wavelength."
    },
    {
        "video_name": "qbCZbP6_j48",
        "Q": "At 11:00, is there a unit that absorption is measured?",
        "A": "Assuming you mean absorbance, the answer is no. There is no unit. To avoid confusion, we sometimes use abs to indicate that it is an absorbance, but there is no actual unit. The reason that there is no unit is that the units all cancel out when the computation is done."
    },
    {
        "video_name": "qbCZbP6_j48",
        "Q": "About Absorbance at 7:57. When Sal says \"The negative log of T\" does he mean you take the logT and multiply it by negative 1 (-1), or does it mean 1 over the log of T (1/logT)? Like how 10^-1=1/10.",
        "A": "He means either (-1)\u00c3\u0097logT or log(1/T). Note: log(1/T) = log1 - log T = 0 - log T = -logT. They both mean the same thing."
    },
    {
        "video_name": "qbCZbP6_j48",
        "Q": "Nearly At 7:56, Sal said that Absorbance is equal to the negative log of T. What does log mean ?",
        "A": "Its a logarithm. That is typically covered in Algebra II. You need to know how to work with logs and exponential functions in order to be able to do General Chemistry."
    },
    {
        "video_name": "UPoXG1Z3sI8",
        "Q": "at like 5:50ish when he's finding the percent of hydrogen, can you just subtract the percentage of carbon to find the percent of hydrogen?",
        "A": "Of course, but it s always good to have a double check which you would have by calculating both independently."
    },
    {
        "video_name": "UPoXG1Z3sI8",
        "Q": "At 3:45, since Sal used atomic weights in his calculations and not the specific atomic masses, shouldn't it be the molecular weight of benzene, and not the mass?",
        "A": "Yes, he should really have referred to the molecular weight of benzene."
    },
    {
        "video_name": "6vy5CX6vK0I",
        "Q": "What was the word he said at 7:33 when he was talking about the diabetic person? \"I've drawn for you now someone with a ( ), which is diabetes, meaning they have some medical condition.\"",
        "A": "Comorbidity. Comorbidities are any other conditions that a person has at the same time as the one you re interested in."
    },
    {
        "video_name": "6vy5CX6vK0I",
        "Q": "At 2:52 he talks about RNA. What is RNA?",
        "A": "RiboNucleic Acid"
    },
    {
        "video_name": "6vy5CX6vK0I",
        "Q": "Why would an elderly person be higher risk? 8:08",
        "A": "They are weaker than young men or women."
    },
    {
        "video_name": "24vtg9Ehr0Q",
        "Q": "At 12:09, David says we have to break up the diagonal vector in order to substitute it into the horizontal and vertical equations. Couldn't we just solve the horizontal and vertical equations - at which point we would have a diagonal vector - and add the 45N vector to the sum of all the Forces?",
        "A": "As far as I have understood your question, No you cannot do that because Force is a VECTOR QUANTITY and it must follow the vector laws of addition. To do that we have to take suitable axes and break the diagonal vector into FcosTheta and FsinTheta accordingly and proceed further."
    },
    {
        "video_name": "24vtg9Ehr0Q",
        "Q": "At 9:22 what does the a in the graphic represent? I understand ay and ax but a?",
        "A": "a is the overall acceleration not the horizontal or vertical acceleration."
    },
    {
        "video_name": "24vtg9Ehr0Q",
        "Q": "at 13:50. Instead of using sin and cosine to find the portion of force which acts upon each axis, can we divide 45 Newtons by 90 and multiply by 30 for the x axis and by 60 for the y axis?\n\n45N/90 times 30 = force in the x direction\n45N/90 times 60 = force in the y direction\n\ndoes this work? It seems simpler that using cos/sin.\nthanks :)",
        "A": "It might work in this problem, but if the angle was something a little harder, you would need to use sin and cos."
    },
    {
        "video_name": "24vtg9Ehr0Q",
        "Q": "2:34. So if we have negative force, which would be negative? Mass or acceleration?",
        "A": "which one of those is a vector?"
    },
    {
        "video_name": "24vtg9Ehr0Q",
        "Q": "Why are the Horizontal and vertical accelerations not squared when using pythagorus equation? At 9:20",
        "A": "Eventually, the total horizontal force and total vertical force will be squared and added and then square rooted (it that is a word) to equal the total force in a diagonal direction, but he was not going to go through the whole problem yet."
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "I know that it is a long question but PLEASE PLEASE PLEASE ANSWER...\u00f0\u009f\u0099\u008f\u00f0\u009f\u0099\u0087\u00f0\u009f\u0099\u008f\u00f0\u009f\u0099\u0087\nIf we replace the balloon Sal draws in around 5:50 with a round glass flask and make holes in it in different levels and insert tubes with the one open side facing upwards and then we press the piston, the water spurting out of the tubes will rise upto the same level. Why?",
        "A": "Because when an external pressure is applied to a fluid the pressure is transmitted throughout the fluid undiminished. Hence when you press the piston the fluid or in this case water spurts out of the holes and all reach the same level....."
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "At 0:49 you divide F1 by A1. I do understand P=F\\A but I don't understand how you can 'just' divide F1 by A1? Can you please explain this.",
        "A": "Thanks Lucas, I was considering A1 x D1 as a given (volume) and couldn t understand how you could divide F by A and not doing anything on the otherside but obviously Sal is considering, which he also says, F1 x D1 and he is re-writing this one. Makes sense. Thanks for the explanation."
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "0:37, how can the Dinput equal to Doutput?",
        "A": "They aren t equal. The volume input and volume output would be the same. Area x Distance = Volume. In this case, the area changes between the two, therefore so does the distance. However, volume stays the same. Pressure also stays the same. What changes is the force."
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "At around 3:25, he establishes that P1V1=P2V2. I was just wondering if this had anything to do with Boyle's gas law?",
        "A": "PV = nRT. If T is constant, then the ideal gas law becomes PV = constant, which we can say another way as P1V1 = P2V2. In other words, Boyle s Law and Charles Law are both special cases of the ideal gas law."
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "In 6:55, Sal draws that pipey thingy. What is it?",
        "A": "It is a U -shaped tube filled with a fluid.The diameter of the right-hand section is greater than the diameter of the left-hand section."
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "why at 9:10 does the iniatial force divided by the initial are ( 10 / 2) become 20n?\nShouldn't it be 5n?",
        "A": "initial force by initial area is pressure(given as 10 Pa). pressure is force /area therefore force =pressure x area =10 Pa x 2 m^2 =20N"
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "At 1:47, why does Sal multiply and divide by A1 and A2?",
        "A": "The left side of the equation is multiplied and divided by A1 (keeping the same value) and the right side of the equation is multiplied and divided by A2 (also keeping the same value). This is just to convert the relation to another kind of dimension, changing force by pressure."
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "At around 9:58 the denominator is 4m when it should be 4 m^2, but I suppose it doesn't matter since we all get it anyway.",
        "A": "You are right. Thank you for pointing out this mistake because it might help other confused people"
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "At 7:36, Sal mentions area in and area out. How does the area change throughout?",
        "A": "The area depends on the container. Here Sal has shown a kind of a U-tube which has different areas at the ends. Thus the area is different in the in and out conditions."
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "At 10:01 we calculated f2 to b 40n nd f1 to b 20n...my doubt is shouldn't a larger surface area hav smaller force as compared to that of a smaller surface area have large force...or mayB im confused...what's the right way??",
        "A": "perhaps you just feel it that way (actually i felt it in a similar way) but there is no reason why it should be correct. See just remember one thing, the pressures at the same level should be equal and that gives that you get a larger output force where the area is larger."
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "at 5:00 when you discussed that water in a single container could not be compressed butt is int there oxygen in water.",
        "A": "yes, there is oxygen in water, but it is at a concentration of about 10 ppm (million). by the time those oxygen molecules have any noticable effect on the pressure, you d have more problems with the container breaking or stretching due to the pressure. Also, since there is no such thing as a perfect incompressible fluid in the first place (only almost perfect), the oxygen gets lost in the rounding."
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "At 6:00 what if the structure above the balloon is not rigid and is made up of a rubbery substance? What changes will take place in the balloon?",
        "A": "Well, like he said. The pressure are size in the balloon will increase uniformly."
    },
    {
        "video_name": "lWDtFHDVqqk",
        "Q": "At 6:22 Sal says that the liquid is not compressible .but what happens when we are filling the balloon with water and is trying to squeeze it.is it not becoming compressed?",
        "A": "No, it s not. The water moves around but its total volume does not change."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "at 8:03, for l=1 ml has 3 values(-1,0&1) and has 3 kinds of orientation(on x-axis,y-axis & z-axis).Does any of the 3 values of ml correspond to any specific orientation?",
        "A": "No, x, y, and z are all relative. Since they end up being symmetric in the end, it doesnt matter. It just makes it easier to picture."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "at 4:30 in the video it says that for n-1 of 1, l=0 which makes sense but if l=0,1,2,3........n-1 then if n=0 can l=-1 ?",
        "A": "No, n can only be equal to 1 or higher, so l can only be zero or a positive integer up to n-1"
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At 2:20, Jay says that the higher energy level has higher energy associated with it. However, in another video, Sal said that the farther the electron from the nucleus, the weaker the force on it that's why they're the ones to get pick off first.\n\nAren't these 2 statements conflicting with each other?",
        "A": "No, they are consistent with one another. Energy and force are not the same thing. You need to review the concepts of work and potential energy."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At 10:00,is only p-orbital possible??What do the different axes signify?What difference does it make?",
        "A": "The different axes depict the spatial arrangement of the electron clouds. The arrangement of these orbitals actually dictates the orientations and angles of the bonds. To have covalent bonds, the orbital clouds need to overlap, sort of like how Legos can only be connected in certain ways."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "at 3:02 , how does l represent the shape of an orbital?",
        "A": "If you are talking about physically, to help you understand, you can use balloons :)"
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At 3:10: The values of l are dependent on n. And they go on up to n-1. Why not till n?",
        "A": "This is just a rule that we have to accept. I don t think there is any explanation for it but limiting l to n-1 fits the experimental data. It explains why, at the n = 1 level, there is only one orbital (an s orbital) whereas as at the n = 2 level, there is an s orbital plus p orbitals. If l went up to n, rather than n-1, then there would be two types of orbitals at the n = 1 level."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At 1:51 Jay says that n is also called energy level or shell. But we cannot simultaneously determine the velocity and position of an electron, so that should rule out the existence of any well defined trajectories of the electron. Then how can we define these shells as the path the electron follows around the nucleus?",
        "A": "The shells are not paths. They are energy levels. Also, it s not true that we can t simultaneously determine velocity and position. What the uncertainty principle says is that there is a limit to the precision with which both of those quantities can be known simultaneously. Measurements that are less precise than that limit are no problem."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At 2:27, why does energy increase away from the nucleus ?",
        "A": "The electron is attracted to the nucleus. It takes energy to pull an electron away from the nucleus. The further you have to pull the electron from the nucleus, the greater the energy you have to put into the system."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "at 1:17 he says that the first quantum number is called principal quantum number and this is symbolized by n which can be 1,2,3 and so on. if n symbolizes first quantum number how can it be 2,3 and so on. what does n symbolize and what is principal quantum number?",
        "A": "It s essentially the shell that an electron is in"
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At around 5:30 minutes when sir refers to the shape of p-orbital (bow tie shape), should we consider it in vertical form or can it be taken in horizontal form also?",
        "A": "yes obviously the answer is right"
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At 6:29, what is meant by the word 'Orientation\"?",
        "A": "The word orientation refers to the physical location of a subshell for a given atom. In the example with the p orbital there are 3 different subshells, (p x , p y , and p z ) each of these subshells occupies a different space surrounding the atom. If you think about it from the perspective of the electrons, they want to be spread out as far as possible in the space they have because likes repel likes. Thus, the orbitals are spread out as much as possible and oriented in the the x, y and z planes respectively."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At 2:23 khan says abt electron having higher energy level associated with it. how does an electron have an higher energy level associated with it?",
        "A": "The electron by itself does not, but the atom does, and the way it does is by having the electron move against the electrical attraction of the positively charged nucleus."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "What does \"quantum\" mean in \"quantum mechanics\"? Jay referred to Bohr model as \"classical mechanics\" (@0:40).",
        "A": "The quantum in quantum mechanics means that some things come in discrete amounts called quanta. In classical theories values like position and energy need to be able to take on continuous values. In classical physics there is no mechanism to describe why the Bohr model of the atom exists, there is no reason that you couldn t have electrons in any orbit and not just in the ones defined in the Bohr model. When you introduce the idea of quanta of energy this starts to make sense."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "at 10:21, the whole explanation of the mag quantum is finished. But I'm still wondering, how do the integers correlate to px, py, and pz? How does that relate to -1,0, and +1?",
        "A": "The orbitals that come from the Schr\u00c3\u00b6dinger equation with m = -1, 0, -1 are impossible to graph, because they contain complex terms like e^(im\u00cf\u0086), where i = \u00e2\u0088\u009a(-1), m is the quantum number, and \u00cf\u0086 is an angle. But certain mathematical combinations of these orbitals remove the complex terms and give the familiar px, py, and pz orbitals."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At 7:30 - 10:15 it's determined that there is the s orbital and three p orbitals. What does this imply about the location of the electron? Is the electron going to be found in one orbital or is it found in any of those p-orbitals and s-orbital?",
        "A": "You can have one electron on each side. For example, in the s orbital you can have 2 electrons, 2 on each side, and for the p orbital you can also have 2 on each side, but you have to keep in mind that when you are writing the electron configuration, there are 3 sets of p orbitals."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At 11:14 Why does the electron spin quantum number has 1/2 value and not 1?",
        "A": "All fermions have an spin of integer + 1/2. Quantum spin is a measure of the intrinsic angular momentum of a particle."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At 9:12, how do I know which axis to draw the orbital on? Does it matter?",
        "A": "It doesnt matter."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At 4:43 he says that n=2 therefore l= 0, 1 . I understand how he got 1 (n-1) but I don't understand why it can be both 0 and 1? Also, at the part where he is talking about Magnetic Quantum Number, I also don't quite understand how he came to the conclusion of : if l=0 then ml=0 and if l=1 then ml= -1, 0, +1. Please help",
        "A": "l can be any integer from 0 to n-1. It is telling you which orbitals are present for that n value. So if n = 2, l can be both 0 and 1 which correspond to 2s and 2p orbitals. ml can be integers from -l to +l including 0. If l = 0 the only possible value for ml then is also 0. If l = 1 then now ml can be -1, 0 and +1 It s more a matter of this is just how the quantum numbers are defined."
    },
    {
        "video_name": "KrXE_SzRoqw",
        "Q": "At 4:38, he says that if n is equal to two, then l is equal to 0 AND 1. Before that, he said that if n is equal to one, then l is equal to just 0. So would I be correct if I say, \"If n is equal to two, then l is equal to n and 1?\" I am sorry if it sounds confusing.",
        "A": "If n=2 its not equal to 0 and 1, it just means that the principal quantum number n=2 contains subshells that are named l=0 and l=1. n=1 contains a subshell named l=0. Its important to remember that the l=0 subshell that is WITHIN the n=1 shell is not the same as the l=0 subshell WITHIN the n=2 shell or any other shell"
    },
    {
        "video_name": "XQTIKNXDAao",
        "Q": "1:55? What in the world is that sound?",
        "A": "Coffee maker taking apart."
    },
    {
        "video_name": "XQTIKNXDAao",
        "Q": "7:40 when the metal expende he stop the circuit, whow do the circuit come back again ?",
        "A": "The metal expands due to heat and since its in the bimetallic strip it bends and opens the circuit. With the circuit open it stops producing heat and allows the bimetallic strip to cool down again which un-bends the bimetallic strip allowing contact again."
    },
    {
        "video_name": "Q-aGAX27SIo",
        "Q": "At 3:01 you began discussing homologous structures as evidence for evolution. My biology teacher said that these structures are controlled by different DNA. How does that work?",
        "A": "You d have to elaborate on what your biology teacher said, because that s pretty vague. All DNA on everything living on this planet is different. There is not a single pair of living organisms that have identical DNA... So please elaborate on your teacher s wordage."
    },
    {
        "video_name": "ITA1rW5UraU",
        "Q": "@ 3:28 Sal says we will go into the math for this a little bit later. Where is the video where he does go into the math for this?",
        "A": "He might of forgotten to finish off the math. Managing and creating this whole site is hard and takes alot of time. Keep going on Sal!!!"
    },
    {
        "video_name": "ITA1rW5UraU",
        "Q": "I`m sorry but just to recap, Force is equal to acceleration multiplied by mass?\n\"3:28\"",
        "A": "You can refer to Newton s Second Law of Motion whcih states that the Force applied to an object is proportional to the rate of change in momentum; F=(mv-mu) / t F=m(v-u) / t and since a=(v-u)/t you simply get F=ma Hope i helped!"
    },
    {
        "video_name": "J2L-X2sUigs",
        "Q": "How is the fourth quadrant (shown at 11:15) not defying the second law of thermodynamics as it is a reaction in which entropy is decreasing and takes energy. If this were possible then the entropy of the universe would be fluctuating, no? It would mean that somethings could happen that lower the entropy of the universe",
        "A": "The fourth quadrant is not defying the second law of thermodynamics because it is using up free energy. A spontaneous reaction that decreases the entropy of the universe is not possible, as it violates the second law. Remember that the entropy of the system can decrease, as long as it compensates for that by increasing the entropy of the surroundings (universe). One way of doing so is by sucking up free energy. :)"
    },
    {
        "video_name": "KPh60w6McPI",
        "Q": "At apx. 4:20, shouldn't this proceed via Sn1 since it's a secondary alcohol?",
        "A": "We aren t given enough information to make a good decision. The tosylate is an excellent leaving group, so I could predict SN1. But Br\u00e2\u0081\u00bb is a fairly good nucleophile, so I could predict SN2. If we knew the solvent, we could make a better prediction: polar aprotic = SN2; polar protic = SN1. Overall, I would predict SN1 based on the excellent leaving group."
    },
    {
        "video_name": "KPh60w6McPI",
        "Q": "At 3:30, unlike reactions involving primary alcohol and tertiary alcohol, the secondary alcohol first reacted with TsCl and pyridine to make it a better leaving group. Is it possible to remove the hydroxyl group without first reacting the alcohol with TsCl?",
        "A": "It is not possible to remove the OH group itself. You have to react the OH with something to make it a better leaving group, but it doesn t have to be TsCl."
    },
    {
        "video_name": "ixRtSV3CXPA",
        "Q": "At 2.35 Sal wrote delta U + P delta V = 0 which is true for adiabatic expansion only, not for adiabatic compression. Using this equation he obtained another expression at 10:44 which should again be true only for adiabatic expansion. Then at 12:54 how could it be applied for process D to A which is adiabatic compression ?",
        "A": "Why is it true for expansion but not compression? The only difference between the two is the sign of delta v."
    },
    {
        "video_name": "ixRtSV3CXPA",
        "Q": "Is it fair to say that 1/T * deltaT has no unit?\n\nSame goes for 1/V * deltaV\n\nBecause at 6:51 Sal claims that the sum of those to zero but if they do have a unit one would be a unit of volume and one would be a temperature, adding those to up and expecting them to be 0 would be weird right?",
        "A": "Because the units of delta T and the T are same,if you put the units in 1/T*delta T then the units will cancel themselves.So,it has no unit.Same goes for the 2nd expression!"
    },
    {
        "video_name": "ixRtSV3CXPA",
        "Q": "From about 13:02 I didn't understand how Sal changed the Ts and Tf to T1 and T2. Please help.",
        "A": "Yeah, S stands for start and F stands for finish, so in the segment he was working with (B to C), the start value is T1 and the finish value is T2, and in the other segment (D to A), the start value is T2 and the finish value is T1. Check the graph yourself to confirm that (at 11:30)"
    },
    {
        "video_name": "ixRtSV3CXPA",
        "Q": "Could someone explain to me why at 5:56 Sal is able to divide the 'work done' term by nRT, specifically by T? If it were an isothermal process then I would understand how since T isn't varying we can just cancel it but in an adiabatic process such as this, T is a variable so it is not erroneous to divide through by it in this way ? (He has written pressure as a function of volume AND temperature NOT just volume, as would be the case when dealing with work done in an isothermal process).",
        "A": "I believe he can divide throughout by a constant or variable, as long as he does it equally on both sides of the equation (which he does) There is no physical significance to what he is doing, it is simply a mathematical way of rearranging the equation to make the integeration possible (seperating V and T terms) Make sense OK??"
    },
    {
        "video_name": "ixRtSV3CXPA",
        "Q": "At 2:07 time, it was glazed over that P is considered to be constant in the quasi-static process. Why is obvious that P needs to be kept constant? What about the system makes it clear that pressure has be considered constant, except to allow the proof to work out?\nI am noting that when adiabatic systems were originally discussed in an earlier video, pressure was shown to vary, thus these approximations have been difficult to conjure for the purposes of making thermodynamic proofs. Much thanks.",
        "A": "Although in an infinitesimal change P is almost constant , when Sal substituted P as a function of V and T he was not keeping P constant. So, when integrating the equation the functionality of P is brought in through P = nRt/V. P is not constant in the total adiabatic process. Over an infinitesimal interval deltaP approximately equals (nRT/(V+deltaV)-nRT/V) which is much smaller than deltaV"
    },
    {
        "video_name": "JoGQYSTlOKo",
        "Q": "At 1:36 he says, that pH= -log[H3O+]. Can someone explain why?",
        "A": "In an aqueous solution (i.e. when water is the solvent), each of the H+ (protons) donated by an acid will combine with a water molecule to form H3O+. Consequently, in water H+ and H3O+ are used interchangeably."
    },
    {
        "video_name": "JoGQYSTlOKo",
        "Q": "at 8:25 why do we add the volume of the base to the volume of the acid for the new volume??",
        "A": "Because the reaction occurs in an aqueous solution, the volume is additive."
    },
    {
        "video_name": "JoGQYSTlOKo",
        "Q": "at 8:36, why is the new volume of H3O+ 30ml??\nI dont understand y r v adding the volume of the base?",
        "A": "Because this is an aqueous solution, volume is additive."
    },
    {
        "video_name": "JoGQYSTlOKo",
        "Q": "at 7:30, is that an ICE Table?",
        "A": "An incomplete one, but yes."
    },
    {
        "video_name": "qkLzAXUP_K0",
        "Q": "At 6:22, \"3d electrons have higher energy than 4s\". Could you define energy more clearly? I thought that in general the further away an electron is from the nucleus the more energy it must have, in order to resist attraction from the positive nucleus (as described in earlier video). Are we talking about kinetic energy here? Do the electrons in 3d move faster than the ones in 4s, even though electrons in 3d are closer to the nucleus?",
        "A": "Attraction to the nucleus is not the only factor at play in determining the energy. There is also repulsion by the electrons that are already there. The 3 shell is crowded by the time you are ready to start filling the d subshell. It s easier to go to 4s first."
    },
    {
        "video_name": "qkLzAXUP_K0",
        "Q": "At 6:14, Sal says that the outermost electrons will be in the (4S^2) sub-shell, and the electrons with the highest energy will be in the (3D^6) sub-shell, so I am unable to understand that, first of all, why aren't the outermost electrons in the last sub-shell (ie. 3D^6), and secondly, how come the electrons with highest energy in different sub-shell than the outermost electrons?",
        "A": "4s fills before 3d. That s just how it is. There is no rule that says 3 has to fill up completely before 4 starts. Nature does it the way nature wants to do it, not the way that seems simple to us. Nature fills the lower energy levels first. 4s is lower energy than 3d."
    },
    {
        "video_name": "qkLzAXUP_K0",
        "Q": "at 0:40 what is a S subshell?",
        "A": "In every electron shell electrons are further organised into subshells. The very first (innermost) shell only holds 2 electrons, and they belong to the S Subshell you asked about; there are no other subshells in the first main shell. However, the other shells can contain more than 2 electrons. The first 2 still go into the S subshells, but they re called S-2, S-3 etc. to make the names different from the first one. The extra electrons in every electron shell go into subshells other than the S."
    },
    {
        "video_name": "qkLzAXUP_K0",
        "Q": "At 5:15, why doesn't Sal write the configuration for iron like this? 1s2, 2s2, 3s2, 4s2, 2p6, 3p6, 3d6",
        "A": "We list the orbitals in increasing energy"
    },
    {
        "video_name": "1V9wVmO0Tfg",
        "Q": "At 7:33, I don't understand what Sal means by saying kilometers per second per mega parsec.",
        "A": "If two points are a megaparsec away from each other, they will be moving away from each other at about 70 km/s. So, if they are 5 megaparsecs away from each other, they will be moving away from each other at 350 km/s. So, the further they get from each other, the faster they move away from each other."
    },
    {
        "video_name": "1V9wVmO0Tfg",
        "Q": "At 3:09 Sal says something about a 4 dimensional sphere. What are the four dimensions and are there more dimensions after that?",
        "A": "You know of the three spatial dimensions. Try to imagine one more added on top of those three. That is the 4th dimension Sal is referring to in this video. There may be even more spatial dimensions than four, but we currently have no way to test that."
    },
    {
        "video_name": "1V9wVmO0Tfg",
        "Q": "If I took an inflatable balloon with some dots on it and blew some air into it, would that be a good visualization of the expansion of the universe like Sal talks about at around 2:45?\nThanks ;) !",
        "A": "That is the most common experiment to help people visualize the expansion of the universe. It shows that not only is the universe expanding, but also how that expansion is accelerating."
    },
    {
        "video_name": "1V9wVmO0Tfg",
        "Q": "at 8:57 what is a mega parsec",
        "A": "Mega means one million. So that s a million parsecs. A parsec is a measure of distance, equal to a little more than 3 light years."
    },
    {
        "video_name": "bFj3HpdC4Uk",
        "Q": "At 11:15, the resonance structure of the sulfur compound looks like it has 10 electrons in its octet instead of 8 (double bonded to an oxygen, a lone pair, and two methyl compounds). Why would this happen?",
        "A": "Sulfur doesn t have to follow the octet rule, lots of elements in the third period and below don t have to follow it."
    },
    {
        "video_name": "bFj3HpdC4Uk",
        "Q": "At 8:30, shouldn't the electrons from the lower-left oxygen be attacking the carbon, and the pi bond going to the oxygen? Jay has it set up so that the pi bond goes straight to the C-O sigma bond.",
        "A": "No, it must be the \u00cf\u0080 electrons that move. Otherwise the upper right carbon would be pentavalent."
    },
    {
        "video_name": "bFj3HpdC4Uk",
        "Q": "at 9:01, what keeps that negative oxygen from the already cleaved intermediate from earlier from just forming DMSO earlier on? It would seem counter intuitive to cleave, then reform, then cleave again. Not that nature always follows the easiest path given, but is there any specific reason that extra oxygen bonds to the carbon at 6:50 rather than just leaving to form DMSO?",
        "A": "There is no DMS present during the reaction. The usual solvents is a mixture of methanol and dichloromethane. DMS is added for reductive workup only after the molozonide has formed.. It is easily oxidized to DMSO by the molozonide and any excess ozone."
    },
    {
        "video_name": "bFj3HpdC4Uk",
        "Q": "My prof mentioned that the first step of the reaction that forms molozonide at 2:55 is a 1,3 dipolar cycloaddition. Can someone elaborate on the 1,3 dipolar part?",
        "A": "The 1,3-dipolar part is referring to the ozone (O3). A dipole is a compound that is neutral overall, but carries a positive and a negative charge (like O3). The 1,3 part is because, if you number the oxygens in ozone, it is oxygens 1 and 3 that react in the cycloaddition."
    },
    {
        "video_name": "bFj3HpdC4Uk",
        "Q": "At 8:51, why does sulfur attack oxygen?",
        "A": "Oxygen-oxygen bonds are weak and therefore unstable, which is why they can be attacked by the sulphur (from DMS)."
    },
    {
        "video_name": "bFj3HpdC4Uk",
        "Q": "At 3:50, why would that happen?",
        "A": "Any arrangement with three O atoms in a row is unstable. That is why ozone, :O-O=O:, is so reactive in the first place. The five-membered ring formed by the reaction of ozone with the ring, called a molozonide , is still highly unstable, because it still has 3 O atoms in a row. It could decompose back into ozone and the original alkene, but if it decomposes as in the video, we get more stable products \u00e2\u0080\u0094 carbonyl groups and only two adjacent O atoms."
    },
    {
        "video_name": "bFj3HpdC4Uk",
        "Q": "At 13:28 can we produce an alkane from wolf kishnerr reduction or clemensen reduction",
        "A": "Yes, but the point of the reaction is to produce carbonyl groups from the alkene product."
    },
    {
        "video_name": "bFj3HpdC4Uk",
        "Q": "How are the those resonance structures at 1:33 if they are the same structure just flipped, doesn't a resonance structure have to be a different configuration of electrons?",
        "A": "They haven\u00e2\u0080\u0099t flipped. The O atoms are still in the same place. Only the positions of the electrons have changed. That\u00e2\u0080\u0099s why they are resonance structures."
    },
    {
        "video_name": "bFj3HpdC4Uk",
        "Q": "At 12:25, Jay skips writing out the whole mechanism for the reaction again, understandably. However, he fails to mention how the side attached to only one oxygen would 'flip' over to repel the partial negative and formal negative charges on the oxygen. How could the 'flip' occur if all the carbons are connected? Do they break off and then reattach to the ring or is there another mechanism? Thanks.",
        "A": "What needs to happen is that the negative charged oxygen needs to attack the other carbonyl, nothing really needs to flip there s free rotation around the single bonds."
    },
    {
        "video_name": "yAQSucmHrAk",
        "Q": "The wave arcs drawn at around 1:30 are constantly bending up towards the surface. In the video on refraction, Sal showed how increasing density caused a bend toward the surface. However, once the wave is moving parallel to the curvature of the earth, why doesn't the angle decrease or stop? On the second half of the arc, while it is moving closer to the surface of the earth, the density is decreasing, so why doesn't the wave start curving the other direction?",
        "A": "If the wave started curing in the other direction it would mean the layers of density would have reversed, the layers would be getting more dense. But they don t. Going up, the wave would be traveling into less dense material, thus, turning toward the left. At first I didn t get it either, but I drew it out the way Sal did in Refraction of Seismic Waves. You should also try it!! It all made sense to me after that. Good luck!!"
    },
    {
        "video_name": "yAQSucmHrAk",
        "Q": "Shouldnt the waves in a \"uniform earth\" move in straight lines? Why are they curving back to the surface at 1:30?\n\nI understand why they curve when they change a medium.",
        "A": "i think he meant, uniform as in with no liquid core tho made with the same material that would still get denser as we go down"
    },
    {
        "video_name": "yAQSucmHrAk",
        "Q": "why are the waves travelling in arcs from one point to the other? Why don't they travel in a straight line? aka see time stamp 2:11.",
        "A": "disregard again! Should ve just watched the vid previous to this!"
    },
    {
        "video_name": "QaOtsYouet0",
        "Q": "Would it be possible, to use the equation in its form at 1:33, to get the cell potential with a temperature different to 25 \u00c2\u00b0C/298K? So not under standard conditions.",
        "A": "Yes. You just have to replace the temperature (in Kelvin) and since F and R are constants they stay the same"
    },
    {
        "video_name": "eM7arWJJ3zk",
        "Q": "At 6:44, does the leading strand use a RNA primer too? Or is it just DNA polymerase that comes in and starts working 5' to 3'?",
        "A": "Yes, the leading strand uses a RNA primer to start off the new DNA chain, and then DNA polymerase will come in."
    },
    {
        "video_name": "eM7arWJJ3zk",
        "Q": "I know that we also have a polymerase I and polymerase III. How is polymerase I different than polymerase III mentioned at 7:02?",
        "A": "DNA pol 1 has 5 to 3 polymerase activity in addition it has 5 to 3 and also 3 to 5 exonuclease activity. it can read and correct errors in both direction while DNA pol has only 5 to 3 exonuclease activitty. further, DNA pol 33 is the main enzyme for replication while DNA pol 1 used for replacement of RNA primer added by primase enzyme by DNA nucleotides."
    },
    {
        "video_name": "eM7arWJJ3zk",
        "Q": "Around 6:36, Sal says an RNA primer binds to the 5' end of the leading strand. Why is this RNA primer involved in DNA replication?",
        "A": "The DNA-polymerase can only add nucleotides on an existing strand of DNA, so the primer (located at ori - origin of replication) fakes a DNA strand with a couple of RNA nucleotides. It s just to get things going. The primer is always broken down and replaced by DNA at the end of the replication process."
    },
    {
        "video_name": "VIdvlSFDl2c",
        "Q": "At 1:38 in the video, the instructor says to leave the LED lights OFF to just test the motor. I have watched this whole series of videos over and over, and as far as I can tell the ONLY connection to the positive pole of the battery When the DPDT switches are not pressed) is THROUGH the LEDs. If that part of the circuit is interrupted (lights off) how can any current flow through the motors? Please indicate how the motors get positive current without the LEDs getting current.",
        "A": "maybe they are separate moters."
    },
    {
        "video_name": "L-0FkEPPdXE",
        "Q": "shouldn't sal wrote, 4p6 4d10 5s2 5p6 at 2:03 instead of 4p6 5s2 4d10 5p6 for the xenon configuration ?",
        "A": "The way that Sal wrote the configuration is the standard order to right the configuration. However, many chemists would be fine with your order, too, since both describe which electrons are where. If you re taking a chemistry class you should double check with your teacher, though, to see if they re okay with you using a different order."
    },
    {
        "video_name": "L-0FkEPPdXE",
        "Q": "Sal writes at about 4:00 Nd: [Xe]6s^2 4f^4. Could you write Nd: [Ba] 4f^4 too? Or do you just substitute in a noble gas and then start at the beginning of the next period? Thanks in advance!",
        "A": "When using this shortcut, you must start with a noble gas and work from there."
    },
    {
        "video_name": "Bt0mz4mGddk",
        "Q": "1:58, Im sorry, what does the \"d\" stand for, I haven't seen the equation that way before.",
        "A": "The d is the sign of the derivative. In this case, the d is part of the d/dt , which means change of ___ with respect to time . In this case, since it is d[A]/dt, it means the change of the concentration of A with respect to time, or the rate of how fast the concentration of A is changing as time goes on."
    },
    {
        "video_name": "VXLSTd_dlKg",
        "Q": "At 6:20, What is the centromere made of?",
        "A": "Centromere is the central (in case of metacentric chromosomes) part of the chromosome which is made up of: An Electron dense material in centre and two kinetochores which surround the electron dense material"
    },
    {
        "video_name": "VXLSTd_dlKg",
        "Q": "At 5:09, why does Sal refer to the strand of DNA as a chromosome? I thought it was still in chromatin form? And if it hasn't formed a chromosome yet, would the duplicated strand still be called a sister chromatid?",
        "A": "He is correct but inside genetics there chromosomes and inside chromosones theres dna and im not 100% sure what is inside dna :) Hope this helps :)"
    },
    {
        "video_name": "VXLSTd_dlKg",
        "Q": "At 5:41, are chromosomes and chromatids the same thing?",
        "A": "A chromatids are just pairs of chromosomes. If they split, then they would be called separate chromosomes."
    },
    {
        "video_name": "VXLSTd_dlKg",
        "Q": "At 3:01 to 3:06 of the video I have a question: Does the number of chromosomes determine the complexity of the organism?",
        "A": "No, not at all. Adders-tongue, a plant, has 1260 chromosomes whereas humans have 46. A few more examples; pineapples 50, elephant 56, potato 48. We wouldn t consider potatoes more complex than we are at all. Chromosome count has nothing to do with complexity, intelligence, or any other way we try to rate ourselves and other species."
    },
    {
        "video_name": "SqE5S1qS7ZU",
        "Q": "Is it actually called 2,3-DPG? The notation at 6:18 says that it's called bisphosphoglycerate, so wouldn't it be BPG?",
        "A": "The prefixes bis- and di- are often used interchangeably. Bisphosphoglycerate seems to be more common, but diphosphoglycerate would also be correct and both abbreviations would be correct."
    },
    {
        "video_name": "gqrh8wbPXVE",
        "Q": "At 4:23, how did carbon suddenly get another proton and become Nitrogen. Without an electron, it should Stay a Carbon isotope and become an ion (+1), right?",
        "A": "In beta decay, one of the down quarks that composes a neutron (two down and one up quarks) in C-14 decays via the weak interaction into an up quark, leaving a proton (two up and one down quarks). The W\u00e2\u0081\u00bb boson from the weak interaction quickly decays into an electron and an anti-neutrino."
    },
    {
        "video_name": "enr7JqvehJs",
        "Q": "At 6:51,why did he put *\"-\"* before J?",
        "A": "The pattern for calculating the determinate is +,-,+. Because J is the second component, you subtract it from the determinate. The other two components (I, K) are added to the determinate."
    },
    {
        "video_name": "enr7JqvehJs",
        "Q": "At 1:25 the i unit vector comes up.\nIt is a silly question - but why does it have like half a triangle on top of it...soon the j, and k have it too. (I bet I've missed something somewhere.)",
        "A": "That s just how we indicate that it is the unit vector. As opposed to, say, the square root of -1."
    },
    {
        "video_name": "enr7JqvehJs",
        "Q": "at 7:57 where did that minus come from",
        "A": "This comes from the definition of a determinant for a 2x2 matrix, if you have a matrix | a b | | c d | then the determinant for that matrix will be ( a*d - b*c )"
    },
    {
        "video_name": "enr7JqvehJs",
        "Q": "Sal said at 5:08 in the video that he is going to do a proof can some one tell me where is it?",
        "A": "It s in the linear algebra playlist."
    },
    {
        "video_name": "FaOOx6IZxV8",
        "Q": "At 1:59, why does the bromine have a formal charge of +1 after it bonds with the two carbons? It has 8 valence electrons doesn't that make it negative?",
        "A": "No, formal charge = valence electrons - non bonding electrons - bonding electrons/2 7 - 4 - 4/2 = +1 Think about it another way, a neutral bromine has 7 electrons. This one has 2 bonds and 2 lone pairs. If the bonds were broken equally and we gave 1 electron to each atom of each bond, bromine would now have 6 electrons (2 from the 2 broken bonds and 4 from lone pairs) 7 - 6 = 1"
    },
    {
        "video_name": "FaOOx6IZxV8",
        "Q": "4:02 Why the halide ion formed won't act as a nucleophile and attack the hydrogen from the H2O added to the compound?",
        "A": "The halide ion formed can act as a base and attack the hydrogen from the H2O added to the compound. But H2O is more likely to attack because: 1. The H2O molecules vastly outnumber the X- ions. 2. H2O is a much stronger base than X-."
    },
    {
        "video_name": "FaOOx6IZxV8",
        "Q": "I'm pretty sure I understand this, but I just want to make sure. 2:12, Jay says that the halogen attracts the magenta electrons closer to it. Wouldn't it attract both sets of electrons closer to it and you would have a choice of which carbon the water would attach. If I'm wrong and for some reason the magenta electrons are more attracted to the halogen than the other set, then why?",
        "A": "You are correct. In this case the substrate is symmetrical, so the halogen attracts both electrons equally."
    },
    {
        "video_name": "FaOOx6IZxV8",
        "Q": "3:00 Why wont Br- attack the partially positive C instead of H-OH?",
        "A": "It can. Indeed, Br- is a stronger nucleophile than water, because of its negative charge. But there are so many more water molecules present than bromide ions that the chances of attack by a water molecule are much greater than by a bromide ion. For example, the concentration of water may be about 55 mol/L The concentration of Br2 may be about 0.1 mol/L. and only a very small fraction of these exist as Br- ions at any given instant."
    },
    {
        "video_name": "FaOOx6IZxV8",
        "Q": "At 06:34, you mentioned that the the carbon on the right is a partial carbocation. I'm a little confused on how that carbon, as well as the one on the left, have partial positive charges.",
        "A": "The Br atom has a formal positive charge. Also, it is a fairly electronegative atom. The Br wants electrons! Is gets some of this electron density by drawing the electrons in the C-Br bonds closer to itself. This decreases the electron density around the two carbon atoms and gives them a partial positive charge."
    },
    {
        "video_name": "FaOOx6IZxV8",
        "Q": "At 6:40, How is a tertiary carbon electrophile undergoing an SN2 reaction? Wouldn't the other 3 Carbons block it with Sterics? If there were a Hydrogen present instead of that Methyl group I would see no problem. Thanks for the videos!",
        "A": "Don t forget that that at point, the bonds are highly strained.The tertiary carbon is almost trigonal planar, so it s easy for a nucleophile to attach."
    },
    {
        "video_name": "H3XT-YU2n0c",
        "Q": "At 4:34, Hank says some of the pollen may find its way to a female cone.\nIsn't the chances of this happening very low? How are there so many pine trees when the chances of fertilization are low?",
        "A": "Yes, the chances are very low. But they have large numbers on their side. Something may only happen 0.01% of the time, but if you try it a million times, you re likely to achieve the result."
    },
    {
        "video_name": "I6LsWSXx8fg",
        "Q": "at 13:25 does Sal mean to write conformation?",
        "A": "He does, well spotted!"
    },
    {
        "video_name": "I6LsWSXx8fg",
        "Q": "This may be a \"which came first- chix or egg\" question, but it really is sincere:\n\nat 13:22 is the molecule considered \"anti-configuration\" primarily because it is 1) in the most stable position? or 2) because a methyl group forms a 180 degree angle with another methyl group?\n\nP.S. please don't say both",
        "A": "I promise not to say both. . Oops, I just said it. Anyway, 2 is correct. Anti means opposite . The anti configuration happens to be the most stable."
    },
    {
        "video_name": "I6LsWSXx8fg",
        "Q": "at around 13:15 you say, \"... so it's further away from other methyl groups...\" when you were referring to the other carbons in the ring. They aren't technically called methyl groups right?",
        "A": "yea you re right, he meant CH2 groups. remember the ending -yl only refers to groups off the main chain"
    },
    {
        "video_name": "dFFzAP2OZ3E",
        "Q": "4:5. If the molecules spread from one corner of the box into the whole box, no increase in entropy happens? So, the fact that the molecules very rarely gather together back into the corner has nothing to do with the second law?",
        "A": "Sal states that entropy is only defined for systems in equilibrium. Surely, a box with all particles in a corner is not in equilibrium. If so, it doesn t makesense to say that the particles spreading into the entire box increases entropy. Still, that the particles won t surely reduce themselves back to the corner may have a relation to the second law: if the particles did, you could perhaps put a wall and get two systems whose added entropy is lower than thhe original s."
    },
    {
        "video_name": "vzID7ds600c",
        "Q": "At 3:30 why is the pressure upwards at the bottom of the cube greater than the pressure at the top?",
        "A": "If the body submerged in the liquid is a 3 dimensional object, the object clearly has a height. We know that the liquid pressure is directly proportional to depth. Since the depth at the top is lesser than the depth at the bottom, the pressure at the bottom of the cube is greater than that at the top of the cube."
    },
    {
        "video_name": "vzID7ds600c",
        "Q": "At 8:00, to summarize Archimedes principle- for every submerged object the weight of water displacement equals the object's weight? Is that right? Thanks.",
        "A": "No. Nothing was talked about the object s weight. It summarizes saying that the buoyancy force acting on the submerged object is equal to weight of displaced liquid, which depends only on the volume of the object. That is not the only force acting in the object though, there is also the object s weight but it was not mentioned so far."
    },
    {
        "video_name": "vzID7ds600c",
        "Q": "Refer 00:34 & 02:25 sec in Fluids(Part 5). We know pressure varies as (density*g*h) for a jar with liquid to level h (and no objects). Please educate me on how the pressure at each point of the jar will be effected if a cube denser than the water is suspended (and immersed) in the water using a thread. Is the profile of pressure in the jar w.r.t the height of the jar going to remain unaffected with the introduction of objects into the liquid. If so why? Please explain me mathematically.Thank you",
        "A": "what IS Archimedes Principle?? i dont understand"
    },
    {
        "video_name": "vzID7ds600c",
        "Q": "7:24: When Sal shows that the volume of the cube is also the amount of liquid displaced, is this true with any/every object or does this just so happens in this example? Please answer, Thank you!",
        "A": "Yes, whatever is the volume of object immersed in liquid ( be it the whole object or a part of it) , that is the amount of liquid displaced. Its true for any object of any shape."
    },
    {
        "video_name": "vzID7ds600c",
        "Q": "Hi, I think that the pressure is scalar. So that the pressure has no direction. He should to say \"force of pressure\" (at 1:46)? Is it right? Thx, P.",
        "A": "pressure is a tensor quantity ; neither scalar or vector , a coordinate-independent geometric quantity."
    },
    {
        "video_name": "vzID7ds600c",
        "Q": "At 8:08, Sal says the net upward force of submerged objects equals the weight of liquid displaced. Would it be accurate to say this applies to gases too since we're talking about weight, and possible compression wouldn't change weight?",
        "A": "That is correct. Archimedes principle is a law of ideal fluids. Fluids aren t just liquids, they also include gases."
    },
    {
        "video_name": "vzID7ds600c",
        "Q": "At 0:50, Sal says that even though the pressure is dependent on depth, it acts in all directions. How does this make intuitive sense? Also, if pressure is dependent on acceleration due to gravity, it shouldn't exist in the upwards direction at all (since gravity acts downwards).",
        "A": "WHen you are standing on the ground do you feel pressure against your feet? This is because you are being pulled down by gravity but the ground is pushing up with an equal force. In liquid you have the same thing, the liquid above the depth is being pulled down by gravity but the liquid below it has to have provide enough pressure to hold it up."
    },
    {
        "video_name": "vzID7ds600c",
        "Q": "at 6:42, would it be the density of the fluid instead of saying the density of the liquid ? or is it arbitrary ?",
        "A": "It would be a little more correct to say fluid instead of liquid , but it s a very minor distinction. After all, all liquids are fluids, but not all fluids are liquid. And the main difference between liquids and other fluids is that liquids are incompressible. But that particular characteristic has no impact on this example, so there really is no problem in using one or the other."
    },
    {
        "video_name": "vzID7ds600c",
        "Q": "at 1:10 u said that on every particle of water equal force is exerted cos of which it doesnt move. but arent the molecules of any fluid always in motion?",
        "A": "They are in Brownian motion, but that motion is random (and very small) so it cancels out and doesn t really matter for a problem like this."
    },
    {
        "video_name": "vzID7ds600c",
        "Q": "at 6:37 when (dpg)d^2, it should be equal to d^3pd^2gd^2. why it is only d^3 pg",
        "A": "He s multiplying and not adding and that s why the distributive property that you are talking about won t work. For eg-when I say 2^2(5*4*2)=2^2(20*2)=4*40=160 160 is actually 8 multiplied by 20 ie 2 cubed multiplied by 5*4 That s what he s doing when he says d^2(dpg).He s actually multiplying d*d*p*g*d ie d^3pg. I hope this helps.:):):)"
    },
    {
        "video_name": "vzID7ds600c",
        "Q": "At 6:33, why does sal keep on saying p instead of rho?",
        "A": "Because he doesn t want to say rho"
    },
    {
        "video_name": "Y0eWnOZpSpQ",
        "Q": "At 2:26, so a volcano or a mountain would form at a subduction zone?",
        "A": "Yes, volcanoes form at the subduction zone because its like a the mid ocean ridge. When two plates collide and one subducts below the other there is usually a small space that magma can squeeze through, thus creating volcanes"
    },
    {
        "video_name": "uCW6154hPkc",
        "Q": "Although Jay points out only one beta Carbon at 01:28, actually there is another beta Carbon left to the alpha Carbon in the drawing, but since it doesn't matter which one to take, the other one is not mentioned, right?",
        "A": "Yes, there is another beta carbon in the ring, and the methyl group is also a beta carbon. It does matter which one you choose. You could remove H atoms from any of these, but you would get a different product in each case (actually, you get a mixture of all three products). Jay just showed you the mechanism that gives the most stable, and therefore the major, product."
    },
    {
        "video_name": "uCW6154hPkc",
        "Q": "Why is there no stereochemistry for isopropyl group at 06:33? I don't understand Jay's explanation about the reason.",
        "A": "There is stereochemistry. He is really saying that the isopropyl group is no longer pointed up or down. Since it is on a double-bonded (sp2 hybridized) trigonal planar carbon, it is neither up nor down but in the plane of the ring."
    },
    {
        "video_name": "uCW6154hPkc",
        "Q": "7:50, the preference depends on the base right, if it is sterically hindered or not? And here, we are just assuming the base is not sterically hindered?",
        "A": "The base is ethoxide ion, so it is not sterically hindered. However, the preference comes from the requirements of the E2 mechanism The important point of this video is that E2 eliminations must be anticoplanar. In cyclohexanes, E2 eliminations must be trans-diaxial even if the cyclohexane ring has to assume an unfavourable conformation to do this."
    },
    {
        "video_name": "uCW6154hPkc",
        "Q": "At 7:06 Why does the isopropyl group become flat and planar and no longer pokes out at us?",
        "A": "Because the location of the double bond has changed. The carbon that has the isopropyl substituent is now involved in a double bond, so is sp2 hybridized and trigonal planar. That is, the isopropyl substituent will now be in the plane of the page. The cyclohexene on the left has the isopropyl group coming out of the page because in that structure the carbon with the isopropyl substituent is not involved in a double bond so is sp3 hybridized and has tetrahedral geometry."
    },
    {
        "video_name": "uCW6154hPkc",
        "Q": "at 9:21, jay said the leaving group should be axial. Why must it be axial?",
        "A": "For the chlorine group to form anti peri-planer position with the hydrogen it must be in axial position.Also since in the equitorial position the co planerity is not achieved by the chlorine and the hydrogen atom.hence the axial position is required."
    },
    {
        "video_name": "uCW6154hPkc",
        "Q": "At 11:20, is the product from the reaction with the minor conformation of menthyl chloride. This means that the minor conformation is being used up to make the product. This puts stress on the equilibrium between the minor and major conformations, so that equilibrium should shift toward the left. Does this continue to the point where there is no longer any of the major conformation?",
        "A": "Essentially, yes. This is an example of Le Ch\u00c3\u00a2telier s priinciple in action. As you remove one component in an equilibrium, the system responds in such a way as to relieve the stress by producing more of that component."
    },
    {
        "video_name": "1aJBToJrlvA",
        "Q": "At 10:00, I thought that the cells couldn't live without mitochondria because they wouldn't get the energy to do stuffs.",
        "A": "anaerobic cells can live without mitochondria because they need no oxygen for respiration they get energy from the glycolysis process which occur in cytoplasm not in mitochondria."
    },
    {
        "video_name": "1aJBToJrlvA",
        "Q": "refrence to 1:00 aren't there a membrain AND a cell wall?",
        "A": "In plant cells, there is both a cell membrane and cell wall, but in animal cells, there is only a cell membrane."
    },
    {
        "video_name": "1aJBToJrlvA",
        "Q": "Referenceing 10:55, if it is true that all mitocondrial DNA can be traced back to a single \"Eve\" mitocondria and it doesn't combine with \"father\" mitocondrial DNA, does that mean that all mitocondria in the world are the same? How do variances occur?",
        "A": "Variances occur through mutation"
    },
    {
        "video_name": "1aJBToJrlvA",
        "Q": "At 10:00-10:05, if cells didn't have mitochondria before, then how did they produce energy?",
        "A": "I am not an expert but I think they might have used glycolosis before cells had mitichondria. I have read about glycolosis and it produces a net 2ATP without mitichondria. It is possible they used another method of course."
    },
    {
        "video_name": "1aJBToJrlvA",
        "Q": "At 8:50what is rRNA?",
        "A": "rRNA is ribosomal RNA, essentially RNA specifically generated to produce ribosomes. It is produced and stored in the nucleolus, a region within the nucleus."
    },
    {
        "video_name": "1aJBToJrlvA",
        "Q": "seedless vascular plants? what are they 5:00",
        "A": "They are mainly ferns and horsetails, but most people aren t very familiar with them. They have vascular tissue so can transport nutrients more efficiently than the non-vascular plants (mainly mosses) so typically grow larger. They tend to be in wet environments because they have swimming sperm instead of pollen which is more common in plants with seeds (gymnosperms and angiosperms)."
    },
    {
        "video_name": "1aJBToJrlvA",
        "Q": "At 4:00, what about Pseudopods?",
        "A": "There two arm-like projections of an amoeba which they can use to consume food and move."
    },
    {
        "video_name": "1aJBToJrlvA",
        "Q": "what was that thing at around 0:50?",
        "A": "That was a basic diagram of a animal cell"
    },
    {
        "video_name": "1aJBToJrlvA",
        "Q": "At 4:25 Hank references City. What exactly is he referring to by city?\nAnd aren't the centrosomes involved in cell division, so are they not a separate organelle altogether?",
        "A": "He means that the cell is like a city, with sewage, roads and stuff. Symbolically"
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "The probability function that Sal demonstrates around 1:47 gives me the impression that electrons constantly change energy levels \"sharing\" the energy that moves them from each shell for an extremely sort period of time. Is that true? I read similar questions in the comments section but i don't really get the answers... thx!",
        "A": "The electrons stay within their own orbital unless they get energy from outside."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "Why doesn't Sal use 1s1 when he has doing the electron configuration for carbon on 7:20",
        "A": "There are 6 electrons in carbon, so 1s2 is necessary. Because carbon is in the 2nd energy shell, it fills up the 1st. 1s1 wouldn t fill up the first energy shell."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "If we looked at the electron configuration for Boron (1s2 2s2 2p1), would the single electron in the 2p orbital occupy both sides of the \"dumbbell shape\" or only one side? I looked at the chart at 6:00 and noticed the two sides of the orbital don't seem to actually touch.",
        "A": "The 1 p electron would occupy both sides of the dumbbell shape. As these shells are probability distributions in which we can most probably find the electron, that means that there is a very small probability in which the electron is not found within the volume of the orbital. This is when the electron could move from one side of the dumbbell to the other."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "What is Sal doing at 6:22?",
        "A": "He fills in the electrons in each p orbital. Each orbital can consist of maximum two electrons, and the lines represent the electrons, I suppose. If you didn t understand my meaning, please write a more specific question, maybe I can answer it. Hope I could help."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "11:50. why did you put 2p\u00c2\u00b3 ?",
        "A": "period block^group"
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 2:30 ,how can we know that electrons don't orbit around, but they jump, without seeing them?",
        "A": "Indeed, we do not actually know for sure that it even jumps! Most of people only believe what they perceive with their eyes... However, the most accepted theory today is quantum mechanics (an abstract theory I would undoubtedly say), but I am not an expert myself so I don t want to give you false inclines. If you are interested, do some research by your own, because this truly is a bemusing topic!"
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "Given what has been talked about in this video (up to 5:30) and the last video, if you added more electrons (making it a negatively charged \"atom\"/without adding more to the nucleus) would you increase its mass? Considering you are expanding the orbital field and its 99% of the atoms mass.",
        "A": "If you are adding more electrons to an atom, you are increasing its mass, because electrons have mass. If, for example, you added an electron to an oxygen atom, the mass of the atom would increase by only 0.003 %. This is such a small change that, for ordinary measurements, we can say that the mass of the ion is the same as that of the atom."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "AT 9:30\nHow do you know when to stop with the electron configuration?",
        "A": "You stop adding electrons to the configuration when you have accounted for all of the electrons in the atom."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 0:08, Sal says \"newtonian\" or something similar sounding. What does that mean?",
        "A": "Newtonian = according to the laws of motion as understood by Newton."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 4:55, at the table that the atomic orbitals are depicted, there are f orbitals only for n 4 and there are d for n5, p for n6 and only s for n7. Other depictions that i have seen, presen n5 with f, n6 with d and n7 with p, Why is that? Are there different opinions? Are they both correct?",
        "A": "The issue is that there are no elements massive enough to actually have f orbitals occupied in the n=6 or n=7 shells. These orbitals still exist, they re just empty -- though they might have an electron in them if the element is in an excited state."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 3:54, why can't there be three electrons there? Why is there only two?",
        "A": "Each atom can only hold a certain amount of electrons due to molecular structure and what-not."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 12:14, how are you able to check if you have done your electron configuration correctly?",
        "A": "when you add the digits of your configuration u will get the atmic number of the element. for example- 2,4 check=2+4=6 which is the atomic number of carbon"
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At around 2:40, how do we know the probability of finding an electron in a given orbital? I thought we couldn't ever know EXACTLY where an electron is. So how would we collect data to get, say, \"90%\"?",
        "A": "The laws of quantum mechanics tell us how to calculate the probability. The orbital is just a graphical display of that probability function. We never do know exactly where an electron is. We don t develop any of the probability data by checking a random sample of electron locations. It s all math."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 6:30, ok so in the case of p orbitals, how would you identify say p sub x in comparison to y when you could observe it from a different angle and see the other",
        "A": "To add to Just Keith s great answer, when an atom is bonded it is common to define the line connecting the two atoms as the z axis."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 11:25 how do we know that nitrogen has 3 orbitals in the P block? No explanation at how Sal arrives at that because he is going backwards in calculating that? Also for Silicon totally confused at how he arrived at the solution. What is the maximum electrons in a P shell?",
        "A": "N is atomic number 7. Start filling the orbitals 1s2. That leaves 5 to go. Next is 2s2. Now there are 3 to go. Next comes p. So 2p3. Max in the p shell is 6. Max in d is 10."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "at 9:20, what happens to the electrons from the 1s subshell? Do they become part of the dumbbell shape of the p subshell, or are they still in the original sphere position?",
        "A": "They stay where they are"
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "In 10:00, Sal moved Helium from group #18 and moved it beside Hydrogen under group #2 , I understand that he moved it to give us a closer idea, but what is the correct group for Helium?",
        "A": "He only moved it there because it has the same electron configuration as group 2 (s2), but helium behaves as a noble gas not as a metal, so it is in group 18."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 04:44, in the column heading, shouldn't it be - s, px, py, pz? pz came first.",
        "A": "Thank you for clarifying my doubt! :)"
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "at 2:55, what makes electrons go close to the nucleus sometimes, but sometimes far away?",
        "A": "what I think Sal is trying to say is that the electrons fill up a cloud and that if we want to map the area where the electrons are 90% of the time, that they would be within a certain regions. In that region of around 90% probability, you will find electrons close to the nucleus and those that are not so close. As we advance in the course, we will later learn about Electron Shielding, and other effects that explain such positioning, but that will be explained later."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 13:30, when he is giving the electron configuration for Silicon (Si) how does he know to put 6 for 2p? Do you have to just guess or is there something that can indicate this?",
        "A": "Its actually the highest number of electrons that can go into p orbital. There is a rule for that."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 3:49, I still do not get the fact of putting and what does it mean when you put 1s2?",
        "A": "In 1s\u00c2\u00b2, the 1 is the number of the energy state and s is the type of orbital in that state So, 1s tells you that you have an s orbital in the lowest energy state. The superscript tells you how many tells you how many electrons are in that orbital. So 1s\u00c2\u00b2 tells you that you have two electrons in an s orbital in the lowest energy state."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "@6:10, why are P-sub-z orbitals in the vertical direction? I thought that was the y-direction.",
        "A": "When looking at a coordinate plane it appears that the y axis is going up and down but in reality it is flat so it is going back and forth. and the x axis is going side to side. the z axis comes in when talking about 3 dimensional things. z goes up and down."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "What is Tetris? (3:10)",
        "A": "It s a game. Shapes fall from above and u hav 2 make it fall in such a way that a row is completed. If that happens, the row vanishes. the game is over when the stacked shapes touch the top"
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "at 10:38 IS Helium in the s or p block and whats the definition of a shell.",
        "A": "s block- only 2 electrons fill the 1st energy level (1s2). A shell is the energy level that the electrons fill."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 5:50, when he is naming the orbitals, what do the S, P, D, and F stand for?",
        "A": "s , p , d and f are names for orbitals. They were originated from descriptions of the spectral lines resulting from electron spectroscopy experiments on atoms within each of the blocks of the periodic table. They stand for sharp, principal, diffuse and fundamental"
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 12:35 in the video, I have a question about the periods. I understand the 1s1 and 1s 2. I don't understand Li since its in period 2 why would it not be 2s3? Thanks!",
        "A": "Because s shells can only hold two electrons. p shells hold 6, d shells hold 10, and f shells hold 14."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "@ 8:53, sal says that the first electron goes into the Pz orbital. doesnt it go into the Px orbital or does that matter?",
        "A": "The electron can t tell the difference between Px, Py, and Pz. It can go into any of them."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "@8:03 why is it 2p^3 isntead of 1p^2?",
        "A": "There are no 1 p shells due to the space about the atom s nucleus. Similarly, there are no 2 d or 3 f sub-shells."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "I saw the periodic table many times in the lecture and I observed and element(Dont know its name) with symbol Tc being put with a dashed square on it. Look in at 04:21. It's got 43 atomic number.\nI dont understand. Is it an exceptional element or something that was discovered very much later on?",
        "A": "Tc is Technetium , and was only discovered with the help of technology. so yes, it was discovered much later than the elements around it, and, strangely enough, very much radioactive and used in medicine."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "at 1:15, sal says that the first orbital, that is the one nearest to the nucleus has the lowest energy. but wont the closest orbital have the highest ?",
        "A": "This is the potential energy of the electron. The electron is electrostatically attracted to the nucleus, just as you are attracted by gravity to the centre of the Earth. It takes energy to pull an object away from something that is attracting it, so the potential energy is higher as the distance increases and lower as the distance decreases. Electron energies are zero when the electron is at infinite distance from the nucleus and become more negative as the distance decreases,"
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "at 7:10, why are electrons designed to be with specific orbitals?",
        "A": "How can we answer why questions like that? This is how our universe is. Why is it that way? There s no way to know."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 2:00 you say that the electron could be anywhere but they appear to be bounded by these non-overlapping orbitals. Can an electron really be anywhere or is there a limit to how far it can be from the nucleus?",
        "A": "There is no limit to how far it can be, but the odds of it being very far away are so infinitesmally small that you can ignore them because it will basicially never happen in the lifetime of the universe. It s like asking is there any limit to the number of consecutive times you can roll a 6 on a die. There s no limit, but you aren t going to roll 20 in a row in your lifetime."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 1:22, Sal says Helium has two electrons, but doesn't he mean two protons?",
        "A": "In a neutral atom, (i.e. one that does not have an overall positive or negative charge) the number of protons and electrons are the same. Sal was correct to say that Helium has two electrons. The neutral Helium atom has two protons and thus also two electrons."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "Why are Carbons electrons configured - 1s2, 2s2, 2p2.... Why is it 1S2? Isnt the 1 orbital just for period 1... I am talking about 8:31 in the video.",
        "A": "You forgot about Helium from right-side of table with 1s2 configuration. You also may write like this [He]2s2 2p2"
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 4:08 why is Sal plotting the 2s orbital(green dots) on top of the 1s orbital? They have the same diameter? Shouldn't it be bigger in diameter?",
        "A": "Finding an electron is all about probability. If you re-watch the video, Sal mentioned that the probability of finding an electron in the 1s orbital is near to the center of the atom is very high. The 2s orbital is farther away from the center. There might be overlap between electrons in the 1s and 2s orbitals. However, the probability of finding a 2s electron near the center is very low."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 11:58, what is the difference 2s1 and 1s1?",
        "A": "He is describing the electron configuration of lithium: 1s\u00c2\u00b22s1. In 1s\u00c2\u00b2, the 1s shows that we have an s orbital at energy level 1, and the superscript 2 tells us that there are two electrons in this orbital. In 2s1, the 2s tells us that we have an s orbital at energy level 2, and there is one electron in this orbital."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "I was still a little confused about P-orbital filling order, like with carbon and nitrogen.\nShouldn't carbons electron configuration be 1s2, 2s2, 1p2. (see 8:00)\nI just wanted a little clarification on how this should be written.\n\n(same thing with nitrogen, is it actually supposed to be 1s2, 2s2, 1p3)\n\n*also, does it matter whether you specify x, y, or z position of the orbital?",
        "A": "There is no 1p orbital, p orbitals start from the second shell on, so that s why they are 2p. If you need the specifics of why I can go into this but the quantum numbers videos go over this too. If you wanted to be specific you could do 2px1 2py1 2pz0 for carbon and 2px1 2py1 2pz1 for nitrogen, but generally we don t bother. Doing this would get very long and complicated around where we get to the transition metals."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 12:30, when you write out the electron configuration for nitrogen, why must you write 1s^2 2s^2 then 2p^3? Wouldn't just writing 2p^3 imply that all the all the electron shells underneath are \"filled up\"?",
        "A": "No. Because when you go for higher configurations(more number of electrons), the electrons in the lower orbitals are not always filled before the electrons in higher orbitals. e.g. the 4s orbital. After 3s you fill 3p but after that you fill 4s and then 3d. For example, the electronic config. of titanium is 1s^2,2s^2,2p^6,3s^2,3p^6,3d^2,4s^2."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "I do understand how there's the 90% chance of finding the electron in a certain area (2:32), but there is eventually a point where it can't be right? I mean you can't have that electron be 5 light-years away, but you can have it in it's shell. Is there a defined area where the electron physically cannot be?",
        "A": "No, there is not eventually a point where it can t be. There s only a point where it is very improbable to be. The probability of being 5 light years away is infinitesimally small. The probability of being even a few inches away is infinitesimally small."
    },
    {
        "video_name": "FmQoSenbtnU",
        "Q": "At 1:10, I thought there were multiple electrons in the first shell? I mean i am not the chemistry type but that's what I thought. Is that true? Cant there be at least 2 electrons?",
        "A": "Shell n=1 holds a maximum of 2 electrons."
    },
    {
        "video_name": "oeyBfrx5RJY",
        "Q": "At 6:59, why do you not put a three in front of the name? (i.e. 3-propanal.) Are Aldehydes not considered substituents?",
        "A": "The aldehyde is always going to be assumed as being on carbon 1, otherwise it wouldn t have the -al suffix."
    },
    {
        "video_name": "oeyBfrx5RJY",
        "Q": "At 7:46 can we name the aldehyde as isohexanal .. ? help me !",
        "A": "Also, iso refers to a branched carbon group (forming a Y-shape) and does NOT apply to hydrogens. Don t confuse the hydrogen on the aldehyde with a carbon - it s not a methyl group! (If it were, it would be a ketone instead of an aldehyde.) If this is difficult to see just by looking at a structure, it might help to draw C s or dots at each carbon while counting to ensure you re not including any hydrogens."
    },
    {
        "video_name": "oeyBfrx5RJY",
        "Q": "At 7:45 why is it called 2-methyl pentanal instead of 2-methyl pentenal because it has double bond with oxygen so alkene?",
        "A": "An alkene is only a double bond between two carbon atoms. Because this molecule has a double bond between a carbon and an oxygen atom (and there is a hydrogen attached to the double bonded carbon), the functional group is called an aldehyde. The presence of the aldehyde is indicated in the suffix -al."
    },
    {
        "video_name": "oeyBfrx5RJY",
        "Q": "Hi, at 2:50 can you correct my answer please ? IUPAC for \"Cinnamaldehyde\" will be: 3-Benzyl prop-2-eneal m I right ? :D",
        "A": "No. Your first answer is totally wrong (benzyl=/=phenyl) The name requires a sterochemical prefix and so is (E) 3-phenylprop-2-enal or (E) 3-phenyl-2-propenal."
    },
    {
        "video_name": "_eEONOJHnEs",
        "Q": "At 1:08, Sal says that sweat is mainly made up of water, so what are the rest of the constituents of sweat? I know that since sweat tastes salty, it has a salt content (NaCl), but what else it has along with water and salt, no matter how small the quantity?",
        "A": "i found this at Chemistry.about.com... hope it helps! Perspiration consists of water, minerals, lactate and urea. On average, the mineral composition is: sodium (0.9 gram/liter) potassium (0.2 g/l) calcium (0.015 g/l) magnesium (0.0013 g/l) Trace metals that the body excretes in sweat include: zinc (0.4 milligrams/liter) copper (0.3\u00e2\u0080\u00930.8 mg/l) iron (1 mg/l) chromium (0.1 mg/l) nickel (0.05 mg/l) lead (0.05 mg/l)"
    },
    {
        "video_name": "_eEONOJHnEs",
        "Q": "At 2:34, Sal says that it becomes it's gassiest state. Are there more or less gassy states then?",
        "A": "I think he might have meant gaseous instead."
    },
    {
        "video_name": "JgYlogdtJDo",
        "Q": "At 7:36 Sal says that 65% of us is oxygen and at 7:54 Sal then says that 18% of us is Carbon, if you add those percentages up that equals 83%. So what is the remaining 17% of us made out of?",
        "A": "Well the building blocks of life are CHNOPS (Carbon, Hydrogen, Nitrogen, Oxygen, Phosphorus, and Sulfur). So the other parts would be the other things I listed but I believe that hydrogen is right up there with oxygen since the human body is mostly water. Also there is going to be a tiny amount of other elements like gold and iron and other things but they would be a tiny tiny amount."
    },
    {
        "video_name": "-KE7jTXwNYs",
        "Q": "At 0:36, what does Sal mean by \"Kilojoules per mole?\" Is it referring to the amount of energy that was released during the chemical reaction that broke off the phosphate group from the triphosphates?",
        "A": "Joules is the unit of energy. Kilojoules is [the unit of energy multiplied by 1,000]."
    },
    {
        "video_name": "-KE7jTXwNYs",
        "Q": "what does he mean by phosphorylate a molecule at 3:45?",
        "A": "Phosphorylate means to add a phosphate group to the molecule, example adenine di-phosphate to tri-phosphate: ADP --> ATP"
    },
    {
        "video_name": "5eX5WnPDnvs",
        "Q": "At 14:27, you said the speed is 7.23m/s. Isn't that wrong because you are using only the height of the ramp, you did not add the radius for the cylinder. gh would be (4+2)*9.8?",
        "A": "When the cylinder gets to the bottom, the center of mass will still be 2m above the ground. 4m is the change in height."
    },
    {
        "video_name": "5eX5WnPDnvs",
        "Q": "02:56; At the split second in time v=0 for the tire in contact with the ground. How is it, reference the road surface, the exact opposite point on the tire (180deg from base) is exhibiting a v>0? Surely the finite time snap would make the two points on tire equal in v? Unless the tire is flexible but this seems outside the scope of this problem...",
        "A": "Relative to the ground, the contact point has no velocity. If it did, it would be slipping, not rolling. Relative to the ground, the point 180 degrees opposite the contact point has big velocity, because the ball is rolling. Relative to the CENTER OF THE BALL, the contact point and its opposite have velocities that are equal in magnitude but opposite in direction. But we are not looking at the velocity compared to the center of the ball. We re looking compared to the ground."
    },
    {
        "video_name": "5eX5WnPDnvs",
        "Q": "At 5:47 'r' being the radius of the ball,isn't 'rw'= tangential velocity of the a particle on the surface of the ball?And how is that equal to the velocity of center of mass?",
        "A": "Watch the video again, carefully, from 3:22 to 4:28. David explains the relationship between the tangential velocity of the surface of the ball and the velocity of the center of mass."
    },
    {
        "video_name": "5eX5WnPDnvs",
        "Q": "At 14:17 energy conservation is used which is only applicable in the absence of non conservative forces. However, isn't static friction required for rolling without slipping?",
        "A": "Not if the object is already rolling. Static friction is required to apply a torque. (Also, total energy is ALWAYS conserved. MECHANICAL energy (PE + KE) is conserved in the absence of non-conservative forces. )"
    },
    {
        "video_name": "FkDVucEoJpU",
        "Q": "Ok Im confused, at 3:00, when discussing how cannon-bard felt that james-lange theory was inconsistent because you can experience a physiological response of increased heart-rate from running and an emotion not be triggered, Isn't that the same thing as cannon-bard theory saying that physiological response and emotion happen at the same time. My heart racing after running a mile doesn't simultaneously trigger an emotion.",
        "A": "What he is saying is Cannon-Bard believed that you experience raised heart rate and other things without having to have an emotion to back it up, though if you do have an emotional trigger it will happen at the same time though, I m with you he could have been a little more thorough about the whole physical activation and separating it from the emotional activations."
    },
    {
        "video_name": "lsHq5aqz4uQ",
        "Q": "At 11:26, how do we calculate the pH of original buffer solution, ie : 9.33?",
        "A": "We know, pH = pKa + log [A-]/[HA] kA of NH4+/NH3 = 5.6 X 10\u00cb\u0084-10 Therefore, pKa= -log(5.6 X 10\u00cb\u0084-10)= 9.25 Given, [A-] = NH3 [HA] = NH4+ So, for the given buffer solution, pH = pKa + log [NH3]/[NH4+] pH = 9.25 + log [0.24]/[0.20] pH = 9.25 + (7.9x10^-2) pH = 9.25 + 0.079 pH = 9.329 = 9.33"
    },
    {
        "video_name": "lsHq5aqz4uQ",
        "Q": "At 8:48, why is the reaction written out as NH4 reacting with H3O instead of HCL and NH4? How would I know in future problems to first react HCL and water to form H3O and then react the H3O with NH4?\n(In the first example, you were able to react NH4 directly with NaOH instead of reacting the NAOH with H2O first, so why is it different with this example?)",
        "A": "HCl in water exists almost exclusively as H\u00e2\u0082\u0083O\u00e2\u0081\u00ba and Cl\u00e2\u0081\u00bb, so it is chemically more correct to write the equation using H\u00e2\u0082\u0083O\u00e2\u0081\u00ba. In practice, it doesn t matter which equation you use. What matters are the concentrations of NH\u00e2\u0082\u0083 and NH\u00e2\u0082\u0084\u00e2\u0081\u00ba, not the equations you produce these species."
    },
    {
        "video_name": "lsHq5aqz4uQ",
        "Q": "At 1:04, 5:48 and 10:25 why do we use the pka of NH4Cl and not NH3 in the calculations. In other problems, how would we know which pka to use?",
        "A": "NH4+ and NH3 are a conjugate acid-base pair, with NH4+ being the acid. The Henderson\u00e2\u0080\u0093Hasselbalch equation that is being used is defined in terms of pH and pKa. pKa refers to the acid, and hence the pKa for NH4+ is used."
    },
    {
        "video_name": "lsHq5aqz4uQ",
        "Q": "at 8:13 he says that \"this is all in water\", but i thought he said it was in the buffer solution (0.20M NH3 and 0.24M NH4+)?",
        "A": "the brackets indicate that it is in an aqueous sol n., although the proper labeling of NH3 and NH4Cl would be NH3 (aq) and NH4Cl (aq). For practicality, it is assumed in this case, that when there are brackets around a compound (ex. [x] , [y+] ) that they are in sol n.."
    },
    {
        "video_name": "lsHq5aqz4uQ",
        "Q": "How do you know which is the acid and which is the base at 0:29 ?",
        "A": "The acid will usually have one more proton (or H) than the base - this allows it to donate that proton and fit the definition of an acid."
    },
    {
        "video_name": "lsHq5aqz4uQ",
        "Q": "I am trying to solve the problem presented at 7:16 without using the Henderson - Hasselbalch equation. as I understand, I am supposed to divide the end concentration of the products by the end concentration of the reactants, and equal the result to Ka, then solve for [H+]. but the concentration of [H+] at the end of the process is 0. what am I doing wrong?",
        "A": "Hi there, I m not sure where you went wrong but here is a step by step working out using that method: NH4+\u00e2\u0086\u0092H+ + NH3 Ka=[H+][NH3]/[NH4+] 5.6x10-10=[H+][NH3]/[NH4+] Original moles NH3=0.12 Original moles NH4+=0.1 New moles NH3=0.09 New moles NH4+=0.13 Conc NH3= 0.18 Conc NH4+=0.26 (5.6x10-10)/(0.18/0.26)=[H+] [H+]=8.09x10-10 -log[H+]=9.09=pH"
    },
    {
        "video_name": "lsHq5aqz4uQ",
        "Q": "At 4:35, how come the initial concentration of OH- is 0.01M? If the buffer system is NH4+ +OH-->NH3 + H2O, shouldn't there already be some OH- before the sodium hydroxide is even added?",
        "A": "This is the concentration of OH\u00e2\u0081\u00bb before it reacts with the buffer. The concentration of OH\u00e2\u0081\u00bb before the buffer is added is 2.1 \u00c3\u0097 10\u00e2\u0081\u00bb\u00e2\u0081\u00b5 mol/L. This is negligible compared with 0.01 mol/L."
    },
    {
        "video_name": "lsHq5aqz4uQ",
        "Q": "At 5:56 how do we know we put NH3 over NH4+. I thought the henderson-Hasselbaclch equation was pH=pKa +log [Conjugate base] / [Conjugate acid]? Is it because the OH- was the limiting so it has run out?",
        "A": "The log term you have isn t correct, it should be log([conjugate base]/[acid]) NH3 is the conjugate base of NH4+ Does that make sense?"
    },
    {
        "video_name": "6rwoktPmqpY",
        "Q": "At 2:49 it is said nitrogen fixing bacteria are on legumes... Are they only found on legumes?",
        "A": "No, that s just one example. Beans are great, though."
    },
    {
        "video_name": "6rwoktPmqpY",
        "Q": "At 0:51\nIf we are 65% Oxygen and only 19% Carbon, why are we called Carbon-based organisms?",
        "A": "even though we have more oxygen, carbon does more."
    },
    {
        "video_name": "6rwoktPmqpY",
        "Q": "@ 3:00 Did he say kudzu? Whats that?",
        "A": "Kudzu is a group of plants in the genus Pueraria, in the pea family Fabaceae, subfamily Faboideae. They are climbing, coiling, and trailing perennial vines native to much of eastern Asia, southeast Asia, and some Pacific Islands."
    },
    {
        "video_name": "6rwoktPmqpY",
        "Q": "2:10\nHe said a triple bond. So if the plants need it how does it break apart? does it ever break apart?",
        "A": "A. It s like three steel ropes in tug of war. They are strong, but put a crazy lot of people on each side or just cheat and throw acid. and they break. B. Yes, refer up to A."
    },
    {
        "video_name": "vny1qUaToHw",
        "Q": "At 4:13, Sal said that the light bulb is an open system. Is that true? Shouldn't that be a closed system, as only energy is transferred from the system to the surroundings and matter is not? What is \"matter\" in thermodynamics anyway?",
        "A": "I agree with you that a light bulb is a closed system. Electrons flowing in and out to heat up the filament don t make it an open system. Matter refers to atoms and molecules."
    },
    {
        "video_name": "7I3-HAW8rmM",
        "Q": "When you talk about scaling the equations down (6:24), wouldn't that affect the 5 Volts on the other side as well? Does it have to do with the currents being expected to be in mA? Thank you in advance.",
        "A": "The scaling up/down by 1000 happens on just the left side of both equations. Every term on the left side is current x resistance. If all currents are in mA, and all resistance is in kOhms, then 10^-3 from the mA cancels out the 10^+3 from the kOhms, and you are just left with the integer parts. In Ohm s Law, the units cancel like this: kOhms x mA = 1. And for big resistors: MegOhms x uA = 1."
    },
    {
        "video_name": "7I3-HAW8rmM",
        "Q": "I'm having trouble figuring out how the algebra was worked out starting at 2:05. He mentioned how I1 is mentioned twice. How does that turn into -(R1 + R2)I1 ?",
        "A": "i1 appears twice in the top equation (the one right next to the Circled-3). The two terms are -R1 i1 and -R2 i1. You can factor out i1 to get ( -R1 -R2 ) i1 which is the same as - ( R1 + R2 ) i1."
    },
    {
        "video_name": "7I3-HAW8rmM",
        "Q": "At 7:07 why do you multiply the equation by 4? I can see that when you times 2 by 4 it gives you eight and it cancels out, but I don't understand why you would need to.",
        "A": "At this step I need to solve two equations with two unknowns (i1 and i2). The technique I m using is elimination . I took a peek at the two equations to see if I could figure out a way to cause an elimination to happen. As I wrote down the 4, I was thinking (but did not say out loud) that multiplying the top equation by 4 would cause the i2 term to cancel out. And it did!"
    },
    {
        "video_name": "7I3-HAW8rmM",
        "Q": "Don't i need to use Ohm's law to figure it? 10:42 ?",
        "A": "Ohm s Law has already been used during the earlier steps of the Mesh Current Method to derive the mesh currents. By the time we get to 10:42 in the video, all you need to do is add or subtract mesh currents to get the element currents."
    },
    {
        "video_name": "_4VC3IHbuW8",
        "Q": "1:30 It's just like gravitational force isn't it. How the earth is pulling on the moon by it's gravitational force, but at the same time using force to keep it away isn't it?\n(It might seem a little bit confusing but read it over a few times to get it)",
        "A": "There is only the one force! It seems like there must be 2 but there is only gravity. The moon is going around the Earth (orbitting) and it is its speed that stops it from crashing into the Earth. make sense??"
    },
    {
        "video_name": "_4VC3IHbuW8",
        "Q": "At 7:38, would not Q2's field affect the magnitude or effect of Q1's field?",
        "A": "Yes it will. But its not like that it would change the magnitude of the field."
    },
    {
        "video_name": "_4VC3IHbuW8",
        "Q": "At 11:24, isn't he electric field is 2.5 N/C when there's 4C at that point in space?",
        "A": "No. As the magnitude of the test charge, q, changes, so does the force it experiences from Q. If a 2 C particle experiences a force of 10 N; a 4 C particle will experience a force of bigger magnitude at the same position. Twice as big, to be exact. Static electric field of Q at a particular distance will remain constant irrespective of the magnitude of the test charge (provided Q does not move!)."
    },
    {
        "video_name": "_4VC3IHbuW8",
        "Q": "Around 8:20 David says that one can determine the Force on an UNknown charge if the Electric field is known..I dont get how we can calculate the electric force without knowing the charge it acts on..",
        "A": "You misheard what was being said. The instructor says that if somehow you know the electric field (both the magnitude and direction) produced by a charge Q of unknown magnitude, you can calculate the force experienced by a charge of any magnitude of your choosing, placed in that field."
    },
    {
        "video_name": "_4VC3IHbuW8",
        "Q": "At 6:12 how does the charge q2 know that it has entered another charge q1's electric field",
        "A": "The electric field is present in free space. The charge interacts with the field. If there is no field, there is no interaction."
    },
    {
        "video_name": "_4VC3IHbuW8",
        "Q": "If u place a 4n charge in the field the field would become 10/4 why did he say 20n in 11:07",
        "A": "It should not become 10N/4C, you are not calculating a new value for the electric field. What you are doing is calculating the other values, charge and force. He says the electric field is always the same no matter what value the test charge has. If you increase the charge of test charge, then the force also becomes larger, they keep their ratio constant: E = F / Q."
    },
    {
        "video_name": "_4VC3IHbuW8",
        "Q": "at 4:34, why doesn't E1 exert a force on Q1?",
        "A": "He later tells that charge cannot exert a force on itself."
    },
    {
        "video_name": "ibdidr-bEvI",
        "Q": "At 0:50 he says, in the hard way you gonna have 3 different accelerations but then later on in the easy way he says the acceleration is gonna be the same for all 3. Why is there a difference?",
        "A": "I believe I know the answer to this one. The magnitude of all three accelerations are the same... only the directions are changing. This problem is just trying to find the magnitude of the acceleration so we can combine the three objects into a single system."
    },
    {
        "video_name": "RhUdv0jjfcE",
        "Q": "At 2:24, why doesn't he write sj instead of sv because j is used for vertical vectors?",
        "A": "The naming of the variables is just convention, it doesn t change how you do the problem. In past videos he has used Sj, Sv (for vertical), and Sy (as in the y axis). Really you could call it whatever you want but those make the most sense."
    },
    {
        "video_name": "RhUdv0jjfcE",
        "Q": "in 0:12, why does he use theta for the angle above the horizontal? can't he just use x?",
        "A": "yes we can use angle x instead of theta,but using theta makes the problem a little comfortable while solving compared to using angle x"
    },
    {
        "video_name": "RhUdv0jjfcE",
        "Q": "At 0:10, He says that angle horizontal to the S line is Theta. How many degrees does Theta represent? Why does he use the Greek letter Theta instead of just saying the degree number? Is Theta a constant amount or a variable like X?",
        "A": "Theta is reserved as such for an angle. Generally, letters in the english alphabet are used as a constant value (e=2.71828 or i=rt(-1)) or are just used to substitute in values (bob is x years old, John=x+5 years old) but letters in the greek alphabet are generally used for the same functions or values (theta, alpha, beta are used for angles. delta is used for change in )"
    },
    {
        "video_name": "RhUdv0jjfcE",
        "Q": "at 2:37 wouldn't the horizontal be known as vector x?",
        "A": "Technically, the horizontal could be known as vector x or ^i (i hat). In this video, he is labeling the horizontal component simply as S_h."
    },
    {
        "video_name": "N6IAzlugWw0",
        "Q": "During the video, Sal talks about the Scientific Method. Does the Scientific Method include Reproducible? Also, What does Sal mean by Reproducible at 11:26?",
        "A": "Reproducible means that, if you complete an experiment and get a certain set of results, and then give the instructions you followed for your experiment to another scientist, that other scientist should be able to get results really close to yours. Reproducible is essentially a measure of how confident you can be in your findings-- if someone else can verify your results by getting something similar, that lends a lot of credibility to your findings."
    },
    {
        "video_name": "N6IAzlugWw0",
        "Q": "At 6:24, wouldn't it have been better to take water from the regions that Sal was testing (the lake and the ocean) and then conduct the experiment?",
        "A": "Yes, it would definitely have been better to use pond water and sea water. Sal s experiment used distilled water and distilled water with salt in it, rather than actual pond & sea water. Pond water and sea water probably contain all sorts of bacteria and chemicals and particles that could change the freezing point- none of which can be found in distilled water."
    },
    {
        "video_name": "N6IAzlugWw0",
        "Q": "At 9:54,it says that you have to test something out.What if scientists have already did it?",
        "A": "This lesson was very helpful because"
    },
    {
        "video_name": "N6IAzlugWw0",
        "Q": "At 3:37 you say you couldn't test if a fairy was performing magic on the pond. But couldn't you test that with 24/7 surveillance? Common sense suggests that it would be a waste of time, but I don't think it's accurate to say it can't be tested. Doesn't a lot of science attempt to detect phenomenon that can't be observed directly?",
        "A": "This is because even with 24/7 surveillance the fairy might not do it again and even so, it may have happened only on that one day. It might not happen at the same time the next day and is not really something that can be observed or explained using the scientific method. I do understand what you are trying to say though."
    },
    {
        "video_name": "N6IAzlugWw0",
        "Q": "From like 8:00-8:10 when he asks if neither froze or both froze, afterwards he says we should find a different explanation but, shouldn't we try different temperatures and then find a different explanation?",
        "A": "It depends on how saturated the solution is. The more salt the lower the freezing point."
    },
    {
        "video_name": "owA2rUggKoA",
        "Q": "At 6:48, why are you assuming that x is a lot smaller than 0.45. Shoudn't you want to get an exact answer instead of an approximation?",
        "A": "Because this video is called Small x approximation . So, no."
    },
    {
        "video_name": "vFfriC55fFw",
        "Q": "at 6:20 , which resonating structure of the acyl group is more stable ?",
        "A": "The structure with the + charge on C is more stable The structure that puts a + charge on an electronegative O atom is higher energy."
    },
    {
        "video_name": "vFfriC55fFw",
        "Q": "In 3:20, Sal says that the electron from the acetyl chloride is given to the AlCl3, but is the partial charge of the Aluminum strong enough to fully take away an electron from chloride? Isn't it more like a Dipole-dipole interaction?",
        "A": "The AlCl\u00e2\u0082\u0083 is a strong Lewis acid. It is more likely that the first step is the formation of a Lewis acid/base complex: CH\u00e2\u0082\u0083CO-Cl + AlCl\u00e2\u0082\u0083 \u00e2\u0086\u0092 CH\u00e2\u0082\u0083CO-Cl\u00e2\u0081\u00ba-Al\u00e2\u0081\u00bb-Cl\u00e2\u0082\u0083 In the next stem, the complex falls apart to form the acetyl cation. CH\u00e2\u0082\u0083CO-Cl\u00e2\u0081\u00ba-Al\u00e2\u0081\u00bbCl\u00e2\u0082\u0083 \u00e2\u0086\u0092 CH\u00e2\u0082\u0083CO\u00e2\u0081\u00ba + Cl-Al\u00e2\u0081\u00bbCl\u00e2\u0082\u0083."
    },
    {
        "video_name": "vFfriC55fFw",
        "Q": "at 9:30 where does the chloride go since the hydrogen took its electron?",
        "A": "The H doesn t take the electrons. The Cl- removed the H from the ring and formed HCl, and the electrons from the C - H sigma bond went to form the third pi bond in the benzene ring."
    },
    {
        "video_name": "Hqzakjo_dYg",
        "Q": "At 3:18 the number is 13.8629... but when i took ln of 0.5 and divided it by -0.5 i got 1.38629...\nWhy? Shouldn't they be the same?",
        "A": "You are off by a factor of 10 because you should have divided by -0.05 not -0.5"
    },
    {
        "video_name": "Hqzakjo_dYg",
        "Q": "At 6:26, What is \"e\" for? why the alphabet becomes 2.71?",
        "A": "e is a constant in math"
    },
    {
        "video_name": "Hqzakjo_dYg",
        "Q": "What doe kt stand for? 0:18",
        "A": "k = any constant, and t = time."
    },
    {
        "video_name": "Hqzakjo_dYg",
        "Q": "at 3:03 what's the log inverse.",
        "A": "10 to the power of X i.e log (100) = 2 Therefore 10 ^ 2 = 100 which is true :) As a more general rule Log a ( b ) = Log ( a ^ b ) In chemistry log base 10 is used almost all the time so: a will always = 10 edit: Just saw that in this video Log base e is used and not log base 10. i.e the inverse of Log base 10 ( b ) = 10 ^ b and the inverse of Log base e ( b ) = e ^ b Hope that helps :)"
    },
    {
        "video_name": "Hqzakjo_dYg",
        "Q": "At 3:30, Sal say that he is assuming that we are dealing with time in years. He then adds that if it were something else, we would have to convert to years.\nDoesn't the formula work with any time unit as long as we are consistent, and don't switch from one to another?",
        "A": "Yes, but if you have the half-life in years, you have to convert time to years."
    },
    {
        "video_name": "Hqzakjo_dYg",
        "Q": "At 0:58, what is k?",
        "A": "K Is a mathematical constant"
    },
    {
        "video_name": "Hqzakjo_dYg",
        "Q": "at 4:30 what is K?",
        "A": "K is a constant (The German for constant is konstant) in this exponential decay example K is the number you have to times the amount of the undecayed atoms at the end of one year by to get the amount of undecayed atoms at the end of the next year."
    },
    {
        "video_name": "FQFBygnIONU",
        "Q": "At 7:48 Sal says that the molecules are called Ligands, at 6:44 Sal says that the molecules are called Hormones. Are the molecules called differently based on whether we're talking about Endocrine signalling or an Autocrine process?",
        "A": "Ligands are molecules that binds to a receptor. So hormones is also a ligand."
    },
    {
        "video_name": "r7gTH_5XfOI",
        "Q": "Was the reason why you decided to keep the 3-2x denominator term @11:37 (instead of assuming it to be 3) due to the magnitude of the result?",
        "A": "Yes. He didn t explain how he knew before. but 2x was not much smaller than 3 so the assumption would be invalid"
    },
    {
        "video_name": "r7gTH_5XfOI",
        "Q": "At 8:30 why do you multiply the Ksp and the Kf to get a new K for the net reaction? What is the reasoning behind this rule?",
        "A": "You multiply them because you re trying to get the K for the net reaction. Multiply the K s of the individual reaction for your net reaction. If you re reversing one of the individual equations, you d change the K to 1/K."
    },
    {
        "video_name": "y55tzg_jW9I",
        "Q": "At 7:41, Why does a light ray bend in an optically denser medium? Is the time taken for light to travel in different medium same when considering displacement of the incident ray constant in different mediums.",
        "A": "Sal explains why it bends in the video.."
    },
    {
        "video_name": "y55tzg_jW9I",
        "Q": "At 6:43, how can you have vacuum and glass in real life?",
        "A": "good question If you have a bottle (strong one) and you pump all the air out of it, the light will pass from the glass into vacuum as it goes into the bottle OK?"
    },
    {
        "video_name": "y55tzg_jW9I",
        "Q": "I noticed a mistake, at 5:37, when he draws the light/car going from the water medium into the vacuum, the light would bend the opposite way. The left wheel of the car would hit the vacuum first and then move FASTER, not slower, causing the light to refract in the opposite direction shown.\nThis makes sense right?",
        "A": "Actually, I believe what Sal said is correct. You and Sal both are correct that the front left tire would be the first to emerge from the slow medium or mud for our car analogy. It would get a grip on the better surface, the fast medium, which would be the pavement so the left side would be moving faster than the right side so the car would go in the angle that Sal showed in his example."
    },
    {
        "video_name": "y55tzg_jW9I",
        "Q": "At 0:48, instead of \"perpendicular\", shouldn't it be the \"normal\"?",
        "A": "It s referring to the same thing, which is the (dotted) line (that is perpendicular to the other line), and that (dotted) line is called the normal. You get what it means!"
    },
    {
        "video_name": "y55tzg_jW9I",
        "Q": "Around the 11:00, he talks about \"Sin\" and \"Theta\" what is that?",
        "A": "Theta is the angle in question. sin(theta) = opposite/hypotenuse (for the right angle formed by angle theta)"
    },
    {
        "video_name": "y55tzg_jW9I",
        "Q": "At 0:07 , I don't get why sal says \"since the surface is smooth\". I thought that the angle of incidence is ALWAYS equal to the angle of reflection? Because, at the POINT at which the ray hits the surface, it bounces off in the same angle.",
        "A": "Well, the video is about refraction. But still, the same idea holds true. If you don t have a smooth surface, you get a whole bunch of different angles of reflection, and you get a diffuse reflection instead of specular."
    },
    {
        "video_name": "y55tzg_jW9I",
        "Q": "How is n=c/c at 13:47?",
        "A": "When Sal says c/c that means that the speed of light in a vacuum (c) is equal to the maximum speed of light (c) so c/c equals one, and n=1"
    },
    {
        "video_name": "y55tzg_jW9I",
        "Q": "AT 5:57, how do you know if the light will refract left or right, and how big the angle will be?",
        "A": "When going from a rarer to a denser medium, the light always refracts towards the normal at the point of contact. You know the size of the angle by knowing the refractive index and snell s law."
    },
    {
        "video_name": "lVacrVMmJX8",
        "Q": "Between 5:33 and 5:37, Rishi mentions that if we have high levels of CO2, then we have high levels of H+.\n\nHowever, I have learned that as soon as H+ ions are formed, they are taken up by heamoglobin to form heamoglobinic acid, often abbreviated as HHb. This acts as a buffer... then how can the pH get low??",
        "A": "Even the binding of H+ to Hb is an equilibrium reaction. It s not to say that every molecule of H+ is instantly mopped up by Hb. Rather, some H+ will always be in solution. Even water is in constant equilibrium with H+ (simplification)."
    },
    {
        "video_name": "lVacrVMmJX8",
        "Q": "At 1:28, the central chemoreceptors, are they actually regulatory interneurons?",
        "A": "Based on this video, they appear to be only sensory neurons, not interneurons"
    },
    {
        "video_name": "j-iR7puLj6M",
        "Q": "why doe the voltage of the batteries add up at 3:37",
        "A": "It doesn t. If you notice at the end the voltage of the battery is still 11V. We just care about the voltage across the terminals which is 11V however many batteries (of 11V) someone connects in parallel. That s why we model the parallel batteries with a single one, for simplicity s sake."
    },
    {
        "video_name": "j-iR7puLj6M",
        "Q": "At 6:30, don't you need to take the inverse of that expression to get the equivalent resistance of the 2 resistors in parallel?",
        "A": "It is a formula for two parallel resistors that is a simplified version of the parallel resistor equation you are referring to, so it already is the inverse. 1/R = 1/R1 + 1/R2, then 1/R = R2/(R1*R2) + R1/(R2*R1). 1/R= (R1+R2)/(R1R2). Finally, R=(R1R2)/(R1+R2)"
    },
    {
        "video_name": "j-iR7puLj6M",
        "Q": "At 6:30, don't you need to take the inverse of that expression to get the equivalent resistance of the 2 resistors in parallel?",
        "A": "R1R2/(R1+R2) is actually already the inverse of the expression 1/R total."
    },
    {
        "video_name": "j-iR7puLj6M",
        "Q": "At 6:30, don't you need to take the inverse of that expression to get the equivalent resistance of the 2 resistors in parallel?",
        "A": "Hey, Jenna! Good question. The answer is really quite simple. In the law of resistance it says: 1/R = 1/R1 + 1/R2 ... . We can change the equation to be something like this: R = (1/R1 + 1/R2)^(-1). Now we just look inside the parentheses and we got 1/R1 + 1/R2. This can be rewritten to (R1+R2)/(R1*R2), by merging these fractions under one dividing line. So the equation now looks like this: R = ((R1+R2)/(R1*R2))^(-1). By taking the inverse of this we get R = (R1*R2)/(R1+R2)."
    },
    {
        "video_name": "j-iR7puLj6M",
        "Q": "i didnt understand the step at aroun 3:55 where the line is drawn in between the two could some explain it please?",
        "A": "Hello Hemanth, The line is a conceptual tool used to simplify the circuit. Here we can draw the line because the circuits has the same voltage at this point. The upper branch (battery plus resistor) is the same as the lower branch (battery plus resistor). If this was a real circuit the ides still works provided the batteries have the same voltage and the resistors have the same value. Regards, APD"
    },
    {
        "video_name": "j-iR7puLj6M",
        "Q": "i am lost at 7:55 how are the equivalent resistor and r3 in series?",
        "A": "The equivalent resistor is a single resistor representing the parallel combination of R1 and R2, with the special notation R1 || R2. That equivalent resistor has one terminal connected to R3, so therefore they are in series. All current flowing in R3 also flows in (R1 || R2)."
    },
    {
        "video_name": "j-iR7puLj6M",
        "Q": "At 4:16 when two battery's are added they voltage or current will also increased",
        "A": "You re not adding the voltages. Theyre at the same potential as they are connected in parallel"
    },
    {
        "video_name": "j-iR7puLj6M",
        "Q": "At 4:25, the speaker says he can combine the two batteries into one big battery. What if the voltage difference between the 2 batteries were different, what would we do in that case?",
        "A": "Then you ll have to use Kirchoff s loop rule and add up all the voltages and equate it to zero. Your variables will be in terms of i1, i2(current). On solving the different equations you get, the current will be obtained"
    },
    {
        "video_name": "j-iR7puLj6M",
        "Q": "How did you get the 2 at about 7:00 mins RR/2R",
        "A": "We have two resistors in parallel, both with value 1.4 ohms. The formula for two parallel resistors is (R1 x R2) / (R1 + R2). If R1 = R2 (which is the case here) then the formula reduces to: (R x R) / (R + R) The numerator is R^2 and the denominator is 2R. One of the R s in the numerator cancels with the R in the denominator, leaving R/2"
    },
    {
        "video_name": "j-iR7puLj6M",
        "Q": "I am confused about connections of batteries at 3:30. Are not they cross-connected ?",
        "A": "Hello Ozgurcan, No, the circuit is good. Observe that the batteries are in parallel with positive to positive terminal and negative to negative terminal. The batteries work together sharing the load like two people riding a tandem bicycle. Regards, APD"
    },
    {
        "video_name": "j-iR7puLj6M",
        "Q": "At 2:02, taking one battery of EMF 11V and another one, for example 15V or 17V (not the same as the previous battery), how will you find out the value of current in the particular segment?",
        "A": "Use Kirchhoff s laws"
    },
    {
        "video_name": "FGq9-R6Yw18",
        "Q": "At 3:34, why doesn't the negatively charged oxygen take a hydrogen from the acid formed when an alcohol took a Hydrogen from the positively charged oxygen at 2:36?",
        "A": "It certainly can. That distinction isn t extremely important in this example because the reaction is occurring in excess alcohol."
    },
    {
        "video_name": "FGq9-R6Yw18",
        "Q": "At 2:35, can the oxygen that has the negative charge grab the hydrogen, that attaches to the positive oxygen, instead of using another R-OH group to deprotonate the hydrogen? I hope you understand my question. Please help thx",
        "A": "It s quite probable that the same R-OH molecule facilitates the transfer of the proton from the positively charged oxygen to the negatively charged oxygen. Therefore, while Jay shows the deprotonation and the protonation as two distinct steps, they probably merge into one which amounts to the transfer of the proton from one oxygen to the other oxygen, as you suggest."
    },
    {
        "video_name": "FGq9-R6Yw18",
        "Q": "At around 3:40 in the video, when the protination of the negatively charged Oxygen is done, we form an alkyl oxide, can this cause the molecule (the hemiacetal/ketal) to react further? like a substitution reaction?",
        "A": "Yes, the hemiacetal/hemiketal can be protonated and react further to form an acetal/ketal."
    },
    {
        "video_name": "FGq9-R6Yw18",
        "Q": "Are the molecules formed at 10:06 are enantiomers?",
        "A": "Yes, the compounds are enantiomers."
    },
    {
        "video_name": "FGq9-R6Yw18",
        "Q": "Could the first intermediate (2:50) react intramolecularly? The proton would come from the positive oxygen. The negative oxygen would attack the proton and the formerly-positive oxygen would now be neutral. Is that possible?",
        "A": "It s possible but unlikely, because the bond angles would have to be 90 \u00c2\u00b0 in the transition state. Another possible process would be intermolecular transfer between two of the intermediates, but there are so few of them that it is unlikely they would collide with each other. The most likely process is deprotonation by a solvent molecule, because there are many solvent molecules available to collide with the intermediate."
    },
    {
        "video_name": "FGq9-R6Yw18",
        "Q": "At 10:00, you show the two different possibilities formed with the OH and H. How exactly will you know when there will be two conformations or one?",
        "A": "Both possibilities will be formed and can inter-convert so the two anomeric forms will be in equilibrium. Which form predominates depends on the other substituents in the ring so with glucose, for example, the ratio of equatorial OH to axial OH is about 2:1."
    },
    {
        "video_name": "FGq9-R6Yw18",
        "Q": "7:35 Do the hemiacetals act as a base to deprotonate and acid to protonate or are additional reactants involved?",
        "A": "No, the solution is filled with base, first the molecule deprotonate and then after becoming conjugated acid, it shall donate H+ to give the product."
    },
    {
        "video_name": "DRz9B6SHWQ4",
        "Q": "at 2:01, why is it called a breadboard? it isn't made of bread.",
        "A": "I prefer proto board."
    },
    {
        "video_name": "yIWKrQTznXc",
        "Q": "at 9:56 he says a decrease in adrenaline. Does that mean that the adrenal gland is always producing epinephrine?",
        "A": "Yes. The parasympathetic and sympathetic systems work together. In most cases, your body will be functioning somewhere between hyperlazy and hyperalert. Because the systems work on a continuum, the neurotransmitters/hormones for both systems are often present."
    },
    {
        "video_name": "yIWKrQTznXc",
        "Q": "at 4:52 what does dilate mean?",
        "A": "By dilate, it means expansion. For emotions, we are concerned with eye dilation and lung dilation"
    },
    {
        "video_name": "yIWKrQTznXc",
        "Q": "at 3:00, I thought the adrenal glands are named because they are adjacent to the kidneys (renal)?",
        "A": "Adrenal glands are called ad-renal because they sit atop the kidneys. Ad means on top of, and renal means pertaining to the kidneys."
    },
    {
        "video_name": "xgDuzj51tAc",
        "Q": "@5:00 is Cyclohexanoic acid also a suitable name for Cyclohexanecarboxylic acid?",
        "A": "Yes it is \u00e2\u00b8\u0082\u00e2\u00b8\u0082\u00e2\u00b8\u009c(\u00e0\u00b4\u00b0\u00e1\u00b4\u0097\u00e0\u00b4\u00b0\u00e0\u00b9\u0091)\u00e2\u00b8\u009d\u00e2\u00b8\u0083\u00e2\u00b8\u0083"
    },
    {
        "video_name": "xgDuzj51tAc",
        "Q": "7:17 What about the second hydrogen and oxygen in ethanol?\nWont they be forming a hydrogen bond??",
        "A": "They would form a hydrogen bond with another molecule of ethanol"
    },
    {
        "video_name": "xgDuzj51tAc",
        "Q": "@ 8:35, can someone breakdown ... \".. more non-polar a molecule becomes ... it decreases in solubility in water ..\" is it because the non-polar molecule is super stable in its space and doesn't want to bond w/water?",
        "A": "A nonpolar molecule does not have dipole interactions with other molecules. Because water is polar and breaks up molecules using these dipole interactions, water can disolve polar molecules. When a molecule is more nonpolar the interactions between water and the molecule do not occur as easily and it will not dissolve as much."
    },
    {
        "video_name": "xgDuzj51tAc",
        "Q": "At 3:24, why isn't it 2-phenolbenzoic acid instead of 2-hydroxybenzoic acid? Shouldn't the OH for labeled #2 be named a phenol since it's attached to a benzene ring?",
        "A": "Phenol can only be used as a base name not as a substituent like that."
    },
    {
        "video_name": "xoUppFlif04",
        "Q": "at 4:40 i understood with the differentiation of \u00cf\u0089 but why didn't the amplitude get squared?",
        "A": "The amplitude is just a constant. The \u00cf\u0089 was squared because it appeared inside of sin(\u00cf\u0089t), and by the chain rule, (each time we differentiate a sin or cos) it falls out. It might be better if you saw the differentiation with and without A included in the original function: f(t) = cos(\u00cf\u0089t), f (t) = -\u00cf\u0089 sin(\u00cf\u0089t), f (t) = -\u00cf\u0089^2 cos(\u00cf\u0089t). Multiplying by a constant doesn t change the inner working of the derivative, so we just have: x(t) = Acos(\u00cf\u0089t), x (t) = -A\u00cf\u0089 sin(\u00cf\u0089t), x (t) = -A\u00cf\u0089^2 cos(\u00cf\u0089t). Hope this helped!"
    },
    {
        "video_name": "xoUppFlif04",
        "Q": "at 1:36 , isnt it mx(double prime(t))/dt(squared)= -kx(t)?\ncause a=d(squared x)/dt(squared)",
        "A": "There are two different notations for derivatives. One is the prime notation: mx (t) = -kx(t) This is one (correct) way of writing it. m*d^2(x(t))/(dt^2) = -kx(t) is another. They are both saying the same thing, the second derivative of the position function is equal to -kx(t). You can use either one, but don t use both at the same time."
    },
    {
        "video_name": "xoUppFlif04",
        "Q": "How could we resolve the differential equation at 2:42 by analitic methods?",
        "A": "You would need to use the techniques of calculus for solving differential equations. That s a college level math class for people studying engineering, math, or science."
    },
    {
        "video_name": "xoUppFlif04",
        "Q": "At about 4:04, how did 'A cos(wt)' become '-Awsin(wt)'? I mean, I know the '-sin' must have come because of derivative of cosx, but how did 'w' get doubled like that?",
        "A": "I see. Thank you! :)"
    },
    {
        "video_name": "xoUppFlif04",
        "Q": "At 6:55,how did x(t)became equal to Acos(wt)?pls help.",
        "A": "If the restoring force is proportional to displacement and is toward equilibrium, then you will always get x(t)=Acos(wt) (assuming you start your measurement of t at the right point in the cycle)"
    },
    {
        "video_name": "xoUppFlif04",
        "Q": "At 2:20, what does x(t) mean? I saw all the video in part 3 also but I can't understand exactly what it meant.\n\nthank you!",
        "A": "x(t) represents the function that relates x to t, often called x of t . This is similar to f(x) in that it has one x-value for any given t-value, and that function can be graphed. In the case of this video, it is Acos(wt) ."
    },
    {
        "video_name": "xoUppFlif04",
        "Q": "At 4:14,how ACos(wt) differentiated to -Awsin(wt) .Can anybody give me a full description with rules.\n\nThank you for your help!",
        "A": "derivative of cos(x) is -sin(x) derivative of cos(ax) is -asin(ax) by chain rule"
    },
    {
        "video_name": "bVHuI_QpYIM",
        "Q": "At 5:18 why is the heat of vaporization of water 541 calories per gram, when it takes one calorie to raise one gram of water 1 degree. How does that end up being 541 calories to arrive at vaporization?",
        "A": "When you vaporize water, the temperature is not changing at all. It s changing state. That s different from heating liquid water."
    },
    {
        "video_name": "HRuLhY0NAMI",
        "Q": "At 0:50, the video says that the index of refraction is uselly a function of the wave length of light. What is the function?",
        "A": "it means that it depends on the wavelength. or; it is related to the wavelength"
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "Hank mentions at 0:58 that as you grow, your cells aren't getting heavier, you're just getting more of them. When you have a person who isn't growing naturally anymore, but is still gaining weight (i.e. fat), is that still the case?",
        "A": "Oh definitely! The intake of foods into your body is what causes this. Your digestive system stores the fat away in cells called adipocytes. These are indeed cells (full of lipids I believe) and the addition of them due to what you eat and a lack of burning them away is what causes that weight gain."
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "At 1:44 Hank says the nucleus store instructions like how?",
        "A": "The nucleus is full of DNA, which are the blueprints for the cell."
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "~00:15 - The next video does not discuss why we cannot clone ourselves. We have already succeeded in cloning other animals. Why could we not be successful cloning ourselves?",
        "A": "I think he just meant we can t naturally clone ourselves, like cells do."
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "At 9:54 Hank states that cells divide into two identical copies. If that were true, then all our cells would be liver cells, or nerve cells, or skin cells. In fact, all our cells would still be stem cells.",
        "A": "The process you re missing from this is differentiation. Differentiation is the process of stem cells developing into specific type of cells. Stem cells will undergo mitosis to create more stem cells. Some stem cells may differentiate into skin cells. When those skin cells undergo mitosis, they will create more skin cells."
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "At 5:20 - Are microtubules the same as spindle fibres?",
        "A": "Spindle fibres are just aggregates of microtubules."
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "At 5:31, Hank says metaphase is the longest phase of mitosis. Isn't prophase the longest phase?",
        "A": "The first and longest phase of mitosis is called the prophase. It can take as much as 50 to 60 percent of the total time required to complete mitosis."
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "When he mentions \"daughter cells\" (in 9:50), are there such things as \"son cells\" or something like that?",
        "A": "the daughter cells in mitosis don t really have anything to do with gender. it s just a term to call the resulting identical cells. you can think of this as the same thing as when both boys and girls refer to a group of friends as guys instead of girls ."
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "What does mitosis mean at 5:36?",
        "A": "Hank said during the biolography that mitosis is the greek word for thread."
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "at 2:73 hank tells that a single cell having 46 chromosomes split into two to give 2 cells with 46 chromosomes again.how is that possible?i mean each cell should have 23 chromosomes right?",
        "A": "They replicate their genome first and then divide."
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "At 5:29, Hank says that metaphase is the longest phase of mitosis. My biology textbook says that it is actually the shortest phase. According to the book, prophase is the longest phase and may take up to half of the total time required to complete mitosis. Is Hank wrong or is my textbook wrong?",
        "A": "My book says that prophase is the longest also."
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "At 0:14 Hank says that identical twins get half of their DNA from their mom and the other half from their dad... isn't that the same for the DNA of someone who isn't an identical twin?",
        "A": "Yes, but do to a process that happens during Meiosis called cross in-over, where the homologous chromosomes cross over each other and mix genes and change things around, causing people to be different. In the case of identical twins, they would have 2 identical eggs, and they would have chromosomes that matched. Hope this helps! :)"
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "At 9:40 - Is Cytokinesis a part of Telophase or is it a separate phase entirely?",
        "A": "Cytokinesis takes place at the same time as telephase, and for that reason, I ve often seen this explained as Interphase Prophase Metaphase Anaphase Telephase and cytokinesis Interphase (repeats)."
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "Around 4:27 Hank says that chromatin condenses around proteins to form chromosomes. Is there a name for this protein? Do we know much about it?",
        "A": "The proteins are called histones. They are rounded and sphere-like in shape. The DNA is coiled around these proteins."
    },
    {
        "video_name": "X1bmedVziGw",
        "Q": "At 5:28, how is metaphase the longest phase? Isn't most time spent in Prophase (still a phase of mitosis)?",
        "A": "Interphase is the longest phase- it lasts for roughly 90% of cell growth. For some cells, such as brain and spinal cord cells, which do not multiply, they remain in interphase (G1) for the entire lifetime."
    },
    {
        "video_name": "-QpkmwIoMaY",
        "Q": "at 12:10, we could have directly converted 54.5moles of water into Litres by dividing it by 22.4?",
        "A": "Yes, except that since 1982 the IUPAC has defined STP as 0\u00c2\u00b0C and 1 bar, so the molar volume at STP is 22.7 L."
    },
    {
        "video_name": "-QpkmwIoMaY",
        "Q": "At around 2:00, what does Sal mean by 23.8 millimeters of mercury is the vapor pressure?",
        "A": "Millimeters of mercury (mmHg) is a unit of pressure, just like the Pascal (Pa), atmosphere (atm), or torr (Torr). 1 atm = 101 325 Pa = 760 torr = about 760 mmHg"
    },
    {
        "video_name": "-QpkmwIoMaY",
        "Q": "@8:27 why does Sal use the volume's room instead of the 2.00 L of water's volume?",
        "A": "The vapor pressure of the liquid is the point at which as much water molecule evaporating is equal to as much water condensing. the vapor pressure gave 23.8mmHg is the partial pressure exerted by the water molecules in gas and vapor pressure of the room of open container. That s why it is necessary to consider the entire volume. I Hope that useful."
    },
    {
        "video_name": "-QpkmwIoMaY",
        "Q": "At around 9:57, why does Sal use the room temp.'s volume rather than the liquid's volume? How do you know which volume to use? Are we not trying to estimate the number of moles in liquid water that's evaporating from the open beaker system? I was a bit confused. If someone can please answer my question clearly, I would greatly appreciate it.\n\nThanks in advance.",
        "A": "It is the volume of the room that is important. Water will keep evaporating until the vapour pressure of the water in the gas state equals 23.8 mmHg. If you had a bigger room, then more water would need to evaporate to reach this pressure. The volume of water is not important except in terms of whether you have enough to evaporate to reach the vapour pressure."
    },
    {
        "video_name": "-QpkmwIoMaY",
        "Q": "around 8:00 Sal says 'if more water molecules evaporate after this point the we are going to have a higher pressure which will actually favour them going into the liquid state so that will pass the equilibrium which is not likely to happen.'\ncan someone please explain this.",
        "A": "At equilibrium, molecules are leaving and returning to the liquid at the same rate, and the vapour pressure is constant. I m more water molecules then enter the vapour phase, they will be hitting the surface of the liquid at a faster rate. The vapour pressure of the liquid will return to its equilibrium value."
    },
    {
        "video_name": "-QpkmwIoMaY",
        "Q": "at 15:59 sal said not all of it will evaporate only a little bit less than half of it will isn't it a little bit less than all of it will evaporate",
        "A": "A little under a liter will evaporate. There are two liters of liquid water to start with. It s a little bit less than half."
    },
    {
        "video_name": "-QpkmwIoMaY",
        "Q": "At 7:25, Sal said three significant figures, but there was an edit that said he meant to say two significant figures. i thought 760 mm Hg had an infinite number of sig figs since it is a whole number. Am i wrong? and if so, could you explain why?",
        "A": "At one time, 1 atm as defined as exactly 760 mmHg. Thus the 760 had an infinite number of significant figures. Now, 1 atm is defined as 101 325.0 Pa or 101.3250 kPa. Also, 1 Torr = \u00c2\u00b9/\u00e2\u0082\u0087\u00e2\u0082\u0086\u00e2\u0082\u0080 atm. Historically, 1 Torr was meant to be exactly the same as 1 mm Hg. Subsequent redefinitions of the two units made them differ slightly (by less than 0.000 015 %). For all ordinary purposes, 1 atm = 760.0000 mm Hg (7 significant figures)."
    },
    {
        "video_name": "-QpkmwIoMaY",
        "Q": "At 9:05, why do they abbreviate mole?",
        "A": "Mole is very commonly abbreviated to mol. Dropping one letter isn t much, but it does help save space! We can t just abbreviate to m since that is already taken by meters."
    },
    {
        "video_name": "-QpkmwIoMaY",
        "Q": "at 1:10 is mm Hg a form of pressure?",
        "A": "It is a way to measure pressure."
    },
    {
        "video_name": "YUEkOBvJSNg",
        "Q": "What is the name of the original method used to portray the Cyclohexane molecule? 00:20",
        "A": "if you mean before he draws it as tetrahedral then that s called its skeletal structure."
    },
    {
        "video_name": "7PgYb0c_3KE",
        "Q": "Why at 3:22 the screen didnt move? or is it just my computer?",
        "A": "Karl said that it was a picture"
    },
    {
        "video_name": "7PgYb0c_3KE",
        "Q": "0:37 it says below his hand stick n click but the title says click and stick why did he do that?",
        "A": "I am not sure!"
    },
    {
        "video_name": "Zqt3btS1FwE",
        "Q": "Hello, at 5:53, how is the dimensional analysis working?\nWhy is Herz (oscillation per sec) times Js equal to just J?\nShouldn't we be left with oscillation times J??\nWhat am I missing?",
        "A": "Oscillation isn t an actual unit. The units for Hz are 1/s so 1/s times Js gives you just J."
    },
    {
        "video_name": "Zqt3btS1FwE",
        "Q": "At 5:00 David mentions Planck discovering this smallest, discrete amount of energy that light can deliver (6.626 x 10^-34 (J\u00c2\u00b7s)). My question: how did Planck, in 1900 (!), quantify such a small quantity of energy?",
        "A": "He calculated it by making the hypothesis that energy comes only in discrete packets (i.e. the photon). He uses this assumption to calculate the frequency distribution of black body radiation with his constant h being the separation between allowed frequency values and it exactly matched what was experimentally observed."
    },
    {
        "video_name": "Zqt3btS1FwE",
        "Q": "how come plancks constant is sooooo small? 5:43",
        "A": "Planck s constant is the amount of energy a photon gains for each additional oscillation per second of frequency. That amount of energy is small."
    },
    {
        "video_name": "Zqt3btS1FwE",
        "Q": "1:03 I still do not understand how can waves be localised?",
        "A": "They can t. That s why we say that photons are particles of light. There s an inherent irresolvable conflict between the wave view of light and the particle view. You ve put your finger on the most important aspect of the conflict: one of them can be localized and the other cannot."
    },
    {
        "video_name": "Zqt3btS1FwE",
        "Q": "At 4:50 the energy of a photon equal to frequency times Plank's const.\nHow does the amplitude of the wave relate to this equation? The higher the wave the more energetic it is, isn\u00e2\u0080\u0099t it?",
        "A": "Altering the amplitude of the wave will influence the intensity, which doesn t necessarily mean that there s more energy in the wave. We re talking photons here, via EM waves, so we don t relate amplitude to the energy of a light wave in the same way that we would for something else, like sound. We stick with E = hf, i.e. dependent on frequency only."
    },
    {
        "video_name": "Zqt3btS1FwE",
        "Q": "3:37 So if light behaves as on or off, not in between, could there be a binary system based on light and whether not it's there? If so, could there be a computer based on the energy of a photon?",
        "A": "Your ideas are similar to those of a quantum computer. A photon can have pretty much any value energy actually, but when it interacts with bound systems such as an atom, then there will only be certain states allowed . I think this is what you mean by not in between . The quantum computer uses these not in-between states. And, rather than being a binary system, it has much more information per bit . keep up your thinking but do your homework too :) so that you have the info: your way of thinking is useful."
    },
    {
        "video_name": "126N4hox9YA",
        "Q": "At 6:00 pm, I want to clarify that if oxygen is having partial negative charge, it means that electrons are towards it but they are not in its valence subshell?",
        "A": "Correct. Oxygen has a partial negative charge because it is an electronegative element. It draws the electrons in the bond towards it, but they do not reside in oxygen s valence (If it did, it would be wholly negative)."
    },
    {
        "video_name": "126N4hox9YA",
        "Q": "At 2:54, the video shows a bond of 2 carbon atoms, since they would have the same electronegativity you would consider it polar, even though there are 2 of the same atoms?",
        "A": "At 3:16 Jay states that this is a nonpolar covalent bond."
    },
    {
        "video_name": "126N4hox9YA",
        "Q": "In around 0:30, the lecturer drew a diagram of carbon and oxygen that I didn't really get. Can someone explain to me what it means?",
        "A": "He is comparing bonds between atoms to see if they are polar or non polar. This is a pretty big concept in chemistry."
    },
    {
        "video_name": "126N4hox9YA",
        "Q": "At 8:06, why was there 6 electrons surrounding chlorine?",
        "A": "because there are 6 valence electrons in chlorine s outermost shell or valence shell"
    },
    {
        "video_name": "126N4hox9YA",
        "Q": "At 10:23 he says that sometimes you can consider a covalent bond to be an ionic one, but how can it be that subjective? Doesn't it have to be either covalent or ionic?",
        "A": "No, it doesn t have to be one or the other. Ionic bonding is really just polar covalent bonding taken to the extreme."
    },
    {
        "video_name": "126N4hox9YA",
        "Q": "at 10:51, if carbon gained an electron how does it have a negative formal charge? is it because the electron is negative?",
        "A": "Yes. In a neutral atom there are equal protons (+ve charged) and electrons (-ve charged). When carbon gains an electron, the number of electrons in the atom increases where the number of protons still remains the same. It creates an overall negative charge in the atom. * (carbon receives 2 electrons since it has 6 electrons and want two more to reach in stable condition) Hope it helps :)"
    },
    {
        "video_name": "126N4hox9YA",
        "Q": "Around 5:15 it's explained that if the difference in electronegativity is less than 0.5 then it is a nonpolar covalent bond, and if it's greater than 0.5 it is a polar covalent bond. Well what if is exactly 0.5, what type of covalent bond is it most likely to be?",
        "A": "The numbers that Sal quotes are not universal, and will differ slightly depending on the source you check. In reality, all bonding is on a continuum, and the bigger the electronegativity difference, the more polar the bond. So, if the difference was 0.5 then you would have a covalent bond that was slightly polar."
    },
    {
        "video_name": "126N4hox9YA",
        "Q": "At 2:11, he says that carbon is partially positive while oxygen is partially negative. In another video explaining formal charge, he said to subtract the bonded electron amount from the valence electrons in the atom. For oxygen, this would be 6 valence - 6 bonded to it, giving 0. I don't understand why, using EN values it's negative, but using the way explained for formal charge it's 0. Is formal charge something different?",
        "A": "Oxygen is more electronegative than carbon. This means that it will attract electrons more than carbon in a covalent bond. Since electrons are slightly nearer to the oxygens than carbon then the oxygens will have a partially negative charge, which results in the carbon having a partial positive charge"
    },
    {
        "video_name": "126N4hox9YA",
        "Q": "At 6:02 Jay states that the electronegativity of Oxygen is 3.5 and Hydrogen is 2.1 according to the Pauling Scale. Is there a way to figure out the Oxygen is more electronegative than Hydrogen without the Pauling Scale?",
        "A": "You can see where it lies in the periodic table and knowing the left-right top-bottom trends of electronegativity. Alternatively, you can extrapolate the fact that in water, the partial negative charge lies on oxygen to mean that oxygen is relatively more electronegative. (You can also calculate the exact electronegativity, which technically is without knowing the pauling scale values, it is irrelevant to what you are asking)"
    },
    {
        "video_name": "126N4hox9YA",
        "Q": "At 5:58, Sal makes a covalent bond between oxygen and hydrogen, but is it possible? I mean Hydrogen only has one electron, and oxygen has six. Even with covalent bond, oxygen will only have 7 valence electrons.",
        "A": "There are other atoms at the ends of those dashes. He is showing you just the two bonded atoms."
    },
    {
        "video_name": "U1KbM4ffiLg",
        "Q": "At around 6:50, could you not have used the fact that A is large (tending to \u00e2\u0088\u009e) v1= -vo/A = 0. Therefore, vI/R1= -vo/R2. Hence, vo/vI = -R2/R1.",
        "A": "Yes. Once you get the idea of how we are going to exploit the large value of A, you can go back through the equations and find places where you can make the simplification earlier. I carried on with the algebra a while longer until I got (1+A)/A. This is the point where making the simplification is a very strong clear argument, since it is so easy to see that this is close to 1. Once you grasp the idea, you should feel free to use it where you want."
    },
    {
        "video_name": "U1KbM4ffiLg",
        "Q": "wouldn't it be the same if we take the assumption of the gain A to be large on the equation at the 6:50, and derive the same results,i.e. vout=-vin*R2/R1?",
        "A": "Yes, you can use the Large A assumption at any point in the derivation."
    },
    {
        "video_name": "5EU-y1VF7g4",
        "Q": "At 6:35 , sal said at the Micro state is changing gazillion of a second but at last few minutes sal said \"Micro state never changes , I don't get it , please tell me if I am missing something .",
        "A": "He means microstate is defined every instant, not that it doesn t change. On the other hand, we cannot talk about macrostate in every instant since it s not well defined."
    },
    {
        "video_name": "5EU-y1VF7g4",
        "Q": "In 6:51 sal says that ALL macrostates stay the same after a second. Wouldn't pressure increase or decrease because new atoms are not touching or touching the walls of the wall?",
        "A": "I think, that a macrostate would stay the same only if the conditions weren t changed. So as long as nothing changes, all macrostates would stay the same. The pressure would probably generally stay the same because there s only so much area to bounce off of and so many molecules, so generally the pressure would be constant as long as no conditions changed that could lead to a fluctuation."
    },
    {
        "video_name": "5EU-y1VF7g4",
        "Q": "In Sal's diagram at 11:22, the surface area of the block becomes half and the weight of the block becomes half, so should not the pressure remain the same?",
        "A": "Sal s intent was to erase half of the rock, but not half of the plate that the rock was resting on. The force is halved and the area remains constant, so the pressure is also halved."
    },
    {
        "video_name": "5EU-y1VF7g4",
        "Q": "At 15:50, aren't the volume and the pressure supposed to be in different axis (switch between them)? The rock was shortened in half, yet the graph tells us that in the second state the volume got bigger? Maybe I didn't grasp the example very well.",
        "A": "the volume would nautrally increase because the the rock was haved and therefor the pressure exerted on the system would also reduce allowing the gas to expand to an extent thereby increasing its volume"
    },
    {
        "video_name": "5EU-y1VF7g4",
        "Q": "At 17:43 Sal says that macrostates are not well defined during the intermediate stage because they are continuously changing. But don't the microstates of the system change too? like we know that the gas molecules randomly move in any direction ,so their velocities are changing too. That's where we come across the concepts of R.M.S velocities,thermal velocities,isn't it?",
        "A": "Is your question if both the Macrostates and Microstates are not well defined during the intermediate stage? The answer to that is yes, they are both not well defined. When Sal said Macrostates he did not mean to make it seem like he was saying Microstates are not as well."
    },
    {
        "video_name": "5EU-y1VF7g4",
        "Q": "At 11:44, why does the piston oscillates when the rock's mass is halved?\n\nThanks. :)",
        "A": "If you re asking why it goes through a bit of harmonic motion (oscillation), it s because the top has momentum after it is pushed up by the pressure of the gas so that when gravity takes over it is a little above the level that it reaches equilibrium. Then gravity brings it down and gives it momentum so that it is a little below the equilibrium level and the gas pressure takes over again. This process of oscillation continues until damping allows the piston to reach equilibrium."
    },
    {
        "video_name": "5EU-y1VF7g4",
        "Q": "what sal had told at 2:56 , an ideal gas is always single atomic molecule gas?",
        "A": "An ideal gas is modelled as a monatomic gas."
    },
    {
        "video_name": "5EU-y1VF7g4",
        "Q": "Why in 13:06 Sal says the system will get a lower pressure, higher volume and probably lower temperature ? I don't get the temperature part, temperature would remain the same, no? the system just decreases the pressure because it gains more volume, however KE and Velocity would be the same, but once the volume is greater the total force and change of momentum decreases, also decreasing Pressure. Decrease of temperature makes no sense.",
        "A": "The temperature could change, why not, the container that Sal uses was not specifically insulated through heat loss or gain to its surroundings. In any experiment of this sort, if temperature does change then the microstates of the molecules changes too, since temperature is essentially a measure of the heat energy of these states. To ensure that temperature does not change as you suggest you would have to keep the container at some fixed temperature, usually by immersion in a water-tank or some such means."
    },
    {
        "video_name": "Kv-hRvEOjuA",
        "Q": "At 3:09, what is a Balmer Rydberg equation?",
        "A": "The Balmer-Rydberg equation or, more simply, the Rydberg equation is the equation used in the video. 1/\u00ce\u00bb = R(1/i\u00c2\u00b2 -1/j\u00c2\u00b2) It is usually written as 1/\u00ce\u00bb = R(1/n\u00e2\u0082\u0081\u00c2\u00b2 -1/n\u00e2\u0082\u0082\u00c2\u00b2), where n\u00e2\u0082\u0081 < n\u00e2\u0082\u0082. For the Balmer series of lines ( the visible lines in the hydrogen spectrum), n\u00e2\u0082\u0081 = 2. So the Rydberg formula for the Balmer series of lines is 1/\u00ce\u00bb = R(1/4 - 1/n\u00e2\u0082\u0082\u00c2\u00b2)."
    },
    {
        "video_name": "Kv-hRvEOjuA",
        "Q": "According to the diagram at 2:58, energy of photon from n=3 to n=2 is more than energy of photon from n=6 to n=5. Am I right? If right why is that so?",
        "A": "Yes. Because the energy levels are proportional to 1/n^2. So they get closer together as n gets bigger."
    },
    {
        "video_name": "Kv-hRvEOjuA",
        "Q": "3:21 Jay says that the Rydberg constant is equal to 1.097x10^7/m but I've been told it's 2.18x10^-18J. Which one is right? Or are they the same, just with different units?",
        "A": "They can t be the same thing with different units, right, because 1 J is not 1/m. The first number you have is the Rydberg constant. The second one is related, but not the same. It is the Rydberg unit of energy. It is equal to the largest energy transition that can take place in a hydrogen atom."
    },
    {
        "video_name": "Kv-hRvEOjuA",
        "Q": "At 1:00, does the emission of light connect to the emission of a photon? Are they the same thing?",
        "A": "Yes, emitting light means emitting a photon (see my previous answer to your questions). A photon is a particle of light."
    },
    {
        "video_name": "1JudNLK1-Ck",
        "Q": "At 7:50, he says that the glucose has a different way to get into the cells without insulin. What way do the cells use to get glucose.",
        "A": "I think he said that because if there is insuline, glucose will be stock in muscules and liver in a polysaccharide form, the glycogen in animals. So, without insuline, glucose won t be stock in glycogen and there will be a lot of glucose in blood. This glucose with get in cells by some specific transporters, the glucose transporters also known as GLUT."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "Shouldn't be oxidizing water, not oxidizing oxygen at 16:26?",
        "A": "Well, it s the oxygen in water that is oxidized, but you are right, it is not molecular oxygen. The idea is that the reduction of molecular oxygen to water is thermodynamically very favorable (that s why it s called oxidation) and that in the process of photosynthesis the reverse process is happening, which is remarkable."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "Around 00:35 he says that PGAL was the first end product of photosynthesis, but wouldn't the oxygen that was released through the light reactions be the first end product or am I wrong??",
        "A": "By product of photosynthesis, he is probably referring to an organic product created as a result of carbon fixation and that is used in the cell. With this definition, oxygen doesn t count, but you are absolutely correct that it is a product produced by reactions that are involved in the earliest part of photosynthesis."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "at 7:34 it says about lumeu,what is lumeu?",
        "A": "It looks a bit like lumeu, but it is actually lumen. The lumen is the space enclosed by the thylakoid membrane. Protons are pumped into the lumen by the electron transport chain."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "at 11:56 what serves as the carbon source for photosynthesis",
        "A": "The atmosphere and heterophs"
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "at 9:50, what happens if it is dark like it is going to rain, does the sun still go to to the plants???",
        "A": "Yes, sunlight does, but not as much and sometimes a lot less than on a sunny day since a lot of the visible light is blocked by the rain clouds. It gets dark like it s going to rain because there are lots of clouds in the sky, and of course it s these clouds that sometimes actually do rain on us. If these rain clouds blocked out all the light from the sun, then it would be as dark as night, which doesn t happen even on very cloudy days."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "At 13:18, why are the hydrogen protons attracted to the Stroma from the Lumen? Presumably the Stroma is more electropositive than the Lumen. What is happening here?",
        "A": "There s a concentration difference. There are more hydrogen ions inside the thylakoid that outside, if I recall correctly. Because of the rules of diffusion, the ions flood out."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "at 12:28,what is atp",
        "A": "ATP stands for adenosine tri phosphate. It is the energy currency of a cell. It is the form in which our cells generate energy from the food we eat. Hope this helps :)"
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "Where do the H+ in the stroma come from? (16:53 and other places)",
        "A": "H+ are returned to the stroma during ATP synthesis - the Calvin cycle is a regulator of the H+ ions in the stroma (NADP picks up the H+ ions and used in the cycle). For it to first contain H+ ions, all that happens is the disassociation of water (H2O) into a hydrogen ion (H+) and a hydroxide ion (OH-)."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "Around 8:50, Sal shows us the structure for Photosystem II. However, I was wondering if his numbering of these is arbitrary for the sake of differentiating multiple proteins, or if the numbers (I, II) signify different protein types?",
        "A": "The two complexes are different but serve similar roles and so are numbered I & II rather than given completely different names. Which one is I and which is II is a result of the history of their discovery and does not correspond to their function or importance in any way."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "At 12:40 Sal says photophosphorylation but writes photophorylation.",
        "A": "photophosphorylation is correct."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "At 6:59 Sal says 'Phosphobilipid Layer', whereas Wikipedia writes it as 'Phospholipid Bilayer'. Does it make any difference? Apologies if question has already been asked :)",
        "A": "It does not make any difference. Both are acceptable."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "6:05\nWhy did he cut himself off so early?",
        "A": "He wanted to stick to one topic.... He would probably waste time talking about that"
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "At 14:05, Sal says, \"the electrons keep entering lower energy states.\" How does an electron enter a lower energy state?",
        "A": "The electron loses energy when it goes down the electron transport chain because it needs to power the proton pump to create a concentration gradient so that the hydrogen ions can go through ATP Synthase and create energy. If you are asking about this for a class, you shouldn t need to know the specifics of how the electron loses energy because you are only expected to know the overall concept of what is happening. If it is just out of curiosity, I m sure you can find something on the internet about it."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "At around 3:00, he mentions that the chloroplasts appear green because they reflect green, and absorb all the other colors. Then why do some plants appear to to have colors other than green?",
        "A": "Chlorophylls are green pigments in plants. Carotenoids are red to yellow pigments and transfer their energy absorbed from the sun to chlorophyll. There are other pigments as well, but they tend to be found in algae and kelp."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "How is oxidising oxygen equals to oxidising water? I don't understand what Sal said at 15:43 to 15:56. Could anyone clarify?",
        "A": "The substance is oxidised by an oxygen atom, so it has gained oxygen. Oxygen oxidises other substances."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "At 14:30 Sal says photosystem I, but on the diagram it says photosystem II. Is it photosystem I or II at 14:30?",
        "A": "I think it s photosystem II, Sal kind of corrects himself later in the video. hope this helps!"
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "At 6:17 Sal mentions that chloroplasts were once other organisms incorporated into the cell...does that mean genetic material in chloroplasts is different from plant DNA?",
        "A": "Yes, the genetic material in chloroplasts has a different sequence from the genetic material in the nuclei of the plant."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "Around 9:30 or so, he says photons excite electrons in the photosystem II. What does he mean?",
        "A": "It means the photons put the electrons in a higher energy state and their energy fuels the pumping of hydrogen protons into the lumen."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "At 1:08, I thought there are 3 stages of Photosynthesis- capturing energy from sun, converting energy to ATP and NADPH, and synthesizing them into carbs, and these 3 stages occur in 2 different processes- light dependent and light independent?",
        "A": "Trapping energy from the sun and photo-oxidation of chlorophyll pigments, coupled with synthesis of ATP and NADPH happen during the light reaction. Synthesis of carbohydrates such as glucose and fructose happen during the dark reaction/calvin cycle."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "At 12:30 when Sal says this process is called photophosphorylation, does he mean the process of the H+'s going into the pump and creating ATP or does he mean when the H+'s go from the stroma to the lumen? or all of it? thanks",
        "A": "I believe he is referring to the entire process."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "What do you mean by chemiosmosis? (at 13:12)",
        "A": "Chemiostasis is where the energy stored in a hydrogen ion gradient is used to drive cellular work such as the synthesis of ATP."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "At 12:20, where does the ADP and the phosphate group come from?",
        "A": "And they are recycled everytime a ATP is broken down in ADT and a phosphate group. (FUN FACT:Did you know that the A in ATP is the same base as the one found in your DNA? ATCG? and that in your DNA, ATP, TTP, GTP and CTP are used to form the DNA strand?)"
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "At 5:44 what is the little purple thing that Sal drew in the stroma.",
        "A": "That s DNA. Chloroplasts have their own DNA."
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "What does 'non-cyclic oxidative photophosphorylation' mean? (it is mentioned at 13:54.)",
        "A": "Non-cyclic --> Not a cycle Oxidative photophosphorylation --> the synthesis of ATP through photophosphorylation (adding a phosphate group) of ADP **ADP becomes ATP"
    },
    {
        "video_name": "GR2GA7chA_c",
        "Q": "Around 15:30 Sal talks about electrons being stripped from water. Do the Hydrogen protons move with their electron or are these two separated?",
        "A": "Since water is H20, and one H is just a proton + an electron, then they go their separate ways, know its late but maybe helps :)"
    },
    {
        "video_name": "VymVIXcyErI",
        "Q": "Around 4:00, why would Y have a positive charge?",
        "A": "Y had a free pair of electrons and in the new configuration this free pair goes between the C and Y to form a double bond. Therefore there is less negative charge on the Y (because it is shared with the C now). Since the positive charge of the Y hasn t decreased, it becomes positively charged. I hope that explanation makes sense."
    },
    {
        "video_name": "VymVIXcyErI",
        "Q": "At 1:49, I think that there is a third contributing resonance structure involved, with the positive charge localized (in LE theory terms, not in reality) on the carbon bonded to the oxygen atom. Right?",
        "A": "Since the structure you describe is used for simple ketones, I guess it must be valid here. I believe that it is a relatively minor contributor to the overall structure."
    },
    {
        "video_name": "VymVIXcyErI",
        "Q": "(3:00) I think I'm a bit confused with electronegativity and conjugation. If Y is an electronegative atom, wouldn't it want to pull electrons towards it, not give them towards the bond with C to make a double bond? Thanks!",
        "A": "It does, this is why the k value is so large, the Y has very little propensity to give electrons to the double bond. He mentioned this structure because it still contributes to the molecules structure, just not much."
    },
    {
        "video_name": "D5ymMYcLtv0",
        "Q": "At 7:20 , why is it that the red and white flowers produce a pink flower? Shouldn't the flower be either red or white? Since both of the \"parent\" flowers are hybrids, why aren't they pink, like their offspring, instead of red and white.",
        "A": "In this case, he is talking about incomplete dominance. This means that the two alleles that determine color will cause a blending effect in the phenotype (In this example, it makes pink). The parent flowers would have pink as a phenotype, however the genotype of the parents are one red allele and one white allele, that, when combined, create a pink phenotype."
    },
    {
        "video_name": "HbDWBeRJboE",
        "Q": "At 1:05, there should be two sets of non-bonded electrons on the nitrogen in the sodamide anion. As drawn, it is a neutral radical. (Maybe I'm just not seeing the fourth non-bonded electron.)",
        "A": "I think there is. It s fainter than the others, but wasn t there before."
    },
    {
        "video_name": "HbDWBeRJboE",
        "Q": "At 1:19, why is the halide opposite the reacting hydrogen the one that is eliminated rather then than the halide on the same side of the bond?",
        "A": "This is an E2 elimination reaction. The H and the Br must be antiperiplanar to each other. This enables the developing p orbitals to overlap and form the \u00cf\u0080 bonds."
    },
    {
        "video_name": "rV6O3rGulaY",
        "Q": "I was under the impression that Homologous Chromosomes had to do with synapsis during Prophase I of meiosis. At roughly time 8:00 Sal says that the chromosomes he was pointing to were \"homologous.\" Wouldn't those two chromosomes be \"sister chromatids\"?",
        "A": "Actually, he is pointing to two different chromosomes (at 7:30ish).Homologous chromosomes are two different versions (as in A vs. a) of a chromosome with the same function. These chromosomes that he is pointing to just haven t replicated themselves into sister chromatids yet. Sister chromatids are two identical copies of the same chromosome, linked at the centromere (middle portion) immediately after replication. They re shown as the X shapes in the next cell of the flowchart."
    },
    {
        "video_name": "5i8HLmVTcRQ",
        "Q": "At 2:27, Sal states that as x approaches c from the right, f of x approaches the same value it does as x approaches c from the left. I am confused about how this can be the case. It seems to me that the value of f of x as x approaches c from the right should be ever so slightly greater than the value of f of x as x approaches c from the left, as the value for which the function is undefined is between them - to approach the same value it seems you would have to \"jump\" over that undefined point...",
        "A": "You re right in that the value of f(x) as x approaches c from the right is ever so slightly greater than the value of f(x) as x approaches c from the left. If the difference between x and c (on both sides) approaches zero, f(x) will approach f(c), which is the definition of the limit."
    },
    {
        "video_name": "WFnyLmL9t2I",
        "Q": "At 0:32, you said that states of consciousness range from alertness to sleep. What about anesthesia sleep? What about coma? Aren't they less aware than sleep? After all, in both of those states, pain does not cause awakening, right? Isn't that less aware?",
        "A": "Good question. The range of states of consciousness shown here could be seen as a subsection of the entire spectrum. Anesthetic sleep is a deeper than regular sleep, and a state of coma is even less aware than ordinary sleep. There are also states of consciousness more alert than every day awareness, for example during a rush of adrenaline."
    },
    {
        "video_name": "WFnyLmL9t2I",
        "Q": "What about lucid dreaming? Is this what she means at 2:02?",
        "A": "She isn t really talking about lucid dreaming here, just the normal dream state. Lucid dreaming is a different state of consciousness than either awake or dreaming, because you are aware that you are dreaming yet you are still asleep. That awareness means that it is not a state of unconsciousness, it is something else."
    },
    {
        "video_name": "WFnyLmL9t2I",
        "Q": "At 1:26, you say that people can induce daydreaming though light meditation, and at 1:51, you say people can induce drowsiness through deep meditation. What do you mean by that?",
        "A": "well i think it refer to your own training and skill in meditating, because meditating the first day is way different than doing it 3 months periodically."
    },
    {
        "video_name": "dfZ9_vFwCA4",
        "Q": "At 4:00 he says 1 cubic meter is 1000 L. Doesn't this depend on the density of the fluid?",
        "A": "Volume and density aren t the same thing. Density measures how much stuff occupies a specified volume. If the mass is greater for substance A compared to substance B, substance A will have larger density than substance B. This is because density is a ratio of mass to volume. However, both substances will occupy the SAME volume."
    },
    {
        "video_name": "VfpKzwrhmqQ",
        "Q": "I cant wrap my mind around newtons third law.. I exert a force on a block (on a frictionless surface) why does it move if it exerts an equal and opposite force? Is it because of the concept similar to what was stated about the earth and the star (i.e the difference in masses) (at 3:48)? Please be somewhat specific when explaining...",
        "A": "Because the force it exerts on you is ON YOU, and the force you exert on it is ON IT. They don t offset."
    },
    {
        "video_name": "VfpKzwrhmqQ",
        "Q": "Near 5:00, he says the star stays put. I thought it moved ever so slightly in a relatively circular motion.",
        "A": "Relative to what? Earth? Empty space? Please be specific."
    },
    {
        "video_name": "VfpKzwrhmqQ",
        "Q": "The third law states that for every action, there is an equal and opposite reaction. But in 0:55, it says that if the forces are equal and opposite they will cancel out. So does this mean that the third law is wrong or it is an incomplete explanation?",
        "A": "Of course the third law is not wrong. Equal and opposite forces ON THE SAME OBJECT will cancel out , the net force on the object will be zero. Third law pairs are on different objects, not the same object, so they can t net to zero. I push on you and you push back just as hard on me. How many forces on you? One, not two."
    },
    {
        "video_name": "VfpKzwrhmqQ",
        "Q": "at around 7:10 you say it has to be equal and opposite always so what if you applied the force as fast as or faster than light?",
        "A": "There is no speed to applying a force. Are you asking what happens if you apply a force to an object traveling at the speed of light? The Newtonian Laws are not accurate when you start dealing with velocities that are a large fraction of the speed of light."
    },
    {
        "video_name": "VfpKzwrhmqQ",
        "Q": "At 9:35, I don't understand what is the green force. What is that force? Where does it come from? Does it mean that the box is pulling the earth with somekind of gravitationnal field? Thanks",
        "A": "It is the normal force. So, a box on a table does not accelerate through the table because there must be an equal force pushing the box upward as down (because the box is not accelerating or moving downward). The table is not actively pushing the box, but it is simply the force that prevents the box from further moving down"
    },
    {
        "video_name": "VfpKzwrhmqQ",
        "Q": "Who Who Wait! At 4:29, massive object's gravitational should be greater!! How can a small planet exert that much force? How will one planet be closer to the star? Eg. Mercury is closer to the sun than the Earth? SOMEBODY HELP ME!! Physics is *CRUSHIN'* my mind.",
        "A": "the planet s small (relative) mass makes it easier to accelerate which makes it so that even if the forces are the same the planet is the one that moves"
    },
    {
        "video_name": "VfpKzwrhmqQ",
        "Q": "9:15 you said that the force on A (the box) is exerted by the B (the earth)\n= F on A by B (downward)\nthe opposite would be\n= F on B by A (upward)\n\n10:55 you said that the force on A (the box) is exerted by the B (the table)\n= F on A by B (upward) <--- this is wrong. i think this should be downward\nopposite would be\n= F on B by a (downward) <--- same as this, this should be upward\n\nthe box's weight is exerting the force downward,",
        "A": "The weight is not a third law partner force to the contact force of the table. The weight is a partner to the gravitational field pulling on the box. Hope that solves it."
    },
    {
        "video_name": "VfpKzwrhmqQ",
        "Q": "At 5:11, the acceleration is going to be the net force divided by the mass, so will there be two accelerations with one net force? One for the star and one for the planet?",
        "A": "Yes, the same force on different objects of course causes different accelerations for those objects, right?"
    },
    {
        "video_name": "VfpKzwrhmqQ",
        "Q": "At 4:36 is the distance a factor of force?",
        "A": "yes as force is a vector quantity"
    },
    {
        "video_name": "u5NCRHcel_g",
        "Q": "in time 4:32 where can i find a SPDT for $0.05 ?",
        "A": "You might have to buy a few hundred of them to get that price. Ebay is a good place to look for deals on bulk electronics parts."
    },
    {
        "video_name": "CFSHq099Mx0",
        "Q": "at 2:52 he got a positive answer, while at 4:21 he got a negative answer. Why is that?",
        "A": "No, he got the same answer at both; the slope is 0 which is neither positive nor negative. Let s call one point a and the second b. I will get exactly the same slope subtracting x and y at point a from the x and y at point b, as I will subtracting the x and y at point b from the x and y at point a, however the sign of the numerator AND DENOMINATOR of a-b will be exactly opposite that of b-a. Since they are exactly opposite, the sign of the quotient (numerator divided by denominator) will not change."
    },
    {
        "video_name": "CFSHq099Mx0",
        "Q": "At 2:35, I don't understand why -1 and -1 equals 0...",
        "A": "What Sal did: -1-(-1)=-1+1= -1 and 1 =0 What you said: -1 and -1 =-1+(-1)=-1-1=-2 I hope this helps you understand what Sal did!"
    },
    {
        "video_name": "CFSHq099Mx0",
        "Q": "Sal said \"one\" but meant \"negative one.\"\n2:08",
        "A": "That was corrected."
    },
    {
        "video_name": "CFSHq099Mx0",
        "Q": "At 1:00, the lines are parallel right?",
        "A": "yes the lines are parallel"
    },
    {
        "video_name": "CFSHq099Mx0",
        "Q": "sal you make a mistake at 3:47 to have x2 not y2",
        "A": "There is no error. Sal is just labeling the points so you can see which number will be (x1, y1) vs (x2, y2) . The points are correctly labeled."
    },
    {
        "video_name": "CFSHq099Mx0",
        "Q": "at 1:18, what do the triangles mean besides the y/x?",
        "A": "The triangles are the greek letter delta . It is commonly used in math to represent the words change in . So, triangle y / triangle x = delta y / delta x which is shorthand for: change in Y / change in X."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "@10:24 What is \"back-filling\"?",
        "A": "Sal s referring to the relative energy levels of the 4s and 3d orbitals. Intuitively, you might expect the orbitals to fill in the order 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, .... However, the 4s orbital is slightly lower energy than the 3d orbitals and the 4s orbital is filled before 3d. Hence, because the order is not quite what you expect, Sal referred to back-filling the 3d after filling the 4s."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 8:06 Sal says that Tin (Sn) has 4 valence electrons.\nSo you can write Tin (Sn) as: {Kr} 5s^2 4p^10 5p^2\nBut its highest energy electron is in p sub shell. So how is its valency 4?",
        "A": "The number of valence electrons is the number of electrons in the outermost shell s s and p subshell, not just the p subshell. There are two 5s electrons and two 5p electrons, for a total of 4 valence electrons."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "In 4:39, aren't all of the group two alkaline earth metals like beryllium and\nmagnesium reactive ? Helium is not reactive, so why put it there. Hydrogen is with the group one alkali metals because it's very reactive like other group one elements, but helium is very stable.",
        "A": "Helium IS a noble gas, BUT it has the same valence electron configuration as the other group 2 elements, eg ns^2"
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "10:10\nwhy is 4s^2 considered farther out, whereas 3d^6 is considered higher energy? i figured that 4 is the higher energy level?",
        "A": "the energy of orbital doesnot depend on the number beside the given orbitals but depend upon the number of degenerate orbitals the particular orbits have ie d shell has 5 degenerate orbital whereas s shell has only 2 degenerate orbital hence 3d6 has higher energy degeneracy means having same energy level"
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 9:25 why is Fe written out as 3d when it is in the 4th period?",
        "A": "Because the first row transition metals like iron are filling the 3d orbitals not the 4d orbitals."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 1:18, isn't helium part of the s' group electrons not the p' group electrons",
        "A": "Helium is a special case since it only has a single electron shell consisting of just an S orbital the second electron produces a completely full set of orbitals like the rest of the noble gasses."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 1:00 what do you mean by p block and s block? How do you know whats what? How do you fine this out,is there a special way?",
        "A": "if the electronic configuration ends with ns then its a s block element. if it ends with np then its a p lock element. but H is an exception"
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "at 5:00 in the video, he's talking about the electron configuration for Carbon, which is [He] 2s2 2p2 and so it therefore has four valence electrons. But how come we don't count [He] as 1s2 and have those two extra valence electrons? As in why Carbon doesn't have six valence electrons?",
        "A": "The valence electrons are the outermost electrons of an atom. Helium has two valence electrons. Carbon has only four valence electrons because the 1s electrons are inner electrons."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 9:30 Sal says that the furthest electron out is 4s2 but the electron with the highest energy is 3d6. But isn't the energy of the electron -13.6/n^2? n is bigger when the electron is more further out so if the electron is furthest out then because of the formula that electron should have the highest energy level. I am wrong? Or am i misunderstanding something? Also what is the reason why iron's reactivity is low? Sal says it's because of the superficial electron configuration but I can't understand.",
        "A": "You are trying to mix the Bohr model with the modern model. You can t do that. Bohr knew nothing about s, p, d, f."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 9:45, when we do the electron configuration for Fe, I don't understand why you have to back up and fill the 3D suborbital, instead of the 4D suborbital. Isn't it in the fourth period?",
        "A": "The first D orbital, although it looks like it s in the fourth period, is actually known as 3D. So that s why Sal wrote down 3D instead of 4D, which is what you d think."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "I'm confused on orbitals and the electron configurations of transition metals. At about 9:00 in the video, Sal starts to explain the electron configuration of iron. Why does he use argon as a base? Does 4s^2 mean that there is 4 s orbitals that hold 2 electrons? I thought that there could only be 1s^2 or 2s^2. And why does the electron configurations of transition metals look as if they forgot the rules of chemistry?",
        "A": "There is one s orbital in each shell, 4s^2 means the s orbital in the fourth shell has two electrons in it. Sal uses argon as a base because it saves time and space. [Ar] means the same thing as 1s^2 2s^2 2p^6 3s^2 3p^6. Everything on the fourth row of the periodic table has the same configuration as argon, plus some more electrons. In general, transition metals do not follow the same rules that the main group (grps 1-2, 13-18) elements do most of the time anyway, their electron configurations included."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "Why at 5:20 does Sal say that the 2S2 shell is not fully completed? Aren't there only 2 electrons in the 2S shell? You have 3 and 4, then 5 starts in the P group, no?",
        "A": "Listen carefully. He is saying that the outermost shell has not been completed yet. The outermost shell can hold up to 2s\u00c2\u00b22p\u00e2\u0081\u00b6, so carbon wants four more electrons in its valence shell."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "I am having a hard time understanding why carbon has 4 valence electrons instead of 2. At 5:07 Sal explains that we assign valence electrons based on having an uncompleted outermost shell... Can someone please explain exactly what uncompleted means? In my mind it has filled up 2s but not 2p, so it should have the two valence electrons of 2p^2",
        "A": "The outermost shell is 2. In shell 2, there are 2 in s and 2 in p, so that s 4 in the outermost shell."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 9:51 Sal says that iron is known to lose 3 electrons. If it loses 2 electrons they would be lost from 4s2. but what would be in the case of 3 electrons? Where would they be lost from? would it form 3d5??",
        "A": "Correct! The electron configuration of Fe is [Ar]4s\u00c2\u00b2 3d\u00e2\u0081\u00b6. The 4s electrons are lost first, so Fe\u00c2\u00b2\u00e2\u0081\u00ba is [Ar]3d\u00e2\u0081\u00b6. Then a 3d electron is lost, so Fe\u00c2\u00b3\u00e2\u0081\u00ba is [Ar]3d\u00e2\u0081\u00b5."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 5:00, why are you not allowed to write the electron configuration as [Be] 2p2? Is there a certain way you should break up the elements when writing it short-hand?",
        "A": "You always start with the prior noble gas."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "In 7:47 , it is said that Tin has 4 valence electrons . After the donation of the 4 valence electrons of Carbon , it has same number of electrons as helium which is a neutral gas. But Tin gets to 46 number element which is not a noble gas . How is this possible ?",
        "A": "It s something you ll get to later but when a set of d orbitals are full it s generally stable and unreactive. Tin has the electron configuration: [Kr] 4d10 5s2 5p2 When it loses the 5p and 5s electrons it is left with [Kr] 4d10 which is called a pseudo noble gas configuration and is quite stable."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "Why is valence electrons' definition not true with the transition metals? What do you mean by non-main block elements?(0:30-0:40)",
        "A": "Usually you think of valence electrons as those being in the shell with the highest n value. So for the first row transition metals that would be the 4th shell. So most could be said to have 2 valence electrons. But that definition isn t very useful as many transition metals show mamy different oxidation states, the d orbitals also take part in their chemistry. So we can t just use a simple highest shell as the definition for valence electrons here."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "at 0:01, what are valence electrons?",
        "A": "valence electrons are the electrons that are in a way left over once all possible orbitals are flled by the rest of the element s electrons."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 7:38, Sal explained that carbon and the elements in that group has 4 valence electrons. I don't get it, because I think there's 14 electrons in the outermost shell in Sn. It's like skipping/ignoring the groups 3-12. Aren't the valence electrons determined by the groups?",
        "A": "Groups 1-2 and 13-18 are the main group elements (also known as the representative elements). Groups 3-12 are transition metals, which don t follow rules the same way the main group elements do. The transition metals have multiple charges and varying numbers of outermost electrons. This makes them a bit annoying (from a teaching and learning perspective) and exceptional."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "Sal, why wouldn't helium be in group 2 since it has two electrons? you yourself said that was a very reasonable argument you could make. [you talked about it at 4:14]",
        "A": "It could be based off its electron configuration, but it behaves chemically like a group 18 element"
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 5:31, He(helium) was placed in a wrong position ??",
        "A": "he put it there to demonstrate at 4:27"
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 4:07 why are the lewis dot structures for Helium on the side (right) instead of in the middle? Is there a separate video for that?",
        "A": "It doesn t really matter which side the lewis dot structures for Helium is. People just prefer to put it on the right side."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 9:30, I don't understand why you have to backfill into the 3d suborbital. Why is it 3d^6 rather than 4d^6?",
        "A": "That s just what they do. It just so happens that 4s and 3d have about the same energy. 4d is much higher."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "Around 9:40 Sal talks about \"backfilling\". What exactly does that mean? Does that just mean going from 4 to 3, and why do we follow that convention?\n\nAlso, why would the d-block have more energy if the s-block is farther out? Wouldn't the s-block have more kinetic energy if it's farther out?",
        "A": "Backfilling means that 4s fills before 3d. 4s is lower energy than 3d, in most atoms."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 9:12 in the video, he says that the electron configuration for iron is [Ar]4s^2 and then 3d^6. I don't understand this stuff about backing up. Can anybody help?",
        "A": "The 4s and 3d orbitals are very close in terms of energy. Don t think of it as backing up, just think of filling electrons in the next orbital."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "Just a quick question...At 3:47 Sal says that He has two valence electrons but my chemistry teacher says that He has 8 valence electrons. Can somebody tell me why it is like that?",
        "A": "I am not sure what your teacher is teaching you... Helium ONLY has 2 electrons, making them their valence electrons as well. Only the noble gases have 8 valence electrons. Everything else wants to be like the noble gases, because of their stability. And this is why bonds will form."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "at 4:04 it says Helium, under it says 1s2, who di he get 1, s, and 2?",
        "A": "1 = the first shell or first energy level s = the electron is in an s orbital, the first shell only has the 1s orbital ^2 = helium has 2 electrons in that 1s orbital."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 9:33 Sal says that the highest energy electrons are the d electrons but I thought that in the video 'Electron configurations in the d orbitals' the guy said that starting with Sc the energy of the 4s orbitals is higher than the energy of the 3d orbitals. Can someone please explain which is correct? Thanks in advance!",
        "A": "The 4s and 3d orbitals have almost identical energy levels. It varies a bit by element which has slightly greater energy than the other."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 1:31, Sal says that Hydrogen has 1 electron, that being its valence electron. What would happen to Hydrogen once it looses that electron? Will it be changed completley or will it still be Hydrogen?",
        "A": "What element it is is determined by the number of protons. Hydrogen always has one proton. Losing the electron would not change what element it is, just its net charge."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "When we determine the amount of valence electrons, would you consider one group as the outer most shell? For example, when Sai talks about Carbon's lewis dot structure around 5:14 and says that Carbon has 4 valence electrons, he first draws out the electron configuration for carbon, which is [He] 2s2 2p2. In other words, would you consider an incomplete shell, while writing out the electron config, everything after the noble gas?",
        "A": "In many introductory chemistry classes, only the electrons in the outermost s and p sublevels are considered valence electrons. So for some elements, that is equal to the number of electrons beyond the previous noble gas. But for others, like Bromine, it s only the electrons in the 4s and 4p orbitals but not those in the 3d."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "When figuring out valence electrons, why do the ones on the s sub level count? At about 5:45, Sal writes the electron configuration for Carbon as: [He] 2s^2 2p^2. I thought s only needed two electrons to feel stable, so why would 2s^2 count in carbon?",
        "A": "It still has the number 2 in front so it is part of carbon s outer most shell"
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 9:47 in the video Valence electrons and bonding, Sal informs that iron may lose ONE two or three electrons. But when I search in the Internet, it says 2 and 3. Is it true?",
        "A": "Iron can show many oxidation states from 0 to +6 depending on the compound, but yes +2 and +3 are the most common."
    },
    {
        "video_name": "NYtPw0WiUCo",
        "Q": "At 5:18, how come 2s^2 isn't a complete subshell? How many electrons should be in the 2s subshell for it to be complete - I had originally thought 2?",
        "A": "It s not 2s2 which is not complete. It s 2p2 which is incomplete. p subshell should have 6 electrons to be complete but in Carbon, it has only 2. So, it s incomplete where 2s subshell in Carbon is complete."
    },
    {
        "video_name": "Z-yv3Yq4Aw4",
        "Q": "At 5:12 He doesn't say if the esophogus is part of the upper respiratory tract or lower respiratory tract",
        "A": "The esophagus is in the upper part of the respiratory tract."
    },
    {
        "video_name": "Z-yv3Yq4Aw4",
        "Q": "From 9:07 to 9:15, why does the right lung have more lobes than the left lung?",
        "A": "Good question. The right lung has an extra lobe simply because it does not need to make as much space for the heart. The left lung is just not big enough for a third lobe to make sense. Hope this helped!"
    },
    {
        "video_name": "Z-yv3Yq4Aw4",
        "Q": "At 3:40 isn't the spelling oesophagus and not esophagus ( I live in india)",
        "A": "Sulfur vs Sulphur, hemoglobin vs haemoglobin, etc... Esophagus is just the American way of spelling oesophagus."
    },
    {
        "video_name": "VsW6NXZIUCQ",
        "Q": "At 7:40, aren't there 8 molecules instead of 6 (2 reds and 6 greens)?",
        "A": "Yes. Thankfully, it doesn t change the final result. You should use the report a mistake button on the right - they ll probably fix it and give you a badge for paying attention."
    },
    {
        "video_name": "X9ypryY7hrQ",
        "Q": "in 2:04 why it was said that its an sn2 mechanism so we used primary alkyl halide\ni mean cant we say like its an sn1 mechanism so we will use a tertiary aklyl halide?",
        "A": "If the mechanism is SN2 for a given reaction, its always SN2. We can t say, Let s say its like an SN1 ."
    },
    {
        "video_name": "X9ypryY7hrQ",
        "Q": "How can the attacked carbon on the alkyl halide at 2:28 be tertiary and SN2 still occur?",
        "A": "At 0:15 he specified that the halide must be a 1\u00c2\u00b0 halide. He was a little careless at 2:28. He should have shown the H atoms explicitly to show that the halide is primary. I agree that it is confusing. Without the H atoms, the compound looks like tert-butyl halide, but it is really methyl halide, so the reaction really is SN2."
    },
    {
        "video_name": "X9ypryY7hrQ",
        "Q": "@ 2:25, the sodium alkoxide gets converted into an ether. The sodium alkoxide has an ionic bond. Why would it want to lose its ionic bond and form an ether with a covalent bond ??. I thought ionic bonds were much more stable so the sodium alkoxide should also be more stable right ?! :/",
        "A": "Yes, you re right about ionic stability but since they involve bondage of charged species hence they remain disassociated {i.e, separate} in the aqueous polar solvent in which the reaction is being carried out.Thus the alkoxide anion is more susceptible to covalent bonding."
    },
    {
        "video_name": "X9ypryY7hrQ",
        "Q": "At 0:53 seconds, can we also use K metal to deprotonate the alcohol, or is K metal strictly for secondary and tertiary alsohols? and If the alkoxide is tiertiary, can it still undergo an SN2 reaction, or does the alkyl halide only matter since the alkoxide will attack and attach there?",
        "A": "Yes, you can use K metal to deprotonate. Also, if the alkoxide is tertiary, it can undergo an SN2 reaction as only oxygen attacks the alkyl halide."
    },
    {
        "video_name": "_WXndBGQnyI",
        "Q": "At 3:31, why does Hydrogen only have around 1amu (I proton) but Oxygen has 16amu (8 protons, 8 neutrons)? Does Hydrogen have any neutrons?",
        "A": "Hydrogen-1 doesn t have any neutrons and this isotope makes up the vast majority of all hydrogen on earth. There are two other naturally occuring isotopes of hydrogen that DO have neutrons, hydrogen-2 (deuterium) has 1 neutron, hydrogen-3 (tritium) has 2 neutrons."
    },
    {
        "video_name": "_WXndBGQnyI",
        "Q": "At 3:09 you say \"Oxygen, the most typical isotope of oxygen you will see on Earth has 8 protons and it's gonna have 8 neutrons, that's the most typical isotope of oxygen.\"\n\nMy question is, would you generally consider an oxygen atom with 8 protons and 8 neutrons an isotope, even though it is the standard form of oxygen?\n\nI have always thought that isotopes were atoms of elements with more or fewer neutrons than the standard number of protons in the atom.",
        "A": "All atoms are isotopes. An isotope simply refers to the specific mass number of the element."
    },
    {
        "video_name": "_WXndBGQnyI",
        "Q": "Hi. At 3:09 you say \"Oxygen, the most typical isotope of oxygen you will see on Earth has 8 protons and it's gonna have 8 neutrons, that's the most typical isotope of oxygen.\"\n\nMy question is, would you generally consider an oxygen atom with 8 protons and 8 neutrons an isotope, even though it is the standard form of oxygen?\n\nI have always thought that isotopes were atoms of elements with more or fewer neutrons than the standard number of protons in the atom.",
        "A": "All atoms are isotopes. Even if the element has only one isotope it is still an isotope. Isotope only specifies the specific nucleon number of the element."
    },
    {
        "video_name": "IkmM4CPnqF0",
        "Q": "anyone know if they appear in nature evenly? or is one more evident? is the staggered conformation more common then the eclipsed? or vice-versa? at 13:42 he mentions the staggered are more stable, i guess this ? also applies to cis/trans, are the trans more common?",
        "A": "the cis/trans story is different. this has more to do with biology since enzymes are very specific and will only recognize the cis or trans form. a common example includes unsaturated fatty acid chains where in nature, most of these bonds are cis. trans fats are not common in nature and are produced through a hydrogenation process in a lab/factory. This is probably too much information but, in short, it s not something of great importance in an organic chemistry class. it is more for a biology or nutrition class"
    },
    {
        "video_name": "IkmM4CPnqF0",
        "Q": "It is eclisped from 11:26-11:29, not staggered right?",
        "A": "Yes, the hydrogens on the far carbon in the newman projection are, in reality, right behind the front carbons hydrogens making it an eclipsed conformation which is higher in energy."
    },
    {
        "video_name": "IkmM4CPnqF0",
        "Q": "At 0:43 is the shape a Tetrahedral ?",
        "A": "Yes , because all of the bonds are sp3 hybridized."
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "What are the physics of the Late Heavy Bombardment (which Sal mentions at 1:16 )?",
        "A": "It was somewhat like equal forces of gravity towards the asteroids (therefore, the asteroid belt formed), between the gas giants and the inner planets along with the sun. But when Neptune and Uranus moved outwards, the gravity inequalized and bombarded the inner planets."
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "At 9:27, Sal mentioned that oxygen bonded with iron inside oceans. So, what kind of water was it?H2O or something else?",
        "A": "He meant that the iron and oxygen formed rust. The oceans were still water, just rusty water"
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "When I was 7:08 through the video, I noticed on the time line Sal was showing, there was Ca written in the captions. I would like to know what is Ca?",
        "A": "G is a common abbreviation for a billion (Giga-). The a simply is short for ago, so Ga is an abbreviation for billion years ago. Ma is million years ago. Ka would likely be thousand years ago."
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "at 1:02 it is said that life began on Earth after the late bombardment. How did they find out that something like the \" Late bombardment\" actually happened?",
        "A": "I would assume mostly from analyzing the craters on the moon. A large number of them were formed at the same time, by the Late Heavy Bombardment. It could also possibly be necessary to account for the formation of the Earth. If they modeled it w/o the Late Heavy Bombardment using computers and the Earth would have turned out differently than it has, they could infer that the Late Heavy Bombardment had to happen."
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "at about 8:00, I do not fully understand what a stromatilite is.",
        "A": "A stromatolite is like modern coral, except with bacteria. The bacteria deposited small amounts of sediment where they lived in their colony, and over thousands of generations all of the sediment built up into the rocky formation you see in the picture."
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "At 8:20 you say that 3.5Ga is the origin of photosynthesis, but isn't that the origin of anoxygenic photosynthesis? Wasn't anoxygenic photosynthesis thought to have been going on for millions of years before the evolution of oxygenic photosynthesis around 2.5Ga?",
        "A": "The anoxygenic photosynthesis was actually happening about 3Ga because primitive bacteria if you will, started developing parts which would allow anoxygenic photosynthesis"
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "At 9:04, Why all living organisms' life begin from the sea not from land.",
        "A": "The water creates a more stable environment in which rapid changes are rare, allowing organisms to survive any changes that might happen. The water is comparable to a house: a barrier to keep dangerous things out. Like harmful radiation and extreme temperature changes."
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "At around 2:01, why were Uranus and Neptune moving away from the sun?",
        "A": "They are not. But they were originally closer to the Sun than they are now. At some point in the solar systems history, Jupiter and Saturn reached a point known as a 1:2 resonance meaning that for every 2 times Jupiter orbited, Saturn orbited once. This destabilized the orbit of several planets and Uranus and Neptune switched places (Neptune used to be closer than Uranus) and moved farther out."
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "At around 8:17 he says that bacteria evolved to utilize energy from the sun and do photosynthesis, and as a result they could produce oxygen. But you need CO2 and H2O to perform photosynthesis, and both of those are made up of O2. Where did that CO2 and H2O get the oxygen? To me it sounds like he is saying that CO2 and H2o (which has oxygen) was used to make oxygen, so it doesn't really make sense.",
        "A": "There was a lot of oxygen from the formation of the solar system. Also, we got oxygen/water from the madness that formed our planet, primarily from volcanic activity. Volcanoes do spit out water vapor and carbon dioxide, along with all the other fun stuff that kept life from forming for a while before they lowered to nonlethal levels."
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "since this has so much biology, where are the viruses in the web at 4:50",
        "A": "never mind :("
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "At 1:06 why were things hitting earth a lot more than now?",
        "A": "Because after a while, if things are crashing into each other and sticking together, there are fewer pieces left to crash."
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "at 5:15, Sal mentions that bacteria broke off from the common ancestor to their own species. My question is, what are viruses? Are they an offshoot of the bacteria, of one of the Archaea or Eucaryota, or are they their own creature that evolved to prey on all forms of life?",
        "A": "to my knowledge no one really knows. so scientist even suggest they are not living at all as they break many of the things normal living things do. I would think that viruses evoled at the same time as multi cell organisms began to from as without these viruse cannot be successful. but it is just a theory..."
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "at 6:22 how can we exactly know that what organism existed long billions of years ago??",
        "A": "We only know that they existed a while ago, we don t know how long that while ago exactly was"
    },
    {
        "video_name": "nYFuxTXDj90",
        "Q": "At 2:14, Sal says the first prokaryotes are formed. In the Hadean eon, how does this just come into existence? I thought that any 'life' molecule is a very specific combination of the elements carbon, hydrogen, oxygen, and nitrogen... so how does it just appear in such a violent period?",
        "A": "The Hadean epoch ended at about 3.8 billion years ago with the Achaean epoch beginning after that. life is thought to have come about 3.5 billion years ago. The Archaean era, while still violent was calm enough to support oceans which is where life is said to have originated."
    },
    {
        "video_name": "V2vR5_B6C5I",
        "Q": "at 4:26 dr. rishi stops talking about a sneese but he never mntions why we go... \"achoo\" and we scrunch out noses and move our mouthes in a funny position. why do we?",
        "A": "That is called a response. That is the brains natural reflex!"
    },
    {
        "video_name": "jIL333CKE9A",
        "Q": "At 3:36, how do we know that acetone is a Bronsted-Lowry base?",
        "A": "Acetone is a Br\u00c3\u00b8nsted-Lowry base because it has lone pairs of electrons on the O atom that can form a bond with an H\u00e2\u0081\u00ba."
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "@6:32, Would it be the same if the order of the atoms was different? For example instead of H2SO4, Sal wrote it as SH2O4?",
        "A": "It would be the same, but H\u00e2\u0082\u0082SO\u00e2\u0082\u0084 is the more proper notation as the compound is an acid."
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "At 3:50, how did sal convert grams of oxygen to moles?",
        "A": "Divide by the molar mass of oxygen, which you can determine by taking the molecular mass and stating it in grams instead of AMU"
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "At 4:05, why does sulfur smell?",
        "A": "Sulfur, as an element, does not smell. The stinky stuff is hydrogen sulfide H2S, which is poisonous and explosive. When inhaled, H2S stimulates olfactory receptors in our nose and mouth which in turn sets off warning alarms in our brains that say Danger! This stuff is poison! Thankfully it does not take much H2S to set off our alarms, and the concentration at which you can smell H2S is much lower than the concentration which short term exposure can harm you."
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "at 1:21 what formula do you use to figure out the moles.",
        "A": "number of moles = mass / molar mass (n=m/M)"
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "From 0:32 to 0:40, Sal said to assume you have 100 grams of the substance in question. What if the total mass is different than 100, like say, the total would be 250, or 85, 74, or 170, what would you do then?",
        "A": "If you are talking about a percentage-----well that means out of 100 . So whether you have 250 or 74 molecules, 60% still means 60 out of every 100 molecules. The ratio will always be the same."
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "At 1:20, so the percent of each element directly transfers to grams of that element?",
        "A": "IF you are dealing with a 100g sample."
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "Is it necessary to go decimal points for moles? For example, here Sal had 1.02, 2.04, and 4.08 moles for the elements, which gave him a ratio of 1:2:4. I did this problem on my own before finishing the video, except i just rounded to whole numbers, and I still got the same answer.\n\nSo, can I just round to whole if the decimal is not too significant?",
        "A": "You need to respect the number of significant digits provided in the problem. Sal does not do a good job with this, but every chemistry teacher I ever heard about will mark problems down if they don t have the correct number of significant digits."
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "at 1:48 what is hydrogen nucleuses?",
        "A": "Atoms of elements have nuclei at their center. This is where we find protons"
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "At 3:35, would you have to double (or half) any calculations if the Oxygen was molecular (O2)?",
        "A": "we would need to double it if it was O2"
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "at 1:00, How do you calculate the percent (%) by mass of an element in a compound formula?",
        "A": "Take the total mass of a compound and then find out how much mass one element takes up. in water, H20, Hydrogen gives 2 protons, but Oxygen has 8 protons and neutrons. The whole thing has an mass of 18 AMU. So, Hydrogen s percent is 2/18 or 11.1% and Oxygen s percent is 16/18 or 88.9%."
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "At 3:09 in the video, why can we use 16g for the atomic weight of oxygen? Shouldn't we use 15.999?",
        "A": "And in the video, many time he repeated the in this topics its full of assumption and approximation so u can round up"
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "At 1:53, Why does Hydrogen not have any neutrons?",
        "A": "Hydrogen can have neutrons. When it has one neutron it is known as deuterium, and if it has two neutrons it is known as tritium. The most common isotope of hydrogen, however, has no neutrons. With larger elements, neutrons help space out the protons to minimize proton-proton repulsion. However, with hydrogen there is only 1 proton so this is not necessary to help stabilize the atom."
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "at 1:00, How do you calculate the percent (%) by mass of an element in a compound formula?",
        "A": "Take the total mass of a compound then find out how much mass one element takes up. Like in water, H20, hydrogen gives 2 protons, but oxygen 8 protons and neutrons. The whole thing has an mass of 18 amu. So hydrogen s percent is 2/18 or 11.1% and oxygen s percent is 16/ 18 or 88.9%"
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "Isn't it Hydrogen Sulphide which smell of rotten eggs and not just plain sulphur? But Sal says it is sulphur at 4:10",
        "A": "You re right, it is hydrogen sulfide that smells of rotten eggs. Sal was probably just meaning sulfur-containing molecules, rather than elemental sulfur. Many compounds that contain sulfur smell!"
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "at 6:31, can I write H2O4S instead of H2SO4 like Sal?",
        "A": "You can write whatever you want, but the conventional way is the way Sal wrote it."
    },
    {
        "video_name": "sXOIIEZh6qg",
        "Q": "3:50 I am still don't exactly understand how you find the mole of the element with the gram of the element?",
        "A": "You need to take a look at your periodic table and the molecular mass is usually displayed in the box with the element. For instance Oxygen has a molecular mass of 16.00 grams per mole. So if I had 8 grams of Oxygen I would have 0.5 mole of Oxygen. You just take the mass given in the problem and divide it by the molecular mass. 8 grams divided by 16.00 grams per mole (8/16=0.5) the grams cancel each other out and then you just have 0.5 mole."
    },
    {
        "video_name": "9BJ_WWRgtLc",
        "Q": "Is there a delay from a pulling of a rope and the box moving?in other words how long does it take for those fibers to transfer the force across?\nApprox 1:32\nThank You",
        "A": "strictly speaking, the force travels at the speed of sound through the material"
    },
    {
        "video_name": "9BJ_WWRgtLc",
        "Q": "At 2:40, why would it require more force to pull the rope if its mass were to be significant?",
        "A": "Because you would have to accelerate the rope, and that would require force, since the rope has mass and F = ma"
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "why doesn't the 3 divide into the 9 at 3:28, leaving us with 3 atm?",
        "A": "First we multiply P1 and V1, get the single value i.e 27atm.ltr and then divide by 3ltr leaving 9 atm :)"
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "At 10:37 he says \"lets say we want our volume in liters\"... is that totally arbitrary? or is there a reason he chose liters and not milliliters?",
        "A": "Liters is the SI unit of volume."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "At 4:50, when you are re arranging the equation you write, (pv)/t=nR=C ... where did the C come from and what is it?",
        "A": "C is an arbitrary variable introduced to simplify how you write n*R (a constant since R is a constant and the number of particles do not change). So instead of having to write n*R each time in an equation, you can simply represent it with a variable that is more quickly to write, in this case C"
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "So... how do I know what to pick for R? At roughly 10:50, Sal has a list of different R possibilities. Where do those come from and how do I know which one is the right one?\nThanks.",
        "A": "What you use depends on what the units for pressure and volume are: Atmosphere and litres? 0.0821 L atm K-1 mol-1 Pascals and cubic metres? 8.314 J K-1 mol-1 The Wikipedia page on the gas constant has a big ish list of different units if you wanted to see. The point is you use whichever one cancels out the P and V units."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "Do you always have to convert to kelvin? 5:25",
        "A": "yes when completing gas law problems"
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "At 11:20, where did the kelvin go?",
        "A": "The unit of Kelvins gets cancelled as the temperature is in Kelvins, while the constant R has a unit of (L*atm)/(mol*K)."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "at 4:51, why T2? I thought temperature was constant. in that case, then it would be T1 still. I'm confused.",
        "A": "Yes., the temperature was constant. At this pointin the video, he has left the question and is deriving the Combined Gas Law equation, P\u00e2\u0082\u0081V\u00e2\u0082\u0081/T\u00e2\u0082\u0081 = P\u00e2\u0082\u0082V\u00e2\u0082\u0082/T\u00e2\u0082\u0082. You can still use this equation if the temperature is constant. You set T\u00e2\u0082\u0081 = T\u00e2\u0082\u0082. The two Ts cancel, and the equation reduces to Boyle s Law: P\u00e2\u0082\u0081V\u00e2\u0082\u0081 = P\u00e2\u0082\u0082V\u00e2\u0082\u0082."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "at 8:33 will the standard pressure and atmosphere all ways be the same or do they change.",
        "A": "Technically, the standard temperature and pressure is whatever the individual considers the norm. However, if you are given a problem where they describe the scenario as having standard temperature and pressure with no more information regarding those two, you should assume 273K and 1 atm."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "At 03:20 he says he divides both sides by 3, but he actually divides by 3 LITERS, right?",
        "A": "yes that is correct"
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "At 6:47 I completely lost how you got 1500, shouldn't it be 30?",
        "A": "Recall that, when working with ratios (as indicated by the = sign), you cross multiply to solve. In this case, you take 300*5 and 2*T. Then you use algebra to solve for T."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "Is it possible, at 6:10, that the pressure is 5atm when we halfed the container?\n\nHow can P2 be 5, shouldnt it be 2, because how much we reduce the container so much the pressure will be bigger (linear growing)?",
        "A": "In this case, he defined the pressure and volume (both of which changed from time 1), so the calculated value was temperature. Remember that PV=nRT is true for all ideal gases, so if the amount of gas involved doesn t change, nR will be constant and PV/T for the first scenario will be equal to PV/T for the second. If P goes up V must go down OR T must go up OR a combination of the two. If P goes down V must go up OR T must go down OR a combination of the two. If P and V go up T must go up. etc. etc. etc."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "At 1:09, do some particles exert more pressure than others?",
        "A": "Yes particles which are moving faster will exert more pressure than particles moving more slowly."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "Okay, sorry for the stupid question, but I don't get how Sal solved that equation at 6:36?\n3/500 = 5/T2\nHow do you solve an equation with fraction on both sides? I already looked in the algebra section, but I couldn't find anything...\nThanks in advance :)",
        "A": "3\u0140=5/t2,taking t2 to the numerator,t2=(5*500)/3,t2=2500/3,hope u can solve it now!!was that helpful??"
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "From 9:22 on Sal talks about the volume of one mole of an ideal gas at stp. Does this example volume apply for one mole of any of the idel gases at stp? Why do the ones with bigger atoms do not take up more space?",
        "A": "Yes, the volume example works for any ideal gas at stp. By definition, ideal gases are assumed to have no intermolecular interactions, and to not take up any space, so as far as the ideal gas assumption goes, there are no bigger or smaller atoms. In real gases, different gases do have atoms of different sizes, but these atoms are usually far apart, so there is much more empty space than atoms; so, the ideal gas assumption is not a terrible one, but it is still only an approximation."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "At 3:20 when Sal divided both sides by 3L, doesn't the 9L also get divided by 3L, leaving us with 3atm=P2?",
        "A": "No, it should be 9atm, because 9L divide by 3L is 3( liter sign should be gone), and you left with 3atm, which mean 3 times 3atm, 9atm Simple way to explain is follow the equation without any masurements. 9L*3atm=3L*?atm to 9*3=3*? (just with simple math, you can easily get the right atmosphere, which is 9!!!)"
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "In this video, at 4:33, would you say this particular problem being demonstrated is an example of Charles' Law?",
        "A": "Charles Law says P/T=k where P=pressure, T=temperature in K, and k=a constant. What Sal is using is called the Combined Gas Law. It combines both Charles Law and Boyle s Law which says PV=k where P and k are defined as before and V=volume (scientists should really learn to spell constant; it starts with c not k :-) ). So you are partly correct, but not completely. Hope this helps."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "at 2:00 when you lower the volume doesnt the tempratre rise",
        "A": "Either the temperature can rise, the pressure can go up or both can happen at the same time."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "At 4:33 Sal writes down the formula for Charles's Law (V1/T1=V2/T2). Since temperature is in the denominator, What would happen if the temperature is 0 Kelvin. Now we all know that you can't get to 0 Kelvin or the absolute zero, but theoretically what would happen because you can't divide by zero? Because scientists have come quite close to reaching 0 Kelvin, it seems likely that someone will break the law someday. Remember this is only a theoretical question.",
        "A": "Nice question! Well, if there is absolutely no movement, i.e. 0K and not a tiny fraction above, then there is also no volume and so pressure cannot exist, and the property of charge would not exist either - which is one of a few reasons why you cannot get to absolute zero by definition. The thing that creates volume are the interactions between charges and these will continue to move regardless how cold you make something."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "At 10:40 how did you get the three versions of R?",
        "A": "The three versions of R come from using different units. If your problem uses m^3, Pa, moles, and K, you should use R=8.3145. If you are using L, kPa, moles, and K, you should also use R=8.3145. If, however, you are using L, atm, moles, and K, you should use R=0.082 (I used R=0.0821. It has more significant digits). I hope this clears the confusion up!"
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "at 2:00 when you lower the volume doesnt the tempratre rise?",
        "A": "PV = nRT. If volume goes down, doesn t temperature have to fall (assuming P does not change)?"
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "at 11:20, where did Kelvin go? I understand we're solving the problem for Volume, which is going to be in Liter but is it ok to ignore Kelvin after multiplying 0.082 x 273K and just put Liters for the answer?\nIn this part of chemistry, I thought the units are important and you have to either cancel out or carry them as long as they're still in the equation.",
        "A": "You divide it by dividing it by Kelvin"
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "When you talk about Kelvin, are you supposed to say 300 degrees Kelvin, or just plain 300 Kelvin. Sal at 6:44 says degrees Kelvin, but whenever I did my chemistry problems, when I wrote Kelvin, I just wrote 300 K, not 300 (degrees circle thingee) K. Thanks for the help!",
        "A": "Yes, the Kelvin scale is written without the degrees unlike the Celsius scale, so it would be 300 K."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "at 6:21, he uses 1atm x 2m^3=5atm x 1m^3, which is p1v1=p2v2, but they do not equal.... is it wrong",
        "A": "P1V1 is only equal to P2V2 if the temperature is constant. In this problem the temperature is changing, so it is the ratio of P1V1/T1 that is equal to P2V2/T2, as Sal has done."
    },
    {
        "video_name": "GwoX_BemwHs",
        "Q": "at 11:43 in the video where does he get the constant R?",
        "A": "R is a constant, like pi or G for gravity. He has given multiple equivalent values depending upon the units used (like saying 1 m = 100 cm = 1000 mm). Then you can use whichever is most convenient to cancel out units."
    },
    {
        "video_name": "Rd4a1X3B61w",
        "Q": "At 0:25, of which molecule is the model?",
        "A": "any molecule can have a model"
    },
    {
        "video_name": "Rd4a1X3B61w",
        "Q": "In the first 0:05 seconds of the video, there and equation on the top of the screen:\n2h2(g)+O2(g)=2h2O(g) what dose that mean?",
        "A": "It means that 2 molecules of hydrogen gas react with 1 molecule of oxygen gas to form 2 molecules of gaseous water."
    },
    {
        "video_name": "Rd4a1X3B61w",
        "Q": "At 2:15, why does he say we don't want the full answer?",
        "A": "It s hard to say that there even is a full answer. We keep on discovering new things each day. Plus, if we know everything, there won t be the joy of learning and discovering new things. Hope this helps!"
    },
    {
        "video_name": "Rd4a1X3B61w",
        "Q": "At 0:07 what does (g) stand for?",
        "A": "The (g) stands for gas. It tells you that the substance it appears after is in the gaseous state. You also might see (s), meaning solid, (l), meaning liquid, and (aq), meaning aqueous, or dissolved in water."
    },
    {
        "video_name": "Rd4a1X3B61w",
        "Q": "at 0:05 when was that picture taken and what were they doing",
        "A": "The small inset picture is of Marie Curie (1867-1934), the scientist who discovered radium and polonium. This was probably taken after she became famous - maybe around 1903, when she was awarded the Nobel Prize for Physics. The main picture appears to be of a later date and possibly around 1940s/50s. I don t recognise either of the scientists. In neither picture is it possible to define what is being done - perhaps they were staged for the camera?"
    },
    {
        "video_name": "Rd4a1X3B61w",
        "Q": "At 5:46 it says that chemical reactions happen in his mouth for him to talk, does that mean that people who can't talk not have that chemical reactions?",
        "A": "No. If it weren t for chemical reactions, you wouldn t even be alive! ATP, the energy that powers us, is synthesized through chemical reactions, giving you the energy you need to do, well, anything. You are basically made of chemical reactions."
    },
    {
        "video_name": "Rd4a1X3B61w",
        "Q": "So, if atoms are non-living (not sure if they are, can somebody confirm it?), and everything is made of atoms, meaning that we humans are made of atoms. The speaker in the video mentioned at 5:34 how Biology is based on molecular interactions. How do these non-living atoms/molecules interacting create life?",
        "A": "Atoms are non-living, as you say. And, yes, we humans are made of atoms and all biology is based on molecular interactions. But how everything comes together to produce a living organism is clearly too complex to be answered in a comment such as this - in fact, it is an entire subject in its own right. Furthermore, there is much that is unknown and yet to be discovered, including the fundamental question of how the first life forms evolved from basic molecules."
    },
    {
        "video_name": "Jt3-PIC2nCs",
        "Q": "Since Carole says \"It's not really hard and fast rules\" (2:06) about at which age a stage starts, would it be possible that a child develops these stages in a different order or parallel to eachother? For instance, could the development of the 'Formal operational' stage start before a child has a (fully) gone through 'the Concrete operational' stage?",
        "A": "I think that could happen, kind of like how in Freud s Psychosexual Stages of Development a child may not complete one of the Stages, which would then lead to problems later in life. For example, not completing the Anal stage could lead to control issues. However, a child wouldn t be able to skip one of Piaget s stages, since learning and development are on a steady course, so to speak."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "At 01:58 he says that Be cannot complete the acted rule because of the formal charge. What I don't know is that it is important to complete the acted rule or have zero formal charge? (which one is important?)",
        "A": "Be is an exception and does now follow the octet rule (like Boron and Aluminum). Therefore formal charge is more important here."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "At 0:55, Is it always that, a Less electronegative atom should be in the centre ?? If yes, then what about Water molecule ?",
        "A": "Hydrogen is never going to be in the centre as as far as we re concerned here it only forms 1 bond."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "At 0:37, how can beryllium form a covalent bond when it is a metal?",
        "A": "Beryllium is observed to be a little bit different from other elements in group 2. It has a co-ordination number of four, whilst other metal ions are 6-coordinated. Beryllium has 1s2.2s2 configuration, and hence when it gives away ions 2-level orbitals become empty and to contain it, lone pairs are drawn; hence making it very similar to covalent bonding."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "@6:00-6:10 you spoke about carbon (central atom) needing to be given an octet but @5:23 you said that oxygen (terminal atom) is going to follow the octet rule? I dunno why but I'm a bit confused as to how the octet rule applies to terminal and central atoms.",
        "A": "If the atom is carbon, nitrogen, oxygen or fluorine it is highly likely that it will always follow the octet rule in a structure. There are a few exceptions with these four, mainly with nitrogen as it has an odd number of valence electrons so sometimes has 1 unpaired electron in a structure (eg NO, NO2, note these are often highly reactive due to that unpaired electron) Anything else probably can break the octet rule"
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "6:28 Why is the covalent bond between the CO2 double bonded",
        "A": "The bonds have to be double bonded to give the C atom an octet of valence electrons."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "why is a double counted as one electron cloud (6:53)?",
        "A": "Each orbital can hold up to two electrons."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "I still don't get why there are no lone pairs around Be at around 1:59?",
        "A": "Why would there be? Be only has 2 valence electrons and both of them are participating in chemical bonds."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "What I don't see is why you can't just keep layering bonds. What I mean is, take a look at 5:55. You could take two electrons from each oxygen and create a double bond on both sides. But couldn't you (and shouldn't you) keep doing that for the remaining two pairs of electrons on both sides of the carbon??\n\nThank you to anyone who can both understand my question and answer it :-)",
        "A": "Well no, then the carbon atom would have quadruple bonds on both sides and have 16 electrons in its valence shell which is not possible."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "at 2:30, formal charge is explained. Does this mean that the number of valence electrons of an atom equals the number of bonded pairs that an atom will have? (example sulfur has 6 valence electrons, so it will form 6 bonds)??",
        "A": "There are different definitions of valence but, according to one definition, that s exactly what it means."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "at 1:20 where do 4 came from?",
        "A": "That is the number of electrons bonded."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "At 2:41 why is it 2-2 =0 ?",
        "A": "That is the formal charge of the beryllium atom."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "Around 6:10 how do you know that carbon needs a double bond?",
        "A": "Carbon follows the octet rule and so it needs 8 electrons in total to fill its octet; therefore, each oxygen will give up a lone pair as another bond. This will result in each oxygen having a double bond with the carbon. Now, each oxygen has 8 electrons as well as the carbon."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "At 6:52 jay said that a double bond that had four electrons in it was one electron cloud, but i thought an electron cloud could only have two electrons in it with opposite spins. can someone explain this to me?",
        "A": "what jay is talking about is electron density regions, the bond as a unit is an electron density space meaning an area between 2 atoms in the bond where electrons are relatively high in density. In a double bond there is a sigma and pi bond you can break those into electron clouds if you like or you can really define a cloud as a single orbital or orbital lobes. all of this is meant to mean an area where electrons are 90% of the time."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "At 0:38 he says that you can find Beryllium in group two on the periodic table. None of the tables I've looked at have group numbers. How can if find the groups->number of valence electrons on a periodic table?",
        "A": "Almost every periodic table I can see on Google image search has the group numbers on them. Even if there aren t group numbers you could write them in, they go from 1-18. The left most group is 1, the right most is 18."
    },
    {
        "video_name": "97POZGcfoY8",
        "Q": "At 2:30-2:40, why does chlorine have 2 valence electrons when it is actually supposed to have 7?",
        "A": "chlorine has 7 valence electrons and Be is the one that has 2 valence electrons."
    },
    {
        "video_name": "BBIGR0RAMtY",
        "Q": "At 1:21, what is the equivalence point?",
        "A": "It is the equivalence point, or stoichiometric point, of a chemical reaction is the point at which an added titrant is stoichiometrically equal to the number of moles of substance present in the sample: the smallest amount of titrant that is sufficient to fully neutralize or react with the analyte. ..."
    },
    {
        "video_name": "BBIGR0RAMtY",
        "Q": "at 4:23 whats an aqueous solution.",
        "A": "When you dissolve something in a liquid, you get a solution. When you dissolve something in water particularly, you get an aqueous solution. Hope this helps :)"
    },
    {
        "video_name": "BBIGR0RAMtY",
        "Q": "At 2:16, what does he mean by PH?",
        "A": "pH is a scale we use to measure the concentration of H+ in a solution. pH generally ranges from 0 to 14, and is a logarithmic scale (so a pH of 3 vs a pH of 4 represents a 10X difference in [H]). pH is defined as: pH = -log[H+] Because the pH is the negative log, low pH numbers indicate high concentrations of H+."
    },
    {
        "video_name": "BBIGR0RAMtY",
        "Q": "May i know why A- is used as an example of a weak base at 4:06? Is it the anion of the weak base or...?",
        "A": "A\u00e2\u0081\u00bb is just a symbol that is used to represent an unspecified conjugate base."
    },
    {
        "video_name": "BBIGR0RAMtY",
        "Q": "at 9:50, besides looking at the graph converting the ml to kind of the respective pH=5, are there any calculation methods leading to the pH? I mean in exams it's no necessary to have graphs all the time...Please help!",
        "A": "Without the pKa there s really no way to figure out the pH. Titration would be pointless if you could >.<"
    },
    {
        "video_name": "koAFBScR41A",
        "Q": "Can isotopes also be classified by the number of electrons they have? In this video (3:38) Sal gives an example of an isotope, F^18 that is negative. So, would F^18 that is positive be a different isotope? Like if F^18 had 7 electrons instead of 10. Thanks!",
        "A": "Isotopes are defined by how many neutrons they have. F^18 - is the same isotope as F^18 +. Charged atoms or molecules are called ions. F^18- and F^18+ are different ions of the same isotope of Fluorine."
    },
    {
        "video_name": "koAFBScR41A",
        "Q": "At 2:02 when Sal says this is an isotope, doesn't an isotope have a different number of neutrons than protons?",
        "A": "Isotopes are atoms that have the same number of protons but different numbers of neutrons. Sulfur-32 (in this video) is one possible isotope of sulfur, there are many more (sulfur-33, sulfur-34, etc.). These are all sulfur atoms, but have different numbers of neutrons. Every atom is an isotope."
    },
    {
        "video_name": "vXESvgGbSiw",
        "Q": "I thought there wouldn't be stereochem at 5:27 since neither C is truly a chiral center...is the path around the ring not the same for both?",
        "A": "You are generating two stereocentres from achiral reactants. If you start with achiral reactants, the product must be achiral. In this case, the product is a meso compound, because it has an internal plane of symmetry."
    },
    {
        "video_name": "9b8qZ6OHZ5s",
        "Q": "Where did the actual mass at 2:26 2:29 of He come from? How do we know it is the actual mass and not another prediction? I could see many students thinking you used the average mass from the periodic table--maybe you did? A brief word on this would have been helpful rather than throwing out a number.",
        "A": "It has been measured via mass spectrometer."
    },
    {
        "video_name": "9b8qZ6OHZ5s",
        "Q": "That's a lot of memorizing. Where did he get the Energy formula on 7:24 (Time)?",
        "A": "If you are referring to E = mc^2, then this is Einstein s famous equation that came from his Special Theory of Relativity."
    },
    {
        "video_name": "9b8qZ6OHZ5s",
        "Q": "at 10:32 , he mentions that the amount of energy that we have to apply to break the nucleus apart is, 4.53346x10^-12 J right? isn't that a very very small amount? so why cant we break the nucleus by just heating or something like that? wouldn't it give the amount of energy needed to break the nucleus?",
        "A": "Breaking apart a single nucleus doesn t give much of a energy. Normally, anything that we break (in fission for example) is way lot more than one nucleus. If we have to break one mole of nucleus. (though practically not possible to have a mole of nucleus..) it ll take a whopping 2.7 X 10^11 J... Go figure..."
    },
    {
        "video_name": "9b8qZ6OHZ5s",
        "Q": "At 5:15 you said that 1 amu=1.6654x10^-27. How come you didn't use 6.02x10^-23? or how did you calculate this?",
        "A": "1.66054x10^-27 is calculated by multiplying Avogadro s number (6.022x10^23) by 1000, then taking the reciprocal. Reason: 1 gram=6.022x10^23 amu 1000 g=1 kg 1 kg=6.022x10^26 amu Divide both sides by 6.022x10^26 to get (approximately): 1 amu=1.66054x10^-27 kg Hope that helps!"
    },
    {
        "video_name": "9b8qZ6OHZ5s",
        "Q": "At around 5:10, why isn't the conversion factor Avagadro's number (6.022 x 10^-23) divided by 1000?",
        "A": "what is 1/(6.022 x 10^23)? Try it"
    },
    {
        "video_name": "9b8qZ6OHZ5s",
        "Q": "At 4:58, what is a conversion factor?",
        "A": "The number (and units) that you multiply one number by to convert it into a different unit. For example, in the video the mass in atomic mass units was multiplied by the conversion factor to convert the mass into kilograms."
    },
    {
        "video_name": "9b8qZ6OHZ5s",
        "Q": "2:26 How was the actual mass calculated?",
        "A": "It would have been measured, rather than calculated, using high resolution mass spectrometry."
    },
    {
        "video_name": "9b8qZ6OHZ5s",
        "Q": "At 4:04, how is the mass converted into energy? I'm a touch confused",
        "A": "Energy is released when the nucleus forms from its separate parts. That energy has mass and therefore the mass of the remaining nucleus is less than the sum of the parts."
    },
    {
        "video_name": "Ri557hvwhcM",
        "Q": "At about 2:20, Sal talks about using a cross product with a charge and \"other stuff\". Does anyone know what he means? Did he make a video on it? Thanks!",
        "A": "Yes I do, and yes he did. I suggest looking it up; He did a good job explaining it. It s in Magnets and Magnetic Force. He also has the dot product, if you re interested. It s in Electric Motors. Good luck!"
    },
    {
        "video_name": "Ri557hvwhcM",
        "Q": "In 4:48 What does the \"M\" stand for?",
        "A": "This is the Greek letter mu. Mu in this formula is the permeability of the material that the wire is surrounded by. It will change the strength of the magnetic field based on what the material is."
    },
    {
        "video_name": "Ri557hvwhcM",
        "Q": "At 2:50 how was the direction found? And is this derived from Biot Savart law?",
        "A": "Sal is using the right hand thumb rule, where your thumb points in the direction of current and your fingers(when curled around the thumb) show the direction of the magnetic field."
    },
    {
        "video_name": "Ri557hvwhcM",
        "Q": "At 5:02 which law does he use to get B=uI/2piR?",
        "A": "He used Ampere s Law \u00e2\u0088\u00aeB\u00e2\u0080\u00a2ds=(\u00ce\u00bc0)(I)"
    },
    {
        "video_name": "d1XTOsnNlgg",
        "Q": "At around 1:50 the narrator mentions using a \"certain amount of base\"... what is this amount? How do we know the acid has been completely neutralized and is it possible to add too much base?",
        "A": "You use an indicator that changes colour when all of the acid or base has been neutralised. It s definitely possible to overshoot the end point by adding too much though."
    },
    {
        "video_name": "d1XTOsnNlgg",
        "Q": "Hello, at 7:40, the mole is for HCL is 0.243 however by my calculations the answer is 0.0243. please elaborate.",
        "A": "You must have a defective calculator. 0.00486/0.0200 = 0.486/2 = 0.243"
    },
    {
        "video_name": "d1XTOsnNlgg",
        "Q": "At 1:01, the alkali is in the burette and the acid is in the conical flask. Shouldn't this be the other way around?",
        "A": "Generally you have the species of known concentration in the burette and the unknown concentration in the flask. This video is consistent with that."
    },
    {
        "video_name": "uMYAc04D0ak",
        "Q": "I don't get why we are ignoring the force of gravity at 7:00",
        "A": "i didn t watch the video but it sounds like probably he is only considering the horizontal motion of the ball"
    },
    {
        "video_name": "uMYAc04D0ak",
        "Q": "my question is that:\nAt 6:40, it is said that the force of gravity doesn't affect change of momentum in the x-direction. Rather, it only acts on the momentum change in the y-direction. That's fine.\n\nBut, does the gravity has any sort of effect (be it even negligible) on the resultant momentum change? I mean, the momentum that occurs both in the x and y direction?",
        "A": "Not really, just during its flight path. Technically, gravity IS working on the ball in the 20 milliseconds that the ball is in contact with the person s face, but it is negligible. I would instead look at the trajectory as it is coming in and after it bounced off. Basically, gravity does affect total momentum by adding y momentum at a constant rate. Hopefully I wasn t too confusing."
    },
    {
        "video_name": "uMYAc04D0ak",
        "Q": "6:14 - \"There was only one X-directed force at this time\". What about drag? Isn't that a force acting in an opposite direction of the ball because of the floating particles in the air? Why wasn't that accounted for?",
        "A": "We are only considering the time interval over which the ball makes contact with the person s face. This collision happens very quickly. For collisions like this, we typically neglect friction/drag because it doesn t really have enough time to be significant during the collision."
    },
    {
        "video_name": "zk3vlhz1b6k",
        "Q": "So, if I extend what Sal says at 6:53, it seems that we don't have a scientific answer for where cells (or their organic, asteroid, etc. precursors) came from. But, isn't that kind of question outside the reach of science? I may be looking too much into his explanation.",
        "A": "No, it s an active area of research. Most of the steps have been generally worked out and progress is being quickly made."
    },
    {
        "video_name": "zk3vlhz1b6k",
        "Q": "At 1:32 is Sal talking about cork? I thought that was what Hooke was looking at under his microscope.",
        "A": "yes they are the dead cells which are sometimes called the bark"
    },
    {
        "video_name": "dQCsA2cCdvA",
        "Q": "At 0:41 Sal mentions energy conversion. What types of energy do humans convert?",
        "A": "Many different types of energetic conversions happen within and between our cells. I d say one of the most common examples occurs between neurons (brain cells). When a signal travels through a neuron, it s electrical energy, but when that signal needs to cross the synapse to reach the next neuron to propagate, it s converted into a chemical to cross the synapse, and then back into an electrical signal in order to cross through the next neuron."
    },
    {
        "video_name": "dQCsA2cCdvA",
        "Q": "Sal says that a cell is large on the atomic scale at 2:25. So what is the smallest thing on that scale. Also is there a smaller scale than the atomic scale?",
        "A": "Probably because it is a simplified explanation :) I just learned about G1, G2 etc this past semester and I m in 1st year university."
    },
    {
        "video_name": "dQCsA2cCdvA",
        "Q": "At 2:42 it says that HIV viruses are emerging from the immune cells. How do the HIV viruses emerge from the immune cells?",
        "A": "The HIV virus attacks the immune system cells and then leave the cell and continue contaminating the rest of the immune system cells."
    },
    {
        "video_name": "dQCsA2cCdvA",
        "Q": "I don't understand how crystal lattice structures and DNA are related; my science teacher has me confused! At 0:32 and 2:11, Mr. Khan mentioned DNA molecules, but I don't see anything in common... Please help me out!",
        "A": "DNA structure is double-helix structure. If you search DNA in google image, you can see this. This structure has repeated form, so it can call lattice."
    },
    {
        "video_name": "dQCsA2cCdvA",
        "Q": "At 2:54 how are virus facinating?",
        "A": "It just depends on the person. Some people find them very fascinating while others don t."
    },
    {
        "video_name": "dQCsA2cCdvA",
        "Q": "At 2:54 he mentions that viruses are right on the edge of life and non-life. What does he mean by that?",
        "A": "Viruses are kind of considered both living and non-living. They don t have a cell structure (cell structures are necessary for life), but they do have genetic material (genetic material is needed for life) that they infect cells with."
    },
    {
        "video_name": "dQCsA2cCdvA",
        "Q": "At 0:46, do all life forms need energy?",
        "A": "Yes.... say if we didn t get energy then we might not be alive (PROBABLY!)"
    },
    {
        "video_name": "xusMSv2z77s",
        "Q": "Okay, so at 4:30 he says quite truely that Torque is equal to m*r^2*alpha\nThen when he works through the next problem, completely drops the idea of squaring....he doesn't write 9^2*50....JUST 9*50.\nNobody else here thinks thats wrong, so what did I not understand",
        "A": "Alpha is angular acceleration, while a is acceleration. Here a=alpha*r"
    },
    {
        "video_name": "xusMSv2z77s",
        "Q": "At 12:56.\n\nWe still have to consider object three even there is no torque acting on it?",
        "A": "Yes, it s rigidly attached to the other masses, so it s still resisting the acceleration."
    },
    {
        "video_name": "xusMSv2z77s",
        "Q": "At 13:09, how would the units work out from Nm / kg * m^2 s into radians / s^2?",
        "A": "A newton is a kg*m/s^2 and radians are unit-less because they represent a ratio (sort of like a percent)"
    },
    {
        "video_name": "j-rBgs_p-bg",
        "Q": "1:27 does oxidation take place always like that ? It removes a hydrogen and adds one bond with oxygen at the same time ?",
        "A": "Oxygen can be defined as addition of oxygen and/or loss of hydrogen, that s simply the definition. We can also generalise it to be an increase in oxidation number which are theoretical charges of each atom."
    },
    {
        "video_name": "j-rBgs_p-bg",
        "Q": "At 9:12, when the hydrogen is lost from the alcohol, does it take one electron with it? Similarly, does the -OH also take one electron from the bond it is sharing with the Cr with it when it leaves?\n\nI'm assuming that is how they are able to form the water, but I wanted to know if I understood the flow of electrons correctly.\n\nThank you!",
        "A": "Yes, in effect, when the hydrogen atom leaves, it takes one electron with it. And when the OH leaves the Cr, in effect, it also takes one electron with it."
    },
    {
        "video_name": "j-rBgs_p-bg",
        "Q": "at 07:50, it is mentioned that \"Jones reagent is Na2Cr2O7 + sulphuric acid + water\" - could you also use acidified potassium dichromate, and would this still be \" jones reagent\"?",
        "A": "I can t see why you couldn t use potassium dichromate in place of the sodium salt."
    },
    {
        "video_name": "j-rBgs_p-bg",
        "Q": "At 1:12, where did the hydrogen on the oxygen go? Was it just nabbed by a random water molecule passing by?",
        "A": "It was taken by something when it was oxidised, that s what the [O] means"
    },
    {
        "video_name": "j-rBgs_p-bg",
        "Q": "As Sal would put it, the way Jay discusses the reduction of Chromium starting at about 11:50 is kind of \"hand wavy.\" How would I be able to determine on my own that the Oxidation State of the Chromium in HCrO3 goes from +4 to +3? \"There's some other chemistry that goes on...\" does not help me understand this process. It may be beyond the scope of this video, but \"Curious minds want to know.\"",
        "A": "Disproportionations and single electron transfers lead to chromium (V) acid and stable Cr(III) hydroxide. The chromium (V) acid promotes a two-electron oxidation of an alcohol and becomes Cr(III). In simple terms, redox reactions continue to occur until the Cr reaches the +3 stable point. If you look at the periodic table, you will see that the Cr+3 has a filled s orbital which increases stability and is the driving force for these continued reactions."
    },
    {
        "video_name": "j-rBgs_p-bg",
        "Q": "at 07:43 I thought when you use Jones's oxidation, CrO3, H2SO4, acetone you make a carboxylic acid from a primary alcohol?",
        "A": "By using jones reagent , we get RCHO group ie , an aldehyde. Jones reagent is a relatively mild oxidising agent. Only a strong oxidising ahent such as chromic acid (H2CrO4) could oxidise an alcohol to carboxylic acid. The oxidising order is as follows - alkanes -> alcohols -> aldehydes -> carboxylic group"
    },
    {
        "video_name": "j-rBgs_p-bg",
        "Q": "At 9:55 you said that water acts as a base to remove that H.But I don't understand why will water act as a base , when the hydrogen you removed has nearly no acidic character and water is also a weak base?",
        "A": "The hydrogen has become extremely acidic because the electrons from the C-H bond can be dumped into the C-O to eject the chromium as a good leaving group. This extreme acidity make up for water s weak basicity."
    },
    {
        "video_name": "6FP-hLuAlr4",
        "Q": "At 1:00 the video says that we need to choose two different points in the Earth's orbit where it forms an isosceles triangle. How would we know where the two points are?",
        "A": "My best guess - scientists are gathering data on distances, angles, etc. daily on many objects. They can then analyze the data for where the angles match on two days at opposite times of the year"
    },
    {
        "video_name": "FNVvQ788wzk",
        "Q": "At 10:45 he doesn't mention specifically if those cross links that link the fibrin together have a name. Do they? And then the second part of my question is this; what do the roman numerals represent that he starts drawing at 5:37? I understand the whole cascade, but what are the numerals? Are they tissue factors?\nTheories are welcome and thanks in advance!! ;)",
        "A": "The Roman numerals represent different types of proteins, (which can also be called coagulation factors) in the cascade. Tissue factors are a type of protein as well. Each number is a different type of protein which activates other proteins, ie. XI activates IX. He doesn t mention names, just that they are known as Factor X, IX, VII, etc."
    },
    {
        "video_name": "LccmkSz-Y-w",
        "Q": "At 5:50, he says only primary or secondary alcohols work for this SN2 PBr3 reaction, yet he shows a tertiary alcohol. Did he mean for this to represent Methanol?",
        "A": "He is just showing three bonds to the carbon without specifying what is on the other end of each bond. But one of the bonds must have an H at the end."
    },
    {
        "video_name": "LccmkSz-Y-w",
        "Q": "At 2:37 it's kicking off the chlorine. Why does this happen if the O has a +1 formal charge and is more electronegative than chlorine?",
        "A": "The other oxygen has a -1 formal charge, and as it forms a double bond with the sulfur, it kicks off the chlorine."
    },
    {
        "video_name": "LccmkSz-Y-w",
        "Q": "At 1:30, why didn't the bond electrons move to the chlorine atom instead of oxygen atom? Isn't chlorine more electronegative?",
        "A": "Oxygen is more electronegative than chlorine"
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "Around 7:40 you were talking about the size of an AU, is that from the corona of the sun to the atmosphere of earth, from the surface of each, or from the center of earth and the sun?",
        "A": "I think that it\u00e2\u0080\u0099s from the center of the Sun to the center of the Earth. By atmosphere of The Sun, you don\u00e2\u0080\u0099t know exactly where you\u00e2\u0080\u0099re starting from and ending at, so it\u00e2\u0080\u0099s the core of these two celestial objects."
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "9:54 how hot can the sun be?",
        "A": "27 million degrees is how hot it can be But only about 10 million degrees on its surface."
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "In \"Scale of the Large\" Sal says (at 4:30) that the Sun is 100x Earth's diameter. In \"Scale of the Earth and Sun\" he says it's 109x Earth's circumference? Are both correct?",
        "A": "The Sun s diameter is roughly 109 times the Earth s diameter. This means its circumference is about 109 that of the Earth as well."
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "Sorry if this is off topic at all, but at about 7:38 you said \"if it could maintain its velocity somehow\" I'm curious, how does an object lose velocity in space? At least when traveling towards the sun. Is it pulled by earth's gravity? If so, wouldn't velocity increase once it escaped Earth's gravity?",
        "A": "If the object is near a large mass, that mass is going to pull on it and change its velocity."
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "Sal makes the Earth as a little dot and the Sunas a big circle at 5:44 but i thought 109 Earths could fit in the sun, and how many Jupiter's could fit in the Sun?",
        "A": "Sal means that if 109 Earths were placed side by side, then that would be the diameter of the Sun. Millions of Earths could actually fit inside the Earth. About 926 Jupiters can fit in the Sun and if 10 Jupiters were placed side by side, then it would be the diameter of the Sun. Grammar note, when placed in-between the noun and s, the means that you are referring to something belonging to the noun. When placed after the s, adding the means that the noun is plural."
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "1:42\nWhat is the conversion of 40,000km into how many miles would that be?",
        "A": "You can use Google to convert units for you. You can just type in 40,000km in miles and you will get the answer = 24854.8477 miles"
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "6:40 on this video you mention the speed of light. When in college (40 yrs ago) a science prof handed a piece of wire to each of us and said it was the distance light would travel in a pico or nano second, I forget which. The wire weas 11.77 inches in length. My question to you is how did she get that number and which one is correct?",
        "A": "Light travels at approximately 186,000 miles per second, a mile is 5280 feet, and there are 12 inches in a foot. 186000 * 5280* 12 = 11,784,960,000 inches per second. 11,785,000,000 * (10^-9) = 11.785. Ten to the minus 9 seconds is a nano second or one billionth of a second. NB billionth is used here in the US vernacular not UK."
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "At about 1:26 he says that a bullet goes about the speed of a jet. So if humans make a really small, skinny plane will it go faster?",
        "A": "I don t really understand much about this, but, yes it would. Since the plane is skinny and small, it would have less air resistance and would cut through air easier than a bigger plane, thus, increasing acceleration. But, since it is small and skinny, strong winds would probably unbalance it. Hope this helps <3"
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "At 7:36, how would a bullet go straight into the Sun, and without falling?",
        "A": "I think that is just an assumption, if the bullet will travel at constant velocity. Although I already have thought of it somehow, thinking that the sun has gravity. The bullet will certainly increase its velocity as it accelerates towards the sun."
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "At 2:02 Sal says that it would take 40 hours (at the speed of a bullet) to circumference the Earth. Can anyone tell me how much time would it take travelling from North Pole to South Pole (at the speed of a bullet)?",
        "A": "Roughly 20 hours"
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "At 6:20 what is a Astronomical Unit?",
        "A": "An astronomical unit is the average distance between the earth and the sun, about 93 million miles"
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "At about 3:00, Sal states that the Sun is approximately 109 times the circumference of the Earth. Is this measurement including the Corona, or just the Photosphere?",
        "A": "just photosphere"
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "At 3:53, he's talking about the circumference of the sun, but how do you calculate the sun's circumference when it's too hot even to reach next to it!",
        "A": "If you know how far away it is, and you can see how wide it looks from here, then you can calculate the diameter and the circumrference."
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "at around 2:30 what makes a bullet travel so fast and also at 3:47 does the bullet lose it's speed because it loses earth gravitational pul",
        "A": "Bullets travel fast because they are propelled by an explosion inside the gun"
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "around 3:50ish in the video it talks about how big the sun is. how can we actually get the circumference of the sun if its so hot? we cannot go near it because of how hot it is. I mean it heats our Earth up and it would take roughly 17 years to travel at the speed of a bullet/jet how can they know how big the sun actually is?",
        "A": "We know the distance to the sun, and we can measure its size as it appears in the sky. Trigonometry does the rest."
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "what is a medicine ball? He says it at 8:54",
        "A": "Its a ball with sand in it. Its heavy (lbs)"
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "At 6:41 he says if the sun disappeared it would take 8 minutes for us to realize but is the \"speed\" of gravity immediate. If it was then we would realize that there was a sudden loss of gravity on the Earth right?",
        "A": "Gravity also moves at the speed of light. If the Sun were to disappear, we would be in orbit around the point of where the Sun was for another 8 minutes."
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "At 05:14, Sal draws the Earth's comparison to the sun. It doesn't really look as if 109 Earths could fit inside the Sun. There'd have to be more than 500!",
        "A": "It s the Sun s circumference* (therefore the diameter and the radius, as they are directly proportional to the circumference) which is 109 times larger than the Earth s. If you want to know how many Earth-sized objects would you need to fill up the volume of the Sun, you would have to scale that number to the power of three. And that s waaaay more than 500 ;-)"
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "(5:00)is that tiny speck really mercury?\n(mercury looks really big in most satellites)",
        "A": "And the little speck to the left is mars and the one just to the right is Venus."
    },
    {
        "video_name": "GZx3U0dbASg",
        "Q": "Considering the depiction of the solar system at 5:05 - If the sun has so much mass, resulting in a huge amount of gravitational pull, what keeps our planet, as well as the rest of the planets in our solar system from crashing into it and each other? My physics teacher told me that it was because of the centripetal force that each planet maintains, but how could a velocity perpendicular to that of the sun's gravity keep them on the orbital path they are currently on?",
        "A": "Anything that is in orbit stays in orbit because it has sideways velocity that is high enough so that, even though it is constantly accelerating toward the center of the circular path, it never gets any closer to the surface (or, for an elliptical orbit, it gets closer but still not close enough to crash) Sal has a vid on orbits."
    },
    {
        "video_name": "2ZgBJxT9pbU",
        "Q": "At 8:10, why did Sal not add the vector symbol to \"h\", he left it as a scalar? Even though he specified verbally that it is a vector quantity. Also, wouldn't the magnitude, as well as the position of the object in terms of height from the Earth's core, have an effect on its \"Final Velocity\"? (For example: A heaver object would fall faster than a lighter object\")",
        "A": "Heavier objects do not fall faster than light objects unless air resistance is significant."
    },
    {
        "video_name": "2ZgBJxT9pbU",
        "Q": "Why do we find average velocity to determine the displacement at 3:52?\nCan't we take the final velocity alone which is not zero ?",
        "A": "Displacement is not determined by the final velocity, but by the average velocity during the trip."
    },
    {
        "video_name": "2ZgBJxT9pbU",
        "Q": "At 9:12, the formula Vf= -square root2ah, does this formula is only used for the objects that is accelerating downwards?",
        "A": "It works for any object that starts from rest and accelerates with uniform acceleration of a."
    },
    {
        "video_name": "2ZgBJxT9pbU",
        "Q": "I got lost at 6:00 why did he multiply both sides from 2? and where did this 2 come from? is it 2 sec for change in time, I thought we didn't know change in time.",
        "A": "to simplify the formula he took out the common denominator of the two fractions by mulitplying both sides by 2*a (acceleration)."
    },
    {
        "video_name": "2ZgBJxT9pbU",
        "Q": "I'm a little confused about how to recognize when to use particular equations. At around 4:30 Sal asks what the delta t would be. In my head I immediately said \"time final - time initial\", but it ended up being delta t = delta v/acceleration. I'm not sure when to use the equation for a = v/t vs when to use the actual \"change in\" equation (final - initial). I hope this makes sense, or I'm just simply not noticing something. Thank you!",
        "A": "It depends on what your trying to figure out and what you have. In this one we are trying to find final velocity and we are not given time. So using time final -time initial doesn t help unless you are given the time. I think essentially you are trying to get as many unknowns out of the equation as possible. Therefore, because we know the initial velocity and acceleration we use that equation. I hope this helps."
    },
    {
        "video_name": "2ZgBJxT9pbU",
        "Q": "At 9:00, would you be able to modify the final equation as follows to easily apply to other gravity situations: Vf = -SquareRt of 2g*h m2/s2.",
        "A": "Try it and see if it gives you the right answer. If it does, you can do it. Good way to learn."
    },
    {
        "video_name": "2ZgBJxT9pbU",
        "Q": "At ~4:30, isn't acceleration thehow fast velocity changes.",
        "A": "If I understand your question, yes."
    },
    {
        "video_name": "2ZgBJxT9pbU",
        "Q": "in 1:12, sal says there is no air. then wouldn't the person explode?",
        "A": "He wouldn t explode. He would just get asphyxiated"
    },
    {
        "video_name": "2ZgBJxT9pbU",
        "Q": "I know this sounds stupid but I'm not a morning person and it's only 10:41 am, so I having trouble getting my head round why mass doesn't matter. Is it because all the vector/scalar \"quantities\" are relative to each other? I have no idea.\n\nThanks. :-)",
        "A": "Mass doesn t matter because objects all accelerate at 9.8 m/s^2 on the surface of the earth, regardless of how heavy they are (ignoring air resistance) The reason they all accelerate at the same rate is that more massive objects have more gravitational pull on them (weight = mg) but they also require more force to accelerate them (F = ma) so you have F = mg = ma, and m cancels out."
    },
    {
        "video_name": "2ZgBJxT9pbU",
        "Q": "At 9:55, we get to know that final Velocity = square root of 19.6*h. This means that any object would fall with the same speed because the equation does not have m(mass). But this isn't really true. Is it?",
        "A": "Yes it is true (if we ignore air resistance)"
    },
    {
        "video_name": "2ZgBJxT9pbU",
        "Q": "Where does the 2xa come from at 6:30?? Why the number 2?",
        "A": "The 2*a is seen in the denominator on the right side of the equation and so to clear the fraction, both sides of the equation are multiplied by 2*a. As to your question about the origin of 2 in this problem, recall that we include the average velocity. This is found by adding the initial and final velocities and dividing that sum by 2 to get the average. That s where we get the 2 in this problem."
    },
    {
        "video_name": "qdmYAXjhWiw",
        "Q": "At 19:11 we see three equations to solve for power. in equation one P=I^2R this indicates power is directly proportional to current and resistance. However, in the third one we see that P=V^2/R this indicates power is directly proportional to voltage and inversely proportional to resistance, but how is that possible. Did I miss something?",
        "A": "V = IR so both of those are true. P = I^2R indicates power proportional to R, if I is held constant. But if you just increase R and do nothing else, I will go down. P = V^2/R indicates power inversely proportional to resistance if V is held constant."
    },
    {
        "video_name": "qdmYAXjhWiw",
        "Q": "At 9:30 how would you solve for the current going through the 6 ohm resistor and the 12 ohm resistor?",
        "A": "One way to do this is by using the loop equation to find the voltage going through the parallel. Keep in mind, the voltage for the resistors is the voltage that comes out. Because the voltage is the same for both the resistors in the parallel, simply plug the found voltage into the equation and find the currents."
    },
    {
        "video_name": "dlpmllTx5MY",
        "Q": "Thank you very much.\nBut I have a question at 8:55. I cannot understand why the time is 2s?",
        "A": "ah~ I understood! I think I missed it..^^; thank you~"
    },
    {
        "video_name": "AmOO4j0E408",
        "Q": "At 5:27, why are the bases called Adenine, Cytosine, Guanine, and Thymine? Is there a story to that?",
        "A": "I don t really think there is an exact reason why they are termed those names. The names probably stem from a Greek or Latin background, as are most words."
    },
    {
        "video_name": "AmOO4j0E408",
        "Q": "At 9:11 javascript: void 0, Sal Khan said about Chromosomes. He said that Chromosomes are made entirely out of DNA but in actual it is made up of histones. Why he said ?? I didn't quite understand.",
        "A": "Histones are proteins that pack and unpack DNA in order to condense the chromosome. So yes, there are histones, but there are DNA too. I guess he said entirely out of DNA in order to simplify things in this introductory video."
    },
    {
        "video_name": "AmOO4j0E408",
        "Q": "At 7:38, Sal said that the human genome has 6 billion base pairs... isn't a genome a haploid and have 23 chromosomes? so then when you have a full complement of 46 chromosomes...shouldn't it be 12 billion base pairs?\nOr does a genome have 3 billion base pairs? Or am I missing something?",
        "A": "Yes you re correct. I guess Sal is trying to get the concept of general size across. The true definition of genome size is the same as c-value . As you say it is based on a haploid gamete which for humans has 3 billion base pairs (6 billion bases) across 23 chromosomes. In a standard human diploid cell there are 6 billion base pairs (12 billion bases) across 23 pairs of chromosomes, but the official genome size is 3 billion base pairs (6 billion bases). Well spotted."
    },
    {
        "video_name": "AmOO4j0E408",
        "Q": "5:45 Mr. Khan draws a lot more guanine -cytosine pair. Just want to know if the G-C pair is more common than the A-T pair",
        "A": "In most species, the number of A-T pairs exceeds G-C pairs."
    },
    {
        "video_name": "AmOO4j0E408",
        "Q": "How does the human genome have 6 billion base pairs? At 7:51",
        "A": "No. The human genome has around 3 billion base pairs, meaning it contains around 6 billion bases. Hope this helped!"
    },
    {
        "video_name": "AmOO4j0E408",
        "Q": "Around 11:22 it said DNA can replicate itself because the pairs are always the same. How does it know what pair with what?",
        "A": "I believe that it is because Adenine is incapable of bonding with Cytosine or Guanine because of its molecular structure (it might bond with something else, I don t know for sure, but the nitrogen bases are the only things present). The same is true for all the other bases."
    },
    {
        "video_name": "ETdNsO7mKXM",
        "Q": "I don't understand why at 8:07 we are we using a reduced mass instead of say, m1 + m2",
        "A": "In a vibration, it s not just one atom that moves back and forth. The centre of mass must stay in the same place. Thus, as one atom moves in one direction, the other atom must move in the opposite direction. This is what the physicists call a two-body vibration problem, in which the two atoms vibrate about a common centre of mass. The position of the centre of mass is determined by the reduced mass of the system."
    },
    {
        "video_name": "ETdNsO7mKXM",
        "Q": "At 10:05, if increasing the reduced mass results in a lower frequency of vibration, why does an O-H bond stretch at a higher wavenumber than C-H? Shouldn't the increased reduced mass of O-H make the wavenumber go lower?",
        "A": "The frequency also depends on the force constant k, which corresponds roughly to the bond strength. An OH bond is stronger than a CH bond. Its effect overpowers the effect of the reduced mass."
    },
    {
        "video_name": "oxf0LMJTklg",
        "Q": "How likely is it that in the video at ~5:30 the electrons would travel clockwise again and the double bond resonance would stay in one area of the molecule rather than making the full circle? Or in other words, how likely is it that this molecule would complete the full sequence of \"resonance\" back to the starting point without backtracking any number of times?",
        "A": "The electrons aren t travelling at all. We just have to draw the resonance contributors in some order that makes sense. The actual structure isn t part of the time one structure and part of the time another structure. The structure is at all times a single resonance hybrid of all the structures."
    },
    {
        "video_name": "oxf0LMJTklg",
        "Q": "This is a very stupid question but I'm just curious: Can't the resonance backfire? (At 4:10 it is clearly seen that the carbon lone pairs are next to two pi bonds, so it can go either way right?)",
        "A": "Sure. But going the other way would not give us a new resonance structure would it?"
    },
    {
        "video_name": "oxf0LMJTklg",
        "Q": "at 6:10, is the last resonance the same with first molecule starting? thanks though",
        "A": "Definitely very close, however the conjugated double bonds are represented as being in different positions. So technically, yes, they are different resonance structures, but one would say (and they would be right) that the two contribute exactly evenly to the overall structure of the molecule, because the conjugation dictates that the electrons are evenly dispersed along the pi system."
    },
    {
        "video_name": "oxf0LMJTklg",
        "Q": "At 2:13, I do not understand how the lone pair is \"next to\" the pi bond. The oxygen that bonds to the ring is only a single, sigma bond.",
        "A": "if you understand hybridization,then you may know that oxygen is sp2 hybridized and so is carbon,the pi bond and the p orbital of oxygen containing the lone pair of electrons are out of the plane of the benzene ring and next to each other ."
    },
    {
        "video_name": "oxf0LMJTklg",
        "Q": "At 6:30, the first and last resonance structures for the phenoxide anion\u00e2\u0080\u00a6 Are we not allowed to say that those molecules are essentially the same? Because the way we count the carbons in both to try and give the alkenes the lowest possible value, the original structure we'll count from the top going to the left (CCW), and the last structure we'll count from the top going to the right (CW).",
        "A": "these are resonance structures. they are all the same molecule. The observation you are making is true, though. if you flipped the molecule over, it would look exactly the same. Its important to realize, however, that there is no flip implied here, and that these two resonance structures are different. It doesn t seem to matter for this molecule, but when you get into chiral molecules, these structures could be entirely different molecules."
    },
    {
        "video_name": "oxf0LMJTklg",
        "Q": "4:10 carbon with 2 bonds and a lone pair wil give it formal charge of '0'. why is there -1 charge on carbon?",
        "A": "That carbon on the right structure at the bottom of the ring has 3 bonds and a lone pair. There s an implied hydrogen there. Formal charge = valence electrons - non bonding electrons - bonding electrons/2 4 - 2 - 6/2 = -1"
    },
    {
        "video_name": "oxf0LMJTklg",
        "Q": "At 2:54 (and other places as well), how come the carbon has a -1 formal charge? I thought the formal charge should be 0, because it typically has 4 bonds, and it currently has 2 bonds and 2 lone electrons which results in 4-4 = 0.",
        "A": "you forgot that one C-H bond"
    },
    {
        "video_name": "HO1dsNrzE58",
        "Q": "At 7:16, how do you know that 0.0100 is completely canceled out? How do you know that the H30 doesn't remain, but that there is some base left over??",
        "A": "We re working with a strong acid and a strong base. Strong acids have a low pka, meaning they will completely dissociate. We re also working with a strong base. Strong bases have a low pkb, meaning they ll readily accept the protons. Thereby, the reaction goes to completion. If we had weak acids or bases, then you would be correct - some base or acid might be left over."
    },
    {
        "video_name": "HO1dsNrzE58",
        "Q": "At 4:10, how do we know that the sodium ions and the chloride anions don't interact with water to affect pH? Is this experimentally determined?",
        "A": "Yes, this is well established. There is no significant change in the pH due ONLY to sodium or chloride ions being present. But the ions do interact with water in very significant ways, so that a saturated solution of NaCl (aq) has some different properties than pure water, though pH is not much affected."
    },
    {
        "video_name": "LWtXthfG9_M",
        "Q": "At 8:50 Sal says we have 20-30 Trillion Red Blood Cells. I thought we had 10 trillion cells total?",
        "A": "Men have 5 000 000 red blood cells / cubic mm and women have 4 500 000 red blood cells / cubic mm. This way you can find how many red blood cells you have."
    },
    {
        "video_name": "LWtXthfG9_M",
        "Q": "At 13:00, what does Sal mean about the graph with the hemoglobin in an acidic environment? If it was in an acidic environment with CO2, wouldn't the graph just flatline because it has no oxygen at all? Or go up, then down? I didn't think it would just be a little less than the hemoglobin in regular conditions.",
        "A": "It s not like your blood becomes pure acid at these sites. It gets a bit more acidic, and the hemoglobin then dumps some of its oxygen."
    },
    {
        "video_name": "LWtXthfG9_M",
        "Q": "at 11:31, how does the hemoglobin know to let go of the oxygen, doesn't it need to carry it to the capillaries?",
        "A": "When cells that the blood is passing need oxygen, they release their carbon dioxide, which basically binds itself to the hemoglobin. Since the hemoglobin now has some CO2 on it, it knows that it needs to release its oxygen. So the cell releases the CO2, which binds to the hemoglobin resulting in the release of O2 for the cell to use."
    },
    {
        "video_name": "LWtXthfG9_M",
        "Q": "At 1:16, Sal said that the heme group is a porphyrin structure? What does that mean?",
        "A": "A porphyrin is a rigid, square-planar molecule made of four pyrroles (a five-membered ring containing a nitrogen atom) connecting to form a larger ring. The molecule is stabilized by the aromatic character which extends over its entire structure. Source: Google"
    },
    {
        "video_name": "LWtXthfG9_M",
        "Q": "At 4:19 to 4:21, can you spell the term that is used to describe the high concentration of proton?",
        "A": "Acidity. Don t forget, if you click options at the bottom of the video, there is an interactive transcript that you can read."
    },
    {
        "video_name": "LWtXthfG9_M",
        "Q": "@2:10 Sal says about Cooperative Binding i.e. affinity of the Hb for oxygen, it's great but I have also been taught about the formation of Carboxyhemoglobin which said that Hb has about 240 times greater affinity for carbon monoxide.\nSo like why is it that Hb has a greater affinity to bond with carbon monoxide than oxygen?\nIs it a defect in our body or does it have any function?:/:/:/\nThanx in advance,\nMurtuza Abbas.",
        "A": "Humans evolved in an oxygen-rich environment and our pulmonary and circulatory system developed accordingly. They are good at what they do, but they can only handle oxygen. I guess you could call it a defect in the same way that you can call our skin a defect for being so penetrable by knives and bullets."
    },
    {
        "video_name": "LWtXthfG9_M",
        "Q": "Sal talks about carbonic anhydrase at 6:14. Where is carbonic anhydrase located? Is it floating around in the plasma or is it located in a red blood cell?\n\nThank you!",
        "A": "its located in the RBC"
    },
    {
        "video_name": "LWtXthfG9_M",
        "Q": "[8:54] I had been curious to find out how much hemoglobin proteins we have. With some calculation, I found out we have about 8 sixtillion, 100 quintillion hemoglobin proteins! (8,100,000,000,000,000,000,000) I was wondering... why do we need that much?",
        "A": "Protein molecules are very small, and we are very big compared to them. See if you can figure out how many cells we have in our bodies."
    },
    {
        "video_name": "LWtXthfG9_M",
        "Q": "at 5:05, why do the cells release a lot of CO2 into the capillaries if he is running? In order to make more ATP?",
        "A": "No its because he s making a lot of ATP that he produces more CO2. The cell respiration equation goes Glucose + Oxygen --> Carbon Dioxide + Water + Energy (In the form of Heat and ATP). Now you can imagine if this reaction happened faster more CO2 would be produced as a result or the increased amount of respiration going on in the cells. This increase CO2 then goes into the blood."
    },
    {
        "video_name": "fSk1Crn3R2E",
        "Q": "At 7:36, why is the top carbon a chirality center? Aren't there two CH2 substituents?",
        "A": "That doesn t matter, if you have the same group you continue going along each chain until you find a point of difference. In this case, the carbon to the left and right of that top carbon appear to be the same (CHH vs CHH) but when you go 1 carbon over there is a difference (CHH vs HHH)"
    },
    {
        "video_name": "fSk1Crn3R2E",
        "Q": "at 6:25 don't both carbons have an oxidation state of -1, making net -2 o.s in the product?",
        "A": "Yes, each carbon has an oxidation number of -1, but it also has two new H atoms, each with an oxidation number of +1. The net oxidation number of the product is 0."
    },
    {
        "video_name": "fSk1Crn3R2E",
        "Q": "@8:45 seconds. Why are the two compounds enantiomers? I thought they looked like Meso Compounds but couldn't figure out why I'm wrong. Thanks in advance.",
        "A": "They re mirror images. You can see it if you flip the right molecule over. There is no internal mirror plane in the molecule so it can t be meso."
    },
    {
        "video_name": "fSk1Crn3R2E",
        "Q": "At 9:58, and all the way to the end of the video regarding the Alpha Pinene example, are we suppose to always view all bicyclic molecules as spaceships docking into a port and as a result, the existing substituents will push up while the H2 will stick downward? How can you be sure of the stereochemistry of the hydrogens?",
        "A": "In a metal-catalyzed hydrogenation of an alkene, the H atoms are sticking up from the metal surface (the landing area) and the alkene is coming down on top of them, so the H atoms will attach to the under side of the alkene. The bulkier portion of the alkene (the cockpit) will stay on the top, so the H atoms will attach like landing gear to the bottom side."
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "2:45 But if you pushed a box on the ground, it would stop, and that's changing speed, right? The only reason that example worked is because ice has hardly any friction!",
        "A": "Yes if you pushed a box that would change its speed but an unbalanced force doesn t always change the speed as with Sals example"
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "At 1:44 Sal said that \"if the net force on a body is zero, its velocity will not change\". Won't the object be able to accelerate?",
        "A": "The velocity is the speed of something in a certain direction so if the object were to accelerate then the velocity would change"
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "Before 1:00, what are your answers to the four questions?",
        "A": "From top to bottom: True, False, True, False."
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "at 2:05, sal says if it was velocity instead of speed,statment 02 would be right..i donot get.does anyone please explain?",
        "A": "speed is distance covered divided by time. velocity is distance covered divided by time IN A GIVEN DIRECTION. if your speed changes, then it s simple: either the distance you covered changed or the time it took you to cover that distance changed. if your velocity changes, then either your speed changed, or the direction your traveling changed."
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "at 0:30, when sal explains question 3, what does he mean?",
        "A": "Newton s First Law says something about an object in motion stays in motion and an object at rest stays at rest, unless a net force or unbalanced force acts upon the object. I understood the Question 3 @ 0.30 to be one exception to Newton s 1st Law of Motion. The best example of a net force or unbalanced force is gravity slowing objects that are initially moving. I see the difference by thinking about motion in Space vs motion on Earth. Its different but the same Laws are applicable."
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "At 2:05, Just to make sure; speed and velocity are different right? Speed is a scalar quantity and Velocity is a vector quantity right?",
        "A": "Yep! Speed doesn t contain information about the direction, so it would just be something like 45mph. Velocity contains direction and speed, like 45mph south."
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "what does speed is the magnitude of velocity,it does not take in account the direction mean? at 2:11",
        "A": "What he s saying is, speed measures the size of your velocity. It does not matter what direction you re going in."
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "At 3:34 you said that the rope would not change the skater's speed on;y the direction. Is the rope an unbalanced force?",
        "A": "If it changes the velocity vector, it must be, right? Because changing velocity is acceleration, and acceleration requires an unbalanced force."
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "At7:38, what is the difference between speed and velocity?",
        "A": "Hello, remember to check older questions and answers, since this one was asked several times. In short, velocity is speed with direction, like 20m/s northeast. Velocity is a vector, while speed is a scalar."
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "At 4:54, wouldn't the object eventually slow down due to air resistance? Wouldn't friction reduce its speed and eventually make it crash into the planet? That's assuming the planet has an atmosphere, like earth.",
        "A": "Yes, then there would be a force on the object that would cause its state of motion to change The first law says object maintains its state of motion unless a force acts on it"
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "at 2:34 if the ice skater grabs a rope from behind, the ice skater will stop. What does this situation relate to? Because the ice skater will stop, so the speed changed",
        "A": "The word always is also used in this statement which is another reason why it is wrong"
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "at 4:15, for statement 2, if an object has a constant \"speed\" and and a unbalanced force pushes it for any direction, wouldn't the speed of the object increase/ decrease?",
        "A": "not if the force acts perpendicular to the momentary movement"
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "At 2:05, what's the difference between speed and velocity?You said that if statement # 2 had the word \"velocity\" it would be true.",
        "A": "speed does not have direction. Velocity has direction. You can say the car has a top speed of 250 kmph. This does not indicate where the car is going. But while describing direction specific phenomenons, say cyclones, you have to specify the direction . The hurricane is approaching the city with a velocity of 250 kmph or the stream flows due south with a high velocity"
    },
    {
        "video_name": "D1NubiWCpQg",
        "Q": "In 2:19-2:23, Sal mentions centripetal acceleration, centripetal forces, and inward forces. What are those and how do they explain the difference between speed and velocity?\nAlso, could someone explain to me why statement two was false?",
        "A": "Watch sal s vids on circular motion Sal can explain why statement two is false. Listen again."
    },
    {
        "video_name": "FK1s1-OJ5BE",
        "Q": "Ok. I am using Khan as a review tool. In class, we discussed how the mantle becomes more slushy as it becomes closer to the core. 1:08 makes it sound like the mantle is only solid. Am I misunderstanding my teacher or your video?",
        "A": "At 1:15 it specifically states then you have the liquid part ."
    },
    {
        "video_name": "qSFY7GKhSRs",
        "Q": "I am little confused. Change in momentum should be equal to the final momentum minus the initial momentum. If so, shouldn't the answer at 6:14 be -2mv (-mv-mv)?",
        "A": "No, it is 2mv. Intial momentum = mv Final momentum = -mv Change in momentum = mv -(-mv) = mv + mv = 2mv"
    },
    {
        "video_name": "qSFY7GKhSRs",
        "Q": "Around 6:20, Sal mentions that the change in momentum is 2mv. Could someone explain why this is or is there another video that explains this? Please and thank you.",
        "A": "If you go from +mv to -mv, the change is 2mv"
    },
    {
        "video_name": "qSFY7GKhSRs",
        "Q": "At 14:19 it's said that all of the internal energy of the system is in kinetic energy because it's an ideal monoatomic gas. Even in a monoatomic gas, such as He, every atom has potential and kinetic energy \"stored\" within it due to the electrons moving about the nucleus. Why isn't this energy also included as part of the internal energy?",
        "A": "I m guessing that since He is inert, none of these electrons interact with other substances so no transfers of energy occur."
    },
    {
        "video_name": "qSFY7GKhSRs",
        "Q": "14:12 , U = 3/2 PV ?? what happenned to N ?? shudnt it be U = 3PV/2N ?? or rather 3/2 PV = U N i mean we're considering ALL THE PARTICLES in the system right ?? U = 3/2 PV would apply for only 1 particle right ??",
        "A": "Only if you re considering all the particles in the system. You re right, the equation is for just one particle, but this is done just to make things clear for the student. If you ve noticed, most equations and definitions are fashioned this way."
    },
    {
        "video_name": "qSFY7GKhSRs",
        "Q": "13:21 why the one particle of kinetic energy becomes mv square over 2? Did you explain about the Kinetic Energy of the particle at the beginning of the video?",
        "A": "The formula for Kinetic Energy of any object is KE = 1/2 mv^2 This would be covered in a separate video on kinetic energy somewhere."
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "At 11:24, we say that the electrons are localized to the Nitrogen and that they cannot participate in the resonance that some students think may happen (the lone pair next to the pi bond).\n\nMy question is: Why can't the hybridization be sp? This will allow for 2 unhybridized p orbitals which would let the lone pair participate in a double bond with the Carbon to its left.",
        "A": "If the Nitrogen was sp hybridized, that would mean the geometry would be linear. The geometry can t be linear on any atoms in a ring, because rings are always bent. That means that the lone pair has to be localized to keep the geometry of N from being linear."
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "At 5:45, why does Carbon have a -1 formal charge? Isn't carbon bonded with 6 valence electrons? So I did 4-6 to get -2 formal charge?",
        "A": "That carbon has 2 bonds to carbons, 1 implied bond to hydrogen and 1 lone pair Formal charge = valence electrons - lone pair electrons - bonds 4 - 2 - 3 = -1"
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "At 10:38, how do you know the lone pair is in the hybridized orbital, not the unhybridized p orbital? Is it because there is a pi bond bonded directly to the Nitrogen in the ring that this is an exception?",
        "A": "We know that the lone pair is held within a hybridized sp2 orbital because the double bond connected to the nitrogen has a pi bond (i.e. the unhybridized p orbital) which must contain a pair of electrons used to form the double bond. Since there are already electrons in this unhybridized p orbital, the electrons on the nitrogen have no where to go and cannot participate in resonance!"
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "At 5:22, how are you allowed to push the electron onto the carbon to create a lone pair if carbons cannot have an expanded octet? If the carbon is connected to two other carbons and a hydrogen, how can it also have a lone pair attached?",
        "A": "That doesn t expand carbon s octet though, it still has 8 electrons around it. It now has 3 bonds (like you said, 2 to other carbons, 1 to hydrogen) which is 6 electrons, plus the 2 electrons from the new lone pair so that is 8 in total, which is perfectly acceptable."
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "At 8:24 , how come from resonance you jump from 4 pi electrons to 6?",
        "A": "What jump? The entire point of this video to show the nitrogen s lone pair in this molecule is in a p orbital (not a sp3 orbital), so the 2 lone pair electrons are part of the pi system."
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "Whats the aromatic compound at 4:30 called ?",
        "A": "That compound is pyrrole, an important component of chlorophyl, hemoglobin, and many other things."
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "In last example (11:39), I would still assume resonance of pi-bonds in the ring, as in benzene right?",
        "A": "yes, absolutely! the N has the p orbital to fit right in, so it would resonate just as a benzene ring would!"
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "At 6:20, can I assume that the nitrogen's hybridization is a hybrid of SP2 and SP3? So it's a hybrid of hybrids?",
        "A": "No, it would be considered sp2 in both of these structures. For reasons discussed in the video."
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "At 1:00, That seems reasonable, but can't the lone pair of e- on N go to either one of the R groups?\nHow do I just know how the electrons are likely to be delocalized?",
        "A": "The R groups are carbons that already have the max number of bonds, so they can t take part in resonance. To find delocalised electrons, look for lone pairs of electrons one atom away from a double bond."
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "At 11:00 Jay mentions the localization of the lone electrons imposed by the sp2 hybridized orbital. Is this always the case? So, if in a given structure the lone pair of electrons is in a hybridized orbital, are they always considered localized?",
        "A": "I d hazard a yes (at least for these cyclic compounds where the pi system necesitates that the lone pair be in p orbital to be delocalized). I dunno how this might change if d orbitals are involved, for example."
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "At 5:24, how are the magenta electrons able to move onto the other carbon if it already has two other carbons and two hydrogens attached to it (at least I think it does)? Wouldn't it now have ten electrons around it?",
        "A": "You ve miscounted the number of hydrogens, it only has 1 bonded to it in both structures. Left structure: 1 double bond to the carbon above, 1 single bond to the carbon to the right, 1 implied single bond to a hydrogen. Right structure: 1 single bond to carbon above, 1 single bond to carbon to the right, 1 implied bond to a hydrogen and 1 lone pair of electrons."
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "at 0:43 he says that sp3 means trigonal, pyramidal.\nbut arent sp3 hybridized orbits tetrahedral `?",
        "A": "He said trigonal pyramidal and he s correct because you saw that S.N. = 4 , it means that N atom is sp3 hybridised. Since there is a lone pair above N atom , then the shape is trigonal pyramidal not tetrahedral. If there was a sigma bond instead of that lone pair, then the answer would be tetrahedral. Hope that my explanation helped you."
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "The figure shown at 8:45 has p-orbitals for not only the carbons but the nitrogen as well. The nitrogen would not have a p-orbital when the carbons do right? It would have the four sp3 hybrid orbitals instead wouldn't it? 1 for the hydrogen, 2 for the carbons, and one for the lone pair?",
        "A": "The nitrogen in that 5 membered ring (pyrrole) is also sp2. When you have a lone pair next to a double bond, the lone pair will be in a p orbital. This allows the pi electrons to be delocalised. Having delocalised electrons/resonance structures is generally a more stable state, in this case the nitrogen being sp2 also results in an aromatic system which is very stable."
    },
    {
        "video_name": "kQCS1AhAnMI",
        "Q": "at 6:28 when the lone pair electron are located in a p orbital and this orbital participate with one electron forming the pi bond .. where did the other electron go !\nthank you",
        "A": "BOTH the N lone pair electrons formed that pi bond, not just one. Both the electrons in the original pi bond are now on the bottom left carbon."
    },
    {
        "video_name": "JRrm8TfBj1c",
        "Q": "In general, doesn't increasing HR increase both contractility and arterial elastance? If so, shouldn't two lines need to be adjusted at 6:35?",
        "A": "Contractility is the force of contraction, which HR doesn t increase. Great job making it this far, and this was 7 months before this answer was posted!"
    },
    {
        "video_name": "vvf-qiN9SWA",
        "Q": "At 5:40, 2nd derivative for the numerator is d^2i but in the denominator it is dt^2? Why is that?",
        "A": "Good question, I never thought about that before. I don t know why the notation for second derivative is like that, but that s the convention. All the 2 s are just part of the notation. They are not really math operators. You don t square anything when taking a second derivative."
    },
    {
        "video_name": "-J--4Jc5FB0",
        "Q": "At 4:29 why does the H always come first?",
        "A": "The H stands for hemagglutinin, and it comes first simply because of naming convention."
    },
    {
        "video_name": "bqcVNzObJec",
        "Q": "At 0:38, what means by 'some hot loop'?",
        "A": "He said some hot glue , not some hot loop"
    },
    {
        "video_name": "-yCIMk1x0Pk",
        "Q": "I'm confused....I was under the impression that sex cells only performed meiosis and that somatic cells performed mitosis. At 1:40 Sal starts talking about the zygote performing mitosis. Help please!",
        "A": "simply the zygote is the 1st Somatic cell to be formed from the sex cells"
    },
    {
        "video_name": "-yCIMk1x0Pk",
        "Q": "@15:53 Khan says, \".. bunch of eggs\". Isn't it only 1 egg that a female produces?? If they are from previous menstrual cycle wouldn't they lose the ability to fertilize with a sperm?",
        "A": "Fertilization in vitro can really help couples out, but the more you can take the more likely you ll have one or two that will work. This happens in the body too, the majority of fertilized zygotes don t make it naturally in a woman s body for different reasons. So the hormone is added to obtain more eggs than normal for increased chances of success in the laboratory in controlled conditions."
    },
    {
        "video_name": "-yCIMk1x0Pk",
        "Q": "At 15:00, Sal says that the cells can duplicate. How does it get the energy it needs for duplicating (e.g. cells in our BODY use our FOOD to duplicate)?",
        "A": "Yeah, something like that. The body stores energy from glucose and fat in ATP, which is in turn used to divide the cells, among other stuff like the basic functions for life."
    },
    {
        "video_name": "-yCIMk1x0Pk",
        "Q": "At 4:17, what does the liquid that is filling the zygote come from? What kind of liquid is it?",
        "A": "I think it is amniotic fluid."
    },
    {
        "video_name": "R2EtXOoIU-E",
        "Q": "At 1:45 I dont understand .... How is it possible that the noble gases Krypton through Radon have an electronegative charge?",
        "A": "Elektronegativity is not a charge but rather a number which tells us how likely it is for an atom in a molecule to pull electrons to itself. Krypton,Xenon and Radon can bond with other atoms, but you will need a very aggresive reactand like OF2 to oxidize these noble gases."
    },
    {
        "video_name": "R2EtXOoIU-E",
        "Q": "At 2:02, what is partial charge?",
        "A": "A partial charge is a non-integer charge value when measured in elementary charge units. Partial charge is more commonly called net atomic charge. It is represented by the Greek lowercase letter \u00ce\u00b4, namely \u00ce\u00b4\u00e2\u0088\u0092 or \u00ce\u00b4+. Partial charges are created due to the asymmetric distribution of electrons in chemical bonds."
    },
    {
        "video_name": "R2EtXOoIU-E",
        "Q": "At 4:12, if Florine is more electronegative than oxygen, why not we consider it as standard? That is, why don't we say 'Floridised' instead of 'Oxidised' ?",
        "A": "Our atmosphere is 20% oxygen and almost all metals are found naturally in compounds with oxygen (ie oxidised) on Earth. The term was coined before we knew that oxygen was not the only element that could do this."
    },
    {
        "video_name": "R2EtXOoIU-E",
        "Q": "which element is most electronegative? 1:20",
        "A": "Nitrogen, Oxygen and Fluorine are the most electronegative elements but fluorine is the most electronegative element"
    },
    {
        "video_name": "R2EtXOoIU-E",
        "Q": "At 1:49 did Sal mean pretend instead of portend, or is this just another word?",
        "A": "He meant pretend!"
    },
    {
        "video_name": "2QjdcVTgTTA",
        "Q": "(7:47) Why did Sal put the K in 3 dimensions down, when he said that it usually points up",
        "A": "You can put K ANYWHERE. Z usually points up."
    },
    {
        "video_name": "zA0fvwtvgvA",
        "Q": "at 7:14 when sal said orthagonal what does that mean?",
        "A": "Perpendicular."
    },
    {
        "video_name": "zA0fvwtvgvA",
        "Q": "at 2:40 cross product is not used only in 3d. I think it can be used to see if a point is inside a triangle algorithm, which is a 2 d problem.",
        "A": "The cross product of two vectors is perpendicular to both vectors so I am not sure how it could be used in a 2D situation."
    },
    {
        "video_name": "aSTvToHuG3I",
        "Q": "At 1:10, the square of t in denominator is missing. Is it ok or a mistake?",
        "A": "Good eye. I made a notation error by leaving out the ^2 on the bottom of the second derivative term. This is a small flaw, it doesn t harm the derivation."
    },
    {
        "video_name": "aSTvToHuG3I",
        "Q": "I'm sorry, but how do you get the equation at 00:56 (eq. with the green color)?",
        "A": "The green equation at :56 comes from dividing both sides of the previous equation by L. I just noticed an error in the green equation, the denominator of the left-hand second derivative term should be dt^2, instead of just plain dt."
    },
    {
        "video_name": "Z9IWg0pcpqo",
        "Q": "When naming the position of an alcohol, would you follow the same step as you would with a Ketone (video time 2:36)? Name all the carbons, then where the carbonyl is then you know to name that position after the carbon number.",
        "A": "Sure, the method you use for naming is generally the same, just different functional groups change the name. Eg ketones change the base name to anone (like propanone, butanone), alcohols change the name to ol (like methanol, ethanol) There s probably a video on naming alcohols too if you need clarification."
    },
    {
        "video_name": "M5uOIy-JTmo",
        "Q": "8:13:11 If on a P-V diagram one is going horizontally (increasing V, while keeping P constant), from state(1) to state(2), it seems as though this can only be accomplished by adding heat to the system. Can one know if all the inputed heat goes into work to raise the piston (increase V), or is some part used to change the internal energy (raise temperature) ?",
        "A": "if you add heat to a system and if internal energy increases temperature increases as the pressure is constant in your case to balance PV=nRT volume increases so work is done by the system and we know that U=(3/2)nRT from the next videos we can write PV=(2/3)U therefore workdone = two third change in internal energy ... form u=q+w you can calculate what amount will be utilized to increase internal energy what amount will be used to do work"
    },
    {
        "video_name": "M5uOIy-JTmo",
        "Q": "This is more of a math question, but when you're integrating and multiplying by the dv which Sal talks about at 9:10, what are you actually multiplying by? Seems to me like we just write it out in the formula but never actually take it into the calculation. If dv is the width of the rectangle and you don't actually ever multiply by this width you don't really ever get an area as your answer.",
        "A": "when you multiply by dv you are multiplying by the tiny quantity known as dv. If you add up all the tiny dv s you get v"
    },
    {
        "video_name": "M5uOIy-JTmo",
        "Q": "why are we trying to keep our macro states defied? what does it even mean? 5:26",
        "A": "watch the video on quasi-static processes. he goes in depth over the importance of keeping the macrostates defined."
    },
    {
        "video_name": "M5uOIy-JTmo",
        "Q": "I might be thinking in circles over here ... At 5:29 Sal says that the pressure will be lower in the higher volume space because of a larger distance to travel. I understand that intuitively but I keep thinking that shouldn't kinetic energy also play a role? Is kinetic energy lower after moving the piston? I think so. And all things bring equal, does a reduction in kinetic energy also imply a reduction in force?",
        "A": "The average kinetic energy per unit volume is related to the pressure. The total kinetic energy is the same but it is spread over a larger volume so the energy per unit volume is lower."
    },
    {
        "video_name": "M5uOIy-JTmo",
        "Q": "14:55 , sal says Delta U = 0 ( change in int energy = 0 ) then definitely the heat content of the system has changed , because the net work done by the system is not ZERO.\nd Q + Work = d U ...but since d U = 0 , THEN d Q = -Work , so the heat content chages??\nHas it?",
        "A": "Let\u00c4\u0085s start with: \u00ce\u0094U= Q + W (done on the system) If \u00ce\u0094U=0, then Q= -W(done on the system) = W (done by system) As it has been stated previously: Energy cannot be created or destroyed , so the only way to compansate the work done by the system is heating up the system by the amount of heat equal to that work Q = W (done by system)"
    },
    {
        "video_name": "8GQvMt-ow4w",
        "Q": "At 4:58, is the electromagnetic radiation Infra Red radiation?",
        "A": "Electromagnetic radiation includes Microwaves, radio waves, light (infrared, visible and ultraviolet), x-rays and gamma rays. The only difference in these is their frequency."
    },
    {
        "video_name": "8GQvMt-ow4w",
        "Q": "At 5:00,it is said that acceleration of charged particles causes the release of radiation.Then how is infrared radiation released from our body",
        "A": "Through the acceleration of charged particles from our body. The particles accelerate as a result of temperature."
    },
    {
        "video_name": "8GQvMt-ow4w",
        "Q": "At 6:20\nhow can you say that charged particles are being accelerated??\nAren't most of the molecules neutral??",
        "A": "No, the gas molecules (C, CO, H, HCN, O, etc., aka products of combustion ) in a fire are mostly ionized, which is what makes up the glowing plasma which we see as flames. Above the flames, much of the gas is still ionized, but it is radiating in the infrared part of the spectrum."
    },
    {
        "video_name": "8GQvMt-ow4w",
        "Q": "At 6:30, I have 2 related doubts:\n1) How does acceleration of charged particles cause radiation?\n2) It is said that radiation contains energy to produce light. Is light a form of energy?\nI don't get around the concept of Charged particles acceleration-radiation-light",
        "A": "1) accelerating charges radiate. That s just how it works. There isn t really a deeper answer. 2) Light is energy."
    },
    {
        "video_name": "jabo8iTesqQ",
        "Q": "At 0:25, Which material wire is used\n?",
        "A": "I think he is using magnet wire which has a very thin coating on it. if you wanna buy one, you can just call it magnet wire"
    },
    {
        "video_name": "8t-Un7yH99g",
        "Q": "At 8:30, they say that pills on patent will be more expensive. What is the cause of that, and how long does the patent last?",
        "A": "Drug patents last for 20 years. The pills are more expensive to allow the makers to regain the investment made to create the drug. If other companies were immediately allowed to copy the drug, there would be no monetary incentive to do new drug research."
    },
    {
        "video_name": "vVTwzjvWySs",
        "Q": "At 3:40 Why don't we put the oxidation state of the anion?",
        "A": "It s because cations that are transition metals (cobalt being one) can have different valencies whereas anions usually only have the one valency, although there are exceptions when the anion contains a transition metal. (An exception amongst anions is manganate and permanganate, where the manganese has an oxidation state of +7 and +6, respectively. Sometimes there are referred to as manganate(VI) and permanganate(VII), but as they have different names the Roman numerals are often omitted.)"
    },
    {
        "video_name": "vVTwzjvWySs",
        "Q": "at 3:43, when your re-writing the compound name, wouldn't you still have to add the 2 after the roman numeral 3 for cobalt?",
        "A": "That would be redundant information. You know (or at least can easily look up) the charge on the anion and are being told the charge of the cation, so it is simple to figure out the ratio of the two."
    },
    {
        "video_name": "lJDjWZqhpVc",
        "Q": "3:18, when you say Vo, is Vo the same as the gain of the circuit? Also, what exactly is gain?",
        "A": "Vo is the name of the output voltage. Gain is the ratio of the output voltage, Vo, to the input voltage, Vi. gain = Vo/Vin. At 2:00, the gain is represented by the capital A. Vo = A Vin."
    },
    {
        "video_name": "lJDjWZqhpVc",
        "Q": "At 7:17, are BOTH of the power supplies, POSITIVE power supplies?",
        "A": "Hello Wagner, Yes, the power supplies are identical. For example, you could use two 9 VDC batteries. Notice how they are connected at 7:24. The ground makes one power supply appear positive and the other appear negative. The op amp takes advantage of this by alternatively pulling the load to the positive supply and then to the negative supply. Regards, APD"
    },
    {
        "video_name": "z8h7QgevqjM",
        "Q": "in 4:40, could you also use the E and Z configuration for the alkene?",
        "A": "Yes. The name would be (E)-hept-3-enoic acid."
    },
    {
        "video_name": "z8h7QgevqjM",
        "Q": "@2:25, would you not have to assign a chirality for the third carbon on the 3 methyl hexanoic acid?",
        "A": "You would, but it isn t super important when you re just learning naming."
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "at 1:30 sal says that the length remains the same but isn't it a ray and rays don't have a fixed length?\nor is the arrow tip or head of the vector signify its end and the other end or the head are these the correct names for the two ends of the vectors?",
        "A": "The length of a vector represents it s magnitude. it is not a ray."
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "At 6:58 Sal says, \"I'm going to give a very peculiar angle, but I picked this for a specific reason just so things work out neatly in the end...\" and he labels the angle as \"36.8699 degrees\" and says, \"I'm picking that particular number for a particular reason...\"but later he calls the angle 36.899 degrees and does all his calculations using 36.899. What was the \"specific\" and \"particular\" reason that Sal chose 36.8699 degrees if he wasn't going to use that angle in his example?",
        "A": "sin(36.869897645844021296855612559093 ) = 3/5 sin(36) = 0.58778525229247312916870595463907 3/5 (or 0.6) is definitely better than the 0.58778525229247312916870595463907."
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "In this example it was pretty straight forward to see -- at 7:12 Sal breaks vector a up into a horizontal component along the x-axis and a vertical component that is parallel to the y-axis. How did he know not to put the vertical component ON the y-axis? For example, if a plane was turning and banked at some angle theta, and the lift force was perpendicular to the wings, if you wanted to break up the lift vector, how would you know if theta was between the lift and x-axis or the lift and y-axis?",
        "A": "It doesn t matter how you define theta. Just apply the correct geometry trigonometry to it however you define it, and you will be fine. I encourage you to try pursuing exactly the example you were asking about. It will really help you learn how this whole vector component thing works."
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "at 8:00, does the placement of the arrows really matter? In other words, could the horizontal component be pointing left and the vertical component be pointing down?",
        "A": "Yes, it does matter. If the vertical component is pointing down and the horizontal component pointing left, then the reference angle changes"
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "At 7:00 , the angle mentioned is 36.8699. But later on he solves it with 36.899. Which of the angle is to be used for the calculation?",
        "A": "Actually, you use the angle that is given originally which is 36.8699, I believe Sal just made a mistake and didn t realize that he copied it wrong because when you plug 36.8699 into the trig functions, the numbers come out perfectly as 3 &4 and you don t have to round"
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "At 7:00,how do we know which angle to choose,the upper one or the lower one?",
        "A": "Usually, an angle is measured from the positive x-axis in the counter-clockwise direction. So, the angle measured is from the lower one in this case."
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "at 11:00 why does he only do the cosine and sine? Why not the tangent?",
        "A": "We are only concerned with the distance from the axes of the vectors: cosine and sine of theta represent the distance from the x and y axis of the endpoint of the vector respectively. As you learn trigonometry, you ll learn that the tangent function is different than cosine and sine and is somewhat less useful. In this case, taking tan of theta would give you the slope of the vector, the y component divided by the x component, which is less useful that being able to see the x and y components individually."
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "2:04 what can you do to vectors? So can you use translation but not rotation/reflection/enlargement?",
        "A": "I am not a maths teacher, but I do recall that you can do all of the things you mention using matrices."
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "Why at 31:24 do you add vector A to vector B?",
        "A": "to get their resultant vector"
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "At 6:25, what do the 2 lines on either side of the \"a\" he writes mean?",
        "A": "It is a way of indicating the length or magnitude of the vector a."
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "at 7:30, why didnt he take the angle between the y axis and the vector",
        "A": "It is just convention to take the x axis angle, and if you look at the unit circle, angles start at the x axis and go to the y axis too. Its easier for intuition if you take the x axis angle as it helps with the trigonometry."
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "At 7:10 he uses the angle 36.8699 deg and then at 9:27 he uses 36.899 deg. Did he do something to affect those numbers, or did he just transfer them wrong?",
        "A": "Sal dropped a digit here. He meant to keep using 36.8699 degrees, not 36.899"
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "@ 3:27 I dont understand how its a + b = c instead of being the pythagorean c^2 = a^2 + b^2",
        "A": "It s because the vectors don t have to form a right triangle. This is the general formula for displacement"
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "At 6:26, Sal expressed the magnitue of 'a' with double lines , can't we just represent it with single lies?",
        "A": "It is how you represent the magnitude of a vector. Single vertical lines represent the absolute value of a number. So if you want to know the absolute value of number a, you write |a|, if you want to find the magnitude of vector quantity a, you write ||a||. Hope that helps."
    },
    {
        "video_name": "xp6ibuI8UuQ",
        "Q": "At 1:39 what those Sal mean by it does not matter?",
        "A": "It doesn t matter where you put a vector, you will still have the same vector, as long as it has the same magnitude and direction."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 3:53 you said that when a hidrogen fusion happens one of the protons decay into a neutron, losing mass, that becomes energy. How does it happen? And that means protons are a little bit heavier than neutrons?",
        "A": "No, the neutron has more mass, and that s why when it converts to a proton it gives off energy How does it happen? That s one of the things protons or neutrons can do when they are in an unstable nucleus."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 6:42, when Sal is talking about main sequence stars, how do you get different types of Stars like M or O?",
        "A": "very big gas clouds."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 7:14 you implied that if Jupiter was bigger, it could become a star. If Jupiter ran into a cloud of Hydrogen would it be able to gain the mass it needed to become a star? And if so, how would the greater gravitational pull and heat affect our solar system?",
        "A": "Jupiter would have to be about 80 times as massive as it currently is for it to become a small red dwarf star. So it would take a whole lot more hydrogen before that would happen. The greater gravitation would have huge impacts on the orbits of all the other planets! The extra heat would also radically change the surface temperature of all the planets as well."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 1:34 . what does Kelvin mean?",
        "A": "Kelvin is a temperature scale that starts at absolute zero, which is the lowest temperature theoretically possible at which all atomic motion stops and no heat is created. On the Celsius scale, it s about -273 degrees Celsius. A single degree on the Kelvin scale is the same in magnitude as a single degree Celsius."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 6:20 Sal says that the energy from fusion keeps the star from collapsing. Isn't this energy light? If so then how does this energy provide an outward force that prevents the star from collapsing? Thanks",
        "A": "No, but light interacting with matter results in the matter gaining heat and according to Boyle s law, the hotter the object, the volume it takes up. The gravity of the star is preventing the gas from spreading out so the gas puts tremendous pressure on the outer layers. The balance from gravity and the expansive pressure of the heat keeps the star together."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "5:40 What is a nucleon?",
        "A": "A nucleon is one of the particles that makes up the atomic nucleus. There are two types of nucleons: neutrons and protons."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "at 3:55, sal says that one of the fusing protons degrades in a neutron. that would decrease the net charge in the nucleus. does it mean that one of the electrons in combining H atoms gets absorbed in the nucleus to balance the charge? please explain fusion process.",
        "A": "No, the \u00ce\u00b2\u00e2\u0081\u00ba decay process emits a positron, which balances the total electromagnetic charge of the reaction."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "3:09 is ignition and fusion two different things",
        "A": "Fusion ignition is a term for a phase while stars are being born."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 1:38, what exactly is 10 million kelvin?",
        "A": "9,999,726.85 \u00c2\u00baC or 17,999,540.33 \u00c2\u00baF"
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "3:53 why does one of the protons decay into a neutron? surely that breaks charge conservation, and also the idea that protons are extremely stable... unless of course its just a typical beta decay and a positron and neutrino is released, though khan didn't say that so I'm assuming its not that :S",
        "A": "when a proton decays into a neutron, a positron has to be emitted because charge conservation always holds"
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 5:45,why doesn't it become tritium in between this procedure of deuterium into helium?",
        "A": "some tritium might be produced in the same environment, but it decays into deuterium much more quickly. Tritium capturing Hydrogen would produce Helium. it s just far more efficient to fuse two Deuterium."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "at 0:03 how would a cloud of H form?",
        "A": "The hydrogen gets drawn together by gravity."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 1:38, Sal Khan uses the term Kelvin. What is Kelvin? Is it another unit to measure temperature like Fahrenheit or Celsius?",
        "A": "It is a unit of temperature, similar to Celsius in that it has the same degrees, but it is an absolute scale, so 0 K is -273.15 C."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "at 3:52 in the video, he says that one of the proton degrades intoa neutron but why",
        "A": "How long does it take for one of the protons to degrade?"
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "at 5:45, i believe that what you meant was \"nucleus\", not \"nucleon\". if im wrong, please explain : )",
        "A": "A nucleon is a particle that is found in a nucleus which are either the proton or neutron. Sal is talking about the number of nucleons in each nucleus."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "how did the gravity make the hydrogen atoms condense at 1:00?",
        "A": "Well, everything with mass exerts a force of gravity. Even small gas particles. So, if you wait long enough, you re guaranteed to have any kind of gas cloud collapse."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "how come at 5:00 hydrogen forms together?",
        "A": "There is a mix of every element."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "im pretty sure that when he said ,at 3:28, \"nucleuses\". isnt it supposed to be nuclei or am i mistaken?",
        "A": "Yes, that s what he says in many videos. However, the proper plural of nucleus is nuclei."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "at 1:01, how does temperature go up when the hydrogen atoms are coming close to one another?",
        "A": "Gravitational potential energy is being converted to kinetic energy of the atoms, and temperature is the average kinetic energy of atoms."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "So, at about 7:00, Sal said that not all hydrogen things become stars. If they don't become stars, then what happens to them? Do they become planets or do they collapse on themselves?",
        "A": "They become brown dwarfs or planets."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "@ 00:59, what does dense mean??",
        "A": "Density is how much mass an object has in a set amount of volume. So if you had a gallon of air, and a gallon of milk, the milk is said to be more dense. Here on earth things that have more mass weigh more, so you can say that if two things are the same volume, and one weighs more than the other, it is more dense."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "at 4:01 he says \"the resulting mass off the two atoms is less than one [hydrogen] atom\" how is this possible if matter is neither created nor destroyed?",
        "A": "When nuclear forces come into play, matter can be converted to energy and energy to matter. In this case, matter is converted into energy, causing surrounding matter to be converted to energy, and so on for the whole life of the star (this is called a nuclear chain reaction)."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 3:50, Sal mentions that when two hydrogen atoms fuse to form deuterium, one of the protons decades into a neutron. The energy released balances the change in mass, but is there something which balances the change in electric charge? Isn't another particle created? If not, how can charge just disappear like that?",
        "A": "This first stage of the fusion process in the sun releases a positron and a neutrino. The positron is what balances the charge since charge is always conserved as you guessed."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "0:15\nwouldn't gas pressure stop the hydrogen from compressing?",
        "A": "It resists it, but the gravity is too overpowering. If gravity were turned off, the pressure would cause the cloud to expand rapidly again."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At around 3:09 more or less,he says that huge temperature is needed to bring two H atoms close.So how do these H atoms combine to form H2 on earth where the temperature is nothing around 1 million K?",
        "A": "Well in the sun, the hydrogen atoms fuse to get helium. This means the two atom s nuclei fuse together and the total electrons orbit the new nucleus. On earth, the two bond to get a hydrogen molecule."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "at 4:40. what is fusion ignition.",
        "A": "When the star first begins fusing hydrogen into helium"
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 0:59 Sal says the temperature is raising? What is the cause for this rise in temperature?",
        "A": "Nuclear fusion of lower mass elements release more energy than it requires to initiate fusion. Not all of this energy immediately escapes the star, rather it takes several thousand years for light to get out from the center of the sun. This radiation gradually raises the temperature even further from the gravitational energy initially inputting. Once you get to Iron, the energy required to fuse it is greater than the energy it releases, so the temperature cannot increase further from the fusion process."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 2:13 Sal refers to the Coulomb forces.\nWhat are the Coulomb forces?",
        "A": "The Coulomb force, also called electrostatic force or Coulomb interaction is the attraction or repulsion of particles or objects because of their electric charge."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "what's a kelvin compared to clesius?\nTime in Video: 1:35-1:40",
        "A": "0 K = -273.15 C 0 C = 273.15 K"
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "in 0:27, the atoms start moving closer together. Does anyone know how long it takes form stage 1 all the way to when the atoms start fusing together.",
        "A": "It depends on the size of the star. A massive star might be able to start fusing in a few hundred thousand years while a tiny star will take up to a billion years."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 3:42, what happens to the electrons circling the atoms, because one of the atoms degrades into a neutron so what happens to both of the neutrons at this point?",
        "A": "An atom does not degrade into a neutron. Perhaps you mean a proton? Inside a star, the electrons are not circling the nucleus. It s too hot and they have all been stripped away"
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "3:53 - \"proton degrades in a neutron\". Why? How is that happening?",
        "A": "Two protons alone in a nucleus is not a stable arrangement. To become more stable, one of the protons gives off a positron and becomes a neutron. The proton-neutron arrangement is more stable. You can intuitively see why, because the proton and neutron don t repel each other, whereas two protons do."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "at 1:04, how do some hydrogen atoms fuses and can't create helium",
        "A": "It is a matter of timing, when hydrogen fuses into deuterium it is unstable and will fairly quickly break apart into hydrogen atoms again, it takes further fusions to get to helium."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "7:17 does that mean that there is a possibility of Jupiter becoming a star in the future?? i don't mean the near future, but is it possible?",
        "A": "Jupiter would have to absorb all of the other planets 70 times over in order to ignite nuclear fusion."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 4:00 it's stated that a proton plus a neutron have less mass than two protons. But a neutron has more mass than a proton, not less. In fact a neutron will decay into a proton plus energy. Where am I going wrong?",
        "A": "Subatomic particles that are bound together actually have a little less mass than they do when they are separate. This has to do with mass-energy equivalence, and the fact that bound particles have lower energy than unbound ones."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At around 3:50 Sal mentions how when two protons fuse, one of them decays into a neutron, and there is a net loss in mass of the nucleus, releasing energy. But I looked up the masses of protons and neutrons and every site says that neutrons are slightly more massive than protons.\n\nproton: 1.007277 amu\nneutron: 1.008665 amu\n\nSo why isn't there a net gain in the mass of the resulting nucleus if one of the protons is converted into a neutron, something with slightly more mass? What am I missing?",
        "A": "In order for a proton to decay into a neutron, it has to absorb one electron. A neutron is slightly less mass than a proton plus one electron so there is a net loss of mass."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 0:34, or somewhat then, Sal said that all atoms of Hydrogen would be attracted by each others gravitational field. Consequently, Nebulae Compresses themselves. But, if all atoms were attractin' each other, they would have compensated each other's force, resulting in an unstable equilibrium, (or might be stable). Why it isn't so?",
        "A": "Firstly, a nebula has edges, which means that atoms toward the edges feel a larger imbalance of gravitational force towards the center. Secondly, even assuming the nebula is fairly homogeneous, even slight irregularities in distribution are enough for gravity to start clumping atoms together. Similar to balancing a ball at the top of a mound, any slight pull in one direction over another will be enough to get the ball rolling."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "4:30 I always tend to wonder how a fire starts in space in the first place; No air in space, no fire. If you close the lid on most candles, the fire goes out. If there's supposed to be no air in space, how does the fire keep burning? :/ Mystery to me...",
        "A": "Fire has nothing to do with this. The reactions that take place in a star are less like fire and more like the types of reactions that take place in nuclear reactors or atomic bombs. These nuclear reactions do not require oxygen or and reactant of any king besides the substance involved, in this case being hydrogen. When pressure and temperature are high enough, the hydrogen physically fuses together into helium creating a massive amount of energy."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "where would the gravity come from in 0:17?",
        "A": "Gravity is in all matter. So the gravity of the atoms in the cloud attracts each other."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 2:15,what comes before the atoms",
        "A": "Before the existence of atoms, there were subatomic particles. These include, but in no way limited to, protons, electrons, and neutrons. Before those, the universe was a plasma and essentially was too hot for photons to move across the primordial universe. At that stage, the universe was a hot ball of energy."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At around 4:04,Sal says that the loss of mass creates energy.Does this mean that when a object loses mass,this loss of mass creates energy?",
        "A": "why is it that mass is converted into energy? i am trying the formatting tips code tips(x > ) { return; )"
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "whats a kelvin in 1:35 ?",
        "A": "Kelvin is an absolute temperature scale set to the same degrees as Celsius. So, 0 Kelvin = -273.15 C and 0 C = 273.15 K."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 3:03, he mentions how fusion will occur when they are close enough. Exactly how close can are they? The strong nuclear force is only attractive between 0.5 - 3.0fm, so how can fusion occur?",
        "A": "The pressure and the heat in the core of the star is enough to overcome the repulsive coulomb force bringing the atoms close enough to be within the ~2.5 fm needed for the residual nuclear force to bind them."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "So when the hydrogen fuses into deuterium, and that fuses into Helium. I don't get that. First of all, when the hydrogen fused, one proton became a neutron. So when the deuterium fuses, shouldn't the new proton also turn into a neutron, so you have some other isotope of Hydrogen?\n-5:51",
        "A": "No. When deuterium fuses with another hydrogen, the two protons are stable. You ve made a stable isotope of helium. The neutron in there makes a big difference."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "Probably a silly question but I'm just wondering... At 7:12 in the video Sal mentions how Jupiter didn't reach full fusion, but is halfway there. Does that mean that Jupiter could possibly turn into a star if enough hydrogen atoms where around it?",
        "A": "Jupiter is not even halfway there. It would have to engulf all the other planets 13 times over to achieve nuclear fusion."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 4:18, what would happen if it kept collapsing on itself? Could it be possible start a supernova then form black holes or neutron stars?",
        "A": "Good question. I think that it would, because that is what happens to really large stars like red giants. There may or may not be an explosion involved."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "Around 04:00 Sal states that the combining masses of a proton and a neutron are smaller than the masses of two proton. But isn't the mass of a neutron bigger than the mass of a proton? Am I missing something?",
        "A": "Some of the mass is converted to binding energy."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 3:49 Sal talks about coulomb forces. What in the world are coulomb forces? Sorry I feel like i should know this.",
        "A": "Force of attraction or repulsion between electric charges"
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 3:52, Sal said that the proton \"degrades\" into a neutron. He also said that the combined mass of the proton and neutron is less than the mass of the two protons. But don't neutrons have a higher mass than protons?",
        "A": "Yes, neutrons have higher mass than protons. Not sure what you are referring to when you say combined mass of proton and neutron."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "If the atoms fuse together as said in 2:45 wouldn't all of the atoms fuse together to create a whole new element?",
        "A": "Hydrogen atoms fuse to make helium atoms."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "Yes, as above mentioned in the video at 7:19 \"several masses above Jupiter\" If Jupiter would be 11 times more bigger than it is now, it would have been formed into a star. But it actually didn't happened. If it would be like that- a star, we would not be able to reach at this stage.\nTRUE OR NOT?",
        "A": "If Jupiter were to be 13 times more massive, then it would officially be called a brown dwarf but it would need to be multiplied further by 3 before it can actually start fusing hydrogen. If Jupiter were to be a small star, there would still be night, just with an object maybe twice the brightness of the moon in the sky. Life would not be affected greatly by this."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "What does he say at 3:01? To overcome the ..... force?",
        "A": "Coulomb...i.e. electrostatic force"
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "Sal says at 4:06 that when 2 hydrogen atoms fuse one of the protons degrades to a neutron and a \"lot\" of energy is released. How much energy?",
        "A": "Subtract proton mass from neutron mass. multiply by c^2. That s the energy."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 0:28 , you said that the atoms slowly attract to each other. Where is evidence of this? Is it radiation or any space equipment that is used?",
        "A": "The law of gravity has been well established in the 300+ years since Newton came up with it."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At the 0:59, Khan said that the temperature will going up or it's getting hotter, so when that, can the hydrogen atoms getting explode?",
        "A": "no, why would they explode?"
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 0:19, Sal says that gravity would start to effect these atoms and they would start to condense. Since all the Hydrogen atoms have the same weight, where does this gravity come from?",
        "A": "Gravity is something that is proportional to an objects mass, just because they all have the same mass doesn t mean that there is no gravity."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 1:37, Sal said that the temperature of the cloud is nearing 10 million K. Wouldn't the \"cloud\" already be a star by then, even before 10 million K (autoignition temp for hydrogen is 500 something Celsius)?",
        "A": "Hydrogen in stars does not burn. It undergoes nuclear fusion. The autoignition temperature is irrelevant because there is no oxygen."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At around 0:14 Sal says that the cloud would be MASSIVE in both distance and in mass....what's the difference? I think that mass is the amount of space that the cloud (in this situation) takes up and distance would be how long it stretches out...right? So what's significant about the cloud being both huge in mass and distance?",
        "A": "Mass is not the amount of space it takes up, that is volume. A bowling ball and balloon take up the same space but the bowling ball has more mass. Mass is a quantitative measure of how much matter something has."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 2:40 Sal talks about the Coulomb Force. What is it?",
        "A": "The coulomb force is the repulsive force between electrons that orbit atoms. Its the reason why instead of atoms passing through each other, they repel each other and don t let them pass. Its the reason why you can touch things."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "Wait, so Jupiter could have been a star? 7:10",
        "A": "For Jupiter to become a star, it would have to maintain sufficient pressure and temperature to cause fusion. Jupiter is far too massive to be affected sufficiently by a single explosion - there would be fusion around the edges, but nothing long term."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 5:16 he said it becomes Helium, shouldn't it be Tritium?",
        "A": "No, when deuterium fuses with another hydrogen or deuterium atom, the new atom will have two protons, forming Helium-3 or Helium-4."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At 3:50 is told that one of the protons transformt into an neutron, how is this posseble? the mass of a neutron is greater than that of a proton. can someone explain this?",
        "A": "You are forgetting that you also have an anti-neutrino in the reaction. When a neutron decays it produces a proton, electron and an anti-neutrino"
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "What is a Kelvin - how many degrees does he mean?\n1:36",
        "A": "A Kelvin is the SI measure of temperature. The size of a Kelvin is the same as the size of a degree Celsius, but the starting points of the scales are separated by 273 degrees. If you are talking about stars, which involve temperatures from thousands of degrees to tens of millions of degrees, that 273 degree difference is irrelevant. 10 million Kelvin is pretty much the same as 10 million degrees Celsius."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "1:32,why are the atoms becoming hotter",
        "A": "As the atoms get squeezed together, they pick up speed. The gravity that pulled all of the atoms towards this point also imparted large speeds to these atoms. As the atoms get closer and closer, they get faster, and many, many collisions occur which excites the atoms even more. Atoms moving with high speeds is what we perceive as high temperatures."
    },
    {
        "video_name": "i-NNWI8Ccas",
        "Q": "At approximately 3:12, Sal explains how there is a \"fusion ignition\", and there is extremely hot temperatures, at around 10 million Kelvin. If hydrogen can burn so easily, why doesn't it burn in a traditional flame?",
        "A": "In that case, because of the lack of oxygen and earth-atmosphere- If what you mean by traditional flame is fire ."
    },
    {
        "video_name": "fif5ghe8JM0",
        "Q": "How do the goblet cells that Sal mentions at 6:57 help the colon?",
        "A": "they excrete mucus so that the colon is not torn as the fecal matter moves through the organ. they are really well demonstrated in the mayer mucicarmine stain, if you would like to see them a bit better."
    },
    {
        "video_name": "snw0BrCBQYQ",
        "Q": "I'm trying to understand how at 1:22 Sal assumes the graph will be a parabola when there is no ^2 value of \u00ce\u00b8.\nThe ^2 pertains to \"s\" (speed in this case) which is itself technically a coefficient in the given equation (as the equation is not a function of \"s\"). So I'm a bit confused about this assumption?",
        "A": "Horizontal position is a function of time, and vertical position is a function of time squared, so vertical position against horizontal position makes a parabola, as would vertical position against time."
    },
    {
        "video_name": "snw0BrCBQYQ",
        "Q": "@ 3:54 Why is it ok to factor out the (2s^2/g) in the course of finding the derivative of d(theta)? Why do we not take the derivative along with everything else? Do we call the factor (2s^2/g) a constant because its a scalar that just needs to be defined?",
        "A": "Really, Sal is taking a partial derivative \u00e2\u0088\u0082/\u00e2\u0088\u0082\u00ce\u00b8. When you take a partial derivative, all other variables act like constants. So the 2s^2/g out front can be treated as a constant with respect to \u00ce\u00b8."
    },
    {
        "video_name": "snw0BrCBQYQ",
        "Q": "why g = 10m/s^2 not g = -10 m/s^2 at time 09:59 m",
        "A": "According to the referential used by Sal, the scalar of the gravity acceleration should indeed be written as a negative number. Sal was probably thinking of the vector magnitude."
    },
    {
        "video_name": "snw0BrCBQYQ",
        "Q": "At 2:51 Sal states that negative sin of theta is a derivative of cos theta. I firstly do not understand why it is a negative sin as it Is later shown by Sal that positive cos theta is a derivative of positive sin theta. What's the difference? Secondly I don't understand how both sin theta and cos theta are derived from each other. (I have not yet learnt calculus so that may explain my ignorance).",
        "A": "The derivative of sin is cos The derivative of cos is -sin This is basic calculus You can see it if you graph sin and cos and realize that the derivative is the slope Sal probably has a vid about derivatives of trig functions in the calculus section"
    },
    {
        "video_name": "snw0BrCBQYQ",
        "Q": "At 10:00, Sal said that the gravity is equal to 10 meters per second squared. Doesn't that mean that if you jumped off a cliff you would be going up and never landing? Shouldn't it be -10m/s squared?",
        "A": "Obviously the 10 m/s^2 is DOWN. If you say 10 m/s^2 down then that s a vector and it means exactly what it says. If you have defined up as the positive direction then it would be -10m/s^2 if you want to use it in a problem. If you define down as positive then it is 10 m/s^2. Gravity is not negative or positive, it s just downward."
    },
    {
        "video_name": "snw0BrCBQYQ",
        "Q": "At 3:17 would the red cos be negative or positive",
        "A": "The red cos is positive because it s the derivative of the red sin. The derivative with respect to theta of positive sin(theta) is positive cos(theta)."
    },
    {
        "video_name": "snw0BrCBQYQ",
        "Q": "At 4:10 Sal took the derivative(s) of the trig functions. Why doesn't he get the derivative of 2s^2/g ?",
        "A": "He is taking the derivative of d with respect to \u00ce\u0098. The 2s\u00c2\u00b2/g term is constant with respect to \u00ce\u0098 so it can be factored out."
    },
    {
        "video_name": "BNmOZO9hbb0",
        "Q": "At 9:52, how does he know n=1 ?",
        "A": "There are 6 \u00cf\u0080 electrons. 4n + 2 = 6 4n = 4 n = 1"
    },
    {
        "video_name": "BNmOZO9hbb0",
        "Q": "(1:30) A pKa value of 16 indicates a weak acid, doesn't it?",
        "A": "Yes, it does."
    },
    {
        "video_name": "BNmOZO9hbb0",
        "Q": "For the last figure, at 8:16, why does the does the top carbon have a pi orbital. Isn't it only the electrons involved with the pi bond?",
        "A": "The top carbon is positively charged, so is sp2 hybridized. Thus, it has an unhybridized p orbital. This p orbital has the necessary orientation to be involved in resonance structures with the double bonds in the ring. Thus, through resonance, the p_ orbital on the carbocation is part of the _pi system."
    },
    {
        "video_name": "BNmOZO9hbb0",
        "Q": "at 5:30\ny there are three bonding orbitals?\ny not 1 bonding orbital, then 2 non-bonding orbitals?",
        "A": "At 6:00, the horizontal dashed line marks the division between the bonding orbitals below the line and the antibonding orbitals above it. So we see that there are three bonding and two antibonding orbitals."
    },
    {
        "video_name": "BNmOZO9hbb0",
        "Q": "At 9:40 Sal drew in 6 of the orbitals but there are 7 MO. Why did he not fill in the last one?",
        "A": "I think you miscounted. He drew in one energy level at each corner of the inverted heptagon.That make seven molecular orbitals, but he had only 6 electrons to put in these 7 orbitals. They went as pairs into the three orbitals with the lowest energy."
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "At 5:41, will charge always be given number?",
        "A": "Coulomb s Law has 4 variables, F, q1, q2, and r. If you know three of them, you can figure out the fourth."
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "At 8:07 why did Sal took coulomb in the denominator when he was actually multiplying it with the constant 'k'??",
        "A": "If you are trying to isolate K on one side, which is what he is doing, you would divide both sides by C^2 to take it out of that side of the equation."
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "at 10:25 why does the denominator change from 0.5 to 0.25?",
        "A": "He squared 0.5 so it was 0.5*0.5=0.25."
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "At 10:30, since the entire numerator is being divided by the denominator (0.25 in this case) wouldn't we also multiply the whole numerator with the denominator? He multiplies only 45 by 4, and not 10^5. Shouldn't it be something like : (45*10^5 N) * 4?",
        "A": "Sal does exactly what you suggest: (45 x 10^5)/0.25 == (45 x 10^5) x 4 = 45 x 10^5 x 4 = 180 x 10^5 = 1.8 x 10^7"
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "at 4:55 Sal says that electrostatic force at a close range can overcome gravitational force. Why is it so?\nand Why is gravitational force weaker at a close range?",
        "A": "It s not weaker at close range. It s weaker in general. We don t often see this because the electric force is so strong that we don t often find a lot of unpaired charges."
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "At 10:26 why did he divide by 0.25 and not by 0.5",
        "A": "If you look at the formula the denominator is (0.5m)^2, so 0.5^2 is .25. Hope that clears it up."
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "At 10:38 why does he add 20?",
        "A": "He s just doing mental math out loud. He needs to do 45 / 0.25, which is the same as 45 * 4. So when he does 45*4 in his head, he does 40 * 4 + 5 * 4, which is 160 + 20 = 180."
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "Around 4:47 , there is a alphabet 'K\" in coulomb law , is that the constant?\nCan we apply any unit of distance (cm nm m) to this law ?",
        "A": "You have to use the units that go along with the version of k you are using. Usually that means you need to use meters for distance, coulombs for charge, and newtons for force."
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "at 2:50 sal says that we take the absolute value of the charge. now my question here is why did he do that if we got a negative value for force then we know they attract and if positive it'd be a repulsion. so then why take the absolute value at if we can get more info. by the type of force.\ni feel i am missing something in my reasoning.",
        "A": "Instead of trying to track attraction and repulsion with + and - signs, it is much easier to just calculate the force and then look at whether the charges are opposite or same to know the direction of the force. Maybe that s what Sal is doing"
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "at min 7:46, why did he say that we need to end up with only the newtons?",
        "A": "Because the formula is supposed to give you force, and force is in N."
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "At 4:31, isn't gravitational potential energy equal to G (gravitational force) times the product of the two masses and inversely proportional to the distance (r) NOT the distance squared (r^2)?",
        "A": "At 431 he is talking about the force, not PE."
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "I don't understand how you got K. Can you explain what you did a in the video around 6:45?\nHow did you get 9x10^9? Also where did the newtons in the equation come from?",
        "A": "k is a constant. It has been measured by experiment. That constant is where the newtons come from."
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "at 10:30, how did Sal come to the conclusion of 180? I didn't quite understand the calculation process.",
        "A": "45 x 4 = 180 Sal separated the multiplication into two steps: 45 x 4 can be broken up into 40 x 4 = 160, and 5 x 4 = 20, so 45 x 4 = 160 + 20 = 180"
    },
    {
        "video_name": "2GQTfpDE9DQ",
        "Q": "is newtons same as C^2/m^2 at around 8:40?",
        "A": "Not quite. The constant K in the equation has dimensions. It has the job of converting the units on the right side (C^2/m^2) into newtons on the left. The units of K are (n m^2 / C^2)."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "At 11:05, why dont we get one ch3o attacked from above the plane and one from below the plane, as wee did at 8:35.",
        "A": "You do, but at 11:05, you get the same product whether the nucleophile attacks from the top or the bottom, so he shows only the attack from one side. At 8:35, you get different products when the attack is from the top vs. the bottom, so he has to show both directions of attack."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "At about 11:00 - Why does the methyl group leave? Isn't a methyl group a poor leaving group? I'm so confused about the solvolysis reactions!",
        "A": "No methyl group leaves. A methanol molecule adds to the carbocation, then another molecule of methanol removes the proton from the positively charged oxygen to give the final, neutral product."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "i need explanation on how he was able to remove that proton at 7:40",
        "A": "The Br- that left to form the carbocation or a solvent molecule could remove the proton. Proton transfer can happen very quickly, and it s not always easy to determine the mechanism of their removal/addition, and often there are numerous valid possibilities."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "Can't we turn 2\u00c2\u00b0benzylic carbocation into 3\u00c2\u00b0 by hydride shift? 06:07",
        "A": "Not for a benzylic carbocation, there s no H on the neighbouring C in the ring to shift. Note it has resonance stability though."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "At 10:53, when describing the last step of the reaction, could the iodide ion produced from the first step also function as a base, or is the methanol the only molecule that can act as a base to complete the reaction? As shown in the video, the methanol taking in the proton will form a cation, where if the iodide ion took the proton, it would form hydroiodic acid, which is more stable than a cation. So why can't the iodide ion act as a base if it would lead to more stable, unionized molecules?",
        "A": "HI is a strong acid, so I\u00e2\u0081\u00bb is a very weak base. Methanol is a much stronger base."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "At 3:15 when the second H2O takes away the H from the molecule, and then again at 10:50 when the second methanol molecule takes away the H from the molecule, how are we supposed to know that these steps happen? Are H atoms generally taken away like that by a base?",
        "A": "Yes. The H atom on an oxonium ion is strongly acidic (think hydronium ion ). The oxonium ion will donate this proton to any base present in the mixture."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "@ 8:41 why was the H proton removed?",
        "A": "It was removed because oxygen will not want to retain a +1 formal charge. It will give that proton the boot and take its electron. The proton will be taken up by a water molecule. For brevity, he didn t show that part of the mechanism."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "At 11:20 he gives us the final product of the reaction, but what would the byproduct be? Would it form one molecule, or does it form water and a second molecule?",
        "A": "I think the most probable result is for the I- to undergo an SN2 reaction with CH3OH2+ to yield CH3I and H2O. Since the pKa of HI is much lower than that of CH3OH2+ (-10 vs -2.5) there will be essentially no HI formed."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "6:55.... i thought that in an SN1 reaction, the first step is that the leaving group spontaneously leaves.. therefore the formation of the carbocation is not the first step of this reaction, the departure of the leaving group is?",
        "A": "The carbocation is being formed at the same time as the leaving group departs. The departure of the leaving group results in the simultaneous formation of the carbocation and of the leaving group with one more negative charge. You cannot have one without the other."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "What does it mean that methanol, which is an acid (though relatively weak), functions as base at 10:54? How can an acid function as base? Am I missing something?",
        "A": "Many molecules can function as both an acid and a base. The proton on the OH group can be removed by a strong base, so in this case the methanol is behaving like an acid (proton donor). The O of the OH group has lone pair electrons that can accept a proton and form an oxonium ion. In this case the methanol is behaving like a base (proton acceptor). So if the methanol is in the presence of a base, it can behave as an acid. If it is in the presence of an acid, it can behave as a base."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "At 11:12, are there supposed to be any lone pairs of electrons on the Oxygen since we kicked them off in the previous step?",
        "A": "On the one he s just drawing just then? There would be two lone pairs."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "At 10:29 could the nucleophile attack from the other side as well?",
        "A": "Yes, the nucleophile can attack from either side."
    },
    {
        "video_name": "wCspf85eQQo",
        "Q": "9:54 The reaction involves cyclo hexane group now why didn't he convert cyclo hexane into chair form?",
        "A": "What would redrawing it as a chair achieve though? He isn t showing any stereochemistry here, just overall showing what happens in the reaction"
    },
    {
        "video_name": "m6lMGoZTJnc",
        "Q": "at 5:28, is time between one coincidence of perihelion with summer solstice and the next 21000 years?\n\nwhat is the time period for Perihelion precession?\n\nwith respect to an external frame of reference, does the perihelion precession occur in the orbital plane?",
        "A": "One complete orbit of perihelion precession of Earth takes about 100,000 years I don t know the rest........"
    },
    {
        "video_name": "m6lMGoZTJnc",
        "Q": "At 3:45 you explain how every few thousand years our orbit changes, does this mean our orbit might lead us directly into the sun?",
        "A": "No, it does not change that way. Pay attention to the types of changes described in the video."
    },
    {
        "video_name": "IQJ4DBkCnco",
        "Q": "At 2:24, What is haploid?",
        "A": "Human gametes - sperm and eggs are haploids. Haploid is a cell that have just 1 chromosome of each kind in your nucleo. n=23 The antagonism is diploid: n=46"
    },
    {
        "video_name": "IQJ4DBkCnco",
        "Q": "At 3:15, you are dividing a cell that already a haploid cell, I thought that the meiosis also undergoes replication from 2n to 4n and then divides to 2n and then divides again to 1n or n?",
        "A": "No, it just goes from one 2n cell to 2 n cells (Meiosis 1) and then to 4 n cells (Meiosis 2). A germ cell that starts of with 4n would eventually turn into 4 2n cells. The number of chromosomes (or n) only increases when the cell splits."
    },
    {
        "video_name": "IQJ4DBkCnco",
        "Q": "At 5:29, does it mean that all sex cells are germ cells, example: egg cells, sperm cells, pollen, etc. are all germ cells? What is the definition of a germ cell?",
        "A": "A germ cell can be used to mean gametes (which are eggs, sperm, pollen) but a more specific definition would be that a germ cell is a diploid cell that will undergo meiosis to produce haploid cells (which are gametes)"
    },
    {
        "video_name": "IQJ4DBkCnco",
        "Q": "Around 4:40 it mentions that one of the cells will become an egg (if you're a woman). I was told in a previous science course that when girls are born, they are born with all the eggs they will ever have already present in the ovaries. Is this a myth?",
        "A": "Nope. It s true. Females are born with all the eggs they will ever have"
    },
    {
        "video_name": "IQJ4DBkCnco",
        "Q": "At 4:32, the correction states that only one cell will be an egg cell while the rest will be polar bodies. What are polar bodies?",
        "A": "Polar bodies are the result of oogenesis. In oognenesis, you can only create one successful egg and you get 3 polar bodies as the byproduct. They are much smaller than the egg but still contain the same DNA. For every one egg that the body creates, 3 poar bodies are also produced."
    },
    {
        "video_name": "IQJ4DBkCnco",
        "Q": "He keeps referring to a cell having a diploid number, like at 2:07, but what does that mean?",
        "A": "Diploid number of cells means containing a full set of 46 chromosomes (23 pairs). Haploid is half of the chromosomes, which is 23 chromosomes."
    },
    {
        "video_name": "nv2kfBFkv4s",
        "Q": "At 7:13, wouldn't the O- on the left grab the middle H+ proton on the right, instead of the bottom one that was indicated?",
        "A": "Because the molecules are drawn in two dimensions it looks as if the three hydrogens on the nitrogen are different but if you can imagine the molecule in three dimensions the three hydrogen are equivalent. They are equidistant from the hydroxyl group and chemically equivalent. Remember also that the positive charge is on the nitrogen, not on one of the hydrogens."
    },
    {
        "video_name": "nv2kfBFkv4s",
        "Q": "At 7:13 why does the oxygen have a negative charge, according to what I know oxygen already has a partially negative charge.",
        "A": "It gets a full negative charge instead of a partially negative charge by taking the hydrogen s electrons."
    },
    {
        "video_name": "nv2kfBFkv4s",
        "Q": "At 0:20 is '\u00e2\u0088\u009d' the sign for alpha carbon?",
        "A": "Yes, it is the Greek letter for alpha."
    },
    {
        "video_name": "nv2kfBFkv4s",
        "Q": "At 6:15 Sal says the nitrogen end of an amino acid grabs an extra proton to become a positive charge. I thought if you take away or add a proton to an atom it changes into a completely different element. Does he mean electron?",
        "A": "No, what he said is correct. What he means by grabbing a proton is that a hydrogen ion bonds with the amino acid. It is a chemical reaction, not a nuclear reaction."
    },
    {
        "video_name": "nv2kfBFkv4s",
        "Q": "At 5:01 if the Carbon already has a double bound with an Oxygen, how can it still bond with Nitrogen?",
        "A": "Carbon can have 4 bonds, it has one with the other carbon and two with the oxygen. That leaves one left over that it can have with the nitrogen, this is the bond it previously had with the second oxygen- but that was broken to release the molecule of water."
    },
    {
        "video_name": "nv2kfBFkv4s",
        "Q": "As Sal said at 0:37 that a peptide is nothing more than a chain of amino acids, he also said in the last video that amino acids is the main substance in formation of proteins. My question to you is that does that mean that a peptide is a protein?",
        "A": "Many proteins are formed by not only one strand of amino acids, but many. These proteins are also called polypeptides. Therefore if a protein were to contain only one strand of amino acids it could be called a peptide as you have noticed. But since most proteins are not only composed of one chain you cannot call them a peptide, but a polypeptide."
    },
    {
        "video_name": "nv2kfBFkv4s",
        "Q": "At 7:22, 3 lone pairs of oxygen are shown while oxygen only contains 6 electrons in its valence shell. 2 electrons are used to form the bond with Carbon so how was the 3rd lone pair present?",
        "A": "3 lone pairs of electrons = 6 electrons The covalent bond between carbon and oxygen has two electrons, one from the oxygen and one from the carbon."
    },
    {
        "video_name": "KFrhclyEpVM",
        "Q": "At 9:04,\nwhen the units were multiplied ,where did the radian go.\nie:kgX(m^2)X(rad/sec),it became\nkgX(m^2)/sec",
        "A": "radians have no units A radian is just a ratio, like a percentage"
    },
    {
        "video_name": "KFrhclyEpVM",
        "Q": "At 7:40 he says that the angular counterpart of speed v is angular speed w. An earlier video told us the relationship between speed v was radius * angular speed rw, so wouldn't that yield a final equation of I * rw = mr^3w rather than Iw?",
        "A": "v is the velocity of the object while w is the angular velocity of an object(i.e. when it covers a distance theta). Therefore, v= w.r is the relation between v and w."
    },
    {
        "video_name": "Xb05CaG7TsQ",
        "Q": "At 10:30, Sal is drawing a metallic atomic structure. But why are the electrons in the inter-atomic space, and not associated with the orbits?",
        "A": "the electrons are held weakly by the atom of a metal so the electrons escape with minimal energy to form a sea of electrons in-between the atoms. these electrons are called mobile electrons and they form metallic bonds which is the reason why metals conduct electricity and non-metals don t."
    },
    {
        "video_name": "Xb05CaG7TsQ",
        "Q": "07:49 - y don't the air particles transfer heat energy to the ball ?? the air molecules too must be moving abt randomly as a result of various forces right ?? or is it a two way process that both of them exchange temperature (K.E.) until thermal equilibrium is achieved ??",
        "A": "the molecules do move around and i believe that the molecules do transfer heat energy to the ball."
    },
    {
        "video_name": "Xb05CaG7TsQ",
        "Q": "at 2:50 if the ball is in ideal conditions, will it really loose energy while going up.If no then why will it ever come back?",
        "A": "it does loose energy on the way up."
    },
    {
        "video_name": "Xb05CaG7TsQ",
        "Q": "If I were in space (no gravity or friction), and in my hand I have an electromagnet, and in front of me floats a normal magnet... Say I push the other magnet away with my electromagnet, and then pull it back to me by reversing the polarity. I've wasted energy doing this (moving the normal magnet back and forth). Where did that energy go and is the first law mentioned at 0:32 preserved?",
        "A": "If there s no friction, you did not waste any energy. Whatever energy you put in you get back out. The first law is preserved. If there is friction, then any energy wasted is actually radiated as heat."
    },
    {
        "video_name": "Xb05CaG7TsQ",
        "Q": "At 3:00, Sal tells about the friction due to air particles while the ball falls down..Won't the ball rub against the air at the same rate while going upwards? Due to frictional forces ? So the energy i.e.P.E & K.E should be conserved?",
        "A": "Friction works against you both ways."
    },
    {
        "video_name": "_qiFIMviIeM",
        "Q": "At 3:00 what is the reason to suppose that the relationship between the Galilean transformation and the relativistic transformation is simply a scaling factor (gamma) rather than some more complex relationship?",
        "A": "gamma is derived from the assumption that the speed of light has to be the same in all reference frames For a derivation, go to youtube and search for Sixty Symbols Gamma"
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "As can be seen at 10:30, when you say that 0.07 mol of Cl2 is required, is that for both reactants to run out at the same time meaning there will be no limiting reagent or excess?",
        "A": "Correct. 0.07 moles of chlorine gas and 0.0117 moles of phosphorus (P4 is white or yellow phosphorous) is required to have an amount of 0 moles/grams of reactants remaining if the reaction is a 100% yield (very rare). But for the purposes of this video, you are correct."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 9:45, why did Sal multiply 35 by 2 only? Why not multiply the product of that by 6, by following the initial balanced equation ratio?",
        "A": "The 6 represents the number of molecules of chlorine gas. He multiplies by 2 because that is the number of chlorine atoms that make up the molecule of chlorine gas. He already accounted for the 6 when he found out that there was 0.07 moles of chlorine gas. At that point he was solving for the molecular weight of chlorine gas (\u00e2\u0089\u008835 amu x 2 atoms of Cl). Hope this helps. Good Question!"
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "How large is a mole, like in size, of phosphorous? He mentioned a mole as very large around 4:00",
        "A": "1 mole of an element is equal to its atomic mass, so 1 mole of phosphorus is about 31 grams."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "at 6:47,when would stoichiometry be used?",
        "A": "it helps especially when you have to calculate various factors like weight of the substance taken, amount of substance consumed,etc."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "Doesn't rounding off the moles of Phosphorus at about 5:57 effect the final value that we're looking for? Student number 15000886",
        "A": "While Sal is a good mathematician, he is not a chemist. In his videos, he rounds off too much for what is standard practice in chemistry. When doing the calculations, you must respect the significant digits and round accordingly. Most chemistry teachers will mark your problem wrong it you round off too much or too little. The rules for significant digits as used by chemists should be taught in your class. You must round off that way, not the way that Sal does."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 13:33, shouldn't he multiply 6.41 by 4. (because there are 4 molecules of PCl3).\nAm i right?",
        "A": "No because the mass of the product equals to the mass of the combined reactants. That s why the mass of 4 mole of PCl3 is already equal to one mole of P4 and 6 mole of Cl2. You do not need to multiply by 4 again."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "4:00 ish. Are there any specific equations for stoichiometry?",
        "A": "Here are some good ones to know: n x M = m (n is moles, M is molar mass, and m is mass in grams) I found that remembering that equation made rearranging formula around a lot easier. For example 2 moles O2 = (n x M) 2 x 16 = 32 grams OR 32g O2(g) = (m / M) 32g / 16g = 2 moles"
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "at around 13:15 at the end where you are finding the grams of PCl3 produced, why don't i have to divide the 6.41 by 4... is it just asking for the amount of PCl3 produced or is it asking for the entire reaction value?",
        "A": "We had to take the grams of the whole of PCI3 produced so we will not divide. If we had to we would have divided Cl2 by 6."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "In the beginning of the video, at around 0:30, why is Phosphorus \"P4''? I know Chloride is Cl2 because its diatomic, but why is a 4 already attached to Phosphorus? Is that a rule that I haven't learned or something..",
        "A": "the p4 is actually found in nature. so phosphorus is found with 4 atoms in nature i.e. the atomicity of p is 4 and hence you can call it tetratomic. there are many other elements that have more than 2 as their atomicity like sulphur which is octatomic i.e. 8 atoms."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "at around 11:15, Sal says that we should multiply 0.07 by 70. Shouldn't it be 70.096 or 71?",
        "A": "There may be two possible estimations for the reason 1) For easier calculation, he could have rounded the number to 70. 2) The reason why molar mass of Cl2 is 70.096 is that chemists have tried to calculate the average existing rate of chlorine element with atomic mass of 35 amu and 36 amu in natural state. He could have thought that the sample used in the experiment would consist of 2 atoms with amu of 35, resulting in molar mass of 70 amu. :)"
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "do you mean atomic mass (units) at 3:20?",
        "A": "Yes, when Sal is discussing the atomic weight, in this case of for the molecule P4, he is talking about the relative atomic mass in atomic mass units, that is then used to determine the relative formula mass for the molecule in question."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "at 4:22 where did Sal get the number 1.45 from?",
        "A": "That was in the question at the start of the video - he says we are told we have 1.45 g of P4 ."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 12:24, you mentioned 2 ways to solve part 2 of the problem, an easy and a hard way. You chose the easy way, how would you do it the hard way?",
        "A": "But he explained how you could do it the hard way at that time stamp.. You do what he did for the last 10 or so minutes for Cl2 just for PCl3. The steps you would take are the same. Mass P4 -> moles P4 -> moles PCl3 -> mass PCl3"
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At about 10:35, how do you figure out what to put where in the equation? I know, silly question, but still.",
        "A": "At 10:35, he is designing a conversion factor to convert 0.07 mol of Cl\u00e2\u0082\u0082 to grams of Cl\u00e2\u0082\u0082. The equation is: 0.07 mol Cl\u00e2\u0082\u0082 \u00c3\u0097 conversion factor = grams of Cl\u00e2\u0082\u0082. We know that 1 mol Cl\u00e2\u0082\u0082 = 70.91 g, so the conversion factor must be either 1 mol Cl\u00e2\u0082\u0082/70.91 g Cl\u00e2\u0082\u0082 or 70.91 g Cl\u00e2\u0082\u0082/1 mol Cl\u00e2\u0082\u0082 We choose the one with mol Cl\u00e2\u0082\u0082 on the bottom, to make the units cancel. Then, 0.07 mol Cl\u00e2\u0082\u0082 \u00c3\u0097 (70.91 g Cl\u00e2\u0082\u0082/1 mol Cl\u00e2\u0082\u0082) = 5 g Cl\u00e2\u0082\u0082."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 13:33 Sal said the mass of PCl3 is 6.41g but isn't it the mass of 4 molecules of PCl3?",
        "A": "He had 0.0117 mol of P\u00e2\u0082\u0084, so he formed 4\u00c3\u00970.0117 mol of PCl\u00e2\u0082\u0083 = 0.0468 mol of PCl\u00e2\u0082\u0083 \u00e2\u0089\u0088 6.41 g of PCl\u00e2\u0082\u0083. The 6.41 g is the mass of 0.0468 mol of PCl\u00e2\u0082\u0083."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 10:30, you said 'atomic weight. Can we write atomic mass?",
        "A": "We should say atomic mass . Atomic weight is an obsolete term."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "at 6:51 : in order to complete a stoichimetry problem do you have to have a complete balanced equation?",
        "A": "It s ALWAYS an amazing idea to have a complete balanced equation. Regardless of if you re doing a stoichiometry problem or not, I d be hard pressed to find an example where it WOULDN T be beneficial to do it."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "at 4:38, there's something I don't get everytime. 1 mol P4 / 124 g P4 . But why?",
        "A": "If you have 6.022*10^24 (one mole) molecules of P4, they will weigh 124 grams. So there are 124 grams per mole, or one mole per 124 grams. This can be expressed as 1 mol P4 / 124 g P4. Hope this helps!"
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 13:00 minute it is said that only product is PCl3 and as mass shouldn't change we can simply sum reactants to get mass of product. But what if reaction is endotermic or exotermic?",
        "A": "In all regular reactions (whether exothermic or endothermic) mass is conserved. Nuclear reactions are the only exception. :)"
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "I know a mole is 6.02 times 10 to the 23rd, but why Sal said ten to the 20th?(3:11)",
        "A": "He said it very quickly and I couldn t work out exactly what he said. But you are right that Avogadro s constant is 6.02 x 10^23."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 1:30, how can you just multiply the phosphorus and chlorine atoms in the products side? Was it just because the equation was unbalanced AKA incorrect?",
        "A": "Well, it is like having 4cages of penguins and each cage holds 3penguins. How many penguins do you have? 12!"
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 10:22 Sal says Cl 2's atomic weight is 70.96 .Shan't it be molecular weight?",
        "A": "Yeah, I think it should be molecular weight."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 2:11 what does the Stoichiometrical ratio tell us?",
        "A": "Good question! The stoichiometic ratio of any equation tells you what the ideal ratio of any reactant to any product should be. In the example: P4(s) + 6 Cl2(g) ==> 4 PCL3(L) This means that 1 mole of P4 reacts with 6 moles of Cl2 to create 4 moles of PCL3. If you add reactants in these mole ratios there will be no leftover reactants and hence no limiting reactant."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 13:11, how do you know that \"it's gonna react fully?\"",
        "A": "At the very start he has written down 2 questions: Given 1.45 grams of P4: - how many grams of Cl2 are required to fully react? - how many grams of PCl3 are produced? In the video he works out exactly how many grams of Cl2 are required. So he knows it is fully going to react because he worked it out."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At about 3:50, Sal writes 1 mol P4 / 124g. Later he uses it as a conversion factor in an equation. How do you figure out which conversion factor to use, ie how do you know not to use 124g / 1 mol P4 or something? Does it depend on the equation? Thanks in advance!",
        "A": "It depends what do u need. In the problem he needs to convert grams P4 to mol P4. So u need a conversion factor to get mol from grams. If u use 124 g / 1mol u will get g^2/mol, which u dont need in this case. If u need grams from mol, having mol, then u multiply by 124 g/mol P4"
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 11:43, what is meant by molar mass? I haven's reall understood much of this problem. Could someone help me, please?",
        "A": "As the name says, it is the mass of one mole of something! One mole of something is 6.022 x 10^23 of that thing. So if you had just one mole dollars, you could buy many many Bill Gates by that money. Let me ask you a molar mass question now. Question: My mass is 85 kg. What is my molar mass? Answer: 6.022 x 10^23 x 85 kg = 5.11 x 10^25 kg FYI: 6.022 x 10^23 is also known as Avogadro s number."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "at 4:01, how come he wrote mol and not mole?",
        "A": "mol is the SI symbol for the unit mole . SI symbols are never pluralized. The SI style guide says that you use symbols with numbers and words with words. Thus, you write 2 mol or two moles , but not 2 mols or 2 moles or two mol ."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 10:40, why did Sal say he wanted moles in the denominator?",
        "A": "You have to multiply the number of moles by the mass per mole to get the total mass. If you look at the units you have moles * (mass / something) = mass. Something has to be expressed in moles for that equation to make sense."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 7:30, what is the reason for that step? What exactly was the question?",
        "A": "He s working out how many moles of CL2(g) is needed. He did this by multiplying moles of P4 by 6. He includes an extra step of dividing by 1 to cancel out the units, but that extra step is not needed. :)"
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "Wouldn't it just be easier to assume that if this is not a limited reagent problem, that you can just use the 1:4 ratio to calculate the product first? i know it says that you have to figure out how much Cl2 you'll need, but it just seems easier to get the product out of the way first.",
        "A": "It doesn t matter which one you do first. He was asking two questions: (a) How much Cl\u00e2\u0082\u0082 do you need? (b) How much PCl\u00e2\u0082\u0083 is formed? He chose to use the answer from (a) to calculate (b). But you re absolutely right. You can use the 1:4 molar ratio to convert grams of P\u00e2\u0082\u0084 to grams of PCl\u00e2\u0082\u0083."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 10:20 when you find the atomic weight of Cl2 to be 70.9, why don't you then multiply that by 6 because the molecule in the equation was 6Cl2?",
        "A": "The mass of one molecule of Cl\u00e2\u0082\u0082 is 70.9 u, so the mass of 1 mol of Cl\u00e2\u0082\u0082 is 70.9 g. You already used the 6 in the equation to calculate that you need 0.07 mol Cl\u00e2\u0082\u0082. So, if 1 mol has a mass of 70.9 g, then 0.07 mol has a mass of 0.07 \u00c3\u0097 70.9 g =5 g."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "At 4:06 khan forgets to put an \"e\" for mole",
        "A": "Like cm is an abbreviation for centimeter(s), mol is the abbreviation for mole(s)."
    },
    {
        "video_name": "jFv6k2OV7IU",
        "Q": "Why does Sal use the exact averaged chlorine weight at around 9:45 instead of rounding it?",
        "A": "Chlorine has two important isotopes: \u00c2\u00b3\u00e2\u0081\u00b5Cl (75.78%) and \u00c2\u00b3\u00e2\u0081\u00b7Cl (24.22%). You cannot get valid numbers if you just round its average atomic mass to a whole number. Incidentally, though Sal tends to round off his atomic masses quite a bit, you wouldn t be able to do that in real-world applications (such as in Chem lab). It is standard to round the atomic mass to four significant digits (where that is possible)."
    },
    {
        "video_name": "VlaGxYjnoPY",
        "Q": "At 3:16, Sal said that at higher altitudes the speed of sound decreases but I don't understand why. As you higher up, the density decreases so the speed of sound would increase right?",
        "A": "No, generally speaking, low density slows sound down, and high density speeds it up, especially in a gas."
    },
    {
        "video_name": "-MaD9Ycy3a4",
        "Q": "At 6:43 How is the charge of Individual capacitor's charge equal to the charge of the Equivalent capacitor's charge ? Isn't the individual capacitor's charge be equal to 1/4 th the equivalent's charge?",
        "A": "No. That s what the video just told you. Capacitors in series have the same charge."
    },
    {
        "video_name": "2yqjMiFUMlA",
        "Q": "I get most of this, but I don't quite understand why 1 / j = - j (at 7:30 in the video). Why doesn't it work like a normal reciprocal?",
        "A": "1/j Multiply top and bottom by j/j. 1/j = 1/j x j/j = j / (j x j) The denominator is j^2 or -1 1/j = j/(-1) 1/j = -j"
    },
    {
        "video_name": "8shz0KfqkMo",
        "Q": "6:23 What are the so-called \"light-sensitive molecules\" made by the rods and cones?\n\nEdit: Are these the light-sensing pigments that get bleached during periods of high light exposure, causing down-regulation?",
        "A": "I think he s referring to photo-receptors And yes, I think so. This explains how when you re coming out of a dark room into the sunlight, it s hard to adjust for a few seconds. The rods don t like bright light, and I m thinking that there s a pathway to down-regulate the rods and switch over to the cones being activated."
    },
    {
        "video_name": "vkIBypq8MTE",
        "Q": "At 4:04, why did you skip talking about the frequency of the natural response?",
        "A": "Good point. I mentioned the amplitude, but not the frequency. Wish I had."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At 2:10, why is it 3d8 instead of 4d8?",
        "A": "Because the transition metals on the 4th row are filling up the 3d orbitals not the 4d orbitals. The 4s and 3d orbitals are very close in terms of energy."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "@2:10, he says that we have to write 3d8 instead of 4d8, why is he doing that?",
        "A": "Because that is what the electron configuration is. After 4s it goes 3d. 4d is much higher in energy. These numbers and letters have meaning outside of the periodic table, d and f orbitals do not line up with the rows of the periodic table."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At about 4:20, Sal mentions that the energy shell names run like \"S\" \"P\" \"D\" \"F\" \"G\" \"H\" etc - what is the origin and logic (if any) behind the energy shell naming?",
        "A": "The letters come from spectroscopy data, so it s: s for sharp p for principal d for diffuse f fundamental and I believe after this we follow alphabetic rules But, I wouldn t know the history to this naming"
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At 7:46, what is the full name for element Zr?",
        "A": "Zirconium is the name. If you want to find information on the periodic table, Google is quite helpful and easy to navigate. I hope this helps."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At 2:07, you mentioned it is going to be 3d (8) rather than 4 d (8). I did not understand this. As we have moved into 4th line, it should be 4th d8 right not 3 d (8) ?",
        "A": "These numbers and letters have a meaning outside of the periodic table, so 3d is correct. It just so happens that the energy of the 3d orbitals is very close to the energy of the 4s orbital."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "I'm confused about energy states.. If the \"further\" an e is away from the nucleus, the more energy it has, then why would an electron the 3d shell be in a higher energy state than an electron in the 4s shell? Isn't the 4th shell further away than the 3rd shell, thus making electrons in the 4th shell \"higher\" in energy? (This question stems from the breakdown around the 3:05 mark)\n\nThanks in advance!",
        "A": "In a one-electron atom, a 3d orbital is lower in energy than an unoccupied 4s orbital, but there is only a little difference between the energies. The 3d orbitals are more compact than the 4s. As a result, any electrons entering 3d orbitals will experience greater mutual repulsion. These electron-electron repulsions raise the energy of the 3d more than the 4s, and the 4s ends up more stable."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "7:55 why do the highest energy electron matter the most?",
        "A": "Because these are the ones that form bonds so atoms can take part in chemical reactions."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At 0:45, when you were talking about the elements, a question that came to mind was , is there any element that does not follow this electron configuration pattern?",
        "A": "Yes, there are quite a few that don t exactly work this way, but usually they are pretty close. The last few electrons might fill in in a different order than suggested by this method. I think copper is one of those. There are a few others."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At 2:18, I thought you were supposed to subtract 1 from the overall d-block. Thus, for nickel, I thought its d-block would have been 4d7, as it is the eighth element in that part of the d-block.",
        "A": "Actually, nickel s electron configuration is [Ar] 4s2 3d8."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "Hi! At 2:28 minutes in the movie, Sal says that the orbital 4s2 it's the orbital that it is more far from the nucleus and that the orbital 3d8 is the orbital that has more energy. How is that possible? If the 3d8 has more energy it can stay more far from the nucleous so it should be the orbital with the valence elctrons. Hope you can understand my doubt :) I am waiting for an answer.",
        "A": "Have a read of the answer given above by Natalie Bates, it answers your question very well"
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At 5:09, Sal mentions that each sub-shell holds a different number of electrons. However, I learned that every orbital can only hold 2 electrons. Am I mistaken, or is there a difference between an orbital and a sub-shell? Any help would be greatly appreciated.",
        "A": "There is a difference. The subshell can be made up of more than one orbital. s = 1 orbital (x2 = 2 electrons total are possible in this subshell) p = 3 orbitals (x2 = 6 electrons total are possible in this subshell) d = 5 orbitals (x2 = 10 electrons total are possible in this subshell) f = 7 orbitals (x2 = 14 electrons total are possible in this subshell"
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At 4:10, why are the shells named how they are?",
        "A": "A now defunct naming system for the blocks was used to name them. They stand for: s- strong/sharp p- principal d- diffuse f- fundamental/fine Their abbreviations stand, but the next ones to be used will be g, h, etc with no name/long form. This answer repeated from a previous question in which I answered about a year ago."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "What do you mean by the d-orbitals in the 4th period \"backfill the 3rd energy shell\"(2:00)? What is backfilling?? plz help!",
        "A": "It just means that when you use the d block you take the period you are in and subtract 1. When you are in the 4th period, instead of filling 4d you fill 3d and when you get to the 5th period then you fill 4d."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "at 8:07, why is it 4d^2? the period is the fifth so shouldn't it be 5d^2?",
        "A": "After Ca (at. no. 20), the orders of the energy levels get mixed up. 4s <3d and 5s < 4d. So 4d gets filled after 5s."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At 6:57, what makes using the first method of figuring out electron configurations easier?",
        "A": "The way the periodic table is arranged already shows you the order of electron orbital filling. It s easier to use the periodic table to work out electron configurations (which you probably are already looking at when answering such questions) than to construct the diagram in the second method and start from scratch each time."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "at 5:40 ish when he explains the second way, he puts the lines through diagonally. Cant you just go horizontally and fill all those that can be filled in then go to the second shell?",
        "A": "That would work for the first couple of energy levels, but you d run into a problem when you hit the 3d orbitals, since for some atoms the 4s orbital is actually lower in energy than the 3d orbitals, and thus the 4s orbitals fill first. This diagonal filling diagram that Sal has drawn is just a way to remember the order of the orbitals energy levels."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "So, at 5:20, where did Sal get the \"2\" in 1s2? Thanks!",
        "A": "The 2 refers to the number of electrons in that subshell."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At 2:07, why do we have to backfill the third shell? Isn't there supposed to be a 3d^10.",
        "A": "The empty 4s orbital is of slightly lower energy than the empty 3d orbitals so is filled first."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "at 0:15 why do they call them \"blocks\"? like s block, d block, etc.",
        "A": "Because the periodic table coincidentally forms blocks when you organize the elements by characteristics, which also corresponds to electron configurations. The naming of each block s,p,d, and f are just arbitrary terms."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At 2:00, in the d orbital, why do you have to subtract a period from the \"d block\"?",
        "A": "Due to the 3d electrons being at a slightly higher energy state, the 4s electrons are filled first. For convenience, the 3d shell/period is place one row down for this filling of shells pattern. Similarly, the f sub-shell orbital is also place further down than their row number would suggest."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "9:30-\"these are the ones that matter for most chemistry purposes.\" Why?",
        "A": "Chemical reactions involve the exchange and sharing of electrons. The outermost electrons are the most accessible, and are held the least tightly (they are furthest away from the positively charged nucleus, so experience the smallest electrostatic attraction). Thus, they are the electrons that are most easily transferred or shared with other atoms. Since these outermost electrons control bonding and interactions with other atoms, they are the most interesting to chemists studying chemical reactions."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "at 5:10 how many electrons can fit on the f block. and how can you tell how many can fit on there.",
        "A": "The f subshell can hold at most 14 electrons. We know that because there are 7 f orbitals per f subshell, each orbital can hold a maximum of 2 electrons; therefore, the total number of electrons possible in an f subshell is 14. Note: The f block is the area of the periodic table wherein the f subshell is being filled. So, it is not correct terminology to speak of an individual element having an f block. Instead, you should refer to an element having an f subshell."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "2:53 But doesn't what energy state the electron is in effect it's orbital, so that electrons in higher energy states will also be farther from the nucleus?",
        "A": "Yes, but when we do electron configurations we are talking about the ground state, not some arbitrarily elevated energy state."
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At about 4:20, Sal mentions that the energy shell names run like \"S\" \"P\" \"D\" \"F\" \"G\" \"H\" etc - what is the origin and logic (if any) behind the energy shell naming?",
        "A": "From what I can remember, the names come from atomic spectral line studies. S stood for Sharp, P for Principal, D for Diffused, F for Fundamental, and anything after that is alphabetical. Something about the energies and where they fall all the spectral line if I am correct. Don t know much more detail than that"
    },
    {
        "video_name": "YURReI6OJsg",
        "Q": "At 9:48, what are all those crazy and unpredictable things that the electrons are doing? And Why?",
        "A": "They electrons are jumping and disappering because of a principel called the uncertinty principel that say that you cant know the electron position and the speed 100 procent at the same time. Therefor the electron disapper and appear in an another place.Second the electron behaves as wave around the nucleous, and can be two places at the same time. This is confirmed in the double split experiment."
    },
    {
        "video_name": "DnNqe8o0ehc",
        "Q": "at 11:17 Sal says phosphoglycerate. Would that be phosphoglycerate mutase or kinase? Much needed answer for a biology paper. Thanks!",
        "A": "Neither, phosphoglycerate is the small molecule substrate of the enzymes phosphoglycerate kinase and mutase."
    },
    {
        "video_name": "DnNqe8o0ehc",
        "Q": "At 9:40, does he mean carbon or oxygen or carbon and oxygen?",
        "A": "He means Carbon molecules!"
    },
    {
        "video_name": "DnNqe8o0ehc",
        "Q": "At 06:33, he mentions the 5/6 PGAL molecules being recycled into RuBP. In my textbook, it says it's triose phosphate that gets recycled -- are PGAL and triose phosphate the same thing?",
        "A": "Yes, they are."
    },
    {
        "video_name": "KWf5kSNPbz0",
        "Q": "at 0:30, why does it work with only one battery? I thought you needed two for it to work?",
        "A": "Cause 1 battery is enough volts to power 2 motors, no need for a second 1."
    },
    {
        "video_name": "17EhKw2tsu4",
        "Q": "Why does he write the 'i', at 3:00? Does it stand for conventional current?",
        "A": "The letter i or capital I is the variable name we use for current."
    },
    {
        "video_name": "17EhKw2tsu4",
        "Q": "how come currents repells voltage at 0:46to 0:49",
        "A": "It s better to say the thing getting repelled is negative charge. This happens according to Coulomb s Law. Negative charges are repelled by other negative charges. The negative terminal of the battery connected to the right end of the wire corresponds to a large excess of negative charge. So any electrons already in the wire are pushed towards the left."
    },
    {
        "video_name": "V9JixmPdeuk",
        "Q": "Would someone please remind me why (at 3:59) the electron pair shifts first to the carboxylate oxygen and then back to reform the double bond to kick out the Cl- rather than the Cl- just leaving as the carboxylate attacks the carbonyl carbon of acetyl chloride?",
        "A": "It s just a formalism really. When we get more experience we will show that all happen in one step."
    },
    {
        "video_name": "V9JixmPdeuk",
        "Q": "11:00 where did the Lithium go",
        "A": "It just stayed in solution as lithium ions and is a spectator so it isn t important to know anything about the lithium in the reaction other than the fact that it became lithium ions."
    },
    {
        "video_name": "uQTG0x4qaYs",
        "Q": "What is this video even about? 0:00, 2:53?",
        "A": "It is about the laser part of a dvd player"
    },
    {
        "video_name": "yAFphxolU5w",
        "Q": "What does 'grok' mean? 3:03, 6:15",
        "A": "Robert Heinlein fist used the term grok in his Novel Stranger in a Strange Land. Grok essentially means to have a deep, comprehensive, thorough understanding of a concept. Perhaps the grok in this context would mean that by now it should be second nature."
    },
    {
        "video_name": "yAFphxolU5w",
        "Q": "Are there different ways to write the answer? At 8:36, he just jumps to say it is -3sinxcos^2x. Couldn't it also just be written as -sinx*3cos^2x?",
        "A": "I was also a bit confused first because my intuition before he solved it told me it should be 3(cos(x))^2 * - sin(x) (derive outer function first, then inner), and it s true. It s basically just the how multiplication works which gives it a certain flexibility. 2*5*3 is the same as 3*2*5 etc."
    },
    {
        "video_name": "yAFphxolU5w",
        "Q": "8:50 how did sal multiply 3cosx by the -sinx",
        "A": "You can see this expression this way: 3 * cos(x) * -1 * sin(x). And remember, multiplication is commutative, so you can just multiply the terms in front of the functions and you end up with -3 * cos(x) * sin(x)."
    },
    {
        "video_name": "0XkyAPXq55k",
        "Q": "At 4:14 Sal writes 1 - 0.30, shouldn't it be 1 + 0.30 since velocity v is negative?",
        "A": "A few seconds later, he changes it to a plus (and so then agrees with you). However, I think the 0.3 should be a 0.35."
    },
    {
        "video_name": "0XkyAPXq55k",
        "Q": "At 4:15 how comes 0.5 times 0.7 = 0.3 instead of 0.35?\nWas Sal keeping one significant figure in this calculation?\nBut if this is the case I don't see why the results of other calculations have had 2 significant figures.",
        "A": "Sal made a mistake. You are correct, it should be 0.35, and so the final answer should be 0.89c, not 0.92c."
    },
    {
        "video_name": "54JkvkIT0Dc",
        "Q": "At 5:48, does Hank mean that we are cutting 8000 hectares of trees a day worldwide or only in the Amazon forest ?",
        "A": "By 800 you mean 8000. Hank was talking about the entire world."
    },
    {
        "video_name": "54JkvkIT0Dc",
        "Q": "what does biodiversty mean at 4:40",
        "A": "The variety of life in the world or in a particular habitat or ecosystem."
    },
    {
        "video_name": "54JkvkIT0Dc",
        "Q": "At 7:10 Hank talks about the effects of overirrigation and how the salt in the water builds up in the soil, but if that is so how is it possible that in egypt irrigation has been used for thousands of years as the main source of watering crops and as it looks seems to be still working?",
        "A": "in egypt the salt is not allowed to build up due to floods.But now as we do not allow natural flooding salt is starting to build up"
    },
    {
        "video_name": "54JkvkIT0Dc",
        "Q": "What's Hank saying at 7:24? i'm struggling to decipher",
        "A": "The vegetation and lack of salt prevent deserts so when cattle eat grass and the water has more salt desertification occurs."
    },
    {
        "video_name": "54JkvkIT0Dc",
        "Q": "At 4:01, does the \"we\" mentioned refer to the US alone or to the world as a whole?",
        "A": "I think it refers to the globe as a whole as he then compares the figure to the output of the global economy."
    },
    {
        "video_name": "qLU0X154wlE",
        "Q": "0:45 Sal says angular momentum quantum no. writing l, which is azimuthal quantum no. Is this way correct too?",
        "A": "Yes both are the same."
    },
    {
        "video_name": "qLU0X154wlE",
        "Q": "At 4:43 how do you know how many p orbitals there are? Why is there only one s orbital?",
        "A": "Scientists have determined how many orbitals are within each sublevel. Theres only one s orbital because theres only one way to orient a sphere (which is the shape of the orbital). No matter how you rotate it, its still the same sphere. p orbitals are dumbell-shaped. Scientists have determined that nature allows for three orientations of dumbell-shaped orbitals."
    },
    {
        "video_name": "IDQYakHRAG8",
        "Q": "At 3:08, Sal said that like charges repel. If I placed an item of a positive charge inside a cube where all the walls are positive, would my item float?",
        "A": "Nice question. Assuming there are no ther forces such as gravity... First thing to mention is the faraday cage: There is no charge inside a conducting container. But I think your answer is reasonable; in my view; assuming all charges are equal, it would work its way to the centre"
    },
    {
        "video_name": "IDQYakHRAG8",
        "Q": "At 6:15, why is the balloon grabbing electrons, not protons, from your hair?",
        "A": "Because the electron is much more smaller than proton and neutron. It s very easy to move. Imagine the fat person as the proton and neutron and thin person as electron. So the thin person would run or travel faster than the fat person, right? Then when the balloon rubs on hairs, the free electron moves from hair to the balloon cause the hair to stand up. Hope it helpful."
    },
    {
        "video_name": "IDQYakHRAG8",
        "Q": "At 9:46 when Sal said that 1C=6.24 * 10^18 why this value only? Any specific reason or formula?",
        "A": "The formula is Q= n.e or charge is equal to the number of electrons times the charge if 1 electron . To find the number of electrons in 1 C of charge we divide 1 from the charge of 1 electron , i.e, 1.6 *10^-19 . After solving it comes out to be 6.25*10^18 electrons"
    },
    {
        "video_name": "IDQYakHRAG8",
        "Q": "At 6:19, Sal said that the balloon grabs electrons from our hair, but doesn't our hair grabs electrons from the balloon?\nAnd why hair becomes positively charged and the balloon negatively charged?",
        "A": "The balloon holds onto its electrons more tightly"
    },
    {
        "video_name": "IDQYakHRAG8",
        "Q": "At 06:17, Sal uncle says, \"The balloon gains electrons\", so can't the hairs gain electrons? Why only the balloon?",
        "A": "Because of their atomic and molecular structure, some materials are more like to receive electrons and others are more like to give them away. When you rub the balloon on your hair, the rubber of the balloon (more like to receive electrons) will remove some electrons from your hair (more like to give it away). It has to be with their atomic and molecular structure. The energy needed for the electrons to jump from your hair to the balloon is given by the movement (rubbing)."
    },
    {
        "video_name": "IDQYakHRAG8",
        "Q": "Some confusion from a physics newcomer. At 10:00, Sal explains that magnitude-wise, 1 C is approximately 6.24 x 10^18 protons, but that if you frame it using the electron, it's the opposite - but he then describes it as 6.24 x 10^18 electrons. If (+) e is equal to the charge of a proton and (-) e the charge of an electron, wouldn't e be negative if you're framing it using electrons? Or is it all arbitrary because charge is charge is charge regardless of whether you're talking positive or negative?",
        "A": "e is, by definition, 1.6E-19 C The charge on an electron is -e The charge on a proton is e"
    },
    {
        "video_name": "IDQYakHRAG8",
        "Q": "At 7:04, Sal said that the tip of our hair acquires positive charge and the root part gets negatively charged . As we know that the hair tip is nearer to the negatively charged balloon , why doesn't hair tip acquires negative charge ??",
        "A": "Charging a material by rubbing simply transfers or redistributes the charges in that material. Here, when we rub a balloon with hair positive charges are redistributed within your hair, specifically at the tip.This is because of a property (electron affinity) of hair and the balloon. In this case the balloon has more electron affinity than hair which results in redistribution of positive charges at the tip of the hair and not the electrons."
    },
    {
        "video_name": "IDQYakHRAG8",
        "Q": "at 4:30 he starts showing the nucleus and that has protons so why don't they repel each other?",
        "A": "they do. but there is a stronger force holding them close (The nuclear strong force)"
    },
    {
        "video_name": "IDQYakHRAG8",
        "Q": "At 10:08 well how can we exactly define Coulomb ?",
        "A": "A columb is the charge on any two objects (on both of them) which when kept one meter apart exert a force of 8.85*10^9 Newtons on each other."
    },
    {
        "video_name": "IDQYakHRAG8",
        "Q": "@10:43 If 1C=6.24*10^18e. Then shouldn't 1e= 1/(6.24*10^18)C... I mean the reciprocal of the whole thing. How come e=1.60*10^19?",
        "A": "What is 1/(6.24E18)? Do it on your calculator."
    },
    {
        "video_name": "IDQYakHRAG8",
        "Q": "At 4:31 you state that protons have a positive charge. You also said that positive charges repel each other resulting in an acceleration of the two charges away from each other (at 3:09 in the video). The question is why do the two protons you sketched do not repel each other so that the nucleus would be ripped apart?",
        "A": "They do repel each other, so that means there must be another force at work holding them together. That force is called the strong nuclear force, and it is very strong but has a very short range, about equal to the diameter of a proton."
    },
    {
        "video_name": "NUyYlRxMtcs",
        "Q": "At 4:58, is there a reason to write OH- instead of HO-, or is there no real difference between writing one or the other?",
        "A": "No, they mean the same thing"
    },
    {
        "video_name": "NUyYlRxMtcs",
        "Q": "Where did you get the number 1x10-7 moles from? 6:40",
        "A": "It comes from experiment. The concentration of hydronium ion in pure water is 1 \u00c3\u0097 10\u00e2\u0081\u00bb\u00e2\u0081\u00b7 mol/L."
    },
    {
        "video_name": "NUyYlRxMtcs",
        "Q": "At 4:49, why is it OH- instead of HO-?",
        "A": "That s just how hydroxide is commonly shown as a formula."
    },
    {
        "video_name": "NUyYlRxMtcs",
        "Q": "At 9:12, how is it that moles of H3O is 1x10-7 instead of being 6x1016. I thought the previous value was the Molarity of the H3O?",
        "A": "The 6 \u00e2\u0080\u00a2 10^16 is the number of molecules of H3O in the liter of water, not the number of moles in that liter."
    },
    {
        "video_name": "NUyYlRxMtcs",
        "Q": "At 10:09 and 10:19, when Sal says, \"one of them\" does he mean for every one mole of hydronium or for every one molecule of hydronium there is 560,000,000 H20 moles/molecules? What unit is he using for the ratio, moles or molecules?",
        "A": "Either. It s a ratio, so it works for both."
    },
    {
        "video_name": "cM-SFbffb7k",
        "Q": "At 10:30, shouldn't the name be bicyclo[3.2.1]6,8-dimethyloctane due to the Alphabetical order to be followed in IUPAC Nomenckature ?",
        "A": "The video is correct. This is because the core structure is bicyclo[3.2.1]octane which contains two methyl substituents. So 6.8-dimethyl is used as a prefix to the name of the core structure."
    },
    {
        "video_name": "cM-SFbffb7k",
        "Q": "Hello!I wold greatly appreciate help with the numbering at 2:10 please. How come the 7 and 8 are considered the second longest? Thank you!",
        "A": "In this compound , there are 2 rings . First (2,3,4,5) , Second (7,8) .First Ring has 4 carbons between Bridgeheads and second has 2 carbons in between. Since 4>2 , 2,3,4,5 ring is the longest and 7,8 ring is the second longest."
    },
    {
        "video_name": "cM-SFbffb7k",
        "Q": "At about 10:00, couldn't you name the molecule 2,8-dimethylbicyclo[3.2.1]octane, because you can start from the closer bridgehead carbon instead of the one in the back?",
        "A": "I like your thinking. If we were naming non-bicyclic compounds, you definitely would want the lowest number on each substituent. However, when naming bicyclic compounds, you must start at the bridgehead and move toward the largest ring. In this case the ring with 3 carbons and not 2 (or 1 for that matter). Notice that Jay does pick the numbering system with the lowest number possible (6 rather than 7)."
    },
    {
        "video_name": "cM-SFbffb7k",
        "Q": "Near 10:00, why is the methyl group attached to Carbon 8 not counted in the z/shortest path? Are substituents just not included in [x,y,z]?",
        "A": "No they re not included in that. Not sure why you think it should be? The shortest path from one of the bridgehead carbons to the other doesn t go up to the methyl group and down again."
    },
    {
        "video_name": "cM-SFbffb7k",
        "Q": "At 5:00 doesn't it just take 1 cut sinds its a 3 dimentional molecule (figure).",
        "A": "No, it takes 2 cuts, even when you think of the molecule in 3 dimensions. If you have a model kit, try building this molecule and breaking any one bond to convince yourself that you would still be left with a cycle."
    },
    {
        "video_name": "ZAgQH2azx3w",
        "Q": "at 1:14, How did he know to identify the magenta carbon as SP3 Hybridized? Thanks!",
        "A": "The carbon atom is directly attached to 4 other atoms: O, C, C, and H. The four bonds correspond to sp\u00c2\u00b3 hybridization."
    },
    {
        "video_name": "ZAgQH2azx3w",
        "Q": "At around 5:00, we're drawing the 3D structure of the two outer carbon atoms. But as these are both sp3 hybridized, would it be safe to assume that they have free rotation from the central carbon perspective? So wouldn't both outer carbons be rotating constantly (And the hydrogens around them)?",
        "A": "You are correct - the particular orientation Jay draws is just a matter of convenience. Normally we don t show the hydrogens, which allows us to completely ignore the rapid spinning of the outer carbons. Note also that in some more complicated molecules you can limit the free rotation of sigma bonds by steric hinderence (parts of the molecule get in each other s way)."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "at about 3:45 atomic mass units are defined in terms of 1/12 of the mass of carbon 12, and at about 5:30 you define carbon 12 in terms of atomic mass. isn't that circular? am i missing something or being pointlessly pedantic?",
        "A": "Yes, carbon can be represented in terms of atomic mass because one amu is 1/12 of carbon mass, so carbon 12 amu is 12. That is because carbon itself is also an atom."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "How much exactly does an electron add to it's mass? 6:57",
        "A": "9.10938356 \u00c3\u0097 10-31 kilograms"
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "At 7:40 - 7:42 when Sal says \"...the weighted average of the masses of all of the carbons,\" does he mean carbon isotopes?",
        "A": "Yes that s what he means"
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "At about 7:25, Sal said that the amu of both a proton and 1 neutron was 1 amu. I thought you would add both the proton and the neutron to get the amu. So, just checking, shouldn't the amu be 2?",
        "A": "The mass of a proton is 1 u. The mass of a neutron is 1 u. The mass of a proton and a neutron together is 2 u."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "Do a neutron and proton have the same atomic weight? re: 6:56",
        "A": "nope they are not equal in weight 1.6750 x 10-24 g neutron 1.6727 x 10-24 g proton commonly due to a little variation in their weights they are considered to be of equal mass"
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "At 1:25, what do you use to measure newtons and what is the short for newtons? For example, (kilograms is kg)",
        "A": "N a spring scale is one way"
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "Why is Carbon-12 used as a reference for amu ? What are the problems that the chemists faced when they used Hydrogen (protium) as the reference, as you had mentioned at 4:19 ?",
        "A": "It is simply easier to use carbon-12. You can create a perfect block of carbon-12 and measure/calculate the number of atoms in it, but this is impossible to do with hydrogen."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "I am having a hard time understanding this,\nAt 5:36 what does Carbon 12 and carbon 14 have to do with atomic weight?",
        "A": "C12 has a different mass from C14 The atomic weight of carbon is the average mass that you get when you take the mix of C12 and C14 (and other isotopes) as they are found in nature."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "At 4:19, what are the reasons why people see carbon-12 as the benchmark instead of hydrogen-1 ?",
        "A": "Carbon doesn t accidentally explode when you are trying to work with it, for one thing."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "In 0:46 you said inertia, what is it?",
        "A": "inertia is how much a object resists movement. basicly it is how easy it is to start and stop an object"
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "Do you weight differently depending on how high you are in earth's atmosphere? He says in ( 2:23 ) space you weight around zero newtons, but what about, for example, on Mt. Everest? Or in a high-flying plane?",
        "A": "Yes, you weigh very slightly less when you are in an airplane or on Mt Everest. Very slightly. Astronauts on the space shuttle experience a force of gravity that is only about 8% less than on earth. They feel weightless because they are in free-fall orbit, not because they are far from earth."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "Are pounds a measure of mass or weight? At 1:44, he only talks about grams, kilograms, and Newtons.",
        "A": "Scientists don t use pounds, they use the SI system, in which the kilogram is the standard unit of mass and the Newton is the standard unit of force (and therefore weight) you can google pounds to learn more about their role in their system of measurement"
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "In 4:35, Sal says that Carbon is measured at 12amu. Why is that? I know that its atomic # is 6. And that if the atomic # changed, so would the element. SO my question is how did he find the amu? DId he just multiply the atomic # by 2?",
        "A": "Atomic mass includes protons, neutrons and electrons, not just protons. C12 has 6 neutrons. To a close approximation, you have 6 amu of protons and 6 amu of neutrons, giving 12 amu."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "At 6:29, he talks about electrons. So how much do electrons weigh? Do the number of electrons on a large base effect an element?",
        "A": "1 electron has about 1/1800th of the mass of a proton. So even in atoms with 100+ electrons they still aren t adding much to the mass compared to protons or neutrons."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "What's the symbol Sal draws next to 1 proton and 1 neutron at 7:00?",
        "A": "Is means approximately equal to and looks like a squiggly equal sign: \u00e2\u0089\u0088"
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "At 5:57, I don't get why carbom 14 has 6 protons? Rather than 7 protons and 7 neutrons?",
        "A": "If you go back to the previous videos of this chapter, Sal said that number of protons in a atom define the element. If the carbon isotope have had 7 neutrons and 7 protons then it would not have been carbon itself! it would been nitrogen! Hope i didn t confuse you. :) ^_^"
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "At 6:09, the video mentions isotopes of carbon. How, exactly, does one isotope end up with more neutrons than another?",
        "A": "Mostly that s just how they are when they are created by the stars, which create all the elements heavier than helium."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "At 5:20, when Sal is calculating the atomic weight of carbon, he only uses Carbon-12 and Carbon-14. Why doesn't he include Carbon-13, even though there's a lot more of that than Carbon-14?\n\nEdit: This isn't the first time. Sal (and a lot of chemistry books/websites) completely ignore Carbon-13. Why?",
        "A": "They are making a mistake. The standard atomic mass of carbon is from carbon-12 and carbon-13. Carbon-14 exists in far too much of a trace amount to impact the standard atomic mass."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "At 7:40, how is it possible to measure all the carbon atoms on Earth and how do we know that there are no more carbon isotopes?",
        "A": "It isn t possible. We use sampling."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "At 4:20, the video talked about how carbon-12 is the benchmark for atomic mass units. Is there a reason why they chose carbon over another isotope? For example, if they took carbon-14, and divided the total mass by 14, would that still be around an amu? Also, I'm trying to figure out the difference between mass number and atomic mass. Is atomic mass the more specific mass of all the protons and electrons in an atom, while mass number is rounded to integers in atomic mass units?",
        "A": "the vast majority of carbon is C12. C14 is unstable. Mass number just tells you how many protons and neutrons are in the nucleus."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "At 3:25 Sal says that an A.M.U. is very small fraction of a gram. How many A.M.U.'s equal a gram?",
        "A": "6.02*10^23 AMU / gram Recognize that number? If not, you will see it again if you do more chemistry. It s called Avogadro s number."
    },
    {
        "video_name": "NG-rrorZcM8",
        "Q": "At 6:58, I am a little confused. Does 1 proton and 1 neutron adds up to 1 amu or each proton and neutron is 1 amu?",
        "A": "Each of them are about 1 amu."
    },
    {
        "video_name": "XEPdMvZqCHQ",
        "Q": "at 1:30 he doesnt draw the lone pairs on the condensed structure. do i always leave out the lone pairs on the condensed structure diagram?",
        "A": "yes, that s the idea of the condensed structure"
    },
    {
        "video_name": "XEPdMvZqCHQ",
        "Q": "At 6:12, he drew C5H13O! Why ?",
        "A": "Because he finished drawing the top molecules with the C3H8O formula. Now he made up a new condensed formula and turned it into a structural formula."
    },
    {
        "video_name": "XEPdMvZqCHQ",
        "Q": "At 3:57, does it matter if I attach CH_3 group to the given carbon instead of H_3C?Would it mean that the given C is attached to H?I have many people with different views on this.Please answer.",
        "A": "-CH3 and H3C- are both the same thing. Both represents a carbon with 1 free valency and 3 other valencies eaten up by 3 H atoms. There are no difference at all between this two radicals."
    },
    {
        "video_name": "XEPdMvZqCHQ",
        "Q": "In the first example, extrapolated from the condensed version, the OH connects directly to the Carbon but in the second example, also derived from the condensed version(4:23), the CH3 on the right side connects to the the Oxygen and not the Carbon. How do you know which atom to bond it to?",
        "A": "Carbon can form four bonds with other atoms. In the second example, the carbon had already formed four bonds. The oxygen had only formed one of a possible two bonds, so the CH3 bonded to it."
    },
    {
        "video_name": "FTO0lS1Q9QI",
        "Q": "At 6:13, why did you multiple the whole equation by (10^-3m/1L)?",
        "A": "1 litre = 1 x 10^-3 cubic metres."
    },
    {
        "video_name": "FTO0lS1Q9QI",
        "Q": "At about 4:43 she defines joules in terms of other units(kgm2/s2) please could you tell me how she derived that?",
        "A": "It s one of the definitions of joules... The energy transferred to an object when a force of 1 Newton acts on that object a distance of 1 metre. J = N * m N = kg * m * s^-2 Therefore: J = kg * m^2 * s^-2"
    },
    {
        "video_name": "FTO0lS1Q9QI",
        "Q": "I do not understand how, around 5:15, the instructor multiplies L by (m^3/L) instead of ((m/10)^3/L). A liter is a cubic decimeter, not a cubic meter. I apologize if I am missing something.",
        "A": "10^3 L= 1m^3. Therefore 1L=10^-3 m^3"
    },
    {
        "video_name": "0cDmgCcX_nU",
        "Q": "At 3:20, I don't understand why ay=0? Also at 7:05 he says that is the tension of the rope but what happened with m (yellow)?",
        "A": "Ay is 0 because there is no acceleration up or down, therfore the forces of Fg and Ty must balance eachother out. When finding the total tension, m was multiplied out by multipling both sides by m. M multiplied by 0 is still 0."
    },
    {
        "video_name": "0cDmgCcX_nU",
        "Q": "At the end of the video (11:50) , the speed you calculated was 2.38ms, yet when I did this in my calculator I got 4.7m, I checked to see if I did it wrong but I don't think I have.",
        "A": "Hi! You ll want to make sure your calculator mode is set to degrees and not radians. I yielded the answer 2.38 m/s in degree mode and 4.7 m/s in radian mode. Hope this helps!"
    },
    {
        "video_name": "umvNQj-zmq4",
        "Q": "at 1:15, what is eccentricity?",
        "A": "Eccentricity also just refers to how squished an ellipse is. A circle has eccentricity of zero. An ellipse is between 0 and 1"
    },
    {
        "video_name": "dbdVMThH1n8",
        "Q": "@7:19, should it not be an addition reaction rather than an Sn2?",
        "A": "Sal isn t saying this is a SN2 reaction, just that we have seen similar mechanisms with nucleophiles attacking the partially positively charged carbonyl carbon in SN2 reactions."
    },
    {
        "video_name": "dbdVMThH1n8",
        "Q": "Where did the Hidrogen come from at 5:36 , where the oxygen grab a electron to a positive hidrogen ?",
        "A": "If you check the reaction arrow underneath it says cat. H2SO4. That means there is acid present so a source of H+."
    },
    {
        "video_name": "dbdVMThH1n8",
        "Q": "7:01.... hang on, didn't Sal say that the ethanol was already protonated by the sulfuric acid? Therefore, how can we have an unprotonated ethanol now floating about, which can stage a nucleophilic attack on the carbonyl carbon? Sulfuric acid is so strong that I thought all the ethanol would have been protonated by now?",
        "A": "Remember that sulfuric acid is a catalyst and is present in small amounts. It protonates every ethanol molecule it can until it is all used up. But ethanol is the solvent and is present in much larger quantities (the excess reactant, if you like), so there is lots of unprotonated ethanol still remaining in the mixture."
    },
    {
        "video_name": "dbdVMThH1n8",
        "Q": "At 14:13, wouldn't the H30+ leave via an SN1 mechanism vs. the SN2 mechanism shown in the video? Carbon #1 of the carboxylic acid derivative would have 5 bonds after being given an electron from the ethanol attacking the top O-H group, which is not possible.",
        "A": "The mechanism shown at 14:13 in the video is not an SN2 displacement. It is an E2 elimination."
    },
    {
        "video_name": "dbdVMThH1n8",
        "Q": "At 6:55 you say its a carbonyl carbon. Isnt that a carboxyl carbon? Or can it called carbonyl ?",
        "A": "A C=O group is called a carbonyl group, whether it is part of an aldehyde or a ketone or a carboxyl group."
    },
    {
        "video_name": "dbdVMThH1n8",
        "Q": "At 07:31 why does the carbonyl carbon have a partial positive when the oxygen attached to the carbonyl also has a positive? To me it seems like it should be opposite. I guess I can see maybe its because the oxygen is pulling the electrons from the carbon due to it having a positive charge? But two partial positive charges next to each other doesn't seem right... Am I missing something?",
        "A": "There is nothing wrong with two partial positive charges on adjacent atoms. In fact, it is a more stable arrangement than having a full positive charge on only one atom. The structure at 7:31 is a resonance hybrid of the contributor shown and a second contributor with a positive charge on the C. The delocalization of charge stabilizes the hybrid, and the major contributor is probably the one with positive charge on C (the less electronegative atom)."
    },
    {
        "video_name": "i9rfWOAEplk",
        "Q": "At 1:32, Jay mentions steric hindrance. I am not sure what steric hindrance is, could someone please explain it to me?",
        "A": "Sterics just refers to big bulky groups that block access to certain sites on the molecule"
    },
    {
        "video_name": "tzqZsPjHFVQ",
        "Q": "8:25 Why do we square the equation? I get that each person has two alleles, but then why don't we just multiply by 2?",
        "A": "If you multiplied by 2 then it wouldn t be equal to 1 anymore."
    },
    {
        "video_name": "tzqZsPjHFVQ",
        "Q": "At 9:23, does he mean 1 - 0.3?",
        "A": "Hank said that p + q = 1. Since q equaled 0.3, then p is 1 - 0.3. That s how Hank got the 0.7 as the value of p."
    },
    {
        "video_name": "tzqZsPjHFVQ",
        "Q": "at 8:33, where did the 2pq come from?",
        "A": "When you square the equation p+q, you have to take (p+q) times (p+q). If you understand algebra, then you know that will equal p^2 (which represents p squared) + pq + pq + q^2. This is the same as p^2 + 2pq + q^2. It s basic algebra, but I see where the confusion may have occurred since Hank didn t actually explain the process."
    },
    {
        "video_name": "tzqZsPjHFVQ",
        "Q": "~2:00 he mentions allele frequency. Remind me, what is an allele?",
        "A": "An allele is a variation of a gene. For example, a gene that causes hair to be a particular color in a species might come in a variety of versions that lead to a variety of possible hair colors. Each version of the gene would be an allele."
    },
    {
        "video_name": "L677-Fl0joY",
        "Q": "Does hogged mean share at 12:30?",
        "A": "Hogged in this sense means that the Nitrogen is more likely to have the electron that is shared between the Nitrogen and the bonded Hydrogen. The Nitrogen wants the electron more because it is bigger, meaning it has a bigger positively charged proton nucleus that wants to be balanced out with as many negative electrons as it can get. This gives the Nitrogen a slight negative charge and the Hydrogen a slight positive charge since the electron is more often found orbiting around the Nitrogen."
    },
    {
        "video_name": "L677-Fl0joY",
        "Q": "At around 5:27, why does Sal put prime on the constants? Is it just a common notation?",
        "A": "Yes; it s a common notation. It s usually used to differentiate between the carbon atoms in different molecules. Other than that i don t see any specific reasons on why we use primes"
    },
    {
        "video_name": "L677-Fl0joY",
        "Q": "At 9:09, Sal says that the Ribose is deoxy because it is missing the 2' carbon's oxygen. Isn't the deoxy Ribose also missing the 1' oxygen by being bonded to the nitrogeneous base and the 5' and 3' oxygens by being bonded to the phosphates?",
        "A": "No. The deoxyribose and phosphate group share an oxygen atom."
    },
    {
        "video_name": "L677-Fl0joY",
        "Q": "At 5:46, what is a prime?",
        "A": "It s a common notation usually used to differentiate the carbons from other atoms. He didn t have to use them here."
    },
    {
        "video_name": "L677-Fl0joY",
        "Q": "at 12:05, what is the symbol Sal is writing for 'partial' ?",
        "A": "That s a lowercase delta symbol! You may have seen an upper case delta symbol, a triangle \u00ce\u0094, which signifies a change in something. A lower case delta is \u00f0\u009d\u009b\u00bf and signifies partial ."
    },
    {
        "video_name": "L677-Fl0joY",
        "Q": "why is he using the accents when numbering the carbon atoms in the ribose(around 6:30)?",
        "A": "To correlate them with the carbon atoms in the ribose molecule just above."
    },
    {
        "video_name": "L677-Fl0joY",
        "Q": "At around 3:59, it is stated that the reason that the oxygen maintains its negative charge and does not gain a hydrogen is because it is so electronegative that it takes the hydrogen's electron and then lets the hydrogen go. Why doesn't the oxygen form an ionic bond with the hydrogen?",
        "A": "Ionic bond basicaly isn t a bond, for example NaCl is Na+ flying somewhere and Cl- elsewhere. H+ ionts are probably either flying around and making acidic environment or they are just trapped by other molecule."
    },
    {
        "video_name": "L677-Fl0joY",
        "Q": "at 11:47 how was Nitrogen bound to the Hydrogen in the first place?",
        "A": "Nitrogen needs 3 electrons to complete it s outer shell to have 8 electrons on it. At 11:47 you can see that Nitrogen has 1 covelent bond with a Carbon atom and 2 covelent bonds with 2 Hydrogens. That s 3 electrons, exactly as many as Nitrogen needs to feel satisfied . Also Nitrogen is more electronegative than Hydrogen, so the electrons that they share will spend more time around the Nitrogen and because of that Nitrogen will become slightly negatively charged."
    },
    {
        "video_name": "L677-Fl0joY",
        "Q": "At 12:45 Sal draws the connection between the two rungs. But I noticed the first rung has only two connections (the Thymine and Adenine bonding) and the second one (Cytosine and Guanine bonding) has three connections. Is there some pattern the DNA follows, or is it because of the different types of bonding?",
        "A": "That is how the two molecules bond to each other. A and T have 2 hydrogen bonds while G and C have 3 hydrogen bonds."
    },
    {
        "video_name": "L677-Fl0joY",
        "Q": "In 11:23 can thymine and cytosine be purins and adenine and guanine pyramidines or are they always in the form shown in te video?",
        "A": "thymine and cytosine are always pyrimidines while guanine and adenine are always purines"
    },
    {
        "video_name": "L677-Fl0joY",
        "Q": "At 11:31, I saw that you classified cytosine as a pyrimidine and guanine as a purine, but I leaned in my bio class that they it is the opposite. So I was just wondering if cytosines are a purine or pyrimidine since purines = pyrimidines.",
        "A": "Adenine and guanine are purines while cytosine and thymine (and uracil) are pyrimidines. This differentiation is made because purines are much larger than pyrimidines since they have 2 sugar rings instead of 1, which is what pyrimidines have. And, yes, you are right when you say that purines pair with pyrimidines. If purines paired with purines, the diameter of the DNA strand would be too large and if pyrimidines paired with pyrimidines it would be too small."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "Around 1:26 Sal talks about molecular hydrogen and molecular oxygen, and how this is the type of atom(s) you would Typically find in nature. But why is that?",
        "A": "If atoms do not have a full outer shell of electrons, they are very unstable and will be extremely reactive. If you ever have a bucket filled with oxygen atoms, they will without a doubt react to form molecular oxygen. Elemental compounds only exist for noble gases."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "Around 7:07 Sal wrote \"10N\" indicating those two charges.\nWhats the meaning of that?",
        "A": "omg thats amazing"
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 3:55, the sidenote in the video says that Sal said \"rigorously\" but meant \"vigorously\". Is there any difference in meaning to the 2 words?",
        "A": "There is a big difference in the two words. Rigorously means doing something in an extremely thorough and careful way (as when, say, applying your brain to solving a chemical problem). On the other hand, vigorously means strenuously and with energy - in this, case he was referring to how the molecules were moving around."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "2:48, Why isn't the product 2 (H^2 O^2)? If the answer was 2 (H^2 O) wouldn't it have to be 2 ( H O ) because the subscript is not there for oxygen so it wouldn't also not be there for hydrogen?",
        "A": "I really don t understand what you re saying here. What we observe from this reaction is that water is formed, not hydrogen peroxide. Water has one oxygen atom and two hydrogen atoms, hence the formula H2O."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 7:41 Sal said that if you add more products what does Sal mean by \"products\"?",
        "A": "The products are whatever substances are being formed by a chemical reaction."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 5:20 Sal said that rocket burn liquid hydrogen and oxygen. Is water produced in or out of rocket?",
        "A": "Water is produced where the reaction occurs: inside the rocket."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "at 4:30 what is the difference between compounds and molecules?",
        "A": "Molecule is used for covalent structures only. Compound is used for anything that is made up from atoms. NaCl, H2O, CO2 are all compounds but only H2O and CO2 are molecules."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 8:23, Why H2CO3 is an important molecule?",
        "A": "Sal explained it in the video. It s found in blood (?), soda, oceans, and much more!"
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At about 4:50 what is the difference between a molecule and a compound? Is it just that a compound has two elements?",
        "A": "A compound has 2 different elements but a molecule can have 2 of the same"
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 7:50, are you describing a ion that can be an acid or a base? Because an acid molecule is a molecule that discharges a hydrogen molecule, and a base is a molecule that accepts a hydrogen molecule.",
        "A": "Yes, that is exactly what he is doing. HCO\u00e2\u0082\u0083\u00e2\u0081\u00bb can act as an acid and give up a proton to form CO\u00e2\u0082\u0083\u00c2\u00b2\u00e2\u0081\u00bb. Also, HCO\u00e2\u0082\u0083\u00e2\u0081\u00bb can act as a base and accept a proton to form H\u00e2\u0082\u0082CO\u00e2\u0082\u0083."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "at 6:45 he said* an imbalance of electrons or protons* but earlier in a previous lesson he said if there is an increase in proton then the element changes. please explain am not very good in chemistry",
        "A": "An imbalance relies entirely on the amount of electrons. If there are more electrons than protons, than it s an anion, and if there are more protons than electrons, it s a cation. The proton is just the other side of the scale , what you use to measure the net charge of an electron."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "so im a bit confused. @ 4:37 why was the equation of 2h2+02 turned out to be 2h2o? what happened to the 2 behind oxygen?",
        "A": "During the reaction, diatomic oxygen (O2) is actually split into O+O (the same happens to the diatomic hydrogen (H2). These two oxygen atoms then react with the four hydrogen (H+H+H+H) to form two water molecules (2 H2O)"
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "What is \"net charge,\" mentioned around 6:50 in the video?",
        "A": "The net charge is the difference between the number of positive charges and the number of negative charges. The ion has one more electron than it has protons, so it has a net charge of -1 or a net negative charge ."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "So, if you were to only have enough molecular hydrogen for a ratio of 1:1 with the oxygen, would you just get a reaction with less energy released, less water, and some leftover oxygen?",
        "A": "Molecular hydrogen would be H2 so it would fully react, however I think you mean atomic so it that case you would be correct."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 1:16 Sal says molecular hydrogen and atoms and 2 molecules of molecular hydrogen. Could somebody please clarify these terminologies?",
        "A": "Molecular hydrogen is made up of two or more hydrogen atoms. In the video, Sal says two molecules of molecular hydrogen which means two different molecules of molecular hydrogen."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 7:38, if you provide more reactants, how does it go towards the right hand side and vice versa?",
        "A": "If you have more reactants, there will be more collisions between them, and you will form more products. The position of equilibrium moves to the right."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "In 6:37 Sal refers to HCO3 as a bicarbonate ion. I understand where the carbonate comes from, but why is it called bicarbonate?",
        "A": "The prefix bi in bicarbonate comes from an outdated naming system and is based on the observation that there is two times as much carbonate (CO3) per sodium ion in sodium bicarbonate (NaHCO3) and other bicarbonates as in sodium carbonate (Na2CO3) and other carbonates.[5] The name lives on as a trivial name (Wikipedia)."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 7:35, how is it going back and forth and is it simultaneous?",
        "A": "Yes, reactant molecules are forming products, and other product molecules are forming reactants simultaneously."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 3:30 is the energy being released from the hydrogen bonds splitting and then re forming with the oxygen?",
        "A": "Just remember breaking bonds takes energy, making these releases it. So it takes a little energy to break the hydrogen bonds but once it starts it releases a lot of energy."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 6:10 Sal Khan says electrolysis. What is electrolysis?",
        "A": "Electrolysis is when a current is passed through a solution of ions. A chemical reaction can occur that wouldn t otherwise occur."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "at 4:15 why does he says \"bouncing around like crazy, aren't atoms supposed to stay in the same spot?",
        "A": "No, atoms are constantly moving, they don t stay in the same spot because of the amount of energy in them."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 7:33 Sal said that the reaction is in equilibrium , what does it mean ?",
        "A": "The reaction is proceeding in the forward direction and in the reverse direction at the same rate. (at least, most of the time!)"
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "The video discusses irreversible reactions at about 6:20. When would an irreversible reaction need to be reversed/ Why would it be reversed?",
        "A": "Take, for example, your body. Your cells are constantly making proteins for when they need them. Let s say you eat a meal and your blood sugar raises. You need to make insulin to lower your blood sugar. So your cells go through the reaction of making that insulin. Well, that insulin needs to be broken down because your blood sugar won t always be raised from that meal. If reactions in your body couldn t be reversed, it d be bad news."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 7:37 Sal says the Hydrogen might have lost an electron. How does that happen?",
        "A": "By another atom coming by that needs another electron in its outer electron shell. It picks it up from the hydrogen to complete its outer electron shell."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "at 8:30 he says that there is carbonic acid in soft dring so is he saying that its basicaly syrup and carbon?",
        "A": "Soft drinks also contain preservatives and flavorings, as well as at least one sweetener of some type, carbonic acid, and water, as you said."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "at about 4:45 what is the difference between compound and molecule",
        "A": "A molecule is a group of two or more chemically bonded atoms. A compound is two or more combined elements. A compound is composed of different elements"
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "3:33\nIf they are always bouncing around, does that mean that hydrogen and oxygen are going to combine and water will fall from the sky??",
        "A": "No, because there are no free hydrogen atoms or free oxygen atoms bouncing around in the sky. But if you take some hydrogen gas, and add some energy when it is in the presence of oxygen (i.e. light it with a match), you will get water. This is a common high school chem demonstration."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "At 3:06, Sal talks about how he just made 2 water molecules. take a look at the left picture, then take a look at the right picture. notice that the 2 oxygen molecules have split, does this mean that oxygen molecules preform mitosis like chromosomes?",
        "A": "The process is quite different. In mitosis, the chromosome creates a perfect copy of itself. In the reaction. an oxygen molecule splits into two halves."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "When Sal says this at 8:07 he means the bicarbonate, but wouldn't it make more sense to say energy?",
        "A": "At that point in the video he s talking about a thermodynamic equilibrium and the language he s using is correct."
    },
    {
        "video_name": "TStjgUmL1RQ",
        "Q": "What does Sal mean at 4:18 when he says that the more energy you apply to the system...? What type of energy is he talking about?",
        "A": "He s referring here to kinetic energy arising from the movement of the molecules. If you heat up a system, then the molecules move faster and have more kinetic energy. There is also energy stored in chemical bonds. Energy needs to be put it to break bonds, but energy is released when new bonds are formed. The amount of energy put in and the amount released are usually not the same which is why chemical reactions are often accompanied by temperature changes."
    },
    {
        "video_name": "qWR7SBsnLwo",
        "Q": "What is meant by the initial and the final positions here?I don't understand that part about a specific part taken by an object at 1:34 minutes in the video...Can anyone help me please?",
        "A": "He is referring to the displacement (straight line path) since W=Fs so it doesn t matter what the distance of the path is, only as he says, the initial and final positions"
    },
    {
        "video_name": "vcStzn55MG0",
        "Q": "At 3:13, does it always go clockwise?",
        "A": "No, there is a technique discussed in the previous magnetism videos that you can use to find the direction of the current called the right-hand rule (RHR). The next video on this playlist called Len z Law explains how to determine the direction of the current using the RHR."
    },
    {
        "video_name": "MDh_5n0OO2M",
        "Q": "At 3:55, how does he get trisubstituted for the top one?",
        "A": "In 1-methylcyclohexene, the atoms that are directly attached to the alkene carbons (C1 and C2) are C3, C6 and the methyl group. That makes the alkene trisubstituted."
    },
    {
        "video_name": "MDh_5n0OO2M",
        "Q": "at 7:57 how did he make 2 hydrogens instead of one as now carbon has made 5 bonds its incorrect right ?",
        "A": "The \u00ce\u00b2\u00e2\u0082\u0082 carbon atom (a CH\u00e2\u0082\u0083 group) had three hydrogen atoms. When one of those H atoms was eliminated to form the alkene, the \u00ce\u00b2\u00e2\u0082\u0082 carbon became C-1 of 2-methylbut-1-ene, and it still had two H atoms remaining on it. Remember that bond-line structures do not normally show the H atoms that are attached to carbon unless you want to draw special attention to them."
    },
    {
        "video_name": "MDh_5n0OO2M",
        "Q": "At 1:14, why was the C-OH selected as the alpha carbon?",
        "A": "This was selected as the alpha carbon because it is here that the carbocation is formed when the OH group leaves."
    },
    {
        "video_name": "MDh_5n0OO2M",
        "Q": "will I be safe if I just chose the (Bata) with the lowest number of H? Is that the same thing? at 4:20",
        "A": "Yes, you would be safe. The carbon with the lowest number of H atoms has the largest number of substituents. So that is the same thing."
    },
    {
        "video_name": "aj34f2Bg9Vw",
        "Q": "at 0:42, isn't it dm3 not litres?",
        "A": "1 dm3 has the same volume as 1 liter. It does not matter which units you use as long as you are consistent"
    },
    {
        "video_name": "aj34f2Bg9Vw",
        "Q": "Why do you muliple by the 2 for the mole? I dont understand the formula being used. The time is 3:40 on the video. Thanks",
        "A": "while 50 mL Semi molar solution is mixed with 100 mL Desi molar solution what is the resultant concentration of the mixture solution? can anybody help me solve it?"
    },
    {
        "video_name": "JMsqu236bZo",
        "Q": "At 6:46, how are ketones different from ethers",
        "A": "The way I think of ethers is an Oxygen with a R group on ether side ( R-O-R ). Ketones C=O where the carbon has two R groups attached and a double bonded Oxygen. Lets compare. with ketones the O only is attached to 1 R group, while with Ethers the O is attached to 2 R groups."
    },
    {
        "video_name": "JMsqu236bZo",
        "Q": "At 8:03, should it be named 'methylethylketone' or 'ethylmethylketone'? Generally, the 'ethyl' group would take precedence going by the alphabet but does it hold true with ketones too?",
        "A": "This is a problem with rules and convention. The rules say that it should be ethylmethylketone. Unfortunately, it is often called methylelthylketone. In the U.S. it is sold as a paint solvent with the name MEK . When writing formally, such as in a textbook or research paper, it should always be written as ethylmethylketone."
    },
    {
        "video_name": "JMsqu236bZo",
        "Q": "At 5:49 why isn't the molecule called cyclohexanal? What is incorrect about naming the molecule in that manner?",
        "A": "When the -al ending is placed on a name like in pentanal, what it really says is pentan-1-al, and since carbon 1 can t be in an aldehyde group and still close the ring, the name cyclohexanal is nonsense."
    },
    {
        "video_name": "JMsqu236bZo",
        "Q": "At 4:25 why does the carbon of the phenyl get numbered 1? Looking at the previous examples, I would have expected the 1st carbon to be the one double bonded to the oxygen as that is the aldehyde.",
        "A": "This is because the entire aldehyde is considered its own part, and is not included in the numbering of the ring system. The aldehyde is on the 1 of the ring system."
    },
    {
        "video_name": "JMsqu236bZo",
        "Q": "At 5:08, it the methoxy substituent the same as a methyl ether substituent? Or are they different?",
        "A": "They are the same. The correct IUPAC name is methoxy, which is a methyl ether substituent."
    },
    {
        "video_name": "JMsqu236bZo",
        "Q": "At 2:21 you named the cyclic structure as phenyl but at 3:47 you called the cyclic structure as benzene.\nSir,why this difference?",
        "A": "The name of the compound C\u00e2\u0082\u0086H\u00e2\u0082\u0086 is benzene. As a substituent group, the C\u00e2\u0082\u0086H\u00e2\u0082\u0085 group is called phenyl."
    },
    {
        "video_name": "JMsqu236bZo",
        "Q": "At 4:30, why do you start numbering the molecule from the phenyl group clockwise, without first giving the carboxyl group number 1?",
        "A": "There is no carboxyl (-COOH) group, I guess you may have meant the carbonyl (C=O), which is part of the aldehyde group? When numbering around a ring (including an aromatic ring), the numbers refer to the ring carbons and are used to give the positions of substituents attached to the ring."
    },
    {
        "video_name": "JMsqu236bZo",
        "Q": "At 4:55, why do you use the suffix -xy in order to name the substituents of the phenyl group? Such as 4-hydroxy or, 3-methoxy. Thanks!",
        "A": "-oxy is used when an oxygen atom is attached directly to an organic compound (e.g. in an ether bond)."
    },
    {
        "video_name": "Qcd0IhW_85I",
        "Q": "At 2:17, F does equal to ma. But the acceleration of a circular motion is just the resultant of centripetal acceleration and tangential acceleration. Does electron has tangential acceleration or is it zero?",
        "A": "F=ma for straight line acceleration. For circular motion, you need to adjust the formula (until you have something that looks completely different) and then use angular momentum/velocity/acceleration."
    },
    {
        "video_name": "Qcd0IhW_85I",
        "Q": "At 9:11, what does he mean by r1 ? I get that 1 is some integer, but what does it indicates the radius?",
        "A": "It indicates that it is the first permitted orbital radius."
    },
    {
        "video_name": "Qcd0IhW_85I",
        "Q": "at 0:35 jay says that the electron in this example moves counterclockwise and that made me think in what direction do electrons orbit. so what direction do they orbit in.",
        "A": "Electrons do not in reality orbit the nucleus. This is the major flaw of the Bohr model. While the Bohr model does explain some properties of the atom, it fails in many aspects. The electron, rather than orbiting the nucleus, exists more as a standing wave in a cloud-like structure we call an orbital."
    },
    {
        "video_name": "Qcd0IhW_85I",
        "Q": "At 8:59, why does n=1? and what does \"quantize\" mean?",
        "A": "There are discrete orbitals electrons reside within (or around) a nuclei. n = 1 is the state of lowest energy, closest to the nuclei."
    },
    {
        "video_name": "Qcd0IhW_85I",
        "Q": "at 4:57,how did we know that there should be h/2*pi.",
        "A": "From my understanding this is the expression the Niels Bohr came up with for the quantised angular momentum The classical expression for Angular momentum is L = mvr and this is equated to Bohr s quantised expression h/2pi"
    },
    {
        "video_name": "Qcd0IhW_85I",
        "Q": "at 0:15 the tutor says \"even the bohr's model is not reality\" what does he mean from it?",
        "A": "Just what he says. The Borh model is not a correct representation of what really happens inside the atom. We have replaced it with a modern model that more accurately describes our observations of how atoms behave. The new model is the quantum mechanical model. (Of course, no model is reality, that s why they are called models)"
    },
    {
        "video_name": "Qcd0IhW_85I",
        "Q": "At 5:00, how did Bohr come to the comparison between angular momentum and n.(h/2pi)? And what does the h/(2pi) stand for?",
        "A": "The h/2pi is the reduced planck s constant and is pretty common in quantum mechanics (along with something divided by 2pi)."
    },
    {
        "video_name": "Qcd0IhW_85I",
        "Q": "At 6:15, he canceled r^2 and r which were denominators. Can you do that?",
        "A": "Better review your algebra. Of course he can multiply both sides by r. You can always do the same thing to both sides of an equation."
    },
    {
        "video_name": "Qcd0IhW_85I",
        "Q": "Can we assume that the electron goes clockwise at 0:44 ?",
        "A": "Yes, the equation will still work."
    },
    {
        "video_name": "PSl6IXUCNuw",
        "Q": "@1:17 what is E. coli?\nThanks",
        "A": "E. coli is a bacterium commonly found in the intestines of humans and other animals, where it usually causes no harm. Some strains can cause severe food poisoning, especially in old people and children."
    },
    {
        "video_name": "5yQlaMy4T2w",
        "Q": "At 5:40, he says it absorbs \"heat\"... On previous videos, heat is defined as the rate at which molecules collide with each other. Therefore, how is heat absorbed?",
        "A": "If fast moving particles in object A bump into slow moving particles in object B, then the particles in A slow down and the particles in B speed up and thermal energy has been transferred from A to B."
    },
    {
        "video_name": "kY8FEq0teOs",
        "Q": "What are the exact numbers at 5:25?",
        "A": ".. I imagine it can vary. On average it is probably +40mV."
    },
    {
        "video_name": "kY8FEq0teOs",
        "Q": "From 7:50 Why would Potassium [K] want to get out from the positively charged environment?",
        "A": "Potassium exist as ions in this environment, they are positively charged. 2 things could account for them to want to get out from the positvely charged environment. 1. Positive charges repel eachother and 2. Processes of difussion wants higher entropy and therefore K+ will leave."
    },
    {
        "video_name": "kY8FEq0teOs",
        "Q": "At 4:30, when Sal says \"let's say that the sodium channel opens at -55mV,\" WHY exactly would a sodium channel in real life open at a certain mV threshold?",
        "A": "The channels are what we call voltage gated . On the protein, there s a part of its structure called the voltage sensing domain. The charges on the amino acid side chains interact with the membrane potential. When the voltage sensing domain reaches a certain voltage, the interaction between the charges and the membrane potential makes the protein change shape, which then opens up the channel."
    },
    {
        "video_name": "kY8FEq0teOs",
        "Q": "At 8:35 Sal says the axon gets back to its equilibrium state and indicates that this would be the original -70 mV. This confuses be because the opening of the gates should decrease the potential difference between the inside and the outside, and it should require some work to build the 70 mV difference again?",
        "A": "When he says the membrane potential reaches equilibrium its actually because the K+ voltage channels close at -80mV so resting potential can be restored and go back up to -70mV."
    },
    {
        "video_name": "8VOcnNekTjw",
        "Q": "@1:14 What causes the quartz crystal to respond this way to either mechanical or electrical action? Is there a name for this effect? Are there other materials used in elsewhere or existing in nature that exhibit this property?",
        "A": "This behavior exhibited by the crystal is called the pizeoelectric effect. There a number of materials other than quartz that can exhibit this effect. This effect is influenced by the molecular structure of the crystal, the shape it is cut into and the orientation of the cut through the crystal. There are also other factors that can influence the oscillation."
    },
    {
        "video_name": "8VOcnNekTjw",
        "Q": "@ 0:01, i dont get it",
        "A": "A microcontroller is another word for an embeded computer. It is a controller suited to applications in appliances and cars. I believe modern cars have many microcontrolleres. You do not see these devices becuase they are under the hood and monitor the engine. A microcontroller can monitor oxygen level, position of the cam shaft, fuel pressure, and even change the fuel/air mixture based on the driving conditions."
    },
    {
        "video_name": "oFGkYA_diDA",
        "Q": "Is that picture in the beginning of the video, sal, the one to the right? At 0:06?",
        "A": "yeah, that is him. except, it s obviously not him now"
    },
    {
        "video_name": "ou9YMWlJgkE",
        "Q": "At 4:20, Sal had written acceleration m/s^2.Why m squared ?\nIt is confusing.",
        "A": "no worry; it just means the units of acceleration are change in velocity per second and velocity is m/s so we get m/s per second. And we can write it as m/s/s or m/s>2 ok??"
    },
    {
        "video_name": "ou9YMWlJgkE",
        "Q": "At 7:34 Could you explain this: \"The rate of change of linear momentum is directly proportional to the impressed force,and takes the direction in which the force acts\"?",
        "A": "All this is saying is force equals change in momentum over change in time: F=delta_p/delta_t And the direction which a force is applied is going to be in the same direction in which the impulse is generated."
    },
    {
        "video_name": "ou9YMWlJgkE",
        "Q": "At 3:34 Sal says that in the formula F=ma, 'm' is the constant of proportionality but if a body is travelling at the velocity of light then, its mass will constantly change with respect to time so how can it be true that mass is constant?",
        "A": "Mass will be constant as far as you are able to tell in just about any experiment you are able to do. It took 300 years after Newton for anyone to be able to measure the change in mass that occurs when objects travel at speeds that we never, ever experience. Unless you start traveling around at speed in excess of about 1/4 the speed of light, you are not going to be able to measure a difference in mass, so you can treat it as a constant except when you are doing problems in certain advanced physics classes"
    },
    {
        "video_name": "ou9YMWlJgkE",
        "Q": "@6:69 Mr Sal says the more mass you have the more you accelerate. does it imply to a freely falling body which is only under the influence of gravity",
        "A": "Gravity does not depend on mass of the object.. Technically, a feather & a stone dropped from the same height should reach the ground at the same time.. The reason why this doesn t happen is due to air resistance.. Which slows down the feather due to its shape..."
    },
    {
        "video_name": "ou9YMWlJgkE",
        "Q": "@ 1:15 what is a vector quantity?",
        "A": "It s a quantity, that can be represented by a vector. If you don t know what that is, imagine it to be like an arrow - it has a magnitude and a direction. Something like velocity, force, momentum, acceleration, electric field, etc.. Things that only have a magnitude, but no direction are called scalar quantities - or just scalars ( normal numbers). Physical quantities that are scalars are for example: mass, temperature, charge, volume and speed (the magnitude of the velocity-vector)."
    },
    {
        "video_name": "ou9YMWlJgkE",
        "Q": "At 2:04, Mr. Sal mentions the formula. Why couldn't it be A = F x M as in Acceleration = Force times mass? Because if you apply Force to a Mass, it Accelerates.",
        "A": "hmm... i think you re right! thanks for the answer!"
    },
    {
        "video_name": "DNoDJlB0ygU",
        "Q": "At 1:48 what is he writing? What is the -1 exponent for? What is the greek symbol he keeps refering to? Thanks! :)",
        "A": "The Greek symbol is theta . It is a symbol often used in math to denote an unknown angle. The -1 exponent he adds to the Tan means it is the inverse tangent. If you need more information on this, I recommend reviewing the videos and practice problems on Trigonometry. I hope that helps :)"
    },
    {
        "video_name": "SFd8mcpP3TU",
        "Q": "At 4:33 , Jay says that the halogen can give its pair of electrons to form a pi bond but if that halogen is so electronegative (as Jay says previously) then why does it give them anyway? Is it because electrons are highly active and doesn't stay where they are, so they are forming the pi bond even for a slightest moment?",
        "A": "Cl isn t giving away its electrons. It is partially sharing its electrons through the \u00cf\u0080 system. This delocalization of electrons lowers the total energy of the system despite the unhappiness of the Cl at losing some of its control over these electrons."
    },
    {
        "video_name": "SFd8mcpP3TU",
        "Q": "2:30, R-groups( in this case -CH3) are electron-donating, isn't the electron density being donated to the ring through the bond between the ring and the methyl group?",
        "A": ". When the benzene ring attacked some of electrophiles [ ether ortho or para position ] the carbon where methyl group is attached will be in it a positive charge [ due to resonance ]. When we arrived to this situation , both methyl group and benzene will share with each other one electron [ from any hydrogen that attached to methyl group ], or to be more specific, they will share the positive charge. This process make alkyl groups as activator groups. Good Luck."
    },
    {
        "video_name": "SFd8mcpP3TU",
        "Q": "At 5:35, Jay says induction must be more important than resonance. Is induction in general a more important factor than resonance when determining the affect of a substituent on electron density in a benzene ring? Or does he mean that in that induction is stronger than resonance in this specific example?",
        "A": "It s difficult to always predict whether induction or resonance will be the more important factor. In this specific case induction is the more important effect, but in other cases resonance may be more important. Often, we look at experimental results and induce which factor must have been more important, induction or resonance, based on the results (so it s an after-the-fact explanation, rather than a prediction)."
    },
    {
        "video_name": "SFd8mcpP3TU",
        "Q": "From 7:30 to 8:00, Jay says that Chlorine and Nitrogen have approximately the same electronegativity. Is there an authoritative (or even reliable) electronegativity chart (or period table showing electronegativity) that is worth memorizing? When I googled for it, different charts have different electronegativity values for the same element.\n\nAlso, how is the electronegativity value calculated in the first place?",
        "A": "I don\u00e2\u0080\u0099t think any electronegativity chart is worth memorizing, but you should memorize trends in electronegativity values. There are different charts because different people created their own scales. The one most commonly used is the Pauling electronegativity scale. He developed it using differences in bond energies and then scaling the numbers so H had a value of 2.2 and F a value of 4.0. There have been some minor adjustments since then."
    },
    {
        "video_name": "ii-KBKSODek",
        "Q": "In the SpaceTime Graph for Sally at 7:30 Sal says that light will take 1 second to go from Sally to the space ship and another second for its reflection, so it means that Sally is also travelling at the speed of light which contradicts his previous saying and in the SpaceTme Graph in his frame of reference that Sally is travelling at half the speed of light, so I`m a bit confused as to what to take in to consideration.",
        "A": "Speed of light is regardless of any frame of reference. Though Sally is moving at half the speed of light, the emitted photon is still travelling at the original speed of light. Sally is still moving with 0.5c (not sure that cleared what you asked still hope that helped =>...)"
    },
    {
        "video_name": "ii-KBKSODek",
        "Q": "At 7:18, would the fact that both \"Sally\" and the person in front of her in the spaceship are moving at half the speed of light affect the time it takes a photon to travel back and forth once between their ships?",
        "A": "No, because they are NOT moving at half the speed of light. Inside their ship, they are not moving relative to one another, and when they look out the window, stuff is rushing past them at half the speed of light."
    },
    {
        "video_name": "ii-KBKSODek",
        "Q": "What does the -1s imply on the vertical time axes ?(at 7:00)",
        "A": "1 second prior to where t=0 is defined for the time axis."
    },
    {
        "video_name": "ii-KBKSODek",
        "Q": "At 12:00 , how does the path of the photon being 45\u00c2\u00b0 with respect to both pairs of axes indicate the speed of light being 3*(10^8) m/s in all frames of reference?\n\nDoes it mean that if I drew the diagram myself, I would just find that 1 second in either frame of reference's vertical axis maps to 3*(10^8) m in the horizontal? If so, how doesn't that contradict the idea that space and time aren't absolute?",
        "A": "It is not the scale of the axes that give you the indication of space and time not being absolute it is comes from how distance is defined. In normal Euclidian space when you have 2 dimensions the distance is defined by the pythagorean theorem as sqrt(x^2 + y^2). In Minkowski space the distance or space-time interval is defined by sqrt(x^2 - (c^2 * t^2))."
    },
    {
        "video_name": "aHlOp5nYs28",
        "Q": "Sorry if this is a stupid question.Can't tension be present in any long structures,other than ropes?Like maybe if i make an iron rod and use it to pull a car?]\nAlso,at 6:59 can't the box pull the wall?\n\nThanks.",
        "A": "Yes, tension can be present in an iron rod, for example. Though an iron rod will tend to have a significant amount of mass, so pretending it s massless might be a little less accurate. The box could theoretically pull down the wall, yes, if the wall was really poorly built. We re assuming that the wall and ceiling are stable."
    },
    {
        "video_name": "aHlOp5nYs28",
        "Q": "At 10:07, why didn't David use trigonometry to solve for T1y? I don't understand why not.",
        "A": "just try it, nicholas is right"
    },
    {
        "video_name": "aHlOp5nYs28",
        "Q": "Forgive me, I haven't taken a math or science class in over a decade. When solving for T1y at 10:45, we add 30N to both sides. Why do they not cancel each other out?",
        "A": "they cancel each other but when added to 0 it doesn t get canceled . so, there is another 30 N left in the L.H.S."
    },
    {
        "video_name": "aHlOp5nYs28",
        "Q": "15:26 Why doesn't David just use the Pythagorean Theorem here?",
        "A": "He could. There are usually more than a couple ways to solve a problem. As you can see using Pythagoras sqrt(60^2 - 30^2) = 30*sqrt(3) as in the video. However it makes sense that he used trig because it s good to be consistent when making calculations; T1 could only be found using trig."
    },
    {
        "video_name": "viLpmZtQYzE",
        "Q": "At 9:47 David says the axis will exert some force but no torque. Why will the axis exert force in the first place?",
        "A": "Newton s third law."
    },
    {
        "video_name": "viLpmZtQYzE",
        "Q": "What if the ball after hitting the rod did not stop and went on straight with a speed of +2m/s. Is this situation the same as described at 13:25?",
        "A": "so i guess instead of using v=8 m/s in the conservation of momentum equation, we would use the speed the ball lost. In your case as an example we would use v=6 m/s since that is the speed the ball lost when it transferred the momentum."
    },
    {
        "video_name": "6IS_M6CX7FE",
        "Q": "Around 7:00 Sal states that density dependent factors are biotic. How is food and water shortage biotic - it is not living (for water).",
        "A": "I believe it is a biotic factor because it supports living things. But I m not quite sure myself. It is possible that Sal made a mistake(we all make them). But I am quite sure that it is a factor for biotic organisms and that s why it is a biotic factor ."
    },
    {
        "video_name": "rPim-9WSg6g",
        "Q": "At 6:40 it is claimed that the cell potential will be lower because there are fewer oxygen molecules available. Is that really correct? Even if there would be only one oxygen molecule available to oxidize (2) zinc molecules, the potential (being an intensive property) should still be the same as under STP. The difference to STP is that the electric current (number of transferred electrons over a unit time frame) would be very small.",
        "A": "Pressure isn t the same however. So it wouldn t be standard STP. Think of cell potential in relation to Q. Q would be larger due to lower concentration of reactants. This in turn would lead to lower instantaneous cell potential. E = E* - .592/n * log Q. Log Q increases as Q increases which decreases E."
    },
    {
        "video_name": "MtwvLru62Qw",
        "Q": "around 11:40 in ths video..............the alpha corbon in the ring has zero hydrogen's attached to it............which is lesser than 2 .........why we dont form a double bond there...........??????????//",
        "A": "If there are no hydrogens attatched to a beta carbon, there s no source for one of the electrons needed to make the double bond. Also, the beta carbon on the ring would have a double bond and three single bonds, and you can t have more than four total bonds on a carbon."
    },
    {
        "video_name": "MtwvLru62Qw",
        "Q": "Just want to make sure my IUPAC memory is intact. The name of the compound Sal mentions at 0:05 will be \"1-methyl-1,(1-iodo-1-methylpropyl)cyclopentane\", right?",
        "A": "The correct name is 1-[(1R)-1-Iodo-1-methylpropyl]-1-methylcyclopentane. Remember, the iodo comes before both methyls because of the alphabetic naming convention. Also, don t forget the sterochemistry on the side chain!"
    },
    {
        "video_name": "PFcGiMLwjeY",
        "Q": "At 6:00 he says we would have a perpetual motion machine if something was as efficient as carnot cycle. But in the carnot cycle, aren't we doing work on the system at the points C to D? So we are inputting some energy, when we compress the piston. How is a perpetual motion machine?",
        "A": "mmm good point. However, surely a perpetual motion engine might require energy from us, as long as it returns the same amount of energy back to us in some stage of the cycle...??"
    },
    {
        "video_name": "PFcGiMLwjeY",
        "Q": "at 10:00 he says the system generates heat and so it will be given to surrounding...so it must be -Qf right...",
        "A": "Ya I thought of the same thing.. Shouldn t it be dU=Q-(W=Qf) ? i.e. Q goes into doing displacement work + frictional work or heat"
    },
    {
        "video_name": "PFcGiMLwjeY",
        "Q": "Sal says at 3:45 that a reversible process is quasi-static and frictionless, but can't a system that is quasi-static but with friction also be reversible if the exact amount of energy that the system loses to friction is put back in?",
        "A": "If you have to put in energy to replace what was lost, it s not reversible."
    },
    {
        "video_name": "PFcGiMLwjeY",
        "Q": "Why does the W cancel out at 11:30 ? If friction was there when the system expanded how can the work done on the system be the same as the work it did on the surrounding before?",
        "A": "It appears to me that he is looking at the same isotherm, so even if you go back in the other direction ( from A to B, or from B to A) the area under the curve, or the work, will be the same, so they will cancel out."
    },
    {
        "video_name": "PFcGiMLwjeY",
        "Q": "At 6:04 sal said that the system should be frictionless. But I thought that with the same quasi-static increase or decrease, the friction is of the same value. Right? Because friction is essentially related to how much distance the piston moves. So if the distance x is increased (expansion) by a Z amount of quasi-static change, when decreasing(compression) with the same value Z, the friction would be the same right? So this is applicable in real life right?",
        "A": "Friction is not a conserved property. So losses due to friction are irreversible. You lose that energy as heat. So saying the system is frictionless is a necessary simplification."
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "At 9:35 , why sal said that 1s orbital of hydrogen is overlapping sp2 orbital ?\nIt should be overlapping sp3 orbital . Right ?\nPlease correct me where i am wrong .\nThank you.",
        "A": "No. At this point in the video, we are dealing with the carbon atom in ethene. Here, the carbon atom is sp\u00c2\u00b2 hybridized. So the hydrogen 1s orbital is overlapping an sp\u00c2\u00b2 orbital of the carbon."
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "at 6:32, why does the hybridization become 2SP2 ?",
        "A": "The C atom needs three orbitals for bonding. So it takes one s orbital and and two p orbitals to make the new hybrid orbitals.Since these orbitals must all be equivalent to each other, they end up as s(1/3)p(2/3). If we multiply the superscripts by 3, we get s1p2 or sp2. The bottom line is that we use just enough atomic orbitals (one s and two p) to create the three new bonds."
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "At 3:10 ,Sal drew ethene with a double bond in carbon so doesn't ethene have a triple bond in it ?",
        "A": "No. Ethene has a C-C double bond (1 sigma bond + 1 pi bond), ethyne has a C-C triple bond (1 sigma bond + 2 pi bonds)"
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "isnt it ethyne? not ethine? @ 13:13\nalso for the second pi-bond in ethyne, does both the lobes of the p orbital have to overlap? or will it just be one half of the lobe lyk for ethene?",
        "A": "In both ethyne and ethene each p orbital overlaps another in two places."
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "At 11:50, if on addition of sigma and pi bonds, the overall bond between the two carbon atoms gets stronger, then why is it that double and triple bonds are easier to break? Like in addition reactions..",
        "A": "bond strenght and reactivity are two different things. The electron density in multiply bonded atoms is greater and therefore they are more vulnerable to attack by electrophiles or electron seeking species."
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "At 13:24, Sal shows the last line or bond of the C-C bond in ethyne to be a sigma bond- by convention don't take the middle line?",
        "A": "I m not aware of any convention like this. You can choose any of the three lines as the \u00cf\u0083 bond."
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "13:04 why pi bond stops structure from rotating? is it now too heavy or something??",
        "A": "Think of it this way, suppose, you have an axle of a toy car or something. If you hold one wheel and try to rotate the other, it will spin, right? Now, suppose you add another axle connecting the two wheels, so now, you can t hold one of them and rotate the other. Hope that helped!"
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "At 6:20 in the video..why does 1 electron from the s orbital and 2 electrons from the p orbitals combine? Why does the other p electron stay by itself?",
        "A": "The first bond between two atoms is always a sigma bond and the other bonds are always pi bonds.... and a hybridized orbital cannot be involved in a pi bond. Thus we need to leave one electron (in case of Carbon double bond) to let the Carbon have the second bond as a pi bond. Hope that helps! :)"
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "At 6:10, why does he count the number of bonds instead of the number of electron groups?",
        "A": "He could have counted the number of electron groups instead. Since covalent bonds involve a pair of shared electrons, we can think of each atom in the bond being associated with one of these electrons. Since carbon has 4 covalent bonds in this molecule (and no lone pairs), it has 4 bonding electrons. That is, you get to the same place either way. Be careful to only use this method when lone pairs electrons are not involved though."
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "Just to clarify- when he draws the overlapping P orbitals at 11:15, the orbitals overlap each other in two different places (on either side of the C-C sigma bond). Wouldn't each overlapping area be its own bond? Or is a pi bond actually the combination of the two separate bonds?",
        "A": "You re right that the two overlapping areas together make up a single pi bond. Be careful not to think of this as two separate bonds though. It is a single (pi) bond comprised of the side-on overlap of adjacent p orbitals. It only appears as two separate entities because there is a node in the middle (just like a p orbital has two parts (a positive phase and a negative phase) separated by a node, but it is a single orbital)."
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "why at 5:07 sal writes 2sp^3 not sp^3?",
        "A": "The 2 represents the 2nd shell."
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "at 6:25, Sal says that the s orbital mixes with two of the p orbitals. What does he exactly mean by \"mixing\". And if they do \"mix\", how does it happen?",
        "A": "The mixing is a mathematical term that relates to the equations coming from the Schr\u00c3\u00b6dinger equation. Its as if, in algebra, you add three equations together and then divide by three to get three new equations. The new equations are identical in every except that, when you plot them, they point in three different directions. As an analogy to baking a cake: If you mix 1 part of white batter and 2 parts of red batter and divide into thee parts, you get three pink cakes."
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "At 06:08, Sal gives an example of ethene. So, is the compound weaker now it has 3 sigma bonds and 1 pi bond whereas in the case of methane, it contains 4 sigma bonds?",
        "A": "Ethene is not weaker than methane . Pi bonds tends to be stronger than sigma bonds. Hope that answers ur question :/"
    },
    {
        "video_name": "lJX8DxoPRfk",
        "Q": "At 13:23 , how sal said that the first bonds are sigma bonds and the hybridization is sp ?\nPlease explain .\nThank you.",
        "A": "At this point in the video, we are dealing with the carbon atom in ethyne. Here, the carbon atom is sp hybridized. So the hydrogen 1s orbital is overlapping an sp orbital of the carbon to form a C-H \u00cf\u0083 bond. If carbon is directly attached to four other atoms, as in CH4 or H3C-CH3, the C is sp\u00c2\u00b3 hybridized. If carbon is directly attached to three other atoms, as in H2C=CH2, the C is sp\u00c2\u00b2 hybridized. If carbon is directly attached to two other atoms, as in HC\u00e2\u0089\u00a1CH the C is sp hybridized."
    },
    {
        "video_name": "ONBJo7dXJm8",
        "Q": "At 7:35, why do you have to time the volume when finding the probability?",
        "A": "because it determines how much stuff is in there so it affects the chance that the molecules will collide"
    },
    {
        "video_name": "ONBJo7dXJm8",
        "Q": "1:27 you say the rate of the forward reaciton is equal to the backward reaction but the concentrations can be different. Wouldn't that mean that the rates are different because one reaction is happening more often than the other? By saying equal rates you give the impression that they are both happening as much as each other giving equal concentrations.",
        "A": "its alright i understand now i was just thinking about it the wrong way"
    },
    {
        "video_name": "ONBJo7dXJm8",
        "Q": "At 9:10, if you want to find the reverse rate for the equation, do you have to subtract it to the Keq?",
        "A": "No. The Keq for the reverse reaction should be the inverse of the forward reaction, or keq(rev) = 1/keq(for). So the reverse rate would just be the inverse Keq."
    },
    {
        "video_name": "ONBJo7dXJm8",
        "Q": "at 10:10 will the reaction rates always be the same?",
        "A": "They will be, at EQUILIBRIUM. That is what that word means - balanced ."
    },
    {
        "video_name": "ONBJo7dXJm8",
        "Q": "The rate-forward and rate-reverse equations look like the rate-law equation (ex. 5:45), but here we use stoichiometric coefficients as exponents, as opposed to rate-order exponents. Is it fair to say that we assume a high(ish) order? Perhaps second or higher?",
        "A": "The explanation is complicated but, the exponents in equilibrium reactions are the stoichiometric coefficients. The orders of reaction don t matter. For nonequilibrium reactions, you must determine the exponents (reaction orders) experimentally."
    },
    {
        "video_name": "ONBJo7dXJm8",
        "Q": "at 10:00, can K ever be cancelled by a negative K ?",
        "A": "The is no positive or negative sign in from of the K s - those are subscripts, denoting the K for the forward reaction (Sal call this K sub +) and the reverse reaction (K calls this K sub -) - don t confuse yourself!"
    },
    {
        "video_name": "ONBJo7dXJm8",
        "Q": "My chemistry teacher gives us the equation as Keq=[products]^m/[reactants]^n, does this work the same as 0:50 ? or are there different exponents?",
        "A": "Same thing I guess. It s just the concentration of the products raised to their respective stoichiometric coefficients, divided by, the concentration of the reactants raised to their respective stoichiometric coefficients."
    },
    {
        "video_name": "ETzUpoqZIHY",
        "Q": "what is beta or whatever when he said it in 3:29 and in 6:50 ???????",
        "A": "Theta, that circle with a line through the middle, is one of the conventional Greek letters used to represent an angle. In this case, it s the angle between looking straight up, and the apparent position of the star we re looking for."
    },
    {
        "video_name": "ETzUpoqZIHY",
        "Q": "Considering the representation of the Earth's rotation, when Sal says that the observer is on the Equator (at 1:40), he actually should say \"on the Greenwich Meridian\", right?",
        "A": "No. The Greenwich (or prime) meridian is an arbitrary human construct and has no mathematical/physical significance (in other words, the prime meridian could pass through Washington DC and it would not make a difference). By taking measurements on the equator, it simplifies a lot of the math involved."
    },
    {
        "video_name": "ETzUpoqZIHY",
        "Q": "at 6:20 he said its symmetric but if the star was shifted it may not be right?",
        "A": "The star will not move enough during six months to make any difference at all. The night sky looks pretty much the same now as it did hundreds of years ago"
    },
    {
        "video_name": "ETzUpoqZIHY",
        "Q": "I need help understanding this starting at 2:35... how can we measure how far objects in space are from us?",
        "A": "We measure them using calculations involving the angles presented in the video. The unit that we get from doing this is called a parsec, or the PARallax of one arcSECond. The other videos will explain this more."
    },
    {
        "video_name": "ePwN6rHhNyU",
        "Q": "Just to be clear, the perfusion of both lungs is increased in the example at 8:15, right? If the flow is increased over the same unit of tissue, like when the same lung flow goes from 2.5 to 3 L/min, then the perfusion is increased.",
        "A": "Yes, there is still 5L of blood flowing to the lungs in total. When 1/2 of the left lung is removed (that is, 1/4 of the total lung mass), there is now the same total amount of blood flowing in per minute (5L/min), so the total flow to the lungs as a whole is the same as before. Perfusion is defined as flow divided by amount of tissue, so it will be higher, because the same flow is serving a smaller amount of tissue, so more flow per tissue, which means higher perfusion."
    },
    {
        "video_name": "ePwN6rHhNyU",
        "Q": "@6:25, should he say that those (1kg and .5kg) are the masses, not the weights?",
        "A": "Yes you are correct, however in medicine patient s weight is measured in kg, because medical measurements are on the metric scale (unless you are in pharmacy)."
    },
    {
        "video_name": "a16uKH2K7gM",
        "Q": "Why does at 2:44 he says the power only flows in one direction?",
        "A": "He said that b/c he was talking about DC ,or Direct Current."
    },
    {
        "video_name": "UlJmX9SLGRw",
        "Q": "6:05... why does he refer to the \"non-polar\" parts of an ether molecule, when it has already been taught to us by Sal that ethers are already non-polar due to their absence of partial negative and partial positive charges?",
        "A": "Sorry, but ethers are polar compounds. They just aren t as polar as alcohols, for example..The O atom is highly electronegative, so it has a partial negative charge, and the C atoms next to it have a partial positive charge. The hydrocarbon parts of the molecule are nonpolar."
    },
    {
        "video_name": "54c_oPpTTP8",
        "Q": "I think you should specify that the \"tea spoon\" (at 6:29), necessary for a normal amount of salt per day, also includes salt that you get from all the food/drink that day. For example, I'm looking at mineral water bottle which has ~200 mg Na/liter , so if you drink 2 l/ day you already got ~17%* of the Na recommended daily or ~27% for a low salt diet.",
        "A": "i agree alot i can`t live without salt"
    },
    {
        "video_name": "54c_oPpTTP8",
        "Q": "At 2:55, how does he end up with 40%? Is it some kind of math?",
        "A": "Yes. He took the atomic mass of Na and divided it by the molecular mass of NaCl. I checked his math, and it looks right. I hope this helps :)"
    },
    {
        "video_name": "AznXSVx2xX0",
        "Q": "At 9:08, where are we pulling out the negative 1? I don't see where it is being pulled out from. Also, why did i and j flip? Is that where the negative comes from? Since we reversed the j^2 and i^2, it became negative and that is where we are pulling the negative sign from? Thank you.",
        "A": "Yes, there is a -1 outside the parenthesis (which cancels the -R) and another inside the parenthesis that flips i and j."
    },
    {
        "video_name": "AznXSVx2xX0",
        "Q": "At 0:27, What does it mean by an atom is in ground state?",
        "A": "An atom in its ground state is when it has the lowest possible energy. The alternative is that it gains energy from light, in which case electrons just jump energy levels, and then an atom is said to be in an excited state."
    },
    {
        "video_name": "AznXSVx2xX0",
        "Q": "At around 8:00, isn't when you use Rydberg's constant, the whole equation equal to the energy of the photon? Rydberg's constant is not a different constant/planck's constant and the speed of light. R = 2.1799x10^-18. Then you have to solve energy for wavelength. Am I missing some really easy algebra or something? I think all of the constants are mixed up.",
        "A": "The formula at 8:00 is a wavelength equation. It uses 1/\u00ce\u00bb, so you must use the Rydberg constant in the correct units: R = 1.097 \u00c3\u0097 10\u00e2\u0081\u00bb\u00e2\u0081\u00b7 m\u00e2\u0081\u00bb\u00c2\u00b9. If you use the energy equation with \u00ce\u0094E on the left hand side, you must use the Rydberg constant with energy units: R = 2.1799\u00c3\u009710\u00e2\u0081\u00bb\u00c2\u00b9\u00e2\u0081\u00b8 J. The Rydberg constant is a different number when it is expressed in different units."
    },
    {
        "video_name": "AznXSVx2xX0",
        "Q": "The emission of an electron results in the emission of photon, producing light. (1:30) But what happens to the atom when it absorbs the extra enegry in absorption? At what point will emission occur?",
        "A": "It is not the emission of an electron that produces a photon. A photon is produced when an electron releases energy by moving to a lower energy level within the atom. When the atom absorbs energy, it re-emits it in a random direction, almost instantaneously. The exact time delay is random, but always so fast that it seems pretty much instantaneous."
    },
    {
        "video_name": "AznXSVx2xX0",
        "Q": "At 1:30, he says that the electron falls back to ground state. How come it falls back to ground state but if it started off at the third energy level it wouldn't fall back to ground state,",
        "A": "It always falls eventually to the lowest available energy state."
    },
    {
        "video_name": "AznXSVx2xX0",
        "Q": "At 2:25, how is the energy of the photon equal to the difference of E3 and E1? How are the two energy levels related to the energy of the photon?",
        "A": "when the electron is in the excited state (E3) then it has more energy. It is vibrating faster here. Then it decides to lose some energy. And so it stop vibrating so much and lets some energy go. So its energy might drop to E1. where does the energy go? The energy produces or becomes a photon. How much energy does this photon have? the same amount of energy that the electron lost = E3 - E1 ok??"
    },
    {
        "video_name": "AznXSVx2xX0",
        "Q": "What do you mean by pulling out -ve one in 9:12 ?",
        "A": "He factored out a -1 from the term in parentheses, making (1/j\u00c2\u00b2 - 1/i\u00c2\u00b2) into -(1/i\u00c2\u00b2 - 1/j\u00c2\u00b2)"
    },
    {
        "video_name": "AznXSVx2xX0",
        "Q": "I don't understand how he got the step at 9:31 ......\nHow did the negative signs change to positive?",
        "A": "I think he was multiplying, and when you multiply two numbers, you get a positive number."
    },
    {
        "video_name": "AznXSVx2xX0",
        "Q": "At about 1:15, the speaker says that the electron at a higher energy level will eventually fall back down to its ground state. What causes the electron to fall back down and release energy in the form of a photon? Or is it some sort of random phenomenon?",
        "A": "It is spontaneous. The electron will lose energy by losing a photon."
    },
    {
        "video_name": "AznXSVx2xX0",
        "Q": "Are Ej and Ei direction dependent? At 4:01, Ej and Ei became E3 and E1 (respectively) because the electron \"fell down\" to a lower energy level. However, if the electron \"jumped to\" a higher level, would Ej andEi become E1 and E3 (respectively)?",
        "A": "No. Ej is always the higher e level and Ei is always the lower e level."
    },
    {
        "video_name": "AznXSVx2xX0",
        "Q": "At 2:33 where do the photon come from ? Wasn't an electron\nPhotons and electrons confuse me a lot",
        "A": "There are photons everywhere around you lol"
    },
    {
        "video_name": "qLMsZKx_a8s",
        "Q": "At 0:16, he says the dome is opaque, but wouldn't that mean that no light can pass through?",
        "A": "I believe opaque means semi-transparent so light can pass through"
    },
    {
        "video_name": "qLMsZKx_a8s",
        "Q": "0:38 what is zinc?",
        "A": "Zinc (Zn) is one of the transitional metals located on the Periodic Table of Elements. Basically, it s a metal that is useful in industry for its resistance to rusting/oxidation."
    },
    {
        "video_name": "qLMsZKx_a8s",
        "Q": "At 2:03, why do the manufacturers not make the light less intense instead of putting the layers of plastic in between?",
        "A": "if the light was less intense then when you covered it it might not show even without the first layer of plastic. if you didn t cover the LED then you would have nothing to press on and it would look ugly."
    },
    {
        "video_name": "qLMsZKx_a8s",
        "Q": "what's that app called at 3:30?",
        "A": "I don t know exactly which one he is using but for android i use one called ElectroDroid it has a lot of reference materials not just resistor stuff as well."
    },
    {
        "video_name": "qLMsZKx_a8s",
        "Q": "0:18 what is opeca crillic",
        "A": "He says opaque acrylic. Acrylic is the type of plastic, and opaque refers to a substance that doesn t allow light to pass through."
    },
    {
        "video_name": "qLMsZKx_a8s",
        "Q": "At 2:18, how long does it take for the springs to wear out?",
        "A": "It varies on how much you use it of course, but it s safe to say it wont stop springing for years."
    },
    {
        "video_name": "fY_ejjMRYg0",
        "Q": "at 4:00 why doesn't the carboxylic acid get turned into an acetal?",
        "A": "Only aldehydes form acetals. Ketones, as in this case, form ketals. Carboxylic acids form neither. In fact, under the mild conditions of ketal formation, they don t even form esters with the ethylene glycol. That s why you can protect the ketone while you do a reaction on the carboxylic acid."
    },
    {
        "video_name": "fY_ejjMRYg0",
        "Q": "I still don't understand the explanation of the purpose of using thioacetals instead of acetals mentioned at 6:15? Why use one instead of the other?",
        "A": "If you use an acetal, then there would be no way to change the ketone to an alkane. A thioacetal is used because there is a reaction to easily get rid of the ketone functional group. Also, you can use Thioacetals as a protecting group as well, but only Thioacetals have the reaction with Raney Nickel to go from Thioketal or Thioacetal to Alkane."
    },
    {
        "video_name": "fY_ejjMRYg0",
        "Q": "At 4:28, shouldn't it be called cyclic ketal? It has R groups on both sides of the carbonyl carbon. Thanks :-)",
        "A": "Yes, it is a cyclic ketal. Some chemists use the word acetal to refer to both acetals and ketals."
    },
    {
        "video_name": "fY_ejjMRYg0",
        "Q": "around 4:00, can the acid protonate the hydroxyl portion of the carboxylic group, thereby forming a water molecule which may become a leaving group?",
        "A": "Yes, it can, and it will form an oxonium ion which is a good leaving group. But it won\u00e2\u0080\u0099t leave, because the carbocation that would form is highly unstable."
    },
    {
        "video_name": "fY_ejjMRYg0",
        "Q": "At 8:10 when talking about reducing the Cyclo-dithiol acetal and adding the two hydrogen groups, is this all done by Raney Nickel?",
        "A": "Yes, Raney Nickel is the only reagent needed to react with the cylic thioketal to produce an alkane. Raney Nickel is also good for the hydrogenation of Alkenes and Alkynes."
    },
    {
        "video_name": "Mz2nDXElcoM",
        "Q": "6:06 I dont really understand what you where looking for that gave you \"49 root of 3 \" ??",
        "A": "You were looking for the Fg which was perpendicular to the inclined plane. Since to find the angle you would need to multiply 98 by cos(30), it would give you the value of Fg which was perpendicular to the inclined plane. Cos(30) can be expressed as Root 3/2 and 98 multiplied by Root 3/2 is the same as 49 multiplied by root 3. Hope that helps!"
    },
    {
        "video_name": "Mz2nDXElcoM",
        "Q": "In 7:19, how could the wedge exert a force upward? Isn't the ice preventing it from doing that?",
        "A": "From Newton s Third Law, the block and wedge are exerting forces upon each other. Gravity is pulling the block downward onto the wedge. The wedge prevents the block from accelerating straight down, which means there is a force exerted by the wedge on the block."
    },
    {
        "video_name": "Mz2nDXElcoM",
        "Q": "At 1:48: I know that downward acceleration is negative, upward acceleration is positive. But why isn't the force of gravity acting on an object primarily negative, even if the object isn't accelerating downward, since the direction of Fg is always downward?",
        "A": "Positive and negative are arbitrary. Gravity is not negative or positive, it just points down. If you want to call down positive, g is positive. If you want to call up positive, g is negative. You can do whatever is convenient as long as you are consistent."
    },
    {
        "video_name": "Mz2nDXElcoM",
        "Q": "at 2:46, what exactly is the meaning of component of force of gravity that is perpendicular to that plain in the video and I did not understand why are 2 vectors\nconsidered ? would be grateful if anybody could explain sir .",
        "A": "The two vectors are considered because the normal force acts perpendicular to the plane. In order to simplify the problem, it helps to shift the coordinate system to match the direction of motion. When solving these problems, the X-direction is usually considered to be parallel to the plane. The Y-direction is perpendicular to the plane. I hope that helps!"
    },
    {
        "video_name": "DRtLXagrMHw",
        "Q": "At 1:08,how can a meteor just be able to wipe out all the dinosaurs and make all of them extinct? Was the meteor big?Wide?Dense?Did the meteor wipe out the herbivores which made the carnivores die ?Please HELP me?",
        "A": "The meteor that hit Earth was about 5 miles in diameter. We don t really know how dense it was. If it was a comet, it would be made mostly of ice but an asteroid would be mainly silicate material (rock) and metal making it much more dense. The impactor kicked up millions of tons of dust which covered the planet. Many plants died out and large herbivores died out which also killed large carnivores. Small animals did not need as much food and were able to survive."
    },
    {
        "video_name": "DRtLXagrMHw",
        "Q": "At 2:50 Sal says \"Jesus' Birth\" but how can you say Jesus ever existed when there is no actual evidence he was real.",
        "A": "We know Jesus actually existed. there are many documents referring to him and he does appear in the Roman census. What people keep fighting about is if he was divine or not."
    },
    {
        "video_name": "DRtLXagrMHw",
        "Q": "0:27 Sal says that he could not get better pictures because they were copyrighted by NASA. Because this is for educational proposes wouldn't he be able to take the picture even if it is copy righted?",
        "A": "He doesn t say that the image is copyrighted by NASA, he says that it is the best non-copyrighted image he could find and that it s from NASA. NASA is a publicly funded organization, so anything they create is public domain. NASA can t copyright any imagery they create. I think what he was saying was that more interesting diagrams were copyrighted by their respective creators for example National Geographic might have had a really cool image but he couldn t use it because National Geographic copyrighted it."
    },
    {
        "video_name": "DRtLXagrMHw",
        "Q": "At 4:57 when Sal says \"the last land dinosaurs became extinct\" would he be indirectly be referring to birds??",
        "A": "The birds survived"
    },
    {
        "video_name": "CbgWfnqLJI0",
        "Q": "You said the double bonds are cis (3:40) they look trans to me",
        "A": "The H atoms are on the same side of each double bond. Same side = cis."
    },
    {
        "video_name": "CbgWfnqLJI0",
        "Q": "From 1:15 to 3:30, the inside substituents (in the video hydrogen atoms) go up and the methyl groups go down. Can the opposite happen (methyl groups going up and the inside substituent going down)? It seems if we flip the product over (or look at the opposite side of the molecule), the opposite is also possible.",
        "A": "Yes, both additions are possible"
    },
    {
        "video_name": "CbgWfnqLJI0",
        "Q": "At 1:12, why is it trans and not cis?\nThank you,\nEster",
        "A": "The hydrogen atoms are on the opposite side of both double bonds, so both of them are trans."
    },
    {
        "video_name": "CbgWfnqLJI0",
        "Q": "why no enantiomer at 3:29?",
        "A": "The carbon atoms in the adduct formed at 4:50 are not chiral."
    },
    {
        "video_name": "CbgWfnqLJI0",
        "Q": "at 4:38, when you drew the product, you started with a 5 carbon cyclo diene and then made a 6 carbon cyclo diene structure.... how were you able to add an additional C to the new product made?",
        "A": "The 5-carbon ring is still there. It consists of atoms 1, 2, 3, and 4 of the cyclopentadiene plus the atom in the middle of the product. The new 6-carbon ring consists of atoms 1, 2, 3, and 4 of the cyclopentadiene plus the alkyne carbons of the dimethyl butynedioate."
    },
    {
        "video_name": "zR4OaoiNkA0",
        "Q": "I wonder how borane reduces the carboxylic acid to an alcohol, without reducing the ketone, given that the mechanism has to pass through an aldehyde/ketone (see video at 5:00).",
        "A": "BH\u00e2\u0082\u0083 is a much weaker reducing agent than LiAlH\u00e2\u0082\u0084. But it still reduces carboxylic acids to aldehydes, and it reduces aldehydes to alcohols. However, carboxylic acids are so much more reactive than aldehydes that you can control the conditions and stop the borane reduction at the aldehyde stage."
    },
    {
        "video_name": "zR4OaoiNkA0",
        "Q": "At 1:30, I can understand if the hydride attacks the carbonyl carbon, but if it instead attacks the hydrogen of the alcohol group, it'll form H2 gas, which itself would be pretty dangerous as it might cause an explosion. This was the exact same reason Sal gave in an earlier video where he said that LAH can't be used directly with water because it'll result in an explosion. So won't there be an explosion in this case? Thanks :^)",
        "A": "Yes, the reaction with water is explosive. However, in the LAH reduction of a carboxylic acid, a solution of the acid in an ether solvent is slowly added to a solution or stirred suspension of LAH in the same solvent. The solutions can be cooled and the rate of addition controlled to keep the reaction slow enough that the hydrogen does not explode."
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "Sal said that the acceleration is -9.8m/s^2. if this is talking about the ball going upwards and then coming back down, that means that it first decreases in velocity and the increases when it comes back down, right? Then how does the velocity decreasing part hold strong (time-14:20)? it would be nice if you explained that as well. thank you! :)",
        "A": "The velocity is decreasing the whole time. It starts very positive, begins decreasing immediate, then goes to zero at the top of the arc, then goes negative and then more and more negative. Direction matters for velocity. The speed starts positive, goes to 0, then increases again. Speed is the magnitude of velocity (you can think of it as the absolute value)."
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "At 5:55 Sal says that the displacement is zero when the initial velocity is 19.6m/s. It is counter intuitive to me as to how an object can have a velocity but no displacement. Shouldn't the velocity also be zero at time 0 to start with and reach 19.6 at 1 second? Could someone please explain?",
        "A": "When Sal says initial velocity is 19.6 m/s he means that the object has a state of motion of 19.6 m/s but has covered 0 metres. Why? Because, no time has passed. If say, 1 second passes, the object s displacement would obviously be 19.6. Think of it this way: You have a car at rest and you hit the accelerator suddenly. The speedometer shows some high value whereas you have hardly travelled a metre forward."
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "At 14:11, my understanding is that when delt t = 4 the object is back where it started from. Is this correct?",
        "A": "yup at the ending of the 4th second the object is back in its initial position. that s why we get displacement = 0 at t=4."
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "At 14:30, Velocity is introduced in the position graph, all the vectors are vertical, shouldn't it be tangent to the trajectory?",
        "A": "That plot is only showing the position in one dimension (up/down), so you can only get the velocity (magnitude and direction) of the projectile in that dimension. In this case, the slope of the y vs t graph is giving you the magnitude of Vy and direction (+Vy for up direction or -Vy for down direction)."
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "At 1:31, shouldn't we call the 'displacement\" vs time graph the \"position\" vs time graph? After 2 seconds, the ball is actually going downwards. On the so-called \"displacement\" vs time graph, however, the ball looks like just going 9.8 meters upwards. The graph is confusing, at least to me. Should we correct the name, or is there anything wrong with my thinking?",
        "A": "Displacement vs time is completely fine as long as our delta S is not very very small. When your displacement differential is very small you can shift to position. In displacement vs time graph never misunderstood time axis as a road of the object road is the displacement axis. Now observe it carefully, object is at the halt by 2 sec. and going down between 2 to 3 sec. Good luck"
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "14:11\nvelocity is not decreasing all of the time during the flight. It decreases while the projectile is moving upwards, reaching zero velocity at maximum height. Velocity starts to increase as the projectile travels downwards. In fact, just before it hits the ground the projectile will reach its maximum velocity and will have maximum kinetic energy. Right?",
        "A": "What Sal means by decreasing all of the time during the flight is that the velocity is becoming more negative. The speed of the object does just what you described. The speed is high, slows to zero, and then increases until impact upon the ground. However, because velocity is a vector and changes sign to indicate direction, Sal considers a negative velocity as less than a positive velocity. Thus, the velocity continues to decrease even as the magnitude rises in that negative direction. Hope that helps!"
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "At 12:27, Sal says the ball has a displacement of 14.7m at 3s which is the same as when it was when the ball was going upwards at 1s. How can the displacement of 14.7m have the same sign when the ball is going upwards or downwards? Shouldn't displacement indicate the direction? Is it because it is expressed as the magnitude of displacement?",
        "A": "Sign of displacement would only change if you buried the ball down 14.7m below the earth. The ball in this scenario is simply retracing it s motion once it begins to fall down."
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "@05:54\nHow is it possible for something to be moving at 19.6m/s and haven't moved at all. It means that, it ISN\"T moving at 19.6m/s.\nJust wondering... And thank you for the videos. Been burning the midnight oil for 3 days in a row now. I even read my exam papers in your voice now!",
        "A": "If you throw a ball upward, it will come back to the ground, crashing with some speed, right? But how far is the ball from the ground, when it comes all the way back down? On average, it doesn t seem to have moved. Forces acting on it, changing its speed, but it still moved back to its initial spot."
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "When you indicate the values on the V(t) graph for t=3s and t=4s points (\"11:29\" and \"13:05\") you actually indicates velocity values about -11 m/s (lower than -10) and -21 m/s(lower than -20) accordingly instead of -9.8 and -19.6 values that we have in the table. Is it some inaccuracy in the graph building or accidental small mistake? Thanks in advance.",
        "A": "It s just a small mistake, he got confused thinking the velocity here must be a little less than -20 m/s so he putted the dot a little down the -20, not realizing that the module there is actually greater, not smaller =) Just a little distraction ;)"
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "in 14:50 he said the first line colored blue which is the displacement s he is saying it is velocity",
        "A": "No, it s displacement over time, which is velocity,"
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "At 14:00 all the graphs look just like derivative graphs in calculus, can they be derived by using derivatives instead of going through the whole process that sal uses in the video?",
        "A": "Yes, there s lots of stuff that can be done more easily in physics if you know calculus - that s why Newton (and Leibniz) invented it!"
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "At 14:05, Sal said that the velocity was constantly decreasing? What if we assume our convention to be up(negative) and down(positive)? Wouldn't the velocity-time graph have a positive slope, meaning that the rate of change of velocity is positive? Isn't that wrong?",
        "A": "The slope would be opposite, but so would the definition of what that slope corresponds to in the real world, so the answer would not change."
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "I am a little confused at the term \"negative acceleration\"; I thought that when you slow down whilst still in the positive direction, you would inadvertently begin to decelerate. As in - 9.8 m/s at the point, 3:51.",
        "A": "Negative acceleration doesn t mean slowing down . It means accelerating in the opposite direction of what you defined as positive. If you throw a rock upwards (positive is up), it will suffer a negative acceleration, that firstly cause it to slow down and stop and secondly to accelerate downwards."
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "What was happening at the end when arrows were being drawn starting at 14:49? Were the arrow complying to the rules of the graph or were they just being used as a visual aid? Was it showing instantaneous velocity and instantaneous acceleration? Why was it done using arrows rather than points?",
        "A": "The blue arrows are to show the velocity at that moment and the orange ones are to display the constant acceleration, and yes the arrows were just for visual aid."
    },
    {
        "video_name": "T0zpF_j7Mvo",
        "Q": "why did Sal take initial velocity =19.6 m/s at 10:10. Shouldn't it be 9.8 m/s ?",
        "A": "No. The initial velocity is a result of the force you throw the projectile upwards with. Sal explains this at 1:15, I think. 9.8 m/s will be the velocity(after 1 s) if you release a projectile, with only gravity acting on it. As for the value, he calculated it in an earlier video."
    },
    {
        "video_name": "Xpgsg-fY4DY",
        "Q": "At 7:15, why cant we consider a component of the gravity vector that points toward the center?",
        "A": "At 7:15, the gravity points tangentially to the circle, which is perpendicular to the centripetal direction. Therefore, the component of gravity vector the points towards the center has magnitude of 0."
    },
    {
        "video_name": "Xpgsg-fY4DY",
        "Q": "at 2:54..why do the normal force pointing downward instead of upward?",
        "A": "because the track is exerting a downward force, not an upward force."
    },
    {
        "video_name": "Bdd1Lu5jN08",
        "Q": "At 8:58, the narrator said \"height\" with the ending sound of \"width\". I'm used to saying \"height\" as in \"kite\". Which pronunciation is correct?",
        "A": "The pronunciation of height as in kite is correct."
    },
    {
        "video_name": "CIyAs0bxeoI",
        "Q": "Why does he say at 10:42 'respectively'?",
        "A": "That s b coz on addition of hydrgen(H), NAD becomes NADH and FADH becomes FADH(2)"
    },
    {
        "video_name": "CIyAs0bxeoI",
        "Q": "At 1:27, Did Hank mean to say \"equals energy\" instead of \"plus energy\"? why would you spend energy to make it?",
        "A": "No, he meant that the glucose and oxygen, through cellular respiration, produced 6CO2, 6H2O, and energy."
    },
    {
        "video_name": "CIyAs0bxeoI",
        "Q": "At 3:14, what happened to that hydrogen atom after it left the water molecule?",
        "A": "The hydrogen ions (H+) move down a electrochemical gradient, basically going from high concentration of H+ to low concentration. This produces energy for ADP to bond with a phosphate group to make more ATP."
    },
    {
        "video_name": "CIyAs0bxeoI",
        "Q": "at 5:30 hank said that you need 2 nadh to power cellular respiration , but when your first cellular respiration happens where does the 2 nadh come from?",
        "A": "at 5:10 hank says that in addition to 4 ATPs formed two pyruvates and two NADH are also formed at the end of glycolosis. this is wher the two NADH comes from."
    },
    {
        "video_name": "CIyAs0bxeoI",
        "Q": "At 2:32, I cannot understand about ATP and ADP.\nHow are they different? How are they similar? How come ADP is made from ATP?",
        "A": "ATP stands for adenosine triphosphate. ADP stands for adenosine diphosphate. ATP can be formed from ADP by the addition of a phosphate group, and vice versa."
    },
    {
        "video_name": "CIyAs0bxeoI",
        "Q": "Is the process around 7:45 (when the three carbon chain is converted into a 2 carbon chain which binds with Coenzyme-A to form acetyl coA) oxidative decarboxylation?",
        "A": "It is. And if you are wondering where does the energy go, I advise you to look up what all co-factors participate in the reaction here."
    },
    {
        "video_name": "iEKA0jUstPs",
        "Q": "At 0:50 the H is added to the left carbon in the products, why not the right? is there a rule?",
        "A": "We write H-X from left to right, so he put the H on the left hand carbon. But it is an arbitrary choice. It doesn t matter which carbon gets the H."
    },
    {
        "video_name": "iEKA0jUstPs",
        "Q": "At 08:04, it is said that there is no rearrangement possible. Why ?",
        "A": "Not impossible, just not beneficial. There is no possibility of achieving a more stable carbocation thru rearrangement (only a different secondary) so it does not occur."
    },
    {
        "video_name": "iEKA0jUstPs",
        "Q": "At 9:22 Jay draws a 3D structure for the corbocation.He specifies that it's an sp2 hybridized and has a trigonal planar structure but at the same time he draws the hydrogen with a wedge and the methyl group with a dash which isn't a trigonal planar strucuture, how is that possible?",
        "A": "That s because we are looking at it edge-on instead of from the front. If you look at a triangular slice of pizza edge-on, one corner will be closest to your eye (a wedge), on corner will be furthest away (a dash), and one corner will be half-way between ( a solid line."
    },
    {
        "video_name": "iEKA0jUstPs",
        "Q": "From 3:28 to 4:30, based on Markovnikovs rule, would the tertiary carbocation be the major, and the secondary carbocation be the minor?",
        "A": "I ve understood that Markovnikov s underlines that the carbocation forms always to the most stable Carbon - while the elektrofile (H+) goes where the most hydrogens are. However according to Zaitsev s rule, in elimination you would get Major and Minor (usually Cis-Trans) products. Then again, I m not a master in chemistry yet :)"
    },
    {
        "video_name": "iEKA0jUstPs",
        "Q": "At 11:50, with the last example, would the steric hindrance from the methyl group mean that the mixture would be a bit less than racemic, meaning you would have a major and minor product still?",
        "A": "The carbocation is flat and both sides are equally exposed. If you have trouble seeing this and you have a model kit, it could be helpful to use that \u00e2\u0080\u0093 just remember that carbocations are sp2 hybridized and so you want to make one of the three bonds a double bond to get the geometry right."
    },
    {
        "video_name": "iEKA0jUstPs",
        "Q": "At 6:15, what happened to the original hydrogen that got attached to the right of the double bond? Does the hydride shift not occur with an existing hydrogen on the ring?",
        "A": "The original H atom from H-Cl got attached to C-2 of the ethenyl group and stayed there. It didn t move. The H atom on C-1 of the ring then shifted to C-1 (the positively charged carbon) of the side-chain. So yes, the hydride shift does occur with an existing hydrogen on the ring."
    },
    {
        "video_name": "iEKA0jUstPs",
        "Q": "At 3:28, why is a tertiary carbocation more stable than a secondary one? Does Jay explain this in other video?",
        "A": "Hyper-conjugation. Mentioned in carbocation."
    },
    {
        "video_name": "iEKA0jUstPs",
        "Q": "At 5:19, where did the hydrogen come from?",
        "A": "The hydrogen came from the HCl."
    },
    {
        "video_name": "iEKA0jUstPs",
        "Q": "At 10:44 why does he use words like methyl going way & hydrogen coming, I mean that does he mean that the bond is getting weak & strong respectively?",
        "A": "He is referring to the three dimensional structure of the molecule. The CH\u00e2\u0082\u0083CH\u00e2\u0082\u0082, C, and Cl are all in the plane of the paper. The CH\u00e2\u0082\u0083 is on a dashed bond, so it is furthest from our eye. It is going away from us. The H is on a wedge, so it is closest to our eye. It is coming toward us"
    },
    {
        "video_name": "iEKA0jUstPs",
        "Q": "at about 4:00, isn't there a hydrogen missing from the aromatic ring? Or did he just leave it out?",
        "A": "There is no aromatic ring. Remember that in bond-line structures you do not show the H atoms on C unless you want to draw attention to them. He just didn t bother to show the missing H atoms."
    },
    {
        "video_name": "iEKA0jUstPs",
        "Q": "At 6:56, are there two enantiomer products of the reaction?",
        "A": "No. Because the carbon atom is not chiral. It has a Cl and an ethyl group, but the two ring CH2 groups are equivalent to each other. That gives you only three different groups, while you need four different groups for the carbon to be chiral."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "At 5:00 you named the tert-butyl structure 1,1-dimethylethyl. Why is it not 2-methylpropyl? Does the 1st number have to come at the place where it connects to the chain?\n\nThanks!\nLoving the Videos and the site!\nNels",
        "A": "The longest chain must start with the carbon attached to the ring. The longest chain in a tert-butyl group is only two carbon atoms long (ethyl). Its name must be 1,1-dimethylethyl."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "At 2:22 example 1 can we name it 4(propyl-heptane ) instead of 4(isopropyl-heptane)",
        "A": "In isopropyl it s bonded to the main chain on the middle carbon, in propyl it s bonded on the end. So no you can t just call it propyl."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "At 1:35 sal said and wrote 1-methylethyl but it should be arranged in alphabetically . and should be written as 1-ethylmethyl ?",
        "A": "The rule is first take the longest carbon chain (Ethyl is longer than Methyl) then add substituents. 1-ethylmethyl is incorrect, just a 1-heptapropyl would be incorrect."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "at 5:03 why cant I connect the far left carbon to the far right carbon and consider that chain to be a propyl? and call it 2-methylpropyl?",
        "A": "The bonds pointing downward that have a squiggly line over them are representing that the group is connected to a larger base chain/ring (something with more than 4 carbons) and everything you see is a substituent as a whole. So that carbon on the bottom is connected to a theoretical array of other carbons, and is not actually a methyl (-CH3) group."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "At 01:03 isnt the longest chain in the alkyl molecule made up of 3 carbons rather than 2? Plz help.....",
        "A": "You have to start from the carbon that is attached to the main chain. You can go out only one more carbon from there."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "At 9:30, What's R?",
        "A": "R generally refers to a non-specific side group. For instance, it may be CH3, the rest of a hydrocarbon chain, or any number/combination of atoms. We call it the non-specific R because we are not so interested in what it represents, rather what it is attached to (in this case the CH3)."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "at 3:44 he says it to be 1,1-methyl propyl. Can't it be called 1-dimethyl propyl? Answer me please.",
        "A": "he said 1-methylpropyl it is just one methyl group attached to the propyl"
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "At about 6:00 Is there any logic behind the use of the -am(yl) suffix and the neo- prefix?",
        "A": "Amyl is the common name for a 5-carbon group. Pentyl is the *IUPAC name.* Just as iso means a methyl group on the next-to-last carbon, neo means two methyl groups on the next-to-last carbon."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "At 6:00, why is it 3-mythlbutyl and not 2-mythlbutyl? Isnt it shorter to name it from right to left, or is there a rule to start from the center in case of branched off substituents?",
        "A": "The rule is that the numbering of a side-chain starts with the carbon attached to the main chain numbered as C-1."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "at 6:00, could it also be it be named 2-methylbutyl since you can number the chain so the methyl group can be located on the 2 carbon instead of the 3?",
        "A": "Not as a substituent. Because we re assuming it s attached to a bigger molecule, you start numbering from the carbon that s bonded to the parent chain."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "At 4:46, I don't quite get it... how can we know where to start and which is the main carbon chain? It seems to be propyl..",
        "A": "What he s doing at that time is showing what the groups look like coming off a parent chain. The zig zagged lines is the parent chain, it doesn t matter exactly what it is."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "at 4:48, why couldn`t we name the substituent as a 2-methyl-propyl?\nthanks",
        "A": "Because the longest continuous chain contains only two carbon atoms. So you must name the group as a substituted ethyl group."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "Please help me understand why at 9:00 the compound name isn't 2-cyclobutylbutane, thank you!",
        "A": "IUPAC recommends using the ring as the parent chain over an alkyl chain"
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "Shouldn't the molecule on the top left at 4:01 be called 2-methyl butane?",
        "A": "The squiggly line bit indicates that this group is part of a bigger molecule, but we don t care exactly what it is, because he is just introducing this to you as a group and how you name it."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "At 4:58, shouldn't it be named 1,1-bis methyl ethyl, since it is systematic naming?",
        "A": "1,1-dimethylethyl is the systematic name of the group. If you had two of these groups, you could have a name like 1,1-bis(1,1-dimethylethyl)cyclohexane."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "8:55 why did Jay say its sec butyl. I think it should be tert butyl",
        "A": "Well it is sec butyl not tert butyl. It s sec butyl because it is bonded to the SECOND carbon of the butyl group. It is not tert butyl because that is a straight butyl chain, tert butyl would need the carbon labelled 3 to be bonded to the carbon labelled 1 instead."
    },
    {
        "video_name": "joQd0qVnX4M",
        "Q": "At 12:39, isn't the C is bonded to 2 other Cs, considering Jay said we ignore the ring?",
        "A": "The C is directly bonded to two other carbons. We don\u00e2\u0080\u0099t count the carbons that are not directly attached to it."
    },
    {
        "video_name": "X-m0JDCw6TE",
        "Q": "At 4:51, the last part of the brain the temporal cortex is discussed. Later on, this is called the temporal lobe. Are these terms 'lobe' and 'cortex' are used interchangeably ? or are they referring to different things?",
        "A": "In anatomy, a lobe is a clear anatomical division or extension of an organ (as seen for example in the brain, the lung, liver or the kidney) that can be determined without the use of a microscope at the gross anatomy level. While the cortex (Latin for bark, rind, shell or husk) is the outermost layer of an organ."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "@16:58 he says GTP is guanosine triphosphate, but in our book it is written guanine triphosphate??",
        "A": "They re really both the same thing, you can also say Adenosine Tri-Phosphate or Adenine Tri-Phosphate for ATP. Whichever your teacher favors is probably the best one to use."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "@ 13:02 - \"Every NADH in the ETC produces 3 ATPs\" - in terms of the NET effect, are the 2 NADHs produced during gylcolysis going to NET fewer ATP molecules as they will need to be transported across a membrane to go into the ETC?",
        "A": "No, the NADH that goes to the ETC from the Glycolysis is shunted through the membrane of the mitochondria and there it is used either as a NADH or FADH2 depending on the electron shuffle. But the valve remains the same ( 3 ATP?NADH and 2 ATP/FADH2). This is what my Biology book shows at least."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "how is 3 molecules of carbon dioxide released when only two carbons are lost? 6:51-7:15",
        "A": "No third carbon is released the formation of acetyl CoA from 1 pyruvate"
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "At 3:10 he says one inner membrane is crista and many inner membranes is cristae but one fold in the inner membrane is crista and many folds (plural) are cristae",
        "A": "Hi ff142! The folds of the inner membrane are called cristae."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "At 8:28, Sal says that FAD is oxidized to become FADH2. I thought oxidation meant losing hydrogen atoms, not gaining them (OIL RIG). Could someone explain this to me because it is not making much sense. Thanks.",
        "A": "See its OIL RIG ; Oxidation is loss of electron and gain of proton. Reduction is gain of electron and loss of proton. --Proton can also be considered as hydrogen."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "Where does the oxalicetic acid come from at 6:28",
        "A": "It s just floating around along the area and helps with the reaction."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "What causes some cells to not produce the theoretical 38 ATPs as a result of this cycle? Sal mentions this at 13:51.",
        "A": "Only super efficient cells can produce 38 ATP according to Sal. Some cells may not be so efficient whereas some cells won t have the stipulated amount of glucose to produce 2 pyruvate molecules or maybe this number is just the maximum number and that which can be used for studies and exams."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "At 8:32, Sal says FAD is oxidized to FADH_2 in the Kreb's cycle. Does he mean FAD is REDUCED to FADH_2? In order to reduce FAD, two electrons are transfered along with 2H+ to FADH_2/",
        "A": "you re right. sal meant reduced , not oxidised. NAD and FAD are only reduced, not oxidised."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "Ask a question about krebs / citric acid cycle\nBe as specific as possible!\nRemember, you're asking the entire Khan Academy community for answers.\nMention a time from the video, like \"5:28\", to help people understand what your question is about.\nPlease don't report problems or mistakes here. Send in a problem report instead.",
        "A": "Answer this question"
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "At 3:30, the \"outer compartment.\" Is that also known as the inter-membrane space?",
        "A": "Yes. The outer compartment is also called the inter-membrane space because it s between the inner and outer membranes, hence, the name INTER-membrane space. Hope this helped show you how you were right!"
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "In the image from 15:30 to 17:47 why are there two 6 carbon groups shown (Aconitase and Isocitrate dehydrogenase)?",
        "A": "Between the different reactions that occur during the Krebs cycle, different products are made. It just so happens that citrate is the first step of the citric acid cycle, therefore being the most prominent."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "At around 7:15 - 7:20, Sal mentions how two iterations of the Krebs Cycle would be responsible for oxidizing the entire glucose molecule to six molecules of carbon dioxide. Does this mean that all the glucose that enters our body is eventually turned to carbon dioxide?",
        "A": "Eventually, yes. But only the carbon in the glucose. Other parts of the glucose are still there."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "I read in my textbook that fats can be turned into glycogen in the liver. But, here at 14:41 , Sal said that fats can be turned into glucose in the liver and the annotation correcting him said that they can be turned into energy in the liver. Which one is correct ?",
        "A": "Glycogen a very large molecule made up of a lot of individual glucose molecules linked together. So those two terms are synonymous. In humans, only some fats, and some parts of fats can be converted into glucose, and the rest goes to other molecules. Since glucose can be broken down to create ATP, which is our body s energy currency, in a way all three statements are correct."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "when the acetyl-CoA forms, then does it forms in both of the pyruvates? 89:32",
        "A": "Yes. Each pyruvate becomes an acetyl-CoA. Hope this helps :)"
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "At 8:15, when citric acid is being transformed back to oxaloacetic acid, it says that an ADP is turned into an ATP. How does this happen if there isn't a phosphate on citric acid?",
        "A": "Inorganic phosphate (usually labelled Pi) floating around the cell is added to ADP to form ATP."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "In my biology book it says that in the glycolysis stadium 4 ATP's are made, and not 2 ATP's like mentioned in the previous video. So in the entire glycolysis and kreb cycle there are 6 ATP's made in total and not 4 ATP's like Sal says at 12:21.. Im terribly confused right now, WHAT IS THE CORRECT ANSWER? D:",
        "A": "it is right like sal said. in glycolysis you invest 2 ATP to get 4 ATP. So you have 2 ATP net / Glucose in the glycolysis."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "at 05:40 where do we get the 4 carbon molecule, oxaloacetic acid, from????",
        "A": "If you go to around 7:55 you ll see how the Krebs cycle is literally a cycle. Oxaloacetic acid is really a product of the Krebs cycle that is reused at the end of the cycle in order to keep it going."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "At 4:01 he begins describing pyruvate oxidation saying that the Kreb's cycle often gets credit for this process but he describes it as if it's a separate step. Technically speaking would it be a part of the Kreb's cycle, glycolysis or is it an inbetween step that isn't a part of either?",
        "A": "Pyruvate oxidation is technically its own step called the transition phase. Glycolysis creates two pyruvate, and the Krebs cycle starts with Acetyl-CoA; the missing part in between would be the transition phase."
    },
    {
        "video_name": "juM2ROSLWfw",
        "Q": "At 12:31 I am confused by how we got 10 NADH. Were there two NADHs produced for every one pyruvate? I thought Sal said that each pyruvate produces one NADH.",
        "A": "This is an energy summary portion so the breakdown is: 2 NADH are from glycolisis 2 NADH for pyruvate to acetylCoA 3 NADH per acetylCoA in the cycle so this gets times 2 for 6 6+2+2=10 total"
    },
    {
        "video_name": "CqsYCIjSm9A",
        "Q": "Hi, at minute 7:33, the derivative of r to the minus 2, should be: minus 2 r to the minus 1. But I can not find the button to submit my fix, probably because I'm working from an IPAD.",
        "A": "Sal is taking the Integral of r^-1, What you are trying to do is take the derivative. Also the derivative of r^-2 would be -2r^-3 (You have to subtract 1 from the negative 2, not add one to it)"
    },
    {
        "video_name": "CqsYCIjSm9A",
        "Q": "At 4:45, Sal says that the force on q2 is equal to (kq1q2)/r^2 away from q1, and that we only have to exert the same force magnitude in the opposite direction (towards q1) in order to do work through the distance q2 moves. But then what is holding q2 at the original position in q1's field? Surely there has to be some force to cancel the force pointing away from q1 while q2 is at rest in the field of q1?",
        "A": "Nothing is holding it. It could be traveling away at the time we apply the force that moves it in q1 s direction. Once we apply a force equal but opposite that exerted by q1 it starts moving in q1 s direction."
    },
    {
        "video_name": "CqsYCIjSm9A",
        "Q": "at 5:14 you said the differential work will be negative as we r applying force in the opposite direction of the field then why is it that the total work done comes out to be positive as its also done in opposite direction of the field?",
        "A": "without going into the calculus part... see the external force driving the particle towards the field and the displacement of the particle is in the same direction, hence the work done by the external force is positive (cos0=1)... we can even see it through calculus"
    },
    {
        "video_name": "CqsYCIjSm9A",
        "Q": "at 5:07 sal shows the work done to move the test charge radially inwards by delta r is (-ve) .. but isn't the work done is supposed to be positive? i get the idea that the integral comes as a positive value thus indicating an increase in electric potential energy but when he takes dW shoudn't it be also positive as it is taken inwards ? What a PARADOX !",
        "A": "Like charges repel. You have to push on the moving charge in the same direction that you are moving it. Work = F*d, and if F and d are in same direction work is positive."
    },
    {
        "video_name": "CqsYCIjSm9A",
        "Q": "Around 1:40 Sal said that the field lines are weakened as we get far from the variable charge. Why did he say that although he said in other videos that it is constant and he proved it ?",
        "A": "You are mistaken about what you think you saw in other videos. I bet you are thinking of a video where he found the field from an infinite electric plate. That s pretty different from the field of a point charge."
    },
    {
        "video_name": "CqsYCIjSm9A",
        "Q": "At 9:8 you said it was the amount of work, which was and the calculated relation which was also the potential difference. If it's work it's unit is Joule and if it's potential difference it's unit is volt but you used both for the same relation. Please help me out here",
        "A": "electric potential is work per unit of charge. It s measured in volts, or joules/coulomb (those are the same thing) Electric potential energy is just energy, so it s measured in joules."
    },
    {
        "video_name": "O9RPGJcAfJk",
        "Q": "at 6:06, I thought you use the letter of the group name and ignored what goes before like iso and sec but Khan underlines iso making it seem like you include the iso when determined the order to name the structure?",
        "A": "To determine what order you name the structure, all prefixes are ignored except for iso."
    },
    {
        "video_name": "O9RPGJcAfJk",
        "Q": "3:55 the systematic name is (1-methylethyl). How do I know which one to name first. I would have used the alphabetic order which would be the (1-ethylmethyl).\nThank you very much",
        "A": "The side chain is an ethyl, but then there is also a methyl coming off of the first carbon of the ethyl. That is why it is 1-methylethyl"
    },
    {
        "video_name": "O9RPGJcAfJk",
        "Q": "At 8:15, is there supposed to be a dash between ethyl) and decane in\n5,6-diethyl-5-(1-methylethyl)decane?",
        "A": "No, there is no hyphen between letters and parentheses."
    },
    {
        "video_name": "NYI0bJK64MY",
        "Q": "The resonance structure formed at 8:20, can this also be called acetic anhydride? Thanks :-)",
        "A": "Yes, they are resonance structures contributing to the actual structure of acetic anhydride."
    },
    {
        "video_name": "NYI0bJK64MY",
        "Q": "At 9:40 Sal says that acyl chloride doesn't form a resonance structure because Cl is very electronegative. This satement contradicts the other resonance structures where Oxygen gets a partial positive charge but yet more electronegative than Chlorine and the same with Nitrogen which has the same electronegativity. So how is this explanation valid for not having a resonance?",
        "A": "The lone pair electrons on the Cl are in a different shell than O (n=3 and n=2) this leads to poor interaction of the valence electrons and reduces the stability."
    },
    {
        "video_name": "sZG-zHkGR4U",
        "Q": "In this video, at around 2:22 when the stack of bricks are lifted in the air, we say it has gravitational potential energy and so we use mgh.\nWhat about in the previous video, when the bear was being pushed at an angle? Would the vertical component give it some amount of gravitational potential energy as well? Why don't we take that into consideration?",
        "A": "If you find the upward vertical force then the force will be perpendicular to the displacement(movement) of the bear, thus no work is done. This is why you cant find the vertical component in this case, but the horizontal component which is parallel to the movement of the bear."
    },
    {
        "video_name": "sZG-zHkGR4U",
        "Q": "in 3:08 he says that teddy bear does not do any work at all but if u keep on holding it then it brings the weight down or if im wrong then the teddy bear on lifting the wieght he applies force right then it applies force now i cant be wrong if im still wrong plz explain me if u can",
        "A": "To lift the weight up, work is done because you re applying a force on the weight and the weight travels upwards. But to keep the weight lifted up, you re no longer moving the weight. It s stationary. All you re doing is opposing the force of gravity by applying and equal and opposite force with your hands. But the weight is not moving, so no work is done."
    },
    {
        "video_name": "sZG-zHkGR4U",
        "Q": "At 1:23 when skater hits the wall and lose his energy - it's kind of applying force to the wall, right? and just because of some force opposite - similar to friction and friction itself - the bricks don't move. How do we know what was that force? is it possible to calculate it with the laws we know to this moment?\nEven knowing the work done by bricks is 2500J and that W = FdcosA which will be in this case W=-Fd, we can't calculate because d is 0. How do we find the force?",
        "A": "You need to know how long it took to come to a stop after contacting the wall. Then you can use F*t = m*v Or you need to know how far the skater s center of mass traveled after contacting the wall - while the skater was getting smashed. d is not zero because the skater does not stop instantaneously. Then you can use W/d = F You can make reasonable estimates for either of those items."
    },
    {
        "video_name": "T2ZnxsBWU3E",
        "Q": "At 0:08, where are the wires in the left hand bottom corner coming from and where do they go to?\nWhere is the video about wiring up the sound module to the Arduino?",
        "A": "The sound module gets wired up the same way as the camera using the extra two transistors/resistors in this video. There is no video just for wiring the sound module."
    },
    {
        "video_name": "T2ZnxsBWU3E",
        "Q": "at 2:17, what is a transister?",
        "A": "Its what turns on and off."
    },
    {
        "video_name": "Kq0Er6JBMmc",
        "Q": "I have this same Arduino as in the video, and it came with a 9-volt battery and adapter, but at 2:49, the video states that the Arduino takes 5 volts. 4 volts is a big difference, so why can the Arduino run on 9 volts and not explode or do something else bad?",
        "A": "I think he means it needs a minimum of 5 volts. However, I would send 200 volts through it, it might really explode!"
    },
    {
        "video_name": "Kq0Er6JBMmc",
        "Q": "Wait, so we need a camera, a hairdryer, a taplight, and an alarm clock. Is there something I am missing? And do we need the brands that he shows? And at about 4:30, he starts using alligator clips. Is all the wiring safe for a 7 year old to do?",
        "A": "You should look at the parts list. it is SO helpful."
    },
    {
        "video_name": "QhiVnFvshZg",
        "Q": "What is a hemoglobin? 0:36",
        "A": "Hemoglobin is the thing in your blood that allows the transport of oxygen and gives your blood the red color. Structurally, it has four globin molecules (proteins) together and each of these can carry one oxygen molecule. It will also have a heme-group (iron) which is a transition metal and like other transition metals gives color."
    },
    {
        "video_name": "QhiVnFvshZg",
        "Q": "At 5:03, I was told that the arteries are highly oxygenated, but in the video it isn't. So is it true that arteries carry the blood then AWAY from the heart and Veins carry blood to the Heart?",
        "A": "It is not necessary that arteries carry oxygenated blood and veins carry deoxygenated blood. Exceptions are there such as pulmonary artery carrying deoxygenated blood from the heart to the lungs for purification while pulmonary vein carrying oxygenated blood from lungs back to the heart."
    },
    {
        "video_name": "QhiVnFvshZg",
        "Q": "at 7:21 sal told that the blue coloured inferior venacava IS A VEIN ? but in the last video he told that the blue pipes are artery and carry deoxygenated blood.",
        "A": "Arteries carry blood away from the heart, veins carry blood to the heart. Whether the blood is deoxygenated or not doesn t matter when identifying arteries and veins."
    },
    {
        "video_name": "QhiVnFvshZg",
        "Q": "At 2:55 Sal was explaining how the aviola are (indirectly) connected to our mouth. I was wondering, if one were to have an non-lethal, acute traumatic event to the throat, would that cause the aviola to distend further into the lungs? Or would the connective tissues holding it keep it from doing so?",
        "A": "If you are just referring to physical force on the throat, then no. The connective tissue does keep it in place. If you are talking about increasing the pressure in the throat then it might cause distension in the alveoli."
    },
    {
        "video_name": "QhiVnFvshZg",
        "Q": "At 13:58, the blood completes a circulation in about a minute. Then, how fast does it go?",
        "A": "Blood leaves the heart at a rate of about 5 litres per minute. The cross sectional area of the aortic valve is about 3 or 4 square centimetres. That gives a linear glow rate of about 1.4 metres/second. Blood is a non-Newtonian fluid so all the simple dynamic equations can t be applied with complete confidence, plus in the smaller vessels the rate of flow in the centre and at the edges is significantly different - cells tend to be in the middle of the stream where flow is faster, with fewer at the edges."
    },
    {
        "video_name": "QhiVnFvshZg",
        "Q": "at 04:01, what are those bean like things?",
        "A": "Sorry; misunderstood the question. The bean-like things are the kidneys."
    },
    {
        "video_name": "QhiVnFvshZg",
        "Q": "At 9:28, when Sal says the blood enters the right atrium, what hole is he drawing the arrow on? In other words, what vein does the blood come from?",
        "A": "it is the Vena Cava."
    },
    {
        "video_name": "QhiVnFvshZg",
        "Q": "At 7:26, Sal says that veins are deoxygenated. But in the left hand corner, he labels the arteries as deoxygenated and the veins as oxygen. Which is which?",
        "A": "The pulmonary veins are oxygenated and the pulmonary arteries are deoxygenated. Aside from those, arteries are generally oxygenated and veins are generally deoxygenated. At the time point you specified, he is specifically referring to the Superior vena cava, which carries deoxygenated blood."
    },
    {
        "video_name": "QhiVnFvshZg",
        "Q": "At 5:02, Sal says there might be oxygen. Is there always a little oxygen, or is it occasional?",
        "A": "The only thing I found around 5:02 was that Sal said that the blood is deoxygenated. Well the oxygen is never fully deoxygenated, it always has some amount of oxygen."
    },
    {
        "video_name": "QhiVnFvshZg",
        "Q": "12:00 Where does the aorta pump blood to?",
        "A": "The aorta pumps blood through the systemic circulation; it is the first step before the blood travels throughout the rest of the body."
    },
    {
        "video_name": "QhiVnFvshZg",
        "Q": "At 6:09 Sal says Inferior vana cava what is it i mean what is the function and the difference between superior vena cava and Inferior vena cava.",
        "A": "The superior vena cava ( vena cava means hollow vein , named in ancient times) is the vessel that brings deoxygenated blood to the heart (veins, generally, have deoxygenated blood in them) from the upper part of the body. The inferior vena cava brings deoxygenated blood from the lower part of the body"
    },
    {
        "video_name": "QhiVnFvshZg",
        "Q": "10:29: How does the pulmonary artery turn into a pulmonary vein?",
        "A": "The pulmonary artery branches off into tiny capillaries. These capillaries rejoin to become the pulmonary vein."
    },
    {
        "video_name": "xlJYYM5TWoA",
        "Q": "2:36, Khan solves for v1 and v2, by either multiplying or diving either side by 2, and doesn't end up with 1/4v1 and 4v2?",
        "A": "you confused which one he did with each. he multiplied 1/2v2 by two, and 2v1 by 1/2, changing the R value to 2R for v2 and 1/2R for v1."
    },
    {
        "video_name": "xlJYYM5TWoA",
        "Q": "i did not understand how the heights cancel out, isn't the right side at a higher height relative to the height from the left. I dont understand the concept behind that please help at time approximately at 3:43 to 3:52 its said that h1=h2",
        "A": "The midpoint of the pipe on either side is level in this example. Therefore, h1 = h2 and the terms cancel out when you subtract them from each side. Alternatively, you can give them the value of 0m and the term \u00cf\u0081gh on either side equals 0."
    },
    {
        "video_name": "xlJYYM5TWoA",
        "Q": "in 2:18. does that mean that, v1 times A1 = v2 times A2 (Law of Continuity) is the same as R1 = R2 ? (Incoming Flux = Outgoing Flux)",
        "A": "I think you are right."
    },
    {
        "video_name": "SJ3TKg_3PD8",
        "Q": "At 6:44 you say you can use the Jones reagent to oxidize Phenol; however, it does not have a Hydrogen attached to the Alpha Carbon. How is this?",
        "A": "Ah, but the oxidation of phenols by Jones reagent follows a different mechanism from that for the oxidation of alcohols."
    },
    {
        "video_name": "SJ3TKg_3PD8",
        "Q": "2:02 All the steps I understand - except this one. Why does this happen? What provokes it?",
        "A": "Ethanol is oxidized from an oxidation state of -1 to +1 (4- 5 for the alpha carbon in the alcohol and 4-3 in the alpha carbon in ethanal). The first hydrogen in the alcohol is first removed while leaving its electrons. These electrons then create a double bond with the alpha carbon. This would give the carbon 5 bonds, so the hydrogen is removed maintaining both electrons from the bond. This H- ion then bonds to the NAD+ ring displacing the electrons within the ring. This effectively reduces the NAD+ to NADH."
    },
    {
        "video_name": "-Aj5BTnz-v0",
        "Q": "At 7:12, why exactly can carbon let go of one of the bonds?",
        "A": "That s a property of carbonyl (C=O) groups that you will come across a lot if you study chemistry."
    },
    {
        "video_name": "z8M4EciPpYI",
        "Q": "at 1:16, the name isomer means what? i don't understand",
        "A": "In chemistry, isomers are essentially different arrangements of the same chemical formula."
    },
    {
        "video_name": "z8M4EciPpYI",
        "Q": "6:00 what are the names of the two enantiomers?",
        "A": "(R)-Bromochlorofluoromethane and (S)-bromochlorofluoromethane (left and right structures, respectively)."
    },
    {
        "video_name": "z8M4EciPpYI",
        "Q": "1:56 . if the molecules are structural Isomers , but in a different shapes , are they the same kind of atom or does that make them different ?",
        "A": "The atoms are the same, and the number of each atom in the molecule is the same."
    },
    {
        "video_name": "z8M4EciPpYI",
        "Q": "what did you mean by biologically active eat 6:32?",
        "A": "I didn t hear that"
    },
    {
        "video_name": "X_3QAB3o4Vw",
        "Q": "How long does it take the star to cycle as shown at 4:30?",
        "A": "Different Cepheid variable stars have different periods, ranging from one day to one hundred days. Polaris, a well-known star that happens to be a Cepheid variable, has a period of four days."
    },
    {
        "video_name": "X_3QAB3o4Vw",
        "Q": "2:20- What makes ionized helium and doubly ionized helium still...well, helium, instead of some other element? Do some elements become other elements when ionized?",
        "A": "Until you change the number of protons, you still have helium, regardless of what you do to the number of electrons. That s true for all elements."
    },
    {
        "video_name": "VH0ciDED3wM",
        "Q": "@8:58 when is a methyl shift possible, like under what conditions would a methyl shift occur?",
        "A": "So that a more stable carbocation is formed. If a methyl shift causes the + charge to be moved from the 1\u00c2\u00b0 carbon move to the 2\u00c2\u00b0 or 3\u00c2\u00b0 carbon ; or from the 2\u00c2\u00b0 carbon to 3\u00c2\u00b0 carbon then it will occur"
    },
    {
        "video_name": "Bt6V7D5av9A",
        "Q": "At 5:38, you said that you can go around the circuit either way, but wouldn't the rises and falls be opposite? so do you have to go a certain direction like positive to negative or negative to positive?",
        "A": "No, you get the same answer either way. Try it. You just have to make sure that when you trace the loop, if you go through a resistor in the same direction as you ve defined the current, you call it a voltage drop, and if you go the other way, you call it a voltage increase. Likewise if you trace the loop through a battery from - to + that has to be a voltage increase, and if you go backward through a battery it has to be a decrease. It won t matter which way you go around the loop!"
    },
    {
        "video_name": "Bt6V7D5av9A",
        "Q": "I thought it was called 'Voltage Drop' not 'Voltage Fall' in 4:33.",
        "A": "These terms mean the same thing. For consistency in the video, at 4:33, I probably should have used the term sum of the voltage drops . I think I said falls because that seems more clearly to be the opposite of rises ."
    },
    {
        "video_name": "Bt6V7D5av9A",
        "Q": "At 1:50, why is the voltage rise (-)10 Volts ?",
        "A": "There is always a voltage drop across a resistor. So he said that there is a drop of 10V. Then in other words he said that there is a rise of -10V. So rise of negative voltage is equivalent to drop of 10 V. Its just a different way of saying it."
    },
    {
        "video_name": "ALLSsIDhFdU",
        "Q": "At 0:49, why does intermolecular hydrogen bonding between water molecules weaken the intramolecular O-H bonds?",
        "A": "Bonding can be thought of as sharing electrons. If a molecule has a hydrogen bond then some of the electron density in the intramolecular O-H bond moves into the intermolecular O-H bond\u00c2\u00a7. By necessity, this means that the intramolecular bond has been weakened. \u00c2\u00a7note: The number of electrons isn t changing, just where they hang out ."
    },
    {
        "video_name": "a_Y9wBQ610o",
        "Q": "At 4:13 , So if they have the same number on both sides they have NO gradient ?",
        "A": "Yes exactly, gradients exist when there is a difference between both sides."
    },
    {
        "video_name": "a_Y9wBQ610o",
        "Q": "At 1:33, does the concentration gradient work the opposite way as well, from low to high concentration?",
        "A": "Yes, it can in some cases. This however is not diffusion or passive transport. This is called active transport and does not occur naturally. It requires energy to move against the concentration gradient."
    },
    {
        "video_name": "NdRl1C6Jr5o",
        "Q": "Great video! One question: at 10:31, he points out the alpha carbon in the ring structure. This carbon would have two H atoms bonded to it, which would make it achiral, is that correct? I think my misunderstanding may be based on whether chirality is a prerequisite for the reaction here...Thanks for any help!",
        "A": "Chirality is not a prerequisite for keto-enol tautomerism. All you need is an \u00ce\u00b1-hydrogen."
    },
    {
        "video_name": "NdRl1C6Jr5o",
        "Q": "why has the red carbon in the reaction with a base a negative formal charge (5:19), and in the reaction with acid a positive formal charge (2:49) ? they both have 3 bondings..",
        "A": "No, the carbon at 5:19 has four bondings: (1) to the R group; (2) to the aldehyde group; (3) to the lone pair; and (4) to the remaining (but unshown) H atom."
    },
    {
        "video_name": "NdRl1C6Jr5o",
        "Q": "3:17 Why does the H2O act as a base, rather than a nucleophile?",
        "A": "All Lewis Bases are nucleophiles. All Lewis acids are electrophiles."
    },
    {
        "video_name": "NdRl1C6Jr5o",
        "Q": "2:15 A base shares it's nonbonding electrons with a proton, while a nucleophile will attack an electron-deficient atom other than a proton. So why does water act as a base and not nucleophille ?",
        "A": "he shows the acid-catalyzed mechanism (start with H3O+), so water/hydronium acts as an acid only. the base-catalyzed mechanism would require a base, such as -OH (shown at 5:00)"
    },
    {
        "video_name": "4frpZ4Q0q58",
        "Q": "At 5:20 he says you count the charges per second, which equals current. I heard, however, that current is measured in Coulombs per second. Could you measure it in fundamental charges (or electrons) per second, or should you always convert it to Coulombs first?",
        "A": "The general definition of Current is charge/second. The standard unit of current is Amperes, which is defined as coulombs/second. If you need to express your answer in amperes, then you count the number of fundamental charges going by and convert that to coulombs. For super tiny currents, it is entirely reasonable to say something like, the current is 625 electrons per second. Everybody would understand what that means."
    },
    {
        "video_name": "4frpZ4Q0q58",
        "Q": "At 1:48 ,if the electrons are moving in the left direction then why the direction of the current is opposite the flow of electrons?",
        "A": "The current arrow points (by convention) in the direction positive charge moves. Electrons have a negative charge, so when they move we say the current arrow points in the direction they are coming from."
    },
    {
        "video_name": "4frpZ4Q0q58",
        "Q": "1:40 why would electrons and currents going opposite of each other? either that or I just didn't pay attention.",
        "A": "Hello Andrew, No worries, you are paying attention! There are two ways of looking at current. There is the conventional flow where current flows from positive to negative. We also have electron flow where current flows from negative to positive. This is just a model. Both directions are valid and you will see both in the literature. Most authors and equipment manufacturers assume conventional flow. Regards, APD"
    },
    {
        "video_name": "KfTosrMs5W0",
        "Q": "at 1:16 jay says \"concerted 8 electron mechanism \" what does it mean and in which video does he introduce the word \"concerted\"",
        "A": "A concerted mechanism is one in which all the bond breaking and bond formation occurs at the same time. If you watch the next 30 s of the video, you will see that he uses four electron arrows. Each arrow represents the motion of two electrons, so this is an 8 electron mechanism . Since all the movements happen at the same time, this is a concerted 8 electron mechanism . This may be the first video in which he uses the word concerted , but SN2 displacements and E2 eliminations are also concerted reactions."
    },
    {
        "video_name": "KfTosrMs5W0",
        "Q": "At 11:54 what are the names of the result products?\nThanks",
        "A": "trans-cyclohexane-1,2-diol"
    },
    {
        "video_name": "KfTosrMs5W0",
        "Q": "4:50 since neutral water has a certain amount of hydronium ions can water be used without an acid as a source of h3o+?",
        "A": "It s an acid catalysed reaction and the acid speeds up the reaction. It would proceed much more slowly at neutral pH."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "At 11:11, why are the three pairs of electrons per F atoms in BF3 not counted as lone pairs?",
        "A": "At 11:11, he is talking about the electron pairs around boron. Boron has no lone pairs, but it has three bonding pairs, for a total of six electrons in its valence shell."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "at 3:22, what does he mean about the electron density and the shortening of the sp2 bond? please explain, i am confused...",
        "A": "Electron density refers to the probability of finding an electron in a given volume of space. An s electron is more likely to be found closer to the nucleus than a p electron, so it has a greater electron density closer to the nucleus. As sp\u00c2\u00b2 orbital has \u00e2\u0085\u0093 s character, while an sp\u00c2\u00b3 orbital has \u00c2\u00bc s character, so the electron density in an sp\u00c2\u00b2 orbital is greater closer to the nucleus. I.e. an sp\u00c2\u00b2 bond is shorter than an sp\u00c2\u00b3 bond."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "At 6:42, he says p-orbitals can overlap side by side, but why?\nThey don't overlap directly.",
        "A": "Yes, they do overlap directly. They are much closer together than shown in the diagram."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "From 9:50 onwards structure of BF3 is explained, we see that fluorine has a steric number of 4 does that mean fluorine is SP3 hybridized?",
        "A": "Yes, we would say that F is sp\u00c2\u00b3 hybridized, although we can t prove it, because we can t measure the angles between the lone pairs."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "At 10:56 jay tell us that boron has one unhybridised p orbital.how does he know that?",
        "A": "Every atom has three p orbitals in a given shell. Boron used two of them to form the sp\u00c2\u00b2 hybrid orbitals, so the third p orbital was unhybridized."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "6:40 why does Jay draw lines connecting both sets of the p orbital lobes when only one set of lobes has electrons?",
        "A": "Actually, those are not two sets of orbitals, but each one is a complete orbital. Jay has drawn the electrons both in the upper half, but they can be in any part of the probability cloud of their orbital."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "~11:57 he said that the Boron in the BF3 can accept a pair of electrons and function as a \"Lewis Acid.\" What is a \"Lewis Acid\"?",
        "A": "A Lewis acid is anything that can accept a pair of electrons."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "at 7:20, is there only one pi bond, or would that be showing two pi bonds (One for each overlap)?",
        "A": "There s just one pi bond there. Both the top and bottom are part of the same pi bond."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "At 5:32 why does carbon atom have 2 purple orbitals when it has only one unhybridised electron left?",
        "A": "The purple orbitals are the unhybridized p orbitals. There is one on each carbon, and each each orbital contains the one left-over electron from each carbon atom"
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "at 2:10 when we took the 2s orbital and promoted it and then we left a P orbital unhybridized why didn't we take the 3 valence electrons the 3 in the P orbital and left the unexcited s unhybridized?",
        "A": "If we did that, we would predict three bonds all at 90 \u00c2\u00b0 to each other. We know experimentally that the bond angles must be 120 \u00c2\u00b0, so that can t be what happens."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "How do i know all the distances between the elements, as in the minute 8:00?",
        "A": "If you want an accurate bond length, it is best to look it up. The point Jay is getting across is that a double bond between two carbons is shorter than a single bond between two carbons. That is the important message here."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "I thought a Pi bond created a double bond, but there is also a sigma bond, so why is there not three bonds? Around 5:00 to 8:30, C2H4.",
        "A": "A double bond consists of 1 sigma bond + 1 pi bond. The pi bond looks like it would form two bonds, because the orbitals are above and under the carbon atom and so they overlap twice (above and underneath the c-atom), but this counts as just one pi bond. Is that what you mean?"
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "at 1:16 how did he assume that 2s orbital has only one electron and the other is in p orbital",
        "A": "We know from analysing different molecules that carbon always has 4 bondings when bonding. Therefore there have to be 4 different orbitals with one electron in each of it. The hybridisation modell was mede to explain this phenomenon. (Sorry for my english, I m not a native^^)"
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "At 1:33 the sp2 hybridisation occurs from a state of sp3 hybridisation - surely the starting state should be a normal carbon atom where there are 2 electrons in 2s and 2 electrons in 2p",
        "A": "It does not start from a sp3 hybridisation. If it would, all electrons would have to be drawn at the same height (=energy level) which is not the case. It s just a mather of fact that carbon always has it s valence electrons in four different orbitals when bonding. So this is where our thoughts start."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "at 3:15, he says that there is more P character in the sp2 hybrid orbital than in the sp3 hybrid orbital. but the sp3 hybrid orbital has 75% P character and the sp2 hybrid orbital has 67% P character, then how does the sp2 hybrid orbital have a higher P character? please explain. I think he meant that there is more S character in sp2 than sp3.",
        "A": "Right on! There is more s character in an sp\u00c2\u00b2 orbital than in an sp\u00c2\u00b3 orbital."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "i am still not clear about how can we obtain 2 un-hybridized orbitals for each carbon atom when there are 3 hybrid orbitals present with it? as shown in part 5:32 of his video",
        "A": "Each carbon atom has one unhybridized orbital, but the molecule has two carbon atoms, so the molecule has two unhybridized orbitals that can overlap to form the \u00cf\u0080 bond."
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "at 11:52, what did he say? It can accept an electron pair and function as a ?",
        "A": "Lewis Acid it accepts a proton in this case fluorine"
    },
    {
        "video_name": "ROzkyTgscGg",
        "Q": "At 11:54 , it is explained how BF3 functions as a lewis acid . Are all lewis acid follow the same principal ?",
        "A": "Yes if you mean that they will have an empty orbital that can accept a pair of electrons"
    },
    {
        "video_name": "gluN2wxqES0",
        "Q": "At 2:37, it should be sqrt(81+16) = sqrt(97)",
        "A": "Yes, It should be. They put a box in the bottom right to show the correction."
    },
    {
        "video_name": "gluN2wxqES0",
        "Q": "At 2:58 indicates 9^2 is 36? I am calculating 81?",
        "A": "Sal made a mistake, it is 81 instead of 36."
    },
    {
        "video_name": "gluN2wxqES0",
        "Q": "at 2:54, 9 squared is actually equal to 81, meaning the problem equals the square root of 81 plus 16, giving you a final answer of the square root of 97",
        "A": "Sal made a mistake right at that point. 9 squared equals 81, but he said 36... A pop up appears at that point to admit the mistake. You may see this kind of mistake in few videos. Please follow the procedure and send them a notice. So they can fix it."
    },
    {
        "video_name": "gluN2wxqES0",
        "Q": "At 2:40, Sal says that 9^2 is 36, but it should be 81",
        "A": "This is a known problem in the video. There is a box that pop s up on the video stating the error and the correct solution."
    },
    {
        "video_name": "gluN2wxqES0",
        "Q": "At 3:56 could you not simply the square root?",
        "A": "Sal didn t want to simplify the square root. Its your choice to simplify or not simplify. It doesn t affect the answer."
    },
    {
        "video_name": "QKXeZFwFPS0",
        "Q": "at 5:13, why is the force of tension smaller than gravity, what if you have the assumption that the 3kg box doesn't move?",
        "A": "The force of tension in the rope is smaller than gravity because the block is moving. In order for the block to move downwards there must be a net downwards force in the vertical direction, which means the force of tension must be less than the force of gravity. If the 3kg block was not moving then there would be no net force in the vertical direction. In order to have no net force, the force of tension must be equal to the force of gravity."
    },
    {
        "video_name": "QKXeZFwFPS0",
        "Q": "10:56 we are treating the modulus of vectors since we are equating x and y directions. Then why is the -ve sign appearing? Is it nescessary at all?",
        "A": "Yes it is, but only in the y direction. It is convention that down is negative, that is why the acceleration due to gravity is normally written as -9.8m/s^2. What you are equating here is the magnitude of the acceleration, not the direction of it."
    },
    {
        "video_name": "O_nY1TM2RZM",
        "Q": "When he explains the equation at 5:50, isn't the explanation backwards? Shouldn't the Weber\nfraction be the incremental threshold over the background intensity?",
        "A": "He gave the correct equation in 5:05, he just made an error when reexplaining in 6:02."
    },
    {
        "video_name": "O_nY1TM2RZM",
        "Q": "In class, when it comes to Weber's Law, my teacher says, \"We perceive differences on a logarithmic, not a linear scale.\" (or something like that) What does it mean? I mean, how does it have to do with the linear graph at 7:26 since my teacher uses the word \"logarithmic\"?",
        "A": "ok but originally, did Weber and Fechner talk about linear or logarithmic relationship? Or maybe they assumed that the x axis was logarithmic?"
    },
    {
        "video_name": "O_nY1TM2RZM",
        "Q": "3:30 The sensory neurons that Ronald mentions, are they afferent neurons?\nAlso, what kind of receptor is being used (i.e. position or touch mechanoreceptors)?",
        "A": "Arent all sensory neurons afferent."
    },
    {
        "video_name": "rKoIcgBM4Vg",
        "Q": "does the inner shell get any bigger than 2:10?",
        "A": "The inner shell is largest when there is one electron in it (i.e. H). Once the inner shell has two electrons in it (i.e. when the inner shell electron configuration is 1s2) then it doesn t get any smaller."
    },
    {
        "video_name": "rKoIcgBM4Vg",
        "Q": "At about 8:20, does it have to be the outer shell electron that was lost? I know that it's the easiest to remove but will you always see it that way (that the outermost shell loses the electron to form a cation)?",
        "A": "Any electron could be lost, but then a higher energy electrons will tend to emit a photon and jump down into the hole left behind. Therefore the net result will be as if the outermost electron had been removed."
    },
    {
        "video_name": "rKoIcgBM4Vg",
        "Q": "Something I dont understand is when we're writing down Cl^- (chloride anion), why do we write it as [Ne] 3s^2 3p^5 (10:12) instead of just simply writing the next noble gas, Argon if they should have the same electron configuration (18 electrons). Thanks!",
        "A": "Because it s not a noble gas. It may have the e- configuration of a noble gas but we know that it s a chlorine ion because its protons are same in number. It s not an neutral atom rather it s an ion. Moreover, the atoms of noble never become an ion as they don t need any e-. So you can t write that. :) Again, you made a mistake in writing the e- configuration of Ne. It will be 2s2 2p6."
    },
    {
        "video_name": "rKoIcgBM4Vg",
        "Q": "At 1:10, who decided that we should put elements in the order they are in, and what language do all of the elements come from?",
        "A": "They are in order of increasing number of protons in the nucleus which makes sense, but Dimitri Mendeleev is credited with developing the first modern periodic table. The symbols are mostly from their English name, but some come from Latin."
    },
    {
        "video_name": "rKoIcgBM4Vg",
        "Q": "At 6:43, wouldn't the inner shell electrons also repel the outer shell electrons even more because of Coulomb's Law?",
        "A": "of course they would repel, which will form shield, stopping outer shell electrons to collide with protons this is called shielding effect"
    },
    {
        "video_name": "rKoIcgBM4Vg",
        "Q": "At 10:17 you mention that the Cl anion has a larger atomic radius than the neutral Cl atom. Would this be because the Zeff decreases from + 7 to +6? Or is it just solely because of the electron repulsion?",
        "A": "Cl anion now has one electron more than protons, so it now has fewer protons per electrons. Therefore the pull per electron is weaker, and the ion is larger."
    },
    {
        "video_name": "TitrRpMUt0I",
        "Q": "At 7:20 How does decomposition occurs in deserts",
        "A": "In deserts decomposition is done small bugs and some micro organisms living there"
    },
    {
        "video_name": "TitrRpMUt0I",
        "Q": "Would humans be apex predators at 6:06",
        "A": "Yes, because there s no one to feed on them except decomposers."
    },
    {
        "video_name": "TitrRpMUt0I",
        "Q": "At 4:18 to 4:29, Mr. Khan didn't fully explain why, or what I think so. Is it only 10% that gets transferred, because the animal/plant needs some of the energy? (sorry if it has a bit of weird wording)",
        "A": "Actually the average value of energy transferred is roughly 10%."
    },
    {
        "video_name": "q_Q9C1Ooofc",
        "Q": "At 2:32, It wouldn't be possible to have a 6-ethyldecane since you would start from the end nearest the ethyl group on both. So there are two 5-ethyl groups",
        "A": "So the 5-ethyl was named so because it is closest to the end, and the 6-ethyl is NOT 5-ethyl because you have to keep counting from the end you used to count the first ethyl group? Is that why it is 5,6-diethyldecane rather than 5,5-diethyldecane? I m not sure I explained it well enough..."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "At 1:49 David says that in insulators electrons cannot move freely due to low energy! But what if we provide them the energy required by them to move freely?? Will then the insulator can act as a conductor??",
        "A": "He doesn t say lack of energy. He says they don t have the right energy levels and bands . You can t change that by adding energy. The energy levels are determined by the atomic and molecular structure of the material."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "At 9:15, Aren't the +ves of the rigth rod be more attracted to the -ves of their own than the -ves of the left rod?\n\nThat is, the electrostatic force between +ves & -ves of the same right rod would be greater than that between +ves of the right rod and -ves of the left rod. So, the vector sum would have a net attraction to the -ves of the same rod! Am I wrong?",
        "A": "You are correct that the positive charge on the conducting rod will attract the negative charge, but only if that force of attraction becomes greater than the positive charge on the insulating rod. This is why the induced charge will be less than the amount of excess charge on the insulating rod. Essentially a charge will be induced in the conducting rod to cancel out the E field within the conductor created by the insulating rod."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "How to you put charges to the insulator 4:10, if these charges (electrons) cannot move inside? those charges would stay at the surface, wouldnt they?",
        "A": "Wherever you put them, that s where they will stay."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "you say at 10:47 the negetive charges get out of there .But if u see in a magnet if u put the S pole with the N pole u cant say that they will repel just because there is a south pole at the other end.so what do u mean to say ?",
        "A": "In the magnet example, you can say they will repel. Happens that this force is a function of distance, therefore the repulsion is weaker than the attraction, and the net force is attractive. Coming back to the charges subject, keep in mind that each charge is interacting with every other charge. Starting from a system balanced with an equal number of protons and electrons, you can bring a charged rod and it will interact with every charge of the system. Some electrons will naturally flow away due to repulsion."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "How do we charge something negatively (like charge the metal rod at 7:11) ?",
        "A": "Yes, you are right but we must remember that we can only interfere with electron not protons and add or subtract them as required"
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "At 11:00, David talks about having a \"ground\" for electrons to move to. But wouldn't the \"ground\" (ex: metal) also have delocalized electrons that would repel the electrons from the cylinder, preventing them from moving?",
        "A": "The entire wire is made of metal. The electrons repel each other and space themselves out accordingly. If you connect this piece of metal to another piece of metal, that doesn t change."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "At 1:54 he talks about how insulators do not have the right energy levels for electrons to transfer. I thought electrons transferred based on how many valence electrons were in the atoms. What kind of energy is he talking about anyways? Is it fixed based on the material or fluid based on added electricity or something? sorry if i sound incompetent, this is not my forte",
        "A": "The flow is based on potential difference and yes obviously it would be based on the material depending on material s allowance for the electrons to flow."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "@1:19 he says that the positively charged center can move around in fluids ?\nHow is that possible?\nIsn't the stability of the atom compromised?\nPlease explain!\nThanks in advance!",
        "A": "not quite sure if my answer is relevant but... The atom can become polarized without the stability become compromised. For example in the presence of charged plates, the electron cloud might be drawn one way or the other so that the centre of positive charge is no longer coincident with the negative centre and, therefore, neutralisation of the atom can not take place. The atom aquires a positive and negative pole . does that make sense? help??"
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "Why doesn't what happens with induction at 9:00 happen to touch at 8:00?",
        "A": "You mean the electrons on the neutral rod moving away and the side closer becoming positive as the rods get close? That does happen, he just doesn t draw it. And in BOTH situations, some of the extra electrons on the negatively charged rod will move toward the induced positive side of the neutral rod. He doesn t mention that for either case."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "At 8:40 , why don't the negative charges which all gather on the far side of the second rod move away from each other? Wouldn't they be further apart if some stayed where they were?",
        "A": "Because it is an insulator, and charge cannot move around on an insulator. That s what it means to be an insulator: charge can t move."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "at 8:30, when the rods are closer to each other , how positively charges can move to one side of rod ? Aren't they fixed at a position ?",
        "A": "Yes they are. However, we often say for simplicity of visualizing a scenario that positive charges move as a trick. They don t actually, we just have the negative charges moving the in opposite direction, but its simpler to consider the problem like this."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "At 1:49 David refers to why some materials conduct current more easily than other and mentions bands and energy levels. Can someone give a more detailed description?",
        "A": "he probably is talking about valence electrons and the atoms kinetic energy. So if something has a lot of valence electrons it could affect how well the electrons can move. And if the atoms are moving a lot, it might also affect the way the electrons move around."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "Why the negative charge leave at 10:44? Aren't them attracted by the positive charge in the rod?",
        "A": "Nope, because the no. of electron are more then the no. of proton which leds to more repulsive force then the attractive force hence they are not attracted by positive charge. Also the protons are very much deep inside the atom but the electrons are on the outer side."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "At 10:24, wouldn't the attractive force between charges on the right move the positive charges to the right too?",
        "A": "The positive charges are the nuclei, which can t move."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "at 11:25 why would the electrons in ground want to go back to the rod once we take away the second rod? I mean if ground is large enough wouldn't the electrons just spread out as much as possible even if we take away the second rod and don't cut the wire.",
        "A": "they d want to go back the same reason they wanted to leave."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "the question is at 10:46 he was talking about the grounding effect on a conductor. His example said that the ground took some of the electrons from the conductor so the net charge became positive and it will attract the other conductor. My question is whether it is possible\u00e2\u0080\u00a6",
        "A": "yes it is possible.... have u heard about earthing? it is done in the same way... only -ve charge is taken down by earth because +ve experiences attraction force from the other -ve charged conducter... the excess of electrons get removed, not all of them hence creating a net charge +ve.. similarly it can be done for +ve charge if the other conductor placed closer is +vely charged:)"
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "At 11:50 the second rod is connected to the ground by a wire? Can electrons travel through a wire or any material. If so why can't they come back again to the rod?",
        "A": "yeah electrons can travel through wire and they didn t come back because the wire was cut."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "At 12:27, if the balloon can stick to the ceiling through an excess of electrons that push the electrons in the ceiling further away so that it's more attracted to the protons in the ceiling, then what makes it fall down eventually?",
        "A": "Charge leaks off the balloon into the air, particularly through water molecules in the air."
    },
    {
        "video_name": "ZgDIX2GOaxQ",
        "Q": "At 6:05, the video says positive charge would reside at the outside edge of a conductor, but I though positive charge couldn't move and only negative charge could move in a conductor?",
        "A": "Protons don t move, but protons are not the only way you can have positive charge."
    },
    {
        "video_name": "rkntp3_cZl4",
        "Q": "At 5:39, how do you calculate k?",
        "A": "You don t calculate it, you measure it."
    },
    {
        "video_name": "rkntp3_cZl4",
        "Q": "At 3:45 David says that when voltage decrease the capacitance increases and same at 5:28 but in 1st lecture on capacitance he told \"the only way to change the capacitance of a capacitor is to alter the physical characteristics\" and here iam stucked !!",
        "A": "changing the dielectric is changing the physical characteristic"
    },
    {
        "video_name": "7udSVZ7bDb8",
        "Q": "At 3:05\n\nWhy are hydrocarbons hydrophoc if Carbon is much more eletronegative than Hydrogen?\n\nWouldn't it nab H's electrones and then gain a partial negative chrage transforming it in a polar molecule?",
        "A": "Carbon is not significantly more electronegative than hydrogen. They have roughly the same electronegativities, which is why a carbon-hydrogen bond is nonpolar. Also important to note, the polarity of a molecule depends on its asymmetry, not the polarity of the bonds. Hydrocarbons tend to be nonpolar because they tend to be symmetrical molecules, which is why they are hydrophobic. I hope I helped!"
    },
    {
        "video_name": "7udSVZ7bDb8",
        "Q": "At 1:04 why did Sal call the secod of the secondary structures 'anti-parallel'? They seem parallel to me.",
        "A": "Anti-parallel usually refers to strands that are parallel, but going in the other direction. -----> <----- The above arrows are running anti-parallel"
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "Why at 11:20 Vi-1/2Vi equals to 1/2Vi ?",
        "A": "But what is this 1 ? Is it the Vi because Vi is 5, not 1."
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "At 7:22, when he said the time was still 5 seconds, was that just a mistake or am i missing?",
        "A": "No it is 4 seconds!!"
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "At 6:40, can I calculate the area as a trapezoid? Will that work?",
        "A": "Try it and see. Good way to learn."
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "At 7:22 you said that the base of the triangle is 5. But the base of the triangle is actually 4.",
        "A": "he actually says by mistake that base of the triangle is 5 but afterwards he corrects it. have nice day :)"
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "what is the correlation between distance traveled and area.(5:56)How is it that distance is the area under the curve. I dont get that part. Someone please answer.",
        "A": "Sal explains it in the previous video."
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "Starting from 6:02 Sal breaks the integral of the velocity-time curve into two parts. But wouldn't it be easier if he used the formula A(trapezoid) = 1/2(side 1 + side 2)*height, which in this case would be 1/2(v1 + v2)*deltaT? One step versus two steps.",
        "A": "Yes, but it is easier and more clear in his way, since in physics math is not the biggest challenge"
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "at 9:49, so this formula only works if your initial velocity is not 0 and that it has constant acceleration and that it has that specific shape?",
        "A": "At 9:49 he is still going through the explanation and derivation of the generic equation of S = \u00ce\u0094t((Vf + Vi)/2)"
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "Is there another way to figure out total displacement besides breaking down the area under the line into rectangles and triangles like he does at 8:25?",
        "A": "Another way is to use the formula where S = displacement, u = initial velocity, v = final velocity, A = acceleration and t = change in time: S = ut + (1/2)A(t^2). Yet another way is to calculate the Arithmetic Mean of velocity [i.e (v + u)/2] and S = Arithmetic Mean of velocity x Time"
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "At 6:03, it was said that distance covered was the area under the 'curve'. However it appeared to be a straight line. Is there something I'm missing?",
        "A": "A straight line is just a special case of a curve."
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "At 1:35 why did sal use a triangle before the t?",
        "A": "The \u00e2\u0088\u0086 means change and is frequently used in physics. I think the t stands for time, so what he was trying to say is change in time meaning how much time has passed for a certain process to complete or for a certain event to take place."
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "Sal says we're only operating in one dimension at around 0:38 . How could this be? Isn't one dimension lines? Aren't we normally operating in three dimensions? Or does one dimension have another meaning for it.",
        "A": "In this problem, they are operating in one dimension - all the motion is in a straight line."
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "Is the formula mentioned at 5:40 just the formula for acceleration but algebraically manipulated? Also, are there other applications for average velocity where acceleration does not have to be constant, or does acceleration have to be constant every time one calculates an average velocity?",
        "A": "You can always calculate average velocity but the way you calculate it gets more complicated if the acceleration is not constant."
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "At 12:18 Sal gave us the second formula, but isn't this formula just the area of a trapezoid but with different variables?",
        "A": "It just so happens that the area of the trapezoid in question corresponds to the area under the line on the velocity vs. time graph. Note how the bases are of magnitude v_i and v_f. The height of the trapezoid (distance between bases) is \u00ce\u0094t. Thus, these are just 2 ways of looking at the same problem of finding area under a curve--1st, decompose that area into smaller and easier to calculate parts OR 2nd, use a known area formula if that area is a particular shape (like a trapezoid)."
    },
    {
        "video_name": "MAS6mBRZZXA",
        "Q": "What does Sal do in 11:05 where he says that v(i) and 1/2 v(i) will simplify to 1/2 v(i)?",
        "A": "He is using algebra to simplify the expression in the parentheses. There is: ( v(i) + 1/2 v(f) - 1/2 v(i) ) So he looks at the two v(i) terms and notes that x - 1/2 x = 1/2 x is always true. For example, 10 - (1/2)*10 = (1/2)*10, since 10 - 5 = 5. If you subtract half of something, you are always left with half of the original. So he has: v(i) - 1/2 v(i) = 1/2 v(i) and thus ( v(i) + 1/2 v(f) - 1/2 v(i) ) = 1/2 v(i) + 1/2 v(f)"
    },
    {
        "video_name": "qzUE0ey1yQA",
        "Q": "what will happen if I place a neg- charge in between two positive charge and keep on reducing the distance between them? at 7:00 pm",
        "A": "Net electric field between them is zero. It does not matter if you reduce or augment the distance."
    },
    {
        "video_name": "qzUE0ey1yQA",
        "Q": "At 6:47, how when the direction is right it is +ve , when left it is -ve ?",
        "A": "Great question, The + sign indicates the field ( at some point ) is pointing to the right. The - sign indicates the field ( at some point ) is pointing to the left. This comes in incredibly handy when using mathematics to work out electric field problems. Instead of writing the electrical field at this point is pointing to the right or this electrical field is pointing to the left at this particular point we can express both statements with one of two symbols."
    },
    {
        "video_name": "XtAB9GOfLiM",
        "Q": "min 2:04, why times 2?",
        "A": "It has to go up and down."
    },
    {
        "video_name": "XtAB9GOfLiM",
        "Q": "At 1:45 , Sal takes g=10m/s^2 .If we use sign notations(+ve for up and -ve for down), wouldn't the value of g be -10m/s^2 ?",
        "A": "If you call up positive, then g is negative. If you call down negative, g is positive. You can define direction however you like."
    },
    {
        "video_name": "XtAB9GOfLiM",
        "Q": "At 5:19 , how the component of x specified the speed we are travelling in ?",
        "A": "My video is limited to 4:12. How do you get 5:19?"
    },
    {
        "video_name": "XtAB9GOfLiM",
        "Q": "Why do we use Sv instead of S to find the hangtime at about 2:30? Also how did the video get so clear all of sudden???",
        "A": "if thrown at the same velocity - Brian.ruiz Just to clarify everybody, he actually means same vertical velocity"
    },
    {
        "video_name": "p9NzhA3E-70",
        "Q": "Why don't you draw in the ch2 molecules on the neuman projections? At 8:21, for example, and I think it also occurs in the chair conformation drawing somewhere.",
        "A": "There are not CH2 groups in the molecule, only hydrogen atoms. However, I don t know if there is a reason that the instructor left off the hydrogen atoms."
    },
    {
        "video_name": "vuGpUFjLaYE",
        "Q": "At 7:47, why do we use kilograms instead of grams for the mass in the kinetic energy equation?",
        "A": "Because we expect an answer in joules, and if we use grams then we won t get joules, we ll get kilojoules."
    },
    {
        "video_name": "vuGpUFjLaYE",
        "Q": "At 1:25, there is a usage of the term photoelectron, what does it actually mean?",
        "A": "An electron that is ejected from the metal surface by a photon that hit the surface"
    },
    {
        "video_name": "vuGpUFjLaYE",
        "Q": "At 7:40, how does Sal know the mass of the electron? Is it constant for all atoms or did he derive it in a special way? If he derived it please include how he did it in the answer.",
        "A": "It s the same for every atom. The mass is something that s known, you can look it up in a book."
    },
    {
        "video_name": "0na0xtIHkXA",
        "Q": "at 10:07 Jay mentions the angle between lone pair and F to be 120. Should this not change since the lone pair will push the 2 F in the equatorial position towards each other and away from the lone pair?",
        "A": "The bond angles in these videos are always ideal bond angles if there were no interactions between atoms/lone pairs, they won t be exactly what is observed in a real molecule."
    },
    {
        "video_name": "0na0xtIHkXA",
        "Q": "At 2:02, Sir Jay said that there are only 5 electron clouds. Why are the lone pairs of electrons around each of the Chlorine atoms not counted as electron clouds? Don't they also affect the shape of the molecule? Do correct me if I am wrong.",
        "A": "When we draw structures of molecules such as these we only care about the electron pairs around the central atom because that will tell us the overall shape of the molecule! The electron pairs around chlorine do not change the shape"
    },
    {
        "video_name": "0na0xtIHkXA",
        "Q": "Wait... at 1:03, Jay said that Cl fallows the octet rule..\nWhen I looked at the periodic table, Cl is in the 3rd period.\nI thought that elements in the 3rd period don't fallow the octet rule..... What happend? can someone explain to me why Cl is fallowing the octet rule even though it's in the 3rd period please....?",
        "A": "Cl does follow the octet rule! it is not until the following period when the d orbitals (beginning of transition metals) are used when things are a little different."
    },
    {
        "video_name": "0na0xtIHkXA",
        "Q": "1:18 to 1:25 How the phosphorous have 10 electrons not 8?",
        "A": "Elements in period 3 and on don t have to follow the octet rule"
    },
    {
        "video_name": "KwNe9x0eChs",
        "Q": "At 4:27 and 4:45, plants use the sun's energy to fix the CO2 molecules, or convert them from their gaseous to solid form. However, gases are said to be at a higher energy state than solids(which is why their molecules are far apart ). So shouldn't energy be released during fixation, and not required/used of the sun?",
        "A": "It s not taking gaseous CO2 and turning it into solid CO2, it s breaking the bonds and using the carbon in other molecules (such as glucose), the energy is required for the reactions. Also, even if that was the case, if you have a process that is lowering the energy of a substance the process might still require energy for the process (e.g. freezers)."
    },
    {
        "video_name": "wwjtuZ5vTvc",
        "Q": "If one were to try what Sal suggested at around 3:25, wouldn't the Buoyant force add to the normal force on the scale and cause you to weigh more? Isn't it the same situation as weighing oneself on an upward accelerating elevator?",
        "A": "Remember that normal force corresponds to your weight. Thus if you are in water, not only your weight will decrease, but your normal force as well."
    },
    {
        "video_name": "wwjtuZ5vTvc",
        "Q": "At 7:52 Sal equates the Buoyant force to the Weight of the object.Isn't the buoyant force equal to the loss in weight?Or is it equal to the apparent weight of the object?This is confusing me.Please help.",
        "A": "Sorry for such a delayed response, but i hope my answer helps. Buoyant force or upthrust on an object is equal to the loss in its weight when immersed in a fluid. So, when you subtract this loss in weight from original weight, u get a quantity called apparent weight."
    },
    {
        "video_name": "wwjtuZ5vTvc",
        "Q": "At 6:00 Sal wants to know the volume of object submerged in the fluid tutorial David explains the concept of specific gravity and tells us how to find the volume of object submerged using specific gravity concept which gives the answer in a fraction of a second... i want to know can i use this concept wherever this type of question is being asked....",
        "A": "Yes you can. You will notice that Sal got the same result as David did. Sal just elaborated the process."
    },
    {
        "video_name": "wwjtuZ5vTvc",
        "Q": "At 6:15, Sal said that the buoyant force must equal the weight of the object. But, depending on the block's density, the forces don't have to be equal, right? Or are they simply Newton's Third Law forces?",
        "A": "They must be equal IF the object is floating. If it s sinking, they re not equal! Buoyant force and weight are NOT 3rd law pairs."
    },
    {
        "video_name": "wwjtuZ5vTvc",
        "Q": "in the very end of the video at 9:13, is the density of the balsa wood over the density of water the same as the specific gravity of the balsa wood? In this case because the fluid is water?",
        "A": "The specific gravity of a substance is a dimensionless number and is simply the ratio of the densities of the substance to that of the reference material. So long as the densities for the substance and the reference are in the same units, the units measured are not important. WRT water the specific gravity of a substance is still a dimensionless number."
    },
    {
        "video_name": "wwjtuZ5vTvc",
        "Q": "Why is buoyant force equal to the weight of the object \"on land\" minus the weight of the object in the water (0:40 seconds in video) ? I thought buoyant force was just the F bottom - F top of an object in water. Could someone help explain buoyant force a little better? Thank you!",
        "A": "Maybe You Missed the fact that Weight (mass * Gravitation) = Force (mass * Acceleration)."
    },
    {
        "video_name": "wwjtuZ5vTvc",
        "Q": "at 1:36 what about the force from upwards the water have its weight on the top of the cube",
        "A": "it is not to be considered"
    },
    {
        "video_name": "7ZLjzp7thNI",
        "Q": "at 1:32, correct me if I'm wrong, but isn't the concentration of water 55.56 M, not 1?",
        "A": "We consider H2O(l) as 1 because it is not going to be relevant in finding our pH or pOH. To find pH and pOH we are only worried about aqueous solutions, meaning we throw them into water. So if we are throwing them into water, the Hydrogen Ions begin to mix making more H2O(l) molecules. This is why we excluded H2O (L) in the ICE tables as well."
    },
    {
        "video_name": "AE1pkp-_cKE",
        "Q": "And whats the reason of using mercuric-sulfate? I thought theres a intermediate with the mercuric-sulfate before the enole, which looks like kind of an \"epoxide\" but just with mercuric instead of oxygen, so the H2O can attack from the opposite site to form the enol form, like at 2:45.",
        "A": "Hmm... i d say there s missing a step between the alkyne and the enol-intermediate, because to form the enol you first must have a nucleophilic attack from the alkyne pii-bond, which then makes the carbon able to get attacked from the water."
    },
    {
        "video_name": "05qDIjKevJo",
        "Q": "Thank you so much for this, however, I don't understand the argument from 19:12. The distance between the Sun and the Earth doesn't change, right? So how is it, that more energy from the sun is dissipated in winter just because of the angle?? Am I stupid?",
        "A": "Take a flashlight and point it strait at the ground, the beam of light will produce a circle of light. Now point the flashlight at the ground at an angle, the beam of light is not producing an ellipse. The amount of energy being produced by the flashlight is the same regardless of the angle but the area of the ellipse if greater than the circle so you have the same amount of energy spread out over a larger area so the intensity is less."
    },
    {
        "video_name": "05qDIjKevJo",
        "Q": "1:08 Why does the earth's axial tilt change over large periods of time?",
        "A": "It s a wobble just like you see in a toy top. Things that spin do that if there is any slight imbalance in the forces acting on them. The earth is not a perfect sphere, and the moon and the sun tug on it to get it to line up just the right way. That tugging is similar to the gravitational pull that a top experiences."
    },
    {
        "video_name": "05qDIjKevJo",
        "Q": "Referring back to the part of video 3:52-4:15, my science teacher said the distance is 23.5 degrees, not 23.4 degrees. Who is right? Or are both measurements wrong?",
        "A": "Most likely, both are right. It depends on how accurate the measurements were, when they were taken, things like that."
    },
    {
        "video_name": "05qDIjKevJo",
        "Q": "I found out that the Sahara desert, evry 10,000 years, changes climates from a desert to a tropical forest because of a tilt in the Earth's axis. But at 4:55 Sal says that it takes like 41,000 years to go from 22.1- 24.5. Is this a different kind of tilt?",
        "A": "I don t know that much about that cycle but the Sahara desert has been a desert for millions of years."
    },
    {
        "video_name": "05qDIjKevJo",
        "Q": "at 00:46 that is a circle.You said \"close to a circle\"",
        "A": "The earth s orbital path is not exactly a circle. It s actually slightly elliptical."
    },
    {
        "video_name": "7BgiKyvviyU",
        "Q": "At 6:51, Jay says we should do a lot of excercises in our textbook. Did he recommend any textbook for beginners, or does he mean a general orgchem textbook?",
        "A": "He was pretty well telling u to do your homework. :D"
    },
    {
        "video_name": "7BgiKyvviyU",
        "Q": "Around 8:44 I don\"t understand what does the stability of whats left have to do with the leaving H+?",
        "A": "The more stable a conjugate base is the strong the acid is due to the equilibrium favoring the forward reaction a little bit more. This is relatively speaking. A non organic example are the halides, where the iodine anion is more stable than the flourine anion leading to a difference in the pKa of HF (3.2) and HI (-10). This is carried over to resonance structures, if your conjugate base has a resonance structure it s charge is delocalised and the anion is resonance stabilised, making it s corresponding acid stronger."
    },
    {
        "video_name": "7BgiKyvviyU",
        "Q": "at 0:56 why does the oxygen have a negative charge?",
        "A": "If an oxygen has 1 bond and 3 lone pairs then it has to have a formal charge of -1 A formula for formal charge that may be worth memorising is: Formal charge = valence electrons - lone pair electrons - bonds Plugging in the above numbers: 6 - 6 - 1 = -1 As for how it s formed, you could think that it was acetic acid but now it has lost a H+, which would leave behind a negative charge on the oxygen, as it now has 1 extra electron it otherwise wouldn t have."
    },
    {
        "video_name": "7BgiKyvviyU",
        "Q": "~ 3:48:\nSo the point in drawing a resonance structure is to better describe the real bonding relationship between two atoms in molecules like the one described? If I take the single bonded oxygen in the left molecule for instance, because oxygen has a free atom valence electron count of 6 but in the molecule above has 7 electrons, its formal charge is -1. So the double bond REALLY is dispersed evenly between the two oxygens and the connected carbon in nature. No?",
        "A": "Yes, the double bond really is dispersed evenly between the two oxygens and the connected carbon. Each C-O bond is really a one-and-a-half bond."
    },
    {
        "video_name": "7BgiKyvviyU",
        "Q": "At 7:08 Jay says that we cannot form the resonance structure of ethoxide ion because it already has formed 4 bonds but even the acetate ion has 4 bonds then why does it form a hybrid?",
        "A": "One of the bonds on that carbon in acetate can be pushed into the other oxygen as a lone pair as shown earlier in the video. You can t do that in ethoxide."
    },
    {
        "video_name": "7BgiKyvviyU",
        "Q": "Why at 1:19 does that oxygen have a -1 formal charge? It was my understanding that oxygen's atomic number was 8, and that particular oxygen has 7 electrons. Therefore, 8 - 7 = +1, not -1. Can anyone explain where I'm wrong?",
        "A": "You re right to say that an oxygen atom has 8 electrons, but only 6 of them are valence electrons. Non-valence electrons aren t shown in Lewis structures."
    },
    {
        "video_name": "7BgiKyvviyU",
        "Q": "At 2:33 is that a traditional or official way to draw a partial double bond? Thanks.",
        "A": "That s the traditional way that partial double bond character is shown yes."
    },
    {
        "video_name": "7BgiKyvviyU",
        "Q": "Which atoms on the periodic table can exceed an octet of electrons? at 5:49 he says that carbon can't.... Why is that?",
        "A": "Carbon cannot exceed an octet due to it s position on the periodic table. Carbon is a second row element, and it is a violation to exceed the octet for a second row element. Carbon only has 4 orbitals in its valence shell, each orbital can either form a bond or hold a lone pair. The sum of bonds and lone pairs can not exceed 4. Therefore you cannot have 4 bonds and a lone pair as this equals 5 orbitals and Carbon can only have 4 orbitals. I hope this helps you to understand."
    },
    {
        "video_name": "8wIodo1HD4Y",
        "Q": "At 7:46, Jay says \"decreasing at rate of -1.8\u00c3\u009710^-5 M/s\". Wouldn't that mean N2O5 is increasing at 1.8\u00c3\u009710^-5 M/s?",
        "A": "As the reaction proceeds , the reactant is converted into the product(s). hence in this reaction the conc. of N2O5 is reducing/disappearing."
    },
    {
        "video_name": "8wIodo1HD4Y",
        "Q": "At 7:16, how did you get 1/4?",
        "A": "2:O for every 1:N"
    },
    {
        "video_name": "8wIodo1HD4Y",
        "Q": "At 5:40, how is the product of (9.0x10^-6)4 = 3.6x10^-5?? I understand that (9.0)4=3.6 but why is (10^-6)4=10^-5??",
        "A": "Have you heard of scientific notation before? it s really important to know what it is. 9.0 x 10^-6 means the same thing as 0.0000090 0.0000090 * 4 = 0.000036 You do not apply the *4 separately to both things."
    },
    {
        "video_name": "8wIodo1HD4Y",
        "Q": "At 8:38, if in the question the rate was given in terms of NO2,\nWould the rate, then be:\nChange in Conc. of N02/Change in time= 4*(Change in Conc. of O2/ Change in time)=\n-2(Change in conc. of N2O5/Change in time)?",
        "A": "Yes, that s right. The concentration of NO2 is increasing at four times the rate of increase in concentration of O2 and at twice the rate of the decrease in N2O5."
    },
    {
        "video_name": "9zoS5WGsmpc",
        "Q": "At 8:50 how does FAD reduce into FADH2? It should be just FADH right?",
        "A": "well yes Q is the coenzyme Q, an antioxidant called ubiquinone with vitamin-like properties, and QH2 is the reduced form of the coenzyme. I have to acknowledge that Sal is right in using QH2 as the electron carrier. Although textbooks teach that FADH2 does this, it is not correct. FADH2 remains bound to the Succinate dehydrogenase (complex II) and as such it does not carry the electrons to the next complex III (cytochrome bc1); QH2 does. Therefore, QH2 is the true electron carrier."
    },
    {
        "video_name": "9zoS5WGsmpc",
        "Q": "At 2:09, NAD+ --> NADH. Where does the proton come from? Is it just floating around in the cytosol?",
        "A": "Hmm... well, this might be a bit late, but to explain in the best way I can, there s a lot of H+ (hydrogen ions) that are drifting around. They re the reason why the electron transport chain is needed in the first place, in order to bring in more hydrogen ions against the concentration gradient. The hydrogen ions are very high-energy, so they react with NAD+ to make it become NADH. I m very sorry if that didn t help, I m rather new to this topic myself. I just hope it helped a little. ^^"
    },
    {
        "video_name": "zjIVJh4JLNo",
        "Q": "At about 8:00, Sal is talking about how the salt will dissolve into the water at a higher temperature. Would the salt fall back into crystalline form as the water temperature dropped?",
        "A": "Yes, but only to a very small extent because NaCl is very water soluble."
    },
    {
        "video_name": "zjIVJh4JLNo",
        "Q": "At 3:22 if an element has a + sign on top of it does it add or subtract an electron?",
        "A": "It is positive, so it adds an electron."
    },
    {
        "video_name": "zjIVJh4JLNo",
        "Q": "At around 6:30, he says chlorine instead of chloride. Does it matter? I've always thought one was a compound.",
        "A": "chloride is the negative form of chlorine. It becomes chloride when it forms a bond with another element"
    },
    {
        "video_name": "zjIVJh4JLNo",
        "Q": "At 3:09 Sal says that water is in it's most compact state for water, which i understand but what if water freezes faster than it can crystallize? Does this happen? I'm just curious. And, is that what happens when you go out on the lake and it just froze over but the ice is clear?\nThank you,\nScott",
        "A": "Water cannot freeze whithout crystallizing, so they will always line up and take up more space. Clear ice can be caused by many factors including the purity of the water, movement and air incoorporation into the water, and lack of snowfall during formation."
    },
    {
        "video_name": "zjIVJh4JLNo",
        "Q": "At 6:25, does it matter whether you say chlorine or chloride",
        "A": "Yes, they are not the same thing. Chlorine refers to the element in its unreacted state (Cl\u00e2\u0082\u0082). Chloride refers to it in its reacted state (usually the Cl\u00e2\u0081\u00bb anion state). So, no, the two terms are not interchangeable."
    },
    {
        "video_name": "zjIVJh4JLNo",
        "Q": "at 6:50. How does a liquid break the chemical bond between NaCL?",
        "A": "NaCl(l)\u00e2\u0086\u0092Na+ + Cl-"
    },
    {
        "video_name": "zjIVJh4JLNo",
        "Q": "At 9:30 to the end. Gases by definition cannot stay in the same place unless in a tank. How is gas dissolved into a solvent? I'm assuming it has to be directly released into the solvent? Also how does a gas become soluble when the temperature goes down? Is that in context of keeping the gas in, or putting it in?",
        "A": "Do you ever drink soda? Those bubbles are carbon dioxide that was dissolved in the liquid."
    },
    {
        "video_name": "zjIVJh4JLNo",
        "Q": "At about 2:33, when all the oxygen ions are being attracted to the hydrogen ions, why does that not count as a bond (like in Lewis structures) even though the ions are attracted to each other?",
        "A": "because the bonds are weak and they are attracted by hydrogen bonds which is a different type of bond making it weaker than normal bond. Oxygen becomes slightly negative, while Hydrogen becomes slightly positive."
    },
    {
        "video_name": "zjIVJh4JLNo",
        "Q": "wait what? at 3:30 or so, does he say sodium is larger then chloride? 'it becomes smaller as you go to the right of the table\" dont they get larger? the AMU goes, up, the amount of protons/nuetrons/electrons go up... but they get smaller? im totally confused now.",
        "A": "The element Na is really bigger than Cl (radius size increases from the right to the left on the periodic table remember?). But when we are talking about ions, Cl- is bigger than Na+ cause Cl gets an electron and it\u00c2\u00b4s radius gets bigger than Na+ with minus one electron ;)"
    },
    {
        "video_name": "zjIVJh4JLNo",
        "Q": "At 11:09 when Sal talks about shaking a bottle.. does shaking ONLY physically move the molecules around or does it also giving them kinetic energy in the same way increasing temperature would do?",
        "A": "You are also increasing kinetic energy, but so small an amount. This will become heat, but again so tiny it s fair to assume it s an immeasurable amount."
    },
    {
        "video_name": "AJGrSWxPrd8",
        "Q": "At about 5:00, Jay says that anti-coplanar/anti-periplanar conformation is necessary for an E2 reaction. Why is this the case?",
        "A": "That is because the developing p orbitals must be parallel in order to form the \u00cf\u0080 bond of the alkene."
    },
    {
        "video_name": "tQcB9BLUoVI",
        "Q": "At 2:56, why and how does the particle change its momentum after bouncing off the wall of the container ?",
        "A": "The momentum changes because the particle is suffering an impulse as it collides. At this model, every collision is elastic, which makes sense, since in the macroscopical world non-elastic collisions lose energy in the form of heat and sound, two things that a microscopical collision can t create, by definition. Thus, when the particle hits the wall, it simply bounces back."
    },
    {
        "video_name": "tQcB9BLUoVI",
        "Q": "at 9:22 you said that particle wont even hit the wall . hows that possible ,it always has velocity",
        "A": "He says that it won t hit the wall in the same amount of time, meaning that it requires more time for the particle to get to the other wall when the walls are farther apart."
    },
    {
        "video_name": "tQcB9BLUoVI",
        "Q": "This may sound very weird but @1:33, what is the meaning of elastic bumps and no loss of momentum. How do molecules of gas have momentum? Isn't that for solid objects only?",
        "A": "you could think about the gas molecule as being a solid object . Certainly, this is how they are visualised in the kinetic theory of gases. they have mass and velocity, so they have momentum. Equally, their collisions are assumed to be perfectly elastic, so no KE is lost during the collision. OK??"
    },
    {
        "video_name": "tQcB9BLUoVI",
        "Q": "At 1:12, what does the word ricocheting mean in that context?",
        "A": "it means bouncing off . Like a bullet bouncing off a rock or wall."
    },
    {
        "video_name": "tQcB9BLUoVI",
        "Q": "At 1:13 what is the meaning of 'ricocheting'?",
        "A": "It means to beat a surface and continue moving in a different direction"
    },
    {
        "video_name": "tQcB9BLUoVI",
        "Q": "At 0:25 where Sal talks about gases being compressible and liquids incompressible, what does that mean? I mean the incompressible part.",
        "A": "In classical mechanics, liquids are usually considered incompressible for sake of simplicity. Liquids have very high resistances to pressure. Incompressible is the opposite of compressible."
    },
    {
        "video_name": "tQcB9BLUoVI",
        "Q": "3:13 I don't understand why he speaks of a change in time and uses delta. Can someone give an example perhaps? What is the difference between t and delta t, and p and delta p?",
        "A": "If you move from A to B, \u00ce\u0094v = vB - vA vA === velocity at A vB === velocity at B"
    },
    {
        "video_name": "-vp1X7_u3KU",
        "Q": "At ~1:24, it is mentioned that there are different types of \"taste buds\" (i.e. fungiform). Aren't those papillae? From my understanding, papillae and taste buds are different, as taste buds are contained within papillae. Can somebody clarify or correct me if I am wrong?",
        "A": "You are right. Taste buds are the common name, papillae is the term used by science folks. Like big leg bone versus femur . People love words, they are tasty!"
    },
    {
        "video_name": "-vp1X7_u3KU",
        "Q": "At around 12:00 Ronald mentions NaCl receptors on \"sweet\" cells. Is there anything that this is specifically referring to (perhaps an experiment or some kind of physiological condition)? Or is this merely a suggestion?",
        "A": "It s possible to add conditional activation of part of the DNA that encodes for a salty receptor in transgenetic animal models. Using varying techniques to measure brain activity (e.g. EEG), you could see which areas of the brain are activated that would traditional respond to sweet cells with a salty diet."
    },
    {
        "video_name": "JxOMm24lSR4",
        "Q": "At 2:50, I'm not sure there is a need to consider the integral. The bounds of the integral go from 0 to t, so it doesn't seem valid to evaluate this integral for the current value given for times less than zero.",
        "A": "You are correct. I used the integral with lower limit of t=0 to evaluate what happens before t=0. For the time before 0 I should have used the general version of the integral expression where the lower limit is t= -infinity. There is a more about this in the previous video Capacitor i-v equation at 5:00."
    },
    {
        "video_name": "9R9vPcPyUwo",
        "Q": "At 0:30, how expensive were DVD players when the DVD technology was just new?",
        "A": "Originally, they cost... ahem $700 to $1,200. Yeah. That much. Then I suspect the price drastically went down."
    },
    {
        "video_name": "4tcLzVJdxZs",
        "Q": "I didn't catch why we have 2 moles of electrons at 5:35?",
        "A": "Zn(s) loses 2 electrons when it is oxidized to become Zn2+ (aqueous). Those 2 electrons travel the wire to the cathode and the Copper ions (Cu 2-) in solution pick them up (get reduced) to copper solid. So for every mole of Zn(s) that s oxidized, 2 moles of electrons are stripped to travel the wire and reduce one mole of copper ions."
    },
    {
        "video_name": "b_qDLacdkFg",
        "Q": "At 5:26 while showing the stereochemistry of the product he shows the CH3 going away from us but while making the structure he shows it in the plane. Is it a mistake or is it like that only??",
        "A": "I think it s because it doesn t really matter if the CH3 is pointing back as using wedges and dashes for lines is used to indicate stereochemistry. In this case the carbon attached to that CH3 is not chiral (as there are two CH3 s attached to it) so using dotted line or normal line makes no difference"
    },
    {
        "video_name": "b_qDLacdkFg",
        "Q": "At 6:20, why are the methyl groups which are shown to have stereochemistry going into the page or coming out of the page on the left side not represented as such on the right side?",
        "A": "It doesn t matter for the left side because it is not a chiral carbon."
    },
    {
        "video_name": "b_qDLacdkFg",
        "Q": "At 6:34 you say that the products are enantiomers. How do you know the product isn't a racemic mixture like in hydration and hydrohalogenation? Does it have to do with the fact that it is a concerted reaction and doesn't utilize a carbocation?",
        "A": "This reaction will produce a racemic mixture, which is defined as having equal amounts of each enantiomer. In other words, you can t have a racemic mixture without having enantiomers!"
    },
    {
        "video_name": "_-I3HdmyYfE",
        "Q": "At 4:05, wouldn't the stereochemistry of the product be inverted because of the Sn2 mechanism?",
        "A": "Yes, it would, but you can t see it, because the carbon being attacked at 4:05 is not chiral."
    },
    {
        "video_name": "_-I3HdmyYfE",
        "Q": "at 4:20, the alkylated species sholud have an inverted configuration because of Sn2 mechanism, rite?",
        "A": "I was going to ask the same the thing. Yes, I believe that s an error. Backside attack always results in inversion. The lone hydrogen drawn with slashes/dashes should swing around & end up where the leaving group (X) started out."
    },
    {
        "video_name": "8G0RkfeKfBg",
        "Q": "At 4:17, instead of removing CO2, can the acid undergo dehydration and form an anhydride? If yes then which is the major product, the anhydride or ketone?",
        "A": "You could form an anhydride, but the driving force for the reaction is the exceptional stability of CO\u00e2\u0082\u0082. The anhydride never gets a chance to form."
    },
    {
        "video_name": "oTjTXS40pqs",
        "Q": "The phenomenon at around 1:15 , is it called the superposition (when waves overlap each other) ?",
        "A": "Yes that s the superposition principle (for wave interference in this case). When you have two or more waves in the same space you can take the sum of their individual amplitudes at each point to construct a net resulting wave just like in the video."
    },
    {
        "video_name": "oTjTXS40pqs",
        "Q": "at 13:05ish, how do you use the path length difference if you have a different kind of diagram? such as two waves perpendicular to each other? would the Pythagorean Theorum be the delta x?",
        "A": "Yes, I am pretty sure that is correct although there are rarely real life situations where that would happen with sound. Just make sure you angle the final wave outward not inward."
    },
    {
        "video_name": "oTjTXS40pqs",
        "Q": "At around 1:01 David talks about a cosine wave? I looked at the respective answer that Jishnu V Nair gave but how is trigonometry associated with this? In Physics?",
        "A": "This is the same as asking how is algebra associated with physics, or how is calculus associated with physics. Math is a tool that pervades physics."
    },
    {
        "video_name": "oTjTXS40pqs",
        "Q": "What does David mean when he talks about a Consine Wave? 1:01\nWhere can I learn more about it?",
        "A": "Cosine wave: its a 2D representation of 2 varying quantities. It has a phase difference of pi/2 when compared to Sine wave. At a given instant, Cosine wave lags/ leads the sine wave by pi/2. You can know more in detail (formulae, representation etc.) about it in the internet under the topics: Trignometry, sinusoidal waveforms. Cosine and Sine waves are pretty important mathematical representations finding applications in various subjects."
    },
    {
        "video_name": "Mx7KM-k2MMo",
        "Q": "Ref 10:02- If OH(-) trumps F(-) in terms of nucleophilicity in a protic solvent, then why isn't the same logic applicable for OH(-) and I(-)?",
        "A": "Remember, in protic solvents, the more electronegative anions are more solvated (blocked) by the protons in protic solvents. This makes the more electronegative anions worse nucleophiles in protic solvents. Fluoride is more electronegative than the oxygen in hydroxide, therefore OH->F-. Likewise, oxygen is more electronegative than iodine and I->OH-."
    },
    {
        "video_name": "Mx7KM-k2MMo",
        "Q": "At 4:31 how could F- be a strong base than I- as i understand basicity means giving away electrons but F- is very electronegative? .. i feel that there is a contradiction",
        "A": "HI is a stronger acid than HF."
    },
    {
        "video_name": "Mx7KM-k2MMo",
        "Q": "@12:00 How about Basicity for those last two examples? which one is a better base? Thanks!",
        "A": "oh(-)..i think"
    },
    {
        "video_name": "Mx7KM-k2MMo",
        "Q": "At 4:03 on picture fluorine accept one electron and became fluoride, but text say fluorine swipe one electron away?",
        "A": "They mean the same thing. To swipe means to steal or to take away. Both mean that the F atom gets an extra electron."
    },
    {
        "video_name": "MS7x2hDEhrw",
        "Q": "at 02:53, he says that there was signifigant life. how long ago was this?",
        "A": "In the Vendian Period (about 650-542 million years ago), there was the Ediacara Fauna , the first large multi-cellular soft-bodied organisms."
    },
    {
        "video_name": "MS7x2hDEhrw",
        "Q": "Sal talks about how ozone begins to be created in the atmosphere so that it stops uv rays from coming in. 2:35. Doesn't't the ionosphere do this also? It was created before the ozone layer, it is made from the earth's core. Is it just that it is not strong enough, or is there another reason?",
        "A": "The ionosphere doesn t actually stop UV radiation. You must have read something wrong about the ionosphere. Meteors burn down in the ionosphere (maybe fully, or maybe partially) but it doesn t stop UV radiation. That s why animals didn t come out of the sea until the ozone layer was formed."
    },
    {
        "video_name": "MS7x2hDEhrw",
        "Q": "At 5:05, Sal talks about the cambrian period.\nI've also heard about the precambrian period, but he didn't talk about it. So I'm wondering, when was the precambrian period?",
        "A": "The precambrian period is the time from the beginning of Earth (4.6 billion years ago) to the beginning of the cambrian period, which was around 540 million years. This is about 88% of geologic time up until now."
    },
    {
        "video_name": "MS7x2hDEhrw",
        "Q": "3:20 why did it say that animals were there first? I know that algae was first...",
        "A": "Well algae was first in watery area but bacteria is considered life and life in animal ways."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 4:51, a movement to the north is 5 km in 1 hours.\nIf the road (toward the north) is like 's', so movement toward the north is 5km but another rest amount of movements (other direction such as east, west) is long, long..,\nso Shantanu ran very very fast.., do we have the value of evaluating his(?) average speed??\n\nthank you for reading this article and sorry for my English ability.",
        "A": "If you had a function which described his motion, it would be possible to calculate his average speed."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "at around 2:17 you said that the symbol for displacement was s with an arrow on top but my teacher told me that the symbol for displacement is delta d with an arrow on top. Why did you use d with an arrow on top? Which one is considered right?",
        "A": "The letters don t matter. Focus on the concepts."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "36 seconds in a hour?? at 9:00",
        "A": "he said 3600 seconds in an hour 60* 60 = 100(6*6)=100*36= 3,600. I think he worded it as thirty six hundreds which was probably where you got that, but it is the same 1s 3600"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "2:25 speaking of why use an S for displacement.... perhaps its for speed plus a direction (thus the arrow on top)",
        "A": "speed in a particular direction is the definition for velocity but good try :)"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 2:10, you said that displacement was written as S with an arrow on the top but in the article it says that x with a triangle before it is used to represent displacement. So, which one should I use?",
        "A": "S with a arrow above i is always displacement and s without any line is direction"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 11:23, Sal put an equal sign saying that Shantanu traveled 1.39 meters per second, but he rounded that number. Shouldn't he put an approximation sign instead?",
        "A": "no. All physical measurements are rounded to some extent or other. WE would never be able to use the equal sign. Instead scientists use significant figures to indicate the appropriate degree of precision. Sal has vids about it."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 3:35, you wrote r= distance over time as d over t, but wouldn't you use a different letter for distance for just the reason you use S for displacement?",
        "A": "Typically yes, but it is much easier to explain a concept using letters which make sense."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "at the end 11:36, if velocity is direction and speed, shouldn't the answer be 1.39m/s North?",
        "A": "You are right. Usually we dont worry about direction unless it has been asked for in the quesiton. Only magnitude"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "1:50\nis displacement also delta y?\nif not the what is delta y",
        "A": "displacement can also be delta y, depending on what dimension you are working in. If you dealing with the y direction, then displacement would be delta y. If you are working in the x direction, it would be delta x."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At around 2:20, Sal explains that D is used to indicate a derivative. I'm just starting Calculus, but we've always used f'(x) to represent the derivative of f(x). Am I missing something or might there be another explanation?",
        "A": "D is described as the function that differentiates some other function. It s usually used with differential equations. D of f(x) is f (x)"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 7:17, why multiply by 1000?",
        "A": "Because you want to convert to meters from kilometers. One kilometers is equivalent to 1000 meters"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 0:15 he drew an arrow over the S, but wasn't the arrow used only for vectors? That S is a scalar so it hasn't any direction, then why does it get an arrow?",
        "A": "s is displacement, and it is a vector."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 11:35 you gave the scalar 1.39 meters/second. I was wondering if the vector would be 1.39 meters/second north?",
        "A": "Yes, you can call it that. But, we normally choose a 1D/2D/3D axis system., as per requirement. So if the north south direction is taken as x axis, you would say 1.39 i m/s"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 2:05, you said the reason why \"S\" is used instead of \"d\" is because it will be used for something completely different in calculus. But at 3:01, you just used \"d\" for distance. Can someone please explain why ?",
        "A": "Hi Jeremy, This is just a way of shortening the word, distance. Displacement was shortened to an s because that is its official term, whereas there isn t really one for distance. I hope this makes sense(?)"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 1:56 isn't distance/displacement supposed to be x? That's how my teacher explained it to me, at least.",
        "A": "Letters don t matter. Focus on the concepts."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "For this situation, we are assuming that his velocity was constant within the hour he was travelling. (3:46) Would the solution be different if his velocity was changing?",
        "A": "no since the overall time he takes does not change nether does the average velocity"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 12:40 pm I will like to know how to you would calculate the average velocity when given two distances without direction.",
        "A": "You can t. You need to have need to have two distances with directions and two times. So you add two displacements on the numerator and divide it by total of two times from where the rest at 0 and the final time. Therefore you get the average velocity. I hope that helps!"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 3:03, why is \"d\" used for distance if it also stands for something in calculus? I thought that another letter had to be used because of that.",
        "A": "t does not matters what letter u choose to represent distance.but for displacement u have to put an arrow above that letter."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "AT 3:03, you use \"d\" as distance. Couldn't distance and derivative get mixed up since they share \"d\"",
        "A": "They certainly could. However, since this is an introductory video to basic mechanics, it isn t too important - it s mainly to get the intuition behind the equation."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "at 4:44, Sal mentions the letter Delta. What does the inverted Delta mean?",
        "A": "Inverted delta is the vector differential operator (del) del = d/dx + d/dy + d/dz"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "Is there a specific difference between Time and Change in Time? 4:16",
        "A": "consider the equation for speed; = distance travelled / time taken or change in distance / change in time ok?"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 5:11, Sal said we treat 5km/1 hour like a fraction. Shouldn't it be treated as a ratio?",
        "A": "What is the difference?"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 3:25, Sal mentions whether velocity, displacement, rate and distance are vector or scalar quantities. (I know which one is which! :D) But he did not mention whether time is vector or scalar. *Which one is it??* Thanks if you answered!!",
        "A": "Time is a scalar. This is because you cannot say that I moved 5 minutes left . To be a vector quantity it usually needs to able to be represented in 3 dimensional space."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 2:11, he briefly talks about the derivative operator. Could someone please tell me more about that?",
        "A": "Watch sal s vids on calculus."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "at 7:35 why do the km get cancelled out?",
        "A": "Because if you multiply the two fractions, the km/km divides out and becomes 1."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "Is displacement the shortest distance from a body's initial point to its final point, or is it distance with a direction, like Sal says at 0:54?",
        "A": "You are absolutely correct. displacement is the smallest path form the initial point to the final point."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 6:30, he said something about magnitude (the one circled in green). What is this \"magnitude\"?",
        "A": "magnitude means size... eg The magnitude of my speed is 24 m/s A vector has both magnitude (represented by the length of the line) and direction (given by the direction that the arrow points)"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 3:29, if derivative operators is d, then why is d being used for distance?",
        "A": "Ah the joys of using different notations for the same thing. The derivative operator is usually \u00ce\u0094 or \u00ce\u00b4 (greek capital and lower case delta). The delta is the greek equivalent of d which means that when people have limited ability to use the green letters, or are just using shorthand they use d instead of the greek d. It is I admit not ideal if you are using d for your distance as well, which is why s is frequently used as a measure of distance, or x sometimes."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "6:00 I have a problem with Sal stating that the rate is 5km/hr, because the problem stated displacement (5km north) not total distance, which could be any number. For example lets say the rouad was a winding road the distance might be 10km, but the displacement would be 5km. Average speed cannot be calculate when given a displacement.",
        "A": "You re absolutely right. In this problem, I think Sal is assuming a perfectly linear path (the math right now seems to assume 1-dimensional motion). If it were a winding road, you d have to find the true distance traveled, which would amount to integrating the speed with respect to time for the duration of the trip. Distance = Integral(speed)*dt He s probably sparing us the calculus in this simple example."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "at 2:05, Sal Khan explained why we use (s) instead of (d). I do not know too much about derivatives, but why wouldn't (d) work? Because wouldn't you put an arrow above the variable to show it is a vector quantity? Or is an arrow also put above the (d) for derivative?",
        "A": "Hi, It is really just for the sake of simplicity. Even with the arrow, it would be too complicated and people would get confused."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "Why does he say d/t on 3:29 instead of s without an arrow",
        "A": "s is usually used for displacement, so writing s without an arrow is technically incorrect. You must write d without an arrow as that is the scalar quantity."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "11:11, So the answer for average velocity is 1.39 m/s, what would the average velocity be if it went back to it's starting point at the same distance and time?",
        "A": "If there s no displacement, the average velocity was zero"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 11:32, Sal put 5000/3600=1.39. However, he rounded the number, so shouldn't there be that rounded equal thing (squiggly equal sign)?",
        "A": "no, because if we did that we would always use the squiggly thing in physics, since every measurement is rounded in some way. Instead we use the concept of significant figures."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 11:35, shouldn't he have written 1.39 meters/second NORTH? Because doesn't that indicate the difference between speed and velocity?",
        "A": "At 11:35, he does not write 1.39 meters/second NORTH,because he mentioned around 6:06 that he was gonna figure out the average SPEED not VELOCITY."
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "What is the basic difference between velocity and speed, as mentioned in the video at 3:41?",
        "A": "velocity is the speed in a given direction and speed does not have direction"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 8:55, Sal mentions \"an intuitive gut check.\" Any explanations? :P",
        "A": "Knowing that an inch is WAY smaller than a mile... If you were to drive to your friend s house and, for whatever reason, felt the need to know how many inches you had driven you could calculate it. But by intuitive gut check he means you take the answer that you calculated and ask if that seems correct . If in the above situation you d gotten 2 inches, you d intuitively know that you HAD to have driven far more than that. It s just about looking at your answer and thinking Hm. Is this logically even possible?"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 4:16, Khan mentions delta, which means \"change in {something}.\" Can someone tell me what \"change in...\" means? For example, sometimes people say the slope is just x/y but in upper-level math, people say that it's change in x/change in y. What does adding \"change in\" mean?",
        "A": "Slope is not x/y, it is change in x over change in y How do you calculate slope of a line between (5,2) and (20, 5) You do (5-2)/(20-5) That s change in x over change in y"
    },
    {
        "video_name": "oRKxmXwLvUU",
        "Q": "At 9:15, I don't quite understand how the dimensional analysis works, can someone please help explain?",
        "A": "Units behave like algebraic quantities. Only like units can add together: 1.00m + 54 cm = ? 1.00m +0.54m = 1.54m This is just like 1x +54 y = ? -> cannot be simplified 1x+.54x=1.54x Similarly, products of units behave the same way 2.00m / 1.00s = 2.00 m/s Because of this, units can cancel 2.00m/s * 4.00s = 8.00m Compare this to 2x/y * 4 y = 8x"
    },
    {
        "video_name": "Rr7LhdSKMxY",
        "Q": "At 0:45 you said that Electronegativity is specifically for covalent bonds but why did you use Sodium and Chlorine as an example? Sodium and Chlorine are bonded ionically not covalently.",
        "A": "Whether something is considered covalent or ionic depends on the difference in electronegativity between the two atoms. If the electronegativity difference is so large instead of the electrons being shared we say they are transferred to the more electronegative atom."
    },
    {
        "video_name": "Rr7LhdSKMxY",
        "Q": "At 4:48, Sal says that Oxygen atoms are more -ve and Hydrogen are more +ve.\nIs that why Water is Polar?",
        "A": "That s correct. In water, the H atoms are positive and the O atom is negative. The water molecule as a whole has a positive end and a negative end, so water is polar."
    },
    {
        "video_name": "Rr7LhdSKMxY",
        "Q": "8:09 but didnt that compensate with the fact that more protons mean more attraction?",
        "A": "As you go down the group the shielding from the growing number of electrons increases. Therefore, although the charge in the nucleus is increasing, its effects are not felt so much. Furthermore, the outer electrons are further away from the nucleus as you descend down the group, and this also reduces the attractive force that they feel from the nucleus."
    },
    {
        "video_name": "Rr7LhdSKMxY",
        "Q": "at 7:00 Sal writes \"more electronegative\". If he had written more electron affinity would that be true as well? Do those two trends mirror each other on the periodic table?",
        "A": "They are of course related. The electron affinity is defined by the energy that is released when an electron is added to a neutral atom. It is a specific property of an element. For the electronegativity different definitions exist. Electronegativity describes mostly how much atoms attract electrons in a bond."
    },
    {
        "video_name": "Rr7LhdSKMxY",
        "Q": "Is there anything called Electropostivity? this question struck my brain at 6:30.",
        "A": "No, electropositivity does not exist."
    },
    {
        "video_name": "Rr7LhdSKMxY",
        "Q": "8:04 Doesn't an electron far away from the nucleus have more potential energy, therefore being stronger attracted to the nucleus?",
        "A": "Yes more PE, no not more strongly attracted. Not sure where you got that link."
    },
    {
        "video_name": "Rr7LhdSKMxY",
        "Q": "At 8:56, though it is said we don't normally consider noble gases' electronegativity, but is it correct to say noble gases are more electronegative than the others?",
        "A": "Force applied by nucleus to pull electrons is called electronegativity. Noble gases are considered to be more electronegative if we are talking about force applied by nucleus to pull its own electron. But due to complete octet (except helium) it can t gain electron so we may say it has low electronegative value."
    },
    {
        "video_name": "Rr7LhdSKMxY",
        "Q": "Is hydrogen an exception to the rule described at about 7:05 (electronegativity increasing to the right) because, similarly to the elements in column 17, just one more electron will make it noble (plus the electrons are so close to the positive nucleus)?",
        "A": "Hydrogen has a lot of exceptions because of the fact that it only has one electron. It can pretty much bond with anything no matter the charge."
    },
    {
        "video_name": "Rr7LhdSKMxY",
        "Q": "[0:23] \u00e2\u0080\u0098Electronegativity and Electron affinity are so closely related to each other that in general, if something has a high electronegativity, they have a high electron Affinity...\u00e2\u0080\u0098 For the element [N], the electronegativity is far away from the electron Affinity. Same problem with [Be], [Mg] and much more electrons ... !?",
        "A": "Sal s statement is too generalised. There are plenty of exceptions to the rule that electronegativity and electron affinity are closely related. Electronegativity refers to the tendency of an atom in a bond to attract the electrons in the bond to itself. Electron affinity refers to the energy change when an electron is added to a neutral atom in the gaseous state. So the two parameters have very different definitions. Electron affinity can be measured whereas electronegativity is calculated from other parameters."
    },
    {
        "video_name": "mMCcBsSAlF4",
        "Q": "At 2:48, Sal drew the two cells together. Are they still one cell or are they just touching?",
        "A": "They are separate cells. But depending on what type of cell they are they may or may not remain connected. Spermatocytes for example stay connected during meiosis."
    },
    {
        "video_name": "XZWoMXVANww",
        "Q": "at 8:48 why did you not include Chlorine into the equation?",
        "A": "See the chloride ion as the conjugate base of HCl, which is a very strong acid. Since a very strong acid has a very weak conjugate base, the chloride ion don t really take protons from water, thus it does not affect the pH of the solution, we thus do not include it into the equation. You may also refer to the previous video. This is similar to the reason why the chloride ion (and the sodium ion) in NaCl does not affect the pH of the solution."
    },
    {
        "video_name": "cP8iQu57dQo",
        "Q": "So at 7:46, when he says that cholesterol helps the fluidity, is that what makes the cell membrane \"fluid mosaic\"?",
        "A": "Yes, because cholesterol helps to regulate the membrane s fluidity at different temperatures. If the membrane were too get too fluid or not fluid enough, it would not be able to function correctly."
    },
    {
        "video_name": "gsu4gjrFApA",
        "Q": "At around 4:19 in the video, can someone please explain to me how he got that value for the oh ion.",
        "A": "The value of 0.20 M for [-OH] is defined in the question at the top of the screen - Calculate the pH of a 0.20 M solution of NaOH ."
    },
    {
        "video_name": "gsu4gjrFApA",
        "Q": "At 4:09, where did that H30 come from?",
        "A": "[H\u00e2\u0082\u0083O\u00e2\u0081\u00ba] is the same as [H\u00e2\u0081\u00ba] because the H\u00e2\u0081\u00ba ions released by an acid will bond to the H\u00e2\u0082\u0082O and become H\u00e2\u0082\u0083O\u00e2\u0081\u00ba. A plain hydrogen ion won t stay alone for long."
    },
    {
        "video_name": "gsu4gjrFApA",
        "Q": "At 1:24, if we have 0.030 M of HNO3 and 0.030 M of H3O, does this mean we also have 1 M of NO3?",
        "A": "You would have 0.03 M of NO3-"
    },
    {
        "video_name": "gsu4gjrFApA",
        "Q": "At 1:10 he says that the concentration of HNO3 is the same as the concentration of H3O+. How come we can make this assumption.\n\nThanks",
        "A": "Remember,Strong Acids dissociate completely. That means, in the reaction, almost every HNO3 dissociate into H3O+. so, conc. of HNO3 is equal to conc. of H3O+"
    },
    {
        "video_name": "gsu4gjrFApA",
        "Q": "At 0:00, is there a way of calculating if an acid is strong or weak?\nThanks",
        "A": "If you re given a Ka value in the question it s a weak acid. Memorise the common strong acids: HCl, H2SO4, HNO3, HBr, HI, HClO4 (there may be a few more depending on what book you use)"
    },
    {
        "video_name": "gsu4gjrFApA",
        "Q": "At 0:49 ,he says if. Is there any case where the water doesn't accept the proton from a strong acid. if at all?",
        "A": "Under these circumstances, the proton will always end up on the water molecule to give H3O+."
    },
    {
        "video_name": "gsu4gjrFApA",
        "Q": "At 3:50, would the concentration of the sodium cation be 0.20 M as well?\n\nThank you",
        "A": "Yes, indeed. You get as many moles of Na\u00e2\u0081\u00ba as you get of OH\u00e2\u0081\u00bb."
    },
    {
        "video_name": "gsu4gjrFApA",
        "Q": "At 3:15, I'm confused as to why we can't just calculate pH as -log(.2) like the first one. What is the difference between the two questions?",
        "A": "pH is for (H+) ions and pOH is for (OH)ions. pH+pOH = 14"
    },
    {
        "video_name": "gsu4gjrFApA",
        "Q": "At 5:07, he said the concentration of H3O is 5.0x10^-14. Can someone explain how he got that?",
        "A": "But his steps are shown right above that... What part don t you follow? It s really basic algebra. 1.0 x 10^-14 / 0.20 = 5.0 x 10^-14"
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "At 0:25, he says that Carbon (C-6) is in the fourth row so it has four electrons. So does that mean Silicon (SI-14) has four too because it is in the fourth row?",
        "A": "yes, it certainly does in many cases. as you know that the modern periodic table is based on the atomic no.. , their arrangements in the valence shells as well which shows the same no of rows.as silicon has at no:14 (2,8,4). so, it lies in the 4 th row."
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "What is CH3 NH2 ( at 2:31 ) ?",
        "A": "If I m not mistaken, Methylamine is the name of CH3NH2"
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "At 3:18 why is it okay to add to electron dots on one side?",
        "A": "I m having trouble understanding your question, could you explain further? Are you talking about the 2 lone electron dots on top of the nitrogen?"
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "At 5:44, how do you know that the carbons will have a bond together. Also how do you know that there is a single bond between both carbons? Couldn't there be a double bond as it will still not have a full outer shell without the Hydrogens?",
        "A": "You can t have an H atom between the two C atoms. That would give it 4 valence electrons. There might be a double bond, but you start with a single bond as a trial structure. If that doesn t work, then you might try a double bond."
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "At 3:30 he refers to checking on the octet rule. What is the octet rule?",
        "A": "Many atoms like to have 8 electrons around them, especially carbon, nitrogen, oxygen and fluorine. A Lewis structure involving these atoms will almost always show 8 electrons around them (though there are exceptions!) Some elements do not obey this rule but you will learn about them in time."
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "At 5:36 Sal writes out the Lewis Dot Structure for C2H6 and it works out perfectly... Let's say there were 7 Hydrogens instead of 6. Where would the 7th Hydrogen go? Would it tag onto one of the Hydrogens?",
        "A": "C2H7 does not exist or does not work out. Two carbons can only have 6 hydrogens (look at the Lewis structures)."
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "How do you know where the pair of electrons go? For example, at 3:26, nitrogen has 2 on top, but how can you be sure the pair of bonded electrons don't go on the right?",
        "A": "It doesn t actually matter where you place the bonded electrons vs the lone pair (the two on the top) as long as it vaugely makes sense. You could have switched their places at it would have been fine, convention is just to put the lone pairs at the top."
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "How do we know which atoms are bonded to which? At 2:41, for CH3NH2 why will Carbon be bonded to Nitrogen?",
        "A": "There is no other choice. H atoms can form only one bond each, so they must be on the outside. The C and the N are on the inside and have no choice but to bond to each other."
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "At 1:24 to 1:26 its said that the eight electrons around carbon make it stable. When making the bonds is the number eight what I should be looking for identifying whether it is stable or not ?",
        "A": "For atoms from B to Si, you should look for 8 valence electrons. For atoms after Si, 8 electrons are still good, but the nonmetals can expand their octets and use 10 or 12 valence electrons."
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "at 4:34 can the dots in the lewis dot structure be drawin in any order?",
        "A": "Yes it doesn t matter"
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "1:00 \"I know that two valence electrons equals one singe covalent bond.\" Does this mean that when the there are unbonded/unpaired electrons on both the central and axial atoms, there are going to be a sharing of a number of electrons?",
        "A": "If I understand your question correctly, then no, not necessarily. Atoms will share electrons in covalent bonds to achieve an octet (i.e. 8 electrons; or in the case of H and He, 2 electrons), but both the central and the bonded atom may still have unbonded electrons that they don t share with each other. Radical species (i.e. molecules or atoms with unpaired electrons) can also exist where an unpaired electron is not shared with its neighbouring atom(s)."
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "AT 2:38, what is the common name and iupac name of the compound CH3NH2?",
        "A": "Methylamine is the most common name, the IUPAC name is methanamine."
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "How do you know that there are 3 hydrogens on a carbon? 2:47",
        "A": "It is in the molecular formula. the CH3 implies that there are 3 hydrogens on that carbon. since Carbon ususally forms 4 bonds, and H only makes 1, there is only one obvious way to attach the molecules together in the given order. Therefore CH3NH2 represents the molecule drawn at 2:47"
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "At 3:11, Nitrogen is left alone without anything bonding with its two valence electrons, does this make it unstable?",
        "A": "It would be extremely unstable and highly reactive. But if you continue watching it isn t left like that, he s just drawing it in stages."
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "at 1:07, why do we connect them ?",
        "A": "Since the C and each H are sharing two electrons, they are covalently bonded. Covalent bonds are represented by lines, so Sal has replaced each electron pair with a line to indicated which atoms are bonded."
    },
    {
        "video_name": "BIZNBfBuu1w",
        "Q": "At 4:41, why doesn't Jay continue drawing the valence electrons around the oxygen? Is it impossible to have the hydrogen bond below the oxygen as opposed to next to it? Why?",
        "A": "Again, it doesn t really matter in terms of the structure, he could have put the H on any of them. Those 3 positions on the O are identical if you consider the molecule in 3D."
    },
    {
        "video_name": "xKJ3txXIuQk",
        "Q": "AT 4:56 what is inthe middle of the t-cell and the red blood cell ?",
        "A": "could be some mess, like a cell that died a while ago, but some other cells haven t tidied it up yet"
    },
    {
        "video_name": "xKJ3txXIuQk",
        "Q": "At 4:20 why is the HIV or t-cell spikey and more rounder than the red blood cell?",
        "A": "a red blood cell is biconcave so it can carry more oxygen around the body."
    },
    {
        "video_name": "xKJ3txXIuQk",
        "Q": "About 2:00 he said hemoglobin. What is hemoglobin?",
        "A": "Hemoglobin is the substance in red blood cells that carries oxygen."
    },
    {
        "video_name": "xKJ3txXIuQk",
        "Q": "At 5:50, Sal said that RBC has 250 million molecules of Haemoglobin but,\nRBC carries only 4 molecules fo Haemoglobin at a time.Why Sal said RBC has 250 million molecules of Haemoglobin ?",
        "A": "All cells are made up of molecules, which are composed of atoms. Red blood cells do contain 250 million haemoglobin molecules, with each one being able to carry 4 oxygen molecules at a time. Hope this helped!"
    },
    {
        "video_name": "mAdUjImDx58",
        "Q": "When he says survival of the genetics 7:46, does he actually mean survival of the fittest?",
        "A": "He meant the survival pf genes :)"
    },
    {
        "video_name": "OIkD1viOMWk",
        "Q": "11:35 Does the mirror equation work if the object is between the center of curvature and the focal point?\n\nWhat about if the object is closer than the focal point?",
        "A": "Yes. The mirror equation works for all kinds of mirrors. It works in all cases,only if the medium around it is same. Sign convention must be used."
    },
    {
        "video_name": "OIkD1viOMWk",
        "Q": "So, at about 6:25 he finds the magnification equation, and at 11:13 he finds the mirror equation. Can't both of these equations be used to find lengths and heights for lenses as well?",
        "A": "I believe the signs change in case of lenses. Check out the next few tutorials."
    },
    {
        "video_name": "ks1B1_umFk8",
        "Q": "At 5:25 Why dont you use the formula V=kQ1Q2/r? instead of KQ/r",
        "A": "Because your equation will give you potential energy, which is different from potential. potential = potential energy per unit of charge."
    },
    {
        "video_name": "w3IKEa_GOYs",
        "Q": "At 4:14 I looked at the picture and i wondered, are there \"super-stars\" that are maybe 2 to 5 times as large as other stars like our sun?",
        "A": "There are stars that are a hundred times as large as the sun."
    },
    {
        "video_name": "w3IKEa_GOYs",
        "Q": "At 2:27 it was stated that we are watching image of nebula which is 7000 light years old. How can we get the latest image?",
        "A": "The image we get is the latest image to arrive here. You can t get tomorrow s newspaper today."
    },
    {
        "video_name": "w3IKEa_GOYs",
        "Q": "At about 2:39, Sal states there was a supernova in the Eagle Nebula and we won't be able to see the effects of it for another thousand years. Does that mean that particles from a supernova explosion move faster than the speed of light??",
        "A": "No, the event occurred about six thousand years ago. It will still be about 1000 years before the effects are seen."
    },
    {
        "video_name": "w3IKEa_GOYs",
        "Q": "At 5:25, what is the difference between nebulae and planetary nebulae? Thanks",
        "A": "A planetary nebula is not exactly a nebula, but rather a partial supernova remnant."
    },
    {
        "video_name": "w3IKEa_GOYs",
        "Q": "Are the three pictures shown at 0:30 actual photographs?",
        "A": "Yes, those are actual photographs."
    },
    {
        "video_name": "w3IKEa_GOYs",
        "Q": "Isn't it that if a star explodes, it forms a black hole, and i think that is a dwarf star that forms it. And if something is seven thousand light years away, as said on 7:04-7:07, how many years would it take for you to get to it. And since its so far away, how did a person get a picture of it.",
        "A": "Sometimes, No, Seven thousand years by light travel and millions of years by modern space probes, And it is so large that we can easily see it."
    },
    {
        "video_name": "w3IKEa_GOYs",
        "Q": "At 1:00, it says something like 7 light years and at 1:10 it said something like 4 light years what are light years?",
        "A": "A light-year is a measure of the distance light can travel in one year: 9.46 trillion kilometers. It is a measure of distance NOT of time. Hope that helps, Isabel."
    },
    {
        "video_name": "w3IKEa_GOYs",
        "Q": "At 4:55 how are blue stars more massive than red stars? Wont the energy that they emit dissipate over a larger area, thus cooling temperatures? Blue has a short wavelength so it should be really hot. So should't blue stars be smaller?",
        "A": "For main sequence stars, as they become more massive, the stars must burn hotter to counteract the extra gravitational force. Hotter burning stars are bluer than cooler burning stars."
    },
    {
        "video_name": "w3IKEa_GOYs",
        "Q": "So if I understand correctly, a blue star has a shorter life span than a red star [circa 5:00]. What is the life span difference between the two? Also, was the blue star always blue or like the red star, was it once yellow?",
        "A": "Would a human be affected any way by a blue star differently then from a yellow sun (i.e. Vitamin D and tanning UV s)?"
    },
    {
        "video_name": "w3IKEa_GOYs",
        "Q": "1. At 1:51, if those nebulae made of hydrogen then how they are colorful if hydrogen is colorless?\n\n2. the nebula is created from the helium flash or from supernova?",
        "A": "1. maybe the UV rays, x-rays, gamma rays, or tech color 2. supernova"
    },
    {
        "video_name": "0CQ5ls3Uc2Q",
        "Q": "Are nitrogenous bases basic because the nitrogen atoms have an extra lone pair? (at 1:32 Sal talks about how they can be used to attract hydrogen protons; according to the Bronsted-Lowry acid-base theory, base are proton acceptors.)",
        "A": "What you say is correct."
    },
    {
        "video_name": "0CQ5ls3Uc2Q",
        "Q": "why do they use at 6:35 the ' for the primes?",
        "A": "That is simply the notation used for primes . It is so that when you read it, you know it is prime. Kind of like measurements. If I told you I had 5, you would wonder 5 what? So we label it. 5 apples. 5mg. However you want to label it. So they just write the apostrophe for labeling purposes."
    },
    {
        "video_name": "0CQ5ls3Uc2Q",
        "Q": "@ 2:15, what does Sal mean when he meant that the Phosphate group is \"deprotonated\"?",
        "A": "lost a hydrogen"
    },
    {
        "video_name": "0CQ5ls3Uc2Q",
        "Q": "At 2:26 Sal said that the sugar backbones are are acidic, but the rungs of the ladder are basic. Won't that make DNA neutral?",
        "A": "no, the bases effect masked by the formation of hydrogen bonds between complementary bases. the acidity is due to presence of many phosophate attached to the sugar which make the over all structure acidic."
    },
    {
        "video_name": "0CQ5ls3Uc2Q",
        "Q": "At 1:33, when Sal says that nitrogen is really electronegative and can take up more hydrogen protons, does he mean hydrogen electrons? Thanks!",
        "A": "Sal is talking about hydrogen bonding. With hydrogen bonding, the hydrogen in question is covalently bound to an electronegative atom, such as oxygen or nitrogen. This electronegative atom pulls electron density away from the hydrogen giving the hydrogen a partial positive charge. If there is atom close by that has a lone pair of electrons, such as an oxygen or nitrogen atom, then these lone pair electrons are attracted to the hydrogen, giving rise to a hydrogen bond."
    },
    {
        "video_name": "391txUI76gM",
        "Q": "At around 2:39, I just realized that in Sal's other videos (because Sal pointed out the unit of measurement) , for the equation to the gravitational constant, he used G=6.67*10^-11 kg*m^3/s^2, but then in this video, he used m^3/kg*s^2. Could somebody please enlighten me?",
        "A": "The correct units would be m^3/(kg*s^2)"
    },
    {
        "video_name": "391txUI76gM",
        "Q": "in which cases G is not constant? which sal said at 2:07.",
        "A": "As far as we know, G is constant everywhere, but we don t know that to be true for certain. g = GM/r^2, so it varies depending on the mass and radius of a planet."
    },
    {
        "video_name": "391txUI76gM",
        "Q": "At 0:26-0:33, you said that two objects are attracted to each other because they have masses, does that mean light which has no mass are not affected?",
        "A": "Light has no rest mass, but it s never at rest, so that doesn t tell you much. Gravity affects energy, and light is energy, so gravity affects light. Newton did not realize this, so it s not part of his law of gravitation. It s covered by Einstein and general relativity, though."
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "This may be a basic maths question but I'm confused. At 8:30 why does Sal turn the at over 2 into 1/2 a multiplied by t squared? The t was over the 2 meaning it was only half t, the same as the a. So you can't then go on to say t multiplied by t is t squared because the first t is only a half t.",
        "A": "Ah. It makes sense when you show the working out. It has baffled me for a couple of days this. Many thanks."
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "At 2:33, there is a formula in blue which I don't understand. Why is there a BIG G and 2 m with an r sq. ?",
        "A": "That s the law of universal gravitation."
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "Hey guys. Wondering...at 8:30, is't the calculation actually (delta t) times 1/2 (delt t)?",
        "A": "Any way is the fine. It will still give you the right answer :)"
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "At 1:10, I don't understand what Sal means when he says, \"If you want acceleration on Earth, you divide force by mass.\" What is \"acceleration on Earth\"?",
        "A": "This is the acceleration due to gravity, usually denoted g Newton s second law shows it as F=m a (F & a are vectors which hopefully are shown as bold if the asterisks work). g is the value of a on earth due to gravity and it s direction is toward the center of the earth."
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "At 7:07 I don't understand why Sal split the term up into two fractions over two to be added together.",
        "A": "In this problem we do not know what the final velocity is. Instead, we know that Vf = Vi + a*t The term was split up into two fractions to group together and simplify what is being multiplied by t."
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "At 2:15, what's the difference between G (gravity) and g (gravity)?",
        "A": "G is the gravitational constant in Newton s formula for gravitational force F = GMm/r\u00c2\u00b2. g is the acceleration caused by the gravitational force. F = mg = GMm/r\u00c2\u00b2 so g = GM/r\u00c2\u00b2"
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "At 1:34, Sal shows that the acceleration is equal to g, but isn't g the magnitude of acceleration, not the acceleration itself?",
        "A": "No, g is gravitational field strength, or the acceleration due to gravity. It is a vector. The magnitude of that vector is 9.8 m/s^2 on the surface of the earth."
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "At 2:04, how is the acceleration of an object equal to its gravitational force?",
        "A": "objects fall at 9.8 m/s^2 near the surface of the earth the force on an object is 9.8 N/kg"
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "A question - at 6:03, when Sal is deriving final velocity, writing it as the sum of initial velocity and the product of acceleration and the change in time...I was just wondering, if we derive a formula for a projectile that has been thrown, wouldn't the final velocity equal zero?\n\nor we consider the final velocity at that very impact, with which the projectile fall back to earth?",
        "A": "The final velocity is the velocity at the moment the object reaches a certain height. We don t care that there happens to be ground there that the object is going to run into."
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "at 1:12 sal says F=m*a does that mean that something with a mass of 1000 kg. traveling at a constant velocity of 120 km / hr (with no acceleration) does not have any force\n(F = 1000 kg / 1 * 0 km / second^2 = 0 kg * km / second^2)?",
        "A": "velocity is not acceleration. F = ma means that an object traveling at a constant velocity has no net force on it (the sum of the forces on it are zero)"
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "1:08 Is \"acceleration on Earth\" different from the acceleration we learned in the previous lessons (a= v/t)?",
        "A": "no, acceleration will always be the change in velocity over time. However, the force of gravity will be different on different planets etc, so the acceleration will also be different in those locations: Force = mass * acceleration F = ma (mass won t change)"
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "At 1:03 he said 9.8 m/s2.But i looked it up on google and it said it was 9.78 m/s2.Who is right?",
        "A": "It really varies depending your altitude and local density of the crust of the earth around you. Also because the earth is not a perfect sphere it is about 9.780 at the Equator to about 9.832 at the poles."
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "At 8:40, shouldn't it be (delta t^2) not (delta t)^2? Because although Sal put the ^2 in the same colour as the (delta t), clearly grouping the two together, the ^2 could apply to the whole line, meaning; s*Vi*delta t+1/2*a*(delta t) * s*Vi*delta t+1/2*a*(delta t). Instead of s*Vi*delta t+1/2*a*((delta t) * (delta t)). Or something like that. :-P",
        "A": "\u00ce\u0094t\u00c2\u00b2 is the same as (\u00ce\u0094t)\u00c2\u00b2. (Delta)t^2 is the same as [(Delta)t]^2 . Just imagine \u00ce\u0094t , (Delta)t , as a single unit rather than two different units. The \u00ce\u0094 delta just means change in , it s used as an abreviation. The rest of your question is a bit confusing. :s"
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "when Sal tells at 3:49 that up is positive and down is negative should it be he other way round,because the work done is against gravity so up will be negative and the 'g' will be positive since its in the same direction?",
        "A": "That is a good question. In physics and engineering for example, it doesn t actually matter which direction you select for positive and negative. They are conventions you can set yourself and become stated assumptions . As long as you are consistant in your direction conventions, it is entirely acceptable to select either direction to be positive. I hope that helps."
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "At 7:27, why you have multiplied the whole equation by change in time in this video?",
        "A": "the part he was solving for before that was only the average velocity he just put that part back into the original equation that looks like s = v \u00e2\u0088\u0086t, which would be multiplying by time on the right side of the equation hope that helped"
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "Sorry that this is such a base question but at 7:07 why did Sal divide the a times delta t term by 2?",
        "A": "The delta t at the end was the same delta t at the end of the equation for finding Displacement. Notice how he put all that stuff in brackets? Thats because all the terms in brackets is how we are finding average velocity."
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "2:54\nWhat if you don't assume gravity is constant how would you calculate displacement then?",
        "A": "The gravity is always constant, as long as you are in a fixed position"
    },
    {
        "video_name": "wlB0x9W-qBU",
        "Q": "At 2:35, what Sal means is that since the radius of the earth is ~6,371 km and we're throwing an object up, say, 0.1 km, that is such an insignificant addition to the distance, that it doesn't really affect anything, except at the super precise level, right?",
        "A": "Quite so. In the gravity lessons you will find that large distances (around the same magnitude of the radius of the earth) change gravitational force and cause differences."
    },
    {
        "video_name": "6BR0Q5e74bs",
        "Q": "@ 5:00 , can the bond be attached to any carbon on the cyclopentyl ?",
        "A": "Yes! All carbons on cyclopentane are equivalent, so it doesn t matter which one the main molecule is attached to (if cyclopentane also had a functional group, this would not be true!)"
    },
    {
        "video_name": "6BR0Q5e74bs",
        "Q": "Can we write the organic compound discussed at 6:30 as 4,4,7,10-tetramethyldodecane?",
        "A": "You could and it drawing it does give us the same structure, but it doesn t use the lowest numbering possible for the groups. So the two numbering schemes we have are (3,6,9,9) vs (4,4,7,10), the numbering you use is based on the first point of difference, 3 is lower than 4 so we use 3,6,9,9. I ll add, there s an often quoted but not correct rule regarding numbering where you add the numbers up, this is not an actual rule. The lowest number at the first point of difference is the rule."
    },
    {
        "video_name": "6BR0Q5e74bs",
        "Q": "@ 8:29 Can't we name the other carbon 1 and finally obtain isobutyls?",
        "A": "He could have named any carbon 1 and still have obtained 1,3-di-tert-butylcyclopentane."
    },
    {
        "video_name": "6BR0Q5e74bs",
        "Q": "what would the scientific name have been at 9:50 if there was only one methyl group on each of those 2 ethyl groups, i am really confused with that last one and i think that there should be only one methyl group on each ethyl group with respect to the question.",
        "A": "It would have been 1,3-bis-(1-methylethyl)cyclopentane."
    },
    {
        "video_name": "6BR0Q5e74bs",
        "Q": "At, 6:40, shouldn't it be 3,3,6,9-tetramethyldodecane? With that numbering you run into the two groups first.",
        "A": "Do you mean 4,4,7,10-tetramethyldodecane? Maybe you miscounted."
    },
    {
        "video_name": "6BR0Q5e74bs",
        "Q": "At 9:50 Another systematic name Could be 1,3-diIsobutylcyclopentane?",
        "A": "No, an isobutyl group is different from a tert-butyl group."
    },
    {
        "video_name": "6BR0Q5e74bs",
        "Q": "8:26 > when numbering the carbon chain of the substituent (in doing the SYSTEMATIC approach, not common name approach), is it a general rule to number carbon '1' as being the carbon connected to the parent group?\n\nIn completing the 'answer' before seeing Sal do it, I had trouble knowing which carbon was carbon '1', on the methylethyl substituent, and thus didn't know where to put the two methyl groups.\n\nthanks! lauren :)",
        "A": "Yes, carbon 1 of the substituent should always be the one that is connected to the main alkane in terms of the systematic approach."
    },
    {
        "video_name": "6BR0Q5e74bs",
        "Q": "at 5:04, shouldn't the structure for cyclohexdecane be a 16 sided polygon?",
        "A": "No, because the internal angles of a regular hexadecagon are 157.5\u00c2\u00b0. The tetrahedral bond angle is 109.5\u00c2\u00b0. so a flat 16-gon would be horribly strained. The bonds pucker out of the plane of the paper to get bond angles of 109.5 \u00c2\u00b0. It is impossible to draw a correct flat structure for cyclo-16, so the diagram in the video, while not correct, is closer to correct than that of a regular 16-gon."
    },
    {
        "video_name": "6BR0Q5e74bs",
        "Q": "At 3:25, when the connection to the middle carbon is made to the ring, does that count as another carbon connection? Or is that connection just there to form the Y shape? I'm confused... I see that the connection has to be made for the isopropyl group to be attached to the ring, however does it serve as another carbon attachment? Or is it just solely there to be used as a connection to the cyclohexadecane?",
        "A": "The connection is not another carbon but simply a connection to join it to cyclohexdecane. If we name them in the other way one purple line and one yellow line makes the ethyl and the other purple line being the methyl. Which in other words is isopropyl. Hope this answered your question. :)"
    },
    {
        "video_name": "6BR0Q5e74bs",
        "Q": "at 9:54 can we name tert butyl as neo butyl too ? becoz neo means 2 branches right?",
        "A": "You cannot name tert-butyl as neobutyl. You can only use the neo- prefix with substituents with at least five carbons. neo means, if you re interested, there is something situated on the primary carbon attached to to a t-butyl group. you will only use the term neopentyl and in no other context. There is not neohexyl, neobutyl...etc....Only a 5-carbon substituent can have the neo-structure."
    },
    {
        "video_name": "eQXGpturk3A",
        "Q": "At 0:18 seconds, he uses the term \"meniscus\". when would use this during biology?",
        "A": "It s good when you want to measure liquids in a measuring flask. To make sure it s accurate, you line up the bottom of the meniscus to the measurement marking so your measurements are more consistent. Thinking of ways you would do it in a biology lab, could be for practically anything."
    },
    {
        "video_name": "eQXGpturk3A",
        "Q": "At 6:49, would the same happen with mercury? Or would it be the opposite? What would happen?",
        "A": "No. Because mercury is a pure elemental substance (and therefore nonpolar), it does not demonstrate the same process as the polar water molecule."
    },
    {
        "video_name": "eQXGpturk3A",
        "Q": "At 5:24, does the water have a meniscus in the container (if it was glass or a polar substance)?",
        "A": "I believe Sal s implying that the tub is made of plastic, which is very nonpolar and does not attract water. Either way, the forces between the water and the tub are less important than the attraction of the water to the glass tube."
    },
    {
        "video_name": "eQXGpturk3A",
        "Q": "At 2:47, Sal says that silicon is less electronegative than hydrogen. I thought electronegativity increases from left to right, so how is this possible? Or am I missing something?",
        "A": "You are completely right and what you state is the general trend but there is one tiny technicality missing: hydrogen is sometimes drawn in group 1, and sometimes it isn t because it is so different to the other group 1 metals. On the periodic table, you will sometime see hydrogen removed from group 1 for this reason. Hydrogen is just a bit of an exception, even in electronegativity: hydrogen is 2.20 while silicon is 1.90."
    },
    {
        "video_name": "eQXGpturk3A",
        "Q": "at 2:34 Sal says that there are two oxygen atoms per silicon atoms then why are there six oxygen atoms and only two silicon atoms?",
        "A": "He just hasn t drawn every atom in that picture. The top 2 oxygens would be bonded to another silicon, which would be bonded to 2 oxygens, etc."
    },
    {
        "video_name": "UrfLAlk2b_8",
        "Q": "at 2:48 you say \"treating systems as a single object.\" So is the acceleration of the system equal to the acceleration of the sliding block.",
        "A": "Yes. Because the blocks are connected by a string that does not stretch, they must have the same acceleration -- they move in a completely synchronized way. That also means that at each moment, they also have the same displacement and velocity (though as he mentions at the end of the video, any vector pointing to the RIGHT for the mass on the table will become a vector pointing DOWN for the hanging (falling) mass)."
    },
    {
        "video_name": "UrfLAlk2b_8",
        "Q": "At 9:00 you adjust for friction and multiply the mass 5 kg X the Normal Force. But earlier in the video you said that we should exclude forces which are perpendicular to the motion. Why is that?\nBy they way, this is a great video! Thanks for creating it!",
        "A": "Normal force is perpendicular to the motion of the system, but friction is parallel to the motion of the system. However, we need to know the normal force in order to calculate the force due to friction."
    },
    {
        "video_name": "UrfLAlk2b_8",
        "Q": "At 6:00 ~, he says to add up the external forces, and the force of friction and force of gravity are grouped together. How come the force of friction can be subtracted from the force of gravity? Isn't the force of friction a horizontal force and the force of gravity a vertical force?",
        "A": "We re considering the two blocks as one system, like one object. So we don t really pay much attention to the vertical and horizontal directions. If you notice, in this video, David doesn t put the subscripts x and y for the frictional force and the gravitational force as it isn t significant in this method. :)"
    },
    {
        "video_name": "IuBoeDihLUc",
        "Q": "If matter is anything that has mass and volume... and mass is nothing but amount of matter in a substance ( 0:30 ), what is really matter and mass?",
        "A": "Mass is a measure of an object s resistance to acceleration, or a measure of the force the object will feel in a gravitational field. Matter is not a clearly defined term in physics. It is commonly used to refer to observable physical objects and the particles that comprise them."
    },
    {
        "video_name": "IuBoeDihLUc",
        "Q": "At 0:37, Sal says that mass is the amount of matter in an object. Isn't density a measure of how much \"stuff\" is in the given amount of an object? I'm confused.",
        "A": "Density is how tightly the mass is packed. It s mass per unit of volume. Mass is just how much stuff there is, regardless of how much space it takes up. 1 kg of lead has the same mass as 1 kg of feathers, but it s a lot more dense."
    },
    {
        "video_name": "IuBoeDihLUc",
        "Q": "At 0:34, does this mean that matter in solid state has more mass than matter in liquid state, because more molecules are packed in a solid as compared to a liquid?",
        "A": "Not exactly more weight. it means that Matter can exist in all the 3 states but the conversion of states does not change the weight/mass of the material. It may cause the density of the material to change but not the weight. For eg: 1 cup of liquid water weighs the same if it is frozen or if it is vaporized.(Even water vapor has some weight)"
    },
    {
        "video_name": "IuBoeDihLUc",
        "Q": "At 4:24 Why does it look like Sal did 9.8 meters over seconds squared instead of 9.8 meters per second squared?\nSee it looks like 9.8 .... m\n......................................... __\n.......................................... s\u00c2\u00b2 .......instead of 9.8 m/s\u00c2\u00b2\n\nSorry if the way I did this is confusing at all, the dots are nothing important for the question. I just lined it all up with them.",
        "A": "Those are the same thing."
    },
    {
        "video_name": "IuBoeDihLUc",
        "Q": "At 1:27 Sal is describing the difference between mass & weight. Regarding mass he states an object with greater mass accelerates slower, and an object with less mass accelerates faster for a given force. We know that is not true for a free falling objects if we can ignore air resistance. In this case, the force of gravity is the given force. He also makes this same statement in \"Would a brick or feather fall faster\" video. In that video, he goes on to contradict himself. Can someone clarify this?",
        "A": "The force of gravity on a more massive object IS greater. That force is m*g. But it also takes more force to accelerate a more massive object. That force is m*a When an object is in free fall with no air resistance we have mg = ma m divides out and we have g = a So objects accelerate at the same rate, g"
    },
    {
        "video_name": "IuBoeDihLUc",
        "Q": "2:06. How do you find something's mass? Every time you weigh something, you're just finding it's weight. How did you calculate your mass without finding weight first?",
        "A": "as long as we are on the surface of the earth, weight corresponds to mass by a factor of 9.8 N/kg So we can easily determine mass by weighing things Note that not ALL scales find weight. Some do measure mass. See if you can think of a scale design that would measure mass (i.e. it would give you the same answer on the moon as it would give you on earth)"
    },
    {
        "video_name": "IuBoeDihLUc",
        "Q": "8:15\nif pounds are a way of measuring weight , why does our weight keep changing even though we measure the weight in the exact same place?",
        "A": "Weight is mass times acceleration due to gravity . If ur mass increases then that will have a directly proportional effect on the weight. As u said that u r measuring weight in the exact same place , the weight change should be the result of mass change"
    },
    {
        "video_name": "IuBoeDihLUc",
        "Q": "At 7:29 he says, \"a nuclear reaction inside of you.\" How is there a nuclear reaction inside a person?",
        "A": "There isn t one. He was just making a stament about how some object would loos mass over time."
    },
    {
        "video_name": "-82UNytyrCQ",
        "Q": "At 2:00, is there a specific place on the wire where the node is, if you were connecting two components with a wire, where would the node be, or is it just saying they are connected if they share a node.",
        "A": "There is no specific spot on the wire where the node is. A node is the whole junction, since every spot on the wire is at the same voltage."
    },
    {
        "video_name": "-82UNytyrCQ",
        "Q": "At 6:31 the first loop is illustrated as extending to the end of the second mesh, but I don't understand why that is. My intuition is that there would only be 3 loops, but we were told to take the 3 loops given to us and add it to the number of mesh to get the loops. Why is this? It doesn't seem intuitive to understand, and I don't feel as if I was given really any information regarding what a loop is or even how to adequately count them. Thanks!",
        "A": "Think of the meshes as minimal examples of loops, in the sense that they don t contain smaller loops inside of them. I think you may see that the meshes, being already covered, weren t also listed as loops until the end, where we we re told to count those three meshes in as well when counting loops."
    },
    {
        "video_name": "-82UNytyrCQ",
        "Q": "At 6:10, doesn't the mesh just depend on how you draw the circuit, or is it actually there.",
        "A": "It s both. In the circuit at 6:10 you could draw the circuit with the two branches on the right reversed (with the two series resistors towards the inside). That changes what each mesh arrow travels through, but there are still three meshes. I m pretty sure you can t redraw a circuit and get a different number of meshes."
    },
    {
        "video_name": "-82UNytyrCQ",
        "Q": "What is the difference, if there is one, between branches and elements? At 4:44, he talks about them as it they are the same.",
        "A": "The term branch comes from graph theory (a sub-field of mathematics). A branch is the connection between the nodes of an arbitrary network. In electronics, circuit elements create the connections between nodes, so branches and elements mean the same thing when talking about circuits. In electronics, we ll say branch voltage or element voltage and mean the same thing."
    },
    {
        "video_name": "VxEIsPOdOLw",
        "Q": "5:31 \"Prostacyclin ... is a peptide\"\n\nIs it mistake?",
        "A": "Prostacyclin is a prostaglandin member of the eicosanoid family of lipid molecules. It inhibits platelet activation and is also an effective vasodilator but I do think tht is a mistake as well saying that is a peptide"
    },
    {
        "video_name": "Hp46Y6j_Qdc",
        "Q": "At 0:42 what is a header pin connecter?",
        "A": "It is a connector that allows you to attach wires to the pins that protrude from the arduino and motor controller."
    },
    {
        "video_name": "emdHj6WodLw",
        "Q": "It appears that the answer to thei problem at 6:20 would not change if we threw the object down at 30 m/s as opposed to up. How can the be?",
        "A": "If you throw up an object with a certain velocity it will come back with the same magnitude and since the initial velocity is squared in the formula we do not take the direction into account."
    },
    {
        "video_name": "emdHj6WodLw",
        "Q": "Why is the denominator 2a in the formula (4:07 in the video) ? B/c change in distance = average velocity times change in velocity divided by acceleration? So when did accerleration multiply itself by 2.? Is it because average velocity is equal to acceleration times time?",
        "A": "because its the average velocity for change in distance"
    },
    {
        "video_name": "emdHj6WodLw",
        "Q": "At 5:42, Sal said that using the + sign was very important. But wouldn't 30 m/s and +30 m/s be the same thing?",
        "A": "Yes, but he meant it is important to use + and not -."
    },
    {
        "video_name": "emdHj6WodLw",
        "Q": "At 5:00, wouldn't you need to have a constant acceleration for the equation to be true?",
        "A": "You do have constant acceleration in free-fall."
    },
    {
        "video_name": "emdHj6WodLw",
        "Q": "When you multiply expressions at 3:20 (Vf+Vi)(Vf+Vi) why is does it become Vf^2 + Vi^2 why does it not become VF^2 +Vi^2 + 2VfVi?",
        "A": "soz didn t see that it was a difference of squares."
    },
    {
        "video_name": "emdHj6WodLw",
        "Q": "At 8:30 Sal calculates the Velocity final = 104m/s (down)\nIf a gun fired a penny straight up from the bottom of the cliff at 104m/s (up) would it achieve the same height as the penny Sal threw from the top of the cliff?",
        "A": "Please reread the question Harsh. Darren is correct; in his example the height achieved would be the same. However Darren, please note that a real bullet would be travelling much faster than this."
    },
    {
        "video_name": "emdHj6WodLw",
        "Q": "I thought Avg Vel: Change in position/ Change in time? I'm kind of confused with the part at the 2:00 mark.",
        "A": "Cool, got it guys Thanks!"
    },
    {
        "video_name": "emdHj6WodLw",
        "Q": "Is Vf, which is solved for around 6:13 a vector or a scalar?\n\nAs far as I can tell, it looks like it could never be negative.",
        "A": "Hi, remember when Sal said velocity is a vector? that means Vf is always a vector too. But a lot of the time you cant notice the difference between vector and scalar. Just remember that if you know velocity then that means you should know the direction something is going. speed alone (which is scalar) can t tell you that."
    },
    {
        "video_name": "FEF6PxWOvsk",
        "Q": "At around 1:48 he says something about beta... can someone explain what he means??",
        "A": "He was talking about the Weak Nuclear force which is responsable for radioactive decay. In early atomic theory the first two types of radiation were called Alpha and Beta radiation. As it turns out Alpha radiation is basically 2 protons and two neutrons which is a helium nucleus and Beta radiation is an electron. So when Sal refers to Beta+ and Beta- radiation at that point he is refering to electrons and anti-electrons (positrons)."
    },
    {
        "video_name": "FEF6PxWOvsk",
        "Q": "at 2:56 what is a electron anti-neutrino",
        "A": "A fundamental particle with no charge, 1/2 integer spin and very small mass. It is in a class of particles called leptons."
    },
    {
        "video_name": "FEF6PxWOvsk",
        "Q": "At 2:13, we were told about one of quarks of neutron flips and turns into a proton. Then, what is quarks basically?",
        "A": "Quarks are fundamental particles that interact via all four of the fundamental forces."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "Sal introduces Group 11 at 0:25 and I can't help but notice that this is the column that contains copper, silver, and gold. We know their colors: reddish-brown, grey, and yellow. Roentgenium [Rg] aka #111 is also in Group 11. What does that mean and what color is #111?",
        "A": "Actually, there is. All elements have colours, maybe except noble gases. it depends on the types of allotropes of an element"
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "6:03 he said about \"back filling the d block\" What does that mean?",
        "A": "As an example, it means that the 4s level fills (usually) before 3d, even though you might expect 3 to fill completely before 4 starts to fill. It just doesn t work that way. Nature doesn t have to follow rules that seem simple to us. There is a simple rule that nature follows here, though: it always fills the lowest energy levels first. 4s is slightly lower energy than 3d in most atoms."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "At 5:25, did he mean 3d1?",
        "A": "Yes he did. There is a note on the bottom right of the video that says this."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "At about 0:46 you talk about the F-block elements, I would like to know if there is a name to specify those particular elements such as there is for Alkali Metals or Alkaline-Earth Metals?",
        "A": "The top row are commonly called the lanthanides and the bottom row are commonly called the actinides"
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "at 6:40 Sal says that there are 4 valence electrons in the outermost shell of that particular group . Does Pb has the properties of carbon?",
        "A": "Not really, because lead is a metal and carbon is a non metal."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "At 3:30 what is meant by nirvana ?",
        "A": "this stable state that the atom is in once it has a full outer electron shell :-) The goal of atoms is to fill their outermost electron shells because that will increase their stability most"
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "At 6:01 how did he get the 2 on Sc: [Ar] 4s2 3d1",
        "A": "That tells you how many electrons are in an orbital. So Sc: [Ar] 4s2 3d1 means that Sc has the electron configuration of argon (that is what [Ar] means) PLUS 2 electrons in the 4s orbital and 1 electron in a 3d orbital (hat is what 4s2 3d1 means)"
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "At 3:00, it talks about the valence of Li and I do not understand what it means by 2s'",
        "A": "Electrons are arranged in electron shells and subshells. Lithium s electron arrangement is 1s2 2s1. That means that it s valence electrons (outer most electrons) are the 2s1 electrons: second shell, s-shell, 1 electron"
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "At 3:00, why does it say Li: [He] 2s1? What does 2s1 mean?",
        "A": "I ll go over it all any maybe it will make it a bit more clear Li: means that this is the electron configuration for lithium [He] means that we use the electron configuration of helium as a base, helium is 1s^2 2s^1 means that there is 1 electron in the 2s orbital. The 2s orbital is the second orbital that gets filled in atoms. If you re still curious check out the videos on electron configuration."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "At 2:30 Sal names the first group as the alkali metals, but then at 4:10 he names the second as alkaliNE metals. What is the difference? Why can't they both be alkali or both be alkaline?\n\nThanks",
        "A": "first group is basically THE ALKALI METALS which are very reactive and cannot exist freely. second group is THE ALKALINE EARTH METALS which are found in the earth s crust."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "Why does Sal put a different element into brackets? like at 2:49",
        "A": "That element in square brackets represents the electron configuration of the noble gas from the previous row. It saves significant time and space with larger atoms rather than writing out the full electron configuration each time."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "At 5:25, why isn't it 4d1 instead of 3d1 for scandium since it is in the fourth period?",
        "A": "Because that is the actual order in which the subshells fill. 4s and 3d have nearly the same energy level, so that is why they fill the way they do. If you re asking why didn t we just number the 3d the 4d since the 3d fills in the 4th period, the answer is that number actually means something -- it is used in some equations that are above this level of study. So, the number just are what they are: we cannot renumber them."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "6:11 I don't understand why it's 3d, because it's on the 4th period, so shouldn't it be 4d like the s and p block?",
        "A": "There isn t a good answer other than it just happens like that. The 3d orbitals are at about the same energy as the 4s orbital, and the 4d orbitals are at about the same energy as the 5s orbital. Things start to get complicated when you go further down, the rules don t always work very well."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "At 6:15 he started writing Sc:[Ar] 4a23d1. What does that mean? Can someone please explain this to me?",
        "A": "watch sal s vids about electron configurations"
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "on 5:16, how did you get the letters? I t is really confusing me.",
        "A": "if you re talking about this, Sc= [Ar]4s^2 then it is the electronic configuration of Scandium. to be precise Sc= 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^1 if I am not wrong. :P Remember the aufbau principle? you just go on filling."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "Alright, so at around/leading up to 6:29, Sal's talking about Sc, and he says there are 2 electrons in the outermost shell, then catches his mistake and corrects 3s to 3d. He then talks about backfilling.\n\nIt's a little unclear to me, are there 2 valence electrons here (as he seems to indicate), or are there 3 valence electrons here (following the pattern set by the previous examples where it appears the electrons following the Noble Gas in the Noble Gas notation are the valence electrons)?",
        "A": "There are two valence electrons. The d subshell is an inner shell, not a valence shell, so the electrons in a d subshell are not valence electrons."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "How do people count [He]2s1 at 2:58 ?",
        "A": "Best bet would be to check out the videos on orbitals and electron configurations. That notation means that a lithium atom has the electron configuration of the noble gas of the row above (helium) and one electron in the 2s orbital."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "What does Sal mean by the \"octet rule\" at 3:26? In fact, what is the octet rule?",
        "A": "The octet rule states that elements gain or lose electrons to attain an electron configuration of the nearest noble gas. or As per Wikipedia: The octet rule is a chemical rule of thumb that reflects observation that atoms of main-group elements tend to combine in such a way that each atom has eight electrons in its valence shell, giving it the same electronic configuration as a noble gas."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "At 5:05, why is Sc equal to Ar?",
        "A": "It isn t. The electron configuration of scandium is the same as the electron configuration of argon plus the two 4s electrons plus a 3d electron."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "at the time 0:55-1:00, why didn't you circle the hydrogen element because they are in the same group?",
        "A": "Hydrogen is a sort of special in a way at its properties, considering it has only one proton and one electron."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "0:16. I noticed how Sal numbered his groups, and it looked unfamiliar. I looked at a table from my 7th grade physical science book, and it numbers the groups differently. The order is (from left to right): 1A, 2A, 3B, 4B, 5B, 6B, 7B, 8B (this is for three columns), 1B, 2B, 3A, 4A, 5A, 6A, 7A, 8A. Which is correct?",
        "A": "You ve stumbled upon the reason why the groups were changed to 1-18!! In the US and Europe the older group labels used the same letters and numbers but they referred to different groups! So both ways were correct depending on what country you are from, but we should all be using 1-18 these days."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "I didn't understand why isn't hydrogen considered as an Alkali Metal at 2:21?",
        "A": "H doesn t fit anywhere in the Periodic Table. It is a gas, not a metal, but we put it in Group 1 because it has one valence electron."
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "At 4:43 the person says something about D-block.\nWhat is D-block?",
        "A": "a block that is in the d cell"
    },
    {
        "video_name": "UXOcWAfBdZg",
        "Q": "0:49. Why are elements 57 - 71 and 89 - 103 not numbered?",
        "A": "Because # 57~71, lanthanides and #89~103,actinides are suppose to be part of the main table, they are supposed to go between the metals and transition metals (as shown by their atomic number), the reason they are put down as a seperate island is because A4 or letter papers cannon be printed with an extra long periodic table, they are often numbered, but rarely used, that s why Sal did not number them, because the chance of encounting one of these elements in a general chemical question is almost 0%"
    },
    {
        "video_name": "NRFPvLp3r3g",
        "Q": "@6:51, I did not understand the cyclopentane part.. it has C-5, H-10 so isint it an Cycloalkene EVEN THOUGH there arent any DOUBLE BONDS? ( As only pentene has C5H10..) Thanks!",
        "A": "C5H10 has 1 degree of unsaturation. A double counts as a degree of unsaturation, so you re right, this could be the formula for pentene. However, every ring also counts as a degree of unsaturation. In this case, the degree of unsaturation comes from a ring so it s cyclopentane."
    },
    {
        "video_name": "NRFPvLp3r3g",
        "Q": "at 1:10'i did not understand the concept",
        "A": "The first four prefixes were invented to preserve names of organic substances that were already in use. The others are mostly derived from Greek words and tell how many carbon atoms are in the compound."
    },
    {
        "video_name": "NRFPvLp3r3g",
        "Q": "At 3:07 he said it right still someone made it wrong .. he said it as iso and someone made it as ico ... IN internet in searched in google then there are results for iso but not ico ... then which of the these are correct?",
        "A": "He said iso, but its really eicosa. The prefix for 20 is eicosa-, so a chain of 20 carbon atoms is eicosane."
    },
    {
        "video_name": "NRFPvLp3r3g",
        "Q": "at around 4:10 sec Sal gives us some examples of naming the organic compounds, he gives eg of hexane, heptane etc. So i would just want to know that is this name only applicable for carbon-hydrogen molecules or single bond combination of carbon with other atoms also (eg chlorine).?",
        "A": "Anything that ends in -ane indicates a base chain of carbon-carbon single bonds. Substituents (eg chlorine) are named by the number of the carbon they are on and the name of their functional group. For example, CH3CHClCH2CH3 would be 2-chlorobutane. This also applies to branches off the main carbon chain. For example, CH3CH(CH3)CH2CH3 would be 2-methylbutane."
    },
    {
        "video_name": "NRFPvLp3r3g",
        "Q": "When you said (at around 3:50), its an alkane because there are no double bonds, what does that really mean? If it was a double bond, what would it be called?\nWhat is the real difference between a hydrocarbon vs a saturated hydrocarbon",
        "A": "If there were a double bond, it would be called an alkene. You can add two H atoms to an alkene and convert it to an alkane, but you can t add any more H atoms to an alkane. An alkane is said to be saturated. an alkene is unsaturated."
    },
    {
        "video_name": "NRFPvLp3r3g",
        "Q": "Are the hydrogens in 5:43 given in the question or should i know them myself?",
        "A": "Well usually for alkanes, you should know how many hydrogens there are. You can use the formula C(x)H(2x+2). Specifically in 5:43 you should know how many hydrogens there are because Carbon needs four bonds to have 8 valence electrons and without the Hydrogen it only has two bonds, 4 electrons. So you know there must be two more bonds for each with a hydrogen."
    },
    {
        "video_name": "Z9Jh-Q59xso",
        "Q": "@ 11:30,whose electron cloud is likely to distort, iodide or carbon..??",
        "A": "iodide because polarisability is directly proportional to the size of anion. therefore iodide being more bigger will have higher distortion. (Acc to Fajan s rule)"
    },
    {
        "video_name": "Z9Jh-Q59xso",
        "Q": "At 2:38,why is cyanide ion less stable than sulphide ion",
        "A": "only a fool would say that. did he say that?"
    },
    {
        "video_name": "Z9Jh-Q59xso",
        "Q": "@10:38, Sal says that Iodide is more polariziable. I get that this is different from being more polar, but can anyone tell me exactly HOW they're different?",
        "A": "go popo 56:28"
    },
    {
        "video_name": "Z9Jh-Q59xso",
        "Q": "At 6:47, according to IUPAC norms, the name of the compound CH3-CH2-O-CH2-CH3 should be ethoxy ethane, right?",
        "A": "Yes, that is correct."
    },
    {
        "video_name": "Z9Jh-Q59xso",
        "Q": "At 3:35, How can oxygen take an electron like that? After forming a hydroxide ion will the bond pair of electrons be pulled by the hydrogen since oxygen has excess?",
        "A": "It is convenient to think of protons by themselves in solution. Actually, something else, perhaps a water molecule, has given an electron to the proton from the other side so that the oxygen could have its original electron. In the OH bond the electrons are still closer to the O atom, because O is more electronegative than H."
    },
    {
        "video_name": "Z9Jh-Q59xso",
        "Q": "@12:43 \"Fluorine is far more electronegative, fluoride is more basic; it will be more stable if it able to form a bond with something than iodide; iodide is pretty stable...\"\n\nThis statement seems illogical. It is not clear why fluorine being more electronegative makes it more basic and thus more stable when forming a bond (sharing its electrons). Anyone have any ideas?",
        "A": "Ummm, fluorine is too electronegative so it won t depart because it s highly unstable as a lone ion. Iodine on the other hand is the conjugate base of hydroiodic acid; a very strong acid....and since the conjugate base of a strong acid is very weak, it is very stable by itself and makes an excellent leaving group. It s a great nucleophile because it has more space to accommodate negative charge, thereby stabilizing its ability to share electrons."
    },
    {
        "video_name": "Z9Jh-Q59xso",
        "Q": "@2:00 when another electron is added doesn't fluorine get an octet and then if it gains wouldn't it be fluoride anion!?",
        "A": "Yes, that is true. Atoms that accept electrons from other atoms pass to anion and that other in cations , in this case fluorine get the octet and it be in flouride anion"
    },
    {
        "video_name": "TDoGrbpJJ14",
        "Q": "At 12:39, can more than two bacteria be swapping DNA? Could there be a large group all swapping at the same time?",
        "A": "Yes that is possible but that does not happen every time"
    },
    {
        "video_name": "TDoGrbpJJ14",
        "Q": "Starting at about 3:35, Sal mentions that we have about 20x number of bacterial cells as we do our own cells, and how they are equal to about 10% of our mass. What is the recycle rate of this mass? How often do we digest/excrete/etc this mass?",
        "A": "It s a constant process. I m afraid I don t know how long it takes to completely replace all of it, but it s constantly in a state of flux. The same is true of the non-bacterial or human cells, and going further, the same is true of the individual atoms within all those cells. I suspect the rates are not uniform, either. I do know that every atom in the human body is recycled in a period of several years."
    },
    {
        "video_name": "TDoGrbpJJ14",
        "Q": "At 14:19 it says antibiotics will kill bacteria, but will it also kill the \"good\" kind of bacteria we human have a lot of it?",
        "A": "Yes, take antibiotics only when you need them- and the right kind, in the right quantity. Bacteria with different chemical compositions will be affected differently by different antibiotics."
    },
    {
        "video_name": "TDoGrbpJJ14",
        "Q": "AT 1:45 he says you mix milk and bacteria to make yoghurt, but surely there should be other ingredients and it could also be bad for you if you have bacteria in your body just from eating yoghurt??",
        "A": "Agreed - but yeast is not a bacteria, it is a eukaryotic organism!"
    },
    {
        "video_name": "TDoGrbpJJ14",
        "Q": "At about 12:50, it is some mentioning about bacterial conjugation. My question is; over what range of time does this conjugation take place?",
        "A": "Conjugation takes minutes. The entire genome of E. coli would take about 100 minutes to transfer via conjugation"
    },
    {
        "video_name": "TDoGrbpJJ14",
        "Q": "Are bacterias that move fast? Like at around 10:20 Sal talks about how bacterias have flagellas to help them move, so I was wondering if when they move, they are fast?",
        "A": "That depends on the bacterial species present. Some move faster than others"
    },
    {
        "video_name": "TDoGrbpJJ14",
        "Q": "8:22 What's the difference between Bacteria and Archaea?",
        "A": "they are the most primitive organisms and are known as living fossils they don t have mucopeptide substance in their cell wall they can survive in harsh conditions"
    },
    {
        "video_name": "TDoGrbpJJ14",
        "Q": "At 10:56, Sal says that bacteria use \"Mutatioins\" to multiply. How often to mutations occur?",
        "A": "The rate is roughly .003 mutations per genome per generation in prokaryotes"
    },
    {
        "video_name": "TDoGrbpJJ14",
        "Q": "At 9:39 Sal states that it replicates it's nucleoids,when it does, does it split immediately?",
        "A": "Yes. After the DNA is copied, the cytoplasm and membrane immediately start to grow enough to separate the two DNA strands from themselves."
    },
    {
        "video_name": "jYR0RlMft8Y",
        "Q": "At 0:35, do all enzymes look similar or closely similar to this?\nOr, can they look a little different?",
        "A": "At 5:31, Sal said that enzymes DO NOT usually look like this in biological systems. Anyway, all enzymes are different."
    },
    {
        "video_name": "jYR0RlMft8Y",
        "Q": "Can hexokinase only catalyze one pair of substrates at a time? The diagram Khan shows us in 2:15 seems like the hexokinase is very large compared to the substrates (Glucose and ATP) so are there multiple places in the enzyme that multiple substrates can \"lock on\"?",
        "A": "I m not sure this applies to hexokinase, but in some enzymes, there are allosteric sites. Inhibitors can go in allosteric sites and prevent substrates from binding. If you still don t get this, just look at the competitive and noncompetitive inhibition videos. Hopefully this all made sense."
    },
    {
        "video_name": "HcpPPEG2wFg",
        "Q": "at 2:56 you said that SN1 and E1 are out, why are they not being considered?",
        "A": "1) The SN1 and E1 mechanisms require the formation of a carbocation. 2) For the molecule shown (1-bromopentane), you can only get a primary carbocation. 3) Primary carbocation are unstable (explained in earlier videos). Jay mentions this for SN1 starting at 0:12."
    },
    {
        "video_name": "HcpPPEG2wFg",
        "Q": "At 0:53, why will SH act only as a nucleophile and not as a base?",
        "A": "He says SH is a weak base since its conjugate acid is pretty strong."
    },
    {
        "video_name": "HcpPPEG2wFg",
        "Q": "At 3:19, in the third example, why was E2 ruled out?",
        "A": "Because nucelophilic character of OH- is high compared to base character. Hence SN2 outweighs E2"
    },
    {
        "video_name": "garegCgMxxg",
        "Q": "At 1:33 David says that 2 whole revolutions is a displacement of 720 degrees. But isn't displacement the OVERALL change in angle? In this case, it would be 0.\n\nStraight after that, he said that moving 270 degrees is actually a displacement of 90 degrees. Isn't that contradictory to what he said about the 2 revolutions? If 270 degrees is actually a displacement of 90, then 2 revolutions is a displacement of 0",
        "A": "Displacement is measured from your starting point. 90 degrees clockwise is the same displacement as 270 degrees ccw. You can t tell the difference between them if you didn t see how you got there."
    },
    {
        "video_name": "garegCgMxxg",
        "Q": "At 2:35 David mentions about the Right Hand Rule, I still couldn't understand how does the thumb tells us the direction? Thank you",
        "A": "Just bend your fingers of RIGHT HAND and try to place it over the small part of perimeter of the circle. Now try to curl the finger in the direction where particle is moving, direction of your thumb in this process will give u the answer. For an example, If ball is moving clockwise, you will curl your finger clockwise and your thumb will try to go inside the screen or paper and that is the answer."
    },
    {
        "video_name": "garegCgMxxg",
        "Q": "At 1:19 when Dave says that if you were to revolve around the circle twice, the displacement wouldn't be two revolutions. Why would it not be zero because it ended at the same place it started?",
        "A": "Because he is not talking about displacement, he is talking about angular displacement."
    },
    {
        "video_name": "garegCgMxxg",
        "Q": "At 10:40 Dave said that the Final Angular Velocity is 1.57 rad/sec. Should the Final Angular Velocity be Pie/4 = 0.7853 per the formula mentioned earlier in the video. Could anyone please let me know why was 1.57 rad/sec calculated?",
        "A": "See, if the angular velocity was constant during the whole journey, the answer to final angular velocity (angular velocity at the end point) will be 0.7853 rad/sec. In this case, angular acceleration would have been 0. However, here the angular velocity is given to be increasing, so it increases from 0 to 1.57 rad/sec (which was given in the question). This ensures that there is some angular acceleration."
    },
    {
        "video_name": "KsdZsWOsB84",
        "Q": "At 7:04, why do those electrons kick back onto the oxygen when the silicon is fine with five bonds? Also, if a bond does have to be removed, why the Si-O bond, why not the bond to the tertbutyl for example?",
        "A": "The electrons kicked back to the oxygen because when you protonated the molecule, the oxygen picked up a hydrogen, giving it a +1 charge. While silicon is okay with five bonds, oxygen is very electronegative, so it wants its electrons back. That s also why it wouldn t be the tertbutyl bond that leaves, because the bonds to silicon aren t the issue, the bonds to oxygen are."
    },
    {
        "video_name": "KsdZsWOsB84",
        "Q": "At 6:40 why isn't the formal charge 6valence e's - 5bonds = +1 ?",
        "A": "Silicon has 4 valence electrons as it is in group 14."
    },
    {
        "video_name": "KsdZsWOsB84",
        "Q": "At 2:44, how do we lose HCl in the process? Don't we just lose a Cl? Where did the H come from?",
        "A": "The H comes from the OH, he underlined it. In the video he said the mechanism for removing HCl was complicated, which means it s really not worth worrying about exactly how it happens."
    },
    {
        "video_name": "6FTiHeius1c",
        "Q": "This may be a dumb question, but at 5:38, when he is talking about something's velocity jumping from one value to another instantaneously being unrealistic, wouldn't one thing hitting something else, for example a baseball bat hitting the ball cause the ball's velocity to jump instantly?",
        "A": "hi, in case of instantaneous velocity you can t get the exact instant. its just a convention. we find it using avg velocity between extremely small time interval. lets say an intreval between 5.00000000001 and 4.99999999999999 can be used in such cases. that is why we use limits there."
    },
    {
        "video_name": "6FTiHeius1c",
        "Q": "At 5:55 Sal giggles as his example assumes an Instantaneous acceleration to double something's speed; Light travels at nearly half speed through a diamond (ok, a little less than), and of course at c through a vacuum, with presumably an instaneous acceleration. Sal's example makes sense if it is measuring a beam of light which travels through a diamond for the first two seconds, then a vacuum from then on.\n\nThis diamond would have to be 299 792.458 kilometers thick.",
        "A": "well even if the light analogy breaks down with given math, we couldn t experimentally determine the rate at which light accelerates between the surface of the diamond and vacuum. This is due to nature but our inadequacy is modeled by Heisenberg s uncertainty principal. basically it s too low mass and moving way too fast to know velocity and position at the same time. TLDR: Only mathematically impossible."
    },
    {
        "video_name": "A8xO92u0VFs",
        "Q": "at 2:30 how does the doctor know he needs a pressure of 115? In our case someone else has already measured it, but to determine the pressure without knowing it before which pressures do you use?",
        "A": "The procedure for checking blood pressure is to inflate the cuff to a point where is cuts off the pressure entirely. The examiner will listen using the stethoscope and slowly release pressure while watching the sphygmomanometer (gauge that tells you mmHg). Once the examiner can hear sounds, this will be your systolic blood pressure, once the sounds go away and continuing to release pressure, this number will be your diastolic."
    },
    {
        "video_name": "nFSMu3bxXVA",
        "Q": "The example with skater at 10:16 freaks out. What if the object could be infinitely large - then its angular velocity would be 0 and it would be in rest. So is it related to the other objects in rest even though they don't have infinite mass?\n\nThis trick is hard to understand. Isn't the skater is doing some kind of work with her arms because they are connected to the body? isn't it like friction?",
        "A": "Work can never be done by a centripetal force because the force is always perpendicular to the displacement, and W = FScos(theta)"
    },
    {
        "video_name": "nFSMu3bxXVA",
        "Q": "Around 1:40, Sal mentions that if there is no impulse or net force, then there will be no change in momentum. That much makes sense. But Sal also mentioned that without impulse or net force, there is a conservation of momentum.\n\nIsn't momentum always conserved in a system, especially if it is an isolated system? That is, the momentum that is lost by one object is gained by another?",
        "A": "momentum is always conserved in a closed system If you apply an impulse from outside the system then the momentum in the system will change If you take a broader view of the system to include that source of impulse, then still momentum was conserved."
    },
    {
        "video_name": "efQjBTPhS_0",
        "Q": "at 1:48, how do you sawdor?",
        "A": "You heat up the wires that you want to solder to with the iron and then you apply the end of the coil of solder to the wires and let the solder wick around the connection."
    },
    {
        "video_name": "efQjBTPhS_0",
        "Q": "I would have thought that the hot glue @4:25 would/could possibly damage the boards circuitry, why doesn't it? and should one use a special hot type of glue?",
        "A": "It does not damage it because it is not touching any part of the circuit that would make it faulty. Also, It can take the heat. Hopes this helps. For more info, Google is your friend."
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "What is a mole of a molecule? (At 1:54)",
        "A": "Approx. 6.022 x 10^23 molecules"
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "At 9:34 Sal says a buffer is a solution of a weak acid in equilibrium with its weak conjugate base\nI thought a weak acid has a strong conjugate base?",
        "A": "A strong base has a conjugate of what is considered a negligible acid , while a strong acid has a conjugate of a negligible base. However, a weak acid will have a conjugate of a weak base and vise versa. Think of it as a three tier, interchangeable system, where the degree of variation on one side forms the degree of variation on the opposite side"
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "I'm confused... On 12:46, shouldn't it be \"plus\" instead of \"minus\"? Thank you.",
        "A": "In general, log(X*Y) = logX + logY But remember, Sal took the NEGATIVE log of the equation. So what he did was -log(XY) = -[logX + logY] = -logX - logY but he skipped the middle step. Sometimes he also writes it this way: -log(XY) = -logX + -logY"
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "Why would increasing the OH at 5:22 make the pH go up? I thought that the pH was directly linked with the hydronium concentration and more OH would raise the pOH?",
        "A": "OH is called hydroxyl . If we increase the OH we will certainly decrease the concentration of hydronium. So , as. A result the pH will rise and the substance will become basic in nature"
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "at 1:55, what is a mole of a molecule?",
        "A": "Approx. 6.022 x 10^23 molecules"
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "Is the NaOH in the beginning (1:45) example added to the products or reactants side of the equation?",
        "A": "He s adding NaOH to the mixture which might be in a beaker. The mixture consists of HA, H* and A- (plus water) in equilibrium. However, the HO- ions react with the H+ ions which are on the right hand side of the equation. Neutralising the H+ ions leads to more HA dissociating into H+ and A-."
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "at 1:55, what is a mole of a molecule?",
        "A": "Approx. 6.022 x 10^23 molecules"
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "At 2:51, shouldn't it be 'weak acid' instead of 'weak base'?",
        "A": "Yes. The HA is a weak acid."
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "If HA dissociates into H and A then doesn't it go right back into producing more HA and the process continues or does Le Chatelier's principle stop the reaction once the HA Dissociates.\n(Min 3:15)",
        "A": "Yes, it does, but the process doesn t stop. Rather, the opposing reactions occur at the same rate. We have a dynamic equilibrium, in which the concentration of everything stays constant, even things are going back and forth all the time."
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "Hello, at 9:32, Sal said a buffer is just a solution of a weak acid with its conjugate weak base. But isn't the conjugate base of a weak acid, strong base? Thank you!",
        "A": "No, it s not true that the conjugate base of a weak acid is always a strong base."
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "At 1:10, how is it possible to stress a equilibrium?",
        "A": "you could increase the volume of one of the components to stress the equilibrium. Or you could increase the pressure. And so on....... There are many ways to do that."
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "At 2:51, how is HA a weak base ? Sal mentions at the start of the video (0:21) that it was a general way of writing an acid. Please help ! !",
        "A": "He misspoke. It s a weak acid."
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "What is a mole of a molecule? (At 1:54)",
        "A": "1 mole of a molecule is 6*10^23 molecules of that substance"
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "at 0:35 Sal says that all of this is in a aqueous solution that is a water solution mixed with something, right? if not can someone tell me what it really is.",
        "A": "Correct, aqueous means dissolved in water. In this case, the acid being dissolved in water"
    },
    {
        "video_name": "LJmFbcaxDPE",
        "Q": "In the equation Sal talks about at around 3:50, if the HA were to go down, why doesn't the A-on the other side go down as well? Wouldn't both the H+ and A- on the right side go down initially?",
        "A": "no, only the H would go down because it gets eaten up by the OH- in order to make water. In response to the H going down, the HA goes down in order to maintain equilibrium by increasing H+. SInce H+ and the A- are on the same side, they both increase. A- doesnt go down at first because OH- only reacts with H+. I hope this helps :)"
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "At minute 4:50 Sal says the work is equal to 6N times 2 meters. But the force needed to go against the field would be -6N right? Wouldn't it be -6N times (finish distance minus start distance)...so 1 meter minus 3 meters...which gives W= -6N x (1m-3m) = 12 J.",
        "A": "Work is force*displacement If the force is in the same direction as the displacement, work is positive If it s in the opposite direction, work is negative"
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "at 4:34 you said that we have to apply 6N force from the left side in order to overcome the electric force from left i.e. 6N\nSo my doubt is how the charge would move?\nwon't it come to rest if we apply equal force from both sides?\nhow does it gains the uniform velocity?",
        "A": "You would need to apply a tiny bit of extra force to accelerate it. You would also need to apply the same force, in the opposite direction, to make it stop. Think of balance forces as enabling a constant velocity to persist, not to bring an object from rest to constant velocity."
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "At 2:00 why is the electric field constant at any distance from the plate? Isn't the electric field weaker farther away, because it is inversely proportional to distance squared?",
        "A": "No, the inverse square law you are trying to apply is for a point charge or a sphere. An infinite plate is neither of those."
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "In 7:43 , the ' mgh' law regarding gravitation was given . Why do we use charge instead of mass in electric potential ?",
        "A": "because charge feels a force in an electrical field mass feels a force in gravitational field OK?"
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "Sal said at 5:46 that electric potential tells us how much work is required to move per unit of charge from one place to another. So if I increase the charge of the charged particles to a hundred coloumbs, will the electric potential remains the same?",
        "A": "Yes electric potential will remain the same.By definition, it is the word done per unit charge to bring to a position with respect to a charge, from infinity, or in other words, the word done to bring a UNIT POSITIVE CHARGE from infinity to that position.Thus, electric potential is a property of a POINT(or position) in an electric field.-So it depends only on the electric field at the point"
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "at 4:32,\nSal said the plain will always apply a force of 6N\nBut as the charges draw close value of r square will reduce.\nHow will force stay constant?",
        "A": "That was a question that was making for myself right now, but remember the definition of Electric field, which really is the electric force experienced by the charge. It means that, by definition, the force will always be constant because the particle is in a uniform electric field and the charge of the particle does not change."
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "In 0:59 Sal says that Electrical Potential Energy is associated with a charged. I've look for that statement in some textbooks and I've found that this is not the case, but EPE is associated with position. He confuses EPE and Electric Potential (voltage). Could anyone bear out this?",
        "A": "All potential energy is associated with relative positions. But that s not the only thing that matters. Electric PE of a system depends on the charge and the position of that charge relative to other charges. Similarly Gravitational PE depends on the mass and the position of the mass relative to other masses Electric potential is electric PE per unit of charge."
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "at 4:58 ....you wrote that w=12 j...how can you say that? as force is continuously changing....we have to use calculus....plz explain",
        "A": "Who said so? Force acting is constant and so the calculus is not required."
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "At 5:15,sal said that the particle that is closer to the plate will have higher potential energy but isn't potential energy affected by height? Isn't it that the particle at a higher distance will have more potential energy? Why is it that the particle closer to the plate has more potential energy?",
        "A": "We are talking about electric potential energy, not gravitational. Gravitational PE is mgh. Electrical potential energy is qV."
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "(4:47 to 4:58 and 6:30 to 6:35)W=4N.2C=8J or 8NC\nV=W/charge=8NC/2C=4N\nDoes that mean votage is a kind of force?",
        "A": "force=Electric field*charge , work done=force*distance , voltage=work done/charge..... IE-voltage=electric field*distance now you see force(electric) and and voltage are clearly different it was just the numbers which turned out to be same"
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "At 6:02, Sal says that Electric potential is how much work is done per unit charge. That is the same definition given in my book for electromotive force. Are electric potential, potential difference and electromotive force all the same? Or, are there any differences?\n\nThanks in advance.",
        "A": "the electomotive force is potential difference but it is in this case it is that potential difference in the cell in an open circuit please vote if you like !"
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "At 5:00 shouldn't it be -12 J because we are moving in the \"Leftward\" direction?",
        "A": "There is no -12J, energy is always positive, only vectors have - and +."
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "@1:56 sal said let the electric field be constant\nmy question is that in previous video , we saw as we move away from source of field, the field decreases, but how come as we move away from infinite plate of charge, the field remain constant ? why there is no decrease in its value as we move away ?",
        "A": "Because the plate is infinite That s a lot different from a point charge Think about what an infinite plate looks like as you move further away from it. Doesn t it always look the same no matter how far away you are?"
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "At 7:03 Sal says that electric potential is independent of charge of the particle. How is that true, if to calculate the electric potential, we divided the work by the charge of the particle? Doesn't that mean that electric potential is in fact dependent on charge of the particle?",
        "A": "It s true BECAUSE we divided the work by the charge. If we didnt do that, then what we would have would be potential energy, which IS dependent on the charge of the particle, because it takes more work to move a bigger charge. If you want to adjust for that, you have to do work per unit of charge, which is what potential is."
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "at 6:40, since we divide by the charge, can we define electric potential as electric field times distance?",
        "A": "Yes! V=E.d where V is potential , E is field and d is distance."
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "At 2:05 Sal says that the electric field vectors should all be the same length, how is this true? I thought the strength of the electric field is always dependent on how far away the charge is?",
        "A": "That s for a point charge. This is an infinite plate. Different."
    },
    {
        "video_name": "zqGvUbvVQXg",
        "Q": "At 8:15, did you say \"Voltage\" when you meant to say \"Electric potential energy\"?\nIf not, I'm confused.",
        "A": "No, he meant potential difference. Voltage is the term electrical engineers tend to use when they mean potential difference. Physicists say potential difference."
    },
    {
        "video_name": "oHvLlS_Sc54",
        "Q": "At 10:50, he mentions 'Eukaryota' and said that this was of organisms with complicated cell structures. He also said you get Prokaryota (not sure of the spelling) which is the opposite. Would an amoeba be an example of this?",
        "A": "Amoeba is actually listed under the Kingdom Protista in the Super Kingdom Eukaryota."
    },
    {
        "video_name": "oHvLlS_Sc54",
        "Q": "At around 2:58, Sal mentioned that till this day, people are still debating whether this method is the best way to classify organisms. So may I know what is the other method(s)? Thanks.",
        "A": "When everything was classified as either animalia or plantae, biologists already knew that there were microscopic life. It was just that bacteria, photosynthetic protists, fungi and plants were considered as plants. But that was oversimplifying things which was why Robert H. Whittaker proposed a five Kingdom classification. :)"
    },
    {
        "video_name": "oHvLlS_Sc54",
        "Q": "At 1:45, how many different kingdoms are there and which has the most creatures?",
        "A": "Technically it would be Bacteria, but I wouldn t get too hung up on taxonomic ranks since the whole system was thought up before Phylogenetics was even around."
    },
    {
        "video_name": "oHvLlS_Sc54",
        "Q": "What was that box and scribbles that Sal was drawing when talking about viruses and whether they are alive or not? (11:32)",
        "A": "viruses have DNA/RNA they don t have a cell body like bacteria and fungi.. so they can t be treated with antibodies. and they only multipy in another body."
    },
    {
        "video_name": "oHvLlS_Sc54",
        "Q": "Does \"Kingdom\" mark the top most tier in classification or do we have any other term that encompasses \"Kingdom\" too?\nAlso, at 4:21 Sal refers to humans as Homo sapiens but I'm sure that I've come across \"Homo sapiens sapiens\" somewhere... So what's the correct nomenclature in this case... Please help!",
        "A": "Domain is greater than Kingdom."
    },
    {
        "video_name": "oHvLlS_Sc54",
        "Q": "At 5:42 would the bonobos be in the Hominini Tribe (and maybe even the Pan genus with the chimpanzees) ?",
        "A": "Yes you are correct bonobos (Pan paniscus) are in the Hominini tribe and the Pan genus"
    },
    {
        "video_name": "oHvLlS_Sc54",
        "Q": "At about 9:00, Sal said \"Let's go to the phylum.\" I'm wondering... what's a phylum? I don't know that much...",
        "A": "Phylum is a category or rank chosen to describe the position of a particular group of living beings (animals or plants) in the Linnaean hierarchy. For example - Phylum : Chordata. This means, the phylum which is a rank includes all animals which are chordates which possess a distinct notochord"
    },
    {
        "video_name": "oHvLlS_Sc54",
        "Q": "At 7:24 why are humans in the order primates?",
        "A": "Our DNA matches closely to that of other primates. If you observe closely, all have a striking resemblance. Hands with thumbs, developed brains, relatively flat face, ....."
    },
    {
        "video_name": "Nk2q-_jkJVs",
        "Q": "5:26 what does that mean",
        "A": "Well, I hope I m answering your question by explaining this: to graph what s happening with the mass, x(t) resembles the mass position at time t . So, when it s supposed to be gaining kinetic energy, it s slope is much, much greater, as opposed to when it stops moving at either A or -A , when it s slope is 0. Sal explains it well when it s accelerating or decelerating so I ll leave my answer as that... hopefully this helps."
    },
    {
        "video_name": "Nk2q-_jkJVs",
        "Q": "in 5:24 what does the formula next to the diagram mean?\nF=Kx,and X(t)=?\nalso does Cos stand for the cosine",
        "A": "The force applied by a spring on a mass is given by the model F_spring = \u00e2\u0080\u0093kx. k is called the spring constant and it is a measure of the stiffness of the spring. x is the displacement of the spring from its equilibrium position. The negative sign indicate that the spring applies a force in the opposite direction of the displacement. x(t) is the displacement from equilibrium as a function of time. And cos is the cosine function."
    },
    {
        "video_name": "0r_JWUkXEkY",
        "Q": "At 13:51 what happens to the D^2?",
        "A": "D is the same as R, so he just uses algebra to cancel them out"
    },
    {
        "video_name": "0r_JWUkXEkY",
        "Q": "In the final question around 15:26, why didn't you cancel out the 4s in the top and bottom of the second velocity equation?",
        "A": "The goal is to compare the two scenarios. The first station s equation has a 4 in the denominator. We, therefore, factor out the 4 in the numerator of the second station s equation, leaving us with a factor of 2 in front of the first station s equation."
    },
    {
        "video_name": "632MAqIB14E",
        "Q": "Why, at 2:26, didnt the negative O take a hydrogen from the water group attached to the carbon?",
        "A": "Because water itself is one of the best leaving groups. So it will wanna leave if there is a base nearby in the surroundings."
    },
    {
        "video_name": "632MAqIB14E",
        "Q": "At 6:31, you said that the more electrophilic the carbonyl carbon gets, the more the nucleophile will attack it. Could you please elaborate on this?",
        "A": "Electrophiles increase the positive charge on carbonyl carbon by taking the electron density towards them. Thus the bond between Carbonyl Carbon and the oxygen becomes more and more polar. This increases the reactivity of Oxygen with hydrogen (hydration) as Oxygen becomes more negative and is easily able to react with hydrogen."
    },
    {
        "video_name": "632MAqIB14E",
        "Q": "1:17 is it true that when we have a weak nuclephile we have to first protonate the oxygen to get a plus formal charge on carbon (so that it is more electrophilic)",
        "A": "Yes it is true in order to occur the reaction."
    },
    {
        "video_name": "632MAqIB14E",
        "Q": "2:04 Could you have intramolecularly transferred the hydrogen on the left to the negative oxygen on the right or can you only transfer hydrogens connected to atoms that share a bond and these are too far?",
        "A": "In most case, the reaction does not undergo the intramolecular process you mentioned. The molecule is more likely to bump into a water molecule with the right position with the enough activation energy. And the intramolecular process you mentioned, if possible, would form a 4 atom ring with the angle of 90. The significant angle restrains suppress such process from happening. :)"
    },
    {
        "video_name": "632MAqIB14E",
        "Q": "At 2:00, the direction from which H2O attacked seems confusing to me. Wouldn't there be steric hinderance due to the R group? What I think is that it should attack from between the R group and the hydrogen. Please explain if I'm wrong (most probably :p)",
        "A": "The H\u00e2\u0082\u0082O is attacking perpendicular to the plane of the aldehyde group."
    },
    {
        "video_name": "632MAqIB14E",
        "Q": "At 4:05, would the product not also be an acetal? Or are hydrates a form of acetals?",
        "A": "No, the product at 4:05 is not an Acetal. Acetals have two -OR groups that branch off the same carbon, at 4:05 the carbon has two -OH groups. To make an acetal, you need to use alcohols. Check out the video on acetals for more info!"
    },
    {
        "video_name": "z_KX8Xpxa-c",
        "Q": "At 8:00, isn't it supposed to be a graph of the inverse of linear mass density? If m/L is the linear mass density, instead of the function f(m/L), shouldn't it be f(L/m)?",
        "A": "Yes he messed up but the answer is still no because in the function the inverse of linear mass density is still under the radical."
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "At 7:35 Sal forgot to place a positive charge on oxygen (water acting as nucleophile donating its lone pair to the carbocation being the electrophile)...However, I have just finished watching this video and he corrected himself at the end of it....At first it looked confusing....",
        "A": "Sal seems to be good at that sometimes. Well spotted :)"
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "At 7:28 Sal says that one electron is moving over ( the orange one) yet he draws a double barbed arrow. I am confused because I can't understand if he is moving just one electron ( only the orange one) or a pair of electrons( orange+ green). Could some one clarify?",
        "A": "It should be a pair of electrons (for example, orange and green) that forms the bond between the oxygen and carbocation, not just the single orange electron."
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "In the last part (10:13) , Sal was a little result orientated, Can we reason it like follows?\n\nHydrogen is more electro-positive than Carbon so Oxygen can attract an electron from hydrogen a little easier than it can from Carbon. Thus we can say that the oxygen attracts the electron from hydrogen which forms a partial positive charge on Hydrogen and thus, the hydrogen attracts an electron from the water molecule around.",
        "A": "I agree with your reasoning. It s also important that the acidity of the molecule is higher than that of the surrounding solution for the proton to move to a water molecule. Protonated alcohols have a pKa of -2.5 so that s a driving part of the last step."
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "At 0:41,from where the Hydronium ions came?\n(please never mind my bad English)",
        "A": "They come from a strong acid (usually sulfuric acid) that is added to the water to generate hydronium ions. H\u00e2\u0082\u0082SO\u00e2\u0082\u0084 + H\u00e2\u0082\u0082O \u00e2\u0087\u008c H\u00e2\u0082\u0083O\u00e2\u0081\u00ba + HSO\u00e2\u0082\u0084\u00e2\u0081\u00bb"
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "At 0:35 Sal should not have put a dash between methyl and pent- since the dash is only used to distinguish between numbers and words.",
        "A": "You are correct, he should not have put a dash there."
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "At 11:28, I didn't get why that oxygen has positive charge?",
        "A": "Formal charge = valence electrons - lone pair electrons - bonds For that oxygen: FC = 6 - 2 - 3 = +1 You can otherwise just remember: An oxygen with no formal charge has 2 lone pairs and 2 bonds An oxygen with a +1 formal charge has 1 lone pair and 3 bonds An oxygen with a -1 formal charge has 3 lone pairs and 1 bond"
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "At 6:41, it is said that the equation is in equilibrium. How do we know that a particular equation is in equilibrium? How can we generalize the idea of equilibrium while solving such problems?",
        "A": "The reaction is only said to be in equilibrium because a catalyst (H+) is involved. You do not ever have to calculate the position of equilibrium or Q or Keq for elementary processes such as these in organic chemistry. (not as an undergrad anyway)"
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "At 9:15 why does Sal not put the positive charge on the oxygen atom?",
        "A": "It should be there. Apparently, he forgot."
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "at 0:32 why is it not named 2-methyl-2-pentdiene, instead of 2-methyl-pent-2-ene?",
        "A": "Diene would mean that there are two C=C double bonds. The 2-ene doesn t mean two double bonds; it means that the one double bond is between carbons 2 and 3."
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "At 2:15, you can see Sal changing the water into the H30. When he does that, he takes away two electrons for the other hydrogen to be added. Why is this? Shouldn't there be three remaining electrons, and not two? Thanks.",
        "A": "Note that it is a hydrogen ION (so it has 0 electrons) added to water not a hydrogen atom (which has 1 electron), the oxygen lone pair of electrons has become the covalent bond that he highlights in green in a few seconds."
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "3:26 you say the 2 carbon will be secondary, but it will actually be a tertiary carbocation, and the 3 carbon would be secondary, not primary. Love the videos.",
        "A": "C2 will be tertiary because it s bonded to 3 carbons. (C1, the C from methyl, and C3) C3 will be secondary carbon because it s bonded to 2 carbons (C2 and C4)"
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "Why doesn't attack OH- the carbocation at 07:05 ? it is much simpler, isn't ? I mean in the end of the theory Sal tells us there is a free proton left and this will bind to the OH- anyway, right? Would be the same result...",
        "A": "Because there are so many more water molecules than hydroxide ions. In a 1 mol/L of HCl, for example, the concentration of water molecules is about 55 mol/L, while the concentration of hydroxide ions is about 10\u00e2\u0081\u00bb\u00c2\u00b9\u00e2\u0081\u00b4 mol/L. The chances of the cation colliding with a water molecule are about 6 .x 10\u00c2\u00b9\u00e2\u0081\u00b4 times more likely than of its colliding with a hydroxide ion."
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "At 3:18 Sal says that the carbon bonded to two methyl groups and doubly bonded to a carbon is secondary.I think it should be tertiary carbon. Am i correct?",
        "A": "the carbon bonded to the two methyl groups is a secondary alkene carbon. i think thats what Sal meant...."
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "At 8:31, shouldn\"t Sal have put a positive charge on the oxygen atom?",
        "A": "Yeah, there should be."
    },
    {
        "video_name": "O_yeKo6-qIg",
        "Q": "At around 7:25, he mentions that a single electron is being transferred to the carbocation, and this forms a bond. If that were true, wouldnt the arrow be a single-headed instead of double-headed?\nAnd I was always under the assumption that this was a two-electron transfer, hence, the double-headed arrow is needed.",
        "A": "It would be two electrons going, as the carbocation has only six electrons in its three bonds, and needs another two electrons."
    },
    {
        "video_name": "2WS1sG9fhOk",
        "Q": "At 8:24 Sal said that m multiplied by v^2 divided by 2 is equal to how much work imputed to get into that velocity. But, work done is force muliplied by displacement.\n\nDoes that prove that these both can be used as the equation to derive work",
        "A": "Yes, every term in physics seem to have more than one formula, mostly due to difference in units, (i.e. Work can be expressed in calories, Calories (kilocalories) and Joules)"
    },
    {
        "video_name": "2WS1sG9fhOk",
        "Q": "At 0:38, I thought work is a push or pull. Or is that force?\nAnother thing, how do you find the final velocity?",
        "A": "Work is defined not as a push or a pull, but as displacement. If it was a push or a pull, Gravity would be considered work, even though it s not necessarily moving anything."
    },
    {
        "video_name": "2WS1sG9fhOk",
        "Q": "At 2:12, you write the unit Newtons? What are Newtons? How was it created? Then also at 3:04, you say the unit joules. What are they? How was it created? Thanks!",
        "A": "Newtons are the unit of force Joules are the unit of energy If you don t know these things you need to go back to earlier physics videos, starting perhaps with Newton s laws of motion"
    },
    {
        "video_name": "7Zin8hG9Nhw",
        "Q": "At 1:57, Jay cites Newton's Second Law of Motion. Isn't this classical physics? So, even though they disagree on the nature of light, you could say that quantum physics is (at least sometimes) derived from classical physics? Is the blackbody experiment and wave-particle duality the only thing that classical physics can't explain (and that quantum physics can)?",
        "A": "Yes, it s classical physics. The Bohr model is a classical model, with quantization forced on it. Bohr did not know about quantum mechanics. It came later, partly in order to explain why his model worked."
    },
    {
        "video_name": "psLX080RQR8",
        "Q": "at 1:14 what is a Haber process? What does it mean?",
        "A": "Haber process is the process of producing ammonia with N2 and H2, as in the equation shown in the video."
    },
    {
        "video_name": "psLX080RQR8",
        "Q": "at 11:00 why did it go from moles to molecules?",
        "A": "Because he used the Avogadro constant to convert it"
    },
    {
        "video_name": "psLX080RQR8",
        "Q": "At 0:30, so you haven't been using real numbers?",
        "A": "He has been using a, b, c and d in almost all the Reaction Rates videos so far. Hope this helps :)"
    },
    {
        "video_name": "psLX080RQR8",
        "Q": "At 11:00 in order to find the probability shouldn't we multiply the (0.15 molecules/dV) times the volume of each hydrogen molecule?",
        "A": "The volume of a hydrogen molecule is a funny thing. The molecule is flying about in its space. There are 0.15 molecules in this tiny box at any one time. This means about every 7th time you look into the box, there would be a molecule inside. At absolute zero, the molecules would stop moving and several molecules could be stacked in that box. There would be no reaction because there would be little energy to cause the reaction. I hope this helps conceptualize the situation"
    },
    {
        "video_name": "psLX080RQR8",
        "Q": "At 7:13 when you are multiplying p(h2), does the subscript 2 affect any of the probabilities ?",
        "A": "No, the subscript is help defining what the molecule is (two hydrogen atoms covalently bonded). It is this molecule (H_2) that is a reactant. However, what will affect the probability is the coefficient in front of H_2 for the reaction (3). You need 3 molecules of H_2 in the area of one N_2 in order for the reaction to occur, and thus influences the probability."
    },
    {
        "video_name": "D1eibbfAEVk",
        "Q": "At 7:29, Sal says that you can get earthquake activity outside of plate boundaries. I understand how the islands are forming, but why are there earthquakes?",
        "A": "try rubbing two pieces of sandpaper together and listen to the sound it makes. next try crushing a soda can, or a plastic bottle. Now imagine doing that as big as mountains. the crinkling and grating sounds are what causes earthquakes."
    },
    {
        "video_name": "D1eibbfAEVk",
        "Q": "Around 6:30 Sal was saying that Kawaii was so small, because it had been eroded over the years. Does this mean that eventually Kawaii will erode so much that it disappears?",
        "A": "Yes, that happens to everything, eventually. Even Mount Everest."
    },
    {
        "video_name": "D1eibbfAEVk",
        "Q": "Did Sal say \"pluves\" or \"plumes\" at 1:52? And if so, why would they be plumes and not a steady stream of magma circulation? What causes these inconsistencies?",
        "A": "Hot mantle material can flow faster than it can rise because of buoyancy. This causes the typical mushroom shape of plumes."
    },
    {
        "video_name": "D1eibbfAEVk",
        "Q": "What is a Hot spot? (2:27)",
        "A": "A hot spot is a place in which geological activity frequently takes place, such as a hole in the crust or a volcano."
    },
    {
        "video_name": "D1eibbfAEVk",
        "Q": "At 3:24 does sal mean million or billion?",
        "A": "I believe he means million"
    },
    {
        "video_name": "PZIoFD_Z73M",
        "Q": "At 4:19, what does it mean to have 10% uncertainty?",
        "A": "Let s say you measured something, and your answer was 4, but there is 10% uncertainty in your answer. This means that your answer might be as much as 10% off, or it might be perfect. You just don t know. The actual value of your answer might be anywhere between 3.6 and 4.4 (since 0.4 is 10% of 4). Another way to write this is: 4 \u00c2\u00b1 0.4. Hope this helps!"
    },
    {
        "video_name": "PZIoFD_Z73M",
        "Q": "At 5:06, why do we multiply the equation by .1 when there is a 10% uncertainty? Don't quite see the reasoning or the logic behind doing this.",
        "A": "The equation for delta p (the uncertainty in the momentum) is equal to mass*velocity*delta v (the uncertainty in the velocity). Sal says 4:16 that the uncertainty in the velocity is 10%. You have to convert the percent to a decimal in order to use it in the equation above, and 10% is equal to 0.1; hence, delta p = mv(0.1)."
    },
    {
        "video_name": "PZIoFD_Z73M",
        "Q": "at 2:40 they say that less the uncertainity of the position of electron,more the uncertainity of momentum of electron can anybody please explain me why?",
        "A": "Because in both experiments and theoretically, you cannot know both the velocity and position of an electron. This is because electrons act like individual particles and like waves (kind of like a wave of light)"
    },
    {
        "video_name": "4HkbFUoX0iI",
        "Q": "at1:05, how can tape stop wires from shorting out?",
        "A": "Tape does not conduct electricity, so it prevents the wires from shorting out."
    },
    {
        "video_name": "F9n3hLnwwc0",
        "Q": "She mixed up identical twins raised together and identical twins not raised together on the graph at 6:38.",
        "A": "While Carbone is correct about the individually administered IQ tests, you can find plenty of online tests. Take several and see what you get. Then average them, throwing out outliers first. This answer is likely not to be more than about 15 points from a score you would make on a test administered by a psychologist (I am one). Try the Mensa website among others."
    },
    {
        "video_name": "I6wzan4hNc4",
        "Q": "4:40 why didn't he drew the molecule has he did on the left.",
        "A": "Because doing it that way helps you picture the relative energies of the bonds."
    },
    {
        "video_name": "I6wzan4hNc4",
        "Q": "At 9:06, he says that cyclotetratraene with the two +1 charged carbons is aromatic. I would disagree. Wouldn't this not be aromatic since there is not a continuous set of overlapping p-orbitals? Correct me if I'm wrong.",
        "A": "The positive charges represent empty p orbitals. They are still overlapping the filled p orbitals."
    },
    {
        "video_name": "vEsAtC9d_MQ",
        "Q": "5:59 \"push the electrons out\" it should be push them inside the Thylakoid membrane",
        "A": "Yes, I think that s what Sal meant. When electrons move from a higher energy state to a lower energy state, they give off energy. That energy is then harnessed to pump the H+ ions from the stroma into the thylakoid lumen."
    },
    {
        "video_name": "vEsAtC9d_MQ",
        "Q": "At 5:00, why is the end product of P680 oxidizing H2O written as 1/2O2 + 2H+, and not something like O + 2H+? Isn't writing 1/2 of O2 redundant, since it would just equal out to one oxygen atom, or is there a significant reason for this notation?",
        "A": "It is written as 1/2 O2 because the oxygen given out during the process is in the form of molecular oxygen and not nascent oxygen. By nascent oxygen, I mean monoatomic oxygen which is unstable. O+ is monoatomic and O2+ is diatomic. So theoretically, you are absolutely correct, but practically speaking, 1/2 O2+ is not the same as O+"
    },
    {
        "video_name": "KL0i1RSnpfI",
        "Q": "at 4:30, we know that the inner core is solid but how could we possibly know it is made up of iron or nickel if we didn't see the materials face to face?",
        "A": "The heaviest elements should be at the core, with the lighter stuff floating on top. Iron and nickel are the most plentiful heavy elements."
    },
    {
        "video_name": "KL0i1RSnpfI",
        "Q": "at 0:46 sal explains wich wave does what,but i am intrasted how do they go through solids",
        "A": "One atom bumps into another atom. This is how the energy is passed through solids. Just visualise Newtons Cradle."
    },
    {
        "video_name": "KL0i1RSnpfI",
        "Q": "So, at 1:51, what is the materiel that the s waves can't travel through? like, what kind of liquid?",
        "A": "any kind of liquid, water, magma....you name it, and if its liquid, YES."
    },
    {
        "video_name": "ZQnGh-t25tI",
        "Q": "At 9:20 Why do we take the angle 120?",
        "A": "Because this is the angle between the lever arm, r, and the force, F. Remember that the direction of the lever arm is very important. It always goes from the axis of rotation to the point where the force is being applied. Because the force and the lever arm are both vectors, you must place them tail-to-tail to find the angle between them, which is 120 degrees."
    },
    {
        "video_name": "ZQnGh-t25tI",
        "Q": "6:30 so the difference between work and torque is that W=Fdcos\u00ce\u00b8 and \u00cf\u0084=Fdsin\u00ce\u00b8?",
        "A": "its a very common misconception In work, d is the distance traveled because of the force. In torque, d is the distance from the pivot point of the point on which the force is applied. Other than that, if you understand vectors,work is a scalar, torque is a vector. Hope it helps."
    },
    {
        "video_name": "ZQnGh-t25tI",
        "Q": "At 1:46 - why is it important that we consider \"r\" a vector? It seems like only the distance is important, rather than also considering the direction.",
        "A": "If you have to combine multiple torques, you are going to need to have the direction of each one."
    },
    {
        "video_name": "ZQnGh-t25tI",
        "Q": "At 9:20, the angle was said 120 degrees, so the used angle in the sine function is sin 120. Can't we use sin 60 directly? using the 60 degree angle you can directly find the magnitude of the force exerted to make Torque, right?",
        "A": "Yes, you can though it wouldn t be right. But you ll get the same answer because sinx=sin(180-x)"
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "At 5:52, Sal drew 6 electrons for the oxgen atom, but aren't there supposed to be 8 electrons around the nucleus?",
        "A": "It s true that there are 8 total electrons in an oxygen atom, but 2 of them will be in the first shell (1s), which leaves six electrons in the second shell (2s and 2p). Since oxygen only has the two shells, this is the outermost shell, containing the valence electrons, which is what we care about when talking about bonding. This is why Sal only drew the 6 valence electrons."
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "At 5:22, isn't hydroxide written as OH and not as HO?",
        "A": "It s usually written as OH, but I think it can be written the other way around as well. You ll see it sometimes if you do organic chemistry, when you draw the structural formula it can be written in this way. Another example is with methyl groups, CH3, sometimes get written as H3C. I don t think it makes it incorrect, it s just a different way of displaying it."
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "at 1:12, what is oxidation state?",
        "A": "Oxidation state is not losing an electron. The oxidation state of an element is that elements most common charge when it becomes an ion. In an Oxidation/Reduction reaction, oxidation has an entirely different meaning. When an element is oxidized it loses electrons, meaning it becomes more positive."
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "At 3:53, I always thought Hydrogen was part of the Alkali metals, so why does he say it can be a Halogen?",
        "A": "Hydrogen can also be considered a halogen because, like the other halogens (fluorine, chlorine, etc.), it only requires one additional electron to complete its valence shell. (In Hydrogen s case it would have the same configuration as helium)"
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "At 2:20, Sal mentions Oxygen is in group 6. He means group 16 I assume, unless there is something I'm missing. I know that's being super picky. Also I don't see a video on Transition metals and their complex assigning of oxidation states - I found on youtube, just curious! Big fan.",
        "A": "In the old days that was referred to as group 6, now it is group 16. Sal probably learned the old way, as I did."
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "My question has to be with the LiH molecule (at 4:20): if you add water to LiH, it produces hydrogen, wich oxydation state is 0, and LiOH. So, if hydride is oxydized, who is getting reduced in this case? Thanks",
        "A": "One of the H atoms in H2O is being reduced from 0 to +1"
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "at 2:11 what are so other thing that could happen?",
        "A": "A good example would be when Halogens are combined with oxygen. For example, in ClO4-, Oxygen still has an oxidation number of -2 but chlorine actually has a number of +7. Another example is with ClO3-, where Chlorine has an oxidation number of +5"
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "At 5:22, isn\"t hydroxide written as OH and not as HO?",
        "A": "Its according to the Electronegativity"
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "At 3:53, I always thought Hydrogen was part of the Alkali metals, so why does he say it can be a Halogen?",
        "A": "Hydrogen is part of the alkali metals, but it can act like a halogen. This is because it is essentially just a proton. It can be happy either losing its sole electron and having empty electron shells, or it can be just as happy gaining an electron and having a full 1s shell. By happy I mostly mean stable."
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "At 6:50, Why doesn't the oxygen take the electron from other hydrogen as well?? wouldn't it prefer doing that since then it will have noble gas state??",
        "A": "Oxygen already has a noble gas state of 8 valence electrons in HO\u00e2\u0081\u00bb. Certainly, H\u00e2\u0082\u0082O is lower in energy than HO\u00e2\u0081\u00bb, but HO\u00e2\u0081\u00bb can easily exist by itself."
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "At 5:36, why did he move the hydrogen to the other side of the periodic table?",
        "A": "Hydrogen is a noble gas, same as the elements in group 18 (the other side). It also assumes many of the same characteristics as the group 18 elements."
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "at 5:25 mins. into this video you state that we should now imagine Hydroxide and you write it out HO, but Hydroxide is a polyatomic and it is written OH, there is no HO! (Correct?)",
        "A": "The two are interchangeable. Think of it as a geometric model (O)--(H) (H)--(O) these two molecules are identical, simply rotated 180 degrees. HO is commonly written when it is a functional group on the left side of a written atom. for example, methanol: H HO-C-H H"
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "At 0:55, will Sal have to write '1+' instead of '+1' if he has to follow the convention of writing oxidation numbers? Or is there no such convention?",
        "A": "The convention is that the sign is written first for oxidation states. For actual charges, the sign is written after the number. However, this convention is not a strictly adhered to. Some people use it the other way around and some don t use it at all."
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "At 5:49, I thought oxygen had 8 electrons not 6?",
        "A": "Oxygen has 8 electrons in total but 2 of them are core electrons that aren t shown in dot structures, but the 6 valence electrons are."
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "at 4:16 if hydrogen fits there then why didn't they put it there?",
        "A": "Because while it does technically fit next to helium, hydrogen displays the same chemical properties as elements such as Lithium, Sodium, Potassium, etc., and it has the same valence electron configuration"
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "1:27 Do metals ever form covalent bonds?",
        "A": "What helps me is I have draw a border between the non-metals, metaloids, and metals and then write in big letters the types of bonds each type can have. Non-metal + Non-metal bonds are always covalent aka they always lend electrons to fill each other s needs to be stable. Since they are lending, it doesn t take much to seperate them again unlike Ionic bonds (strongest type of bonds)"
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "At 8:21 in the video doesn't there need to be 2 hydrogens in order for the charge in the oxygen to be 2-?",
        "A": "The hydroxide anion has an overall charge of -1 Oxidation numbers sum to the charge, so if H is +1, then O has to be -2 in order for both to sum to -1"
    },
    {
        "video_name": "CCsNJFsYSGs",
        "Q": "3:12 Sal said \"Group 5 elements\" but meant \"Group 15 elements.\"\nIsn't group 5 elements correct?? Aren't there only 8 groups max?",
        "A": "There are 18 groups (not counting the f-block). In an older system, the groups were labeled with a number and a suffix of either A or B. But there were inconsistencies in how these groups were labeled so a more straightforward system has been developed. In the modern system, the groups are numbered 1 through 18 with no A or B suffixes. So, the modern Group 15 was previously called Group 5A or VA (or Group VB in Europe)."
    },
    {
        "video_name": "fpq0eICjuSI",
        "Q": "7:04... Is one of these three resonance structures more stable than the other two? Just taking a quick glance at it, I can see that the \"odd one out\" is the middle one, in which the two double bonds are opposite one another (as far away from each other as possible), whereas in the other two diagrams, the double bonds are a bit closer to each other.",
        "A": "You are correct. The middle structure has isolated double bonds. The other two structures have conjugated double bonds, and the conjugation makes them more stable than the isolated bonds."
    },
    {
        "video_name": "iqRTd1NY-pU",
        "Q": "At 3:14, why are there two veins that are connected to the heart (superior and inferior Vena Cava), but only one artery (Aorta)?",
        "A": "It s just how humans were designed I think. Or probably because of the immense pressure the aorta receives."
    },
    {
        "video_name": "iqRTd1NY-pU",
        "Q": "At 6:33, what is the purpose for increasing resistance? Is it to slow down the blood and allow the cells sufficient time to absorb the oxygen and nutrients?",
        "A": "You ve got it! Increased resistance allows the gases time to exchange between the cells of the body and the bloodstream."
    },
    {
        "video_name": "iqRTd1NY-pU",
        "Q": "At 3:05, How is the inferior venacava inferior when it is so big?",
        "A": "Inferior does not refer to size - it refers to location. The superior vena cava brings blood from the head, arms, and upper body to the heart, and the inferior vena cava brings blood from the legs and lower body back to the heart. In anatomical terms, superior is closer to the head and inferior is closer to the feet."
    },
    {
        "video_name": "iqRTd1NY-pU",
        "Q": "At 6:53 the capillaries are introduced but never really expanded upon, so what is a capillary (or capillary bed) and what is its function?",
        "A": "Capillaries are the thinnest blood vessels and they actually deliver oxygen and food and remove carbon dioxide and urea from tissues. All the others are pipes toward or away from capillaries."
    },
    {
        "video_name": "iqRTd1NY-pU",
        "Q": "Are those bands of Elastin protein which Rishi drew on the smaller artery at 6:04",
        "A": "Those bands he was drawing are specifically smooth muscle. However, I would have to assume that small arteries and arterioles do have elastin simply because they are arteries that have to deal with high pressure. Hope that helps :)"
    },
    {
        "video_name": "iqRTd1NY-pU",
        "Q": "at 0:20 its not the ventricle its the atrium",
        "A": "Actually, it is from the atrium, then to the ventricle, then out."
    },
    {
        "video_name": "lsXcKgjg8Hs",
        "Q": "At noon the temperature in Alaska is 6 \u00c2\u00b0F. If the temperature drops 4 \u00c2\u00b0F per hour, what will the temperature be at 9:00pm?",
        "A": "If I m right, it s actually a maths problem of Arithmetic Progressions. First term(a): 6; common diff.(d):-4; no. of terms (hours) (n): 9 An=a+(n-1)d Ans = 6+(9-1)(-4) =-26 F"
    },
    {
        "video_name": "lsXcKgjg8Hs",
        "Q": "at 9:04 where does the joules sign go after the 125,340??",
        "A": "Sal does mention it s in Joules, should have written it but didn t. However, after making it an equation by adding = 125,340 the dimensions can cancel out, because we just want to know X in grams. But you re right, this is not the expression for Habs."
    },
    {
        "video_name": "lsXcKgjg8Hs",
        "Q": "at 2:09, what is joule?",
        "A": "A joule is the unit that energy is measured in. It is named after a scientist who s surname was Joule."
    },
    {
        "video_name": "lsXcKgjg8Hs",
        "Q": "Just to clarify, we are looking at the temperature of the original 500g of water right? because as the ice absorbs heat, it would melt as well, so then the volume of water in the cup will increase.\nAt 7:35",
        "A": "How are you going to keep the original 500g separate from the new water from the melted ice?"
    },
    {
        "video_name": "lsXcKgjg8Hs",
        "Q": "at around 10:10 he says the number of grams is \u00e2\u0089\u0088 354.02 g but a conversion calculator and my understanding got me to believe that the number of grams of ice needed was 13555804.989889. can someone please work this out for me?",
        "A": "i think its in the metric units so its changed to nanometers"
    },
    {
        "video_name": "lsXcKgjg8Hs",
        "Q": "Why doesn't Sal write J/gK after he solves the heat taken out at 3:48?",
        "A": "He is multiplying 4.178J/g.K *500 g * 60 K(K and degree C can be used interchangeably in this problem). The units of g and K in the numerator and denominator cancel and so we are left with only J. Hope this helps :)"
    },
    {
        "video_name": "u1eGSL6J6Fo",
        "Q": "At 11:26 how exactly was he able to get the shape of the sp3 orbital?",
        "A": "The shape comes from the solutions to complicated quantum mechanical equations. He happened to know what these shapes are."
    },
    {
        "video_name": "u1eGSL6J6Fo",
        "Q": "At 15:54 it's mentioned that the presence of sp3 orbitals explains the particular angle between the molecule's bonds, how exactly does this work? Thanks in advance.",
        "A": "There are 4 ps3 orbitals. Each orbital wants to get as far away from the others as possible (negative charges repel each other). Therefore the most optimal shape is the tetrahedron."
    },
    {
        "video_name": "u1eGSL6J6Fo",
        "Q": "At about 9:10, we replace a spin-down electron in the 2s orbital with a spin-up electron in a 2p orbital.\n\nWhy does/can the spin change? I assume it's not arbitrary.\n\nThanks!",
        "A": "The sp3 hybridized model is an excited state so it does not apply to Hund s rule (which only applies to the ground state). Since p orbitals are at a higher energy level than the s orbital, their orbitals can be filled with the spin changed."
    },
    {
        "video_name": "u1eGSL6J6Fo",
        "Q": "Why did Sal say at 1:25 that the first electron is spinning in one direction and the next electron is spinning in the opposite direction? Why do the electrons spin in opposite directions and why do they spin anyway?",
        "A": "This is a result of quantum mechanics. All elementary particles have an intrinsic quantum mechanical property that we call spin . The electrons aren t really spinning, because they are waves, but they behave as if they were spinning on an axis. Another result of quantum mechanics is that no more than two electrons can reside in the same orbital, and they must have opposite spins ."
    },
    {
        "video_name": "u1eGSL6J6Fo",
        "Q": "in 10:39 why is it written sp3 in place of2s and 2px etc? and why 4 lobes are drawn 12:49?",
        "A": "It is written as sp3 because the s and p orbitals of carbon hybridize to combine the 2s and 2p orbitals before it bonds with, say, hydrogen. This explains the shape we observe for methane. Its molecular geometry is tetrahedral. It does this so that the electrons in all the C-H bonds are as far apart as possible. There are 4 lobes because there are 4 hybridized sp3 orbitals."
    },
    {
        "video_name": "u1eGSL6J6Fo",
        "Q": "At 2:43, why does the 2px orbital have a dumbell shape?",
        "A": "Basically, because this is the shape of the wave function that comes from the Schrodinger equation. The d orbitals have very strange shapes, as do the f orbitals."
    },
    {
        "video_name": "u1eGSL6J6Fo",
        "Q": "04:54 What are spins in the context of electrons as waves ?? as particles they are just opposite rotations of the electrons around their own axis right ??",
        "A": "It is complicated! Here s a quotation from a paper: The spin of the electron and other charged particles arises out of the quantum wave structure of matter. Spin is a result of spherical rotation in quantum space of the inward (advanced) spherical quantum wave of an electron at the electron center in order to become the outward (retarded) wave. Wave rotation is required to maintain proper phase relations of the wave amplitudes. (I don t understand it either!)"
    },
    {
        "video_name": "u1eGSL6J6Fo",
        "Q": "04:30 THis is called Hund's Rule/Pauli Principle/Aufbau Principle right?",
        "A": "Hund s rule: electrons occupy orbitals singly before pairing up. Pauli exclusion principle: no two electrons have the same electron configuration (two electrons in the same orbital are spin up or spin down). Aufbau principle: electrons occupy lower energy levels orbitals before higher energy orbitals."
    },
    {
        "video_name": "u1eGSL6J6Fo",
        "Q": "At 4:25, why do the electrons try to \"separate\" themselves into the three p orbitals instead of filling the px orbital first? Is there a specific reason why, or is it just one of those things were the electrons just \"wants\" to separate themselves?",
        "A": "The electrons all have a negative charge, so they repel each other. They will all go into separate orbitals in order to avoid each other. Once there is no other choice, the fourth electron is forced to share an orbital with its repulsive partner."
    },
    {
        "video_name": "u1eGSL6J6Fo",
        "Q": "At 1:10 sal says that the 2s orbital is around the 1s orbital,doesnt that mean that the electron which goes into the 1s orbital will also be going into the 2s orbital?",
        "A": "No, the 1s orbital is inside the 2s orbital, but the locations where a 1s electron is most likely to be found are where a 2s electron is least likely to be found. The orbitals occupy the same space, but a 1s electron always knows it is 1s. It doesn t suddenly say, Oops, I m in the space where a 2s electron could be. I think I ll become a 2s electron . We would have to give it more energy to become a 2s electron."
    },
    {
        "video_name": "TsXlTWgyItw",
        "Q": "At 3:42, what is the difference between solvent and a solute?",
        "A": "A solvent is the substance doing the dissolving (Ex: Water is the solvent when dissolving salt) and the salute is the substance being dissolved (The salt in salt water)"
    },
    {
        "video_name": "TsXlTWgyItw",
        "Q": "from around 5:34, why does Sal keep saying carbon di oxide instead of carbon monoxide?",
        "A": "I think he should be using carbon monoxide as well."
    },
    {
        "video_name": "TsXlTWgyItw",
        "Q": "at 7:00 sal said carbon dioxide where as it should be carbon monoxide",
        "A": "Yes, that got corrected :)"
    },
    {
        "video_name": "TsXlTWgyItw",
        "Q": "At 2:59, how do you know that you are putting borontrifluoride into water and not putting water into borontrifluoride.",
        "A": "It doesn t matter which one you do. You will end up with the same mixture. However, it is easier to bubble a gas into water than to add water to a gas."
    },
    {
        "video_name": "TsXlTWgyItw",
        "Q": "at 5:16, why is carbon ignored in the second example?",
        "A": "The concentration of solids is always equal to one."
    },
    {
        "video_name": "TsXlTWgyItw",
        "Q": "Just clearing this up but at around 8:14, does Sal mean to put K(r)/K(f)?",
        "A": "Sorry, but he did it correctly. He showed that Kf[H2]^3[N2] = Kr[NH3]^2 When you rearrange this to put products over reactants, you get Kf/Kr = [NH3]^2/{[H2]^3[N2]} \u00e2\u0088\u00b4 Keq = Kf/Kr"
    },
    {
        "video_name": "TsXlTWgyItw",
        "Q": "At 8:39 in the work you show on the screen you show the cancellation of Kf/Ku. How did these cancel out?",
        "A": "It s not cancelled. He divided both sides by Ku to create Kf/Ku. Because these are arbitrary constants, dividing them by each other gives another arbitrary constant, which we can redefine as Ke. Thus Kf/Ku = Ke purely by redefining our constants."
    },
    {
        "video_name": "TsXlTWgyItw",
        "Q": "So, at 5:15, you said that BF3(g) + 3 H2O(l) <-> 3 HF(aq) + H3BO3(aq) is a heterogeneous equilibrium, what is an example of a homogeneous equilibrium?",
        "A": "they are heterogeneous because the reactants as well as the products are in different states or phases of matter.. we have gas, pure liquid, and aqueous acid.. example of homogeneous equilibrium is 2H2(g) + O2(g) --> 2H2O(g) where the water is in the gas phase. all of them in the gas phase."
    },
    {
        "video_name": "TsXlTWgyItw",
        "Q": "Around 7:00 water bonded with carbon and formed CO, how is that possible? Oxygen has a negative 2 charge and Carbon has a positive 2 charge. Did Sal make a mistake?",
        "A": "Since carbon has a positive valency of 2 and oxygen has a negative valency of 2, one atom of carbon and one atom of oxygen bond together to form one molecule of carbon monoxide. Carbon also also has a valency of 4+. This carbon atom will bond with 2 oxygen atoms, since they both need 2 atoms each. When this bonding occurs, carbon dioxide will form."
    },
    {
        "video_name": "TsXlTWgyItw",
        "Q": "At 3:15, how do we know that water is everywhere?",
        "A": "Because it is the solvent, which is a liquid in which a solute can dissolve in. In order for this to occur, you must have a considerate higher quantity of solvent (in this case water) than you do solute, thus it s considered to be everywhere. Think of it as air in the Earth s atmosphere. There is so much more air relative to how much space that you and I physically take up, that we consider it to be everywhere."
    },
    {
        "video_name": "akm5H2JsccI",
        "Q": "At 6:08, when he was drawing the Lewis Structure for chlorine, aren't you supposed to start the dots on the right side of the elemental symbol, and not at the top?",
        "A": "There is no rule that says you have to start at a specific side. It doesn t change anything."
    },
    {
        "video_name": "akm5H2JsccI",
        "Q": "At 0:25, valence electrons are shown by the 1a-8a system. Does this mean 3-12 have no valence electrons?",
        "A": "Not at all, but the number of valence electrons used by each atom can vary from one compound to another."
    },
    {
        "video_name": "akm5H2JsccI",
        "Q": "At 3:51, is sodium chloride salt? And also what does dioxide mean in things like silicon dioxide and carbon dioxide?",
        "A": "Sodium chloride is what we call table salt - it is the salt that we put on food. (However, in chemistry, the word salt has a more general use and refers to the product that comes from reacting an acid with a base. Sodium chloride is just one example of a salt - there are many others.) Di means two and oxide means oxygen , so therefore dioxide means two oxygens . So silicon dioxide is SiO2 and carbon dioxide is CO2."
    },
    {
        "video_name": "akm5H2JsccI",
        "Q": "How do you do the calculations at 1:00? 1s^2, 2s^2, stuff like that? And what can you know from those calculations? Is it just counting the number of electrons in each cell? (my guess)",
        "A": "This tells you which electrons are in which orbitals. The orbitals are filled in accordance with the Aufbau principle, taking account of Hund s rule. There are videos on the Aufbau principle and Hund s rule if you ve not already seen them. Knowing where the electrons are is important for understanding chemical bonding and also for predicting the shape of molecules."
    },
    {
        "video_name": "akm5H2JsccI",
        "Q": "At 0:54, what does the P mean in the electron configuration? I get the other parts.",
        "A": "The P stands for the P orbital."
    },
    {
        "video_name": "akm5H2JsccI",
        "Q": "at 0:54-1:00 how did you find the valence electrons",
        "A": "My answer to this is above"
    },
    {
        "video_name": "akm5H2JsccI",
        "Q": "At 2:30, how did CL become 1s^2 2s^2 2p^6 being that is in row 3?",
        "A": "So that means that the six noble gases will have 8 electrons in their outer most energy level."
    },
    {
        "video_name": "akm5H2JsccI",
        "Q": "Why do we ignore groups 3 - 12 like he said at 0:22?",
        "A": "Because the valence electrons aren t as well defined for them. There is a different electron counting method we use for transition metals, the d electron count."
    },
    {
        "video_name": "akm5H2JsccI",
        "Q": "For Chlorine done at: 3:18\nHow come he adds up the 3S2 and the 3p5 together to make 7 valence electrons?\nWouldn't the final layer/shell be 5 since 3p was the last and it had 5 electrons?",
        "A": "Because that s what its valence electrons are. For main group elements they re the ones in the highest energy shell not just the highest energy orbital. 3s and 3p are both in the third shell. A good thing to keep in mind is that the number of valence electrons for s and p block elements is equal to the ones or units place of the group number. Cl is in group 17, the ones place is 7, so it has 7 valence electrons!"
    },
    {
        "video_name": "akm5H2JsccI",
        "Q": "at 5:13 : the Na+ cation is stable because it resembles the configuration of a noble gas Ne. Is it not that stable mean neutral?",
        "A": "No, neutral means that the atom has the same number of protons and electrons, while stable means that the atom is happy with its arrangement of electrons."
    },
    {
        "video_name": "akm5H2JsccI",
        "Q": "at 0:50 what do u mean by 1s2 2s2",
        "A": "how much electrons in each shell"
    },
    {
        "video_name": "R5CRZONOHCU",
        "Q": "at 0:32 what are those double line he puts near the f mean?",
        "A": "The double lines mean the magnitude of the force, which is the size of the force vector. Essentially this is just the absolute value of the force, we are only concerned about its size and not its direction."
    },
    {
        "video_name": "R5CRZONOHCU",
        "Q": "At 3:25 where does Fg/m2 go. Sal just seems to ignore it. Why does he do so",
        "A": "Remember Newton s Second Law? F=ma. You can rearrange the equation to obtain a=F/m. In this case, the F/m is F_g/m_2, so he just replaced that expression with a_g. Every time you see that acceleration, just know it s the same thing as F_g/m_2. Hope this helps!"
    },
    {
        "video_name": "R5CRZONOHCU",
        "Q": "Can someone explain in more detail what the difference is between ag and ||ag|| at 3:17? And also how |ag| fits into this?",
        "A": "Ag is the acceleration due to gravity. it is in vector form. ie it denotes magnitude and direction. and |ag| is the magnitude of the same acceleration. it does not involve the direction."
    },
    {
        "video_name": "R5CRZONOHCU",
        "Q": "at 2:15 Sal mentioned that force and acceleration are scalar quantities. why so?",
        "A": "I believe Sal was referring simply to the MAGNITUDE of the acceleration and force. As you can see in the video, he puts 2 bars around acceleration and force, signifying that its the magnitude of the force and acceleration."
    },
    {
        "video_name": "E-q9JpkGc-8",
        "Q": "Hi at 8:55, you wrote down the equation for parallel resistance. I understand resistance should be lower with more paths to flow but I am trying to understand how you derived the equation to calculate this.\ncould you explain how you arrived at 1 over (1/ R1 + 1/R2...)?",
        "A": "Conductance is the reciprocal of resistance, so C = 1/R. Conductances in parallel add up the same way as resistances add up in series, so that Ct = C1 + C2 + ... + Cn. If we observe that C1 = 1/R1, C2 = 1/R2, and so on, then we can substitute in 1/Rt = 1/R1 + 1/R2 + ... + 1/Rn to find the resistance in parallel."
    },
    {
        "video_name": "E-q9JpkGc-8",
        "Q": "5:15, if we cut the system of vessels before the vessel with R=5, the resistance in the other 2 will drop to 0 because there would be no flow in them, am i right? so the Rt equals R=5.. So its not always greater it can be same as well... correct me if im wrong, i just wanted to discover a situation that doesnt match theory :p",
        "A": "From Poiseuille s law, R = (8\u00ce\u00bcLQ) / (\u00cf\u0080r\u00e2\u0081\u00b4). The most important variable in this scenario is Q, which is flow through the vessel. If we imagine that a clamp has been put on the vessel before those segments, then flow is completely stopped. Because Q = 0 in this scenario, R must therefore also be 0 to satisfy the equation."
    },
    {
        "video_name": "E-q9JpkGc-8",
        "Q": "At 9:32 How did the common denominator become 30 in the parallel resistance?",
        "A": "The lowest common denominator is the smallest number that is divisible by all the denominators of the fractions that you need to add up. In the example of the video we needed to add up 1/2, 1/6 and 1/5, so you need to find the smallest number that is divisible by 5, 6 and 2, which is 30."
    },
    {
        "video_name": "2lcaBPLLoLo",
        "Q": "If the yo-yo is at top of the circle (at around 6:42) and both tension + gravity is acting as the centripetal force, how is it (or rather what is it) that is keeping the yo-yo from collapsing towards the center of the circle?",
        "A": "The centripetal force (both tension and gravity at the top) keeps the motion circular. The yo-yo will collapse towards the center only if the gravity is greater than the tension."
    },
    {
        "video_name": "2lcaBPLLoLo",
        "Q": "At 0:48 , when the object is at the top, let us assume that the velocity is 4.0 m/s.\nNow, what would be the speed of the object when it circles down to the bottom? It should be faster due to gravity. How do we solve for it?",
        "A": "Use conservation of energy"
    },
    {
        "video_name": "2lcaBPLLoLo",
        "Q": "I thought it was a yo-yo not a ball at 0:44",
        "A": "It is he just misspoke. He also said ball at 4:27"
    },
    {
        "video_name": "f8GK2oEN-uI",
        "Q": "What does ''difinitly'' in 0:08?",
        "A": "The word is definitively. When Sal says none of this stuff has been definitively proved, he means that scientists are not sure whether or not all this stuff actually happens, he is just telling you about the ideas that people think are most likely to be true from what we know at the moment."
    },
    {
        "video_name": "f8GK2oEN-uI",
        "Q": "In the cycle mentioned at 2:47 can water ever escape the cycle through evaporation or anything else?",
        "A": "Yes, that little experiment is just a little thing to understand what s going on. But yes, evaporation could happen."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "When Sal is using the calculater (8:21) he puts in 4 divided by .082 divided by 303.\nShouldn't it be : 4 divided by .082 multiplied by 303??\n\nIf not, could you explain why?",
        "A": "In the example 4 is divided by 0.082*303. So you either divide 4 by 0.082 and then by 303 or you have to write 4/(0.082*303). If your calculator doesn t have brackets, you re stuck with the first method."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "So at 6:12 where Sal showed the R constants, I noticed the one for mass is m^3. Why is that and how would we \"cancel out\" the mass as a unit with that in the way?",
        "A": "m^3 is actually a volume measurement (cubic meters). The mass part of the ideal gas equation is in the n , which is for the number of moles (so n is mass/molar mass)."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 7:13 how would you cancel out the L if it is in the numerator in both sides? Would you \"divide\" by L?",
        "A": "The L in the gas constant is actually in the denominator of the overall equation, so it can cancel with the L in the volume. Recall that, to solve for n in the gas law, n=PV/RT, so R is in the denominator. Hope that makes sense!"
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "at 7:02 you said \"303 degrees kelvin\". I was told kelvins arent associated with degrees",
        "A": "You are right! It s 303 kelvins or 30 degrees Celsius, not 303 degrees kelvin."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 3:38, you state that the molecules are stationary when the temperature is absolute zero. However, isn't this incorrect because of quantum mechanic principles?, i.e. the heisenberg uncertainty principle.",
        "A": "He mentions a few times in the videos that temperature is the measure of average kinetic energy of a substance. Therefore if the temperature is at absolute zero that would have to mean there is zero kinetic energy. Thus the molecules would be stationary. However if we were to try and measure or observe these stationary molecules, we would be putting energy into the system. either by light, electron microscope, etc. Which in tern would put the system in a non zero energy state."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 1:31 sal tells us that the pressure on the outside of the ballon is 2 atm...but shouldn't we be using the pressure inside the balloon?...if sal is correct please explain to me the concept behind using the pressure of the outside rather than the pressure inside the balloon",
        "A": "the pressure inside has to be equal to the pressure outside, otherwise the balloon would expand or contract."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "at 2:00 I converted from degree to kelvins, atm to Pa and L to m^3 but got a different answer, what did I do wrong?",
        "A": "Did you use the correct units for R? n = PV/(RT) = 2\u00c3\u009710\u00e2\u0081\u00b5 Pa \u00c3\u0097 2\u00c3\u009710\u00e2\u0081\u00bb\u00c2\u00b3 m\u00c2\u00b3/(8.314 Pa\u00c2\u00b7m\u00c2\u00b3K\u00e2\u0081\u00bb\u00c2\u00b9mol\u00e2\u0081\u00bb\u00c2\u00b9 \u00c3\u0097 300 K)= 0.16 mol This is the same answer as in the video."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 4:00 why does he put 273. 15 degrees Celsius?",
        "A": "that is the conversion between degrees celsius and kelvin. these scales have the same unit magnitude (1C=1K) but a different 0 point. 0C=273.15K/-273.15C=0K"
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 2:08, what units is pressure given in?",
        "A": "Pascals or Newtons per square metre are the best to use, but other measurements are Barr, mm of Hg (mercury), atmospheres, and so forth. As long as you convert to SI units (or units derived fron the base SI units) and apply the appropriate value for R, it all works fine. The most important thing is to make sure your units are consistent!"
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 0:06, Sal talks about the ideal gas equation that states that pressure times volume is equal to the number of molecules times a constant (R) times the temperature, i.e. PV = nRT. But in the workbook I refer to at school, it says PV = RT, i.e. pressure times volume equals a constant (R) times temperature. What happened to the number of molecules?",
        "A": "That workbook should indicate: Pv=RT Where volume is the specific volume and R(often with a subscript) is the specific gas constant, which is unique for each gas. Taken together, the specific volume and the specific gas constant incorporate the information for determining the number of molecules. So, in this form, the ideal gas law is using different variables than in the PV=nRT form. There are a number of other forms of the ideal gas law, just so you know."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "Thank you for these videos! They are very helpful!\n\nMy question;\nAt the time 8:23, Sal gets his calculator and divides 4 /.082 / 303 and gets the answer .16 moles.\n\nMy question is; did Sal divide the denominator instead of multiply?\n\nI'm quite sure that I'm wrong but just in case, I don't understand why he divided all the numbers rather than multiply the denominator before he divided the numerator?\nSo, should it have been 4/.082*303 = 24.846 moles?\n\nThank you!",
        "A": "Dividing PV by RT or dividing PV by R then dividing by T is exactly equal. 4/0.082 = 48.78 then 48.78/303 = 0.161 0.082*303 = 24.846 then 4/24.846 = 0.161 0.161 = 0.161"
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "at 8:50, what did he mean buy number in traditional sense?",
        "A": "6.02 x 10^23 molecules: Isn t that Avagadro s number?"
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 8:30 why is Sal dividing by 303 instead of multiplying .082 * 303 as the total denominator?",
        "A": "The math works out either way. In the same way that 1/4 can be expressed as 1/4 or 1/(2*2) or 1/2/2, 4/24.846 can be expressed as 4/24.846 or 4/(0.082*303) or 4/0.082/303. In the end I suspect it was just to avoid having to calculate the denominator beforehand and then remember that value when another method would work just as well."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 2:59, do you still use Kelvin for Fahrenheit?",
        "A": "You can convert from degrees F to Kelvin and back by using the appropriate conversion factor."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 8:49, the note says Sal only has 1 significant figure to work with so it is 0.2 rather than 0.16. How do you know the number of significant figures- why is it 0.2 and not 0.16?",
        "A": "in multiplication/division the result will have as many significant figures as the number with least significant figures. 2.342*0.123=0.288066=0.288 . in addition/subtraction the result will have as many decimal places as the number with least number of decimal places. 3.456+245.21=248.666=248.67"
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "Sal said \"6.02 times 10 to the 23 and then you will have the number\" at 8:59.\nWhere did this \"6.02 times 10 to the 23\" come from and whats traditional sense?",
        "A": "6.02 x 10^23 is called Avogadro s number. It s how you convert between moles and the number of atoms or molecules or whatever. So 1 mole of hydrogen gas has 6.02 x 10^23 hydrogen molecules (a very very big number)"
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 5:49, when Sal is explaining on how to get the Kelvin value, I'm pretty sure there is no degrees Kelvin and you just write it as Kelvin.",
        "A": "You re right. That question has already been answered."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 8:25, when we are doing the math to calculate the moles, you say \"4/.082/303\" The drawing and the steps before this show it to be 4/.082x303. Which is the correct way?",
        "A": "When he says 4/.082/303 he means that you take 4/.82, and divide by 303. They both total to the same thing, though. Nice point."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 7:05 Sal cancels out the units and is left with /mol.\nCan someone explain why that is \"per mole\" and the process of getting that to the other side of the equation?\nThanks.",
        "A": "Well the per mole is 1/mol or mol^-1. In order to get rid of this unit on both sides, we must multiply both sides by mol. Therefore, the left side gets a unit of mol and on the right side the mol x 1/mol cancels out to give the value of 1 and thus leaves no units on the right with the unknown letter n (for number of moles)."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "4:50 what is the difference between kelvin and Fahrenheit?",
        "A": "The Kelvin scale starts at absolute zero, where there is no thermal energy. Water freezes at 273. The size of a Kelvin is the same size as a degree C. The Fahrenheit scale sets 32 as the freezing point of water and 212 as the boiling point. Fahrenheit degrees are smaller than Kelvins."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 7:45, why is Sal writing \"mole\" as \"mol\"?",
        "A": "mol is the accepted abbreviation for mole, like m for meter"
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "In 3:31, it is stated that Kelvin's minimum is 0, or -273 degrees C. I wonder what is it's maximum?",
        "A": "Absolute hot is 1.416785(71)\u00c3\u009710^32 kelvin. This is what s called the Planck Temperature. It s theorized that this is the temperature at which matter becomes so excited that it breaks down into pure energy and is no longer matter. The first moments of the Big Bang are thought to be this temperature. String Theory uses what s called a Hagedorn Temperature (10^30 K) because at the sub atomic level, physics gets weird. Hope this helped!"
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "He says that he is dividing by 303 at 6:00, but on the screen it says he is multiplying. Help?",
        "A": "Sal set up his equation so that .082 and 303 are in the denominator, so both are dividing the numerator. If you remember the formula is P times V equals n times R times T. To find n it is n equals P times V divided by R times T. It is n=(PV)/(RT) like Sal wrote and calculated, not n=((PV)/R) * T. Either way, assuming you realize that 303 is part of the denominator, they are is equivalent. 4/0.082 = 48.78 then 48.78/303 = 0.161 0.082*303 = 24.846 then 4/24.846 = 0.161 0.161 = 0.161"
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "why we used the pressure on outside of balloon at 1:34.......shouldn't we use the pressure of the gas inside the balloon in the equation?",
        "A": "It depends on the malleability of the material the balloon is made out of. If the ballon did not change volume/shape at all, only the inside pressure should be considedered. If the balloon can change to any volume, only the outside pressure needs to be considered. However, since balloons have neither a constant or potentially infinite volume, we should probably take into account both the pressure inside and outside of the balloon."
    },
    {
        "video_name": "erjMiErRgSQ",
        "Q": "At 7:02, he mentioned degrees Kelvin. To my memory, Kelvin is not measured in degrees. Is this correct?",
        "A": "yes, it s a common mistake, but the Kelvin is a unit like the kilogram or the second. It doesn t really make sense to say 300 degrees Kelvin, but people do it anyway. 300 Kelvins would be the right way."
    },
    {
        "video_name": "sd0BOnN6aNY",
        "Q": "At 7:54\nis diffuse reaction the same as diffuse reflection?",
        "A": "Reaction is chemistry mate."
    },
    {
        "video_name": "sd0BOnN6aNY",
        "Q": "At 10:20, Sal is saying that the light reflected on the lake is reflected to the paddle from where we see it. So then, how do you see the sun on the lake?",
        "A": "The sun on the lake is the result of the light directly coming from the sun and reflecting off the lake. The lake acts like a smooth mirror."
    },
    {
        "video_name": "sd0BOnN6aNY",
        "Q": "from 06:24 to 10:09 you have talked about diffuse reflection and have suggested that in diffuse reflection the incident ray is not equal to the reflected ray. but when we take into point the angle of the surface and compare it to both of the other factors doesn't the property work?",
        "A": "Diffused reflection is observed when the surface isn t smooth. This means that at different points on the surface, the normal is different. The angle of reflection is STILL equal to the angle of incidence at THAT POINT and the NORMAL is unique at THAT point in the surface."
    },
    {
        "video_name": "sd0BOnN6aNY",
        "Q": "Question on the point at 8:15- 8:20: In the case of specular reflection, the original 'image' is always preserved? Why don't I see a reflection of my light bulb off the walls or desks (any flat surfaces)? Is this because microscopically these surfaces are also jagged and result in diffused reflection?",
        "A": "Yes that is correct. If you were to put a shiny/reflective object near the bulb, you would see its specular reflection."
    },
    {
        "video_name": "eEJqaNaq9v8",
        "Q": "In regards to 0:30, the kinetic molecular theory states that particles only move in straight lines. The video says they move in rotation and curves. Is this an error?",
        "A": "They travel in straight lines until they collide with another particle or the walls of the container."
    },
    {
        "video_name": "FBoYigdnLKQ",
        "Q": "@3:38 Low blood pressure (hypotension) is mentioned. What causes it? Is it more dangerous to have than hypertension?",
        "A": "Several things can cause hypotension. For example, if you were injured and were bleeding a lot, your BP can drop below normal ranges. Another example is if your heart gets injured (perhaps from a heart attach) and it can t pump out enough blood then your BP can drop. A third example would be from taking too much hypertension medications, so that you lower it too much. If your BP drops low enough then not enough blood gets to your brain and you can pass out and even die."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "at 3:03 sal uses sin .... can we use cosec also ?",
        "A": "Try using it and see if it works. If it works, you can use it."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "at 3:02 why did sal take sin why not cosec ?",
        "A": "Simply because most people are far more familiar with and competent with sin, cos, and tan than they are with cot, sec, and csc. Look at calculators. They rarely have buttons for cot, sec, and csc, because to get those you just take the reciprocal of one of the others."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "I'm very confused: at 3:30, when Salman is calculating the height of the ramp for plugging into the potential energy, why is he using the height of the ramp? Shouldn't he use the length of the ramp, because that's the way that the rider is travelling and thus has the potential to cover?",
        "A": "Potential energy from gravity comes only from the height."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "07:31 how does 38.455 J become 38.5 kJ?",
        "A": "Sal rounded it: 38,455 J ~ 38,500 J 1,000 J = 1 kJ 38,500 J = 38.5 kJ"
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "There is something wrong from 6:55 onwards..!!\nWork done by friction is FScos(theta) => 500 X 60 X COS(180) => -(500X60)\nRight..!!\n\nAt 6:56 when we take Work done by friction to the initial energy, so he adds it as the work done as it is negative..!! So the -ve work done becomes +ve..!!\nRight..!!\n\nThen shouldn't the value of the work done at 7:40 be (+ve 60 X 500) instead of being -ve..!! As sal already took it as -ve and changed the sign when he brought it to the initial energy..!!",
        "A": "work done = FScos(theta) ,you are correct their. But since since the force is just at 180 degrees it is the same thing as FS. Actually sometimes they even teach W = FS initially and when they introduce vectors then they introduce the actual formula. Friction is -ve because it is in the opposite direction. You can contact me for further clarification my email id is informjaka@gmail.com"
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "Is the rider going up the hill or down the hill. 1:42.",
        "A": "Going down the hill."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "is this equation \"TMEi + Wext = TMEf\" the same equation as the one he used in 6:44 ?",
        "A": "it should be, because external forces are non-conservative (i.e friction)"
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "At 6:48,How Is Initial Energy + Work done by friction = Final energy",
        "A": "The system is losing energy to friction, and work is energy entering or exiting the system. Therefore, we will be adding all the work done on the system to the initial energy to achieve the final energy. In this problem, work done by friction will be negative because the force is applied in the opposite direction as the motion, and will decrease the initial energy, giving us a Final Energy result that is less than the initial energy because energy is lost to friction."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "I didn't get that last move in the equation at 8:35 when you got 38,5 - 60 . 500 to be 8455 positive...",
        "A": "Pay attention to the units. Sal said before 8:35 that 38455 J is nearly 38.5 KJ. Thus, without converting into Kilo Joules, we can directly solve for the final kinetic energy. 38.5KJ or 38455 J-60.500 J = 38455J-30000J = 8455 J. Hope this helps :)"
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "At 5:00 Sal says that when friction comes into play, all the forces are not conserved. Okay, I can see how that works, since the energy is transferred into heat, but that doesn't mean that energy is not conserved; At 0:14 he says that energy is lost to friction. Did Sal just contradict himself, or am I missing something here?",
        "A": "Energy is only conserved in a closed system. In a closed system the energy lost to heat is still part of the system so the total energy is conserved. But if you have an open system and the heat escapes the system it is not conserved."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "At 4:05 Sal put the height in metres, but I was lead to believe that it should always be in km if the weight is in Kg. Is that correct (in the video) or should the height be in kilometres?",
        "A": "There is no requirement to use any particular units as long as you calculate correctly with them."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "at 7:45 why do you not calculate friction as a horizontal distance but times it with the 'hypotenuse\" distance",
        "A": "Friction is not always horizontal. It is opposite to the direction of sliding. Since the bike in this case is sliding down the slope, we have to calculate it as a force with the distance of the slope. Hope this helps!"
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "guys i know this might sound vague but at 6:05 Sal says Ei=Ew+Ef but at 7:18 he says Ei+Ew=Ef .now we cant do that by transposition.Can any please tell me which equation is right??",
        "A": "Both are, it all depends on how you define the work. The work is the change in energy so when it is on one side of the equation it can be looked at as a negative change in energy but as positive on the other."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "At 7:44 wouldn't Work equals force times distance times cos(\u00ce\u00b8) ?",
        "A": "We don t use cos(theta) because the force of friction is already parallel to the direction of displacement. You only have to do that if the force is not parallel to the direction of the displacement, because only parallel forces can do work on an object."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "At 7:27 why the sum of final energy is equals initial energy plus work of friction eventhough the quantity of work of friction is negative? Could we just wrote the sum of final energy is equals to initial energy minus work of friction? Work of friction always negative, right?",
        "A": "Yes, you can do that. He wrote it very generally so it would fit the equation well whether the added work (by friction or anything else) was positive or negative. If the work turned out positive, then you just add the value. If it was negative, then you just add the negative value. Remember, subtracting values is just the addition of a negative value."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "At 7:35 sal wrote 60 \u00c3\u0097 500 ( F d ) but to find the accurate answer we need to use the formula \" F d cos \u00f0\u009d\u009a\u00b9 \" . Using this :-\nEi-W=Ef\n38455 - ( 60 \u00c3\u0097 500 \u00c3\u0097 cos 5\u00c2\u00b0 )\n38455 - ( 30000 \u00c3\u0097 cos 5\u00c2\u00b0 )\n38455 - 29886 ( Rounded up )\n8569 J\n8569 = \u00c2\u00bd m v^2\n8569 = 45 v^2\n8569/45 = v^2\n190.4 = v^2\nv = \u00e2\u0088\u009a190.4\nv = 13.7 m/s\nThough the answers at last are the same but isn't this a better way of doing it?",
        "A": "Here you wouldn t need to use the cosine of the angle because the force of friction is already parallel to the direction of displacement."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "At 8:09 , Why dosent he use cos (theta) while calculating the frictional force on the biker??\nThank You",
        "A": "We don t use cos(theta) because the force of friction is already parallel to the direction of displacement. You only have to do that if the force is not parallel to the direction of the displacement, because only parallel forces can do work on an object."
    },
    {
        "video_name": "YvacYWgygaA",
        "Q": "At 6:24, Sal mentioned negative work occurring when force and displacement are in opposite directions. Will this always be the case, or do you just use the average force of all forces acting on the object(ex: Fg, Ff, Fa)",
        "A": "Yes and No. The work will be negative if you are acting in the opposite direction and you are using VECTOR quantities, but if you are using SCALAR quantities, the values will always be positive, since scalar quantities ignore direction."
    },
    {
        "video_name": "T4iRq8_ZmAA",
        "Q": "Hello, at 9:31 you say to check for shorts, is there a way to do that other than visually?",
        "A": "You can use a multimeter. Check to make sure connections only connect where the copper traces on the bottom join two or more points."
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "At around 2:30-2:40 he says that potential energy turns into kinetic energy. But doesn't kinetic energy mean the energy something has when it is moving? So then if , when the object hits the ground how come its potential energy is 100J?",
        "A": "That s the energy when it just hits the ground. Obviously after that the KE goes to zero."
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "At 2:36, it is said that the object's energy is all converted into Kinetic Energy, but doesn't Kinetic Energy mean an object is in motion? When the box is on the ground though, thee box isn't in motion anymore so why is it still called Kinetic Energy? Or did it mean that RIGHT before the box hits the ground it is all converted into Kinetic Energy?",
        "A": "Right before it hits the ground it s all KE. Obviously after it is at rest it has no KE anymore."
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "At around 8:52 for the second portion of the problem where the block is at 5m instead of 10m, if it is moving why is there final potential energy? Why isn't all the energy at this point just Kinetic?",
        "A": "I think he says pretty clearly that the block is still 5 m high, which means it still has half the potential energy it did when it was 10 m high. PE doesn t get to 0 until the block is on the ground."
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "at 9:36, does the law of conservation mass only apply when the object has stopped moving, because at the beginning there was 100J and halfway through there was 50J....",
        "A": "o, makes sense, thanks"
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "At 7:30 wouldn't changing color also utilize energy?\nThen how can u calculate?",
        "A": "That was a joke he made. But if it were a really hot object, sliding down some frictionless slope and cooling off rapidly as it slid down, even then the mechanical energy (kinetic+potential energy) would still be conserved but the total energy of the block which includes the mechanical as well thermal energy would continue to decrease due to radiation loss to the surrounding environment."
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "At 4:02, Something is mentioned about the kinematics formulas. Now if I were to use these formulas to calculate the velocity, wouldn't then the acceleration -9.8m/s/s be used since the object is falling?",
        "A": "Signs are arbitrary. If you define up as positive, then g is -9.8. If you define down as positive, then g is 9.8"
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "At 5:28, won't the object slide of the second bump?",
        "A": "Were assuming it won t slide off. Sal just drew it like that."
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "So essentially i did exactly what sal did at 3:36 however i used both methods: the kinematic way and the energy way. For the kinematic way i get v^2=196 and for energy way as sal stated we get v^2= 200 now i realise the difference is not huge with V=14ms/s for the kinematic way and V=14.14m/s for the energy way, but why does the difference exist if both methods can determine the same results?",
        "A": "I have the same problem where the 200 coming from?"
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "At 2:22, Salman said all of gravitational potential energy is converted into kinetic energy. But in real \"Kinetic energy is energy possessed by the virtue of the motion of an object\". When at ground, the object has no height so it does not have any gravitational potential energy neither it is moving so it should also not have any kinetic energy. What energy it has, when it is in rest?",
        "A": "When it just touches the ground, it has all kinetic energy. Then contact with the ground causes the object to come to rest. The kinetic energy is now zero. The potential energy is also zero. But energy is conserved, so where did it go? Thermal energy. The ground and the object and the air are all a little bit warmer after the object lands."
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "So around 8:46 Sal says that the PE and KE are equal so that means that the KE and PE are alternate?",
        "A": "In that scenario, when the object is halfway down, the potential energy has the same magnitude as the kinetic energy."
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "At 0:58 what is m times g times h",
        "A": "gravitational potential energy."
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "At : 2:49\nWhere did the joule came from? It supposed to be like \" m * g * h \" which it's should be something with kg's and m's. Instead it popped joule out of nowhere. Can someone solve this?",
        "A": "A joule is a newton*meter. A newton is a mass*acceleration. So the mg portion gives you a newton. Then when you multiply that by a height in meters, you get a newton*meter, or a joule. To be honest, I m not sure where the video is where he first explains this..."
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "When at around 8:45 when Sal makes g=10, is it really supposed to be 9.8 and he is rounding?",
        "A": "The value of g is 9.8 m/s^2.Sal has taken it to be 10 m/s^2 to make the math easier."
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "I don't understand why the velocity found is to the right, at 7:00.",
        "A": "Sal finds the velocity by applying the formula for Kinetic Energy. That formula is KE = 1/2MV^2. Using the numbers in the video, this works out to 14.1 m/s to the right, because the block is sliding that way due to the slide. He remarks that it is the same number as if we had dropped it straight down, because he s trying to make a point that the total energy of the system doesn t change- ignoring friction, you can always turn Potential Energy into Kinetic Energy, and visa versa, no matter what the direction is."
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "at 4:28 the velocity will come exactly 14m/s because g=9.8\nif we take g=9.8 we'll get exactly 14m/s",
        "A": "???????"
    },
    {
        "video_name": "kw_4Loo1HR4",
        "Q": "How did he get 200 Joules, where or how did it comes about? 03:50",
        "A": "he multiplied both sides of the equation with 2 to get rid of the 2 on the left hand side!"
    },
    {
        "video_name": "2h09oj26_H0",
        "Q": "At 2:30 you say that it does not cause the flu. I understand that. But does it cause flu like symptoms?",
        "A": "If so much milder then the actual Flu."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "Around 13:40 he says that cellular respiration produces 38 ATPs, but in my textbook I was taught 36 ATPs. Is this wrong?",
        "A": "Cell respiration can make a maximum of 38 ATP, but two ATPs are used to turn glucose into pyretic acid (glycolysis). Sal s just telling you how much is made, because in reality, that s how much is really made! He s discounting the fact that two ATP is used, but yes, the net gain is 36, and the total is 38."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "at 13:11 he said FAD-> FADH\nbut isn't it FAD-FADH2\nand at 12.20 he said NAD+ -> NADH\nbut isn't t NAD+ -> NADH + H+",
        "A": "The first one is correct but the second one is not. You are adding 2H you only want one so we can get NADH"
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "Near the beginning of the video, starting at about 1:07, Sal refers to 1 mole of glucose, and goes on to state that 1 mole of glucose generates 38 ATP. Does he mean to say that 1 molecule of glucose produces 38 ATP?",
        "A": "No, a mole is not a molecule. A mole is 6.02*10^23 of whatever you are are counting. So a mole of glucose is 6.02*10^23 glucose molecules."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "at (13:19), what's the difference between FADs and NADs?",
        "A": "they serves similar function as in carrying electrons to the electron transport chain. FADH2 delivers it s electron to complex II of the electron transport chain while NADH transport it s electron to complex I. This is why FADH2 and NADH has different ATP value. :)"
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "6:26 Why arent all the Glucose Carbons in a hexagonal ring in the middle with all the Oxygens around the outside?",
        "A": "THats an isomer of normal glucose..."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "in 12:38, sal said \"And those are actually what drives the Electron transport chain.\" what does he mean by \"those\"? NADH or glycolysis and Kreb's cycle?",
        "A": "In that moment, he s referring specifically to the NADH s. Specifically, it s the high energy electrons in NADH that drive the electron transport chain."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "What is a mole? Is it short for molecule?\n\nAlso, at around 3:22, he says it produces 38 ATP's. Isn't it 36? Two from glycolysis, zero from the formation of acetyl coenzyme A, two from the Krebs cycle, and thirty-two from the ETS for a total of 36?",
        "A": "No, mole is not short for molecule. A mole is a number , like a dozen, but many orders of magnitude larger. It is 6.02*10^23."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "At 1:00 to 1:20, Khan said we were producing energy, but how is that gross? I don't get why",
        "A": "It s complicated. However, you can think like we are burning gas to produce energy (like in a car), but in our body the gas is the association to the glucose that is produced in the photosynthesis process, so the glucose is broken in smaller pieces that releases energy."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "At 7:20, Sal explains that glycolysis requires 2 ATPs, but generates 4 ATPs. Is the 4 ATPs generated including the 2 ATPs? Basically, are 6 technically generated, but 2 are used up? Or are 4 technically generated, but 2 are used up?",
        "A": "Your second intuition is correct. With 4 generated, 2 are used up, meaning there is a total net gain of 2 ATP. All the best."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "4:25 wouldn't the amount of ATPs depend on the amount of energy or glucose?\nbecause if you absorb tons of energy and you get 38 you couldn't get that many with a small amount, right?",
        "A": "Glucose can be used to get a certain amount of energy per particle in a perfect world. Sal said that there was 38 ATP per mole of glucose in a perfect world. Therefore, yes it would, but Sal was talking about a specific amount."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "At 13:13 isn't it FADH2 not FADH?",
        "A": "Yea, it s suppose to be FADH2... :)"
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "At 1:00 to 1:20, Sal said we were producing energy, how is that gross?\n( He said it was over gross!)",
        "A": "A gross oversimplification means it s EXTREMELY oversimplified, not that it s gross as in nasty."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "At 9:28, why is glycolysis anaerobic and the others are?",
        "A": "Oxygen isn t needed in the chemical reactions that occur during that stage of cellular respiration, while it is needed in the others."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "at 12:23 whats NAD and how do you get it to NADH",
        "A": "Oh so reducing NADH you have to add hydrogen and two electrons"
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "(at 3:21) it says in the equation 38 ATP, in the equation I am learning in class I have 36 ATP, which is false and why?",
        "A": "it varies, depending on the efficiency of the cells."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "At 3:22, in cellular respiration, is it necessarily always 38 ATPs that a cell produces?",
        "A": "Nope! ATP is actually lost in several places: first of all, ATP is used during respiration to actively transport materials into the mitochondria matrix and inner membrane. Secondly, some H+ leaks across the proton gradient so not all the H+ is converted into ATP (this will make sense once you ve watched the Chemiosmosis/Electron transport chain part.)"
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "at 13:12 isn't FADH actually FADH2?",
        "A": "Try watching the video without the big screen. It actually says that."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "Hi, at 10:35, so there are three stages in cellular respiration?",
        "A": "Yes, as Sal said, there are 3 stages: glycolysis, Kreb s cycle (or citric acid cycle), and electron transport chain. :)"
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "6:33 there are actually 11 hydrogen particles shown in the diagram! Where is the last one?",
        "A": "He said he found the picture on Wikipedia and copied & pasted it on his video, right? Perhaps a chunk of the picture got cut out when he copied it."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "At about 2:15-2:30 ish, he is writing down a chemical reaction for cellular respiration. He then goes on to write that carbon dioxide and water are the products of such a process... Considering that combustion reactions have a product of carbon dioxide and water, does this mean that cellular respiration is a combustion reaction? That doesn't make sense does it?",
        "A": "respiration is like combustion because the reaction and products (including energy) are very similar. However the high temperature required to carry out a heat-mediated combustion reaction is too much to support life (at such high temperatures, proteins will break down). so i guess the body opted for enzymes so the reaction could occur at fairly low temperatures :)"
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "At 12:25 you say Hydrogens get added to NAD+ to become NADH. I'm assuming you mean H+ ions or protons. Am I right?\nThanks a lot for the vid.",
        "A": "From what I understand: NAD+(NAD and 1 proton)takes an electron from H2, thus becoming neutralized - NADH (NAD and 1 proton and 1 electron). The H2 (2 protons and 2 electrons) molecule then becomes a Hydrogen ion (2 protons and 1 electron). So, NAD+ was reduced to NADH and H2 was oxidized to H+. Thanks a lot."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "At 7:40, what does he mean by net ATP's?",
        "A": "Net refers to the overall, after adding or substracting. So if it is net then it refers to the total amount."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "What is sucrose? 5:40.\nOr however you spell it.",
        "A": "Okay. Thankyou very much"
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "At 4:55, if Hydrolysis is to break up a molecule using water... then why isn't glycolysis the method to break up a molecule using glucose?",
        "A": "Glucose is what is being broken down."
    },
    {
        "video_name": "2f7YwCtHcgk",
        "Q": "At 3:40 you say that the energy produces 38 ATPs, but my textbook says 36. Which is right?",
        "A": "Different Biology textbooks will tell you different numbers.In reality, I don t think anybody s sure how many ATP. Generally, it s a number between 36-38. Just ask your teacher which number she prefers. I would go with my textbook, just because that s what your teacher probably uses to write the exams."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "At 2:10, when the resonance structure was created, the original Carbon that had a lone pair that moved, does this mean that according to the bond line structure that a Hydrogen would also attach to it in the resonance structure? Because in the second structure, this would mean that the Carbon only has 3 bonds to it.",
        "A": "Yea there s an implied hydrogen there. That carbon has 3 bonds and a lone pair in the left structure, and 4 bonds in the right structure. Count them again."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "At 06:00 in the video the carbon atom that has + net charge on it does not follows the octet rule. With all the covalent bond it gets up to a total of 6 electrons which is incorrect. And at 06:39 the resonance structure carbon atom on the left is also doing the same. I think in both cases there should be a lone pair of electrons. Somebody please answer??",
        "A": "And to answer your question about the octet rule , just remember that having less than 8 electrons isn t the problem, it s having MORE than 8 electrons that breaks the octet rule! A carbocation is a carbon with a positive formal charge, which indicates less than 8 electrons (6 to be exact). This does not violate the octet rule!"
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "at 6:00 why that molecule has a positive charge? I thought that H atom contributes positive charges while C atom contributes negative charges on molecules.",
        "A": "That + charge tells us that that carbon is missing a hydrogen atom"
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "At 3:17, why is it that we cannot have another bond on the Nitrogen? There are no lone pairs attached to it, so are we just assuming it's there, or are we assuming that there are two Hydrogens attached to it?",
        "A": "the N in this molecule has 4 covalent bonds. (2 single bonds, 1 double bond, total of 8 electrons) If it were to make any more bonds, it wouldn t be obeying the octet rule."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "How do you know the second example at 2:40 is nitromethane? Why is it a methane if there is a double bond?",
        "A": "That s its name. -NO2 is called a nitro group, so CH3NO2 is nitromethane. Same way that CH3Cl is chloromethane. The double bond is in the nitro group not on the carbon."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "Hello, I still don't get why in the first example (0:19) C has formal charge of -1. Please explain it to me!",
        "A": "A carbon that has 3 bonds and 1 lone pair HAS to have a formal charge of -1 Formal charge = valence electrons - lone pair electrons - bonds For that carbon, it has 2 lone pair electrons and 3 bonds so: 4 - 2 - 3 = -1"
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "At 2:21, I just wanted to know why the carbon does not have a positive charge, it only has 3 bonds? Is it because the electron pair was originally from that particular carbon?",
        "A": "If you re confused, remember this: A carbon with a formal charge of 0 has 4 bonds A carbon with a formal charge of -1 has 3 bonds and 1 lone pair A carbon with a formal charge of +1 has 3 bonds and no lone pair So if you mean the carbon that originally had the lone pair, it has an implied bond to hydrogen that isn t shown as we do not generally show C-H bonds in these structures."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "at 3:17, why can't the nitrogen have 5 bonds?",
        "A": "5 bonds would mean that nitrogen is exceeding the octet rule which it does not do. The octet rule says that atoms generally react in such a way as to have 8 electrons in their outer most shell. The common elements that will always follow the octet rule where possible are C N O and F."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "Can someone refresh me on (3:00) why nitrogen cant have 5 bonds? I thought it had 5 valence electrons",
        "A": "i think how you are seeing it is that the 5 bonds around the nitrogen give it its 5 valence electrons, which IS what it wants. However, nitrogen is in period 2 of the periodic table and therefore can only have 4 total bonds (aka 8 total valence electrons). Just draw out nitrogen with 5 bonds coming off of it and I think you will be able to visualize what I am saying."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "At 5:00, how do we know that since the carbon has a +1 charge, it has 3 bonds? Wouldn't that mean it doesn't have a full octet? If that was the case, it should have 2 valence electrons to make the full octet, right?",
        "A": "The carbon must have a vacant orbital to have a + charge. That leaves three other orbitals for bonding. The carbon then has 6 valence electrons, and it needs two more to complete its octet."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "At 4:35 does that mean that they are both the major contributor?",
        "A": "Yes both are the major contributor"
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "Sorry, probably kind of a stupid question, but at 0:10 , why does carbon have a pair of lone electrons in its valence shell, instead of only one e-? It has 3 covalent bonds (to the two neighbouring Cs and the H) therefore, shouldn't that mean that there is one lone electron left since it is sp3 hybridized and there are 4 electrons in valence shell all together? Or is that reasoning wrong?\n\nThank you for the amazing videos otherwise, I'm just completely new to chemistry!",
        "A": "Electrons always exist in pairs. Check out the videos on Hybridisation ."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "At 8:25, the central carbon atom in the resonant structure has only 3 bonds. Its octet isn't complete. How is this possible?",
        "A": "It is still a valid resonance structure, octets do not have to be complete for it to be valid. It does mean it will not contribute much to the overall structure of the molecule, but that resonance structure does show us why carbonyl carbons are reactive towards nucleophiles. Jay explains this further on in the video."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "Why do we use formal charge and not oxidation state at 5:40?",
        "A": "Remember that formal charge and oxidation numbers are really just electron bookkeeping tools. Oxidation numbers assume every bond is 100% ionic and formal charge assumes every bond is 100% covalent. Obviously neither of these are perfect, but formal charge is more ideal for lewis structures as they are almost always covalent in nature."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "At around 3:08 , he says that nitrogen can't have five bonds because of its position on the periodic table. I can't quite comprehend this. Isn't nitrogen placed at the top of group 5A having five valence electrons. Having five v.electrons, I can't see how it should have any problem making five bonds.",
        "A": "Nitrogen is too small to bond to 5 other atoms. Elements in period 3 and on can have 5+ bonds (eg. PCl5), but not elements in period 2. 4 bonds is the most we see nitrogen form."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "In the example started at 4:47, it seems to me that both resonance structures are indeed equal, only flipped...Am I wrong?",
        "A": "Hey, You are right- both resonance structures are similar. However, in resonance structures, we try to spread the electrons over different atoms (even if they are of the same element), so this is OK. I believe you are confusing resonance structures with structural isomers, as the two examples given at 4:47 are resonance structures, but are not structural isomers."
    },
    {
        "video_name": "UHZHkZ6_H5o",
        "Q": "At 6:14, why don't you make the lone pair in the center carbon?? Why did you move the left pi combination to right one right away?? because when the carbon has negative charge, it isn't stable?? Thank you!",
        "A": "You are partly right. If you moved the lone pair to the centre carbon, you would get CH\u00e2\u0082\u0082\u00e2\u0081\u00ba\u00e2\u0080\u0094CH\u00e2\u0081\u00bb\u00e2\u0080\u0094CH\u00e2\u0082\u0082\u00e2\u0081\u00ba. This has three separate charges and is a very high energy contributor. If you move the electrons to the other side of the central C atom, you get CH\u00e2\u0082\u0082\u00e2\u0081\u00ba\u00e2\u0080\u0094CH=CH\u00e2\u0082\u0082. This is equivalent in energy to the first structure, and it has only one positive charge. You want to avoid separation of charge when drawing resonance structures."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "At 4:42 the compound is named as 7,7 dibromo oct-5-yn 4-ol but shouldn't it be named as 2,2 dibromo oct 3-yn 5-ol ?\n\nI understand that the former name is given keeping in mind -OH as the functional group but by giving the latter name we will have the least sum of locants (ie, following lowest sum rule) .\n\nWhen should the lowest sum rule be dropped ?\nThank You in advance.",
        "A": "I know it s a year late but it s still not answered. The highest priority functional group needs to be given the lowest numbering possible. That is the alcohol group in this molecule. So we need to number from the left side as that gives the alcohol a lower number. Also, there is no lowest sum rule, despite what your teacher may have said."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "At 4:45, can you also just write 7-dibromo instead of 7,7-dibromo?",
        "A": "No. Two substituents require the specification of two positions. Otherwise, it would not be clear if a number had been mistakenly left off."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "5:38 did sal forget to name all of the carbons? he only did 1 2 and 3 he forgot the end points too?",
        "A": "1 meth- 2 eth-3 prop-4 but- 5 pent-6 hex-7 hept 8 oct- 9 non- 10 dec Alkanes -ane Alkenes -ene Alcohols -anol Carboxylic Acids -anoic acid Esters -yl -anoate"
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "So at 6:40, Mr Khan shows us two ways of naming. Which one is the more preferred method in the scientific community?",
        "A": "Both methods are used, but the preferred method is now to use an infix rather than a prefix to indicate the position of the functional group, Thus, the preferred name for HOCH2CH2CH2OH is propane-1,3-diol, because it puts the locating numbers as close as possible to the functional groups. The name 1,3-propanediol (with the numbers used as a prefix) is also acceptable, but it is losing favour among chemists."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "At 4:50, 7,7-dibromo oct-5-yn-4-ol is one way to name the molecule. What's the other??",
        "A": "there is only one name. It is 7,7-dibromooct-5-yn-4-ol or 7,7-dibromo-5-octyn-4-ol. They are equivalent."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "At 3:00, In the first example the compound is name in 2 ways (2-pentanol and pentan-2-ol), but in the 2nd example it is named in one way only and that is, 7,7-dibromo oct-5-yn-4-ol, but How can we name it in the first way as given in the first example?",
        "A": "No actually, you are correct. Saharsh and the last response is fallacious. Good question. the locant (substituent number must precede the substituent to which it refers). so 7,7-dibromo-5-octyn-4-ol is the alternative. the 5 precedes the alkyne, the 4 precedes the alcohol. That s all that matters."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "what is alkyne [3:26]",
        "A": "A carbon molecule that contains 1 Carbon-Carbon TRIPLE bond. since it just might be anywhere in the molecule, and its location (carbons in the middle or at the end) can change its chemical properties that we observe, we specify the exact location at which it is locating. we do so by specifying the number (given consecutively from the nearest endpoint carbon) so that if we wanted to draw a picture of the carbon according to its name, it will have the bonds in the right place"
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "at 6:31, is it \"propan-1,3-diol\" or \"propane-1,3-diol\"?",
        "A": "It is propane-1,3-diol. You drop the final e only when it is followed by a or o, as in propanamine and propanol. So you write propan-1-ol but propane-1,3-diol."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "From 6:39 onward, why couldn't the compound be named 3-hydroxypropan-1-ol?",
        "A": "Well it certainly COULD be named that way, but when we have 2 of the same functional group we use di to indicate there are two of them, even when it s in the parent chain name."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "At 0:23 Sal mentions that R stands for radical. Doesn't R stand for an alkyl group? If the functional group for an alcohol is R-OH, doesn't that mean that a hydroxyl group is attached to an alkyl stem?",
        "A": "A radical is an atom or group that has an unpaired electron. For example.the methyl group, CH\u00e2\u0082\u0083\u00c2\u00b7, by itself is a radical. When it is attached to another atom, the electrons become paired, but it is still a methyl group. Thus we use R to refer to an alkyl group\u00c2\u00b7 In ROH, we have an alkyl group R\u00c2\u00b7 combining with a hydroxyl group \u00c2\u00b7OH to form R-OH."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "At 3:40 in this video why was the numbering done from left as it has not followed lowest locant rule?",
        "A": "The OH takes preference in the numbering. The Br atoms take no preference at all. So you must start numbering from the end closest to the OH."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "6:16 When Do I Put an \"e\" and when Not ?\nI'm Confused !!\nwhen to use... alkane- and when alkan- ?",
        "A": "Yes, it is confusing, The e is dropped when it comes before another vowel (a, e, i, o, u), to make the name easier to pronounce. PropANEol becomes propANol, and propANEal becomes propANal. PropANEdiol stays the same. Think of the spelling before the numbers are inserted, then insert the numbers. Then you get propan-2-ol and propane-1,2-diol."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "@ 3:33 shouldnt the carbon chane attached directly to the triple bond be straight and not like in a synched chain as in a normal C-bond?",
        "A": "Yes of course, everything connected to the two sp carbons will be in the same line with them, but Sal was just being comfortable :D"
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "what is pentanol(2:15)",
        "A": "the prefix penta means - 5, nol is suffix used to denote functional group called Alchohol. Therefore, pentanol is called organic compound with 5 carbons and one OH group. CH3CH2CH2-CH2-CH2OH"
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "would you have the e on the end of the propane for the last example \"6:12\"?",
        "A": "if it could be only one hydroxyl group it can be called CH3CH2CH2-0H 1-propanol Alkane+OL - Sal has don an unwitting mistakes in every case we have to remove th letter E methanol ethanol propanol buthanol ect"
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "why is it called cyclohexanol?i can understand hexanol but why is it called cyclo[5:20]",
        "A": "Think of how it s named like it s shape. 6-sided, so it s hexane. Hexanol because it s got an -OH group on it. The reason it s called CYCLOhexanol is because it s not in a single chain anymore-the two ends we would normally see have attached and turned it into a continuous chain of carbons. That s why it s a cyclohexanol and not just hexanol."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "at 4:12, why is it a oct-5-yne instead of counting the carbons the going in the other direction and it being a oct-3-yne ?",
        "A": "At 3:30, Sal informs us that an alcohol takes precedence [over] all [other functional groups], so we want to start numbering closest to the alcohol..."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "at 5:43, sal miscounted the carbons? Isn't there actually 5, he forgot the ends?",
        "A": "Those ends are attached to the OH groups. They are not carbons so Sal did not miscount the carbons."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "When naming the dibromooctynol around 4:45, it seems like there should have been a chiral center at the 4 carbon, since it was bonded to a propyl, -ol, hydrogen, and butyl. Was that missed in the video, or is there some reason the molecule isn't chiral?",
        "A": "I don t think he does stereochemistry for any of these molecules, just the base name. He hasn t used a wedge or a dash for the OH so it s impossible to tell."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "at 3:30 when Sal starts numbering the \"beast\", how does he come to know that alcohol takes precedence over alkyne and the di-bromo?",
        "A": "Functional groups themselves have a priority which gives it the lowest possible number in the name compared to others. Sometimes branching doesn t matter. Here is a simple list of functional groups in order of precedence (must have the lowest number possible) . Carboxylic acids, ahdehydes, ketones, alcohols, amines, alkenes, alkynes, R=OR=Ph=X. Alcohols beat alkynes and Br (X). Hope that helps."
    },
    {
        "video_name": "nQ7QSV4JRSs",
        "Q": "AT 5:40 is it right to name the 4th example \" 1,3 - dipropanol \"",
        "A": "NO. Because we have two -ol groups and not two propanes. 1,3 dipropanol indicates a 6 carbon compound while we have a three carbon compound with two -OH groups.. Hope this cleared your confusion!"
    },
    {
        "video_name": "_eBq2hRETmA",
        "Q": "At 1:50, when did BP equal CO x SVR? What source could be used to back that up?",
        "A": "By definition SVR = BP/CO, and is in units of mmHg\u00c2\u00b7min/L (or equivalent). It s a derived measure that s used as a conceptual model to understand what happens in some kinds of shock because while we can easily measure BP and heart rate, we can t measure SVR or stroke volume to calculate cardiac output. Being able to relate the clinical features of increased SVR (such as pallor or a cold periphery) to the measurable HR and BP can help us to understand why there is a failure of O2 delivery."
    },
    {
        "video_name": "tlSJCG5DqzI",
        "Q": "At about 5:50, the teacher says that Calcium prefers to lose two electrons rather than gaining 10+6. In that hypothetical second scenario, wouldn't Calcium be gaining only 6 electrons, because I thought that the 3d electrons weren't valence electrons? Sorry if my question is confusing.",
        "A": "The 3d electrons (or some of them) definitely are valence electrons for transition metals. Calcium would still need to gain them before it could gain the 6 4p electrons though."
    },
    {
        "video_name": "tlSJCG5DqzI",
        "Q": "In the example given at 6:27, if Ca has a full octet then shouldn't it be a neutral atom?",
        "A": "Ca\u00c2\u00b2\u00e2\u0081\u00ba has a full octet, but it has 2 more protons than electrons, so it has a positive charge."
    },
    {
        "video_name": "ufzlgOyWBHE",
        "Q": "At 6:55 the three iodine atoms are put along a line. Is it possible to put them in a triangular configuration using three single bonds? This would consume 6 valence electrons. The remaining 16 v.e. could then be spread out so that two of the iodine atoms get 5 each and one gets 6 electrons. Then, two of the atoms would have a charge balance of 0 and the last one would have a balance of -1, same as in the shown case except we do not need to \"break\" the octet rule.",
        "A": "It s an unlikely structure, 3 membered rings have a large amount of ring strain and two of those iodines have an unpaired electron so are radicals."
    },
    {
        "video_name": "ufzlgOyWBHE",
        "Q": "At 4:32 , what if we put the two lone pairs axially? Would not the axial - equatorial angle remain 90 as F atoms face repulsion from top and bottom?",
        "A": "Yeah, but one of the ideas behind VSEPR theory is that non-bonding electrons (i.e. lone pairs) will repel as far from a) one another and b) bonding electrons as possible. If we put em axially, we increase repulsion & that destabilizes the molecule. You ve probably figured this out by now, since you asked about it 8 months ago, but for anyone else wondering there it is."
    },
    {
        "video_name": "uWJRx-5n1yw",
        "Q": "At 1:45, Jay says that Hydride ion in itself is not a good nucleophile but a good base. Can someone please explain it to me? Prof. Ernest Zinck?",
        "A": "Hydride is not a good nucleophile because it has two 1s orbital electrons. As you know, the bond will be formed if the orbitals of 2 atoms match well. But C s 2s orbital is not compatible well with 1s orbital of hydride. So it is not a good nucleophile. It is a good base because for H, whose outmost orbital is 1s orbitals, full 1s orbital is very stable."
    },
    {
        "video_name": "uWJRx-5n1yw",
        "Q": "At 2:58 can the alcohol R''OH give of a proton? wouldn't water act as a stronger acid?",
        "A": "I believe that R OH and H2O are alternatives for the solvent \u00e2\u0080\u0093 so you usually wouldn t have both present."
    },
    {
        "video_name": "uWJRx-5n1yw",
        "Q": "At 11:20 what's the excess product when LiAlH4 reduced the Ester group?",
        "A": "Let R1COOR2 be the ester O || R1-C-O-R2 Then the products will be R1-CH2OH and R2-OH"
    },
    {
        "video_name": "5d63YMUxcS4",
        "Q": "At 5:00, Can 1-Pentene exist, As I have not come across the same in any of the textbooks.",
        "A": "Pent-1-ene exists. It is a colourless liquid with the odour of gasoline and a boiling point of 30 \u00c2\u00b0C."
    },
    {
        "video_name": "5d63YMUxcS4",
        "Q": "2:05 Can you call it oxalane?",
        "A": "oxolane is another name for THF, tetrahydrofuran"
    },
    {
        "video_name": "5d63YMUxcS4",
        "Q": "5:28 Which one does the IUPAC reccomend",
        "A": "IUPAC allows two systems: 1. Epoxy system: e.g., 1,4-epoxybutane. 2. Hantzsch-Widman System: e.g. oxolane (tetrahydrofuran). oxo = oxygen; ol = 5-membered ring; ane = saturated."
    },
    {
        "video_name": "5d63YMUxcS4",
        "Q": "6:39... Isn't it important to label what the number is of the carbons that the \"oxide\" group is joining to? Hence, I would have thought that this molecule should be called \"cyclohexene-1,2-oxide\".",
        "A": "Yes, numbers are required by IUPAC. The IUPAC name is 1,2-epoxycyclohexane. Cyclohexene oxide is a common name."
    },
    {
        "video_name": "5d63YMUxcS4",
        "Q": "Can the oxygen at 2:50 be bonded to carbons that are more than 1 bond away from each other? So that the compound would be named 2,4-epoxyoxide, for instance.",
        "A": "Yes, that is possible."
    },
    {
        "video_name": "WRuatAcd2WY",
        "Q": "At 5:50, how do I calculate the inverse sine without using a calculator?",
        "A": "You definitely do for standard values. For eg. sin (pi/6) = 1/2 so sin inverse of 1/2 is pi /6. Similarly, you can do that for other standard values that are worth remembering."
    },
    {
        "video_name": "WRuatAcd2WY",
        "Q": "4:34---> soh-cah-toa?\nWhat's that??",
        "A": "It s basically a mnemonic S.O.H.{SinA=Opposite side of angle A/Hypotenuse} C.A.H.{CosA=Adjacent side of angle A/Hypotenuse} T.O.A.{TanA=Opposite side of angle A/Adjacent side of angle A} These are the basic trigonometric ratios. I initially got to know this mnemonic from my physics teacher at school. Now I know where it comes from."
    },
    {
        "video_name": "WRuatAcd2WY",
        "Q": "I don't understand 4:34 where Sal uses the unit circle to determine the sin90 = 1. Can someone elaborate on how that works? Or is there another video related to that?",
        "A": "Sal has many vids about this in the trigonometry section."
    },
    {
        "video_name": "WRuatAcd2WY",
        "Q": "At 7:45 in Glass Fiber will the energy of light particles will get to zero or TIR will continue to infinite time in ideal case?",
        "A": "the light will eventually be absorbed by the glass."
    },
    {
        "video_name": "obeGVTOZyfE",
        "Q": "At 0:19, sal says delta Q, isn't it delta U?",
        "A": "yes, you can hear he say while writing change in internal energy"
    },
    {
        "video_name": "obeGVTOZyfE",
        "Q": "5:30, what just to clarify.\nSal is claiming that the System did some work when the volume increased?",
        "A": "If the pressure was constant or if it increased, the system must have done work."
    },
    {
        "video_name": "obeGVTOZyfE",
        "Q": "10:22\nSal, you said that Surface Area times Height is Volume. I don't understand how that works.\n\nThe surface area is 2(pi)r^2 + 2(pi)rh. But if you multiply this surface area by height, you get 2(pi)(r^2)h + 2(pi)r(h^2), which does not equal the volume of the cylinder, which is (pi)(r^2)h.\n\nI'm pretty sure I'm not getting something. And also, does SA x h = V always true?",
        "A": "He meant that cross sectional area x height = volume This result is true only for bodies with uniform cross sectional area. If the cross sectional area is non uniform, then we probably use calculus. In this case, cross sectional area = (pi)(r^2) height = h Hence volume = (pi)(r^2)h Let me give you another example: In a cuboid, the cross sectional area is (length x breadth) And height = h Hence volume = length x breadth x height"
    },
    {
        "video_name": "obeGVTOZyfE",
        "Q": "At 6:10 when Sal defines the pressure and force, why does he assume that the force applied on the ceiling of the piston is constant? As the volume increases by the force pushing the ceiling a distance \"x\", shouldn't the pressure decrease? If the pressure decreases in this process, why doesn't the force also decrease?",
        "A": "Because x is assumed to be a very small number. And over a very small displacement the pressure is assumed to be as good as constant. The math of that is explained in the videos on finding the area under a curve in the calculus section (Riemann sums and definite integration)"
    },
    {
        "video_name": "6YU9HVS2GT0",
        "Q": "At 10:50, the speaker says that we get \"easier\" access to some of the \"high tech stuff\", but what good does that do if people cannot afford it? I've learned that hospital bills are the number one cause of individual bankruptcies in the US.",
        "A": "Interestingly, our lack of health insurance in the United States also contributes to worse healthcare outcomes. Many people who cannot afford healthcare delay receiving services, resulting in the condition being far more difficult to treat or becoming untreatable. This helps explain why the United States has a higher morality rate than many other industrialized nations despite having much of the most high-tech medical equipment in the world."
    },
    {
        "video_name": "SJUd5du0T08",
        "Q": "At 2:55 during the video, when you drew that diagram of Summer and Winter, If the N. Hemishpere is tilted toward the sun, the Northern Hemisphere will have Summer. I understand that, but if Earth is on a tilt, why would light be on the whole Northern Hemisphere? Wouldn't one half of the Northern Hemisphere have no light? Or am I thinking of rotation?",
        "A": "While tilted, the earth is also rotating so that side of the Northern hemisphere is currently experiencing nighttime and will rotate towards the Sun eventually."
    },
    {
        "video_name": "SJUd5du0T08",
        "Q": "At 8:49 are southern hemisphere seasons more severe",
        "A": "Where did you get the timestamp? The entire video is only 4:39."
    },
    {
        "video_name": "SJUd5du0T08",
        "Q": "who found out about perihelion at 1:19",
        "A": "Though he probably didn t name it, it was Johannas Kepler (same as the badge) who discovered the elliptical orbits of planets."
    },
    {
        "video_name": "G0h9Sa_s18A",
        "Q": "At 2:08, why can't I say sine leads cosine by 90 degrees?\n\nWhy do we have to see the waves as if they are moving to the left?",
        "A": "Lead and Lag reflect the relative timing of two signals. If func1 has a peak sooner than func2, we say func1 leads func2. In these graphs, time increases as you go to the right. So a time near the origin happens sooner than a time out to the right on the time axis. If you look at the cosine function, it has its peak at t = 0. The sine function has its peak at t = 90 degrees. That is later than the cosine peak. So we say cosine leads sine by 90 , or equivalently, sine lags cosine by 90 ."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "I seem to recall having previously learned that lightning goes UP from the ground to the cloud, not the other way around as he say at ~4:25. Which way does it travel?",
        "A": "There is often a leader current that comes up from earth and sort of paves the way for the big strike to come down."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "What is that 8-like symbol Sal makes at 6:20 to show the electronegativity of the atoms of an H20 molecule. Please help!",
        "A": "It s a lowercase Greek letter, Delta. It means a small change when used in math or science. In this case the positive and negative charges are minor."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "At 7:43, you say that Nitrogen, Oxygen, and Fluorine are the most electronegative atoms. This is false. The three most electronegative atoms are Chlorine, Oxygen, and Fluorine.",
        "A": "Yes, Fluorine is the most electronegative at a value of 3.98, then oxygen with 3.44 and chlorine with 3.16. Nitrogen is not far off of Chlorine with 3.04 on the Pauling scale."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "At 4:35 Sal says that the conduction allows extremely high temperatures...to flow all the way to the ground. What does he mean by this?",
        "A": "Plasma is formed when high temperatures free the electrons from their host atoms for a short time. Lightning is plasma. As the electrons pass through the air from sky to ground, they heat it up so much that some of the molecules in the air around them get ionized. The ionized gas that we see as lightning is the air that the electrons have passed through, getting excited and giving off light."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "At 3:39, what does huge potential difference mean?",
        "A": "It is another way of saying there is a huge voltage between the clouds and the ground."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "6:22 is that why neutrons are deadly to the body because hydrogen is 'essentially just a proton', does hydrogen have any electrons?,\noh & why is it that when the electrons float around nitrogen, it gets a negative charge?",
        "A": "Hydrogen is a proton and an electron, although different isotopes exist (deuterium & tritium) . An extra pair of electrons will give any atom a negative charge."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "at 5:13 im pretty sure gas is the most common due to all the hydrogen and helium gas nebulae and such",
        "A": "I think you misunderstand. Plasma is the most common STATE of matter. Hydrogen and Helium are nothing more than elements, which can exist in solid, liquid, gas, or plasma states. Don t confuse element classification with state of matter."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "At 7:24, I am a little unclear as to what the line underneath the O signifies? Can you explain this .",
        "A": "He was trying to draw two electrons but accidentally connected them into a line."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "At 4:33, Sal tells that the air becomes a plasma state. Wont the element gases get affected?",
        "A": "Yes, as a plasma is ionized gases at much higher temperature, then the molecules would become ionized by the loss of one or more electrons, which would probably cause the atomization of the molecules as well as the ionization. That is all molecules would break into their respective elemental atoms and lose one or more electrons."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "At 8:00, why is it that Hydrogen Bond only arise when H is with N, O, or F only? Why can't Hydrogen Bond arise when H is with Chlorine or another electronegative element?",
        "A": "Chlorine is a bit too large of an element for the intermolecular forces to be as strong. It s bigger due to another shell of electrons being added (as Chlorine is in the row beneath the N, O, and F row) and thus the shielding effect being exhibited (where electrons in the inner shells shield electrons in the outermost shell from the attraction of the nucleus). This causes forces of repulsion that make the atom larger."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "At 2:19, what is plasma, I'm a little confused on what it does?",
        "A": "Plasma is literally Greek for Anything formed ! Plasma is another state of matter. It is when the electrons are literally being stripped off of the atoms. Neon lights you may see are lit by plasma. I used to think that plasma was anything slimy :P ! Thanks for asking!"
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "At 2:10, how do you know if a gas is high pressured or low?",
        "A": "You can measure gas pressure with a device called a manometer, the particles of gas hit a liquid in the manometer and push it up a tube which has a scale where the gas pressure can be read off."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "at 0:45, where is plasma found?",
        "A": "Lots of places. Some examples are in lightning, in solar flares, in the northern lights, in fluorescent lights and inside stars."
    },
    {
        "video_name": "WenwtcuqOj8",
        "Q": "2:45 \"the electrons are bumped off\"...does this mean ALL the electrons of the atom are bumped off, or just one of the outermost, or what?",
        "A": "You don t need to lose all the electrons from the atoms to form a plasma, although that could happen if the conditions were right (eg, a very high temperature)."
    },
    {
        "video_name": "UekR9J31a2o",
        "Q": "From 4:50 on, there is a message saying \"Sal says 'molecules' but means 'atoms.'\" What exactly is meant by that - when did he talk about \"molecules\" when \"atoms\" would have been more correct? I would have said that \"molecules\" was correct.",
        "A": "Helium is a monatomic gas (in STP) and thus is not a molecule but rather an atom. A molecule is a definite group of atoms that are chemically bonded together, helium gas is not a molecule. If it was oxygen gas then yes, it s a molecule due to its being O2 (in normal STP) hope that helps! ^-^"
    },
    {
        "video_name": "UekR9J31a2o",
        "Q": "At 3:48, Sal wrote m for mole. Is this standard/allowed? Or will people confuse this with Meters?",
        "A": "This is not standard, always use mole or it s abbreviation mol"
    },
    {
        "video_name": "oDigu9YxXUg",
        "Q": "At 5:06, why is n=1 and not 6?",
        "A": "H\u00c3\u00bcckel s rule says the number of \u00cf\u0080 electrons must equal 4n+2, where n = 0,1,2, etc. 4n+2 = no. of \u00cf\u0080 electrons The no. of \u00cf\u0080 electrons = 6. 4n + 2 = 6 4n = 4 n = 4/4 = 1"
    },
    {
        "video_name": "20N53khArXA",
        "Q": "In the last example around 7:50 about closure, doesn't that image apply to another Gestalt Principle called subjective contours? He even mentioned \"contour\". Or is it that since I can also fill in the blank and know it's a triangle, the example can be used for both principles: Contours and Closure. An early thanks to those who answer my question!",
        "A": "I would think that they are the same. His definition of law of closure is : objects grouped together are seen as a whole , which makes sense in both subjective contours and closure. In both cases we make invisible connecting lines to make an object appear as a whole."
    },
    {
        "video_name": "uRA8vI2Gzvk",
        "Q": "At 15:34,why does the clay have an ANGULAR momentum before the collision? I thought it was moving translationally?Thanks",
        "A": "Think about it this way: Once the translationally moving mass comes into contact with the rod and sticks to it, its translational momentum will become angular momentum with the same magnitude, so what you are calculating is not it s angular momentum while flying in a straight line, but it s potential angular momentum that it would have if it started moving in a circle."
    },
    {
        "video_name": "uRA8vI2Gzvk",
        "Q": "At 16:51, it says that the gravitational potential energy can be determined using Gm1m2/d. I thought that the formula was Gm1m2/d^2... Or is gravitational attraction different from gravitational potential energy?",
        "A": "gravitational force is not the same as gravitation potential energy Just like mg is not the same as mgh"
    },
    {
        "video_name": "c7Yy9v8dH8s",
        "Q": "what is the animal in 11:25",
        "A": "The animals he mentions in 11:25 are mammals."
    },
    {
        "video_name": "c7Yy9v8dH8s",
        "Q": "At 4:26 he mentions the brain. Does this mean no other animals previously mentioned had a brain? Do insects have a brain, etc.?",
        "A": "Not all animals have a brain. Some have a simplified central nervous system with no actual brain -- just nerve cords."
    },
    {
        "video_name": "c7Yy9v8dH8s",
        "Q": "Hank said at the very end at 11:33 that all mammals evolved from one species. What species was that?",
        "A": "It is some ancestor species. We don t know that species however."
    },
    {
        "video_name": "04gQ0bQu6xk",
        "Q": "10:45 What does he mean by allele?",
        "A": "An allele is an alternate form of one type of gene. (i.e. the gene for eye color could have a brown eyed allele, a blue eyed allele, etc.)"
    },
    {
        "video_name": "04gQ0bQu6xk",
        "Q": "What does Sal mean when he says recommedation at 7:53?",
        "A": "Recombination? During crossover homologous chromosomes swap segments of their DNA. This is called recombination because the two chromatids that swapped DNA are no longer identical to their respective sister chromatid."
    },
    {
        "video_name": "04gQ0bQu6xk",
        "Q": "At around 8:45, Sal is explaining how the two chromosomes swap genetic info/material. How exactly does that work?",
        "A": "Sal meant that they exchange the little section of the chromosomes. Then the genetic information which was in that little section will be changed and form a new chromosome."
    },
    {
        "video_name": "zNAL1R-hZr0",
        "Q": "at 9:03, doesn't rotating the bond create a whole new stereoisomer? i know he was looking for a plane of symmetry to define the molecule as a meso compound but i'm having trouble understanding why you can rotate the bond and still consider it the same",
        "A": "As long as you are breaking no bonds, you still have the same molecule. Rotation about a single bond gives only a different conformation of the same molecule."
    },
    {
        "video_name": "zNAL1R-hZr0",
        "Q": "how can I find the plane of symmetry in the meso compound in 9:30....it is impossible to find it?!",
        "A": "You can t see the plane of symmetry as drawn but, if you rotate the bottom chiral carbon of structure III 180 \u00c2\u00b0 about the C2-C3 bond, you can see a mirror plane midway between the two atoms and perpendicular to the plane of the paper."
    },
    {
        "video_name": "zNAL1R-hZr0",
        "Q": "how is that a chirality center in the video for 0:34 if it doesn't have carbon that is attached to four different groups.",
        "A": "What makes you think it isn t bonded to 4 different groups? Look at the top carbon of the 3 membered ring. It s bonded to a Br, an implied H and two C. But while it s bonded to two C, the rule is that if there s a tie you keep going further until you find a difference. The C to the left is bonded to 2 H and the C below is bonded to 1 Br and 1 H. So there is a point of difference so they aren t the same, that s why it s a chiral centre."
    },
    {
        "video_name": "zNAL1R-hZr0",
        "Q": "at 3:32, how exactly are they superimposable? In other videos, whenever the molecule on the right is rotated to show it's the mirror image of the left, and then set on top of the molecule on the left, it shows they're nonsuperimposable. But here it seemed like the molecules were rotated just to make them superimposable. I don't understand..",
        "A": "You rotate the molecules to see whether or not they are superimposable. In this case, when he rotated the mirror image molecule, the two molecules happened to be superimposable (and therefore identical to each other). It s easier to see if you have the models yourself. You don t need an expensive model kit. Just make one yourself from marshmallows and coloured toothpicks or something similar."
    },
    {
        "video_name": "By-ggTfeuJU",
        "Q": "At 0:27 in the video how could a force that I put on some thing be equal to mine? What I'm getting from this is if I push a slid rule it would push me as far back as I pushed it. I know this isn't true so could you please help me with this? I really don't get it.",
        "A": "The reason that you wouldn t be pushed back as far as you pushed the slide rule is that you have greater mass than the slide rule. You are pushed back a distance proportional to the the slide rule."
    },
    {
        "video_name": "By-ggTfeuJU",
        "Q": "Hey Sal between 4:08 and 4:12 you say that the sand exerts an oppisite force and thats why you are able to walk and not sink .\nThen why is it that in quick sand that you go downwards no matter whichever direction you apply th force in ? (assuming you have walked into the quicksand accidentally)",
        "A": "ohk got it thanx"
    },
    {
        "video_name": "By-ggTfeuJU",
        "Q": "At 7:20, when it talks about the object pushing you backwards as you exert force on it,does the concept also apply to when you're swimming? If I move my hand forward then the rest of me will move backwards. Is this the same principle or am I asking dumb questions?",
        "A": "for example if you slam your hand against the table the force from your hand hits the table and that force hits back, but you don t really feel it"
    },
    {
        "video_name": "By-ggTfeuJU",
        "Q": "At 7:50 how is it possible for the astronaut to return to the shuttle by throwing the object? Wouldn't the object float away itself?",
        "A": "Conservation of momentum says that if the astronaut throws the object in one direction, the astronaut has to float in the other direction. It is just like the recoil when a gun is fired."
    },
    {
        "video_name": "By-ggTfeuJU",
        "Q": "3:58\nIf every object exerts an equal and opposite force on mr then technically I should be able to move anything. Why is it that I am able to move many objects?",
        "A": "I exert a force on an object. The object exerts an equal and opposite force on me. If there are no other forces then both me and the object will be moving in opposite directions."
    },
    {
        "video_name": "By-ggTfeuJU",
        "Q": "At 1:00, if the force you press on the block will cancel because the block will exert an equal opposite force. Then, how will the block move?",
        "A": "You put a force on the block. The block puts a force on you. How many forces are on the block? Just one, right? So it moves. This is a common misunderstanding about the 3rd law. Force pairs in the third law are on different objects, not the same object. Nothing cancels out."
    },
    {
        "video_name": "By-ggTfeuJU",
        "Q": "At 2:06. I don't understand the examples about the block or the desk. How does that work?",
        "A": "at 2:06 , we understand that we can feel pressure exerted against the force given against the block or the desk , which explains the Newton s 3rd Law of motion."
    },
    {
        "video_name": "By-ggTfeuJU",
        "Q": "At 6:25, Sal says \"most massive object\". Why is that? More massive things are harder to accelerate because of inertia right? Wouldn't it be equally efficient to throw a less massive object, faster? Or do i have a wrong idea about inertia?",
        "A": "How fast would he have to throw a peanut? Do you think that would work? What if he could push off against the entire space station? That would work better, right?"
    },
    {
        "video_name": "By-ggTfeuJU",
        "Q": "Two Questions:\n1. Where does the opposite force come from?\n2. At about 7:10, he said that the astronaut had to throw the object. What if he were to wave it around in the air. Would he go backwards, or would he wobble (not the best word choice)",
        "A": "1. It is not possible for one object to push on the other but not feel a corresponding push back on itself. Try it. See if you can push on a chair and not feel a push on your hand. 2. I think you can guess the answer to this. Try."
    },
    {
        "video_name": "By-ggTfeuJU",
        "Q": "at 4:00 the video explained that when we press on the sand, the sand also returns that pressure back but due to gravity we stay in our place.\nthen when gases are released from the rockets,why doesn't the gravity keep them in place and launches them off?",
        "A": "Conservation of momentum requires the rocket to move in the opposite direction of the gas it is shooting out. Gravity is pulling on both of them. If the rocket doesn t expel gas fast enough, it won t be able to produce enough upward force to counter gravity."
    },
    {
        "video_name": "6EdsBabSZ4g",
        "Q": "At 6:56 why sal tells us that earths magnet is fluctuating?",
        "A": "Hmm... alright, let me see The Earth s magnetic field fluctuation as you say, is due to the rotation of the semi molten iron rich material in the core. It is a fluid dynamo in effect. Due to the fluid nature of the conducting/magnetic material there is a wobble in the field. Eventually this wobble is great enough to cause the entire magnetic field to flip or reverse. But, this is simply speaking...it gets a lot more complicated that that."
    },
    {
        "video_name": "6EdsBabSZ4g",
        "Q": "at around 6:37, how did we find out that the magnetic rock changes direction?",
        "A": "The field strength of the magnetic field was measured over the oceans. The part of the field produced by the core was subtracted. The remaining magnetic field is called magnetic anomaly. The stripes that you see on the map are the magnetic anomalies."
    },
    {
        "video_name": "6EdsBabSZ4g",
        "Q": "at 10:00 it would be the blue first right he said it was older than purple?",
        "A": "The blue rock formed before the purple. The blue rock formed at the ridge and then got pushed to either side as the purple rock formed. The same happened to the purple rock as the yellow rock formed. So moving inward, it goes oldest to youngest. :)"
    },
    {
        "video_name": "N_J5SeI_62U",
        "Q": "At 6:08 , vertical forces doesn't effect the normal force this time. Correct me if am wrong , if the box is moving in the vertical direction with Av, there should not be any affect at all. on the contrary there will be affect if the box moves in the horizontal direction with acceleration Ah.",
        "A": "Not sure what it is you are exactly asking but perhaps this will help: In both cases it depends on whether your box is in contact with other surfaces. For example the box can be sitting in a lift and thus experience normal force while the lift accelerates upward or downward. Or, it could be sliding down a horizontal surface, again experiencing normal force among other forces."
    },
    {
        "video_name": "JbLRFt_Z8nI",
        "Q": "around 5:00, how did you already have 0.10M of Ca+2 ions?",
        "A": "You are starting with 0.10 mol/L Ca(NO\u00e2\u0082\u0083)\u00e2\u0082\u0082. This is a strong electrolyte. It dissociates completely in solution. Thus, 0.10 mol Ca(NO\u00e2\u0082\u0083)\u00e2\u0082\u0082 forms 0.10 mol Ca\u00c2\u00b2\u00e2\u0081\u00ba and 0.20 mol NO\u00e2\u0082\u0083\u00e2\u0081\u00bb. Then you add the Ca(OH)\u00e2\u0082\u0082, which produces more calcium ions."
    },
    {
        "video_name": "A4aoLMC1-5Y",
        "Q": "At 0:49, at the schematic diagram what does that circle with an inverted Z mean....and what is Step Response?",
        "A": "The circle is the voltage source. That backwards Z is just a representation of the waveform that it generates. Step response is the reaction of a circuit to a step in voltage, in this case, the reaction of an RC circuit to an instantaneous step in voltage produces a waveform that increases at an exponential decay."
    },
    {
        "video_name": "l3hw0twZSCc",
        "Q": "at 10:00, how shall we decide that we should consider the direction of our middle finger because it seems as though we can take either the field to be going in the screen or coming out of the screen?\nthanks",
        "A": "The field comes out of the screen - there is nothing arbitrary about it. Just watch the video again - Sal explains everything about it. (@ 3:00 for example)."
    },
    {
        "video_name": "l3hw0twZSCc",
        "Q": "I'm really confused about the hand rule (07:00). How does he automatically know that the thumb points to the left?",
        "A": "Hey gilbert Bercero arman is asking about the left pointing thumb not right pointing thumb."
    },
    {
        "video_name": "l3hw0twZSCc",
        "Q": "6:10 what hand is he drawing, left or right?",
        "A": "its the right hand, because he s doing the cross-product thing."
    },
    {
        "video_name": "l3hw0twZSCc",
        "Q": "at 5:19, why do we use the magnetic field strength 'B' of wire 1? shouldn't we use the 'B' of wire 2 since we use I2 and L2?",
        "A": "its the magnetic field generated by wire 1, so it s B1. The magnetic field of wire 2 wouldn t have any effect on wire 2 because wire 2 is exactly in the middle (if that makes sense)"
    },
    {
        "video_name": "7vHh1sfZ5KE",
        "Q": "I don't understand how the potential difference can be the same throughout a perfect conductor. For example, Sal says at 1:24 that \"between elements the voltage is always the same.\" Also, how would you apply Kirchhoff's law, stating that the voltage around a closed loop is zero, if there are no resistors in the circuit and the circuit is only made up of a perfect conductor?",
        "A": "How can the potential difference NOT be the same? If it is different, electrons experience a force and they will flow until it the potential is the same. If you have a battery and you connect the two terminals with a perfect conductor, then very quickly the battery will be discharged and damaged. In the real world there is no such thing as a perfect conductor, so the wire will act like a very very small resistor, but still the battery will quickly be discharged and damaged."
    },
    {
        "video_name": "7vHh1sfZ5KE",
        "Q": "At 8:09 Sal talks about the voltage drop,but i didn't really get it, i mean how can it be possible when the current is constant all over the circuit.\nWell,am pretty sure that am making a mistake somewhere(in some concept)",
        "A": "V = IR. If there s current and resistance, there s a voltage drop."
    },
    {
        "video_name": "7vHh1sfZ5KE",
        "Q": "At 2:45 why are those two voltages the same? I don't understand how the potential differences could be equal if for the volt meter above, it is measuring the voltage from before the current enters the resistors to after? Also, is the electric potential from the positive terminal to the beginning of R1 equal?",
        "A": "Potential difference is something you measure between two points. Points that are connected together will have the same PD as each other."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 8:25 in the video, when Sal is talking about the sec, iso and tert butyl conformations, if it ok to think of it as:\nsec= attached to 1 carbon\niso= attached to 2 carbons\ntert= attached to 3 carbons\nor no?\ni am also a little confused as to to why we still refer to the groups as butyl. Are they still butyl solely because they have 4 carbons in them? Does the fact that they are not in a four carbon chain matter?",
        "A": "no its not actually we give the name iso to any compound when the methyl group is attached to the carbon that is attached to the terminal(ending) carbon,and if two methyl groups are attached to the carbon that is attached to the terminal carbon then it is called the tert group,and if a carbon is attached to a carbon that has all its three remaining valency satisfied with the other carbons then the combined group is called the neo group and if any straight chain is attached though carbon other th"
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 10:56 I would have named it (2-methyl butyl) cyclopentane, why aren't you including that other carbon?",
        "A": "You are counting the ring carbon twice. its part of the parent ring, not the functional group. look at the picture, count the angles/carbons. There are 9 carbons total in the molecule. now look at your suggested name. methyl (1 carbon) butyl (4 carbons) cyclopentane (5 carbons) for a grand total of 10 carbons! which is one too many carbon atoms."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 10:54, instead of writing (1-methyl propyl) cyclopentane, can't we just write 2-cyclopentyl butane?",
        "A": "pentane is longer than butane, so it must be the base name"
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "at 4:27 sal said that the butyl group was attached to 2 other carbons. how? I mean, there is one carbon on the left hand side of the second carbon and 2 carbons on the right hand side?",
        "A": "No. He said that the carbon in the ring is bonded to a carbon that is bonded to two other carbons (that is, two carbons in addition to the one on the left hand side)."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 13:17 Sal talks about how the group would be called (1,1 dimethylethyl)... in the video Organic Chemistry Naming Examples 3 at 8:12, Sal says that for the systematic naming, you use the prefix \"bis\" not \"di\" to indicate more than one of the same group attached to the main backbone. My question is that if there if more than one carbon attached to one point on an alkyl group, is it always the case that the prefix \"di\" is used instead of \"bis\"??",
        "A": "I think sometimes bis- and di- are interchangeable. At least that s what wikipedia said."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 2:50 they have the same molecular formula but do they have the same physical properties ?",
        "A": "That s right. Their atoms are connected in a different order, so they are different compounds and have different properties."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "at 4:23 , Sal says that it's bonded to TWO other carbons, but aren't they THREE other carbons ? i'm confused",
        "A": "He means FROM the main chain, you don t count the cyclopentane. From that first C, you are only attached to 2 other Cs."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 4:53 Sal says that we are attached to a carbon that is attached to other 2carbons but what about the methyl group up their?",
        "A": "That entire butyl group is bunched up together meaning that the methyl group is a part of it. Then that entire butyl group is given a new name of sec. That entire shape is now called sec-butyl. The intuition behind the name is: the one bond that connects the butyl to the cyclopentane is connected two two other carbons within the butyl chain. You can see in the other two types of names that the number of bonds to that one carbon determine the prefix."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 4:15, why sal wrote 'butyl', why it is not butane?",
        "A": "When an alkane is a substituent we drop the -ane and use -yl The parent alkane chain keeps the -ane ending."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 7:22 Sal says that the particular compound is sec butyl but actually it is connected to four carbon atoms ,then it is to be isobutyl right? I am confused in this part!",
        "A": "That one is technically connected to 3 carbons, but with these common names what Sal is talking about is the number of carbons it is bonded on the side chain only. The sec stands for secondary carbon, a secondary carbon is bonded to 2 other carbons. That s what Sal means. Something to remember that may help. In butyl and sec-butyl the carbon chains look the same, they both are 4 carbons long. Butyl is bonded on the first carbon of the chain, where sec-butyl is bonded on the second carbon of the chain."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 12:48, What's the point of writing \"di-\" in front of methyl when we know \"1, 1\" can tell that there are 2 carbon attachments?",
        "A": "It s the other way around. Di- tells you how many there are, and the numbers tell you where they are."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "at 7:30 i didn't understand why you called it iso butyl",
        "A": "Butyl consists of a chain of four carbons with the free bond at the end: CH\u00e2\u0082\u0083CH\u00e2\u0082\u0082CH\u00e2\u0082\u0082CH\u00e2\u0082\u0082\u00e2\u0080\u0094 Isobutyl consists if a chain of three carbon atoms, with a methyl group on the carbon second from the far end: CH\u00e2\u0082\u0083C(CH\u00e2\u0082\u0083)CH\u00e2\u0082\u0082\u00e2\u0080\u0094 It has four carbons, and chemists had to give it a name. It was an isomer of a butyl group. so they called it iso butyl."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 13:32, I do not understand why you have to write 1,1. Please tell me why?",
        "A": "To show to which C atoms the methyls are attached to the ethyl. Consider we said (dimethyl ethyl) cyclopentane, could you be able to figure out if both methyls are attached to the same C atom or each is tied to one of the 2 ethyl C atoms? Without numbers, never!"
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "in the video at 02:55 why don't we consider the longest chain of carbon which starts from the yellow part and has 9 carbon atoms?",
        "A": "We must choose either the largest ring_or_ the longest continuous chain as the base for naming. Since the ring has 5 C atoms and the chain has only four, we must name this as a substituted cyclopentane. A chain must have two ends. If we start numbering from the end of the chain and then continue through the ring, the chain does not end at C-9, because C-9 is connected back to C-5. So the chain ends at C-4.We can t count C-5 as part of the chain because C-5 is part of the ring."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "So iso, sec-, and tert- refer only to the # of bonds to carbons in that alkyl group?\n\nBecause in the tertbutyl example (@6:25) and specifically at 6:48, Sal says that the carbon is attached to 2 carbons but in its entirety it's actually bonded to 3 carbons (1 carbon bond to the cyclopentane ring carbon and 2 more in the butyl). Right? So those sec-, tert-, iso- prefixes only describe the # of carbon bonds not on the main chain?",
        "A": "Short answer. You re right."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 8:36, when Sal wrote butly, didn't he mean Butyl?",
        "A": "Yes, he meant butyl."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "at 1:56 in the video the longest chain is not the pentane instead a octane so how it can be butylcyclopentane.",
        "A": "You count either the longest chain or the largest ring to determine the base name. You don t count them both. In this case the ring is the larger entity, so the compound is named as a cyclopentane."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 7:30 why is iso-butyl not just plane butylcyclopentane if it is only attached to one carbon?",
        "A": "Do you not see how an isobutyl group looks completely different to a normal butyl group? Butyl is 4 carbons long in a chain, isobutyl is 3 carbons long with a methyl group off the second carbon"
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "I cannot understand the difference between tert and sec Carbons.In the above video at 7:04,this man says that tert means a C attached to 3 other Carbons,the same is the case with Sec Carbon,then what is the difference between the two?",
        "A": "Sec-butyl is when the main chain is connected the second carbon of the butyl chain, while tert-butyl is actually a propyl chain with the main branch and another methyl group are connected to the central atom of the propyl chain."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "Around 13:30...when systematically naming the iso-butyl group as 1-methylpropyl, thus giving the entire molecule the systematic name of (1-methylpropyl) cyclopentane....is it necessary to include the 1- or is it implied that the methyl group is on the first carbon in your propyl substituent? as in, could you just say (methylpropyl) cyclobutane?",
        "A": "You always have to give the number of the substituent. The substituent he was naming at 13:30 was the sec-butyl group. Its systematic name is 1-methylpropyl. The systematic name of the isobutyl group is 2-methylpropyl. You need the numbers to tell which is which."
    },
    {
        "video_name": "TJUm860AjNw",
        "Q": "At 13:15 why cant it be named 2-methyl propyl cyclopentane",
        "A": "The longest chain in the group is only two carbon atoms long. it must be a substituted ethyl group."
    },
    {
        "video_name": "_UrfHFEBIpU",
        "Q": "At 9:17 why is the tension in the second wire(not the wire from the ceiling) equal to the x-component of the first wire",
        "A": "The tension of the T1 wire is pulling both upwards and towards the right. It pulls towards the right with a force of 100sqrt(3) N. But, since the object is standing still, then the net force in the horizontal direction must be equal to zero, which means that there must be a force equal and opposite in direction to that force from T1 pulling to the right. This opposite and equal force is provided by the second string T2."
    },
    {
        "video_name": "_UrfHFEBIpU",
        "Q": "6:47 At this point, you know the opposite (T1y) is 100N. Why not save a couple steps and just use Tan \u00ce\u00b8 = T1y/T1x? T1x = 100N/ Tan 30\u00c2\u00ba.",
        "A": "Does that give you the same answer? Then you can do it. There often is more than one way to solve a problem."
    },
    {
        "video_name": "_UrfHFEBIpU",
        "Q": "From 7:00 to about 9, the presenter discusses solving for T1, in the instance that it only takes into account T1y. However, using this reasoning and steps I in question number 1 I did not get the correct answer, at first shot, when completing the \"Force of tension questions\". Additionally, the 3rd hint, given in this problem was in contradiction. I understand that as I am learning it may simply be a misunderstanding on my part, and I am eager to know what it might be.",
        "A": "If we apply a force with some angle it breaks in different vector axis. In X: Fcos[theta] and in Y:Fsin[theta]"
    },
    {
        "video_name": "_UrfHFEBIpU",
        "Q": "does the length of the wire have any implications in its ability to offset the tension in another wire? at 9:38 when Sal compares T2 with the x component of T1",
        "A": "If by changing the lengths of the wires you also change the angle of inclination of the T1 wire, then yes, changing the lengths of the wires would change the tensions of each wires because the tensions depend on the angle of inclination."
    },
    {
        "video_name": "_UrfHFEBIpU",
        "Q": "At 2:32 he says he's adding 2 more strings and then considers the tension in only those 2 strings but what about the first string?",
        "A": "The tension in the original string must equal that of the two new strings because they are connected at the ends, the forces can then be projected onto the x and y axis. T0x=T1x+T2x T0y=T1y+T2y where T0 is the original light blue string"
    },
    {
        "video_name": "_UrfHFEBIpU",
        "Q": "9:07\ndoes the length or material of string matter in tension problems?",
        "A": "The length of a string does not have any role to play on the amount of force acting on the string. What ultimately matters is how high the mass is above the ground, as simple tension, when a mass m is hanging from only one string on a strong support is m*g , where g is the acceleration due to gravity."
    },
    {
        "video_name": "_UrfHFEBIpU",
        "Q": "At 8:44 he says 'sine of 30 degrees is one half' what does he mean by that? That kinda made me confused :(",
        "A": "What he is referring to is the trigonometric function sine. If you are confused, I suggest watching the videos on trig in one of the math playlists."
    },
    {
        "video_name": "_UrfHFEBIpU",
        "Q": "Why does he multiply the cos30 by the T1 200? 9:45",
        "A": "well, he multiplied it to get T2 which is T1*cos30"
    },
    {
        "video_name": "_UrfHFEBIpU",
        "Q": "At 9:12 , where did square root of 3 exactly come from and why was it used?",
        "A": "Cosine of 30 = sqrt(3)/2"
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "sir, at 00:13, why are we looking at 'variation' only within species, can't Natural Selection be applied at a broader lever for example on Genuses, or tribes (going up the ladder).. mammals and non mammals?",
        "A": "Reproduction only happens within a species. Mammals can t reproduce with insects, for example."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "10:25\ndoes it mean that in every single chromosome their is genetic information of every part of body.\nfor example, does the chromosome in my heart cell contain the genes that determine my iris colour.",
        "A": "Yes. Mutations and gametes aside, all of your cells have the same DNA."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "At 9:10, it mentions allele genes. What other types of genes are there?",
        "A": "An allele is not a type of gene. Alleles are a version of a certain gene. Each organism with a diploid genome contains 2 alleles of the same gene inherited from his parents. These alleles can either be homozygous (the 2 genes are the same) or heterzygous (the 2 genes are different)."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "14:30\nare chromosomes in germ cell paired or are they single. scattred anyhow.",
        "A": "If the germ cell is a mature gamete (sperm or egg cell) there is no chromosome pairing because the cell is haploid, having undergone meiosis."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "At 2:14, how could a mutation like that occur? How is it possible that a cytosine could become a guanosine or a row of thymines get deleted? Is it just random, or does something affect them that make that happen?",
        "A": "Mutations can occur in a large variety of manners. Often they are the result of mistakes during DNA replication or chemical damage to the DNA."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "At 14:00, how did Sal get 2^23 combinations?",
        "A": "There are 23 sets of chromosomes and for each set of chromosomes there are 2 possibilities. Thus 2^23 combinations."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "at 8:50, it said \"Sal said hair color but meant hair type\".\nwhy can it not be hair color?",
        "A": "I think he just said the wrong thing, because he continued to talk about hair type afterwords, but which example Sal used doesn t necessarily matter, because all traits, including hair color and type, are inheireted by either the father or the mother."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "Sal was drawing the nucleolus's ( 7:30 ) and stuff so which gender has XX & XY.... sorry i get so confused with the two...??",
        "A": "Hello Layomi, Organisms with an XX genotype are female, due to the absence of a Y chromosome, while organisms with an XY genotype are male, due to the presence of a Y chromosome Remember, anything with a Y chromosome is considered a male. If not, it is considered a female. I hope this answers your question."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "I noticed a correction as 9:32 in the video. The correction says 'Sal says \"gender\" but means \"sex\"'\nWhat is the difference between gender and sex?",
        "A": "In this context, sex refers to the physical traits about being male or female or otherwise -- such things as genetic distinctions, differences in reproductive organs, and sexual dimorphism (in species where that occurs). Gender refers to the psychological, social, and/or culture aspects about sexual identity and roles."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "he lost me at 13:00 , what does he mean 22 to the 23rd, why do u multiply by 22? Dont you just add the 23 male and female chromosomes and you have 46... how many chromosomes does the human body have, 23 or 46.",
        "A": "23 pairs, which is 46. He didn t mean multiply by 22!"
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "At 1:03 Sal says \"these guys\" what exactly are \"these guys\"?",
        "A": "By these guys he means the yellowish species of circle people , which were more fit for reproduction for whatever reason. It can be applied to any species really, it s just in this case he was referring to the circle people ."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "at 13:11 a random (mom or dad) selection of 1 only of each gene. What causes them to be random and not mostly one or the other?",
        "A": "Because it is related to probability (possibility). If you want to understand watch lesson Bonnet square fun I think it will help you."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "At 6:40 and 7:18 Sal said males gave \"Y's\" and females gave \"X's\". Isn't it the other way around?",
        "A": "No; Sal is correct. Females do not have Y chromosomes."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "At 11:25, it is a homologous pair of chromosomes.\nIs every chromosome pair homologous??",
        "A": "In a diploid organism yes."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "In the DNA video, Sal says that alleles are modifications to a gene, or changes to the DNA, but at 9:50, he says they are the versions of a gene?",
        "A": "Alleles are versions of a gene that are slightly different so they are both correct."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "In Sal's \"Introduction to Evolution and Natural Selection\" Video (10:00), he provided an example of natural selection by describing how a variant among flu viruses might become dominant if we use antibiotics irresponsibly. However, most viruses are asexual; so, they can't reproduce sexually. If that's the case, how do these variants come into existence among the population of viruses?",
        "A": "Along with the genetic reshuffling that occurs in sexually reproducing organisms, mutations are also a huge drive for variation and all living things have DNA that naturally mutates over generations. Viruses spread quickly and have much shorter life cycles than animals, so, as a result, genetic mutations among viruses are quick to arise."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "At 11:00 Sal writes germs and then sperm and ova under it. Was he saying that they are germs? Or simply putting the germs and sex cells together to separate them from the somatic cells, as an example?",
        "A": "Sperm and ova are considered germ line cells or simply germ cells. This simply means that they are the cells that will pass on genetic information to the next generation. This differentiates them from somatic cells which are not directly involved in the transmission of genetic information."
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "at 6:56 he draws the chromosomes separate from each other .. so when are the chromosomes X-shaped? Are there 23 chromosome pairs in each of our cell",
        "A": "Chromosomes:A threadlike structure of nucleic acids and protein found in the nucleus of most living cells, carrying genetic information in the form of genes Chromatid:Each of the two threadlike strands into which a chromosome divides longitudinally during cell division. Each contains a double helix of DNA .....:)"
    },
    {
        "video_name": "DuArVnT1i-E",
        "Q": "at 16:50, we talk about combination of male and female DNA which brings more variation. So, is it possible in hermaphrodites?",
        "A": "it works with sexual reproduction and diploid set of chromosomes, when sexual spores are made by meiosis."
    },
    {
        "video_name": "H7nrVDV8ahc",
        "Q": "At 11:28, I don't understand how Sal interpreted the answer as, 1.2 * 10^-4. Why isn't is 1.2 * e^-4?",
        "A": "You misunderstood a common notation: 1.2e-4 means 1.2 x 10\u00e2\u0081\u00bb\u00e2\u0081\u00b4. You will sometimes (not commonly) see this written as 1.2EE-4 To indicate something to the power of e, you would write 1.2exp(-4) which does mean 1.2 * e\u00e2\u0081\u00bb\u00e2\u0081\u00b4"
    },
    {
        "video_name": "H7nrVDV8ahc",
        "Q": "At 11:28, I don't understand how Sal interpreted the answer as, 1.2 * 10^-4. Why isn't is 1.2 * e^-4?",
        "A": "I think the reason for your confusion is that calculators often write 1.2*10^-4 as 1.2e-4 This is just the way that many calculators represent scientific notation the e is not the base of natural logs.....one should read the e on the calculator screen as ....times 10 to the power of...."
    },
    {
        "video_name": "tB4lr5Qh3DI",
        "Q": "I just wonder, is it important that you convert from atm to pascal what calculating Q? it would give a different value for Q, as 1 atm = 101 325 Pa, right? At 3:00, the units we get for Q are [1/atm^2). I have seen people use different values for R, for example: R = 0.08206 L atm / (K mol). When do you convert your units, and when do you use different values for R?",
        "A": "in general it doesn t matter in what units you calculate anything as long as everything is in the same units. in your example you can calculate with everything in atm or everything in Pa"
    },
    {
        "video_name": "ArmlWtDnuys",
        "Q": "At 1:08 he talks about how there is a net of 2 ATP. I know he mentioned this in the last video but I am still confused. What exactly does net 2 ATP mean? How does using 2 ATP that produces 4 ATP end up as net of 2 ATP?",
        "A": "You used 2 ATP, so you can say you have negative 2 ATP. You make 4 ATP, and you can call that positive 4 ATP. -2 plus +4 = +2, or your net 2 ATP."
    },
    {
        "video_name": "ArmlWtDnuys",
        "Q": "Where does the phosphate group come from at 9:13 ?",
        "A": "The inorganic Pi is free floating in the body, not from an ATP, which is why energy is not required in this reaction."
    },
    {
        "video_name": "ArmlWtDnuys",
        "Q": "At around 10:20, how did Sal put a hydrogen atom off the negative oxygen bonded to the 1-carbon? In the Glycolysis page, the image doesn't show this happening. Where did this hydrogen come from anyways?",
        "A": "Throughout the whole process there are H and O atoms being gained and lost. H and O atoms are in excess because glycolysis is happening in aqueous solution (like most metabolic processes) so there is plenty of H and O around at any one point to get involved in the reaction. Technically they should all be depicted for completion s sake, but in much the same way as chemists will sometimes omit the Hydrogens in fischer projections, they are omitted here to make it easier to follow."
    },
    {
        "video_name": "ArmlWtDnuys",
        "Q": "Dehydrogenase enzyme removes hydrogen from a compound, but then how is NAD+ being reduced to NADH when dehydrogenase is used (at 9:01)?",
        "A": "The enzyme dehydrogenase removes 2 Hydrogen atoms from the substrate. Then it delivers 2 electrons and on proton(ie hydrogen ion) to the NAD+, thus reducing it to NADH. The remaining H ion is released into the medium."
    },
    {
        "video_name": "ArmlWtDnuys",
        "Q": "At 1:10, what are NAD & NADH+ exactly ? What do they do ?",
        "A": "NAD is a coenzyme which acts as an electron carrier. It can easily shuttle between its oxidised state (NAD+) and reduced state (NADH)."
    },
    {
        "video_name": "ArmlWtDnuys",
        "Q": "in the reaction sal showed us for glycolysis at 0:02 s it right to show 2 ATP as only a product if it is also used as a reactant?",
        "A": "2 ATP is the net product. Technically, 2 ATP goes into the cycle and 4 ATP is the product."
    },
    {
        "video_name": "cstic6WHr2E",
        "Q": "So @ 1:36 Hank mentioned you could put a sea sponge in a blender and it will reform is there anything simpler that the same rule would apply to?",
        "A": "Sea sponges are basically the simplest multicellular organism. Anything with more structure would have a much harder time totally reforming; it seems unlikely that more complex beings would be able to accomplish this feat. On the simpler side of things, well, you can t really get any simpler than the sea sponge and still be a multicellular organism."
    },
    {
        "video_name": "cstic6WHr2E",
        "Q": "At 6:19, Hank mentions tardigrades. Are they deuterostomes?",
        "A": "yes tardigrades are deuterostomes if your not sure search it up"
    },
    {
        "video_name": "cstic6WHr2E",
        "Q": "at 1:37 why doesn't the actual cells get chopped up by the blender? Wouldnt that kill the actual cell and not let it reform?",
        "A": "The cells are too small to be cut up by any normal blender. They are between 1 and 100 micrometers. That is equivalent to .0001 meters"
    },
    {
        "video_name": "vSsK7Rfa3yA",
        "Q": "At 0:24, isn't Mr. Khan looking for \"sheave\"?",
        "A": "yes. It is not a commonly used word though, so I do not blame him for not using it."
    },
    {
        "video_name": "vSsK7Rfa3yA",
        "Q": "WHY at 5:35 the weight is getting lifted by 1 ft instead of 2ft.....",
        "A": "Because the rope is being shortened on either sides of the pulley."
    },
    {
        "video_name": "vSsK7Rfa3yA",
        "Q": "@ about 10:15\nDoesn't he mean 5 Joules not 5 Newtons?",
        "A": "No he doesn t. Force is measured in Newtons."
    },
    {
        "video_name": "vSsK7Rfa3yA",
        "Q": "At 8:13, how did you know to label the opposite side D/2 and the hypotenuse, D? Is that a trigonometry rule? What if the angle was 60 degrees? How would the lengths change?",
        "A": "yes it is a trigonometry rule IF you find sine of 30* (* means degree) , it ll be 1/2 Therefore the opposite side would be d/2 If the angle was 60* , sine of 60* would be _/3 /2 (root 3 by 2) Then opp. would be _/3 /2 x (d)"
    },
    {
        "video_name": "vSsK7Rfa3yA",
        "Q": "4:37\nwouldn't this length of rope be getting 2 feet longer? Or am I misunderstanding something?",
        "A": "The rope has a fixed length. If you actually measured the rope, it wouldn t be getting shorter or longer, only that specific portion would."
    },
    {
        "video_name": "vSsK7Rfa3yA",
        "Q": "What is the name of that disc thing mentioned at 0:25?",
        "A": "A wheel and axle."
    },
    {
        "video_name": "vSsK7Rfa3yA",
        "Q": "@ roughly 5:30\nWouldn't the rope on the far left go down?",
        "A": "No it wouldn t since it is attached to the ceiling."
    },
    {
        "video_name": "vSsK7Rfa3yA",
        "Q": "Sal remarks at around 9:19 that somethings potential energy equals its mass times height. Shouldn't gravity be included? It seems like if you are lifting something, you would have to apply enough force to counteract gravity and its mass, over some distance.",
        "A": "In the video, Sal uses Weight times Height. Notice he calls it a 10 Newton box , so he is talking about its weight and not its mass."
    },
    {
        "video_name": "vSsK7Rfa3yA",
        "Q": "Sal said At 4:23 every point on the rope's going to move 2 meters to the right, So what makes the point where the wheight attached moves 1 meter ?",
        "A": "I don t think he explained that very well. The attachment of the rope to the ceiling is not going to move at all, although the pulley will move."
    },
    {
        "video_name": "Vq7EOmvU1eQ",
        "Q": "At 1:00 He says chrome wont rust. But my bike is made of chrome and it have very much rust?",
        "A": "or a verry thin layer of chrome plating that is alowing oxygen to get the the base metal (Iron) and form rust"
    },
    {
        "video_name": "Vq7EOmvU1eQ",
        "Q": "At 10:44, it is mentioned that the fan pulls air through past the heater. Wouldn't the fan be pushing air past the heater, to pick up heat on its way out of the funnel?",
        "A": "Hello John, Correct the fan is pushing air past the heater. With this configuration the fan is in a cool air stream. Regards, APD"
    },
    {
        "video_name": "J7-GewgqWUQ",
        "Q": "At 10:46, it is mentioned that substances with a pH > 7 are sometimes referred to as \"alkaline\". Is this where the name of the alkaline metals on the periodic table come from? Are those elements themselves of a lower acidity than others, or is it the compounds made of them? Or is the reverse true, the word from bases derived from the existing name for that group on the table?",
        "A": "The alkali and alkaline metals generally form compounds with the hydroxide ion (OH-), and many of these compounds dissociate completely in solution (into the respective metal and hydroxide). Increasing the concentration of OH- (through this dissociation) also raises the pH, and I m fairly certain that is where the name alkaline comes from."
    },
    {
        "video_name": "J7-GewgqWUQ",
        "Q": "So, the concentration of [H30+] in pure water is 1x10^-2M = 1x10^-7 Moles/Liter\nSo, the ratio of [H30+] to H20 is 1:590000000?\n\nI don't understand why he even went into concentration... what is concentration?",
        "A": "[H\u00e2\u0081\u00ba] = 1 \u00c3\u0097 10\u00e2\u0081\u00bb\u00e2\u0081\u00b7 mol/L. [H\u00e2\u0082\u0082O] = 55.5 mol/L. [H\u00e2\u0081\u00ba]:[H\u00e2\u0082\u0082O] = 1 \u00c3\u0097 10\u00e2\u0081\u00bb\u00e2\u0081\u00b7:55.5 = 1:555 000 000 Concentration is the amount of a constituent divided by the total volume of a mixture, It tells you how acidic the solution is."
    },
    {
        "video_name": "J7-GewgqWUQ",
        "Q": "At 3:48, what is actually the chemical compound of bleach?",
        "A": "Sodium hypochlorite (NaClO) is probably the most common household bleach. It isn t the only bleach though, hydrogen peroxide is another."
    },
    {
        "video_name": "J7-GewgqWUQ",
        "Q": "At 3:28 where Sal talks about orange juice he said that the H+ concentration is 1x10^-4 but I thought acids (as we know orange juice is acidic) where donating protons, so wouldn't it have to be the concentration of Hydronium?",
        "A": "H+, protons, and hydronium ions are frequently interchangeable, actually. The main difference is that protons are spoken of when the Br\u00c3\u00b8nsted-Lowry definition of an acid/base is being used. Hydronium is what you get if you put H+ ions in water, so assuming your solution is aqueous, they are interchangeable. In summary, each term has a correct use which isn t quite the same as the uses of the others, but you use them in problems the same way."
    },
    {
        "video_name": "BM-My1AheLw",
        "Q": "4:41, does lone pair also make bonds sometimes?",
        "A": "Lone pairs can be used to form bonds, for example ammonia (NH3) can form a bond to H+ using the lone pair on the nitrogen, which forms ammonium (NH4+)"
    },
    {
        "video_name": "BM-My1AheLw",
        "Q": "At 5:14, why is the repulsion of the lone pair of electrons stronger than the repulsion from the sigma bond?",
        "A": "The orbitals are what we draw to show the probability of where electrons are. Electrons are always moving (except at absolute zero) and the lone pair are then in a wider orbital. The electrons that are being shared tend to be drawn toward the sharing atoms and thus the bonding orbitals are more oblong."
    },
    {
        "video_name": "BM-My1AheLw",
        "Q": "Starting at 3:25, why is ammonium sp3 hybridization not sp2? I think sp2 should also work",
        "A": "Hybridisation is a way for us to justify the bonding angles and molecular geometry we observe. Ammonia looks like a pyramid, we call it trigonal pyramidal. If it were sp2 it would be flat (trigonal planar) like a BH3 molecule is."
    },
    {
        "video_name": "BM-My1AheLw",
        "Q": "at about 8:34 u say the lone pairs of electrons will repel the bonded electrons and ths decreasing the bond angle. but wont the two bonds having electrons also repel each other",
        "A": "In VSEPR theory (check out Jay s series of videos on this theory), the repulsive effect of lone pair electrons is greater than that of the electrons in sigma bonds. This is because in a sigma bond the electrons are pulled by both nuclei involved in the bond, whereas lone pair electrons are only attracted to one nucleus. The net result is that lone pair electrons are closer to the nucleus of the central atom than those in a sigma bond - being closer they exert a greater repulsive effect."
    },
    {
        "video_name": "BM-My1AheLw",
        "Q": "At 5:30 ish, he's talking about bond angles, but doesn't explain where he gets those exact numbers from. I understand that the lone pair repels more strongly, but where is he getting the exact number from, and how do we know which angle to use?",
        "A": "You can calculate the theoretical bond angles using geometry. The actual bond angles have been measured experimentally by some other researcher. You don\u00e2\u0080\u0099t have to memorize the actual bond angles. If someone asks you for the bond angles in ammonia, you say either 109.5\u00c2\u00b0 or slightly less than 109.5\u00c2\u00b0 . Check what your instructor wants."
    },
    {
        "video_name": "BM-My1AheLw",
        "Q": "At 3:38 , suppose we had only 1 non bonding electron. How do we determine the steric number then? Should I put lone pairs=0 or 1/2 or something else?",
        "A": "An orbital with one nonbonding electron would count as a lone pair."
    },
    {
        "video_name": "BM-My1AheLw",
        "Q": "At 3:37, can someone tell me what actually are lone pairs of electrons?",
        "A": "You know how covalent bonds are a pair of electrons shared between 2 atoms? Lone pairs are a pair of electrons that aren t shared between 2 atoms."
    },
    {
        "video_name": "BM-My1AheLw",
        "Q": "At 5:18: Why does a hybrid orbital containing a lone pair of electrons repel the other hybrid orbitals more than an orbital that is bonded to another atom?",
        "A": "The electrons in a lone pair are held closer to the nucleus than those in a sigma bond between two atoms. The electrons being closer to the nucleus leads to lone pairs having great repulsive effects than bonded electrons."
    },
    {
        "video_name": "BM-My1AheLw",
        "Q": "For the ammonia (NH3) at 4:10, are the lone pair electrons also using an sp3 hybrid orbital even though they are not bonded to another atom?",
        "A": "yes. hybridization lowers the overall energy of the molecule so orbitals will hybridize whenever they have a chance."
    },
    {
        "video_name": "06z7Q8TWPyU",
        "Q": "Does that mean that the space may be expanding faster than the speed of light? (10:13)",
        "A": "Space is expanding at about 70 km/s for each megaparsec of space between two points. So, yes, very distant points are receding from each other at speeds greater than the speed of light. This does not conflict with the theory of special relativity, which simply says that nothing with mass can move through space faster than light."
    },
    {
        "video_name": "06z7Q8TWPyU",
        "Q": "At around 7:41\nhe said that the distance would be 46 billion light years and then another 46 billion light years isn't the universe only 13.7 billion years old soo how could people observe the universe that far already. is this some sort of mistake?",
        "A": "The universe has been expanding ever since the big bang. A photon of light that is just reaching us now, after having traveled, say, for 13 billion years, originated from a point that is now 46 billion light years away."
    },
    {
        "video_name": "06z7Q8TWPyU",
        "Q": "At around 9:23, where Sal is talking about the universe expanding faster than the light can travel, does he mean that the universe is expanding faster than the speed of light? I thought that couldn't happen... Please help...",
        "A": "The laws of physics say that anything with mass can t move faster than light, space has no mass and can thus do whatever it wants. Also, space itself is not moving, at any one point in the universe, the space is standing completely still, nothing is moving in this scenario. Space is expanding, but it is not moving."
    },
    {
        "video_name": "06z7Q8TWPyU",
        "Q": "At 1:28 Sal says that stars have started to form. Does this mean that more mind blowing things are yet to be created?",
        "A": "yes"
    },
    {
        "video_name": "5EWjlpc0S00",
        "Q": "At 1:05, Sal says that the cylinder is on a planet with the same mass as Earth, but doesn't gravitational effects also depend on the cylinder's distance from the center of this planet's mass? In other words, isn't the radius of this planet also taken into account when determining gravitational effects on this cylinder?",
        "A": "Yes. You are right. Gravitational force is equal to G m1.m2 / r squared. I think Sal just meant that Earth had its mass and radius but lost its atmosphere. I don t think he actually referred to another planet."
    },
    {
        "video_name": "5EWjlpc0S00",
        "Q": "9:19 so a Pascal (N/m^2) is actually Kg/s^2m ?",
        "A": "Those are different ways of saying the same thing, just like 3*4 is the same as 2*6 is the same as 12."
    },
    {
        "video_name": "5EWjlpc0S00",
        "Q": "At 0:04 to 0:21, what is meant by \"external pressure\" and \"internal pressure\"? Does it mean pressure of atmosphere and pressure of liquid?",
        "A": "What he meant by external pressure was a pressure exerted by something that s not the liquid itself. So an outside force acting on the liquid. Internal pressure he meant the pressure experienced by the liquid due to its own weight pushing down itself. And yes, the atmospheric pressure would be an external pressure."
    },
    {
        "video_name": "5EWjlpc0S00",
        "Q": "At 9:30, Sal didn't have enough time to expand on the units:\n\nWould the units for pressure be: kg \u00ef\u0082\u009f\u00ef\u0082\u009f\u00ef\u0082\u009f\u00ef\u0082\u009f\u00ef\u0082\u009fx m / s^2 x m^2\n\nSorry, I don't know how insert the multiplication dot or the superscript.",
        "A": "Pressure is force / area. Force could be in newtons. A newton is a kg*m/s^2, so that could get you to the units you asked about."
    },
    {
        "video_name": "5EWjlpc0S00",
        "Q": "I don't really get 2:54 to 4:57 anyone have an answer for that ?",
        "A": "Basically what he is doing is to mathematically express how the thin foil is static. In order for the thin foil to be static as he said, the force exerted by the liquid above = reacting force exerted by the liquid from below. Thus F above = F below, which can be rewritten as mass * g. But I do not know why he wrote Force down = mg, twice. Hope this helped"
    },
    {
        "video_name": "5EWjlpc0S00",
        "Q": "At 7:23, the formula Sal wrote was P=F/A= p*h*A*g/A but why didn't Sal include the A(area) in the formula for pressure at 7:59? Since, at 7:23, both the A's cancel out on the right hand side but there would be still an A on the left side ( P=F/\"A\" ).",
        "A": "I think he is writing P = F/A F = p * V * g, where V = h*A So substituting that in, you get: P = F/A P = pVg/A P = phAg/A P = phg"
    },
    {
        "video_name": "5EWjlpc0S00",
        "Q": "At 2:58, when writing force down why didn't we consider mass of that thin foil...?",
        "A": "For a thought experiment you may just make the thin foil as thin as needed to become negligible. In a real experiment you WOULD need to consider the foil mg term."
    },
    {
        "video_name": "5EWjlpc0S00",
        "Q": "At 2:00, how can a fluid be static? Aren't the molecules always moving around each other?",
        "A": "ok, so even though the molecules are moving, it s static unless it s flowing like in a river?"
    },
    {
        "video_name": "5EWjlpc0S00",
        "Q": "At 3:21, how does Sal get the formula F = M.G??\nPlease answer!\nThanks!",
        "A": "You get F = M * g when you take the law F = M * a and substitute the acceleration due to gravity g for the acceleration."
    },
    {
        "video_name": "5EWjlpc0S00",
        "Q": "at 0:18 Sal mentioned Pascal's principle of pressure. Could someone please tell me what the principle is? I am very curious I also want to know how it affects pressure. thnx.",
        "A": "Pascal s law or the principle of transmission of fluid-pressure states that pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions throughout the fluid such that the pressure ratio remains the same. Hope this helps :)"
    },
    {
        "video_name": "Tmt4zrDK3dA",
        "Q": "Somewhere around 3:40, Sal starts talking about hybrids. What would you get if you married a saint bernard and a german shephard?",
        "A": "A saint Shepard or german Bernard"
    },
    {
        "video_name": "Tmt4zrDK3dA",
        "Q": "2:33 so a lion can breed with a cat",
        "A": "Not with a house Cat no. But a Lion can be forced to breed with a Tiger in captivity since the two species are closely related. This is a cruel thing to do though."
    },
    {
        "video_name": "Tmt4zrDK3dA",
        "Q": "@ 0:22 What is \"Taxonomy\" again?",
        "A": "Taxonomy is a form of classifying species from most specific to most general, or vice-versa. You can think of taxonomy as a mailing address. The order is species, genus, family, order, class, phylum, kingdom, and domain."
    },
    {
        "video_name": "Tmt4zrDK3dA",
        "Q": "At 0:47, Sal talks about grouping things together with other things like it. If you don't say what you're grouping with much detail couldn't you just say all the animals with four legs go in this group or all flower with 6 petals in an other?",
        "A": "Correct. That is why you have to have multiple common features rather than just a randomly selected feature. In fact, some of the original taxonomic classifications from the 18th century have had to be redone for this very reason: animals that were not closely related at all were lumped together based on superficial similarities."
    },
    {
        "video_name": "Tmt4zrDK3dA",
        "Q": "At 2:43 why does sal mean organisms instead of animals?",
        "A": "Because species can refer to other living things like plants as well."
    },
    {
        "video_name": "Tmt4zrDK3dA",
        "Q": "So at 02:41, you have to have the same species, not family to reproduce \"reproducables\" - yes? So the tiger and lion are of the same family. What about the same genus? Where does the line get drawn: genus or species? What is the genus?",
        "A": "A group of closely related species is a Genus and this is just FYI - 1.GENUS 2.KINGDOM 3.SPECIES 4.PHYLUM 5.FAMILY 6.CLASS So, animals need to be in the same class to reproduce, but, same family is also possible, like Sal had mentioned. Then of course you have hybrids...but never mind them...."
    },
    {
        "video_name": "Tmt4zrDK3dA",
        "Q": "So at 4:30 he says that Ligers aren't fertile. So they can't have offspring that are ligers??",
        "A": "If he says that Ligers can t have offspring then they can t have offspring"
    },
    {
        "video_name": "TxwE4_dXo8s",
        "Q": "At 4:30, how did sigma come out of the integration sign? sigma = Q/A, and area of the ring has 'radius' involved, so basically sigma contains a variable with respect to which integration is being done. Please explain it to me. I'll be very grateful :^)",
        "A": "Sigma is defined as the charge density of the infinite plate. Sigma = charge per unit area. So sigma for a particular example might be 1 coulomb per 1 square meter of area. The area variable, A, is a generic area variable, not the area of a particular ring. That means sigma is a constant property of the whole plane, which is why it can be pulled out of the integral."
    },
    {
        "video_name": "TxwE4_dXo8s",
        "Q": "at 10:00 he says the electric field due to the plate is same evry height...\nhow can it be ? in previous videos... he says... the electric field goes on decreasing as we go away from the charge..\nbut then how can be the field be same at at evry height?",
        "A": "I haven t watched the videos in question, but I m fairly certain in previous videos he was not discussing a plate/plane, which if infinitely large produces a constant E-field perpendicular to its surface."
    },
    {
        "video_name": "TxwE4_dXo8s",
        "Q": "At 7:50 ,what is cigma? Will someone clarify",
        "A": "(sigma) \u00cf\u0083 = surface charge density"
    },
    {
        "video_name": "TxwE4_dXo8s",
        "Q": "At around 4:46, Sal takes many constants out of the integral, but leaves the 2r dr in the numerator. The 2 is also constant, and could also be taken out as well, right? Later, around 5:25, Sal makes du=2r dr, but that 2 comes from the u=(h^2 + r^2), as in du/dr = 2r. So, it would seem that the 2 was left in order to substitute out later, right?",
        "A": "This is just one of those things that could take hours to realize if you were trying to integrate this from scratch for the first time."
    },
    {
        "video_name": "TxwE4_dXo8s",
        "Q": "Why h^2 in 5:22 is a constant?",
        "A": "The height is not changing, so it is constant"
    },
    {
        "video_name": "TxwE4_dXo8s",
        "Q": "Theres a difference between the exponent rule and power rule! @6:20",
        "A": "Yaaaay! wonderful question."
    },
    {
        "video_name": "TxwE4_dXo8s",
        "Q": "At 10:54, is he saying he can put a positive test charge at different heights h1, h2, h3, etc where h1 != h2 != h3, and that the net force on that charge due to the electric field of the plate will be the same in all cases? My understanding was that the further the positive test charge is from the plate, the lower the magnitude of the force on that charge due to the electric field will be; but I am not hearing him say that. I'm confused. Will someone clarify?",
        "A": "It s because the plate is infinite. The forces applied to the q charge, as a result from horizontally distant charges on the plate, are close to horizontal, so their horizontal components are big(but cancel out) and their vertical components are insignificant. The higher q is, the smaller the horizontal component gets as the vertical grows(the angle \u00ce\u00b8 grows), the more impact distant charges have."
    },
    {
        "video_name": "TxwE4_dXo8s",
        "Q": "At 3:02, Sal used the intergral from 0 to infinity of the formula. Was it because the plate expand infintely in this case?",
        "A": "Yes, that is because the plate is expanded infinitely....i think..... hehe sorry XD"
    },
    {
        "video_name": "gjKmQ501sAg",
        "Q": "At 1:32 ,it is said that CO2 is put in the blood which is primarily water. Does'nt CO2 react with haemoglobin present in blood cells ?",
        "A": "I know for a fact the carbon monoxide does. Hemoglobin is 200 times more attracted to carbon monoxide than oxygen. This is why it is so harmful to your body when you breathe in the carbon monoxide emitted from exhaust pipes on engines."
    },
    {
        "video_name": "gjKmQ501sAg",
        "Q": "At 5:31, Sal says if a base is added, [H+] will decrease. Why? Because the base reacts with H+?",
        "A": "Exactly! The Base will react with H+. Because the base has a net negative charge (like OH-), it wants to give some of its electrons away to become neutral. The + on H+ tells us that H+ is lacking electrons, so the negative base will really want to donate its electrons to an element that doesn t have enough of them."
    },
    {
        "video_name": "gjKmQ501sAg",
        "Q": "At 1:48 Sal said that carbonic acid is a weak acid, so shouldn't that make the ph value to come down?",
        "A": "If ph is down this would mean it was a strong acid. 7 is neutral. Anything that is near 7 is considered. As you go away from 7 you get stronger acid/bases."
    },
    {
        "video_name": "WThFuJ6wFcM",
        "Q": "At 9:38 what is DDT?",
        "A": "DDT stands for dichlorodiphenyltrichloroethane, or C14H9Cl5. It is an organic compound containing chlorine that was used to kill mosquitos. Unfortunately it is not too good for animals or people either, and can remain in soil for up to 30 years after it is sprayed."
    },
    {
        "video_name": "WThFuJ6wFcM",
        "Q": "I thought that humans have about 50% of deoxyribonucleic acid as a banana, but is it also true with fruit flies, as mentioned at 6:40?",
        "A": "Yes, bananas, apes, flies... they all have similar (to an extent) DNA!"
    },
    {
        "video_name": "WThFuJ6wFcM",
        "Q": "At 2:16 he said that some ideas of the theory of evolution came from geophysics. What are those ideas? I can't see the connection between the two.",
        "A": "The idea is that because of earth moving its crust plates(GEOPHYSICS), beings were able to cross, but got separated when the continents moved away, they had to evolve to survive"
    },
    {
        "video_name": "GtoamALPOP0",
        "Q": "in earlier videos i was told displacement is denoted with a \"S\" so why is this guy saying average speed is \"S\" did i miss something? 6:55",
        "A": "The letters don t matter. Focus on the concepts."
    },
    {
        "video_name": "GtoamALPOP0",
        "Q": "Hey i actually have 3 questions to ask ...\n\u00e2\u0096\u00baQ1 At 32:23 he said the turtle is 3m , but 3m from where ?\n\u00e2\u0096\u00baQ2 When he says that the turtle travels 3m , from where should the distance travelled be counted from, the head or the end..??\n\u00e2\u0096\u00baQ3 When he wrote down the distance traveled by the turtle in 2 seconds i.e 8 m ,isnt it wrong because when we use the distance formula between any two we points the answer as =\u00e2\u0088\u009a68..?? and the points are (2,3) and (4,-5)..??\npls help im confused..",
        "A": "1: 3m from the origin, wherever that is 2: from the origin 3: distance takes account of the path the turtle followed. What you are calculating is what we call the magnitude of the displacement."
    },
    {
        "video_name": "GtoamALPOP0",
        "Q": "Hi, at 12:55, he said to take the absolute value of the instantaneous speed. Why is this?",
        "A": "So at 12:55, he actually said to take the absolute value of the instantaneous velocity in order to find the instantaneous speed. This is because velocity is a vector quantity that is either the positive or negative speed depending on the direction that you are going in. So in order to find the instantaneous speed you take the absolute value of the velocity to make sure the result is positive, because speed is not a vector quantity."
    },
    {
        "video_name": "GtoamALPOP0",
        "Q": "It was said that the value of the graph is the horizontal position at 2:17. Is this a technical term (the word \"value\"), or was that just used to explain?",
        "A": "The Value represents the position of the turtle at a certain time. Different values represent different horizontal positions."
    },
    {
        "video_name": "GtoamALPOP0",
        "Q": "0:50\n\nHow dare he objectify that turtle.\n\nFor a serious question, would I be mistaken in assuming that the position on the graph refers to the speed of the turtle and not the position? Is that for this graph only? Can there be other graphs that refer to speed?",
        "A": "If you look at the vertical axis, it is for position (in metres) The velocity is given by the slope (or steepness ) of the graph ok??"
    },
    {
        "video_name": "GtoamALPOP0",
        "Q": "At 13:10, they said that the turtle could have been moving up and down. Is it possible to indicate it in the graph ? If not, why ?",
        "A": "Because, the graph only shows the dist. Travelled and the time taken, not the direction."
    },
    {
        "video_name": "GtoamALPOP0",
        "Q": "10:55 Does this actually matter? I can't think of a case where you wouldn't get the same answer however you do it. For example:\n\n-5m-3m/4s-2s= -8/2 = -4\nOr\n3m-(-5m)/ 2s-4s= 8/-2= -4\n\nYou end up with the same answer so does it matter which order you use?",
        "A": "that s just basic algebra"
    },
    {
        "video_name": "GtoamALPOP0",
        "Q": "at 5:21 he says distance is always positive and calls the value as +8 and not -8 .....the turtle is moving down right?\ncan someone answer",
        "A": "Distance is always positive, but displacement isn t. Displacement is positive or negative based on which direction the turtle is moving, because it s a vector. The turtle is moving down so the displacement is negative."
    },
    {
        "video_name": "GtoamALPOP0",
        "Q": "At 0:20, in the graph, if I want to find the instantaneous velocity at the t[s] = 4, x[m] = -5.... would I use the displacement taken place from 2 sec to 4 sec or from 4 sec to 10 sec?",
        "A": "In this instance you get to be a bit of a visual cheat. Because the turtle is changing direction, making acceleration 0, the instantaneous velocity is also 0. I answer this question in a bit more detail above. To answer your full question, you wouldn t use either. The slope at that point is 0."
    },
    {
        "video_name": "GtoamALPOP0",
        "Q": "at 1:56 why do they call the y axis x and the x axis t?",
        "A": "this is exactly what at first got me scratching the top of my head..:):)It seems it is the way physicists found to make it complicated!!:):) We ll found out later i guess,but it is a good practice to learn how to read different graphs,with not the standard notation of mathematics..(all this is just me guessing ,take it with a lot of salt5.):):)"
    },
    {
        "video_name": "GtoamALPOP0",
        "Q": "At 4:41, David said the turtle rocketed back to five, but he actually rocketed back to -5.",
        "A": "That s true, that is why the little sign popped up."
    },
    {
        "video_name": "4lbVBMLZ1jQ",
        "Q": "@6:00 why is does propanal have a lower b.p. than acetone? You would think the reverse to be true because its slightly more positive alpha-c would form stronger dipole-dipole interaction than the ketone alpha-c would....also there would be less steric hindrance with the aldehyde to form these interactions. Shouldn't a stronger dipole interaction mean a higher boiling point?",
        "A": "Alcohols all have higher B.P. than the corresponding ketones. As for Propanal vs Acetone the dipole moment of propanal is 2.52 whereas for acetone it is 2.91 so ketones have higher dipole moments than aldehydes because they have the carbonyl more towards the center and thus ketones have higher boiling points than aldehydes that have the same number of carbons."
    },
    {
        "video_name": "4lbVBMLZ1jQ",
        "Q": "At 3:45 Isn't sp2 hybridized carbon (carbon with double bond) more electro negative than oxygen??",
        "A": "It s true that a sp2 hybridized carbon is more electronegative than a sp3 hybridized carbon, but that does not mean that it is more electronegative than oxygen."
    },
    {
        "video_name": "4lbVBMLZ1jQ",
        "Q": "why at 8:53 there is h-bond? Isn't the O of the ketone attached to C? It should not be that readily soluble. Please help me !",
        "A": "Acetone is soluable in water because of the dipole charge that is very similar to how water behaves. Since it is like water the O in ketones typically form H-bonding in water. Also the tip of like dissolves like is very important because polar does not disolve non-polar molecules (ie. oil and water)."
    },
    {
        "video_name": "4lbVBMLZ1jQ",
        "Q": "At 7:01, propan-1-ol is stated to have a boiling point higher than propan-2-ol. Why is this? Don't both have hydrogen bonding?",
        "A": "Both have hydrogen bonding, but propan-1-ol has higher London dispersion forces. Its molecules are linear, so they can approach each other more closely than molecules of propan-2-ol can."
    },
    {
        "video_name": "WbDL7xN-Pn0",
        "Q": "At 6:31, why did a tiny amount of acetate ions react back to give acetic acid, given that they are the conjugate base of a weak acid, that is, it is a strong conjugate base that I assume should react completely with water? Thanks in advance.",
        "A": "The conjugate base of a weak acid is NOT a strong base"
    },
    {
        "video_name": "zAx61CO5mDw",
        "Q": "How do you find the change in angle? the delta theta at 4:22?",
        "A": "delta theta = final theta - initial theta. In this video, initial theta is supposed to be zero. The object traveled 5 revs or 10 pi theta, so delta theta is 10 pi theta."
    },
    {
        "video_name": "zAx61CO5mDw",
        "Q": "At about 4:40,Sal talks about the derivative of an angle.What does that mean?",
        "A": "A derivative is a mathematical way of looking at change. If we have an equation representing the position of an object over time, we can take the derivative of that equation with respect to time, and we will get the velocity of that object. If we have an equation describing some angle over time, we can take the derivative of that and get the angular velocity."
    },
    {
        "video_name": "zAx61CO5mDw",
        "Q": "At 4:50, can you explain how to differentiate d theta/d t ?",
        "A": "theta is called angular distance. so to get angular velocity you differentiate. angular velocity is the change in theta by time. like how you get velocity by ds/dt"
    },
    {
        "video_name": "zAx61CO5mDw",
        "Q": "at 00:57, what are radiants",
        "A": "first of all it is radians not radiants Radians is just another unit to measure angles pi (or \u00cf\u0080) radians = 180 degrees"
    },
    {
        "video_name": "zAx61CO5mDw",
        "Q": "At 0:42, What is a revolution?",
        "A": "Once around a circle is a revolution"
    },
    {
        "video_name": "zAx61CO5mDw",
        "Q": "what is 'd' at 4:53?",
        "A": "I believe it means derivative ; at least that was what he was saying as he wrote down d . It has something to do with the change in omega (the angle) and time. You don t really need to know what d is too understand the video though, so don t feel lost. He was just writing change in omega divided by change in time in a different way using calculus."
    },
    {
        "video_name": "zAx61CO5mDw",
        "Q": "what did sal do at 6:06, when he multiplied w rad/sec * 1 rev/2 pi rad? i didn't get this part",
        "A": "2 pi is one revolution, and the best way I can think of it is that he was using a similar process as in chemistry, canceling out units to get the result. Look back around 1:30, he does it similarly"
    },
    {
        "video_name": "zAx61CO5mDw",
        "Q": "At 4:10, why do we use omega?",
        "A": "That s just the greek letter that has become standard to refer to angular velocity."
    },
    {
        "video_name": "UU366tJPovg",
        "Q": "at 4:53 \"they don't get filtered\" where do the red blood cells and other bigger things go?",
        "A": "After the kidneys remove the toxic wastes, the now pure blood geos back to the heart to be circulated through the whole body"
    },
    {
        "video_name": "1E3Z_R5AHdg",
        "Q": "What type of \"normal force\" is my chair exerting on me?? 1:42",
        "A": "Your chair is exerting a force on you that is keeping you from falling to the ground. That s the normal force. If it weren t there, you d have a force pulling you down - your weight - and no force pushing you up, so you would have unbalanced forced on you and you would accelerate downward, ie fall to the floor."
    },
    {
        "video_name": "1E3Z_R5AHdg",
        "Q": "At 2:16 Sal mentions he is not accelerating, but if we are not moving are we simply accelerating in a constant direction?",
        "A": "No, if you are not moving, you have no acceleration and a constant velocity of 0."
    },
    {
        "video_name": "1E3Z_R5AHdg",
        "Q": "Sal's answer at 5:25 should be written as either -98.1 N or 98.1 towards the Earth, right?!",
        "A": "Yes, but he didn t put that there because he was busy doing the rest of the video."
    },
    {
        "video_name": "_nzaMQKpPLM",
        "Q": "at 11:11 , both are sp2 hybridized C-H bond spectra. So what is the difference between them?",
        "A": "One s an aliphatic substance, while the other is aromatic. It makes a difference in how the bonds stretch, but as he said, that s beyond our level."
    },
    {
        "video_name": "xR4AnXDBnsw",
        "Q": "2:35 Can someone explain to me why there is a \"dx\" next to f(x), and what it does? Is it a label? A symbol? Can I get some examples and an explanation? I can't seem to ever ask the question in the way I want.",
        "A": "dx is just saying that you need to take the integral of the function. Once you do the integral itself, the dx will go away."
    },
    {
        "video_name": "xR4AnXDBnsw",
        "Q": "at 3:47 why does sal take the derivative of x^3?",
        "A": "couldn t one just take the anti derivative right away of x^2 which is very easy?"
    },
    {
        "video_name": "xR4AnXDBnsw",
        "Q": "At 4:31, why Sal din't include the constant c when he takes the integral of x^2?",
        "A": "The constant, c, at the end of the integral, is only required when taking an indefinite integral. In this case, he is taking a definite integral which doesn t require the constant."
    },
    {
        "video_name": "xR4AnXDBnsw",
        "Q": "where does the x^3 at 3:40 come from?",
        "A": "He s working on the antiderivative of x^2. 1/3( x^3) is the antiderivative, since you know integral of x^n = (1/n+1) (x^(n+1)) + C It d be clearer if he done it using the formula."
    },
    {
        "video_name": "xR4AnXDBnsw",
        "Q": "I know another the equation for finding a derivative around the 4:25-4:29 mark. I found it easier and was able to work it out in my head. Depends on how your mind works, anyway. In Year 11 we learn that F(x) = ax^(n+1)/(n+1) for any derivative:\n\nEG:\n\nIndefinite Integral of 3x^2 - 6x: [ax^(n+1) / (n+1)]\n\n= 3x^(2+1) / (2+1) - 6x^(1+1)/(1+1)\n= 3x^3/3 - 6x^2/2\n=x^3 - 3x^2",
        "A": "That method is known as the integral power rule. And you left off the constant: you must add + C or the answer is wrong. Unfortunately, the integral power rule does not always work, so you very often have to use more complex methods. For example, \u00e2\u0088\u00ab (x-1) /x dx = x \u00e2\u0088\u0092 ln |x| + C"
    },
    {
        "video_name": "_ElAuQyw4uA",
        "Q": "Is a photoreceptor a neuron or a nerve? I can't make out which he is saying at 0:29...",
        "A": "A photoreceptor cell is a type of neuron"
    },
    {
        "video_name": "_ElAuQyw4uA",
        "Q": "4:35. Why would cones be responsible for the change in illumination in that example instead of rods. I thought that rods were the ones responsible for detection changes in illumination (like in the dark room example)",
        "A": "Rods are much more sensitive to light, and can respond to single photons. The problem with this is that they saturate easily - if the light is any brighter than a candle, then rod photoreceptors can t recover between signals, so they can t communicate any light information."
    },
    {
        "video_name": "2PM1yc_z4Bk",
        "Q": "At around 5:01, the speaker states that having no reactants gives us a value of zero in the denominator, which makes the reaction quotient equal to infinity. Does that not violate a principle of mathematics: division by zero is undefined? Is this also how it is defined in chemistry, or was the infinity definition invented to make such a calculation possible in a chemistry context ?",
        "A": "Great point, I think I misspoke in the video. We generally assume there is always at least some tiny bit (a molecule or two, perhaps) of the reactants left. That means the denominator is never equal to zero, it just gets very very very tiny so that the quotient approaches infinity but isn t undefined. I hope that helps!"
    },
    {
        "video_name": "2PM1yc_z4Bk",
        "Q": "At 6:13 the speaker said that the shift is to the right but meant to the left, towards the reactants.",
        "A": "I think she was referring to Q being shifted to the right of K on her scale. The result of this will be to drive the reaction to the left, as you say."
    },
    {
        "video_name": "2PM1yc_z4Bk",
        "Q": "At 3:50, why did Kc and Qc get different outcomes? What was the difference between the two?",
        "A": "If Kc and Qc have different values, this means that the reaction has not yet reached equilibrium. Kc only equals Qc at equilibrium."
    },
    {
        "video_name": "MOHzzgQLkk8",
        "Q": "At 3:09, should the CH3 group not be on the bottom left, and the H on the bottom right?",
        "A": "Nope. It is the way you look down it actually is correct."
    },
    {
        "video_name": "sTp4cI9VyCU",
        "Q": "Im wondering about how to see the direction, that is, arrond 7:20, he sais that the direction is south or below horizontal. While iI understand the term below horrizontal, I dont quite get how the thing is going south instead of east.",
        "A": "In that drawing, east is horizontal, and south is down. Any direction between those two is some combination of south and east."
    },
    {
        "video_name": "sTp4cI9VyCU",
        "Q": "at 4:55 when he moved vector to the right, why did he changed its sign form - to +?",
        "A": "You are correct, but he also said that the length/magnitude was equal 29.03."
    },
    {
        "video_name": "sTp4cI9VyCU",
        "Q": "I was just wondering why would Sal caculate the angle of projectile at the 6:28 ?\nI think its the same once the projectile is falling downwards. No matter what hieght does the projectile fall from the groung level, the angle will be the same as angle of launch.",
        "A": "The angle of launch would be equal to the angle at which it strikes the ground only if it falls at the same height. This is because the vertical component of the final velocity would change if the projectile lands at a different height, thus changing the value of the angle formed."
    },
    {
        "video_name": "sTp4cI9VyCU",
        "Q": "In 3:37, Sal said the answer for 29.54-9.8*5.67 = -26.026, but how come when Sal wrote down the answer at 3:41, the answer he wrote down is -26.03? did he do something wrong?",
        "A": "He just rounded to keep up with significant figures."
    },
    {
        "video_name": "sTp4cI9VyCU",
        "Q": "@4:25 why is the velocity constant in the horizontal component ??",
        "A": "The horizontal component is always constant, because there is no acceleration in that direction, so there will be a constant speed. But the vertical component is changing because their is an acceleration due to gravity, g=9.8ms^-2. so the velocity changes."
    },
    {
        "video_name": "sTp4cI9VyCU",
        "Q": "Why does the negative disappear when he moves the 29.03 to the other side at 4:56.",
        "A": "he s looking at the velocity s magnitude, then plugging into the pyth theorem to get the magnitude of the resulting velocity."
    },
    {
        "video_name": "sTp4cI9VyCU",
        "Q": "@2:12, why does the vertical velocity change (from initial to final) but the horizontal velocity does not?",
        "A": "horizontal velocity is constant because there is no horizontal force acting on the projectile (Newton s first law). The vertical velocity changes because there is a vertical force acting on the projectile: gravity (Newton s second law)."
    },
    {
        "video_name": "sTp4cI9VyCU",
        "Q": "At 3:42, how could it be -26.03 negative? 19.74 times 5.67 would never make a negative number.",
        "A": "Watch again. 29.54 - 19.74 * 5.67 is indeed negative."
    },
    {
        "video_name": "sTp4cI9VyCU",
        "Q": "why does he drop the negative sign at 5:07?",
        "A": "In the pythagorean theorem you square the number so it does not matter what the sign is at first , when you square it it will always be positive."
    },
    {
        "video_name": "3XiHrsZNZko",
        "Q": "at 6:20, isn't the \"keto\" more capable of attacking as a nucleophile?\nI thought that the \"enolate\" form is rare anyway, so how can we trust him to do this job?",
        "A": "the enolate ion has a negative charge so it is a better nucleofile. The enol form is more rare but when we catalyze the reaction with a base there is gonne be more of it. And the tautomerisation is an equilibrium so that means that le chatelier is gonne do something, when the enol reacts."
    },
    {
        "video_name": "3XiHrsZNZko",
        "Q": "At 6:22 does the electron attack to that specific Carbon because of its positive charge? I'm referring to the positive charge due to the polarity of the carbonyl bond.",
        "A": "Correct. That\u00e2\u0080\u0099s where and why it attacks."
    },
    {
        "video_name": "3XiHrsZNZko",
        "Q": "At 6:24, it looks like the carbonyl on the enolate ion is about to make the nucleophilic attack, but it's definitely the alpha carbon that makes the bond. Why does he draw a single electron on the carbonyl making the nucleophilic attack?",
        "A": "At 6:24, he is showing that the enolate ion is really a resonance hybrid of two contributors. All the nucleophilic attacks are by electron pairs, not by single electrons."
    },
    {
        "video_name": "3XiHrsZNZko",
        "Q": "If Sal were drawing the lone pairs for this reaction mechanism, at 3:40 would there be a lone pair drawn on the negatively charged alpha carbon? Just wanted to make sure I didn't lose track of any electrons.",
        "A": "Correct, there would be a lone pair drawn on the negatively charged alpha carbon."
    },
    {
        "video_name": "76KtDSfWnkw",
        "Q": "At 9:10 what's hydrolize?",
        "A": "Hydrolyze means react with water . In the video, the cyclic ester is hydrolyzed (reacted with water) to form a cis-1,2-diol."
    },
    {
        "video_name": "76KtDSfWnkw",
        "Q": "At 1:15 what is the driving force in the step where O from OsO4 attacks the alkene??",
        "A": "That s kinetics OsO4 is not very stable so there is a chance something like this will happen."
    },
    {
        "video_name": "76KtDSfWnkw",
        "Q": "At 0:25, you mention that hydrogen peroxide and aqueous sodium bisulfite can be used instead of water and tert-butanol. I understand that H2O2 provides the two -OH groups, but what does the NaHSO3 do? Am I correct in saying that in solution, the bisulfite ion HSO3(-) dissociates to H(+) and SO3(2-), and then the two O(-) groups off of the alkane pick up these protons forming -OH?",
        "A": "The hydrogen peroxide is the real oxidizing agent. Osmium tetroxide works better, but it is very toxic and very expensive, it is used as a catalyst only, along with the hydrogen peroxide. The key intermediate is a cyclic osmate ester. The bisulfite ion reduces the osmium, decomposing the cyclic ester and forming the diol.."
    },
    {
        "video_name": "76KtDSfWnkw",
        "Q": "At 6:14 jay says that the product is non superimposable but I think when we will flip any one of them upside down they are getting superimposed\nplease explain the reason",
        "A": "At 6:14, he says we get only one product, that is, the two structures are superimposable."
    },
    {
        "video_name": "76KtDSfWnkw",
        "Q": "What is the molecular geometry around Os in the osmate ester at 2:16? I think it might be see-saw, but the way Jay drew it suggests otherwise.",
        "A": "I may be wrong, but I would argue that the Os atom is directly attached to four other atoms, so it is sp3 hybridized with a tetrahedral shape. The double bonds come from interactions with various d orbitals."
    },
    {
        "video_name": "76KtDSfWnkw",
        "Q": "At 6:55, when the OH groups are added to the same side, wouldn't the sp3 carbon-carbon bond rotate? I mean I can see how the syn addition on the ring compound is forced into that configuration, but non-ring structure of carbon with sp3 bonding would allow rotation, yes? no? am I crazy and confused?",
        "A": "Yes it would....and depending on the configuration of the alkene, you will get different answers for a syn dihydroxylation, depending on whether it is E or Z. You are not crazy or confused."
    },
    {
        "video_name": "76KtDSfWnkw",
        "Q": "1. What is a dial or diel or whatever Jay says at 2:35?\n2. I have a trouble in recognizing the number of valence electrons. I thought it was 2 since it has 2 electrons in its outermost shell. Maybe I must review general chemistry lectures again.",
        "A": "1. The word is diol , a compound that contains two OH groups. 2. You may have to do that. Only H has two valence electrons in its compounds. Most other atoms have eight valence electrons, i.e., the O atoms in OsO4. Osmium can expand its octet. In OsO4 it has sixteen valence electrons."
    },
    {
        "video_name": "vZOk8NnjILg",
        "Q": "At 8:03, what is tension",
        "A": "The force in the rope"
    },
    {
        "video_name": "vZOk8NnjILg",
        "Q": "7:35 Doesn't a yo-yo actually work the OPPOSITE way? Isn't it centriFUGAL force that keeps it orbiting (the reason it requires a string), while centriPETAL force is what keeps the planets in orbit?",
        "A": "The string is not the same thing as gravity. It has something in common with gravity, which is that it exerts a centripetal force."
    },
    {
        "video_name": "vZOk8NnjILg",
        "Q": "At 10:10, why isn't it normal force? The tires exert a normal force on the car towards the center, does it not?",
        "A": "no, that s not a normal force, it s a fictional force. Normal forces are reaction forces that are perpendicular to the surface of contact."
    },
    {
        "video_name": "vZOk8NnjILg",
        "Q": "At 7:32 Sal said that a yoyo goes in a circular path but doesn't it actually go up and down",
        "A": "It does go up and down or you could swing it in a circular motion about your head just to understand the physics of objects going in circular motion. :)"
    },
    {
        "video_name": "vZOk8NnjILg",
        "Q": "at 8:44 Sal said that the moon doesn't go in the space somewhere random in one direction because of the gravity of earth right.\n\nIf the gravity of the moon can affect the earth and make tides why cant the gravity of the earth (which is approximately 83.3% stronger) doesn't just pull the entire moon in?",
        "A": "Earth does pull the moon in. But the moon is also moving sideways really fast, so even though it is constantly accelerating toward earth, it never gets any closer to the earth."
    },
    {
        "video_name": "vZOk8NnjILg",
        "Q": "I don't understand how Sal drew the change in velocity vectors around 3:00-3:10. Can someone clarify?",
        "A": "That little arrow was just showing how much the velocity changed. The first velocity vector showed what direction the object was going in, and the second vector showed the direction of the object after some time had passed; the arrow he drew was showing just how much the direction had changed. Let me know if this helps!"
    },
    {
        "video_name": "vZOk8NnjILg",
        "Q": "At 10:17 it was said that the force of friction from the tires to the road changes it direction due to which it goes in a circular motion. How does friction make it change its direction (that too perfectly suiting the race track)?",
        "A": "Think about this example: A car on a road, heading towards a corner... its about to turn.... but there is some ice on the track... there is no friction... what will happen as the car tries to turn on the icy road? and why? (Think about newtons first law) let us know if you need more help :)"
    },
    {
        "video_name": "vZOk8NnjILg",
        "Q": "So at about 6:02 Khan says that centripetal and centrifugal forces are very different. What makes this so?",
        "A": "centripetal is toward the center centrifugal would be away from the center"
    },
    {
        "video_name": "gW-7yeUW7-k",
        "Q": "At 0:06 he shows some clip-type-things. where do i get those?",
        "A": "yea you can get them at radio shack they have a magnifying glass that comes with it and a soldering iron holder. it go s for about $15"
    },
    {
        "video_name": "qMc6KOkmjTU",
        "Q": "At 5:42, I dont see why the answer is D. The problem asks for the velocity not the momentum. Since p=mv, shouldnt the velocity equal to p/m which is -20?? Thank you!",
        "A": "Yes, you re right. I just totally forgot to divide by the mass at the end. My bad! I ll add an annotation."
    },
    {
        "video_name": "qMc6KOkmjTU",
        "Q": "Is the answer of the question at 4:30 wrong? Shouldn't we choose C instead of D? We need to consider about the unit of the velocity, right?",
        "A": "Yes, the correct answer is C since you would divide that -40 kg m/s by the mass (2kg) to get the velocity of -20 m/s."
    },
    {
        "video_name": "4qgLz0TF8-M",
        "Q": "At 7:52, why it is told that -(t/RC) has to have no units before we evaluate? What is the reason behind this thing?",
        "A": "The term -(t/RC) is in the exponent. Exponents have to be pure numbers with no dimensions. That s because exponents are really a shorthand notation for a multiplication operation. If I write 2^3 it means multiply 2 by itself 3 times. If I write something like 3^(4 ohms) where there are dimensions in the exponent, I can t make sense of it: multiply 3 by itself 4 ohms times?"
    },
    {
        "video_name": "4qgLz0TF8-M",
        "Q": "at 1:32 you use the formula iC = C dv/dt\n\nsilly question but what is 'd'?",
        "A": "Hello Doug, This is the notation of calculus - d stands for delta or change. You would think of this as a change in voltage per unit time. In English the equation reads The current on the capacitor equals the product of capacitance and the speed at which the voltage is changing. If the voltage is changing quickly the capacitor will have a high current. Regards, APD"
    },
    {
        "video_name": "4qgLz0TF8-M",
        "Q": "at 10:50(probably a stupid question), why don't you also account for current from the capacitor?",
        "A": "We account for the current in the capacitor back at 1:30 where we develop the original differential equation."
    },
    {
        "video_name": "sDZDgctzbkI",
        "Q": "At 6:45 , which product is re stable between those two ?\nAre they both equally stable ?\n\nPlease clear my doubt soon.\nThank you",
        "A": "They are equally stable because both are interacting with a methyl group through 1,3 diaxial strain."
    },
    {
        "video_name": "sDZDgctzbkI",
        "Q": "At 5:20 how do you know that the carbon is becoming sp2 hybridized, doesn't it still have all single bonds?",
        "A": "Jessica, Jay has made previous made videos on general chemistry overview which detail how to work out if an atom is sp1/2/3 hybridised based on it s stereo number (number of sigma bonds) In short, if it has 4 sigma (single covalent) bonds, it is sp3 hybridised (1xS and 3xP), if it is has 3 sigma bonds as is the case here, it is sp2 hybridised (1xS and 2xP). Thus in the example at 5:20 the carbon now has 3 sigma bonds, therefore making it SP2 hybridsed. Hope that helps :)"
    },
    {
        "video_name": "sDZDgctzbkI",
        "Q": "At 0:53 if the bonds of the sp2-bybridized C are all on the same plane why are the bond to 2 H's drawn as wedge and dashes? Am I missing something?",
        "A": "They are all in the same plane. But they are all just dashes only if your eye is perpendicular to the plane. In the video, the plane is drawn as a parallelogram, which tells you that you are looking at it edge-on or just above the plane. That means that one of the bonds must be closest to your eye (a wedge) and the other must be furthest away (a dashed line)."
    },
    {
        "video_name": "sDZDgctzbkI",
        "Q": "At 6:02, what is the stereochemistry of the reactant and what is the stereochemistry of the two products? Is the reactant (R) and the products (R) and (S)?",
        "A": "Reactant is R as you said. The product on the left is S, while that on the right is R."
    },
    {
        "video_name": "sDZDgctzbkI",
        "Q": "Maybe it's a stupid question but, the products are enantiomers \"of each other\", producing a racemate mixture, right? 3:57",
        "A": "Correct. SN1 reactions give racemization at the \u00ce\u00b1 carbon atom. If that is the only chiral centre, you get a racemic mixture. If there are other chiral centres, you get a pair of diastereomers."
    },
    {
        "video_name": "sDZDgctzbkI",
        "Q": "At 7:10, which will be the major and minor products?",
        "A": "the major product is the product on the right (the inversion product)"
    },
    {
        "video_name": "HIQ5hnm61LQ",
        "Q": "At 7:31, two formula appeared:\nx'=gamma(x-beta*ct)\nct'=gamma(ct-bata*x)\n\nI wonder why two formula should use the same constant gamma.",
        "A": "cause it is the lorentz factor that corrects the transformation"
    },
    {
        "video_name": "HIQ5hnm61LQ",
        "Q": "2:30: Can someone explain why alpha = arctan(v/c). I see for the top triangle it works but for the bottom triangle arctan = opp/adj = ct/x = c/v, not v/c. Thanks.",
        "A": "In the last video, they proved that those two angles were equal, which is why they re equal."
    },
    {
        "video_name": "BCH1Gre3Mg0",
        "Q": "at 4:47 to 5:58: Why didn't New York simply stop putting the chemicals in the water in the first place. Surely that would be the best and most efficient solution?",
        "A": "You re right. But still, a wetland would have been necessary to eradicate all the toxic chemicals already present in the water. After all, there s nothing wrong in constructing a wetland. It is just another fixed deposit for the future generation."
    },
    {
        "video_name": "lwO0V5FitAo",
        "Q": "At 7:00 , can I just find the center of mass of the rod. Then plug in the distance to the center of mass and the total mass of the rod into I=mr^2 ?",
        "A": "Try it. Does it give you the same answer? If it does, you can do it. If it doesn t, you can t. Good way to learn."
    },
    {
        "video_name": "J97G6BeYW0I",
        "Q": "At 1:21, Rishi says that blood pressure is force/surface area meaning 115 is the force and 75 is the surface area thus they are in different units, force and surface area. But later on in the video (at 5:51), Rishi says that this is a range, meaning that the 2 numbers are comparable units thus saying that they also are in the same units. Don't these 2 statements then contradict? Could someone please explain this to me?",
        "A": "I agree the way he explained it was a bit confusing, but the 115/75 is a range of pressures. Both the 115 and 75 have units of Force/Surface Area. It does seem like it would make more sense to represent BP as # - # instead of # / #"
    },
    {
        "video_name": "J97G6BeYW0I",
        "Q": "0:08 a fraction? what woah i'm lost here help.....",
        "A": "In this video they explain what the fraction means. The fraction Isa range hat you blood pressure falls into. The largest number is the maximum blood pressure."
    },
    {
        "video_name": "FHptu3tcJnM",
        "Q": "at 3:19 jay placed the hydrogen at the top what if we place it at the bottom would it make any difference or is it more stable to place it at the top ?",
        "A": "It doesn t make a difference. The hydrogen will be removed by the end, making a planar molecule either way. Additionally, since there are no other substituents on the molecule, drawing it both ways would be the same molecule, simply flipped."
    },
    {
        "video_name": "FHptu3tcJnM",
        "Q": "At 10:35, you show the carbocation attaching to benzene such that the red carbon is on top and the green carbon is below. Would it be equally possible for it to attach so that the green carbon is on top and the red one below? The two products would be formed equally since it's equally possible to attack from the bottom or the top? Thanks!",
        "A": "Essentially, yes. Since the group is attached by a single bond, it will be rotating freely, so it doesn t matter whether it attacks one way versus another."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "At 12:20 If the black dwarf is not emitting any radiation or energy and then how could we know that it is already exist ? and how could it be super-dense and don\u00e2\u0080\u0099t has the mass of something ?",
        "A": "We don t know they exist, and they may not exist at all yet, because the universe might not be old enough. This is just the name we give to the remnant of a star that has radiated away all of its energy."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "At 3:15, Sal says that the star is creating energy, but I thought energy could not be created or destroyed?",
        "A": "creating is a bad choice of words The star is converting energy from one form to another. From mass energy to radiant energy."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "Why does the energy being created by these stars go outward? Is light the only kind of energy that is created by stars? Also, if energy is not a particle, how can it push \"things\" outward as Sal mentions at 6:57?",
        "A": "this energy is also heat. and light can also push things a little bit, but what he mainly means is the leftover neutrons, protons, and electrons from fusion."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "At 7:35, I understand that He flash is taking place at the moment of 'ignition' of He, does similar explosion takes place when star is starting fusion of C, O and so on? If yes, are they stronger each time?",
        "A": "It would have to be a crazy big star to make a Oxygen or Carbon flash, and it would be stronger."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "7:53 So in a helium flash, a star can \"explode\" without actually destroying itself, just releasing huge amounts of energy?",
        "A": "Thanks. If it were a binary system (or maybe not system, if there were no planets) then could the helium flash destroy the other star?"
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "at 10:40 Sal says that the sun wont be hot enough. why? pressure or somthing?",
        "A": "The stars that are hot enough tend to be many times more massive than the Sun. Its mass that makes the difference."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "When you said something about plasma 00:18 that it was just a soup of elcectrons and protons, does that mean that there is no nucleas, just the particles?",
        "A": "The nucleus stays intact. It s only the electrons that are ripped from the atom, leaving a positive ion."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "At 2:53 Sal was talking about the hydrogen plasma shell collapsing. Why can't the hydrogen get recycled ?",
        "A": "the hydrogen does get recycled, but at that time, it is not enough to change what is happening"
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "at 3:52 how come the surface of a red giant is cooler than a medium sized star",
        "A": "The amount of energy that gets released is being released from a much larger surface area and a lower energy equates to a lower temperature."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "At 1:46 Sal says the sun was born 4.5 or 4.6 billion years ago. What reference do we have and why are we not sure?",
        "A": "Since we were not here and there is no clock to look at we have to use things like the age of the earth and other objects in the solar system to figure out when it was likely to have formed. We primarily use radioactive processes to estimate how long ago things came together. Since wo do not know all of the conditions that could have affected these processes there are limits on how accurate the estimates can be."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "I know this will sound like a conspiracy theory (either that, or really dumb), but at 1:40, Sal mentions that the sun is getting hotter, and bigger. Could this lead to the temperatures on Earth getting hotter, and messing with the seasons we have now?",
        "A": "In 100 million years, the Sun will only have become 1% brighter. Its increase in luminosity will not affect us in our life time."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "7:50 Sal talks about a He Flash. How often do they happen? Also does this happen with other elements as they are formed in the core?",
        "A": "Yes, thanks, but I m really wondering about frequency of flashes and whether it happens as other atoms fuse."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "1:50 Why is the sun fusing faster now than it was when it was first born?",
        "A": "Its sort of a runaway fusion affect. The core has a higher temperature than required to fuse hydrogen so it can fuse even more. With more fusion, the core is hotter so it can fuse even more hydrogen. This takes a while to add up but it shows after a few billion years."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "At 10:49 I think Khan means that the sun will not get hot enough for heavier elements to form. He makes this clear previously during-> 9:25-9:31.\n\nAbout Helium Flash, I noticed that the whole process was unaffected since Helium fusion continued. My understanding is that Helium and the products of Helium fusion explode and leave a large cavity that then allows the process to continue - H to deuterium to He to He fusion - I imagine the H layer must collapse and become smaller in size. ideas?",
        "A": "i think you re right."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "at 12:21 why exactly would the star be called a black dwarf and/or a white dwarf",
        "A": "White dwarfs glow white from the residual heat of the nuclear fusion it once had when it was a Sun like star. A black dwarf has lost all its heat and thus is pitch black as it does not emit anything at all."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "At 12:15 the very dense dot is a Black Dwarf. I thought it was a black hole.",
        "A": "i think your right"
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "10:38 So if the star doesn't have enough mass to make carbon and oxygen fuse, it kind of reaches a terminal density as a white dwarf at the end of it's life where it can't fuse but gravity is still compressing it tremendously? Why doesn't it continue to condense? At this point is the repelling electromagnetic force stronger than the force of gravity?",
        "A": "At the point where most stars run out of fuel, the electron degeneracy pressure prevents the star from collapsing further. When at that density, the electrons are literally compressed towards the nucleus and are touching it. It would take more mass to break this limit and then you would reach a new limit called the neutron degeneracy pressure where neutrons are literally touching each other. After that, there is the theoretical quark degeneracy pressure and then a black hole."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "Why Does nuclear fusion take place only up to iron? at 06:04.",
        "A": "It has to do with the strength of the Strong Nuclear force. The Strong Nuclear force has a limited distance that is works over. The size of the iron nucleus is the point where the Strong Nuclear force is weakened to the point where to make a larger atom it takes more energy to push the protons/neutrons together than comes out of the process."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "Does gravity for each star come from the fusions making the molecules more dense in 1:10?",
        "A": "No, the total mass of the star doesn t increase as fusion occurs. If anything, it will slightly decrease as some of the mass is converted into energy. Gravity is proportional to mass. Density is not equivalent to mass."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "At 10:01, it is said that all of the fuel (hydrogen and helium) will run out. But at 11:13, he says that a huge cloud of gas surrounds the core. Where does this gas come from and why doesn't it fuse?",
        "A": "The pressure is greatest in the center. (Same is true even for earth). As you move away from the center, the pressure decreases, and at some point it is too low to generate fusion."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "I thought that the fusion of hydrogen was the one to produce the most energy. How come when helium fuses its capable of releasing enough energy to \"break\" the pressure (\"helium flashes\") due to gravitiy by the surrounding matter and hydrogen isnt?? \"07:36\"",
        "A": "You must remember that 2 hydrogen atoms fuse to make a helium atom, but 3 or 4 helium atoms combine to form Carbon or Oxygen. SO even though hydrogen fusion produces more energy per gram, there are more grams of helium. Therefore the helium fusion produces more energy."
    },
    {
        "video_name": "EdYyuUUY-nc",
        "Q": "Also, 1) if the star has hot dense carbon, does it made of diamonds? 2) 12:18 if a black dwarf is black, will the carbon turn to graphite?",
        "A": "Yes the carbon in a white dwarf is under very high pressure and temperature and they are likely in a crystalline structure but this structure is not the same as a diamond. The terms white and black when referring to a dwarf star is basically indicating its temperature not what it made of out."
    },
    {
        "video_name": "1VaKLjvbWZ0",
        "Q": "Hey Willy, I think the web designers need to insert this note at 4:58 saying, \"willy really meant V1\" is unknown not R1..",
        "A": "Oops. Thanks. I will report the slip up."
    },
    {
        "video_name": "gT0IqL1dyyk",
        "Q": "At 10:05, isn't the wavelength of the third harmonic suppose to be 3/2 not 2/3? there's 1 wavelength + 1/2 wavelength",
        "A": "You re correct that for the third harmonic there are 3/2 waves on the string. However the wavelength is only 2/3 of that string. And in this example the string is 10 meters. Thus the wavelength is (2/3)*10 m. I m guessing that you re mixing up the wavelength with the number of waves in this case."
    },
    {
        "video_name": "gT0IqL1dyyk",
        "Q": "4:50 why is the fundamental wavelength chosen by default?",
        "A": "I m no expert, but I would think it relates to energy. Maybe a higher frequency requires more energy, and since the fundamental wavelength has the lowest energy, that is the wavelength defaulted to?"
    },
    {
        "video_name": "BlBQBOb0OQ4",
        "Q": "At 7:44\njay said about steric number what is a steric no",
        "A": "Steric number = number of sigma bonds (or number of bonded atoms) + number of lone pairs From the steric number you can determine hybridisation: Steric number 4 = sp3 Steric number 3 = sp2 Steric number 2 = sp"
    },
    {
        "video_name": "rMuj2MGeIAc",
        "Q": "At 3:02, what does he mean with 'non-living organisms'? If something is not living its not an organism is it?",
        "A": "A virus is an example of a non-living organism. Viruses are sometimes called non-living because they can t replicate their own DNA. They have to invade a cell and use the cell s machinery in order to replicate themselves. As Sal said, viruses are much less complex than living organisms."
    },
    {
        "video_name": "rMuj2MGeIAc",
        "Q": "As Sal says at 8:48 and 9:00 that RNA is able to replicate itself and catalyze actions, How is it possible, I mean how does it do that?",
        "A": "This is possible, and it is the idea behind the RNA world hypothesis (which states that self-replicating RNA gave rise to life on earth). Scientists have synthesized small molecules of RNA called ribozymes (RNA enzymes) that act as enzymes to replicate other strands of RNA. They consist of the same chemical compounds as other RNA s, but tend to have more complex tertiary (non-linear) structures."
    },
    {
        "video_name": "rMuj2MGeIAc",
        "Q": "at 8:53 what is tRNA as Sal said",
        "A": "tRNA (transfer RNA) is a type of RNA molecule. It helps in decoding a mRNA (messenger RNA) sequence into a protein. It translates a triplet nucleotide codon into a specific amino acid, these multiple amino acids accumulate into a protein."
    },
    {
        "video_name": "pAjluTxSYgY",
        "Q": "I am very confused. At 7:35, Sal says that calcium is actually a metal. So does that mean that milk, which contains calcium, actually contains metal?",
        "A": "Yes. And the cereal contains iron, so there s another metal. The amounts of these elements are small, but they are there."
    },
    {
        "video_name": "pAjluTxSYgY",
        "Q": "At 1:31 Sal said that mercury at room temperature is in liquid state.At which temperature is mercury solid ?",
        "A": "Mercury freezes at -37.89\u00c2\u00ba F."
    },
    {
        "video_name": "pAjluTxSYgY",
        "Q": "In 6:52 Sal says that the bottom left corner elements are most likely to share electrons. Don't they \"give away\" electrons rather than share them?",
        "A": "When Jay used the term share , he was using it in the sense of giving electrons away."
    },
    {
        "video_name": "pAjluTxSYgY",
        "Q": "at 3:57 , Sal said that the copper atoms \"share\" their electrons. Does that mean they form covalent bonds?",
        "A": "No, they don t form covalent bonds. Instead you have a sea of electrons, where the electrons are shared between all the copper atoms and are free to roam around. This explains why copper conducts electricity, the electric current arising from the movement of electrons."
    },
    {
        "video_name": "pAjluTxSYgY",
        "Q": "Metallic Character increases down the group and atomic size increases too, hence, the atoms tend to lose electrons to attain a stable, noble gas configuration.\n\nSo according to what Sal said at 6:57, Caesium and Francium are the most active metals, But my teachers have told us that Sodium and Potassium have highest reactivity, so among these which elements have highest reactivity?",
        "A": "Sal was correct when he said about Francium and cesium. The metal reactivity series u learn at school is only of metals most used, so it does not include Francium and therefore Potassium tops the list."
    },
    {
        "video_name": "iorZdz4dsBU",
        "Q": "at 2:30,there are only two significant figures",
        "A": "since 1.69m goes till 1 is a significant figure 6 is one and so is 9. same for 2.09 2 is one 0 is one and 9 is one so there is 6 significant figures!"
    },
    {
        "video_name": "iorZdz4dsBU",
        "Q": "At 8:35 in the video I can't understand why the 113.931775701... was not left at 113 since only three significant figures could be used and the 9 was not part of that (it would be the fourth) but the 9 was rounded up and 114 tiles was the final result.\n\nIt was probably mentioned I tend to miss some key information, I think it has to do with it being to the nearest foot instead of .9 of a foot but other than that I'm not sure if it was a mathematical reason or just to not have .9 of a tile.",
        "A": "this video was so boarding!!"
    },
    {
        "video_name": "iorZdz4dsBU",
        "Q": "I have a similar question as did tiagofischer: from the videos, I understood significant figures as a way to represent the most precise value for a calculation/answer. In the example for the tile-bathroom problem, Mr. Khan rounds up to 114 (8:53). But then why not leave it as 113.9 because it would imply that we were able to measure to the near decimeter as we were when we measured the width of the bathroom (10.1 ft). Thank you and great video!",
        "A": "its 113.9 tiles, not a standard unit of measurement. this context changes everything. Besides, Sal explains that there shouldn t be more significant figures in the answer than in any of the terms."
    },
    {
        "video_name": "iorZdz4dsBU",
        "Q": "at 0:45 Why is the square uneven in the measurements?",
        "A": "It is not a square it is a rectangle."
    },
    {
        "video_name": "iorZdz4dsBU",
        "Q": "At 9:20, why does the answer need to have the least number of significant figures used in the measurements? Aren't we better with information that's more precise?",
        "A": "Let s say you knew half of a certain candy bar weighed 10.0 grams, and the other half weighed 10.03. You need to round it to 3 significant digits simply because you didn t measure the entire thing to that level of precision. To say it was 20.03 is wrong, because you don t know if the first half weighs 10.00, or 10.05, or what. So you need to round it to 3 significant digits to be more accurate."
    },
    {
        "video_name": "iorZdz4dsBU",
        "Q": "At 8:04-8:05, he said that there were 2 significant figures for the number 10.1. If I am understanding this correctly, shouldn't there be 3 significant figures in 10.1?",
        "A": "Yes you are right there are three sig figs..probably an error in video?"
    },
    {
        "video_name": "iorZdz4dsBU",
        "Q": "At 2:47, Sal names 3.5321, the number he gets after multiplying 1.69m and 2.09m a quotient. Wouldn't it be the product, and doesn't the quotient mean the answer you get after dividing two numbers?",
        "A": "Yea it should be. Good job noticing!"
    },
    {
        "video_name": "iorZdz4dsBU",
        "Q": "At 8:27, doesn't 10.7 have 3 significant figures?",
        "A": "Yes, I should think it does... after all, in scientific notation, 10.7 is 1.07 x 10... huh. Maybe he made a mistake?"
    },
    {
        "video_name": "iorZdz4dsBU",
        "Q": "Why a bathroom? 6:57",
        "A": "Sal uses a bathroom as an example. You can use the living room or kitchen or any room."
    },
    {
        "video_name": "Pn5YEMwQb4Y",
        "Q": "At 5:09, what is \"work\"?",
        "A": "Work is the transfer of energy. Energy, scientifically speaking, is the ability to do work. Less-circularly-but-still-scientifically speaking, work is the force required to move the object combined with the distance it is moved. In a formula, where W is work and F is force and D is distance, W=FD. Work is Force in Newtons multiplied by Distance in meters."
    },
    {
        "video_name": "Pn5YEMwQb4Y",
        "Q": "At 1:52, gases are considered a fluid? I just finished a one semester course, and gases and liquids were defined completely differently when we talked about States of Matter. Am I missing something? Liquid and gases are both fluids?",
        "A": "Gases and liquids are two distinct forms of matter. The classification of fluid in physics terms is different than in every-day life- in the normal world, we call liquids fluids, but in physics, a fluid is defined as anything where the particles can flow around each other. Both liquids and gases are hence fluids, as in both of them, the constituent particles can flow and move around each other. Hope this helps!"
    },
    {
        "video_name": "Pn5YEMwQb4Y",
        "Q": "at 5:13 --- why is W(in) = W(out) ?",
        "A": "Because the energy in a closed system has to stay the same."
    },
    {
        "video_name": "Pn5YEMwQb4Y",
        "Q": "At around 5:20 Sal mentions the law of conservation of energy? What is that?",
        "A": "energy is neither created nor destroyed, it just changes form"
    },
    {
        "video_name": "Pn5YEMwQb4Y",
        "Q": "at 1:40 isn't plasma also a fluid like liquids and gases?",
        "A": "Yes, Plasma is a fluid. However, because it is a mixture of neutral atoms, charged ions and free electrons, it responds to and produces electromagnetic forces. There are fluid dynamic equations called Boltzmann equations which include the effects of electromagnetic forces in the normal equations for fluid dynamics given by Stokes."
    },
    {
        "video_name": "Pn5YEMwQb4Y",
        "Q": "Somewhere between 5:10 and 5:30 Sal says to review the videos on work. I would, but I've been confused on how to find them. Where are they?",
        "A": "Type work and energy into the search box."
    },
    {
        "video_name": "Pn5YEMwQb4Y",
        "Q": "at 2:19 Sal says that liquid is incompressible. But what I've read is that 'Gases have the highest compressibility, then Lewis and finally, solids'\n\nI'm confused. Which fact is accurate?",
        "A": "Well if liquids are not compressible then they have no compressibility so there s not really a conflict with the statement you read. In fact liquids are very slightly compressible but only at very high pressures. But we generally treat them as not compressible for purposes of physics problems."
    },
    {
        "video_name": "Pn5YEMwQb4Y",
        "Q": "At 7:43 Sal says that a liquid is incompressable, are some liquids compressable?",
        "A": "All liquids are a tiny bit compressible, but only under very high pressure. We generally say they are incompressible."
    },
    {
        "video_name": "Pn5YEMwQb4Y",
        "Q": "At 2:14, Sal says a liquid is incompressible. But, don't scientists say that some liquids can compress, although we can't notice this?",
        "A": "yes, liquids are tiny bit compressible, but only a little bit, and only at high pressures. You understood what he meant, right?"
    },
    {
        "video_name": "Wl4re38deh0",
        "Q": "at 1:50 what is the really bright star looking thing?",
        "A": "It is likely a close star that happens to be in the shot."
    },
    {
        "video_name": "Wl4re38deh0",
        "Q": "At 1:56 what is that really bright star? Why is it so brighter then others?",
        "A": "I don t think its a quasar. Quasars usually have a bright white color, not orange. It probably is a star that happens to be close to us. A few of those bright dots are big elliptical galaxies. They tend to have a circular shape and they tend to be orange. Also, quasars tend to be far away. We don t even have a good picture of any quasar because it is so far away."
    },
    {
        "video_name": "Wl4re38deh0",
        "Q": "Do you guys think that at 1:45 when Sal is circling that third Galaxy, do you think that it is a cluster? I think it looks like one.",
        "A": "I meant cluster of Galaxies but your right Neil, it could be a nebula, or nova."
    },
    {
        "video_name": "Wl4re38deh0",
        "Q": "At 2:15 if we see both galaxies and stars from hubble telescope, then how come we see all of them in same shape? I mean galaxies are bigger that of stars right?",
        "A": "The galaxies are much, much, much further away than the stars."
    },
    {
        "video_name": "Wl4re38deh0",
        "Q": "At 1:30, when Sal is showing the Hubble image, how could it see those galaxies even if they were light years away?",
        "A": "It just means that the photons we detect from those galaxies has been travelling towards us for years before we detected it."
    },
    {
        "video_name": "Wl4re38deh0",
        "Q": "At 1:51, is the object that is shining brightly (which sort of looks like a compass) a galaxy since Sal did not circle that out?",
        "A": "It s probably either a supernovae, a quasar, or a bright star."
    },
    {
        "video_name": "CNP-dCQ-Cmg",
        "Q": "at 0:34 Sal said the molecules on left have high concentration so they have high probability of moving from left to right. Is that so because there are more molecules on the left side so more number of molecules strike the wall of container and move to the right, and there are less number of molecules on the right so there are less chances of the molecules on the right to strike the wall of the container and move to the left hand side. Is this logic correct?",
        "A": "Exactly, Sal mentions this in the diffusion video as well"
    },
    {
        "video_name": "zGO-MlsdmgU",
        "Q": "0:36... In the last video, Sal called his first example \"epoxybutane\", yet he did not specify which carbons in the hexane ring that the oxygen atom were joined to (nor did he need to, as this is implied). So why now are we expected to specify the 1,2 in epoxyethane? There is no other possible configuration for it so the 1,2 is understood! Am I right or wrong?",
        "A": "The use of numbers is optional when there can be only one possibility, Thus, epoxyethane is correct, but you must use 2,3-epoxybutane because there is also a 1,2-epoxybutane. The preferred IUPAC name for 2,3-epoxybutane is 2,3-dimethyloxirane. The ane tells you that it is saturated. The ir tells you that it is a three-membered ring (think tri ). The oxa tells you that an oxygen atom replaces one of the C atoms. As usual the a is dropped when it comes before a vowel. The O atom is assigned number 1."
    },
    {
        "video_name": "zGO-MlsdmgU",
        "Q": "1:40... \"e\" comes before \"m\" in the alphabet, so shouldn't we mention the epoxy before the methyl?",
        "A": "You name the class of compound before you name the substituents Epoxy is not a substituent. It defines the class of the compound, epoxybutane. Then you put the name of the substituent methyl group before the main name and get 2-methyl-2,3-epoxybutane."
    },
    {
        "video_name": "Q9BtOLDr0vg",
        "Q": "8:35 Isnt standard temprature 25C",
        "A": "No for STP it s 0 C"
    },
    {
        "video_name": "Q9BtOLDr0vg",
        "Q": "1:32 It makes chlorine gas while making metal right? Isn't it dangerous?",
        "A": "Yes, it makes chlorine gas which is potentially dangerous. But it also makes sodium metal, which is potentially dangerous. And, it is carried out using molten sodium chloride at around 800 C, which is potentially dangerous. A lot of processes in chemistry are potentially dangerous, which is why very strict safe working conditions have to be followed to ensure no one gets harmed."
    },
    {
        "video_name": "5gujU2QcGcY",
        "Q": "At 4:00, when Argon gas is added, the total pressure increases. Then shouldn't the reaction move backwards to balance the increase in pressure?\n\nLike when Argon is added the volume available to CO2 decreases so the reaction move backwards\n\nOr like if we add Ar wouldn't it be like the available volume to the other reagents is reduced ?",
        "A": "I completely agree that the volume available to CO2 will decrease. But the total volume of the container still remains the same. That is why at 4:47 she says that we did not change the volume. So addition of Argon will not affect the direction of the reaction. Coming to pressure, the total pressure did increase, but the partial pressure of CO2 did not. And we saw that Kp only depends on the partial pressure of CO2. So pressure too, will not affect the direction of the reaction."
    },
    {
        "video_name": "3yaZ7lkQPUQ",
        "Q": "At 3:52, why did Sal add the 50 m/s onto the 250 m/s instead of another order of operations? Are we allowed to add rates like that?",
        "A": "Yes, why would you not be able to add rates? If you are going 30 km/hr and I tell you to add 10 km/hr to your speed, you go up to 40 km/hr, right?"
    },
    {
        "video_name": "3yaZ7lkQPUQ",
        "Q": "At 3:02, how is the car stationary, but the ground is moving?",
        "A": "The car is in motion at 50 m/s to the left. The observer is in the car, so the car appears to be stationary to the observer, because they are both in the same motion, and the ground appears to be moving to the right at 50 m/s. This lesson is about relative motion, how things appear to observers in various reference frames."
    },
    {
        "video_name": "3yaZ7lkQPUQ",
        "Q": "Did Sal draw the arrow wrongly? (3:59) Because if the plane is supposedly has 50 m/s added to it shouldn't it be going the other direction?",
        "A": "No, he drew it correctly. The plane is still going at 250 m/s from the perspective of a person on the ground, while you are going at 50 m/s from the same perspective, but, because you are in the car it would seem from your perspective that you are stationary, whereas the plane would appear to be going your speed with relation to the ground and its own -- 250 m/s + 50 m/s -- combined. How things appear depends on your point of view."
    },
    {
        "video_name": "oqBHBO8cqLI",
        "Q": "At 9:08,how come the derived formula for the displacement of the object does not include the phase constant? or does it not have an effect?",
        "A": "At 08:06 pm., what I do not understand is under what circumstances is the phase constant equal to zero or -90?"
    },
    {
        "video_name": "oqBHBO8cqLI",
        "Q": "At 2:29 Sal talks about simple harmonic motion. Is there a \"complex\" harmonic motion?",
        "A": "Yes. Complex harmonic motion happens when more than one simple harmonic functions combine. The example we got in class was chords in music. Hitting one key on a piano or one string on a guitar (a note) is an example of simple harmonic motion, playing more than one note simultaneously is a combination of notes (AKA complex harmonic motion)."
    },
    {
        "video_name": "oqBHBO8cqLI",
        "Q": "@7:34, Sal says that the expressions of time period and frequency are independent of the displacement but it is clear that the two expressions are dependent on 'k' which is the spring constant. My question is that isn't 'k' dependent on the displacement 'A' if we go back to the equation F= -kx, if so then the expressions for time period and frequency are indirectly dependent on the displacement. I hope this makes sense and is my reasoning right?",
        "A": "The spring constant k is dependent on the material of which the spring is made and not on the displacement."
    },
    {
        "video_name": "oqBHBO8cqLI",
        "Q": "At 5:24 Sal wrote 2pi*sqrt(m/k). He said that the sqrt (m/k) is the inverse of the sqrt(k/m). Isn't the inverse (k/m)^2.",
        "A": "He probably means reciprocal."
    },
    {
        "video_name": "oqBHBO8cqLI",
        "Q": "At 6:00, Sal said he was assuming no gravity for the equation t=2pisqrt(m/k). Isn't gravity irrelevant in this situation?",
        "A": "Irrelevant in the period."
    },
    {
        "video_name": "HOMn523JBok",
        "Q": "at 9:12 if you dont rotate you can see the images are superimposable right?so why it was said in the video that they are not superimposable ?",
        "A": "Superimposable means a molecule and its mirror image can be placed one on top of the other, and all the molecules/atoms will line up in the exact same configuration. Non-superimposable means the molecule and the mirror of the molecule (like in the video at 9:12) can t be turned in any direction to make their atoms line up. Try using a modeling kit to visualize it for yourself."
    },
    {
        "video_name": "HOMn523JBok",
        "Q": "the enantiomer at 5:37 have chiral carbon atom? and does every chiral molecule is enantiomer?",
        "A": "Yes, every chiral molecule has an enantiomer."
    },
    {
        "video_name": "icNOBmKjDLo",
        "Q": "At 1:54, how is it possible for methanol to have a pKa of 15.54? I thought the maximum scale was until 14?",
        "A": "You re thinking of pH with a max of 14, but even that can actually go higher than 14 pKa is totally fine being higher than 14"
    },
    {
        "video_name": "icNOBmKjDLo",
        "Q": "Do we only calculate the pH of solutions when they're at equilibrium? Lets say we were doing the problem 3:15 and we were told the concentration of acetate ions was also 1M, wouldn't that mean that the concentration of H+ also had to be 1M since the relationship between acetate ions, acetic acid and H30+ is 1 to 1?",
        "A": "If the concentration of acetic acid was 1M then the concentration of H+ would not be 1M as acetic acid is a WEAK acid. Weak means it does not dissociate fully into H+ and acetate."
    },
    {
        "video_name": "icNOBmKjDLo",
        "Q": "at 1:20 how do you calculate pka without a calculator?",
        "A": "pKa implies -log (Ka). If your Ka is in the form (m x 10^n), you can pretty accurately estimate the log without a calculator by putting it into the following equation: n - 0.m"
    },
    {
        "video_name": "CFBKfgGTP98",
        "Q": "At 2:54 Sir said \"if u do heptane you are not finding the longest chain\" ,What does that mean?",
        "A": "He clearly said that he did a wrong mistake in the past, it was an incorrect vision. Actually, the longest chain is made of 9 C atoms , hence, it is Nonane and u saw that there is CH3- bonded at 4th C atom , thus, the nomenclature is 4-Methylnonance."
    },
    {
        "video_name": "prLfVucoxpw",
        "Q": "Why is the equation for force of electric field and the electric field itself the same? (10:30)",
        "A": "There s a root in electric field, its not the same. Electric field is inversely proportional to distance , while force of Electric field is is inversely proportional to distance squared."
    },
    {
        "video_name": "prLfVucoxpw",
        "Q": "hi\nat 8:02 can we consider a disc instead of ring",
        "A": "I think we have to use a ring in order to fix r to one value. We need to fix r in order to fix theta to one value. Fixing theta to one value fixes the field strength of that ring to one value. The full strength of the field then becomes just a summation of the individual fields strengths generated by each concentric ring."
    },
    {
        "video_name": "prLfVucoxpw",
        "Q": "guys look,,,at 9:27 sal mentions that 2*pi*r*dr is the area!!but it should be the volume right>>??\nbecause volume is base area*height.",
        "A": "okay !! I got it,,he used that method by applying calculus!! It s very cool!!"
    },
    {
        "video_name": "prLfVucoxpw",
        "Q": "at 5:32, dont we have to take the resultant? shouldn't it be 2Ecos(angle)?",
        "A": "no, it will be Ecos(theta)"
    },
    {
        "video_name": "prLfVucoxpw",
        "Q": "At 5:30, isn't the y-component of Electric field, E.sin(theta)? Why have we taken it as E.cos(theta)?\nMore precisely, Why should we measure theta from y-axis instead of the x-axis?",
        "A": "It doesn t matter. Measure it from the other axis, then it will change from sin to cos."
    },
    {
        "video_name": "prLfVucoxpw",
        "Q": "At ~9:30, why didn't we take \u00cf\u0080r^2, since there are charged particles everywhere on the ring?",
        "A": "We have to consider the point charges on the ring. So the total charge isn t the area because the ring is assumed to be hollow. So the charge is :- (circumference) * (thickness) * (charge density) Watch the next part and Sal will integrate this equation and obtain a general formula for this situation."
    },
    {
        "video_name": "prLfVucoxpw",
        "Q": "At 5:32, why is the vertical component E1Cos theta, I thought vertical component was found by multiplying sin theta.",
        "A": "That depends on where theta is. In this case, theta is between the y component and the actual field line. So the vertical component will obviously be E1cos(theta)."
    },
    {
        "video_name": "prLfVucoxpw",
        "Q": "At 9:18 why is the width of the circle 'dr'; why does it share the same variable as the radius? Why not 'dx' or 'dw'? What allows us to use 'r' both times? Does it have to do with mstocksl@hotmail.com's question? Thanks!",
        "A": "This is not a notation question. It is the same variable because the width of the ring is the distance from its inner circumference to its outer one along a radial axis. Just as you would call dx to an infinitesimal length, for example, of a rectangle (even though its far from the coordinates origin), you call dr to an infinitesimal length along any r axis."
    },
    {
        "video_name": "LXi5FUMNTjE",
        "Q": "At 3:03 how do you know witch switch is + or -?",
        "A": "Hello Brian, For these small motors you can connect the battery (1.5 VDC NOT 9 VDC) and see which way it turns. If it is the direction you want than you are done otherwise reverse the connections. Regards, APD"
    },
    {
        "video_name": "Hbuh5m3p0x0",
        "Q": "At 6:26 you said that we couldn't have something in between. My question is that if we say n = 1.5,then we have a radii which is 2.75 * R1 . This radii is in between 4 and 9 ? Why is it not possible?",
        "A": "its not impossible, as a matter of fact it does happen like you said. if you go further into learning the modern atomic structure and the Schrodinger s wave equation, you ll see that the orbits consist of various orbitals which are not spheres either. Bohr, at his time didn t have the equipment to measure it accurately, and hence gave the line spectrum theory."
    },
    {
        "video_name": "OpyTJbzA7Fk",
        "Q": "at 6:30 when O-H detaches with C then why does oxygen take back 2 of its electrons? it should take back one since the bond had one electron from C and one from O. Why does O use 2 electrons to form bond?",
        "A": "Atoms don t own the electrons, electrons will move from atom to atom to atom. In organic reactions electrons often move in pairs which I think is because electrons are located in pairs (either in orbitals etc.) and for something to capture the electrons it will be able to capture both."
    },
    {
        "video_name": "OpyTJbzA7Fk",
        "Q": "7:26: Where did all the hydrogen atoms on the left go?",
        "A": "To the aqueous solution that they came in the first place when the dehydration synthesis happened."
    },
    {
        "video_name": "OpyTJbzA7Fk",
        "Q": "At 6:45, does a water molecule have* to be lost in order for the fatty acid and glycerol to bond? Why can't the carbon of the carboxyl group just keep the -OH group and have four bonds?",
        "A": "If that happened, then the carbon you are referring to would have five bonds - a double bond with oxygen, two single bonds with oxygen and a single bond with carbon. That won t happen so water is always lost."
    },
    {
        "video_name": "OpyTJbzA7Fk",
        "Q": "At 9:00, What is R? Sal used it.",
        "A": "it means: Anything that could attach at this spot but would be way to long if I wrote out examples"
    },
    {
        "video_name": "OpyTJbzA7Fk",
        "Q": "at 7:56, what happens to the hydrogen atoms attached to the carbon in the glycerol molecule when it forms triglyceride.",
        "A": "The hydrogen from the glycerol and the hydrogen and oxygen from the fatty acid break off to form a water molecule, allowing the remaining atoms on the glycerol and fatty acid to bond together. The process of bonding through the loss of a water molecule is called hydrolysis and if you are still confused there is a video under the carbohydrate section."
    },
    {
        "video_name": "OpyTJbzA7Fk",
        "Q": "1:34 if glycerol has an alcohol group, would having fat in your body make you drunk?",
        "A": "The -OH group is called an alcohol group. The alcohol that causes intoxication is primarily ethanol"
    },
    {
        "video_name": "OpyTJbzA7Fk",
        "Q": "At 2:48, why is it that most biological systems tend to have an even number of carbons?",
        "A": "I believe the answer lies in the biosynthesis of fatty acids, where carbons are usually added in pairs. Nonetheless, there are fatty acids with odd numbers of carbons."
    },
    {
        "video_name": "OpyTJbzA7Fk",
        "Q": "At 7:43 why could u draw the hydrogens in even if the oxygen already has two bonds?",
        "A": "Some times, a cation of H without electrons can bond with a lone pair from O."
    },
    {
        "video_name": "6M_bjRzyUn0",
        "Q": "At 7:06, David says that the time period isn't dependent on gravity. Then what about the formula for time period which says T= 2*pi* (l/g)^1/2?",
        "A": "your equation is for a pendulum ok??"
    },
    {
        "video_name": "TKGcfbyFXsw",
        "Q": "At about 10:32, there are two centrosomes in the splitting cell, one for each soon-to-be cell. However, when a cell starts cell division, it has two centrosomes. At what point does the other centrosome appear and how?",
        "A": "It appears during interphase in the Synthesis phase (or the S phase). S phase duplicates the DNA in the cell and creates another centrosome for the cell. See the interphase video for more info."
    },
    {
        "video_name": "TKGcfbyFXsw",
        "Q": "At 2:34, Sal says that the nuclear membrane starts to disintegrate. What causes it to disintegrate?",
        "A": "Lysosomes, probably or another self produced enzyme (protein) that its job is to dismantle the membrane of the nucleus..."
    },
    {
        "video_name": "TKGcfbyFXsw",
        "Q": "At 10:52, Sal drew one of the centrosomes inside of the nuclear membrane, and one outside. Which is correct?",
        "A": "The one with the centrosome outside the nuclear membrane is correct."
    },
    {
        "video_name": "TKGcfbyFXsw",
        "Q": "Is the relative position of the structures in the cell at 8:29 accurate?",
        "A": "Sal is only drawing the functional and important per discussion conceptual drawings. There are off course other organelles in the cell that for the purity of the topic being intentionally neglected (Golgi Apparatus, Lysosomes, Rough Endoplasmic Reticulum, et cetera, et cetera, et cetera)..."
    },
    {
        "video_name": "TKGcfbyFXsw",
        "Q": "at 8:36 did not label that part can anyone tell me what that is called just interested (something like connectocord or something?)\namazing video as usual:}",
        "A": "It s called the kinetochore."
    },
    {
        "video_name": "fqe64NjELv8",
        "Q": "The definition of \"recognition\", stated at 0:47, confuses me. Can someone please explain \"recognition\" in more detail?",
        "A": "Well. Recognizition if referred to part of your memory. Identification would be another word, because you need to identifiy things (ex. car, spoon, house) as the environment. It s a iittle complicated to express which of all the things are recognizable (there are colors, things, people). May be (for example) you will not recall right now which is the largest river in Europe, but if someone shows you a list you will remember (or at least, make a deduction)."
    },
    {
        "video_name": "fqe64NjELv8",
        "Q": "At 0:47, what is meant by 'recognition'?",
        "A": "to recognize something like face"
    },
    {
        "video_name": "Wt5EMpUt-_g",
        "Q": "At 2:58, Sal mentions how we take ADP and turn it into ATP. Where does the ADP come from?",
        "A": "ADP is composed of a five carbon atom compound known as ribose, one adenosine molecule and two phosphate groups. It is formed by removing one phosphate from adenosine triphosphate, ATP."
    },
    {
        "video_name": "Wt5EMpUt-_g",
        "Q": "At 3:11, what is NADPH ?",
        "A": "NADPH is a form of NADP. NaDP is an electron carrier whose main role is to accept electrons produced in a reaction. When NADP receives electrons, it becomes NADPH. NADPH releases its electrons for a chemical process."
    },
    {
        "video_name": "vXb0ZvkFkS8",
        "Q": "At 3:06 , he says the \"lower motor neurons in the cranial nerves primarily control the skeletal muscles in the head and the neck.\" But I thought the \"lower\" part meant lower motor neurons control muscles in the lower part of the body, e.g. the legs?",
        "A": "I believe that there are lower motor neurons that control the muscles in the head and neck and lower motor neurons that control the lower portion of our bodies. They are all still lower motor neuron that control different parts of of the body"
    },
    {
        "video_name": "vXb0ZvkFkS8",
        "Q": "Do large and small motor units have anything to do with fast-twitch versus slow-twitch muscles in, say, the legs? The reason I ask is because at 3:22 he implied that muscles with small motor units that synapse at a low number of muscles move faster. So does that mean that fast-twitch leg muscles just have smaller motor units?",
        "A": "I am not positive, but I think it is merely to do with skill. The smaller the motor unit, the less muscle cells being contracted, which means there can be more variation on movement in regards to direction and strength."
    },
    {
        "video_name": "vXb0ZvkFkS8",
        "Q": "At 6:05, Matthew describes fascinations; which are twitches. Sometimes above my eye feels twitchy. Is that part of the fascinations?",
        "A": "Hi Fox Otash, Did you mean fasciculations? Technically, the are involuntary muscle tremors. So, if the muscles on someone s arms were wriggling like a bucketful of worms, that would be a fasciculation. The symptoms you described with your eye are actually very common, and usually occur when people are tired or stressed. I really don t think that the eye twitches completely meet the requirements for a true fasciculation. Hope this helps."
    },
    {
        "video_name": "vXb0ZvkFkS8",
        "Q": "Hello at 5:20 you mention atrophy as being the decrease in size of muscle cells. In my studys we call decrease of size for hypotrophy, while the decrease in muscle cell numbers is termed atrophy. Can you clarify on the termanology?",
        "A": "Hypotrophy is a degeneration of an organ or tissue caused by a loss of cells. Atrophy is a waste away, typically due to the degeneration of cells, or become vestigial during evolution. There is a slight difference, but hypotrophy is the loss of cells, where as atrophy is the decrease in volume of cells."
    },
    {
        "video_name": "P7LKEkcNibo",
        "Q": "At about time 5:55, you divide 4.9 m/s^2 by 4 to get 1.225m. Why is it that the Delta time ^2 does not get divided by 4 also?",
        "A": "(4.9 * x) * 1/4 = (4.9 * 1/4) * x = 1.225 * x Which is an example of the associative law of multiplicatons and divisions."
    },
    {
        "video_name": "P7LKEkcNibo",
        "Q": "At 2:45 I don't understand why we didn't make the change in time negative. Is time a value that can never be made negative or did Sal forget?",
        "A": "T=0 is launch time. How can the thing land before it was launched?"
    },
    {
        "video_name": "P7LKEkcNibo",
        "Q": "At 1:20, Sal says 'remember the acceleration is constant downwards the entire time' What does he mean by that? I'm confused",
        "A": "The acceleration downward is gravity. It is constant, at -9.8m/s^2."
    },
    {
        "video_name": "P7LKEkcNibo",
        "Q": "This is going to sound dumb but at 1:02 you said change of t up is change of t /2. But why must you divide it by 2?",
        "A": "If total time in the air is t, then the time to go up is t/2 and the time to go down it t/2 and it adds up to t"
    },
    {
        "video_name": "P7LKEkcNibo",
        "Q": "At around 1:00 , Sal says that the ball spends half the time going up, and the other going down, but aren't the 2 times, it going up and down, so different as there is an infinite amount of factors acting upon the ball in both situations, such as what comes in its path, wind, weather, where it is caught, and the like that make that assumption of both it going up and coming down the same time, incorrect?",
        "A": "Of course we are not taking all those factors into account because then we could never do anything. We are assuming uniform acceleration, which is pretty close to correct for many situations but not all"
    },
    {
        "video_name": "P7LKEkcNibo",
        "Q": "Just a little clarification, in the end at 6:06, why is the simplified equation of\nS max = 1.225 . (t)^2. Where are the units of m/s^2 ?",
        "A": "You are correct that if he wanted to be completely clear, he should include the units of m/s^2 as part of the the constant. So a better version would be: S max = (1.225 m/s^2) . (t)^2"
    },
    {
        "video_name": "P7LKEkcNibo",
        "Q": "I'm so confused... At 2:20 why didn't Sal divide time by 2 as well, why'd he just divide the 9.8 by 2 and leave time... And at 5:50 why did he not divide (time)^2 by 4 as well ?",
        "A": "So that to make it in General use..the whole point of the video is that to make a general formula for total upward displacement/height, so whenever any person who wanna calculate the height that his ball or anything covers, when he throws it into air can use this formula, if he know the total time through a stopwatch.."
    },
    {
        "video_name": "YYEgq1bweN4",
        "Q": "At 5:20, couldn't we also include the probability for habitable bodies other than planets? E.G. Moons.",
        "A": "We have made a theory on probability of habitable moons also, ex - the theory of life on Europa."
    },
    {
        "video_name": "YYEgq1bweN4",
        "Q": "On 6:20 does Sal mean galaxy instead of solar system??",
        "A": "There s an annotation in the bottom right that clarifies that he does, in fact, mean galaxy instead of solar system."
    },
    {
        "video_name": "YYEgq1bweN4",
        "Q": "4:52 Sal said planets that are good for sustaining life preferably have a rocky core. Do he mean molten core?",
        "A": "He means a solid core Nissan Uddin."
    },
    {
        "video_name": "YYEgq1bweN4",
        "Q": "this is in 0:20, and if a planet orbets a star, with the gravity from the planet, would'nt the star orbet the planet?",
        "A": "Just as the star pulls on the planet, the planet pulls on the star. The star however is a lot more massive and does not move very much from the planet."
    },
    {
        "video_name": "YYEgq1bweN4",
        "Q": "When Sal talks about f subscript l at 7:40, he says he thinks the appearance of life is quite high because of the robustness of life and its ability to appear in extremely unlikely places. However, to me this sounds more like evidence of the tenacity of life - once it is present, it is very hard to dislodge, and not a reflection on how likely life is to appear at all. Any opinions?",
        "A": "You do have a point but after billions of years and a set of good conditions, life is probably inevitable."
    },
    {
        "video_name": "YYEgq1bweN4",
        "Q": "At 9:02 you mention the fraction of intelligent life as 1/10 of all life forms. Isn't it too optimistic?",
        "A": "He is actually not saying that 1/10 of planets have life. He is saying that 1/10 of planets could sustain life, if it were present. Our solar system has eight planets, two of which are habitable. One is Earth, while the other is Mars. Its temperatures range from -80 to 30 degrees F. It has water ice (and likely liquid water under the surface), CO2, and traces of oxygen. So if anything, Sal is being pessimistic on that."
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "4:10 So what different carbohydrates can be produces from photosynthesis?",
        "A": "Carbohydrates are just a fancy name for sugars, starches, saccharides, and polysaccharides. Plants that produce a lot of carbohydrates are sugar cane and potatoes."
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "At 1:19, he mentions the word ATP that is from the cellular respiration video. What exactly is ATP? (I never watched the cellular respiration video...)",
        "A": "It s the energy currency of the cell. Respiration in the cell produces energy, that is stored for a short time in the form of a compound called Adenosine triphosphate (ATP). The ATP is then used by any processes that require energy, for example, active transport."
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "At 10:23, what are photons??",
        "A": "Photons are light particles. As you turn on a light in your house, the light emitted is made up of photons."
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "10:35, so basically respiration is photosynthesis backwards and vice versa?",
        "A": "BASICALLY yes, it differs in more indepth look at the whole process with all the stages but basically at the beginning/in the end you get the same compounds + energy stored in ATP or solar energy"
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "At 7:46 Sal wrote water is taken in & oxygen is given out ligth reactions. But I see that CO2 in not used anwhere.So where it is used?",
        "A": "In the video, he later wrote that the dark reactions use CO2. (Sorry for making the dark reactions, well, dark.) :)"
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "at 11:56,is photosynthesis an endergonic or exergonic reaction",
        "A": "Endergonic reaction"
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "At 5:13, What does the n in the equation stand for?",
        "A": "The n stands for an unspecified integer."
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "so do the plants give us oxygen and we give them carbon dioxide in return? and what, at 1:51 is the key of life?",
        "A": "carbohydrates and oxygen are the key for life. (oxygen for aerobic life form)"
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "At 1:19, he mentions ATP - does anyone know what that means?",
        "A": "Adenosine triphosphate, the molecule which transfers energy around within cells."
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "why the dark reaction and light reaction sounds confusing 7:30 to 8:00",
        "A": "The light reaction is called the light dependent reaction because it requires photons to proceed. Whereas the dark reaction, though occurs in presence of light, doesn t require photons to proceed (It can also take place in the absence of light). So calling a dark reaction a light independent reaction would be more appropriate because a dark reaction might or might not take place in the presence of light"
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "At around 11:43, is phosphate spelled with an e at the end?",
        "A": "Yep, it s there. It looks a bit like phosphat, but the E is kind of curved in with the T."
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "i'm having a confusion......i totally agree with the statement that ATP is one of the products of light-dependent reactions;........i suppose it's also mentioned in the video at 8:10 that NADP is also produced in the light dependent rxn............. but isnt NADP a co-enzyme??....i'm under the impression that NADPH2 IS PRODUCED............",
        "A": "You are correct, but also missed what Sal said. At 8:10 Sal states and writes that in the light reaction ATP and NADPH are produced. He mentions NADP as being similar, and that we learned it in when studying cellular respiration and glycosis. He iterates that NADPH is different though as it has the hydrogen. ATP and NADPH are energy carriers produced (with oxygen) by the light dependent reaction, photosynthesis. I hope this helps clarify what you ask."
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "at 11:05 sal spells phosphoglyceraldehyde as PGal? my teacher says its PGald. can any one tell what's right/",
        "A": "both are. Both of these are valid representations of phosphoglyceraldehyde."
    },
    {
        "video_name": "-rsYk4eCKnA",
        "Q": "At 4:39 when he adds the CO2. the H2O and the photons, shouldn't it be ( C H2 O3)n because if you add the O's, there will be 3, right?",
        "A": "nope, because there s also an O2 molecule on the right-hand side of the equation. There are 3 Os on the left-hand side (CO2+H2O), and 3 from the combination of (CH2O) and O2"
    },
    {
        "video_name": "8x8tA4YPhJw",
        "Q": "at 4:41 how come oxygen is making 3 bonds and has 2 electrons. what about that one electron that wasnt drawn",
        "A": "it is due to dative bond between oxygen"
    },
    {
        "video_name": "8x8tA4YPhJw",
        "Q": "If the formal charge is zero, it means that the compound it's not reactive? what about the oxidation estate?\nAnd another question: Oxygen in nature has 6 valence electrons but then why in 5:25, there are just 2 electrons unpaired?",
        "A": "Both formal charge and oxidation state are bookkeeping tools to help us keep track of where electrons are in molecules. A formal charge of 0 doesn t mean something is stable and unreactive. An oxygen with a formal charge of 0 has 2 bonds and 2 lone pairs. One of these lone pairs can be used to bond with a hydrogen ion. This now means the oxygen has 3 bonds and 1 lone pair and a formal charge of +1."
    },
    {
        "video_name": "8x8tA4YPhJw",
        "Q": "in example at 8:32 oxygen gets a fc of -1,\nfrom previous lessons we know bond between carbon and oxygen is polar.\nbut here oxygen is getting proper -1 assigned. does that mean bond is now ionic?",
        "A": "No, it means that the O atom has a negative charge and can form an ionic bond with some cation."
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "5:51 how is it turning into nitrogen?",
        "A": "A neutron in the Carbon-14 nucleus emits an electron, therefore making it a proton, which turns it into Nitrogen-14"
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "At 2:09, how do you not know when it will decay even if i might a any minute?",
        "A": "Roll a dice. You may roll a 6 at any moment. Do you know when you are going to roll a 6?"
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "During 4:00, Sal says that beta decay would release a electron. If this kept happening, wouldn't everything turn to hydrogen?",
        "A": "No. In beta decay, a neutron turns into a proton. There s no route to hydrogen there."
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "at 1:00 what is half life?",
        "A": "An isotope s half life is the amount of time necessary for the atom to have a 50% chance of decaying."
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "at 4:45 why is the carbon version 14?",
        "A": "Carbon-14 is carbon with 6 protons and 8 neutrons. Carbon-12 has 6 protons and 6 neutrons. The number of protons is the same for every isotope of an element, only the number of neutrons changes."
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "At 10:00, can you give me a basic explanation/definition of half life?",
        "A": "It s the time it takes for half the sample to decay."
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "At 5:39, why do you have to redraw the picture, does the picture really matter what it looks like?",
        "A": "Yes, to some extent. That one half that he drew first is theoretically incorrect. there is a tiny chance that the atoms will decay to look like that. Although it is more realistic to draw the block with little holes here and there. Because the atoms will decay at random times."
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "At 5:17 the video states that indiviudal C14 atoms don't know when to change, and that it's ultimately up to random chance when a certain atom decays. How then are we able to accurately determine a specific rate of decay for a large mass of atoms if each atom's chance of decay is random?",
        "A": "The laws of chance work very precisely when you are dealing with billions and billions of atoms. In the same way, if you flipped a coin a billion billion times, it would come up heads almost exactly half the time."
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "What is the Law of Large Numbers? 7:58",
        "A": "It is a theory that describes the result of performing an experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value; the more trials that are performed, the closer the result is to the expected value. Source: Wikipedia."
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "Example shown at 11:54, how would scientists date the sample if there are, let's say, 12.432448034 grams of carbon-14 left?",
        "A": "There is a formula: A=Pk^(t/T) Where P is the original amount, A is the final amount, t is time passed, and k is the fraction left of the original after T time has passed. If we start with 80 grams of carbon 14 with a half life (so k=.5) of 5740 years, and enough time has passed so that there are only 12.43 grams left, then: 12.43=80(.5)^(t/5740) Solving for t using logarithms will get get you the time passed."
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "At 4:00, I am confused on half-life. What is a half life? How does it affect atoms and isotopes? what happens after a half-life occurs?",
        "A": "An isotope s half life is the amount of time necessary for the atom to have a 50% chance of decaying."
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "I thought half life was only used in algebra 2? 3:03",
        "A": "No it s used in chemistry also"
    },
    {
        "video_name": "9REPnibO4IQ",
        "Q": "At 8:22, why doesn't the n -14 also have a half life?",
        "A": "Nitrogen is stable. Not all atoms decay. C14 does. C12 doesn t. N doesn t."
    },
    {
        "video_name": "zwDJ1wVr7Is",
        "Q": "in 06:19 why does sal write equals 10 newtons? not the weight of it?",
        "A": "10 Newtons is the weight. Remember, weight is a force."
    },
    {
        "video_name": "zwDJ1wVr7Is",
        "Q": "AT around 3:56 shouldnt the equation be sq root of 3 T1/T2=0 i.e. sq rooot of 3 T1 =T2",
        "A": "Well..Isn t it more correct to do 5x / 3x = 0?"
    },
    {
        "video_name": "zwDJ1wVr7Is",
        "Q": "after minute 7:21 the T1 and T2 are being used like the same altought yellows T1 & 2 are the tension on the X axis while Teal T1 & 2 are the tensions in Y axis; is that OK?",
        "A": "He solves the problem as a system of equation. The first equation (T1+ root3*T2=20) describes the forces on the y-axis. The second equation (root3*T1-T2=0) describes the forces on the x-axis. By adding them together he gets a result with is true for both equations. If you d like to now more about system of equations, you ll find some good videos in the math section."
    },
    {
        "video_name": "zwDJ1wVr7Is",
        "Q": "At about 3:35 it says T1*cos (30) = T2*cos(60). How are they equal? I get that they have equal magnitude, but they are vector quantities with opposite directions, shouldn't they have opposite signs (one being negative)? I know I'm wrong because my textbook and teacher do it this way, but I can't seem to get a satisfactory understanding. Thanks.",
        "A": "you know that t1*cos(30)-T2*cos(60)=0 otherwise the thing would be moving. So from there you just flip flop over the equal sign. Also, He explains it around 3:00 if you didn t catch that."
    },
    {
        "video_name": "zwDJ1wVr7Is",
        "Q": "sal talked about the tension on that little wire at @0:23 but while calculating the net force along the y direction sal didn't mention that tension...can anyone please help me on that!",
        "A": "That is because the two y-components of the left and right wire is equal to it. If we call the tension S , then S = T1sin60 + T2sin30 (only in the y-direction, Sy) He also says that S = 10N :) Hope that helped."
    },
    {
        "video_name": "zwDJ1wVr7Is",
        "Q": "Sal, how did you get -20sqrt3 at 8:40? I'm confused about the math portion.",
        "A": "after subtracting the two blue equations, he is left with -20sqrt3."
    },
    {
        "video_name": "zwDJ1wVr7Is",
        "Q": "8:42 How does it go from 10 root three to 20 root three?",
        "A": "Because first he multiplied both sides of the equation by 2 so the 10 changed to 20. Then he multiplied the equation by root 3. Hope that answers your question"
    },
    {
        "video_name": "zwDJ1wVr7Is",
        "Q": "Hey Sal, at 00:47 you say that the point is not accelerating in any direction, how do you know that? like technically isnt it accelerating in the y because the point is not on the pink line parallel to it? like its below the pink line so isnt it accelerating downwards in the y direction.",
        "A": "It is completely balanced. Watch the Balanced and Unbalanced Forces video. The force of gravity is pulling on the box, but the normal force provided by the tension of the two strings are cancelling it out. So there is simply no unbalanced forces acting upon the box, hence it cannot be accelerating in any direction, or moving for that matter."
    },
    {
        "video_name": "zwDJ1wVr7Is",
        "Q": "I understood everything until around 6:10 and it threw me completely off when he used fractions for sine/cosine and started to get rid of the rid of the twos in the denominators by multiplying it by two (and I'm required to use decimals). How do you solve it if you use decimals to once you get to T1sin(30) + T2sin(60) = 10N?",
        "A": "Because sin describes a ratio between the different sides of a triangle sin(30\u00c2\u00b0) is equal to 1/2 (you must know this by heart or use a calculator with sin function). The math topic is Trigonometry, if you d like to know more about this. And in the last step he solves the problem as a system of equation to get a result with is true for both equations."
    },
    {
        "video_name": "zwDJ1wVr7Is",
        "Q": "At 5:58, doesn't T1sin30 + T2sin60 =100? (Because if one draws a force diagram, shouldn't the force in the downwards direction be mg?)",
        "A": "The force down is 10 N. It shows it right there in the picture. The mass of the block must be 1 kg"
    },
    {
        "video_name": "aalANrz7bi8",
        "Q": "At 2:13 Sal says, \"You can type these things into Google...\" and then he showed a bunch of graphs getting closer and closer to f(t). I tried typing the function (the first few terms anyway) into Google, and was just directed to a bunch of sites. Can you be more specific? I would love to try it!",
        "A": "Type one of these lines into Google. f(t) = 3/2 + 6/pi sin t f(t) = 3/2 + 6/pi sin t + 6/3 pi sin 3t f(t) = 3/2 + 6/pi sin t + 6/3 pi sin 3t + 6/5 pi sin 5t"
    },
    {
        "video_name": "OTtI4urvgDg",
        "Q": "12:15 I still dont understand how u could solve it because dont u still have 2 unknowns?",
        "A": "You still need to use simultaneous eauation to solve it, but it would be WAY easier now since you have eliminated all the unnecessary calculations, and it wouldbe faster to solve it"
    },
    {
        "video_name": "_0TqG8-N8DI",
        "Q": "At 6:05, wouldn't it be a hydrate instead of hemi-acetal? There is a OH group, H, and R group but there is no OR group. Infact instead of the OR group, there is another OH group so that makes it a hydrate no? OH, OH, OR, and H groups on a carbon make a hydrate right?",
        "A": "He is talking about the anomeric carbon in the ring structure of glucose. It has an R, an H, an OH, and an OR. That makes it a hemiacetal."
    },
    {
        "video_name": "9K3R6xtO6SU",
        "Q": "What would be the name of the compound formed at 06:50? I'm talking about IUPAC name, just to be clear. Thanks :^)",
        "A": "The preferred IUPAC name is 2-benzofuran-1,3-dione, but almost everyone calls it by its common name, phthalic anhydride."
    },
    {
        "video_name": "9K3R6xtO6SU",
        "Q": "At 1:40, why isn't it the Cl- that leaves the acyl chloride? Why do the electrons get moved onto the oxygen? Is it because O is more electronegative than Cl?",
        "A": "acyl halides always follow the mechanisms of addition then elimination"
    },
    {
        "video_name": "9K3R6xtO6SU",
        "Q": "In 2:00,why don't we kick those electron off the chlorine to form the chloride anion?",
        "A": "Is that not what happens by about the 2:30 mark? Even though we show mechanisms in multiple steps that is for our clarity only, the reaction likely takes place in one very rapid step."
    },
    {
        "video_name": "64GBYOsI9jQ",
        "Q": "3:36-3:46 \"lichens aren't even plants\" - what are lichens then?",
        "A": "Lichens are a symbiotic relationship between Fungi and Algae. The fungi helps in the absorption of minerals and water from the ground while the algae helps in photosynthesizing. But both of them share the food."
    },
    {
        "video_name": "64GBYOsI9jQ",
        "Q": "At 8:04 , Hank mentioned that plants produced unprotected seeds. What are they by definition and how are they unprotected?",
        "A": "The seeds that come from vascular plants are unprotected because they are not enclosed in female tissues. These seeds can be accessed freely by vertebrates, specifically granivores (seed predators), which feed on the seeds of plants as a main or exclusive food source."
    },
    {
        "video_name": "64GBYOsI9jQ",
        "Q": "I thought it was Meiosis that makes gametes. Hank says in 6:32 that plants make it through mitosis. please clarify?",
        "A": "Meiosis is a special type of nuclear division which segregates one copy of each homologous chromosome into each new gamete . Mitosis maintains the cell s original ploidy level (for example, one diploid 2n cell producing two diploid 2n cells; one haploid n cell producing two haploid n cells; etc.). Hope this helps."
    },
    {
        "video_name": "64GBYOsI9jQ",
        "Q": "6:20 hank said mitosis. meant meiosis",
        "A": "yes, meiosis means mitosis also when asexual reproduction reproduce s, it start making spores just like fungus just like a puff ball mushroom."
    },
    {
        "video_name": "64GBYOsI9jQ",
        "Q": "3:36-3:46 \"lichens aren't even plants\" - what are lichens then?",
        "A": "Lichens are composite organisms consisting of a fungus and a photosynthetic partner"
    },
    {
        "video_name": "tnYQ_tQmYrM",
        "Q": "What do you mean, at 1:43, by \"pulling out of voltage source\"?",
        "A": "If you connect something (a resistor, a bunch of resistors, anything that conducts current), it causes a current to flow out of the current source. So pulling current out of a voltage source is just a casual turn of phrase I used to give the sense that the resistors connected to the voltage source are demanding or pulling current. If the resistors were not there, there would be an open circuit, and therefore no current. When the resistors are connected, in a sense they are causing the current to flow."
    },
    {
        "video_name": "AQf3gKTCffs",
        "Q": "4:11, Are there any plants that have both day-dominant and dark-dominant individuals inside of the same species? And if so, would there be a possibility to breed for it?",
        "A": "Most likely not. Due to the natural elongation and shortening of a day or night, the two proposed subspecies would not be able to cross. You could, theoretically, make this happen in captivity, but it would require a lot of effort and would need a lot of gene splicing."
    },
    {
        "video_name": "5CBs36jtZxY",
        "Q": "At 2:45, how does the description of the ionization energy relate to the \"work function\" that was discussed in an earlier video where the photon of light added sufficient energy to remove an electron (not just cause it to move to a higher energy level, but essentially to separate completely. Is the work function just a specific effect of the ionization energy?",
        "A": "Ionization energy is very analogous to work function. Ionization energy usually refers to the energy required to strip an electron away from an atom that is in the gas state. Work function refers to the energy required to strip an electron from a metal to a point just beyond the surface of the metal."
    },
    {
        "video_name": "5CBs36jtZxY",
        "Q": "Around 1:15, Sal discusses how you can take an electron away from hydrogen, and then it will become a positive hydrogen ion. But if the hydrogen ion gained an electron, then its valence electrons would imitate helium, and it should be stabilized that way too, right? Only as a negative hydrogen ion. Why don't we ever hear about hydrogen taking electrons, say from sodium or potassium, like the halogens do?",
        "A": "Hydrogen typically bonds with hydrogen to make H2, the most common form of hydrogen we find in nature. The reason it does that is so that both H atoms have 2 electrons in their s valence shell, which is their outermost shell and which holds 2 electrons."
    },
    {
        "video_name": "5CBs36jtZxY",
        "Q": "When an element loses an electron, does the proton number change as well? At 4:42, when Li loses an electron and becomes He, the proton number is still 3, but the electron number is now 2. Doesn't that imbalance cause a charge?",
        "A": "the proton number does not change because the protons are present inside the nucleus in a very small volume under the action of short range forces which are really too strong. only electrons which are held by electrostatic forces can be ripped off. so when you say that after losing an electron lithium becomes helium, it is not turning into a helium atom but only its electronic configuration resembles that of a helium atom. Indeed the lithium atom will end up with a +1 charge after the loss of the electron."
    },
    {
        "video_name": "5CBs36jtZxY",
        "Q": "At 8:35, Why do the dips occur, is it because, say Beryllium and Boron, Be has a full 's' sub-shell and B starts the 'p' sub-shell?",
        "A": "Yes, you are right. 2p-orbitals are a little further out from the nucleus than the 2s orbital and this reduces the attractive pull from the nucleus. That pull is reduced further by the 2s orbital screening the 2p. Hence it is easier to remove a 2p electron than a 2s electron."
    },
    {
        "video_name": "5CBs36jtZxY",
        "Q": "In the chart shown at 8:40, why is element 80 (Mercury) spike up as if it is a noble gas?",
        "A": "Mercury is the last of the d shell elements in this period, oddly enough that means that the electron being ionized is of of the 6s shell."
    },
    {
        "video_name": "HBi8xjMchZc",
        "Q": "at 1:06, why would it have a stronger attraction?",
        "A": "Because it has more protons pulling the electrons and thus, its size would decrease. Hope this helps :)"
    },
    {
        "video_name": "HBi8xjMchZc",
        "Q": "At 2:32, why is the chlorine atom bigger than the sodium ion? what does this do or change?",
        "A": "Chlorine atom is bigger than sodium as it has completely filled shells ,making it similar to argon. Sodium also by losing an electron behaves as a noble gas that is neon. If ypu know that the atomic size increases down the group then you can make out that neon is smaller than argon , that is why sodium is smaller than chlorine."
    },
    {
        "video_name": "HBi8xjMchZc",
        "Q": "At 2:20, he says \"Clorine ion, cation\". Does he mean anion?",
        "A": "WELL, YES. Sal just forgot and probably told it in a hurry. YOU ARE CORRECT."
    },
    {
        "video_name": "HBi8xjMchZc",
        "Q": "Difference between cations and anions? 2:18",
        "A": "Cations= positively charged ions and anions are negatively charged ions. You can remember them as : A-n-ion --> A negative ion. And cation will be its opposite. :)"
    },
    {
        "video_name": "HBi8xjMchZc",
        "Q": "At 2:20 Sal says \"Clorine ion, cation\". Does he mean anion?",
        "A": "Yes Cations are Positive ions (H+, Na+, Ca2+) and Anions are Negative ions (Cl-, I-, O2-)"
    },
    {
        "video_name": "HBi8xjMchZc",
        "Q": "At 2:13 why is Chlorine bigger than Sodium if its sodium that has a positive charge and Chlorine has a negative charge?",
        "A": "Now,when sodium loses an electron, it has 11 electrons but 10 protons.so, it loses a shell(you may think like that).also,the protons exert a larger force as the distance decreases and the total charge of electrons decreases, which is almost like the protons gain +1 charge than electrons. Now, clorine gains an electron , so , the inside electrons are going to repel more strongly and thus going farther,so, it is going to get larger."
    },
    {
        "video_name": "HBi8xjMchZc",
        "Q": "at 1:55 i always have a hard time understanding this idea if na give one to cl then na will like ne or will it change the whole element so na went to ne ??",
        "A": "Yah she is right kanji is"
    },
    {
        "video_name": "HBi8xjMchZc",
        "Q": "At 0:33, why will Chlorine have a smaller atomic radius, doesn't it have a larger atomic number?",
        "A": "It will have a smaller atomic radius because it has a larger atomic number. Chlorine has more protons and consecutively more electrons. So the electrostatic force of attraction is greater. The protons attract the electrons bringing them closer to the nucleus. Therefore the size of the atom decreases."
    },
    {
        "video_name": "HBi8xjMchZc",
        "Q": "1:37 how does cl and Na have ions",
        "A": "An element like Cl will have 7 valence electrons located in its outer shell before forming a bond. When it forms an ionic bond with an element like Na it is essentially pulling an electron away from the Na. Since electrons are negatively charged, the Cl (which pulled an electron away from the Na) will have a negative charge (and be expressed as Cl-). The Na, which essentially gave away an electron, will then have a positive charge (and be expressed as Na+). Thus you have the ions Cl- and Na+."
    },
    {
        "video_name": "iUrleWSqok4",
        "Q": "At 6:18, Jay said it doesn't matter whether you rotate the front or back carbon. Can anyone explain to me why?",
        "A": "Because you would end up with the same molecular conformations. If you were to flip the whole molecule around, like so what was the back carbon goes to the front, you would have the same molecule."
    },
    {
        "video_name": "iUrleWSqok4",
        "Q": "At 9:36, how can we conclude that the second conformation has a higher energy than the first one? Without calculating and comparing the total energy of methyl-ethyl gauche interaction and hydrogen-methyl gauche interaction, with, that of methyl-methyl gauche interaction and ethyl-hydrogen gauche interaction, how can we tell which one is least stable?",
        "A": "The conformations shown at 9:36 are eclipsed conformations. You do have to add all the individual repulsions to get the total strain energies, but the Et-Et repulsions are the main factor. Assume that the repulsion energies (in kJ/mol) are: H-H = 4; H-Me = 6; H-Et = 8; Me-Me = 11; Me-Et = 13; Et-Et = 15. Which conformer do you calculate to be the least stable?"
    },
    {
        "video_name": "Y3ATc9he254",
        "Q": "At 4:15 he talks about how the molecules are further apart in ice because they don't have an much kinetic energy to continually break their hydrogen bonds and flow past each other (which to me makes sense as to why they stay in one place) but I'm still not clear as to why this would cause them to be further apart and therefor be less dense?",
        "A": "When the molecules join together, they have to join in a certain way that is determined by the shape of the molecule. There s an angle formed between the bonds that link the two H to the O, and that angle just doesn t let the molecules pack together any more closely."
    },
    {
        "video_name": "D5-1qEKtfQ4",
        "Q": "At 8:48 what happens to the proton dissociated from the water molecule and the MgBr + ?",
        "A": "The H from the water is the H in the alcohol, and the OH from the water ends up in the insoluble salt MgBr(OH)."
    },
    {
        "video_name": "D5-1qEKtfQ4",
        "Q": "At 7:45, there is no mention of the chirality center created by the Gringard reagent; after we've formed this chiral center, do we get both sets of enantiomers? Does a gringard reagent usually create enantiomers/diastereomers?",
        "A": "It depends on the reaction. But, whenever you create a chiral centre from an achiral starting material, the product will be a racemic mixture of the two isomers."
    },
    {
        "video_name": "3ROWXs3jtQU",
        "Q": "9:37 Sal said cesium is pretty much the largest common atom one will ever encounter. So is francium (the alkaline metal below cesium) quite rare, then? From the orientation of the periodic table, the radius of the atom gets bigger as one moves down a group, right?",
        "A": "There are about 25 grams of Francium in the entire world. So, you are rather unlikely ever to encounter it."
    },
    {
        "video_name": "3ROWXs3jtQU",
        "Q": "At 5:10, what is a colloid?",
        "A": "A colloid is a mixture that is very similar to a solution. The only difference is that the particles are not dissolved all the way, and the mixture is opaque and cloudy. The floating particles in a colloid never settle out, so it will remain murky."
    },
    {
        "video_name": "3ROWXs3jtQU",
        "Q": "Around 5:10, Sal says colloids are homogenous mixtures but shouldn't they be heterogeneous since the particles are not dissolved which can be seen when a beam of light (or laser) scatters.",
        "A": "They are indeed heterogeneous since they are composed of two different thermodynamic phases, dispersed and the dispersing medium. Here homogeneous refers to homogeneously dispersed, Uniform i.e. equal colloidal concentration every where, not homogeneous phase."
    },
    {
        "video_name": "3ROWXs3jtQU",
        "Q": "When Sal talked about colloids, is it possible to have a solid (suspended) in a solid? Also, 5:38, how can water molecules suspend in air like in fog?",
        "A": "Yes, it is possible to have a solid in solid solution. For example in the case of alloys, it is not a compounds but a solution of its constituents. In fog, the water particles are so small and light that they are held up in the air. These suspended water droplets are called aerosols."
    },
    {
        "video_name": "3ROWXs3jtQU",
        "Q": "At 5:51 i did not understand aerosols",
        "A": "Solid or Liquid mixed in air is aerosol . Vapour in air (fog) is example of liquid aerosol , Dust in air (smoke) is example of solid aerosol . ; )"
    },
    {
        "video_name": "3ROWXs3jtQU",
        "Q": "At 9:52 it should have been 0.26 Angstroms and not 0.026 Angstroms right?",
        "A": "Yes, you are right. A correction for this appears in a box at the bottom of the screen. Angstroms are not part of the SI system of units so shouldn t be used. The SI derived unit is nm and Sal was correct in saying 0.026 nm."
    },
    {
        "video_name": "3ROWXs3jtQU",
        "Q": "At 12:10 Sal refers to liters of solution. what is solution?",
        "A": "A solution is a homogeneous mixture of two substances."
    },
    {
        "video_name": "3ROWXs3jtQU",
        "Q": "at 6:31 what are nanao meters",
        "A": "nano meter is a very small unit of length. it s value is 1/1000000000 meters...... : )"
    },
    {
        "video_name": "3ROWXs3jtQU",
        "Q": "I don't really understand the mole fraction at 10:39.",
        "A": "An example may help. Assume you dissolve 40 g of NaOH in 1,000 g of water. The molar mass of NaOH is 40, so therefore you have 1 mole of NaOH. The molar mass of water is 18, so therefore you have 55.6 moles of water. The mole fraction of NaOH is therefore 1/(1+55.6), which is 0.018. The mole fraction of water is 55.6/(1+55.6), which is 0.982. (Note that 0.018 plus 0.982 equals 1, so all moles are accounted for.)"
    },
    {
        "video_name": "eaQyzkAZGjM",
        "Q": "At 8:30 why Bromine was substituted not florien ?!",
        "A": "It is simply a matter of good, better, best. Bromine and Fluorine are both good leaving groups, but Bromine is a better leaving group than Fluorine. Why? The next video explains. Stronger acids have more stable anionic forms, or conjugate bases. Bromine, further down the Halogen group, is more stable in its anionic form than Fluorine."
    },
    {
        "video_name": "eaQyzkAZGjM",
        "Q": "Around 6:30 you mention that our product is S-2butanol, would we obtain a racemic mixture from this reaction, or all S conformers?",
        "A": "All S. After the inversion happens you can t get any R conformers"
    },
    {
        "video_name": "eaQyzkAZGjM",
        "Q": "at 8:20, why did you just draw the nucleophile as the methoxide anion? Wouldn't the sodium also be bonded to the O-CH3, making it not charged at all?",
        "A": "It would be assumed to be run with a solvent it is soluble in such as methanol, in which case it would be split up into Na+ and OCH3- just like ions in water."
    },
    {
        "video_name": "dnYyMHSSb8M",
        "Q": "4:25, how do you know the half life of Phosphorus 32 ?",
        "A": "I guess it is just something you memorize but it is around 14.3 days"
    },
    {
        "video_name": "tJW_a6JeXD8",
        "Q": "So why do you divide both sides by 20? 13:10",
        "A": "To find lambda, we eliminate all other numbers from the RHS of the equation (that has lambda) : shift all known values to the opposite side to find out the unknown variable."
    },
    {
        "video_name": "tJW_a6JeXD8",
        "Q": "At 1:10 while explaining pulse did Sal say 'perturbation' or something else?What does it mean?",
        "A": "Perturbation is a general term for a disturbance or movement from normal so if the rope or string is normally static and taught, the wave is a perturbation from from normal."
    },
    {
        "video_name": "tJW_a6JeXD8",
        "Q": "at 10:40, u told that Velocity is distance per time, I'm confused because we were taught \"speed\" is distance per time... and \"velocity\" is Displacement per time. does it really matter with the dist. or disp.?",
        "A": "Displacement is a vector and distance is not. Velocity is a vector. So, when we are doing physics, velocity should be displacement over time, but it is common even in physics classes for people to get a little careless about it. Also, in elementary or middle school math classes they often say distance/time because kids that age don t know the word displacement and it is not worth spending time to distinguish between distance and displacement when 5th graders are just trying to learn to do word problems."
    },
    {
        "video_name": "tJW_a6JeXD8",
        "Q": "At 9:31, the teacher says that wavelength is how far a wave has travelled after 1 period... So is that the length of string he's jerking OR the perpendicular distance between two crests, for example?",
        "A": "It s the distance between two successive crests."
    },
    {
        "video_name": "tJW_a6JeXD8",
        "Q": "At 8:55.... Sal stated that the particle's movement is going up...... however since the direction of travel is to the right... wouldn't the particle move towards the trough? Therefore move downwards since in a transverse wave.. the particles move perpendicular to the direction of travel of the wave. ( I maybe wrong)",
        "A": "The direction of the WAVE is to the right. It will pass under the particle."
    },
    {
        "video_name": "tJW_a6JeXD8",
        "Q": "13:14 How do you change cycle/sec which is frequency to sec/cycle (period) ?",
        "A": "1/(cycle/sec) = sec/cycle, right? It s just algebra."
    },
    {
        "video_name": "tJW_a6JeXD8",
        "Q": "At 5:39, I understand the cycles/second. What is another word for cycles per second? And how is the cycles per second important in tasks?",
        "A": "Cycles per second can also be called Hertz."
    },
    {
        "video_name": "tJW_a6JeXD8",
        "Q": "At 12:40, how do I know which equation to use when trying to find the velocity of 100 m/s with a frequency of 20 cycles/second? For example, do I use:\nvelocity=\nwavelength/T or\nwavelength* 1/T or\nwavelength* frequency?",
        "A": "All three of those equations are exactly the same since freq = 1/T. 100m/s = 20Hz * wavelength wavelength = 5m"
    },
    {
        "video_name": "tJW_a6JeXD8",
        "Q": "At 2:03 when Sal says periodic wave, can that also be referred to as transverse wave?",
        "A": "No, periodic waves can be either transverse or longitudinal."
    },
    {
        "video_name": "uUhBEufepWk",
        "Q": "How would you find the concentrations with the impurities? He mentions it quickly at the end from 3:50",
        "A": "You can set up a ratio: How much is observed in the real world with impure substances / How much can I expect without impurities in my perfect-case equation And because they are directly related, it will give you a percent impurity."
    },
    {
        "video_name": "lzWUG4H5QBo",
        "Q": "at 4:21, it looks like he spelled chemical like chemikal. did he? or am I wrong? did I get confused? am I right?was HE wrong? NOT me?",
        "A": "He did not spell chemical wrong. he just put the i and the c very close together. so it appers as a k but if that was the case than where is the i?"
    },
    {
        "video_name": "lzWUG4H5QBo",
        "Q": "In 3:40, does that mean glucose + oxygen = water?",
        "A": "It means that glucose + oxygen in the process of cellular respiration will produce water AND carbon dioxide AND energy in the form of ATP."
    },
    {
        "video_name": "lzWUG4H5QBo",
        "Q": "At 2:54 what is glucose",
        "A": "Glucose is a simple monosaccharide found in plants. It is one of the three dietary monosaccharides, along with fructose and galactose, that are absorbed directly into the bloodstream during digestion. Formula: C6H12O6 Molar mass: 180.16 g/mol IUPAC ID: D-glucose Melting point: 146 \u00c2\u00b0C Density: 1.54 g/cm\u00c2\u00b3 Soluble in: Water, Acetic acid"
    },
    {
        "video_name": "lzWUG4H5QBo",
        "Q": "At about 3:20, they mention glucose input. why do we need glucose in our bodies?",
        "A": "Hi Ed, Glucose is one of the essential elements for respiration in human body cells. Respiration is the process where ATP(adenosine triphosphate, basically it s the energy molecule in the body) is made from glucose and oxygen. If you don t have glucose in your body, you won t have any energy to make your cells function, and therefore your body. Hope this helps, and sorry for the late response!"
    },
    {
        "video_name": "lzWUG4H5QBo",
        "Q": "at 4:27 what does ATP mean?",
        "A": "ATP stands for adenosine triphosphate. It is used as an energy-carrier molecule. Energy for various cellular functions can be released by the removal of one of the three phosphate groups."
    },
    {
        "video_name": "lzWUG4H5QBo",
        "Q": "Just to clear something, What is celluar respiration? I searched \"respiration\" in a dictionary which came up as the act of breathing, But plants don't breathe (at 5:33). So celluar respiration must mean something else. What does it mean? And, at 6:12, Dr. Rishi says plants have enough oxygen to be used by Jack and itself. Do plants need oxygen? I thought they only needed CO2.",
        "A": "Plants use photosynthesis in order to make sugar (glucose), which uses CO2. However, they need to break down the glucose through cellular respiration to use the glucose, which requires oxygen."
    },
    {
        "video_name": "lzWUG4H5QBo",
        "Q": "At 4:45, does that mean that the plant does photosynthesis, then uses the result to make cellular respiration, and causes a cycle, without needing to get the ingredients from the outside?",
        "A": "Photosynthesis produces sugars and fats and amino acids, which are the building blocks of life. Sugars are then used as the raw material in respiration to produce energy. As a result the plant is able to feed itself , although it may need to take in gases like O2 and CO2."
    },
    {
        "video_name": "lzWUG4H5QBo",
        "Q": "At 2:56 was Rishi explaining aerobic respiration?",
        "A": "I m pretty sure that s what was being told"
    },
    {
        "video_name": "lzWUG4H5QBo",
        "Q": "7:41 \u00c2\u00bfWhy are CO2 and O2 not in equal ratios?",
        "A": "Plants have been around a lot longer than animals and there a lot more of them. This means they have had more time to photosynthesise and thus there is more O\u00e2\u0082\u0082 in the atmosphere. However CO\u00e2\u0082\u0082 levels are on the rise due to the wide use of fossil fuels and pollution. Hope this helped!"
    },
    {
        "video_name": "lzWUG4H5QBo",
        "Q": "at 4:20, Rishi mentions ATP. What is ATP and what is its significance?",
        "A": "ATP = adenosin triphosphate = a molecule providing energy for chemical processes. you can find it in any metabolisms in the human body or nature."
    },
    {
        "video_name": "lzWUG4H5QBo",
        "Q": "Around 4:30 how does water and carbon dioxide create a high form of chemical energy.",
        "A": "So the water and carbon dioxide don t make the ATP (high form of chemical energy). As you can see they are located on the right side of the equation, where the products are located, not the reactants. They are made along side the ATP. You make the ATP using glucose and oxygen. It gets a lot more complicated the more we delve into this so I ll leave it at that."
    },
    {
        "video_name": "lzWUG4H5QBo",
        "Q": "At 4:28, Dr. Rishi drew the ATP as one of the things that Jack puts out. Doesn't he need the ATP?",
        "A": "What i m pretty sure Rishi was trying to say is that ATP (adenosine triphosphate) is PRODUCED as a PRODUCT of cellular respiration. After that, the ATP is used to transport chemical energy inside the cells."
    },
    {
        "video_name": "6hjr4n4MCPA",
        "Q": "why did it say \"con\" instead of \"khan\" in the subs at 0:14?",
        "A": "The written transcripts are created by an automagic program. It sometimes makes flubs like this. I sent in a request to fix this one."
    },
    {
        "video_name": "UBlem4dGLGU",
        "Q": "At 0:32 what is ml mean\n/\nmicroleater",
        "A": "Millilitre. Not micro."
    },
    {
        "video_name": "1_OieLuWGyw",
        "Q": "At 1:10, how does the spring go down when there is nothing pushing the spring?",
        "A": "There is a tiny little bump under the bit you press that pushes down on the spring."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "At 6:12, shouldn't he have written that pressure is inversely proportional to v, not 1/v, or is that just how the \"inversely proportional\" notation works?",
        "A": "There s no such thing as a silly question. Especially regarding notation. \u00e2\u0088\u009d means proportional to So regarding Boyles law: P \u00e2\u0088\u009d 1/V This shows the inverse relationship between pressure and volume. If volume increases, then pressure decreases and vice versa (at constant temperature)."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "At 7:40, Sal starts explaining what an ideal gas is but under what conditions are gases most ideal?",
        "A": "Many real gases act a lot like ideal gases as long as the pressure is not too high and the temperature is not too low."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "From 6:05 and further, what vexed me was when Sal mentions how volume decreases corresponds to how pressure increases, I wasn't able to comprehend how that makes P=1/V.\n\nAny guidance?",
        "A": "If P = 1/V, what happens to P if V goes down? Try it with numbers."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "At 1:43, How does bouncing off the sides create pressure, doesn't weight create pressure?",
        "A": "Pressure is a force over an area. If things are bouncing off a surface, they are giving a push, spread over an area."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "At 4:27 what was the constant for?",
        "A": "Constants are almost always there to make the units work. In this equation, you have PV on one side. That s pascals/m^3. On the other side, before the constant, you have nT, which is moles*Kelvin. You need a constant in there to reflect the fact that you chose to use the units you chose. So R goes in there to ensure the right proportionality between PV and nT."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "I have a doubt: At 8:13, If we consider Volume to be zero (v=0), then wouldn't the equation become 0 = nRT ?",
        "A": "It is not the total volume that is considered to be 0, but the volume of each particle. The reason for making this approximation is the volume occupied by each particle is exceedingly small so that almost all of the total volume of the gas is the space between the particles, not the particles themselves. However, if the gas is too cold or the pressure to great or the density too great, then this approximation won t work and we have to use much harder equations of state."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "At 3:00, temperature is the average energy. I dont understand how temperature can be considered as the energy? If it is heat energy, then how can the velocity of the particles be calculated?",
        "A": "The temperature of something is the average amount of kinetic energy released. If a molecule releases a lot of energy, it speeds up, and gets hotter. If it releases a little bit of energy, then it slows down, and gets colder."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "At 5:04, why is there a constant in the equation and what function does it serve?",
        "A": "We have to use constants because we use human invented units of measure. Specifically, pressure \u00c3\u0097 volume is one way of measuring kinetic energy. Temperature \u00c3\u0097 number of moles is another way of indicating kinetic energy. But they use different units, thus it is necessary to have a constant that corrects for this factor. Also, there the gas constant includes the Boltzmann constant which (amongst other things) relates temperature to energy."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "4:35, is temperature always in Kelvin in chemistry?",
        "A": "In chemistry and physics, the mathematics often require using absolute temperature, or the number of degrees above absolute zero. Otherwise, if we compare something at -10 C and +10 C we can get weird answers because of the negative sign. The absolute units for Celsius is Kelvin and the absolute units for Fahrenheit is Rankine."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "at 3:30, How can the kinetic energy be equal when the number is less?",
        "A": "Saying the kinetic energy is equal is another way of saying the temperature is equal. We use the bucket term kinetic energy to cover the vibration, rotation and movement of molecules and can extend that concept to temperature as well. This is an oversimplification but we are talking about macroscopic properties like pressure and temperature. So, to stretch the example, you could say that the one molecules is vibrating 4-times more than the others are individually and so both cases have the same temperature."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "At 1:55,why does PV=nRT?",
        "A": "The equation comes from combining the individual gas laws. Boyle\u00e2\u0080\u0099s Law: V \u00e2\u0088\u009d 1/P Charles\u00e2\u0080\u0099 Law: V \u00e2\u0088\u009d T Avogadro\u00e2\u0080\u0099s Law: V \u00e2\u0088\u009d n If we combine these, we get V \u00e2\u0088\u009d nT/P If we insert a proportionality constant R, V = nRT/P or PV = nRT"
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "at 1:49 why is it impossible to measure ?",
        "A": "Basically, there are too many gas molecules with varying velocities. Each time they collide, they transfer energy and this results in new velocities. Thus, it s impractical to do so."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "at 7:06 you say that R is some constant. How do we find that constant? My teacher said something like .0081 or something, but how do you derive to that value?",
        "A": "You can look online to find the appropriate R value for the units of pressure, temperature, etc. that you are using in the calculation. If you have a chemistry textbook, it probably has a small list of a few R values. Wikipedia (if you choose to trust it) has a very large list."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "At 3:32 he said that the kinetic energies for both the balloons (that is in the balloon that has one molecule and in the balloon that has many molecules) is same.\nHow can the kinetic energy be equal when the number is lesser?",
        "A": "By kinetic energy, he must have meant the average molecular kinetic energy, which then depends only on the temperature and not on the number of molecules. In any case, the total kinetic energy of the gas depends both on temperature and the number of molecules."
    },
    {
        "video_name": "WScwPIPqZa0",
        "Q": "at 2:23 why is temp considered the average energy of a particle.",
        "A": "That s just what temperature is. When the temperature goes up, its because the average kinetic energy of the molecules went up prorportionately."
    },
    {
        "video_name": "ZcZQsj6YAgU",
        "Q": "At 2:07 Why is tension the restoring force? It acts perpendicular to the motion and thus should not affect it.",
        "A": "Forces that act perpendicular to motion don t affect the motion? Where does that idea come from?"
    },
    {
        "video_name": "ZcZQsj6YAgU",
        "Q": "At 12:59,How could acceleration be large when the velocity is zero at the extreme point of expansion of the spring? Isn't acceleration directly proportional to change in velocity?",
        "A": "When an object reverses direction, its velocity is zero at one point of time. Magnitude of acceleration is not directly proportional to velocity."
    },
    {
        "video_name": "ZcZQsj6YAgU",
        "Q": "At 13:10, David says that even though the speed is equal to zero at the when the spring is highly compressed or elongated, the acceleration is the greatest. Is he referring to the acceleration of the spring as it returns to its equilibrium position?",
        "A": "The spring and mass are connected, so wherever the spring goes, so does the mass. Even though the velocity is instantaneously zero when the spring is stretched out (or compressed) to it s maximum, the force/acceleration is at its maximum value (ie. the velocity is changing the most at that point)."
    },
    {
        "video_name": "ZcZQsj6YAgU",
        "Q": "At 3:50 in the video, it is mentioned that even a simple pendulum for very small oscillations is a Simple Harmonic Oscillator. Can someone explain how?",
        "A": "for small oscillations, the force that moves the pendulum back to equilibrium is proportional to the displacement of the pendulum from equilibrium. You can easily find analyses of pendulum motion with google."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "at 0:21, you said that we gain more energy than we put in. And that is our practical experience in all exothermic reactions. But no process can be more than 100% efficient and hence if I put in 5J of heat I cannot get 15J of heat. how is that possible?",
        "A": "The energy that is on the products side is coming not from the heat you put in but from the chemical energy inside the reactants. For a chemical reaction to take place, bonds will be broken and new ones will form. BOND MAKING RELEASES ENERGY, so as a result of the bond making, heat energy is released. In an exothermic reaction, the amount of energy needed to break bonds in the reactants is LESS than the amount of heat released when the bonds in the products are made."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "what is the opposite of exothermic? 0:20",
        "A": "The opposite of exothermic is endothermic. Endothermic reactions remove heat from their environment; i.e., they take heat or use heat from their surroundings. Exothermic reactions give off heat or release heat to their environment."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "At 4:20, how does oxygen take the electrons? Is it just a sharing of electrons..? So O now becomes negative?",
        "A": "higher electronegativity means greater ability of an atom to attract electrons to itself in a chemical bond. oxygen does not take the electrons but because of higher electronegativity the electrons are attracted more to oxygen. so O bears partial negative charge. Also oxidation state of O is mostly 2- . except in OF2 where it is 2+ and in peroxides where it is 1-"
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "At 5:36 explain what procedure Sal is using and what he is trying to find.",
        "A": "He is just writing out half reactions, or the individual parts. He is showing that carbon is oxidized, and that it loses 8 electrons. Later, he shows that the oxygen gained the 8 electrons from the carbon. The half reactions make it easier to see how the reaction works."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "This hydrocarbon @0:06 he is talking about is methane right?",
        "A": "Yes, CH4 is methane"
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "At 8:54, what carbon oxdized?",
        "A": "When a carbon atom in an organic compound loses a bond to hydrogen and gains a new bond to a heteroatom (or to another carbon), we say the compound has been dehydrogenated, or oxidized. A very common biochemical example is the oxidation of an alcohol to a ketone or aldehyde:"
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "If carbon starts out at 4- and becomes 4+, adding 8 e- (as it says at 6:30) why isnt that reduction? isnt reduction gaining electrons?",
        "A": "If the charge is going up 8 (from 4- to 4+), then it is losing 8e-, so therefore it is an oxidation."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "At 5:35, why is there 8 electrons added?",
        "A": "Electron is a negative guy, so if 4 + (-8) would equal -4 . Then C4- ____ C4+ + 8\u00c3\u00a9 Hope That Helps! :-)"
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "At 8:22, why does oxygen gain 8electrons and not gain 2electrons?",
        "A": "one oxygen gains 2 electrons..there were 4 oxygen s so it gained 4*2= 8 electron s"
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "at 4:54 Sal states that carbon has to lose 8 electrons to become 4+ didn't he mean carbon needs to lose 4 electrons?",
        "A": "Since the carbon started with a 4- charge, it needs to lose four electrons to become neutral and then four more to become 4+."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "Why is the carbon double-bonded with the oxygen at 3:04?",
        "A": "Carbon and oxygen are both atoms that follow the octet rule pretty much all the time. The only possible way for that to happen is if there are double bonds between each oxygen and carbon. If you measure the bond lengths experimentally they are consistent with it being double bonds too."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "When Sal mentions how \"all oxygens are created equal, assuming that they are not different isotopes or something like that,\" at 2:26, I didn't quite get what he tries to purpose there. Should it be something to consider or...?",
        "A": "Different isotopes of oxygen have different number of neutrons which could change the electronegativity slightly because the nucleus would be slightly different. The difference would be very small though."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "So just after 9:13, where Sal says that Carbon was oxidized by Oxygen and Oxygen was reduced by Carbon, does that mean that Oxygen could be considered the oxidizing agent and Carbon the reducing agent?",
        "A": "You are absolutely spot on Allison, the reagent being oxidized is reducing agent and the one being reduced is an oxidizing agent.However don t always consider oxygen to be oxidizing agent and hydrogen to be reducing agent as in this example hydrogen is the oxidizing agent : 2Na+H2--->2NaH .Find the oxidizing agent and reducing agent by OIL RIG."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "At 3:39 how does oxygen take 4 electrons of carbon if its CO2?",
        "A": "Glad you figured it out yourself! Something that should be mentioned when learning this is that oxidation numbers are a bookkeeping method we use if we considered every bond to be purely ionic, so if it split up which atoms would take those electrons."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "At 2:53 why is the oxidation state for oxygen zero whereas at 3:36 the oxidation state is negative two. Isn't it always supposed to be negative two?",
        "A": "The oxidation state of anything in its natural elemental state is 0. For oxygen that natural elemental state is the O2 molecule so it has an oxidation number of 0 there. If it were always -2 in every molecule then that would imply oxygen cannot do any redox chemistry which we know isn t true."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "At 8:20, why does oxygen gain eight electrons and not just two electrons? I thought if oxygen went from a neutral charge to a -2 then it would only gain two electrons.",
        "A": "There are four oxygen atoms, each of them gaining two electrons. Take note of the subscript and coefficient."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "the law of conservation of energy states that energy cannot be created or destroyed. at 0:18, Sal said more energy than you put in gets released. So where does that come from?",
        "A": "The forming of chemical bonds releases potential energy."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "When Sal pulls out that scale at 1:41, that's not on the reference tables. How are we supposed to know this?",
        "A": "What reference tables exactly? Do you mean the periodic table with electronegativities on it? I m sure if you need to know the electronegativity to answer a question it will be given, if not you should remember the trend: it increases going to the right and up the periodic table."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "At 1:07 how do you figure out the oxidation number for the elements?",
        "A": "you consider that the number of electrons gained or lost (hypothetically) by the normal configuration of the atom to be the oxidation number of the atom so that it decides the oxidizing character of the atom i.e, the character of attracting the electrons"
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "at 8:43, what tells us that carbon has been oxidezed by oxygen",
        "A": "In this specific reaction: Carbon s oxidation number has increased from -4 to +4 so it has been oxidised Oxygen s oxidation number has decreased from 0 to -2 so it has been reduced So we could say that carbon has been oxidised by oxygen, and that oxygen has been reduced by carbon."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "3:45 were did C get the 8 electrons from ?",
        "A": "C loses 8 electrons to oxygen (Reduction is Loss) If the electrons are on the right side: i.e C4- ===> (C4+) + 8e- then the half equation is saying that Carbon 4- splits up into Carbon 4+ and 8 electrons. This is just showing that Carbon loses 8 electrons. It loses those 8 electrons to oxygen which gains them: 2 02 + 8e- ===> 4 O(2-) Hope that helps! :)"
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "At 7:17, can we write all O2s as 2Os? can we write all molecules as atoms?\nplease help.",
        "A": "Nonono! You write it as O2 so people (and yourself) know that it is existing as a molecule! 2O would mean that both of the O are separated! Just imagine it as, the number on the left is written only when balancing equations. XY means you have X amount of Y molecules."
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "at 3:00 why is carbon dioxide a carbon double?",
        "A": "Carbon atoms tend to share 4 bonds with other elements in a covalent molecule. Therefore in CO2, Carbon double bonds with Oxygen twice to form four bonds. Oxygen also likes to form two bonds between atoms, hence the double bond is the best situation for both atoms in the molecule. Hope that helps :)"
    },
    {
        "video_name": "OE0MMIyMTNU",
        "Q": "at 6:00,\nby GER carbon is gaining an electron therefore it should be reducing",
        "A": "At 6:00, carbon is losing electrons, therefore by LEO it is being oxidized, but it is acting as a reducing agent."
    },
    {
        "video_name": "3RDytvJYehY",
        "Q": "At 8:47, why can't the structure be O=S=O? (Because the formal charge on S and both O atoms becomes zero that way)",
        "A": "I believe he did that to only demonstrate a geometry of a trigonal planar molecule when lone pairs are involved."
    },
    {
        "video_name": "3RDytvJYehY",
        "Q": "1:34 What's the matter, have I missed some videos from playlist?why isn't element in compounds (even in in previous video) not following octet rule?",
        "A": "Were you listening to Jay from about then onwards? He did explain it by saying the formal charges would not be favourable. If you formed a double bond to make B follow the octet rule you would end up with a + formal charge on the most electronegative atom in the whole periodic table... BF3 is electron deficient and boron will readily accept a pair of electrons from another atom or molecule"
    },
    {
        "video_name": "3RDytvJYehY",
        "Q": "At 2:42 Jay said that a lone pair of electrons from a Fluorine could move onto the Boron to fulfill the Boron octet. How would this work since Fluorine would then only have 6 of the 8 required electrons?",
        "A": "He means a lone pair from one F could form a second bond between F and B, not that a lone pair is transferred to B..."
    },
    {
        "video_name": "3RDytvJYehY",
        "Q": "At 2:34, I still don't understand why Boron doesn't follow the octet rule... is it an exception? What's with the charge?",
        "A": "Group 3A elements are an octet exception. They only need 6 valence electrons."
    },
    {
        "video_name": "3RDytvJYehY",
        "Q": "at around 1:45, is it incorrect to make a double bond between one of the fluorine atoms or is either way to do it correct?",
        "A": "Fluorine can only form one bond. It has 7 valence electrons (only needs one electron to complete its octet), and can t expand its octet because it is only in the second period."
    },
    {
        "video_name": "3RDytvJYehY",
        "Q": "I have a somewhat dumb question, but I have to ask: Why do we put lone pairs of electrons into double bonds, as he does at 7:17? Why not one electron from one Oxygen and the other from another Oxygen? Just curious, sorry if this strikes you as dumb.",
        "A": "We do this first to complete every atoms octets and then to minimise what is called formal charge. Electrons don t like being unpaired, they re highly reactive when unpaired. What you re suggesting would leave an unpaired electron on each oxygen right?"
    },
    {
        "video_name": "3RDytvJYehY",
        "Q": "At 3:02, he re-draws the Lewis Dot structure for Boron Trifluoride. Why? Is the first dot structure incorrect? I'm really confused now.",
        "A": "The first one isn t really incorrect, he just drew over it. He does draw the angles more correctly the second time though."
    },
    {
        "video_name": "3RDytvJYehY",
        "Q": "At about 7:20 Sal says that he is moving two electrons from the oxygen to form a double bond with the sulfer. Doesn't a double bond need electrons from both atoms, not just two electrons from one atom? Thanks,\nCalico",
        "A": "Try watching the videos on drawing dot structures."
    },
    {
        "video_name": "3RDytvJYehY",
        "Q": "At 7:45 in the video, why does the sulfur atom still have 2 valence electrons? It formed another bond with one of the oxygen atoms shouldn't it be gone?",
        "A": "S has six valence electrons. It used two electrons to bond with each of the two O atoms, so it still has two unbonded valence electrons."
    },
    {
        "video_name": "3RDytvJYehY",
        "Q": "Isn't boron trifluoride tetrahedral, not trigonal planar as the video states at 4:25?",
        "A": "No, BF3 is trigonal planar; however, if there was a lone pair on the central atom, it would be considered tetrahedral in terms of electron geometry. Check out the video titled VSEPR for 4 electron clouds for examples of tetrahedral molecules!"
    },
    {
        "video_name": "3RDytvJYehY",
        "Q": "at 9:26 i didnt understand why he is saying to leave out formal charges when he is working on VSEPR theory.and i didnt understand why only one oxygen is forming double bond while two double bonds can make the formal charges 'zero'......",
        "A": "I do believe that an SO2 molecule with two double bonds would also be correct. Formal charges would equal 0, and there would be a total of 18 valence electrons"
    },
    {
        "video_name": "3RDytvJYehY",
        "Q": "Why didn't you make two double bonds at 8:32? Wouldn't that structure create more stable structure because the formal charge of sulfur will be 0 and oxygen be 1?",
        "A": "If it was like this: O=S=O the Formal Charge would be +2 on the Sulfur, since it has just 4 electrons (it is sharing 8 electrons with 2 Oxygens, so it has just 4). The Oxygens would have a Formal Charge of zero, since they have [2 lone pairs] and [2 they share with Sulfur], So the OVERALL Formal Charge would be +2."
    },
    {
        "video_name": "jxptCXHLxKQ",
        "Q": "2:17 I did not understand why the ray bent left instead of right",
        "A": "Think of a car, the 2 left tires are driving on the road and the 2 right tires are driving in the mud. The 2 right tires have almost no grip because of the soft and slippery mud. The car bends almost instantly to the right because it has almost no resistance. The same thing is happening when you look at the bottom of the straw."
    },
    {
        "video_name": "jxptCXHLxKQ",
        "Q": "At 2:10, Sal says that the left side is faster than the right side. What about a ray that is coming perpendicular to the ground? Will it stay straight as it goes from water to air?",
        "A": "Yes, it will stay straight. Think of Snell s law: The angle of incidence will be 0 degrees. As you might know, the sine of 0 degrees is 0. So the second side of Snell s law would also have to be equal to 0. We know that the refraction index of the 2nd medium is not zero, and hence we are led to the conclusion that the angle of refraction has to be 0."
    },
    {
        "video_name": "jxptCXHLxKQ",
        "Q": "2:09 . left side of the band moves faster than the right side of the band. and so the band turns right.\nshouldnt the band turn left.. because the left side moves faster.?",
        "A": "Imagine you are in a car, and the left ( driver s side in US ) front wheel turned faster than the right ( passenger s side in US ) front wheel. This is the analogy Sal referred us to in a previous video. Do you see that this would cause a car to turn to the right? The left part of the car moves ahead faster than the right side, and as a result the car turns to the right. Hope that helps!"
    },
    {
        "video_name": "jxptCXHLxKQ",
        "Q": "At 4:19, the straw appears to be bend as well as laterally displaced. What is the reason?",
        "A": "It s all the same reason, it just depends on the angle from which you are viewing it."
    },
    {
        "video_name": "bhxo9Wwr4mA",
        "Q": "Can someone explain how he derived pKeq = pKa acidL - pKa acidR at 3:32? Thanks!",
        "A": "For simpler typing, let s use 1 for acid L and 2 for acid R. Then HA\u00e2\u0082\u0081 \u00e2\u0087\u008c H\u00e2\u0081\u00ba + A\u00e2\u0082\u0081\u00e2\u0081\u00bb; Ka\u00e2\u0082\u0081 H\u00e2\u0081\u00ba + A\u00e2\u0082\u0082\u00e2\u0081\u00bb \u00e2\u0087\u008c HA\u00e2\u0082\u0082; 1/Ka\u00e2\u0082\u0082 (when you reverse an equation, you invert its K) Add the two equations. HA\u00e2\u0082\u0081 + A\u00e2\u0082\u0082\u00e2\u0081\u00bb \u00e2\u0087\u008c HA\u00e2\u0082\u0082 + A\u00e2\u0082\u0081\u00e2\u0081\u00bb Keq = Ka\u00e2\u0082\u0081/Ka\u00e2\u0082\u0082 (when you add two equations, you multiply their K values) Take the negative logarithm of both sides of the equation. pKeq = pKa\u00e2\u0082\u0081 - pKa\u00e2\u0082\u0082 = pKa(acid L) \u00e2\u0080\u0093 pKa(acid R)"
    },
    {
        "video_name": "bhxo9Wwr4mA",
        "Q": "at 5:20 why is the carbanion a greater contributor to the enolate anion's resonance structure? I thought the more electronegative atom was most comfortable with the negative charge.",
        "A": "I think he said that the oxyanion was the greater contributor, but he chose not to draw it."
    },
    {
        "video_name": "bhxo9Wwr4mA",
        "Q": "At 5:45, it's mentioned that the bubbling out of solution of H2 gas causes the reaction to proceed towards the products. Am I correct in thinking this is due to our understanding of Le Chatelier's principle - as the abundance of one species decreases, the equilibrium will attempt to compensate for this by shifting towards the formation of that species (in our case H2 gas)?",
        "A": "You are correct."
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "At about 14:40, shouldn't the tRNA's codon be AUC, not AUG?",
        "A": "What s in the video is correct. The first codon in the mRNA sequence is UAC which matches up with AUG in the tRNA."
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "At 9:48, Sal meant that RNA has Uracil right not DNA?",
        "A": "yes, you are right."
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "at 9:46, why dosen't adenine just pair with thymine in the RNA? Why does it have to bring uracil into the picture? If there is some reason thymine cannot be used in the RNA, why just thymine and not other bases? wouldn't they have also been abandoned for the same reason?",
        "A": "Thymine is methylated uracil. Methyl group makes it more stable and reduces risk of mutation, therefore it is better for DNA. Stability is not important in RNA, as these molecules are short lived, so cells can use uracil to make RNA and don t have to bother with methylation,"
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "11:52 what does the ' U ' stand for ?",
        "A": "Uracyl (the base that replaces Thymine in RNA)"
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "In 7:06 what is deoxy in deoxyribonucleic acid",
        "A": "Doexy- refers that there is less oxygen than in a specified related compound. Ribos is a sugar and if you remove an oxygen in the second position you get deoxyribo."
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "at 14:00 , Where did the aa1 (amino acid 1) came from, and where/how was it formed ?",
        "A": "He just used Amino Acid 1 as an example of multiple codons coding for a molecule."
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "At 10:11 Sal says that Thymine bonds with Adenine, but isn't thymine replaced by Uracil? So shouldn't it be that Uracil binds Adenine?",
        "A": "Yes, Adenine binds with Thymine. But Thymine is also replaced by Uracil in RNA synthesis. Thymine is very similar to Uracil in the sense that the only difference is that it does not have a methyl group on one portion of the nitrogenous base that Thymine does. They are both still pyrimidines and both easily bond with Adenine in a cell."
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "at 8:47, how do you know which strand of DNA to replicate?",
        "A": "There are 2 strands of DNA- a coding strand and a template strand. The template strand is what is translated into RNA, as shown in this video. How do you know which one is the template strand? The DNA isn t just existing by itself, there are loads of markers and signs that tell the enzymes to Start translating here! The template strand will be marked by one of these signals, showing the polymerase enzymes where to go."
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "At 10:10 : Why is it that in the RNA Adenine pairs with uracil, but the Thymine pairs with Adenine still? Why isn't Thymine replaced with Uracil?",
        "A": "In RNA you have G, C, A and U. In DNA you have G, C, A and T. So In RNA the U is like the T and therefore the A of the RNA molecule will match with the Uracil."
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "At 3:10, where does the genetic material to form the replicated DNA strands come from? Like after it splits.",
        "A": "Forwarding the reply from tzfoni.itai on another similar question : The bases are synthesized in the cytoplasm through a rather complex metabolic pathway. They are then transferred into the nucleus where they will be used in the replication process."
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "At 10:11, just to clarify, thymine pairs with adenine, but adenine pairs with uracil? Shouldn't adenine always pair with thymine in that case?",
        "A": "Adenine pairs with thymine in DNA and with uracil in RNA. If you look at the chemical structures of thymine and uracil, you will see that they are almost identical, differing only in the presence of a methyl group, which is not in a position to interfere with the hydrogen bonding with adenine."
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "at 6:19 he says the stand can be sevral thousands to millions but can it be any shorter?",
        "A": "Yes. There are some tiny proteins out there with only twenty or so amino acids. You can calculate that the genes coding for these proteins are probably less than 100bp in length, if there are 3bp per amino acid ."
    },
    {
        "video_name": "6gUY5NoX1Lk",
        "Q": "at 9:04 what makes mRNA",
        "A": "RNA polymerase II makes mRNA."
    },
    {
        "video_name": "jmSWImPs6fQ",
        "Q": "At 10:25 you used the formula dx = vx(t) + 1/2at(sq), why cannot we use the usual formula, v = s/t where v is initial velocity and s is the dx and t = 2.47?",
        "A": "Try it. Can you do it? Why not?"
    },
    {
        "video_name": "jmSWImPs6fQ",
        "Q": "At 3:10 David says that the horizontal velocity in the x direction will always be 5m/s. How is the velocity not being affected due to the acceleration at y direction? As per what the image suggests is that acceleration and the horizontal motion creates a half parabola which finally will end as a straight vertical line, doesn't this mean that velocity at x axis decreases to zero?",
        "A": "In this case, motion in x and motion in y are completely independent. Why? Newton s First. Since the force of gravity is only in the y direction, their is no component of the force vector pointing downwards in the x-direction, so the object s x-velocity remains unchanged, because it is not acted upon by an outside force."
    },
    {
        "video_name": "jmSWImPs6fQ",
        "Q": "At 5:34, how is the acceleration (negative) -9.8m/s^2 ? Doesn't gravity act downwards, meaning, as the ball falls it would have a positive acceleration? Such as 9.8m/s&^2.",
        "A": "If you define up as positive, then acceleration due to gravity is -9.8 m/s^2. If you define down as positive, it s +9.8"
    },
    {
        "video_name": "jmSWImPs6fQ",
        "Q": "9:18 whre did he get that formula,? in the delta y formula is asking to elevate to 2 now doing the root he is decreasing, i dont catch it",
        "A": "It s not a formula. It s simple algebra. -30 = 0 + 1/2 * (-9.8) * t^2 -30 = 1/2 * (-9.8) * t^2 Multiply both sides of the equation by 2, -30 * 2 = (two divided by 2 results into 1) * (-9.8) * t^2 -60 = (-9.8) * t^2 Divide by -9.8 on both sides -60/-9.8 = t^2. And then take square root for t and solve."
    },
    {
        "video_name": "jmSWImPs6fQ",
        "Q": "At 5:29 is the acceleration always -9.8 no matter what?",
        "A": "The acceleration of a freely falling body near the surface of the earth is always 9.8 m/s^2."
    },
    {
        "video_name": "jmSWImPs6fQ",
        "Q": "I'm lost at how the math was done at (9:53), why is the -30 multiplied by 2?",
        "A": "It s simple algebra. -30 = 0 + 1/2 * (-9.8) * t^2 -30 = 1/2 * (-9.8) * t^2 Multiply both sides of the equation by 2, -30 * 2 = (two divided by 2 results into 1) * (-9.8) * t^2 -60 = (-9.8) * t^2 Divide by -9.8 on both sides -60/-9.8 = t^2. And then take square root for t and solve."
    },
    {
        "video_name": "jmSWImPs6fQ",
        "Q": "At 5:35, he says that acceleration is -9.8, but shouldn't it be positive as when you fall gravity acts in your favour?",
        "A": "If you define up as positive, then it is -9.8 If you define down as positive, then it is +9.8 There is no such concept in physics as in your favor"
    },
    {
        "video_name": "st_gQzP4QxY",
        "Q": "at 1:33 Hank was talking about animals breeding and said how the parents need to be able to produce a baby and that baby has to produce more. But, what about in early times (of humans)? We would often marry our brother or sister or cousin and that child would often have birth defects resulting in not being able to have kids of their own... If that was the point of a species then why did our instincts not warn us to not marry our family members? and would that offspring be considered sterile?",
        "A": "You are making some strange and unwarranted assumptions about how early humans mated and about the frequency of birth defects."
    },
    {
        "video_name": "st_gQzP4QxY",
        "Q": "@1:15 hank says we have got big brains so we are intelligent...but elephant , octopus, whale, etc have bigger brains ..?",
        "A": "Its not necessarily big brains that make an organism intelligent. It is how specialized that brain is. For example, the elephant brain you referred to is complex, an amazing work of creation, but it wasn t designed for language and other such super high intensity tasks such as reason. For instance, right now I m using both my language specialization but also my reasoning and logic centers. An elephant might be able to hit random keys on a keyboard, but it wouldn t make any sense to you."
    },
    {
        "video_name": "st_gQzP4QxY",
        "Q": "What does hank mean at 3:58 onwards ?",
        "A": "Well... Prezygotic isolation prevents the fertilization of eggs while postzygotic isolation prevents the formation of fertile offspring. Prezygotic mechanisms include habitat isolation, mating seasons, mechanical isolation, gamete isolation and behavioral isolation. Postzygotic mechanisms include hybrid inviability, hybrid sterility and hybrid breakdown."
    },
    {
        "video_name": "fCdT2z0JcU8",
        "Q": "At 8:24, where did you get the number 0.693 from?",
        "A": "It is part of the half-life constant that he found in the previous video: Half-life of a first-order reaction . He explains how to find the half-life constant in that video."
    },
    {
        "video_name": "fCdT2z0JcU8",
        "Q": "At 6:22, why don't you divide the two ln's? I thought you were supposed to divide when you subtract.",
        "A": "well, he could have written it as ln(0.01/0.05) but his method is more straightforward"
    },
    {
        "video_name": "H-qREcJqUhY",
        "Q": "At 12:43 when he gives the equation, what does BUN mean and why is it divided by 2.8?",
        "A": "At 8:03 he defines BUN."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "At 3:56, can we also write Hepta-2,4-diene?",
        "A": "Yes, that is a slight variation in the system of naming that is allowed."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "At 8:16, why isn't it 1,2,5 trimethyl-cyclo-hexene?",
        "A": "The numbering goes in the direction of the double bond. Therefore the 1 carbon is the one with the methyl group and the double bond, and the 2 carbon is the one counter clockwise to that, on the other end of the double bond."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "At 4:04, wouldn't it be hept-2-4-diene?",
        "A": "The correct name is hepta-2,4-diene. The base name for the main chain is hepta . But the a is dropped when it occurs before a vowel. Thus we get hept-2-ene instead if hepta-2-ene, but hepta-2,4-diene instead of hept-2,4-diene. The numbers are ignored. So in the first case we have a before e but in the second case we have a before d ."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "During the example at 5:30, is there another way to name the heptene group other than hep-2-ene? I was taught to name it as 2-heptene, but am not sure how it would be written with the 2-ethyl group as well. Perhaps \"2-ethylheptene\"?",
        "A": "If you prefer the number out front method, you would name this compound 2- methyl-2-heptene. That method can get somewhat confusing with larger and more substituted molecules, therefore IUPAC suggests embedding the number such as, 2-methylhep-2-ene. Also, It s a methyl group and not an ethyl since it contains only 1 carbon. Hope this is helpful!"
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "Around 7:30 when numbering the cyclohexene carbons: What if nothing was attached to either of the double bonded carbons, which one would you start numbering with?",
        "A": "... such that the number of the nearest group was lowest, so in fact it would be the farther one!"
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "At 5:56, why can't you just say 2-methylheptene instead of 2-methyl Hept-2-ene? It already indicates that the double bond and the methyl are located at carbon 2.",
        "A": "That 2 specifies where the methyl group is but not the double bond is. We need to make sure a name is completely unambiguous."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "At 6:00 2-methyl hept-2-ene why do we write 2-methyl first",
        "A": "by iupac systamatic rules we first indicate longest chain, then denote the position of the functional group in that way that it posses lowest possible number .at last in naming ,substituent whith position written first"
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "In the second example, 2:42 to 3:38, isn't it supposed to be Hept-2,4-diene instead of just Hept-2,4-ene?",
        "A": "ya"
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "At 3:53 is it hept-2,4-ene or hept-2,4-diene?",
        "A": "ohhh ok :)"
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "@4:00- cant we write Hept-2,4-diene,huh??",
        "A": "hepta-2,4-diene whenever you have a di, tri, poly etc., you must put a after your hept in this case. (so not hept-2,4-diene OR hept-2,4-ene) When you write it this way: 2,4-heptadiene, it should clarify."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "At 5:57,can we write 2-Methyheptene instead of 2-MethylHept-2-ene?",
        "A": "No, both 2 s are important. One indicates the location of the methyl substituent and the other indicates the location of the double bond, so both must be included."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "In the last example (6:06 to 8:27), wouldn't the compound be named 2,3,5 trimethylhexene instead of 1,4,6 trimethylhexene? Because the numbers should be as low as possible? The numbers in the first compound sum to 10, and to 11 in the second compound.",
        "A": "you have to count THROUGH the double bond. Amyseh is correct both because the numbers are lower AND the fact that you always count through the double bond."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "Why do we put the numbers in the middle of the word at 3:27?",
        "A": "Do we put the double bond in the inside of a chain[or cyclo] or outside?"
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "at 5:57 if the side will be at any other point except 2 say 7th so will we then have to mention the name of side chain in the formula",
        "A": "If the CH\u00e2\u0082\u0083 group were on C7, the compound would be oct-2-ene. If it were on any other carbon from C3 to C6, you would have to locate the group with a number e.g. 6-methylhept-2-ene."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "at 3:36 shouldn't this be hept-2,4-diene, not hept-2,4-ene?",
        "A": "Yeah...it should be hepta-2,4-diene...."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "at about 3:20 would the name be hept 2,4 di-ene?",
        "A": "It should be hepta-2,4-diene , you may see it as 2,4-heptadiene in some places too. Sal sometimes makes some mistakes."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "At 7:44, we numbered on the side of the double bond and not on that of the substituent. Can we number it the other way round too? And usually, how does numbering in cyclo-compounds work? Do we start the numbering ie put 1 at the point where there is a substituent in alkanes and where there is a double bond in alkenes?",
        "A": "You ve got to start the numbering giving the double/triple bonds more preference. But, if there are functional groups attached, then, start the numbering from there."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "At 5:50 with the example 2-methylhept-2-ene, is that the same as 2-methyl-2-heptene ?",
        "A": "They are both names for the same compound. 2-Methyl-2-heptene is an acceptable name (and the old IUPAC name), but the preferred IUPAC names now put the locating number as close as possible to the functional group. The preferred IUPAC name is now 2-methylhept-2-ene, because it puts the 2 as close as possible to the ene ."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "At 3:10 sal mentioned that there can be2 double bond . My question is that can there be more than 2 double bond?",
        "A": "Not in that molecule, but molecules can have many many double bonds, polyunsaturated fats are an example of this."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "At 6:31 wouldn't the backbone contain 8 carbons? It's the longest chain. In a cyclo do we take the ring even if there is a longer chain of carbons?",
        "A": "A ring will generally take precedence over a straight carbon chain. Review a table of priority when naming compounds."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "at 8:12, sal numbers the methyl groups in a counter-clockwise fashion, resulting in 1,4,6. My understanding was that you should always number substituent groups so that they have the lowest possible number values? so shouldn't it be done clockwise, resulting in 1,2,4-trimethyl cyclohexene?",
        "A": "I believe you must travel through the double bond when numbering."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "Why is the molecule @8:25 not named 2,3,5-trimethylcyclohexene?",
        "A": "The double bond takes priority in the name. Its carbons are automatically numbered 1 and 2. The alkene carbon with the methyl group is automatically number 1, so the other one is number 2. Then you continue numbering in the same direction until you come to the methyl groups at 4 and 6."
    },
    {
        "video_name": "KWv5PaoHwPA",
        "Q": "At 4:32 I've noticed that Sal wrote the compound as H3C-CH2-CH2-CH2-CH=C-CH3-CH3.\n\nIs it a convention to start with H3C and not CH3 because I've seen it many other books but could never figure it out.\nJust wanna know if there is a catch to it or it doesn't really matter?",
        "A": "Thats a good question actually. So, no, it doesn t exactly matter. It is apparently written this way for convenience. In the end, its a methyl molecule."
    },
    {
        "video_name": "S0AJZjt0sEg",
        "Q": "Was there an emergency calculator jump-cut at 5:46?",
        "A": "I think I looked down at my paper notes. It s hard to do mental math when the microphone is live. Usually I m able to hold the cursor still when I glance away, but this time it wiggled enough for you to see the stop/start."
    },
    {
        "video_name": "E34CftP455k",
        "Q": "how is it that the dot product doesnt have direction? (at 02:08) a vector into another vector is just another vector with a direction - ain't that right?!",
        "A": "Taking the length of the vector and you should get what the dot product provides."
    },
    {
        "video_name": "88hs5LcCoX4",
        "Q": "2:47 why does Sal say 'at this point since we're\nchanging our direction, we're not going to be\nstationary relative to the ether'\nWhat is the frame of reference for the ether wind, is the sun moving with the galaxy and the galaxy is moving.",
        "A": "The idea is that the ether, if it existed, would provide a universal frame of reference. Since the earth changes direction constantly as it orbits the sun, the direction of the ether wind , as detected from earth, would have to change."
    },
    {
        "video_name": "88hs5LcCoX4",
        "Q": "At 1:40, Sal said that it wasn't the actual milky way. Then are all the images of the milky way, andromeda, black holes, nebula's etc ... are these all fakes? How do we know that the galaxies are spiral and the black holes are spherical then?",
        "A": "we have lots of real pictures of galaxies other than the milky way. Many of them are spiral. It s not possible to take a picture of a black hole. What shape do you think it could be other than a sphere?"
    },
    {
        "video_name": "88hs5LcCoX4",
        "Q": "At 5:19 when Sal considers a point in the luminiferous ether, does he mean a point in the spiralling Milky Way galaxy? If that were the case the ether wind's direction relative to that point would be opposite to what he said.",
        "A": "in relativity"
    },
    {
        "video_name": "HwkEQfsJenk",
        "Q": "My dad said that CH3COO- inonically bonded with a Na+ is not a salt - then what is CH3COO-?\nI thought Jay(or whoever the teacher is) in 2:35 said\n\"Let's compare this situation to the SALT that's formed from a weak acid and a strong base\"..........",
        "A": "CH3COO- is an anion, forming an ionic bond with Na+. It is a salt (Sodiom acetate.) CH3COO- alone, is just a radical."
    },
    {
        "video_name": "HwkEQfsJenk",
        "Q": "At 4:00, is there any specific reason that acetate ion would react with water and Na ion doesn't?",
        "A": "The Na ion is content being Na+... it does not want to gain an electron because it has a very low electronegativity. Na+ is considerably more stable than solid Na, so it is extremely energetically favorable for the Na+ to remain stable. This makes more sense after redox reactions."
    },
    {
        "video_name": "HwkEQfsJenk",
        "Q": "At about 4:44, after the acetate anion reacts with water and \"produces\" OH- why won't the OH- react back with the free Na+ ion and go back to the beginning of having an acid and base, then split again and so forth?",
        "A": "NaOH is a strong base, so it is unlikely to re-associate in solution. It tends to dissociate in any aqueous solution."
    },
    {
        "video_name": "M7PnxSQedkM",
        "Q": "At 2:40, how do we know its ionic?",
        "A": "Ionic bond is always formed between Metals and Non-metals"
    },
    {
        "video_name": "M7PnxSQedkM",
        "Q": "At 2:24 what is the reducing agent?",
        "A": "A reducing agent gives the electrons to other chemical specie and get oxidized."
    },
    {
        "video_name": "M7PnxSQedkM",
        "Q": "At 1:30, doesn't Sal mean group 6A instead of group 7?",
        "A": "Yes, problem is already fixed"
    },
    {
        "video_name": "M7PnxSQedkM",
        "Q": "At 1:30 why does oxygen always have a two?",
        "A": "Oxygen is a diatomic molecule meaning that it always will exist with pairs. Regarding the structure, each oxygen atom has 2 lone pairs (4 electrons) and 2 bonding pairs i.d. a double bond."
    },
    {
        "video_name": "M7PnxSQedkM",
        "Q": "at 2:14 why is it mg plus 2",
        "A": "And also because it has to Lose 2 (typical metal property) to gain stability :)"
    },
    {
        "video_name": "CnXuSCaCNBo",
        "Q": "At 0:50, what are antibiotics? Do they fight viruses or bacteria?",
        "A": "Antibiotics fight bacteria. To help you remember: the word biotic means living. Bacteria are living, whereas viruses are not."
    },
    {
        "video_name": "CnXuSCaCNBo",
        "Q": "5:45\n\nWhat is a spinal tap and how is it preformed?",
        "A": "A spinal tap is a procedure used to take a sample of cerebrospinal fluid for testing. The sample is obtained via a needle inserted into the spinal canal below the end of the spinal cord."
    },
    {
        "video_name": "CnXuSCaCNBo",
        "Q": "3:03\nWhat does the doctor mean by abnormal movements ?",
        "A": "Abnormal movements are a result of ICP (increased intarcrainial pressure) and could include; Nuchal rigidity- stiff neck Nystagmus, abnormal eye movements Kerning s sign\u00e2\u0080\u0094severe stiffness of the hamstrings causes an inability to straighten the leg when hip is flexed 90 degrees Brudzinski s sign- severe neck stiffness causes patients hips and knees to flex when the neck is flexed"
    },
    {
        "video_name": "TyZODv-UqvU",
        "Q": "At 8:28, Hank mentions that an enzyme breaks the covalent bond of an ATP. It is shown that one phosphate is broken off. What happens to the other adenosine group and 2 phosphates?",
        "A": "A few things can happen: The ADP (as it is called at this stage) can be reenergized through a few different paths, most notably through the Kreb s cycle. ADP in a few cases can also be broken down again, removing another phosphate group and leaving Adenosine Monophosphate. This is far less common than ATP -> ADP because it is a stronger bond."
    },
    {
        "video_name": "TyZODv-UqvU",
        "Q": "In the experiment described at 3:19, why is the salt water diffusing into the pure water? Shouldn't water move INTO the casing as it is a lower concentration inside than outside?",
        "A": "The saltwater is inside the sausage-skin, the lower concentration is outside. And the water isn t moving, the salt is."
    },
    {
        "video_name": "TyZODv-UqvU",
        "Q": "At 6:33, Why sodium and potassium don't pass across the membrane to match the concentrations by simple difusion?",
        "A": "Because certain molecules of substances are bigger than the pores in membranes, so they can t go through."
    },
    {
        "video_name": "TyZODv-UqvU",
        "Q": "At 3:24 isn't cellulose strong enough to keep osmosis from happening?",
        "A": "It s strong enough to keep the cell from bursting because too much water is entering through osmosis. Water is still able to diffuse through the cellulose barrier."
    },
    {
        "video_name": "TyZODv-UqvU",
        "Q": "at around 4:44 he said that there were hydrophilic and hydrophobic parts in a membrane, i was just wondering, what would happen if a water molecule was to get in contact with the hydrophobic parts of the membrane?",
        "A": "This really wouldn t happen but is a good question. Their is a transport protein to transport water from the outside of the cell to the inside and the protein is called aquaporin."
    },
    {
        "video_name": "TyZODv-UqvU",
        "Q": "At 4:58 the aquaporins let water through the cell membrane, can it also lets viruses into the cell?",
        "A": "No, virus particle is much bigger than molecules of water (H2O). Virus is attached to the cell through special receptor(s) and inject its DNA or RNA into the cell."
    },
    {
        "video_name": "TyZODv-UqvU",
        "Q": "At 8:50, you mention the Na ions waiting outside to get inside the cell, but in the demonstration prior, it was said and shown that Na ions are inside the cell and through the NaK pump, it leaves the cells, with K coming into the cell. I'm confused.",
        "A": "Na+ is mostly outside the cell and wants to get in. If a channel opens up that allows Na+ to flow into the cell, Na+ will flow in (this is how depolarization works). But once that Na+ has gotten into the cell, the cell needs to expend energy to pump the Na+ back out. This is done by the Na+/K+ ATPase, which uses ATP to pump Na+ out of the cell and K+ into the cell. This re-establishes the gradient so the cell can remain active and be depolarized again."
    },
    {
        "video_name": "TyZODv-UqvU",
        "Q": "At 0:19 he said selectively permeable. What does permeable mean?",
        "A": "In this context, permeable means that substances can pass through a membrane. A selectively permeable membrane allows only certain substances or certain kinds of substances to pass through. For example a selectively permeable membrane might allow water and any salt dissolved in the water to pass through, but will not allow glucose to pass through. Or a selectively permeable membrane might allow water and very specific kinds of proteins to pass through but not other kinds of proteins or salts."
    },
    {
        "video_name": "TyZODv-UqvU",
        "Q": "Is pinocytosis a type of endocytosis, or are they different ? @10:35\nThanks",
        "A": "Yes, endocytosis is any process that involves cells using energy to absorb molecules, meaning both pinocytosis (absorbing small molecules) or phagocytosis (absorbing large molecules) are subcategories of endocytosis."
    },
    {
        "video_name": "TyZODv-UqvU",
        "Q": "At 5:50 ATP what does it do exactly?",
        "A": "ATP is used to power the cell."
    },
    {
        "video_name": "TyZODv-UqvU",
        "Q": "At around 3:20 you show how salt water diffuses from a sausage casing, where could I find a sausage casing like that? Or what else could I use that would act as a membrane?",
        "A": "You can get something similar called dialysis tubing, that certain substances to diffuse through and trap it, like salt water. You can buy it online."
    },
    {
        "video_name": "TyZODv-UqvU",
        "Q": "How did the white blood cell at 10:30 know that there was a dangerous bacteria outside?",
        "A": "Basically it is a random process: they bump into each other. When near enough, the white blood cell detects the other cell with chemical receptors."
    },
    {
        "video_name": "qpP8D7yQV50",
        "Q": "6:25 Why does that carbon have a negative charge? It has 3 bonds (carbon, carbon, hydrogen) and a lone pair of electrons, so shouldn't it be neutral?",
        "A": "formal charge=#of valence electrons - #of lone pair electrons - #of bonds for carbon=4 - 2 - 3 = -1"
    },
    {
        "video_name": "qpP8D7yQV50",
        "Q": "at 3:42, the two resonance structures ARE NOT enantiomers, right? if you flip one across the mirror plane, you get the other structure but they're just rotations of each other right?",
        "A": "Stuff like chirality and enantiomers refers to molecules where the atoms are rearranged differently... this video is on Resonance structures where the atoms are all in the same place, just the bonds change I think :)"
    },
    {
        "video_name": "qpP8D7yQV50",
        "Q": "at 3:42, the two resonance structures ARE NOT enantiomers, right? if you flip one across the mirror plane, you get the other structure but they're just rotations of each other right?",
        "A": "The two resonance structures are not enantiomers because they do not exist. They are drawings that we use to explain the properties of compounds. The actual structure is a resonance hybrid of the two structures."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "At 9:40, Sal mentioned that magnetic fields are generated by the spin of the electrons and its motion around the protons. How does this work? How is the spin of the electrons and its motion around protons supposed to generate magnetic fields?",
        "A": "What do you mean, how do they do it ? This is like asking how does the earth exert gravity, or how does a positive charge attract an electric charge. THat s what they do. These are fundamental properties of charges. We don t have any deeper explanation."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "At 7:25 Sal says magnetism always comes in the form of a dipole, but after watching this video I read a wikipedia article that said the moon does not currently have a dipolar magnetic field. Would this be because the moon's magnetic field is so weak in comparison to ours? Or something else?",
        "A": "Magnetism always comes in dipoles, but that does not require every planet and moon to have any magnetic field at all. It just says that if they do, they will have two poles, not one. (Scientists recently created a magnetic monopole in a laboratory, so it s no longer accurate to say magnets are always dipoles - but you will never encounter a monopole)"
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "In 9:21, he brakes the magnet into half and still it is a dipole that means, we can still fix it while when we cut it its S first and N the next, and the same thing again, so they will attract each other and come back to its position isn't it ? And please do like my question it makes me happy :)",
        "A": "If I understand what your asking, yeah. If cut a magnet in half, it ll want to pull back together. I ve seen some magic tricks that take advantage of this, making it look like they magically fixed their prop magnet stick after they cut it."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "At 6:49, They said that two North facing magnets facing each repel against each other and one North facing and one south facing magnets attract each other . Why is it so?",
        "A": "There s no way to answer a why question like this. This is what magnets due. Just like masses attract masses and we call that attraction gravity. Unlike charges attract and like charges repel, and we call that the electric force. Like poles repel and unlike poles attract, and we call that the magnetic force."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "at 10:14 Sal says a magnet is created when the electrons line up. how were lodestones' electrons lined up in the first place?",
        "A": "Lodestones were igneous rocks, which means that they were originally lava. When the lava came up through a volcano, the rock took a significant time to cool. In this time, the Earth s magnetic field automatically aligned the domains, or atoms, of the lava. Then, when the rock cooled, the atoms were set in place, making the rock magnetic. This is the process that factories use to make bar magnets today."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "In 08:20 why when a magnet is broken, its parts change into other magnets with two poles",
        "A": "The reason a magnet is a magnet is because it s atoms are all lined up in one direction, so their electromagnetic fields reinforce one another rather than cancel out. When you break the magnet in two, you still have two pieces in which all the atoms are lined up in the correct way, so you still have 2 magnets."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "@9:00 Did I understand Sal correctly? Did he say that electrons themselves have a N/S Dipole? Does this hold true for Protons? I suppose neutrons would have no charge so therefore no pole?",
        "A": "Yes, electrons have magnetic poles and so do protons :) It has to do with a property called spin ."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "At 8:16 Sal Says if we cut the magnett into 2 halfs.They Form 2 new magnets with dipoles. Why Does that happen?",
        "A": "Because the field is directional, think of it like this arrow S-------->N, if you break it in half, you have 2 arrows, that both have direction as well, S---->N S---->N im not sure if it travels from north to south, or south to north, but the idea is the same either way."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "Why is it called a dipole, starting at 7:28 , if it means there are 2 poles? Isn't the prefix meaning 2 \"Bi-\"?",
        "A": "di- is another prefix meaning 2. bi- is from Latin, meaning twice , while di- is from Greek, meaning two ."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "At 4:37, how is the earth a huge magnet?!",
        "A": "It has molten iron in and around its core, which is heated to high temperatures. The earth is like a big magnet because the electric currents that circulate the earth s metallic core create a magnetic field. The circulating electric current creates a dynamo effect like a generator hence the earth s magnetic field."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "At 9:12 Sal said that the magnetic field is generated by the electrons spinning within it. Aren't magnets made up of domains and the arrangement of domains makes something magnetic?",
        "A": "yes, and those two statements are not contradictory magnets are not the same thing as magnetic fields, although magnets HAVE magnetic fields."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "Why are there always two poles of a magnet in the first place? 2:17\nWhy does breaking a magnet in two always create more dipoles - why doesn't it just break into a north and south?",
        "A": "What makes a magnet a magnet is that the electrons in its atoms are aligned and spinning in the same direction. The spin of the electron creates a magnetic field with a north and a south pole. When you break the magnet in half, each piece still has its electrons lined up, and each of those electrons are creating the same north and south that they were generating before."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "So, at about 7:45, Sal introduces the magnetic dipole. What's the difference between a magnetic and electric dipole?",
        "A": "An electric dipole has more positive charges on one side and more negative charges on the other. A magnetic dipole comes from electric currents and not asymmetry of charges."
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "At 10:12 how do you align the spins and the roations of the electrons in order to magnetize a material?",
        "A": "You can put the material in a strong magnetic field, if it is a material that can be magnetized. You can magnetize and iron nail, or even a needle, by stroking it with a magnet in the same direction a few times. Try it!"
    },
    {
        "video_name": "8Y4JSp5U82I",
        "Q": "At 4:10, Sal mentioned something about the South magnetic pole changing positions. Why is that so?\n\nThanks in advance. :)",
        "A": "Every now and then, the North and South poles of Earth switch because the outer core of the Earth is made of liquid nickel and iron, while the inner core (the very center) is made of solid iron and nickel. Because the electrons in the liquid move around with the liquid, the whole core acts as a giant electromagnet. However, this flow sometimes reverses, which causes the North and South Pole to switch place. This is predicted to happen soon (but soon is relative)."
    },
    {
        "video_name": "BTWLwoaNeBA",
        "Q": "At around 2:36, what would happen if you hit the molecules slowly enough, that the bonds wont break? Wont it form a transverse wave in the water?",
        "A": "If you didn t hit the molecules hard enough to break bonds it is not going to matter what type of wave you are looking for because you will never be able to detect it. In theory you are correct but in reality you are not going to transfer any real energy."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 3:41, Sal says the Earth's diameter is 12,700 km long. The Earth's diameter is actually 24,000 miles long.",
        "A": "The diameter of the earth is about 12,700 km. The circumference of the earth is about 40,000 km. That is the difference. Circumference = Pi x diameter Diameter = Circumference / Pi"
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 6:39 he mentions that the nearest star is 4.2 light years away. How much distance can light cover in a year?",
        "A": "Light travels about 186,000 miles per second or exactly 299,792,458 meters/second, in a vacuum. Using simple arithmetic, it should be simple enough to solve. Since we know there are 60 seconds in a minute, 60 minutes in a hour, 24 hours in a day, and a year has 365 days (365.24). I hope this helps a little."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 4:30, can't three earths fit into the red spot? Not 1?",
        "A": "The scale he made isn t exact, it is just to give you an idea of how big it is. Other than that, I really haven t found a reliable source for this or more than one sources that agree on this. In some magazines/websites it says that 2 earths can fit in the red spot, some say three, whereas some say one. I m not really sure about that."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 06:43, why do you think sal went from powers of ten to numbers that were not powers of ten",
        "A": "Because he wanted to get to the point on how large the universe really is."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 6:42, What is the measure of 1AU i.e 1 Astronomical Unit?",
        "A": "An AU is the average distance the earth is from the Sun which is about 149,597,871 kilometers"
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "4:06 is this really how many earths fit into jupiter or just an approximation? I heard somewhere that 100 earths would fit in Jupiter.",
        "A": "It s just an approximation."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "What is au standing for it is visible at 9:00",
        "A": "Astronomical Unit - the average distance from the Earth to the Sun"
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 9:02, Sal says that we can only observe light that left its source 13.7 billion years ago, because that's how old the universe is. Then he says the observable universe is about 93 billions light years across, and the reason that its bigger than 13.7 billion is because the light has been traveling away from us. I'm not sure I understand, how can the universe be 13.7 billion years old and 93 billion light years across without things having been able to travel faster than the speed of light?",
        "A": "Nothing can travel through space at speeds greater than light, but space itself is unconstrained in how fast it can expand. The expansion of the universe accounts for the difference between 93 and 13.7. A photon that started its journey toward us 13.7 billion years ago originated at a point that is now 93/2 billion light years away."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "how do u calculate light years at 10:00",
        "A": "Per Wikipedia: a light-year is the distance that light travels in vacuum in one Julian year (365.25 days). It is about 9.5 quadrillion meters or 5.9 trillion miles."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "In 3:00 the side note in the bottom-right corner said that the Earth had a diameter of 7,900 miles. If human could travel just as easily through terrain as they do through air, how long would it take for one to be sucked into the core? Can someone give me a formula?",
        "A": "no , first of all, we cant go beyond a few kilometres into the crust of the earth. Then even if you go then there is unimanagable heat"
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 1:33 I thought the golden gate bridge is 4200 feet. A mile is 5280.",
        "A": "It was an approximation to make it easier to imagine. it s close to a mile."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "what is even the observable universe? and at 9:12 how could the universe be as large as 93 billion light years if the universe is only 13.8 billion years old?you know there is nothing faster than speed of light. can anyone explain it in more details?",
        "A": "If space were static, then the observable universe would have to have a maximum radius of 13.8 billion years. But space has been expanding. Light that is arriving here after 13.8 billion years came from a point that we estimate is about 46 billion light years away now."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 8:57 he shows the picture of the galaxy and there it is shown that the sun's orbit. Does the sun really orbit? And if it does then what does it orbit?",
        "A": "The sun doesn t orbit, the planets orbit the sun"
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At around 7:40, 7:50, he states that the Milky Way Galaxy is mostly empty space. Does that mean it is COMPLETELY empty? Are there any particles there? I don't mean the Milky Way Galaxy itself, I'm talking about the distances between interstellar space.",
        "A": "Yes, he does mean completely empty - in some regions the particle density is around 5 protons/cubic meter."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "at around 4:30, he drew an earth on Jupiter about the size of the red spot(the storm). But in reality, you could fit all for of the rocky migets into it. If we view earth form that size on Jupiter, it is not nearly 10 of Jupiter",
        "A": "The great red spot is the size of 2 Earths"
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 8:53, why does he say it just becomes playing with numbers and mathematics?",
        "A": "Because it becomes impossible to imagine how large these things are. Things like solar systems, galaxies, and the universe leave the realm of intuition and we can only really describe them with numbers."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 9:11, Why is the age of the universe getting older and older? Could they be wrong?",
        "A": "At 9:11 he is talking about the age of the universe and the size, he is not really talking about it getting older. The age of the universe is increasing as time passes just like your age increases as time passes. They could be wrong but at this point the age of the universe is based on when the big bang occurred. Currently the theories that match observed data indicate that the universe was created in the big bang and that it occurred about 13.78 billion years ago."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 7:44, when you say a vast majority of space is empty, I thought that that \"empty\" space was discovered to be dark matter and energy. Was I misinformed, or is there just way more space between dark matter and energy particles compared to the 83% of it that makes up space?",
        "A": "I think Sal meant empty to be empty of stars, planets, or any kind of visible matter. Technically, empty space really isn t empty. There is dark matter and energy particles, as well as virtual particles."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 4:23 why would the Earth get destroyed if it sits right next to Jupiter?",
        "A": "If we were really close to Jupiter, it s gravity would put us on a collision course and Earth and Jupiter would merge. Destroying Earth in the process. Also Jupiter s radiation would fry any organic material on Earth long before we got there."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "I read somewhere that the Hubble telescope was able to see to the very edge of the universe. But here Sal says that the universe may be infinite at 10:25. I'm confused...",
        "A": "Hubble has been able to see close to the edge of the OBSERVABLE universe (not all the way to that edge, though). The observable universe is defined by the region from which light would have had time to reach us by now. The observable universe is about 90 billion light years in diameter. The total universe is larger than that, and probably infinite although we can t say for sure."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "\"we can only observe light that started leaving from it's source 13.7 billion years ago.\" - 9:00. Does that mean that we cannot see any star that was formed after 13.7 billion years ago? So if we could actually see every single star including new ones, would the sky look more flooded with stars?",
        "A": "We cannot see any object from which light would have taken more than 13.7 billion years to get here. There hasn t been enough time yet for the light to get here."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 06:39, Sal said the nearest star is 4.2 lightyears away. Isn't the sun the nearest star?",
        "A": "Technically you are correct but usually when you say the closet star you exclude the sun. Just like if I were to ask you who is the closest person to you, technically it is you but that is not the intent of the question."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 10:26, how can the universe be infinite?",
        "A": "Think of an ant, crawling on the surface of a beachball. The ant can crawl for an infinite amount of time and never reach the end of the beachball. Same with us, hurtling through space. We may travel forever through our universe and we will never reach the end ."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 9:30 if there are theories of how large the universe is, does that mean there are also theories of what shape the universe is? If so what would that be and why?",
        "A": "The universe is not a definitive shape because it it constantly expanding, due to its subtle redshift. However, it is likely to be a disk-or-circle-like shape."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At about 3:08 in the video he discusses multiplying the section of a continent by 10 to reach about the diameter of Earth. This is confusing me a bit, though. What exactly is he multiplying by 10 to reach that distance? The miles across the picture? The area?",
        "A": "According to the video at 3:22, he is multiplying the miles across the picture."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At roughly 9:42, he said we'd only receive electromagnetic radiation. How can we be sure that the radiation would not be disturbed/dampened on it's way here? For one things dust bends and obstructs light, so how can we be sure of that?",
        "A": "what do ou mean by disturbed/dampened? Lots of stuff happens to radiation as it travels through space. Its path gets bent due to gravity, it interacts with dust and gas..."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 6:13 Sal wrote 1 Au underneath the tiny sun and earth. What does Au mean?",
        "A": "Astronomical unit."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 1:40, when Sal says that that 10 golden gate bridges would be the size of a large city, is he referring to the length?",
        "A": "Yes. When Sal is talking about whenever you multiply something by 10 to get to the size of something else, he is talking about length."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "Starting at 0:22 Sal shares that he is 5'9'', which is roughly approximately around 6 feet or around roughly - without going into the details of the math - around 2 meters ;o)",
        "A": "I am sure Sal would appreciate your two rough approximations since they turn him from 5 9 into roughly 6 7 ! That s going from a point guard to a power forward! And if we approximate the 6 7 which is 79 as roughly 80 , we can get him up to 6 8 ! This is why one needs to be careful with approximations especially if they are all to the high side (which is what you did) or low side. It is usually safer to round off numbers on the high side about as often as to the low side in order to avoid this."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "at 00:58 Sal said that if you were to stand the four wheelers and times that by 10 you will get a tall skyscraper is it exactly ten times or more??",
        "A": "it s rounding up to the CLOSEST amount."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 8:12 how long is an AU ?",
        "A": "An AU is defined as the distance between the Earth and the sun, or about 149,600,000 kilometers"
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "So scale-wise, at around 4:47 a model of Earth was drawn compared to Jupiter & it was about the size of the Great Red Spot. So is that a good measure of scale to use when comparing (namely, was that drawing to scale?)",
        "A": "No, that wasn t drawn to scale, ~3 Earths could fit in the Great Red Spot."
    },
    {
        "video_name": "5FEjrStgcF8",
        "Q": "At 6:13 Sal wrote 1 Au underneath the tiny sun and earth. What does Au mean?",
        "A": "AU stands for 1 astronomical unit which is the distance between the Sun and the Earth. The distance of other planets from the Sun can be measured in AU which gives a helpful comparison between distance of two objects compared to the distance between the Sun and Earth."
    },
    {
        "video_name": "3zt7DWUiaDE",
        "Q": "Does more current mean faster moving electrons or is it more magnitude of electrons?? David mentions at 3:45 that the electrons can't go faster because the current must be same",
        "A": "More current can mean the the electrons are moving faster or that there are more electrons. Current is the amount of charge that passes a particular point in a given amount of time."
    },
    {
        "video_name": "a19T5CX2b-g",
        "Q": "How much blood can safely be drawn from the body at a given time?\n\nI ask because in the situation at 1:30, it's three pints of blood (which I think is just over a litre?)",
        "A": "The average person can withstand a loss of 10-15% of their total blood volume with no clinical symptoms. The specific amount depends on the persons body weight, height and sex. The average adult male has 5-6 liters, while the average adult female has 4-5. When donating blood they take a little less than a pint of blood (about half of a liter) and don t allow people under 110 pounds to donate since this amount would be over 15% of their total blood volume."
    },
    {
        "video_name": "QP8ImP6NCk8",
        "Q": "At 7:40 Rishi says that when CO2 binds to Hb it makes a H+,\nHow is that H+ made when CO2 binds to Hb?",
        "A": "when the carbon dioxide binds to the Hb (hemoglobin), - snatches the oxygen from Hb then that makes H+. co2+Hb02 (switching sign)Hb-coo-+H+ +O2"
    },
    {
        "video_name": "QP8ImP6NCk8",
        "Q": "At 7:19, how does CO2 and HbO2 yield HbCO2, O2, and H+? Where did the proton come from in that equation?",
        "A": "The extra proton is a result of the Bohr effect. There is a video with a more in-depth explanation further in the course."
    },
    {
        "video_name": "QP8ImP6NCk8",
        "Q": "At 14:32, you mention how the amount of CO2 in the alveolus is very low and I am curious to where the CO2 we inhale with the O2 from the outside air goes. Does the CO2 get absorbed somewhere else as air travels to the alveolus? If so where?",
        "A": "The CO2 does not get absorbed into the blood. Main reason being is that the partial pressure of CO2 in blood is higher than the air in the aveoli. This leads to CO2 diffusing out. The opposite is true for O2 so O2 is diffused in and binds to the hemoglobin."
    },
    {
        "video_name": "QP8ImP6NCk8",
        "Q": "At 6:03, does the water affect the oxyhemoglobin because there is oxygen in water molecules?",
        "A": "I don t think so, but I m not sure"
    },
    {
        "video_name": "QP8ImP6NCk8",
        "Q": "So according to what he says at 6:21 , when he calls that compound 'bicarb', if hydrogen is added to some thing , that thing will be called a bi-'so and so' compound, right?",
        "A": "HCO3- + H+ --> H2CO3, which we call carbonic acid. If there s enough protons around in the body for carbonic acid to form, the H2CO3 is quickly converted into CO2 + H2O with the help of an enzyme called carbonic anhydrase."
    },
    {
        "video_name": "xp6Zj24h8uA",
        "Q": "At 6:14.. If malate is stored at night.. How can C4 photosynthesis continue to occur at night if the malate is not converted back to PEP until the next day?",
        "A": "It doesn t. The malate is stored until day, when C4 photosynthesis occurs. The plant won t have to open its stomata since it already has carbon in the form of malate taken in during the night."
    },
    {
        "video_name": "xp6Zj24h8uA",
        "Q": "0:04 - isn't it Carbon from CO2 that becomes fixed? or is CO2 actually fixed?",
        "A": "to answer Kyra, yes and yes. the O2 from the CO2 is released as O2 from the plant AND some is used in the carbohydrates."
    },
    {
        "video_name": "xp6Zj24h8uA",
        "Q": "At 4:50 , how can plants keep their stomata open by themselves? What causes them to keep their stomata open?",
        "A": "Guard cells on the rim of the stomata absorb water to expand and close the hole, and release the water to open the hole."
    },
    {
        "video_name": "xp6Zj24h8uA",
        "Q": "at 5:42, Sal mentions malate and that we learned about it in the last video. I wasn't quite sure what he ment in the last video either. Could someone explain to me what exactly malate is? Please and thank you!",
        "A": "Malate is an intermediary in the Krebs cycle (for respiration) and is a source of CO2 in the Calvin cycle (for photosynthesis). It s also sometimes called Malic acid."
    },
    {
        "video_name": "xp6Zj24h8uA",
        "Q": "6:07 doesn't CO2 assimulation and malate formation actually occur in cytoplasm, not in the chloroplaste?",
        "A": "YOU are absolutely right ! mannazothala .Malate is stored in vacuoles of the mesophyll plant cell. The malate forms in cytoplasm of the mesophyll cell."
    },
    {
        "video_name": "ikFUv-gdNLQ",
        "Q": "How do positively charged ions carry a signal? At 9:22 wouldn't the signal be diluted at the nodes of Ranvier when it just gets flooded with more ions? Is the signal shared?",
        "A": "The signal isn t a horizontal transmission of ions at all, but a wave. There is a very slight time lag (on the order of milliseconds) between the ion channels opening at one node versus the next, and these are all dependent on the membrane potential upstream. If the signal were left to diffusion alone, the ions wouldn t travel more than 1/100th down the length of an axon. But because the ion channels open/close in response to local voltages, the signal propagates very quickly."
    },
    {
        "video_name": "ikFUv-gdNLQ",
        "Q": "@ 7:25, Sal explains that we have gaps in our myelin sheath so that we could \"boost\" the signal with voltage gated channels, so if we intake extra calcium, would it be that this would boost the signal even more?",
        "A": "No, the electrolytes surrounding the neurons are highly regulated and quite difficult to change with diet alone. But if somehow you have excess calcium in your body (hypercalcemia) you will have a myriad of problems - google moans, stones, groans, bones ."
    },
    {
        "video_name": "ikFUv-gdNLQ",
        "Q": "what is the nodes of ranvier 8:48",
        "A": "it is the area between the myelin sheath, it keeps energy in a speeds up the neurons work."
    },
    {
        "video_name": "3BBqL_F9fxQ",
        "Q": "at 7:00 wouldnt the ice break the dam anyway so it looked to me either way the fish would die?",
        "A": "the ice would have to be an extremely low temperature to pierce the dam, freeze the water in the dam, then expand like frozen water does. Like temperatures not found on earth. This is just suggesting the concrete in the dam is at least 6 feet wide (2 meters). which it was probably thicker."
    },
    {
        "video_name": "hTTDI49LhjE",
        "Q": "at 2:23 where is the radius 3 marked? is it in the x-axis or y-axis?",
        "A": "It s marked in every direction except in the negative y direction (because he s only drawing the top half of the graph). Since the center is at the origin, Sal put a radius at the points (-3,0), (0,3), and (3,0)."
    },
    {
        "video_name": "CXCtqBlEZ7g",
        "Q": "I still don't see how, at 3:20, dx represents an infinitely small x. If the derivative is the rate of change, shouldn't dx in the equation represent the rate of change of x?",
        "A": "\u00f0\u009d\u0091\u0091\u00f0\u009d\u0091\u00a5 is not the derivative of \u00f0\u009d\u0091\u0093. It is, as Sal says, an infinitely small change in \u00f0\u009d\u0091\u00a5. The derivative of \u00f0\u009d\u0091\u0093 is (\u00f0\u009d\u0091\u0091\u00f0\u009d\u0091\u00a6)/(\u00f0\u009d\u0091\u0091\u00f0\u009d\u0091\u00a5)."
    },
    {
        "video_name": "J9BWNiOSGlc",
        "Q": "@ 3:17 you explain that \"whenever we push something, it's just the electrons in your hand or the electromagnetic clouds in your hand...\" I was just wondering if that statement applies to everything we touch and hold, like if we're holding/grasping a dumbbell, for example?",
        "A": "Yes, what we think of a touching is basically the interaction of the electromagnetic fields of the electrons in whatever is touching."
    },
    {
        "video_name": "G4CgOF4ccXk",
        "Q": "At 2:07, we should consider the time taken as dt 'coz the velocity changes as soon as the cross sectional area changes but for dt time we can assume that to be constant.",
        "A": "He is simplifying the problem. He is considering the inlet velocity to be constant over the time of interest."
    },
    {
        "video_name": "G4CgOF4ccXk",
        "Q": "Can anybody please explain why at 07:57 Sal ignored time say it is the same at the beginning and at the end of pipe. I mean that the size of the pipe increases and can it affect the time for that volume of water to flow at the outlet?",
        "A": "Time can be ignored because as the size of the pipe decreases the speed of fluid will increase, because the pipe won t expand to hold more liquid in it. So as you add extra liquid to the pipe but the same amount of liquid is going out at the same time, so time is irrelevant. Let s say you add 2 liters in 3s, then 2 liters in 3s leaves. 5 liters in 3s, 5 liters in 3s leaves. The time does not matter, because volume going in=volume going out ."
    },
    {
        "video_name": "G4CgOF4ccXk",
        "Q": "Why did the sal said: time same time? 6:09",
        "A": "he meant that we are considering as much time as the liquid took iniatialy.It means that if the fluid take 2 seconds to cover the initial volume (which sal drew in pink) then we will consider the same 2 seconds for the exit volume. was it helpful?"
    },
    {
        "video_name": "30o4omX5qfo",
        "Q": "in 0:13 where does the cos sign theta comes from",
        "A": "It comes from the fact that when finding what forces do work, only the forces in the direction of the displacement do work. So, we use cosine to take only the component of the force acting at an angle that is in the same direction as the displacement."
    },
    {
        "video_name": "30o4omX5qfo",
        "Q": "In the previous videos of WORK AND ENERGY Sal explained work = potential energy (mgh) ,but in the above video at 2:21 gravitational potential energy is acting downwards due to height of the weights then why is work not being done ?",
        "A": "Negative work is being done by gravity"
    },
    {
        "video_name": "QX2YLR09Q78",
        "Q": "Can you please clarify the \"pressure decreases = velocity increase\" @ 5:00",
        "A": "Let s analyse all the letters, density of water remains constant during the process, the same for height and gravity, but then if something increases (Velocity or Pressure), something else has to decrease (Pressure or Velocity) respectively, or so that it remains equal."
    },
    {
        "video_name": "tR-hzV5Tkxo",
        "Q": "At 2:20 why is the H bonding to the left carbon and not the right? Didn't the electron come from the right carbon?",
        "A": "One of the bonds in the double bond between the carbons is made up of an electron from the left carbon and an electron from the right carbon. If the right carbon gives that electron which makes up the bond to the hydrogen (as the hydrogen is separated from the Cl) The left carbon and the hydrogen will both have a valence electron and create a bond. Or you could think of it as when the right carbon s electron goes to the hydrogen it carries the bond with it"
    },
    {
        "video_name": "tR-hzV5Tkxo",
        "Q": "how do u know when the elements are at positive or negative charge (AT 2:55 & 3:05)",
        "A": "he explains it very directly...when you are neutral and LOSE an e-, then you have a positive 1 charge. if you GAIN an e-, then you have a negative 1 charge. If you don t understand that, you should review formal charges again"
    },
    {
        "video_name": "wvVdgGTrh-o",
        "Q": "At 7:00, one of the nitrogens in the resonance structure has a steric number of 4 meaning it is sp3 hybridised. If that is the case then why does it have a free p orbital?\n\nThanks",
        "A": "Correct. If there were no resonance, it would remain sp\u00c2\u00b3 hybridized. But by rehybridizing to sp\u00c2\u00b2, the p orbital with its two electrons can participate in resonance. This delocalization of electrons lowers the energy of the whole molecule."
    },
    {
        "video_name": "wvVdgGTrh-o",
        "Q": "At 4:01, why did Jay not draw resonance to the carbon on the left?\nWouldn't it be easier to draw the lone pair moving to left, then kick off the double bond onto the next carbon as in pyrrole?",
        "A": "That would be another one of the valid resonance structures for this molecule. If you draw all the possible structures you can see that the negative charge gets spread over the molecule (delocalized). Note however, that the resonance structure with the negative charge on the nitrogen is the most energetically favorable and thus the largest contributor to the overall structure."
    },
    {
        "video_name": "wvVdgGTrh-o",
        "Q": "At 4:00 he talks about the lone pairs resonating to the right. Can the lone pairs also resonate to the left and kick that double bond onto a carbon to make an anion?",
        "A": "Yes, you can. But the carbanion is a higher-energy resonance contributor. The more important contributor has the negative charge on the more electronegative atom, N."
    },
    {
        "video_name": "wvVdgGTrh-o",
        "Q": "When you talk about pyrrole's aromaticity in between 5:00 and 7:00 minute marks, why do you count the carbon with the negative charge on it as sp2? I think it is considered sp3, right?",
        "A": "That carbon would be sp2 for exactly the same reason as the nitrogen being sp2. None of the pi electrons belong to any one atom, they re distributed over all 5 atoms in the ring. It is very favourable energetically for this to be the case."
    },
    {
        "video_name": "wvVdgGTrh-o",
        "Q": "At 1:51, the resonance structure is being drawn so that the Nitrogen would be sp2 hybridized. At that same moment, a lone pair of electrons moved onto a carbon atom. The steric number for that carbon atom would be 4, which translate to sp3 hybridization. This is quite contradicting. Should we exclude/ignore lone pairs of electrons that participate in resonance when calculating steric number?",
        "A": "Yes, that is the takeaway message here If there is a lone pair on an atom next to a pi bond, that atom will be sp2 and the lone pair will be in a p orbital, despite what the steric number would say."
    },
    {
        "video_name": "wvVdgGTrh-o",
        "Q": "At 6:00, the upper nitrogen is sp3 hybridised, and it is part of the main ring, so it should break the continuity of overlapping p orbitals, so how is the resonance structure still aromatic? Thanks a lot",
        "A": "In inidazole both nitrogens are sp2 hybridised. Breaking the aromatic system is not favourable."
    },
    {
        "video_name": "wvVdgGTrh-o",
        "Q": "at 7:00, will the nitrogen change its hybridization state so that it can be more stable?\nAnd between the two resonance contributor will the aromatic one contribute the most?",
        "A": "It doesn t actively change it s state but it is energetically favourable to be sp2 here."
    },
    {
        "video_name": "3NcIK0s3IwU",
        "Q": "At 2:17 in the video, Sal says \"lets see if we can figure out the resistance in these 4 parallel parts\"....how come the 5th resistor (3 ohms) is not considered parallel to them as well? When I did the problem, I did the exact same thing, except I found 1 (4 + 8 + 16 + 16 + 3)? How do we know that 3 ohms is not parallel...is it due to the branching point of the resistors?",
        "A": "For resistors to be in parallel, they have to be connected together at both ends. The fifth one is not. The fifth one is in parallel with the series combination of (the other parallel resistors plus the 1 ohm)."
    },
    {
        "video_name": "3NcIK0s3IwU",
        "Q": "At point 2:52 how come he say 1/2 = 2 ?",
        "A": "He calculated that 1/R (the resistance inverted) is equal to 1/2. He then inverted both sides and came up with R = 2. So, to answer your question, he did not say that 1/2 = 2, only that 2 is the reciprocal of 1/2."
    },
    {
        "video_name": "3NcIK0s3IwU",
        "Q": "At 5:53, Sal said resistors in parallel. But wasn't it actually resistors in series?",
        "A": "Yes, those are in series."
    },
    {
        "video_name": "3NcIK0s3IwU",
        "Q": "Would you please show me how to calculate the voltage across each resistor from (4:20)?",
        "A": "The voltage is obviously the voltage of the battery. See the video before this."
    },
    {
        "video_name": "3NcIK0s3IwU",
        "Q": "At 2:52 , shouldn't it be 10/16?\nAt 2:43, I think he made a mistake by saying 1/4 (the last one) is 1 / 16. Is that correct?",
        "A": "At 2:52 , 4 + 2 + 1 + 1 = 8. At 2:43 , no his last 16 just looks like a four, if you check the math for yourself it s all right."
    },
    {
        "video_name": "3NcIK0s3IwU",
        "Q": "i think that the math at 2:51 is wrong: it should be = 1/R = (4+2+1+4)/16 not (4+2+1+1)/16.",
        "A": "No, but the line above that is wrong. It should be 1/4 + 1/8 + 1/16 + 1/16 It actually is that, but his writing is bad on the second 1/16 so that it looks like 1/4"
    },
    {
        "video_name": "3NcIK0s3IwU",
        "Q": "In 2:45, shouldn't that be 4 + 2 + 1 + 4 instead of 4 + 2 + 1 + 1 ?",
        "A": "His handwriting makes it a little hard to see, but that last fraction is supposed to be 1/16 not 1/4 so when you combine it into one fraction, the numerator should just be unchanged (1) as the common denominator is 16."
    },
    {
        "video_name": "3NcIK0s3IwU",
        "Q": "At 3:00, you use the equation 1/R = 1/4+1/18+1/16+1/16 to solve for the resistance. I take conceptual physics, and in my class, our teacher said to take the product of the resistances and divide it by the sum of the resistances. Is there a better way to do it. Also, is there a particular textbook or source you use in these videos for reference",
        "A": "take the product of the resistances and divide it by the sum of the resistances ... this only works if there are two resistors in parallel, does not work for more than two. For 3 or more, you need to use this formula 1/R = 1/R1 + 1/R2 + 1/R3 + ..."
    },
    {
        "video_name": "3NcIK0s3IwU",
        "Q": "@6:41 how did you just change the resistor to be from 5/2 to 2/5?",
        "A": "its not the resistor its the resistance and the resistance doesnt change from 5/2 to 2/5 here 1/r is 5/2 and hence r=2\u0005..."
    },
    {
        "video_name": "3NcIK0s3IwU",
        "Q": "Im confused at around 2:56, I see that (in the previous video) Sal is essentially just adding up the V/R of each resistor to find the resistance total for that section.\nSo, if I apply my logic where;\nI = V/R (and add up each V/R)\n=> R = 20/4 + 20/8 + 20/16 + 20/16\nR = 5 + 2.5 + 1.25 + 1.25\nR = 10\nIn other words isn't R = 10",
        "A": "sal is not adding up the V/R values, he is adding up 1/R values so 1/4 + 1/8 + 1/16 + 1/4 = 1/2 but 1/2 isnt the resistance 1/2 = 1/R so R=2 this is using the formula 1/R = 1/R(1) +1/R(2) +1/R(3)..."
    },
    {
        "video_name": "5-MM39VCwc0",
        "Q": "wouldnt the molecule you discuss starting at 6:03 have a formal charge of +2? I always thought we considered a double bond as a single entity? Thank you",
        "A": "Regardless, the double bond consists of 4 electrons, 2 of which are given to the nitrogen when considering formal charge."
    },
    {
        "video_name": "5-MM39VCwc0",
        "Q": "what is negative 1 in 7:37",
        "A": "formal charge"
    },
    {
        "video_name": "5-MM39VCwc0",
        "Q": "At 05:14 it's said that \"it's as though Nitrogen has lost an electron\" Where did it go? Did it bond with another atom? Or does it remain as a lone electron in the shell of the original Nitogen buzzing around by itself? Is this why Nitrogen now has a charge of +1? I'm confused! Where do these disappearing electrons go?",
        "A": "It didn t really go anywhere, the lone pair on nitrogen in ammonia picked up a hydrogen ion and formed a covalent bond. The consequence of this is now there is one more proton than electrons in the molecule so it has a positive charge."
    },
    {
        "video_name": "5-MM39VCwc0",
        "Q": "at 5:13 will there not be a co-ordinate bond from nitrogen",
        "A": "Yes, there is a coordinate bond from nitrogen."
    },
    {
        "video_name": "6zV3JEjLoyE",
        "Q": "How did get 3.27 light years @ 7:03? Is a parsec actually 3.268 light years?",
        "A": "Round it up to the nearest hundredth..."
    },
    {
        "video_name": "6zV3JEjLoyE",
        "Q": "at 1:00 what specifically makes an arc second",
        "A": "Basically its 1/3600 of a degree. Imagine yourself outside on a clear night. Extend your arm in front of you and hold out your pinky. That is about 1 degree. Now imagine dividing that space again into 60 equal parts. One of those parts is referred to as an arc minute. Take one of those 60 parts and divide it again into 60 equal parts. An arc second is the size of one of those parts."
    },
    {
        "video_name": "4J-dRA3MGc8",
        "Q": "On about 04:29, it says \"Fee for Service\". What exactly IS the fee for service> Thanks.",
        "A": "Every time the doctor does something, the insurer / patient pays for that particular service (whatever the doctor does is considered a service, such as seeing the patient, testing a mole to see if it s cancerous, removing that mole, etc.)."
    },
    {
        "video_name": "-h_x8TwC1ik",
        "Q": "at 0:31 sal says that we have to find horizontal distance. why can't we use vertical distance?",
        "A": "Because optimal angle means the one that will make it go the furthest, not the highest."
    },
    {
        "video_name": "PSy6zQsk8z0",
        "Q": "at 2:37 he says \"we tried to remove the screws for our tripod mount but one of them is stripped\". what is meant by \"stripped\" in this context (referring to a screw)?",
        "A": "To strip a screw is to grind away the edges. If you look at a screw you will notice the 4-point star like shape, when ever somebody strip s a screw, all they did was Grid it to the point to where it looks like a little whole on top of the screw making it just about IMPOSSIBLE to unscrew"
    },
    {
        "video_name": "PSy6zQsk8z0",
        "Q": "What is the liquid crystal he mentioned at 0:26 and what does it do?",
        "A": "LCD (liquid crystal display) is the technology used for displays in notebook and other smaller computers. Like light-emitting diode (LED) and gas-plasma technologies, LCDs allow displays to be much thinner than cathode ray tube (CRT) technology."
    },
    {
        "video_name": "PSy6zQsk8z0",
        "Q": "At 6:19 what is a JPEG converter?",
        "A": "A JPEG converter processes the image into the common JPEG photo format , which is smaller than the raw output from the camera because it has unused data removed .Some cameras allow you to keep the raw, unprocessed photo"
    },
    {
        "video_name": "PSy6zQsk8z0",
        "Q": "At 5:47, he talks about the filter to filter out infra red light, when I take a photo of a remote in use it shows up in the picture, why does it not get filtered out?",
        "A": "It s possible that the filter he pointed out actually doesn t filter out infrared wavelengths, or that the camera you use doesn t have such a filter."
    },
    {
        "video_name": "PSy6zQsk8z0",
        "Q": "At 2:21, how is the print circuit board manufactured? Is it sort of like an ink jet?",
        "A": "A thin sheet of copper is laid over the plastic, which is then etched into wires that go to the different components on the circuit board."
    },
    {
        "video_name": "PSy6zQsk8z0",
        "Q": "at 0:02 sec why is the first few seconds of the video in fast forward?.",
        "A": "so they can fit more into the video. the video would have to be very long if they did not speed up some parts of it."
    },
    {
        "video_name": "PSy6zQsk8z0",
        "Q": "At 5:50, he mentions \"the bayer filter, which we talked about earlier.\" What other video is he referring to?? I'd really like to check that out.",
        "A": "I think he is referring to part 2."
    },
    {
        "video_name": "mZ-U7Qpkz8Y",
        "Q": "When Jay says that the bonds stretch (2:50), how does he know that the H-C bond corresponds to the wavenumber? Is there another reference?",
        "A": "There are tables of IR absorption bands available for reference. Any organic textbook will have it as well as many sources online. These values have been determined experimentally using known compounds."
    },
    {
        "video_name": "mZ-U7Qpkz8Y",
        "Q": "What are the \"other kinds\" that Jay is referring to at about 0:15?",
        "A": "Another one is bending which can include scissoring, wagging, rocking. Look up vibrational modes on google for more info on these."
    },
    {
        "video_name": "vFSZ5PU0dIY",
        "Q": "At 6:15 it says a polar protic solvent would decrease the likelihood of Sn2 but wouldn't it also decrease E2? I thought both Sn2 and E2 favored polar aprotic solvents.",
        "A": "Being solvated is something worse for the Sn2 because of the hindrance of all those things together, and you can t get bulky things to infiltrate to the nucleus. For an E2 it would be also bad, but not THAT bad"
    },
    {
        "video_name": "vFSZ5PU0dIY",
        "Q": "I'm still confused at 3:05. Why exactly did we have to ignore the heat? It seems to me that it could have easily been an E1 reaction. We have a weak base and a weak nucleophile, so why favor SN1 over E1?",
        "A": "Even with heat, I doubt that there would have been much elimination. Methanoic acid is a very weak base, so I would not expect it to abstract a proton. True, heat favours elimination, and teachers often insert that word as a hint that the major pathway is elimination. In this case, there is very little elimination, so he told us to ignore the heat."
    },
    {
        "video_name": "vFSZ5PU0dIY",
        "Q": "At 9:50, why can't it be SN1?",
        "A": "It can be SN1, but this reaction is always a competition between elimination and substitution. The only nucleophiles present are H2O and HSO4- .Both of these are weak nucleophiles. They are also weak bases, but HSO4- is the stronger base because of its negative charge. So E1 predominates."
    },
    {
        "video_name": "vFSZ5PU0dIY",
        "Q": "4:55... I thought that the only reactions (out of SN1 SN2 E1 and E2) that could occur in a protic solvent were SN1 and E1? So how come he is treating this as an E2 reaction?",
        "A": "You have found the exception to the rule. Ethoxide is a very strong base and will perform an E2 reaction despite the polar, protic solvent. The nucleophile/base and the substrate are more important for determining what reaction will occur than the solvent. Jay even mentions that there will be an SN2 product despite the decreased nucleophilicity of ethoxide."
    },
    {
        "video_name": "vFSZ5PU0dIY",
        "Q": "I understand that a weak nucleophile and base would lead us to decide between SN1 and E1. Then @ 3:05 you say that a polar protic solvent would stabilize a resulting carbocation and you pick SN1 as the mechanism. Wouldn't a polar protic solvent stabilize the carbocation of an E1 reaction just as effectively? They both are favored by protic solvents. Why does the polar protic solvent favor the SN1 reaction over the E1 in this example?",
        "A": "I agree with lacp45, and further wouldn t the protic solvent hydrogen bond the nucleophile, further reducing the nucleophilicity of an already weak nucleophile? Adding heat to the equation - I think this was supposed to be an E1 reaction..."
    },
    {
        "video_name": "vFSZ5PU0dIY",
        "Q": "at 3:40 why cant this weak base favour an E1 mech ??",
        "A": "yes... but i was thinking that the factors... a and b equally favours both SN1 and E1 becoz carbocation forms in both cases... but still SN1 wins.... so basically my Ques. now is can there even be a minor E1 product?"
    },
    {
        "video_name": "vFSZ5PU0dIY",
        "Q": "3:00...So, we are talking about a weak base or a weak nucleophile, in a polar protic solvent, and the alkyl halide is a secondary alkyl halide. To me this suits BOTH an SN1 reaction and an E1 reaction. So why did Jay say that this would occur via an SN1 reaction only?",
        "A": "I was thinking the same thing, but at 8:59 Jay mentions that E1 reactions don t usually occur with alkyl halides. They usually occur with alcohols and a different base, one that is non-nucleophilic, such as the conjugate base of H2SO4. So, a weak, non-nucleophilic base, an alcohol substrate, and high temperatures will favor E1."
    },
    {
        "video_name": "vFSZ5PU0dIY",
        "Q": "Wouldn't SH- be a strong base (0:43)? SH2 is a weak acid after all.",
        "A": "It would not be a strong base. The term strong is reserved for things like NaOH and KOH. So SH- will be a weaker base than those, but stronger than a lot of other substances, like water."
    },
    {
        "video_name": "vFSZ5PU0dIY",
        "Q": "For example at 8:55, why wouldn't the reaction go SN1/E1? You have good nucleophile, good base and even a polar protic solvent.",
        "A": "Usually strong bases and nucleophiles favour E2 or SN2. It is true that polar aprotic solvents promote S1 and E1 reactions however, the strength of the base is the most important factor to consider!"
    },
    {
        "video_name": "E9fw0EU4wpA",
        "Q": "I understood everything except at 2:26, if the molecule has two chiral centers, should not the number of stereoisomers for the molecule be 4, based on the formula 2^n, where n is the number of chiral carbons ? Please explain!",
        "A": "Yes, if a molecule has n chiral centres, it can have up to 2\u00e2\u0081\u00bf stereoisomers. But this reaction is stereospecific. Since the original alkene was trans, it produces only the two trans isomers. It does not produce the other two cis isomers."
    },
    {
        "video_name": "IICR-w1jYcA",
        "Q": "at around 2:00, why not change the order of T-Ta to Ta-T? Then we wouldn't have to deal with a negative k?",
        "A": "Yes, this is how I was inclined to do it as well and I had the same question. I believe he s anticipating later steps in the solution of the differential equation, rather than aiming to write this relationship in the most simplified form."
    },
    {
        "video_name": "IICR-w1jYcA",
        "Q": "At 4:40 Sal starts to integrate, why do the dT and dt terms vanish in the process?",
        "A": "The dT and dt tell you what you are supposed to integrate with respect to, or simply what variable is to be integrated. After you have performed the integration, the dt (or dT) becomes useless and disappears."
    },
    {
        "video_name": "IICR-w1jYcA",
        "Q": "At 5:00, I understand the whole thing about u-substitution, but why wouldn't it be possible to say\n\nintegral of ((T^-1 )- (Ta^-1)) = ln(T) - (Ta^-1)*Tv?\n\nMy intuition tells me that differentiating the result above would in fact lead us to (T^-1) -( Ta^-1). I suppose I am wrong, since I haven't been able to work out the proper solution with this method. But by what rules can I tell that this method of integrating the above is in fact wrong?",
        "A": "The problem you have is that you are making some mistakes with the exponentials when you try to bring the denominator up: 1 \u00e2\u0080\u0095\u00e2\u0080\u0095\u00e2\u0080\u0095\u00e2\u0080\u0095\u00e2\u0080\u0095\u00e2\u0080\u0095 = (T - T\u00e2\u0082\u0090)\u00e2\u0081\u00bb\u00c2\u00b9 \u00e2\u0089\u00a0 T\u00e2\u0081\u00bb\u00c2\u00b9 - T\u00e2\u0082\u0090\u00e2\u0081\u00bb\u00c2\u00b9 T - T\u00e2\u0082\u0090"
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "I am confused, at 4:21 in the video he says average velocity = velocity final (Vf) - velocity initial (Vi) / 2. I thought average velocity = change in position divided by the change in time (xf-xi/tf-ti)? Are they the same or am I missing something?\n\nThanks",
        "A": "They are the same if acceleration is constant. Note that it is (Vf + Vi) / 2, not (Vf - Vi)/2"
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "At 0:21 Sal said \"velocity\" but I'm pretty sure he meant \"vector.\" When you add direction to a scalar (quantity of only magnitude), you get a vector. Velocity is also a vector quantity, so it is easy to get confused.",
        "A": "No, he meant a velocity . This is because Sal was at first giving a Rate, without direction. By adding a direction, he turned it into a Velocity, and yes, a vector too."
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "Why is average velocity taken at 04:06 instead of 72m/s itself?",
        "A": "because there has been a change in velocity since the time it started till the time it took off. Constant acceleration of the aircraft over the runway has been taking place so the velocity throughout the runway of the aircraft is changing with time and to calculate acceleration assuming it is constant ,taking an average of the velocity through its journey is much easier than calculating velocity at each second and calculating the corresponding acceleration (which will be a constant)"
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "At 5:45, Sal gave us the formula for measuring the time using acceleration and the change in velocity. But instead can't we just find the time using the velocity and distance which is t=s/v. why do we have to go through all that?",
        "A": "because the velocity is changing. in this formula t=s/v the velocity is constant."
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "at 4:15, sal says that totl disp=avg *time...\nis this formula always correct or just in the case where initial velocity is zero ?",
        "A": "It is always correct. Average velocity can be calculated between any two points of time, and so change in time is always equal to the amount of time between those two points."
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "At 10:31 wouldn't you square 72 as well",
        "A": "Actually he then noticed he did a mistake and squared it re watch at 10:50"
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "At 13:47, Khan calculated the time, by dividing the final velocity by acceleration (72/33) and ending up with 2.18, but my question is, how come he can do this when the final velocity is in the form of meters per seconds, and the acceleration is in the form of meters per seconds squared?",
        "A": "If initial velocity is 0, we can write v=at =>t=v/a That s why he divided velocity by acceleration. Now to your question. The unit of velocity is m/s and unit of accleration is m/s^2. So, if you divide m/s * 1/(m/s^2) =m/s * s^2/m =s (m and m cancels out and one s cancels another s)"
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "At 4:25, why we can only do that if we are dealing with constant acceleration? How do we find the average velocity if the acceleration is not constant?",
        "A": "Usually that s a really complicated problem. If you re interested in the answer, you should watch the integral calculus videos. I don t think I can explain it with text only, and without writing for at least 2 - 3 hours xD"
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "At 9:14, the formula Sal has derived- does it work if the acceleration is non-constant? After all, the way he found the average velocity is dependent on having a constant acceleration, but just wanted to make sure.",
        "A": "It does not work for non-constant acceleration"
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "At 6:01 what is the difference between initial velocity minus final velocity over two and initial velocity minus final velocity over acceleration",
        "A": "The first one gives you average velocity (if acceleration is constant). The second one gives you the time it takes to change velocity (also assuming acceleration is constant). Look at the units: (m/s) / (m/s^2) = seconds."
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "at 6:42 why was it minus the two squared numbers rather then plus?",
        "A": "Here is an algebraic proof of that: (a + b) (a - b) Distributive Property a(a) + a(-b) + b(a) + b(-b) Simplify a\u00c2\u00b2 - ab + ab - b\u00c2\u00b2 Identity Property of Addition a\u00c2\u00b2 - b\u00c2\u00b2"
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "At 8:36, Sal wrote final velocity and initial velocity as scalar. He should wrote them with arrows, right?",
        "A": "Since this problem can be described in a single dimension of motion you can do the calculation without needing to use vectors. You need to use vectors when you need to deal with more than 1 dimension."
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "At 11:13, Sal says to round to two significant digits. How do I know to round to a particular place? Is there a universal way to round acceleration?",
        "A": "Sal has vids about sig figs."
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "At 3:57 how can the total displacement (S) to be = (average Velocity * the Change in Time)? I couldn't figure it out! Why using the AVERAGE!?",
        "A": "What would you use instead of the average?"
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "what are you doing with acceleration= delta v/ delta t? at around 5:09? im just lost on how you transitioned into that and why you are doing that.\nand can you just refresh what displacement is again?\nthank you!",
        "A": "Khan was just manipulating the original equation which solved for a so that it solved for change in time (delta t). Displacement is distance travelled in a specified direction. It is a vector quantity."
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "At 1:40, Sal wrote vector Vi = 0, isn't it rather vector Vi = zero vector ?",
        "A": "Yes, that s true as well. Velocities are indeed vectors. However, they re essentially just a speed that is in some direction, which is why if you re doing simple 1-dimensional motion, you tend to not write them as vectors, since everything is in the same direction, so the direction component of the vector has no effect on the problem. Usually in physics, you want to break any vectors into their x & y (& z if you re doing a 3D problem) components, and just use magnitudes to do problem solving."
    },
    {
        "video_name": "VYgSXBjEA8I",
        "Q": "AT 11:29 SAL STATES the acceleration to be 33 m/s. The calculaators mistaken. I got 32.4 m/s how could we round off 32.4 to 33?",
        "A": "Me too. Sal rounded 72.222 down to 72, so I did it with a calculator and thats the answer I got (32.4). Sal used the previous answer key hence the difference in answers hope it helps."
    },
    {
        "video_name": "4xtXBsYENzg",
        "Q": "Hank states at 6:11 that macrophages can detect when a cell has gone rouge and 'kill' it but at 7:29 Hank states that Natural Killer Cells are the only phagocyte that can kill other human cells, so is the natural killer the only one that can kill other human cells?",
        "A": "Natural killer cells and Killer T cells (in specific immunity) are the only cells that can kill human cells. Macrophages kill (by engulfing) pathogens, foreign particles."
    },
    {
        "video_name": "4xtXBsYENzg",
        "Q": "At 5:14, How do the leukocytes send signals ?",
        "A": "Chemical responses."
    },
    {
        "video_name": "4xtXBsYENzg",
        "Q": "At 6:14, Hank says that macrophages can kill other cells, but at 7:33 he says that natural killer cells are the only phagocyte in the innate immune system that can kill other human cells. Which is correct?",
        "A": "Macrophages main purpose is not to gobble up all the junk. Its purpose is just to kill whatever freaking heck is inside which wasn t invited. A natural killer cell has TWO main duties. They are to a.GET RID OF USELESS CELLS b. Get rid of bacteria."
    },
    {
        "video_name": "4xtXBsYENzg",
        "Q": "At 13:28, Hanks says that a B-cell \"could be covered in a forest of up to 100,000 different antibodies\". Why does it have 100,000 different antibodies for, like, chicken pox when there is only on kind of chicken pox?",
        "A": "Mutations. Many kinds of one virus can exist. There s more flu types then I can remember, so if it s every virus you ve been exposed to, that d be a lot of antibodies!"
    },
    {
        "video_name": "4xtXBsYENzg",
        "Q": "at 4:35 ish he says why allergic reactions happen. Why don't everyone have the same allergies if this is the case?",
        "A": "Different people have similar, but still different immune systems. Hence, different things will trigger the immune system response, that is, a unique set of allergens will cause person-specific reactions in each person. This means that everyone has different allergies because we aren t all identical."
    },
    {
        "video_name": "4xtXBsYENzg",
        "Q": "At 0:38 he says more germs then in all of the worlds prisons.\nWhat i was wondering was how many different types of germs\nare in a pig pen.",
        "A": "Many harmless types"
    },
    {
        "video_name": "4xtXBsYENzg",
        "Q": "At 11:36, Hank says \"cell wall\" but I think it was supposed to be \"cell membrane.\"",
        "A": "If it is a plant, it is a cell wall, however, an animal cell has none.It has a membrane only.A plant cell, too, has a cell membrane and a cell wall. Vote if this helps!"
    },
    {
        "video_name": "4xtXBsYENzg",
        "Q": "At 6:14, Hank said that macrophages can detect and kill cancer cells, but how come people still has cancer for years? Do the cells multiply too quickly?",
        "A": "Of course, your immune system is good at what it does. Which is why cancer sometimes has an ENORMOUS struggle trying to kill whoever is infected. However, sometimes it never gets the upper hand, which is why people can have it for years on end without dying."
    },
    {
        "video_name": "4xtXBsYENzg",
        "Q": "At 5:03 what protects the brain, spine, and nervous system if the leukocytes cannot go there?",
        "A": "I see, as those areas have lack circulatory vessels, and so have little or no access to lymphocytes. It make easier for graft transplant. Hoping for the answer too..."
    },
    {
        "video_name": "4xtXBsYENzg",
        "Q": "10:30 Do scientists still use the word \"humor\"? I thought that was a totally outdated concept - do \"humors\" just refer to something different today?",
        "A": "The concept of The Four Humors (blood, yellow bile, black bile, and phlegm) and their influence on health and emotion is an outdated concept, and is no longer used. In modern medicine, humors is used to refer to various body fluids. Such as the aqueous and vitreous humors in your eye."
    },
    {
        "video_name": "Jh4UsA2see0",
        "Q": "At 3:55, what exactly is a ecosystem's magic eye? I didn't really get it.",
        "A": "At 4.00, he tells us the thing that makes this ecosystem eye.It isn t really a poster thing it is the way everything blends in to make an ecosystem."
    },
    {
        "video_name": "Jh4UsA2see0",
        "Q": "At 6:43, the Food Web has arrows to the organism who's being eaten. I always learned at school, that the arrow points at the animal who eats, and the arrow starts nearby his food. (Last test, I recieved a lower note becease my teacher considered my food web wrong, but I did it exactly like you did in this video)\nWhat is the correct method to do this?",
        "A": "The same is at my school, we point to the animal that eats. It seems to just depend on the teacher and their preference. Personally, I think the correct method is to point to the organism that is eating."
    },
    {
        "video_name": "Jh4UsA2see0",
        "Q": "At 3:56, you say magic eye... What is a magic eye?",
        "A": "A magic eye was an old type of illusion poster. When you first look at it, it s a jumble of random colors, but if you squint or back away, then it actually turns into a picture."
    },
    {
        "video_name": "Jh4UsA2see0",
        "Q": "at 3:03 Hank says that the animals would be thirsty if there was no stream, wouldn't there be another water source somewhere?",
        "A": "Probably, but I think you re overthinking it!"
    },
    {
        "video_name": "Jh4UsA2see0",
        "Q": "At 1:30 Hank says we'll stop not understanding ecosystems starting \"now\" but the theme music plays and I still don't understand what an ecosystem is :P",
        "A": "A collection of living and nonliving things interacting in the environment."
    },
    {
        "video_name": "Pk4d9lY48GI",
        "Q": "what does ' deprotonated ' mean ? can you expailn it in more detail ? at 8:14",
        "A": "deprotonisation means removal of a proton thats a H+ ion. It occurs in acids due to instability causes."
    },
    {
        "video_name": "Pk4d9lY48GI",
        "Q": "At 9:25 the idea of a \"zwitter ion\" is introduced. But isn't this just another name for a polar molecule? So wouldn't water molecules be zwitter ions?",
        "A": "zwitter ion has a complete charge. while polar molecule has partial charges."
    },
    {
        "video_name": "Pk4d9lY48GI",
        "Q": "As Sal said at 1:43 does that mean that only DNA can form proteins with the help of amino acids and tRNA",
        "A": "No, proteins are formed by the following process: DNA is transcribed to mRNA, which then exists the nucleus and is translated in the ribosomes. The nucleotide sequence on the mRNA is read and tRNA is sent to get the amino acids corresponding to the the triplets. Once the sequence of amino acids is finished, this can either be a protein or part of a protein. Therefore, DNA can form proteins with the help of mRNA, the ribosomes (=rRNA), tRNA and the amino acids."
    },
    {
        "video_name": "peQsBg9P4ms",
        "Q": "At 8:33, we attempt to move the hydrogen from jutting out towards us to going backwards. How do I know which way to turn the molecule/ which molecules move to make the hydrogen go in back? I understand imagining looking at it as if I was behind the hydrogen, but what if the hydrogen was in the plane of the page? Any quick way to know?",
        "A": "when #4 is at vertical position then find the direction of chiral carbon (here we get it in clockwise direction i.e R-rectus) and just reverse it to get answer quickly instead of rotating and taking #4 at horizontal position and again finding the direction of chiral carbon"
    },
    {
        "video_name": "peQsBg9P4ms",
        "Q": "At the end (9:40) how are you able to tell that #1 stays the same but #2 and #3 switch?",
        "A": "Did you listen from about 8:00 onwards? The #4 group needs to be where the #3 group is (facing away with the dashed line). The easiest way to do this is to just rotate around the axis of the #1 bond. #1 stays in place and the other 3 move. Sal makes a good analogy to an umbrella. You can spin an umbrella around its pole and the pole stays in place but the top moves, it s the same idea here."
    },
    {
        "video_name": "peQsBg9P4ms",
        "Q": "At 2:57 when naming the molecule, why is the bottom methyl group chosen instead of the leftmost methyl group? Would it make a difference in the name, since they're both methyl groups and are linked to the same C atom?",
        "A": "Yes, the name would then be 3-bromo-2-fluoro-2,3-dimethylbutane. IUPAC rules state to take the lower set of numbers for the halides, in this case the name which was on the video (1-bromo-3-fluoro)"
    },
    {
        "video_name": "peQsBg9P4ms",
        "Q": "In the last example, around 11:00, I'm still confused when it comes to \"rotating\" the groups. I know that #4 needs to be behind the page, but how do I know which direction the others should be rotated?",
        "A": "If #4 is behind the page, you find the direction that groups #1 -> #2 -> #3 go, if it s clockwise it is R, if it is anticlockwise it is S."
    },
    {
        "video_name": "peQsBg9P4ms",
        "Q": "At 9:00 when Sal begins rotating the groups to get #4 in the back, does it matter what axis I choose to rotate around? I rotated around the #2 group (with Fluoro and methyl) yet still got the sinister position.\nWas this coincidence or is that ok?",
        "A": "No problem .. :) everything is same.... min objective is to bring back th no.4 :D"
    },
    {
        "video_name": "peQsBg9P4ms",
        "Q": "I'm having some trouble throughout these videos of visualizing what Sal means when he draws the molecule \"in the plane of the screen\" (like at 6:37). Could somebody please give me an example of what that looks like 3D?\nThank you very much!",
        "A": "Plane of the screen means its all 2d. When its not 2d, its either virtually jutting our of the screen from the front of it or from the back of it. You know, its like your x, y and z axis . x and y are on the same plane whereas z is 3d, jutting perpendicularly out from the x,y plane. Hope that made it clear."
    },
    {
        "video_name": "peQsBg9P4ms",
        "Q": "At 2:26 Sal says \"bromo-methyl group\". How can it be methyl if only two hydrogens are attached to the carbon?",
        "A": "The methyl group by itself is a substituent. But it has a substituent of it s own, bromine. Therefore it s bromomethyl. Recall isopropyl can be names 1-methylethyl. But we don t say 1-methylethane."
    },
    {
        "video_name": "peQsBg9P4ms",
        "Q": "at 4:00 I would have sworn that back when I originally took organic that the di-prefix does indeed count towards the alphabetical order of things. I of course could be wrong and am looking it up now, but perhaps we could recheck this together.",
        "A": "Nope, di- does not count. Only cyclo and iso, I think :)"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "at 9:44 whats a limiting reagent.",
        "A": "The limiting reagent is whatever we have the least amount of stoichiometrically. The balanced equation tells us how many molecules of each reagent are needed, but it is very hard to measure on a scale down to the exact molecule. So, we often add an excess amount of one reagent and a lesser amount of the other reagent. The limiting reagent will be entirely consumed while a little of the excess reagent will stay behind, unreacted."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 7:21 in the video, when he multiplied .53 by 2 because there were 2 aluminums, would you have multiplied .53 by 5 if it had read 2Al3?",
        "A": "Not quite. If it s 2(Al3) then it s two molecules of Al3, which is 6 aluminium atoms in total."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 5:30, why is 160amu for one FE2O3 molecule allowed to then be considered 160g for one FE2O3 mole? Is that simply something we're supposed to assume/understand? Thanks for the help!",
        "A": "Basically, you can think of the mole as a tool to help you switch between grams and amu. If one molecule of something weighs 160 amu, then 6.022*10^23 molecules (one mole) of that something will weigh 160 g. Hope this helps!"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At around 4:35 why don't we multiply the 27 (Al mass) by 2, seeing that we have 2 molecules of Al (indicated by the coefficient)?",
        "A": "I think it is because alluminum was written as two individual atoms (2Al) instead of a diatomic molecule (Al2). 2Al = 2 individual atoms that weigh 27amu each. Al2 = 1 molecule (containing 2 Al atoms chemically bonded to each other) that weighs 54amu (27x2)."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 6:15 minutes you used iron(iii) in one equation and on the above equation you used iron(ii).",
        "A": "He accidentally wrote Fe3O3 [iron(II)] at one point, and it carried through the rest of the video. But it didn t affect his argument, though."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 6:15, why do we divide the 85 by 160?",
        "A": "Its because 1 mole of iron oxide is equal to 160 g. Thus to find out how many moles are there in 85g of iron oxide we need to divide 85 by 160. Hope this helps :)"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "Is Sal ignoring the rules for significant figures in calculations, or am I just confused about the rules? For example, at 7:05, shouldn't it be rounded to 1.1 moles of Aluminum since .53 only has two significant figures?",
        "A": "Your claim is almost right. Typically, you only change the sig figs at the end of a problem, to insure the greatest accuracy, not at every small steps."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "at 9:03 how do you know when the mass goes over the 28.7 g of aluminum or below it?",
        "A": "Look at the units and make sure they will give you the right units in your answer."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 5:59 Sal says that we have 85 grams of Fe2O3 but at the bottom it says that there is 160 grams of Fe2O3 can somebody please tell me why it is like this??",
        "A": "No, he is saying that 1 mol of Fe\u00e2\u0082\u0082O\u00e2\u0082\u0083 is 160 g."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 6:15, where does he get 160 from to get the moles?",
        "A": "The molecule being Fe2O3. Fe has a mass of 55.8 (he used 56) and oxygen has a mass of 16. 56*2 and 16*3 added together gives you the molar mass of 160"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "at 0:54 how is the reactant and reagant the same thing?",
        "A": "Just rewind a few seconds to 0:34. For most of your elementary chemistry, reactant and reagent can be used to describe the same thing."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "at 2:38 in video, how does one side have 2 iron atoms and the other only have one to start out with? and how does aluminum spontaneously have two on the left side instead of one atom?",
        "A": "Well, in reality, you never have that kind of equation. It is what is called an unbalanced equation and (basically) what scientists do is they look at the reactants (in this case Iron(II) Oxide and Aluminium) and then they look at the products (in this case Aluminium Oxide and Iron(II)) then they look at the amounts of atoms on each side of the equation and balance it so they have the correct equation. The substances and actual reaction never changes, balancing the equation is only to help the scientist."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "i got so confused once you hit 5:30",
        "A": "For any of you who are getting confused, Sal accidentally put at 3 subscript after the Fe at the 5:45 mark. Iron 3 Oxide is Fe2O3, so at that point in the video just think of the subscript 3 to which I referred as a 2. The math works."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 7:10\nWhen calculating moles,why does Fe\u00e2\u0082\u0082O\u00e2\u0082\u0083=2Al?They're both reactants it's not an equation...I'm confused!",
        "A": "He s working out how many grams of Al react with 85 grams of Fe2O3, so to do that you need to use the molar ratio between Fe2O3 which is 1 Fe2O3 : 2 Al This is a very common type of stoichiometry problem."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "at 8:40, why did he multiply for aluminium and divide for Fe2o3?",
        "A": "He had the grams of Fe\u00e2\u0082\u0082O\u00e2\u0082\u0083. To calculate the moles of Fe\u00e2\u0082\u0082O\u00e2\u0082\u0083, he had to divide by the molar mass. Moles = mass/molar mass Once he knew the moles of Al, he had to convert that to grams. That s the reverse process, so he had to multiply by the molar mass. Mass = moles \u00c3\u0097 molar mass"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "at 8:31 mins. Why would you not multiply the Al by 2 since the ratio is 2 to 1?\ninstead of it being 27, should it be 54 amu? and then multiply 1.06 by 54 amu in order to get the end result?",
        "A": "At 8:31, Sal s converting from moles to grams, not moles to moles, so you don t have to worry about ratios."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "at 8:19 how is one mole of aliminium 27 gm and how is 27 equal to 6.02x10^23 ?",
        "A": "The mass of 1 Al atom is 27 u. 1 u = 1.66\u00c3\u009710\u00e2\u0081\u00bb\u00c2\u00b2\u00e2\u0081\u00b4 g. So the mass of 1 Al atom is 27 \u00c3\u0097 1.66\u00c3\u009710\u00e2\u0081\u00bb\u00c2\u00b2\u00e2\u0081\u00b4 g = 4.48\u00c3\u009710\u00e2\u0081\u00bb\u00c2\u00b2\u00c2\u00b3 g. \u00e2\u0088\u00b4 The mass of 1 mol of Al atoms is 6.022\u00c3\u009710\u00c2\u00b2\u00c2\u00b3 atoms \u00c3\u0097 (4.48\u00c3\u009710\u00e2\u0081\u00bb\u00c2\u00b2\u00c2\u00b3 g/1 atom) = 27 g. I.e. the mass of a mole of atoms in grams is numerically the same as the mass of a single atom in unified atomic mass units."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "at 5:54, he says that the mole will be 160g, but the atomic mass unit is also 160. is it that amu is the same thing as the mole or is it that the mole has the same amount as the amu?",
        "A": "Yes, if 1 molecule has a mass of 160 u, then 1 mol of molecules has a mass of 160 g."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 0:06, Sal says stoichiometry,what does he mean by that?",
        "A": "The word stoichiometry derives from two Greek words: stoicheion (meaning element ) and metron (meaning measure ). Stoichiometry deals with calculations about the masses (sometimes volumes) of reactants and products involved in a chemical reaction. Does this answer your question?"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "(6:12) Why was 85g equated to 85/160? Can someone please explain what happened here, thank you!",
        "A": "What he wrote doesn t make sense and he didn t explain it very well. He should have wrote something like: moles of Fe2O3 in 85 grams = 85 g / 160 g/mol"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At about 7:12 he says \"for every one molecule of this, we need one molecule of that.\" I thought Fe2O3 would have an ionic bond and therefore technically wouldn't be a molecule. Am I mistaken?",
        "A": "Technically, you are correct. He should have said formula unit . But chemists often sloppily (and incorrectly) refer to ionic substances as molecules, simply for convenience."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 5:55, why is 1 mole of Fe2O3 100 grams",
        "A": "It isn t, it s approx 160 grams which is what he wrote. 56 x 2 + 16 x 3 = 112 + 48 = 160"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "After 8:45 when you find the grams in 1.06 moles of Aluminium why don't you multiply the grams of Aluminium by 2 if there are 2 Aluminium atoms used in the chemical reaction?",
        "A": "My chemistry teacher answered the same question: 2 was part of the ratio; you already used 2. There is no need to multiply it by 2 again. Does that make sense?"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "In 3:26 he says that he needs 2 aluminums, how does he know that??",
        "A": "Because there are two aluminium atoms in Al2O3"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 2:19, how is that balanced?",
        "A": "There are 2 atoms of Fe, 2 atoms of Al, and 3 atoms of O on each side of the equation, so the equation is balanced."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 5:00ish to get the 160g the atomic mass of Fe was multiplied by 2 and the atomic mass of O was multiplied by 3. So it was 2x56+3x16= 160. But why wasnt the atomic mass for the Al multiplied by 2 since its 2Al?",
        "A": "Because 2Al means that there are two moles of Al atoms, not 1 mole of Al2 molecules. We use the molar mass for the substances involved in the equation and then we can adjust the amount of moles using the coefficients based on how those substance react with each other. In Fe2O3, there are two Fe atoms and three O atoms in each mole of the substance. So its molar mass is the sum of the atomic masses of all its constituent atoms."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "Till 5:49 Sal Sir was using Iron 3 oxide as Fe2O3 but then he uses Fe3O3. Any special reason for that?",
        "A": "He just wrote the wrong thing. He may have been thinking about the fact that the iron atoms in Fe2O3 each have a 3+ charge..."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "Is't the iron oxide at 5:48 so post to be Fe2O3 ?",
        "A": "Transitional metals, like Iron, can have more than one oxidation number (in the case of Iron, 2 and 3) the compound Sal is talking about is Iron Oxide where iron has an oxidation number of three, which is why he is saying Iron (three) oxide. Properly, this is written as Iron (III) Oxide."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "Are aluminum and \"aluminium\", as Sal pronounced in 7:32 basically describe the same metal?",
        "A": "Yes. They are the same thing. Aluminum is preferred in the USA and aluminium is preferred in the UK. I m not sure about the usage in other countries."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 6:24, why does he divide the 85/160. This is different than I learned because I learned: 85g x 1 mol/160 --> .531. I got the same answer as he did but I do not understand how. Are both methods the same or not.",
        "A": "they are the same. your way of doing it is more formal and shows every step. His way is a simplified version. If a teacher asks you to show your work, than your way of doing it would be ideal."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "at 5:56, you wrote fe3o3 instead of fe2o3. did u do it on purpose, pls explain",
        "A": "A text box appears bottom right at 5:43 to say Sal wrote Fe3O3 but meant Fe2O3."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 4:27, is it fine to round the mass of Iron. Does it create any discrepancies in the answer?",
        "A": "It isn t really fine to be rounding like he does in this video. You should use the number as it is on the periodic table and round to the appropriate number of significant figures at the end."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 8:30 Sal says that Al has an amu of 27, and then goes on to say that 1 mole of Al is 27 grams. Does this mean that 1 amu is equal to 1 gram?",
        "A": "1 atom of Al has an amu of 27, whereas 1 mole of Al has a mass of 27 g. So no, amus and grams are not equivalent but are very closely related (via the mole concept). The relationship is that for any atom, if a single atom has mass X amu, then 1 mole of that atom (i.e. 6.02 x 10^23 atoms) will have mass X g. In other words, 1 amu is equal to (1/6.02x10^23) g."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 7:57 in the video why did you rounded the atomic weight of aluminum to be 27 amu when it was actually 26.98 amu. Wouldn't this produce a different result or would this not matter in stoichiometry. I would also like to know the real world application of stoichiometry and how this can help scientists in the real world.",
        "A": "He shouldn t round it, use it as it comes on the periodic table. It wouldn t make much difference here because it s only 0.2 difference, but it s bad practice. Real world applications? If you want to know how much of 2 chemicals to mix together you can use stoichiometry to work that out. Also to work out how efficient a reaction was by calculating the percentage yield."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "Okay. So at around 9:30, Sal explained that Fe2O3 will be the limiting reagent if 2Al is more than 28.62. But if there are less than 85g (which is Fe2O3) but more than 28.62g, is Fe2O3 still the limiting reagent??",
        "A": "ohh wait nevermind. the next video shows it!"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 5:54 isn't Fe3O3 supposed to be Fe2O3? Or is there some sort of balancing thing happening?",
        "A": "Yeah, you are right. He did it by mistake!"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 5:56 why did Sal say Fe3O3 when it should of been Fe2O3? Because a mole of Fe3O3 should be approximately 216 grams. Or do I just don't understand?",
        "A": "A text box appears bottom right at 5:43 to say Sal wrote Fe3O3 but meant Fe2O3."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "sorry if this is a dumb question, but at 2:00, Sal writes a 2 in front of Fe, but when he was writing out the equation, he put the 2 as a subscript. What is the difference and why?",
        "A": "2Fe means 2 iron atoms. Fe2 means 1 molecule of 2 iron atoms bound together."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "at 0:37 what are reagents",
        "A": "It is a substance or mixture for use in chemical analysis or other reactions."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "at 5:55, where did sal get Fe3O3?",
        "A": "thank ya. that confused me greatly..."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 0:39, did you mean to say reactants and products? Because I'm currently taking Chemistry I, and I have never heard my teacher use the word 'reagents'. :o I'm very confused.",
        "A": "Reagents and Reactants are synonyms. They mean the same thing. Reactants are reagents and reagents are reactants."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 2:04, what is the the true meaning of stoichiometry in terms I can understand?",
        "A": "stoichion=element metros=measurement stoichiometry is basically a branch of chemistry dealing with measurement of the amounts of reactants required or products made in any chemical reaction"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 4:54, would O3 not be considered as a diatomic atom, thus doubling the initial molar mass of 48?",
        "A": "The prefix di- means two. An ion with 3 atoms of oxygen in it therefore cannot be diatomic; you mean polyatomic. The ion contains 3 oxygen atoms, each of which has an atomic weight of 16u. 3 x 16 = 48u. Thus the molar mass of the ion is 48g/mol."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At around 5:40, Sal changed Fe2 O3 to Fe3 O3. How did that get there? I probably missed something, but can someone clarify?",
        "A": "I m pretty sure it was a mistake. It should stay Fe2O3."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "I do not fully understand what Sal was saying about the equation after 8:53, can someone please explain, sorry for my ignorance.",
        "A": "After 8:53 He s telling about limiting reagents. If you have more than 26.82 g of Al you still wouldn t be able to get more Al2O3 as you have only 85g of Fe2O3. In this case, the Fe2O3 is the limiting reagent. Likewise, even if you have more than 85 g of Fe2O3, you still wouldn t be able to get more as you only have 26.82g of Al. In this case, the Al is the limiting reagent. Watch the next series of videos on KA. Hope this helps :)"
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "Ok, around 4:00, is it not possible to simply use unitary method to find the molecular weight of Al :\n~ 1 gram of Fe2O3 gives 2 grams of Al\nthus, 85 grams of Fe2O3 will give 2/1*85 ?",
        "A": "The coefficients in a balanced equation represent the ratio of particles. While all moles are the same number of particles, all particles do not weigh the same. It s kind of like comparing apples and watermelons - if you need 1 apple and 2 watermelons, it s not saying the same thing as if you need 1 pound of apple and 1 pound of watermelons. We convert to moles first so that we can compare the number of particles instead - because particles as independent units are what actually react."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "A 5:50, isn't Fe3O3 supposed to be given as Fe2O3?",
        "A": "Yes, you are correct. On some videos they have made corrections with bubbles or script added, but this error seems to persist."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "at 5:46 why did you change the Fe2O3 to Fe3O3?",
        "A": "That was a mistake. He should have left it as Fe2O3. However, he did use the correct molar mass for Fe2O3 even though he accidentally wrote Fe3O3."
    },
    {
        "video_name": "SjQG3rKSZUQ",
        "Q": "At 5:35 in the video, the equation reads Fe3 O3 but shouldn't it be Fe2 O3?? i think you changed it on accident because it goes back to being the original equation. haha",
        "A": "There is a note on the bottom right of the video at this time that says: Sal wrote Fe3O3 but meant 1 mole of Fe2O3."
    },
    {
        "video_name": "9GPuoukU8fM",
        "Q": "At 10:00, isn't that a symmetric molecule so the signal of the C-O bond at 1100 shouldnt be present?",
        "A": "The molecule is symmetric if we assume that the R groups are identical. However, the carbonyls are not symmetric and they are what is giving rise to the absorption around 1100. This is not like the alkene case \u00e2\u0080\u0094 in an symmetric alkene there is no dipole and therefore no signal. That reasoning doesn t apply to an acid anhydride where each carbonyl has a strong dipole."
    },
    {
        "video_name": "9GPuoukU8fM",
        "Q": "Hi guys,\nAt 3:34, the peak at 3300 cm-1 could also be C-H (sp hybridized), how do we know which is which?\nThank ;)",
        "A": "the reason is that: if there is a sp hybridized C-H bond occur, there usually would be a C(triple bond)C occur around 2150 on the spectrum, which there is no such peak in that spectrum."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "At 8:49 would that be written as 4m solar masses?",
        "A": "Yes. 1 solar mass = the mass of our sun"
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "What are electron, nuetron, and quart, degenerecy pressures? Sal mentioned them at 4:46.",
        "A": "Degeneracy Pressures are those forces created by the resistance of particles to be in the same quantum state. For example, you probably know that electrons repel each other. When they get close, they exert a pressure on the other electrons around them. This force is able to stop the gravitational collapse of some stars, but if the mass is too great, then the dying star will form a black hole."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "At about 2:30 Sal mentioned there aren't any intermediate black holes. Why?",
        "A": "We don t know. Its just that an intermediate black hole has never been observed."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "at 4:46 in the video, what does he mean by primordial black hole? the term still confuses me...",
        "A": "A primordial black hole is formed not by the gravitational collapse of a large star but by the extreme density of matter present during the universe s early expansion."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "at 4:50 there is a mention of quark degeneracy pressure what is that?",
        "A": "Degeneracy pressure is the state at which you can t fit any more particles into a volume without forcing those particles into a higher energy state. So, for quark degeneracy, it is the amount of quarks that can fit into a volume before the quarks degrade to a higher energy state."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "At 8:10, what is the name of the black hole in the middle of the Milky Way?",
        "A": "Sagittarius A* is the name"
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "at 3:39, is there such thing as a Ultramassive Black Holes.",
        "A": "Nobody knows. But as far as people know there are no Ultramassive black holes."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "What does the term solar masses mean (this is mentioned around 1:24)?",
        "A": "A solar mass is the mass of the sun so a star with 10 solar masses has 10 times the mass of the sun."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "at about 1:43 solar mass is mentioned? what is it actually? what's its amount",
        "A": "One solar mass is about 1.9891 \u00c3\u0097 10^30 kilograms. The symbol representing this is a lowercase m with a subsript of a circle with a dot in it."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "In 7:10 would the matter be able to skip directly into a black hole form? if not, wouldn't they have to become stars first, and due to the same scenario with background radiation, wouldn't at least some of the photons from the beginning of these enormous stars just now be reaching us?",
        "A": "might jump to a black hole"
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "Somewhere around 7:20 sal says that there is a super massive black hole in the middle of the milky way but if it really existed our milky way would be in the shape of a sphere and not like a disk right but we have pictures and proof that our galaxy is disked shaped so how can we justify that there is a blackhole in the middle of our galaxy",
        "A": "Your assumption is just not right. There s no reason a supermassive black hole would force a galaxy to be spherical rather than spiral"
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "At 3:26, why does the black hole grow as it consumes more matter? Isn't the black hole just a point in space? Where does matter even go after it passes the event horizon?",
        "A": "It s not a point in space. That s just the mathematical prediction of the theory of general relativity, but since that prediction is non-sensical, most physicists believe GR needs modification for the extreme conditions of a black hole. We assume that anything that fall into the black hole stays in there somewhere, in some form, but since nothing comes out we really don t know what happens in there. I suspect we will never know."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "At 6:13 sal said quantum fluctuations. what's that?",
        "A": "quantum, or vacuum fluctuations, are changes in energy that are happening in a vacuum. Based on the Heisenberg Uncertainty Principle, matter and antimatter annihilate each other, causing these changes in energy. Hawking radiation happens due to quantum fluctuation. Say that we have subatomic particles and their antimatter counterparts near the black hole. One of them gets trapped in the black s hole event horizon, eventually being eaten by the singularity. However, the other particle is emitted as electromagnetic radiation."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "At 3:27, Sal mentioned that the Black Holes basically keep growing. So....\nIs there even an end to a black hole?",
        "A": "Black holes do experience what we call evaporation through Hawking radiation but this is a very slow process so until the amount of matter that is falling into a black hole becomes less than the rate of evaporation black holes will continue to get bigger. It is estimated that it will be about 10^100 years before the last of the black holes in the universe evaporates."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "I'm kind of curious. Since black holes - whether the stellar ones or the supermassive ones mentioned in 1:50 - have extremely high gravitational pull, could it be possible that the supermassive ones were made by two stellar black holes being attracted to each other? I mean, I imagine it's not entirely impossible for them to sort of crash into each other and get bigger like that, right?",
        "A": "No, as the video states, supermassive black holes are hundreds of thousands to billions of solar masses . Stellar mass black holes max out at about a few dozen solar masses."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "At 8:30, Sal mentions that there is probably a Super Massive Black Hole at the center of the Milky Way. I thought the sun was at the center! So, is the sun some sort of black hole, or is the sun not at the center of the galaxy?",
        "A": "You have much to learn. The milky way is billions, trillions, possibly quadrillions bigger than the Sun. the sun is on the outer arms of the milky way."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "At 1:28 Sal said mentioned how large a black hole was, he used a measurement called solar masses how large is 1 solar mass?",
        "A": "It is equal to the mass of the Sun, about two nonillion kilograms or about 332,950 times the mass of the Earth, 1,048 times the mass of Jupiter. 1.98892 \u00c3\u0097 1030 kilograms"
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "At 7:37 you said that black holes would become the center of future galaxies. Well, if the black holes have such a powerful gravity then why aren't the celestial bodies of the galaxy being attracted towards the black holes? Stars are fast enough, they can escape the black hole's pull but what about the planets, satellites, etc near the center of the galaxy?",
        "A": "Black holes do not have some new suction power. They are simply have their mass packed into a smaller volume. At distances outside of the event horizon, matter in orbit around a black hole will continue to orbit the black hole, just as it would any other object with that mass."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "at 7:53, you mentioned something called 'angular velocity'. what is that?",
        "A": "angular velocity is actually the number of revolutions in a given time period. Take for instance a point on the circumference of a circle, the number of revolutions per minute would be it s angular velocity."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "At 8:04, might that explain why the Milky Way is in a spiral shape? Is it possible that the black hole is pulling the galaxy in a spiral pattern? Is that even possible since the arms of the Milky Way isn't inside the event horizon?\nThank you!",
        "A": "Yes, it does somewhat. The black hole would be responsible for holding our galaxy in it s shape."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "At 3:27, he said that a black hole would keep growing. Wouldn't that mean a black hole could eventually swallow huge objects like an entire galaxy? If you went into a black hole where would it lead?",
        "A": "A black hole grown quite slowly, a black hole with the mass of our entire galaxy would have a diameter of only 0.314 light years. Once you enter a black hole you fall into the singularity in the center and never leave. It doesn t lead anywhere."
    },
    {
        "video_name": "DxkkAHnqlpY",
        "Q": "You said at 1:00 that black holes are black so if space is black how do you know when your being sucked up by one",
        "A": "you know because you literally get streched and you feel the gravity"
    },
    {
        "video_name": "wD15pD5pCt4",
        "Q": "at 7:02\n\nCan we name it 2-dichlorocyclehenanone or MUST it be 2,2-dichlorocyclohexanone?\n\nThanks",
        "A": "You need the 2,2- since the location of each substituent must be specified."
    },
    {
        "video_name": "wD15pD5pCt4",
        "Q": "At 5:41 would it be incorrect if naming the methyl propyl ketone in the systematic way was 1-methyl-1-butanone to give the functional group the lowest priority rank?",
        "A": "For the systematic name that 1-methyl bit you have is part of the longest carbon chain, so the systematic name is pentan-2-one (sometimes seen as 2-pentanone). So yes that name you have cannot be the systematic name. In general if you end up with 1-methyl in a name you should double check if you found the longest chain correctly."
    },
    {
        "video_name": "wD15pD5pCt4",
        "Q": "At 7:46 could you also call the chain phenylethanone?",
        "A": "Yes. Acetophenone is a common name of the compound."
    },
    {
        "video_name": "wD15pD5pCt4",
        "Q": "what is the difference between a phenol and phenone, there is no -OH group on the molecule being named 7:48",
        "A": "The benzyl or phenyl group that is attached to the acetone scraps its -yl ending and adds the -one ending. Hope that helps! =D"
    },
    {
        "video_name": "wD15pD5pCt4",
        "Q": "At 5:45, would the propyl actual be considered an Ethyl group by the new IUPAC naming system? (2-methyl-3-ethyl-pentan-2-one) ?",
        "A": "No, the IUPAC way is to just name it pentan-2-one. The carbon chain is 5 carbons long, if it were an alkane it would be pentane. Because this is a ketone, this changes the suffix from -e to -one And the carbon of the ketone is the second carbon in the chain So pentan-2-one"
    },
    {
        "video_name": "wD15pD5pCt4",
        "Q": "At 6:00 , in the cyclohexane, each carbon is bonded to only 2 other carbons. Doesn't Carbon need 4 bonds to become stable?",
        "A": "Remember that hydrogen atoms are not shown explicitly in bond-line structures. Any unfilled valences on carbon are assumed to be filled by hydrogen atoms."
    },
    {
        "video_name": "65uG6v7YY2Q",
        "Q": "at 7:57, does the direction of the velocities not matter because the squares will make them positive anyways?",
        "A": "Correct; one could also say that the direction of the velocities does not matter because we are looking for a scalar quantity, that is, kinetic energy."
    },
    {
        "video_name": "65uG6v7YY2Q",
        "Q": "At 11:40, why does he say that the work done is equal to the change of potential energy? Shouldn't it be 1/2mv^2 - mgh because the box starts off with mgh and ends with kinetic energy? Also, what is doing that work on the box, is it gravity?",
        "A": "Work done by gravity converts PE to KE. If you start with PE of mgh and you fall through a distance of h, you have KE that is equal to what the PE was. Energy is conserved."
    },
    {
        "video_name": "65uG6v7YY2Q",
        "Q": "At 11:56, wouldn't the time it takes for the box to fall the same because gravity is the only acceleration that makes the box fall. And the system is frictionless.",
        "A": "No, because the distance traveled is not the same. The length of the ramp with the angle \u00ce\u00b8 is longer than the ramp with angle 2\u00ce\u00b8, so if the boxes are accelerated at the same rate with no friction, the box on the right has farther to travel."
    },
    {
        "video_name": "65uG6v7YY2Q",
        "Q": "At 0:41, he said speed, not velocity, and there is a difference between speed and velocity.",
        "A": "yes. Speed is scalar and ignores direction. Velocity is a vector."
    },
    {
        "video_name": "F7BU1sXtul4",
        "Q": "At 1:47 Sal points out that at time equals 2sec, the 2 spaceships would be 6x10^8 meters away from him. But if the 2nd spaceship starts (when t=0) at 3x10^8 meters, then it doesn't make sense that they would have the same distance from him... Can someone explain?",
        "A": "Sal doesn t say that the two spaceships will be 6x10\u00e2\u0081\u00b8 m after 2 seconds. Sally, the spaceship that was at the origin, will only move 3x10\u00e2\u0081\u00b8 m in 2 seconds, while the spaceship that was already a distance 3x10\u00e2\u0081\u00b8 m away from the origin will further move (2X1.5X10\u00e2\u0081\u00b8 =) 3X10\u00e2\u0081\u00b8 m in those 2 seconds. So in total, that spaceship would be 6X10\u00e2\u0081\u00b8 m from the origin, where Sal is. You can also see this from the graph. By \u00ce\u0094t = 2 s, the two spaceships have traveled different distances. Hope this helps."
    },
    {
        "video_name": "_RTF0DAHBBM",
        "Q": "In the video capillary action and why we see a meniscus, Sal said that water is more attracted to glass than to itself. If the cup in 3:47 is a glass cup, would that affect the bulge?",
        "A": "No, because at that point there is no more glass to stick to since you are stacking water over the capacity of the glass cup."
    },
    {
        "video_name": "_RTF0DAHBBM",
        "Q": "At 3:15 you described about objects being able to float but I dont understand how surface tension can make the object float and when you push it down, it will drown without floating back up.",
        "A": "because surface tension as a force acts only on surfaces, once an object is in the mass of liquid, only forces acting on it are gravity and buoyancy. it can float up if its density is smaller than that of the liquid."
    },
    {
        "video_name": "nJ-PtF14EFw",
        "Q": "at 9:35 jay talks about the energy levels being quantized. what does it mean?\nwhat is a quanta? what is quantum physics all about?\nit'd be very kind if you could give an analogy or an example to understand this concept of quantizing and quantum.",
        "A": "Quantized levels means that only certain values of energy are allowed, because energy comes in little packages called photons. You can have 1 photon of energy or 2, but you can t have 1.5. Quantum physics is all about the consequences of this fact. They turn out to be quite surprising and unintuitive."
    },
    {
        "video_name": "nJ-PtF14EFw",
        "Q": "Why do I have to provide the exact amount of energy so the electron would transition ? At 6:29 that's said",
        "A": "Because there is no in between place to go. Think of it like the rungs on a ladder."
    },
    {
        "video_name": "nJ-PtF14EFw",
        "Q": "At 6:17 What will happen if we provide energy of 10.3eV to the electron?",
        "A": "Nothing. The atom cannot absorb that amount of energy. It can absorb 10.2, but not 10.3. 10.3 would put the electron at an energy level where it is not allowed. It can only jump from one permitted energy level to another. It is never, ever, ever allowed to be anywhere between those permitted levels."
    },
    {
        "video_name": "nJ-PtF14EFw",
        "Q": "At 2:58, Jay says that in the Bohr model, energy is quantized. Is this because, in the derivation of the formula for atoms' radii, we quantized the centripetal acceleration L as rmw sin (theta); r = the vector radius, m = mass, and w = angular velocity? Does limiting the values that centripetal acceleration can have also limit the values that Bohr-model electrons can have?",
        "A": "What do you mean when you ask is it because of what we did in the derivation? it s quantized because it s quantized. To put that into a mathematical model, Bohr assumed that the angular momentum had to be some multiple of h. If you quantize angular momentum you have also forced the quantization of energy."
    },
    {
        "video_name": "nJ-PtF14EFw",
        "Q": "since at time 1:07 in the video the units joules gets cancelled out so from where the eV taken as a unit suddenly?\nit must be unit-less instead",
        "A": "He set up an equation to convert a certain energy value from joules to electron volts, the conversion factor is 1 eV = 1.6E-19 J So when you turn that into an equation you get: -2.17E-18 J * (1 eV / 1.6E-19 J) = -13.6 eV (energy in joules) * (1 eV per 1.6E-19 J) = (energy in electron volts)"
    },
    {
        "video_name": "nJ-PtF14EFw",
        "Q": "At 6:20, if we give more than the specific amount of energy to the electron, say 11 eV what would happen then?\nwould it go to the second energy level or just stay at the first level?",
        "A": "It cannot absorb any amount of energy that is not equal to a permitted amount. That s what quantization means."
    },
    {
        "video_name": "VBcEz8bVbL0",
        "Q": "at 1:14 she says that cortisol is the hormonte to produce fat. I thought that it's a stress hormone.",
        "A": "Cortisol is a stress hormone. It s also an important hormone for your circadian rhythm, it s not just released during stress. Cortisol levels rise every morning, one effect of this is that you get hungry. Too much stimulation from cortisol could lead you to over-eat, which would lead to more fat deposition in your body"
    },
    {
        "video_name": "CFygKiTB-4A",
        "Q": "At 5:40, can you also use pythagorean theorem to get Vay?",
        "A": "yup, you can. i think he just uses sin to be consistent with the Vax although you could just remember that it s a 30-60-90 triangle and say it s 1:sqrt(3):2"
    },
    {
        "video_name": "CFygKiTB-4A",
        "Q": "In the video at 5:25 why you put always square root of 3/2?",
        "A": "Cos30 and sin60 equals square root of 3 over 2"
    },
    {
        "video_name": "CFygKiTB-4A",
        "Q": "at 7:35 momentum in y direction is -10 right how can it be 10",
        "A": "its just the sign convention you can happily make a new one but you must follow it at-least till the end of the answer !"
    },
    {
        "video_name": "CFygKiTB-4A",
        "Q": "When Sal says that the momentum before and after is always going to be the same, (2:08) he means that when the system ends, since both of the sides are the same, everything equals \u00c3\u0098 m/s?",
        "A": "Not exactly. Both sides must be the same, but that doesn t mean the momentum or velocity has to be 0. In this example, the momentum of ball A is 10 kg m/s in the Y direction, and the momentum of ball B is -10 kg m/s in the Y direction. The total is still 0, but the momentums and velocities aren t 0. The values cancel each other out. The same happens with the X direction. Before and after, the total momentum remains at 30 kg m/s."
    },
    {
        "video_name": "CFygKiTB-4A",
        "Q": "how did you find the cosine 30 at 5:15?",
        "A": "He didn t, he just remembered it. There are a bunch of angles that you might want to memorize: sin 60\u00c2\u00b0/cos 30\u00c2\u00b0: sqrt(3)/2 sin 45\u00c2\u00b0/cos 45\u00c2\u00b0: sqrt(2)/2 sin 30\u00c2\u00b0/cos 60\u00c2\u00b0: 1/2 You can get the tan of angles using the fact that tan \u00ce\u0098 = sin \u00ce\u0098/cos \u00ce\u0098: tan 30\u00c2\u00b0 = sin 30\u00c2\u00b0/cos 30\u00c2\u00b0 = (1/2)/(sqrt(3)/2) = (1*2)/(2*sqrt(3)) =1/sqrt(3) = sqrt(3)/3"
    },
    {
        "video_name": "CFygKiTB-4A",
        "Q": "At 4:46-4:45, how come you divided the x-component by 2",
        "A": "hi, actually, it s a basic trigonometric ratio. cosine of the angle = adjacent side / hypotenuse. In the above problem, we can see that adjacent side is Vx i.e. our x-component of velocity and the hypotenuse is 2. Hence, when we substitute these two values in the cosine formula we get Vx/2. Hope this helps! :)"
    },
    {
        "video_name": "CFygKiTB-4A",
        "Q": "7:49 around, shouldnt the momentum be -10kgm/s since it is in th opposite direction",
        "A": "He realizes this at 8:25 and corrects the 10kgm/s to be -10kgm/s for Pby momentum."
    },
    {
        "video_name": "CFygKiTB-4A",
        "Q": "at 7:52, how is the momentum in the y direction for B = 10? If you subtract 10 from both sides, you get -10, but then thats in the opposite direction.",
        "A": "minus represents the negative direction. Mr. Khan forgot to put minus at first but then later on the video he shows minus and talks about it being in the opposite direction."
    },
    {
        "video_name": "CFygKiTB-4A",
        "Q": "Hey guys, just to make sure I got it. At about 03:30, the Piy is equal to 0 because B stands still and A is moving in one dimension? I mean, considering at least what I understood, A won't have a two-dimensional movement(therefore not having a y component) until it hits B right?",
        "A": "Thanks, man :)"
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "Confused. At 6:18, he says you number them so that #1 is starts with the \"lowest\" letter of the alphabet. I thought you make #1 the one that will make the next substituent the next lowest number, and only if there's a tie you then go alphabetically about the next one. Help?",
        "A": "Correct. When you number the atoms in the ring, you make #1 the one that will give the next substituent the next lowest number. So the numbers in the ring are 1, 2, and 7. But when you write the names of the substituents in the ring, you list them alphabetically with the numbers of the carbons to which they are attached. Since butyl < heptyl < propyl (b < h < p), the name becomes 1-t-butyl-7 heptyl-4-prorpylcyclononane."
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "At 5:13 the side group is a heptyl group. If there was a 10 carbon side chain instead of a 7 carbon chain would it be a decyl group? What would the molecule be called in that case?",
        "A": "Yes, it would be a decyl group. The name would be 1-tert-butyl-5-decyl-2-propylcyclononane."
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "@7:25\n\nWhich is the systematic name\n\n\n5-heptyl-2-propyl-1-(1,1-dimethylethyl)cycolnonane\nOR\n1-(1,1,-dimethylethyl)-5-heptyl-2propylcyclononane",
        "A": "Neither answer is correct. The correct answer is 5-heptyl-1-(1,1-dimethylethyl)-2-propylcyclononane. Multiplying prefixes such as the di in dimethyl are ignored in determining the alphabetical order. So you use the order h>m>p (heptyl>methyl>propyl)."
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "can the answer also be 4 sec-butyl tridecane or 4-2 methylpropyl tridecane at 3:28",
        "A": "No. If two different chains have the same length, we must choose the chain with more substituents. The chain you chose has only one substituent. The chain chosen in the video has two substituents, so that one is chosen as the main chain."
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "Why isn't that an isobutyl at around 6:50?",
        "A": "No, that s tert-butyl for sure. Isobutyl looks like a Y . Tert-butyl looks like a W ."
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "at6:43, why is I considered in the case of isopropyl, but t is not considered in ter-butyl, while deciding which functional group comes alphabetically first? please help...im stuck...",
        "A": "The reason is partly historical. The n- and tert- are prefixes. They are modifiers of the name and are separated from it by a hyphen. The iso is not separated by a hyphen and is considered part of the name. You alphabetize according to the names, not the prefixes. Thus, the following groups would be alphabetized in the order sec-butyl, t-butyl, isobutyl, n-propyl Butyl before isobutyl before propyl. But if you have both sec-butyl and t-butyl, s comes before t."
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "At 4:31 y not take the top line chain it has 14 carbon. I mean the 7 carbon chain + the top line of the cyclononane chain + the 3 carbon chain",
        "A": "coz that part would not be a chain .it would be a part of the ring"
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "At 3:16 it is named but why wouldn't the main chain at the 4th position be a butyl? Why is it 3-methyl-4-propyltridecane vs 4-secbutyltridecane? It is a butyl group so why separate the methyl group and ethyl group?",
        "A": "The methyl is at the 3rd carbon and the propyl at the 4th. The are not connected to the same carbon."
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "Can Sal please explain what just happened around at 07:36.?\nI don't get it. How do we do the numbering when we have a cycloalkane as a backbone chain?",
        "A": "if cycloalkane is the the backbone then you still have to number from the longest chain"
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "Is this answer also correct ? @ 3:30 [the first example]\n4-(2-isobutyl)-tridacane\nor\n4-tert-butyltridecane",
        "A": "I would name that one like Sal did."
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "at 6:38 Sal says to ignore tert and sec but according to my text and another text you do count sec tert and iso as far as alphabitizing goes. can someone help?",
        "A": "In my textbook & class we ignore sec and tert, it doesn t really matter though. If you re teacher says to ignore them, then do it. If you end up being a professional chemist, then it won t matter because the same information is conveyed whether or not you use this convention."
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "At 3:15, why not the first compound will have 4-(1-methylpropyl)-tridecane as it's name?",
        "A": "Did you listen to Sal in the video? He explained this from around 00:40 on. When you have 2 parent chains of the same length you choose the one that gives more substituents not less."
    },
    {
        "video_name": "Se-ekDNhCDk",
        "Q": "At 3:55, Sal mentions two candidates for the core structure of the molecule: (1)the heptyl group and (2) the cyclononyl group. The cyclononyl group is larger so he uses it as his core structure. But why didn't he use the tetradecane group that goes from the end of his heptyl group to the end of his \"tert\"-butyl group?",
        "A": "He didn t use it because you can t continue across one side of the ring to make a longer chain. When you ve got a cycloalkane, the naming rules say it s treated as its own group."
    },
    {
        "video_name": "Y5cSGxdDHz4",
        "Q": "At 1:45 Salman Khan told that Avg. Velocity = Change in distance by change in time, But I thought that it was Initial vleocity+ Final Velocity /2? Which one is correct?",
        "A": "Your formula only works when acceleration is constant."
    },
    {
        "video_name": "Y5cSGxdDHz4",
        "Q": "At 7:53 we can only use that method if the acceleration is constant right? So, my question is, how do we find the average velocity if the acceleration is not constant?",
        "A": "You usually need to use calculus for that. You use the same idea, calculating the area under the velocity vs time curve, but calculating the area is more complicated when you don t just have a nice simple shape."
    },
    {
        "video_name": "Y5cSGxdDHz4",
        "Q": "I remember \"Vf^2 = Vi^2 + 2 times a times (delta)d\" from the last lecture (4:41), but here at 0:43, Mr. Kahn has \"Vf^2= Vi^2 +2 times (delta) d\". . . . What happened to that \"times a\"?",
        "A": "He just forgot it! As you can see also in this lecture at 8:00 he writes it again."
    },
    {
        "video_name": "Y5cSGxdDHz4",
        "Q": "In 7:40 he stated that the formula for average velocity is (Vf+Vi)/2 assuming the acceleration is constant , but what if the acceleration wasnt constant and we were dealing with a jerky motion what the formula for average velocity might be ?",
        "A": "Then you would need to calculate the average in a different way, perhaps with calculus."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "At 3:10, Sal said that he is pushing the object upwards with a force of mg up-to a height of h. How is this possible? Gravity is pushing it downwards with a force of mg. The two forces cancel each other and the object will not rise.",
        "A": "Watch the video again. He says a couple of times he is assuming the elevator is moving in constant velocity. That is, there was an initial impulsion, but now the forces only need to cancel out."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "At 6:40 ,\nYou say Potential energy is the product of mass times force of gravity times the height.\nShouldn't it be Acceleration due to gravity?",
        "A": "Yeah. He means that. No doubt about that."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "At 5:16, it is said that the force pulling upwards is equal to the force of gravity pulling downwards, so there is no change in velocity since there is no net force. But then how does the object get pulled up to 100 m is the force pulling is down is equal to the force pulling it upwards",
        "A": "An object moving with constant velocity will have no acceleration but it will displace [ constant(uniform) velocity means equal displacement over equal intervals of time] . Thus it can be moved up to 100 meters above the ground with a constant velocity."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "@ 4:03 is the work done by gravitational pull of the earth is 0 as no displacement of body in downward direction ?",
        "A": "By definition of work, there must be a displacement. No matter how large the force is applied, with zero displacement, then no work is done. Hope this helps."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "@9:05 Sal says that there's some problems with the law of the conservation of energy...what is it?",
        "A": "there are some limitations and conditions"
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "At 1:50 how come the units are Newton metres? I mean shouldn't it be\nKg (m/s)^2",
        "A": "Both things are same Newton meter is in SI system and Kg (m/s)^2 is in fundamental unit."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "At about 3:50, why don't we put the gravitational force to the force diagram?",
        "A": "Gravitational force is self explanatory since it s earth and there is no need to enter 10N/kg but it would have been better if he put though"
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "At 3:08 we are applying force equal to acceleration due to gravity(mg) upwards.\nWhen acceleration due to gravity is pulling something downward, we take g positive, so when we are pushing that box up, why are we not taking g negative as it a vector quantity?",
        "A": "You can define positive and negative however you want. If you define up as positive, then g is -9.8 m/s^2 and an upward force is positive. If you define down as positive than g is 9.8 and an upward force is negative. It doesn t matter, the answer will be the same."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "At 3:17, he said the force going up is mg. Isnt mg=weight which acts vertically downwards?\nSorry if this question is silly, just confused..",
        "A": "Weight pulls down. If the object is not accelerating down, there must be a force equal to the weight pushing up."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "at the problem from around 3:00 is the force the net force or the force applied?",
        "A": "It s the force applied to move the object upwards"
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "at 5:00 ,Sal says the force applied is equal to the force applied by the gravity,HOW IS THAT POSSIBLE,the object would remain at rest [the elevator]?",
        "A": "The key is that the elevator is moving at a CONSTANT velocity. If the forces acting upon an object are in equilibrium (i.e. the vector forces add to zero), than the object isn t necessarily at rest, it is just not accelerating."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "at 6:00 , u explained the workdone by gravity or workdone by external force?",
        "A": "At 6:00, he is explaining the work done by an external force to get it to reach 100m. Once it reaches the top, the work is transferred to potential energy."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "In the first example (around 2:10) I got 122.5 joules. What did I do wrong?",
        "A": "Very interesting, I think Sal just made his first mistake in a video. I calculated it 5 times and it equals 122.5 joules."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "guys at (0:59) he says mv^2/2 is the formula for kinetic energy ,but it is defined as F.S=mv^2/2 so isn't it supposed to be just another formula for work in terms of mass and velocity ??\nI know that its a little vague but its really bugging me out.Thanks for answering.",
        "A": "The reason for the similarity in ideas is that if you have an object and work is done on that object the energy of that object changes by the amount of work done. So if the work is done on an object that starts at rest and the work goes into accelerating the object the kinetic energy will equal the amount of work."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "At 5:14, he says that the force pulling upwards is equal to exactly the force pulling downwards... If it is so, then the elevator should not move at all instead of moving upwards...Isn't it ?",
        "A": "No. Newton s first law says if the elevator is already in motion it will stay in motion if the sum of the forces on it is zero. Force causes CHANGE in velocity, it does not cause velocity. F = ma. If F is 0 then a must be zero, but you can t tell what v is."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "at 5:19 you said that force pulling the elevator upwards is equal to the force pulling downwards, but how can this be possible if the elevator is moving upwards with constant velocity as it is still moving upwards?",
        "A": "If the elevator is not accelerating, the forces are balanced. Objects in motion stay in motion unless an unbalanced force acts on them. To get the elevator accelerating upward, the upward force had to be greater than the downward force. Once it is at constant velocity, the upward and downward forces must be balanced. When it slows down, the downward force must be greater. When it stops, the forces will be in balance again."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "at 4:9 why should we apply more force than acceleration of gravity to move the lift upward?",
        "A": "Well , using Newton s Laws , if we apply a force F such that F - mg = ma ( a=acceleration) in order to move it up then a>0.This implies F > mg."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "At 5:23, if the upward force is equal to the downward force, shouldn't the elevator just not move?",
        "A": "No net force means no acceleration. It does not mean no motion."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "At 5:15 you said there is no net force. if the net force on the object is 0, then wouldn\u00e2\u0080\u0099t the work done be 0 since w=F*d",
        "A": "The net work is 0, and that s why the KE doesn t increase. The work done against gravity is positive and the work done by gravity is negative. If you lift something up at constant velocity you do work on it. Gravity does negative work of the same magnitude. The positive work against gravity increases the PE."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "How is force equal to mgh (m=mass,g=gravity,h=height) as mentioned at 5:50",
        "A": "Newton s second law is F = m * a . here a is acceleration. in case of gravity g = 9.1 m/s\u00c2\u00b2 = a. now moving on to work : Work is equal to W = F * s. We can replace f with ma getting W = m * a * s. In case of gravitation replacing a with g we get W = m * g * s and we can replace s with h getting our equation W = m * g * h"
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "at 1:54 what is a newton meter?",
        "A": "The video creator says in the video that a Newton x meter is just a Joule. That is, a Newton x meter is a measure of work or energy, which is a measure of force through a distance."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "At 5:11, Sal says that the force pulling downwards is equal to the force pulling upwards, yet somehow, the elevator is still moving up with a constant velocity. Wouldn't the two forces cancel each other out, and result in no movement of the elevator?",
        "A": "elevator moves upward due the 1st law of motion that states that a body will remain in the state of rest or uniform motion until or unless any external force is applied.in this case upward and downward force cancel out each other and there is no change in its velocity because a force produces an acceleration and acceleration =0."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "Just need to get cleared up on one thing....\nAt 5:15, if both the force pulling upwards and the force pulling downwards is the same, how does the elevator move up?",
        "A": "If it is already moving, then it will keep moving. If it isn t, it won t. Balanced forces mean no acceleration. They don t mean velocity is zero."
    },
    {
        "video_name": "3mier94pbnU",
        "Q": "At 5:50, if F=ma, and the elevator isn't accelerating, why doesn't F=0 and thus work=0? I get, intuitively, why there is work being done, but based on the equations why does it not work out like that?",
        "A": "F does = 0 The work done by the elevator is equal to the (negative) work done by gravity. So the net work is 0. The work done by the elevator is positive and it appears as an increase in the PE"
    },
    {
        "video_name": "m1PPujngqAw",
        "Q": "At 1:17,he mentions that the air is moving at medium velocity and as he shifts to left at 1:21 and says that the air is moving faster. Why so?",
        "A": "He s just hypothetically saying it because he wanted to use it to demonstrate how density affects the amount of particles flowing through it. Similar to magnetic field lines, there are various densities of them so the more magnetic field lines are in the net the more the change of magnetic flux."
    },
    {
        "video_name": "m1PPujngqAw",
        "Q": "At 7:15, I see that we want the component of the magnetic field that is perpendicular to the surface, So shouldn't the formula for flux involve Cross product? Why is it dot product?(From the next article, What is magnetic flux?) Why am I wrong?",
        "A": "It is dot product because it is a scalar. Perpendicular to the surface is found by going parallel to an area vector that points normal to the surface."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "Around 10:00, give or take, he mentions that the photon has managed to travel 20 million light years in 10 million years. For my own understanding, and maybe that of others, is that basically because the universe is assumed to be traveling at near-light- to light-speed? (Which would mean that the two coordinates are also moving apart VERY quickly, as well, right?)",
        "A": "By travelling I meant expanding. Whoops."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "at around 8:00 I do not get it. If the Universe was only 300,000 thousand years old, how could something be at 30 million light-years distance ?",
        "A": "The universe expanded and is currently expanding faster than the speed of light. In fact, its rate of expansion is increasing. Therefore, having a 30 million light-year distance is not paradoxical in the 300,000 year old universe. (Note that your question says 300,000 thousand years which is really 300,000,000 years or 300 million years in which case your question wouldn t seem like much of a problem but I take it that you meant 300,000)."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "At 2:23, Sal drew the picture of the universe and the point was us. Am I basically in the middle of the universe. Couldn't I be somewhere else instead so the light would take more time to reach me?",
        "A": "You are in the center of your observable universe. Everyone is in the center of his own observable universe."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "2:34 if the light took 13.7 billion years to get to us wouldn't we see the big bang?",
        "A": "Before about 300,000 years after the big bang the universe was too hot for neutral atoms to form, the universe was filled with atoms without electrons attached to them and free electrons. This prevented light from traveling freely so we can t see any details before the universe cooled enough to allow the electrons to bind with the atoms."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "(2:18)are you sure it is a ciricle,Sal?",
        "A": "yes"
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "at 9:50 Sal states that \"Photon says wow i was able to travel 20 million light years in only 10 million years\"\nDoes that mean the photon traveled with the speed grater than speed of light?\nAt what rate the universe is expanding is it more than the speed of light?",
        "A": "I think he mentioned that its not a little bit later later that its not moving faster than the speed of light but the other things are expanding faster than the speed of light so the distance it covered expanded faster than light."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "Yet another question. 10:04 did the photon move 10 mil l.y. but the \"starting point\" moved away from him. Or did he actual move 20 mil l.y.",
        "A": "both"
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "At 6:50 when he is talking about the universe expanding, i was wondering if there was a way we could measure this down to the inch? Put a probe on a planet close to ours and see how much the distance between them grows? Anyone?",
        "A": "The expansion rate is about 70 km/sec per megaparsec of space A megaparsec is over 3 million light years You can do some rough math to figure out about what the expansion rate would be between two planets in the same solar system."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "At 13:00, how is the light ever going to reach us if we're going to keep stretching apart and always be huge amount of distance away from the photon of light that is trying to reach us",
        "A": "Some light will never reach us"
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "At 9:20, he said that the space is stretched as the universe expands. Does that mean that the amount of matter in the universe now is always going to be the same amount of matter no matter what given time?",
        "A": "Basically yes, but it s probably more accurate to say that the total amount of energy and matter will remain the same in the universe (since you can convert matter to energy and vice versa). The amount of stuff in the universe shouldn t change despite the universe stretching."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "At 11:14What was the temperature of the universe when the big bang occurred?",
        "A": "We cannot calculate the temperature at the instant of the big bang but at about 10^-43 Seconds the observable universe had a diameter of about 1 plank length (10^-35 meter) the temperature was about 10^32 K. At about 10^-6 Seconds the temperature would have cooled to about 10^12 K. At about 4 minutes the temperature has dropped to about one billion (10^9) K and this is when the temperature is low enough to allow quarks to combine into protons and neutrons."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "If 13.7 billion years ago we were closer to the object emitting the light at 4:05, the universe was expanding and we were pushed away, does that mean since the light was chasing us the entire way, that the universe has slowed its expansion enough for the light to catch up with us?",
        "A": "No, because one of the properties of the expansion of the universe is that things that are far away are moving away from you very fast, but things that are close by are not moving away from you so fast. So light that started very far away from us could be stuck and unable to reach us, because it is moving away faster than it is traveling toward us. But light that is close enough to have gotten traction on its way to us will reach us eventually."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "At 1:35, Sal talks about how light from a distance 13.7B years away could reach his eye. If so, why can't we SEE that light? Why can't we see infinitely far if we have this light from insane distances, or even closer distances, reaching our eyes?",
        "A": "You can see it, with a telescope. It s too faint for your naked eye. Also, a lot of light that is in the sky is in frequency ranges you can t see with your eyes."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "Okay, one more question (sorry for all the questions, this is just a very confusing video for me). At 13:55, how can we figure out that something that we see as 13.4 billion light years away from us is actually 46 billion light years?",
        "A": "We can tell the distance by measuring the red shift of the light to see how fast the emitting object is receding from us. Then we just apply the Hubble constant to determine the distance."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "At 4:30, Sal draws a 300k years old universe. He then draws the path that a photon is to travel. He says that the path is 30M light years long. If we look at the ratio between the radius of the sphere and the path, the radius is roughly 1.5 times as long as the path. That means that the distance from the center of the 300k years old universe to its outer edge is 45M light years. How has the matter at the edge of the universe traveled 45M light years in just 300k years?",
        "A": "Nothing is to scale here. I hope that s an answer."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "Around 14:02, Khan says that the object that is expanding away from us is about 46 billion light years away from us. How can that be possible if the universe has been around for only about 13.7 billion years?",
        "A": "It expansion is faster than light."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "at 1:18 he writes 13.7 billion years, isn't it really 13.8?",
        "A": "Yes, the most recent estimate is 13.8. This vid was made before that."
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "At 6:20, when he is talking about cosmological time and \"primitive states\", couldn't we relate that to alien life forms? If we just look in a telescope, we will only see their primitive state, while there could actually be beings out there. (Because of how long it takes the photon to travel) So probes are necessary to observe their current state. Is anyone tracking me?",
        "A": "I get your point. Considering the scale of the universe and the fact that we can t really communicate with other galaxies or stars (due to the astronomical distance}, it is unlikely (not impossible) to send probes to such distant galaxies and get their transmissions back. Through a telescope, we might see what they looked like x millions years ago, however, even if we send a probe, to receive a signal from it will also take x million years. Good thinking!"
    },
    {
        "video_name": "6nVysrZQnOQ",
        "Q": "On 11:04, he talks about the space expanding but is there a consistent pattern for the expansion?",
        "A": "Space everywhere expands at a rate of about 70 km/s per megaparsec of space between two objects"
    },
    {
        "video_name": "mJt1QI-yDG4",
        "Q": "At 3:04, he mentions that cyclohexene is the correct answer, but doesn't that mean that there is a double bond in the ring? I can't see any double bonds, and so I thought that molecule that he circled was cyclohexane?",
        "A": "See how there is a second line on the inside of the ring between two carbons? That is the double bond."
    },
    {
        "video_name": "mJt1QI-yDG4",
        "Q": "From 3:30~4:30, why does C=O bond have a higher signal intensity than C=C bond?\nAlso, is it true that the more of a particular bond (e.g. C-H bonds) a molecule has the greater the signal intensity of that bond? That is what I learned from Questions and Answers section under \"Symmetric and asymmetric stretching\" video.",
        "A": "The C=O bond has a greater change of dipole moment during te stretch than the C=C bond does. You need a change in dipole moment for IR absorption to occur. The more bonds of a given type, the greater the intensity of the absorption."
    },
    {
        "video_name": "dVYYh8C80zo",
        "Q": "At 2:26, arc length can also be found using a degree.\nso why do we use radians if we can find arc length by using\n(Theta/360)*2pi*r? I mean why do we have to use radians?",
        "A": "The calculation becomes simpler without the need to incorporate the conversion into the equation. Although it would still be perfectly fine to analyze the problem using degrees if this was the only reason. In the end, it is really up to you."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "At 12:50 what are those cloud like clusters?",
        "A": "Yes they were nebulae"
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "At 12:15, how can we take images of our galaxy when we have just explored some part of our solar system?",
        "A": "We can only take images of our galaxy from our particular vantage point within the galaxy. You can see it with your own eyes if you go out on a clear night with few lights around. Any pictures you see of the Milky Way showing it from outside are merely illustrations."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "at 11:10, did he mean universe or galaxy?",
        "A": "k thats what i thought"
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "What are black holes made of in 9:20? Dark matter? And If there is a black hole in the center of the milky way, which star died to create it?",
        "A": "Black holes aren t made of anything, exactly. The only matter we know of is in the accretion disk, and the light comes from gases swirling around it. The black hole (called Sagittarius A*) at the core of the Milky Way is called a supermassive black hole, because the stuff around it has the mass of 3.3 million of our suns. So it is unlikely that a dying star caused it. Hope this helps."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "When you talk about a black hole in the center of most galaxies, 9:05, wouldn't that mean it's sucking up the stars near it? As in someday it will suck up our solar system?",
        "A": "that s a relief. thanks!"
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "So is sal saying in 5:20 that when we look at certain stars through a telescope that the actual star would not be were we see it, but that it has already moved from it's spot to somewhere else and we simply can't see it in it's real place because it's so far away?",
        "A": "The best way to think of it is, we see the universe as it was, not as it is. We know nothing can travel faster than light. For Example, Alpha Centauri is 4 light-years away, really that means we are looking at Alpha Centauri as it was 4 years ago, because that light is just now reaching us. Items in space to move as well but we are still constantly seeing it how it was."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "At 12:14 what is that really bright star near the bottom-left hand corner? It is much brighter than many of the other stars.",
        "A": "It s a planet... most likely Jupiter."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "My question is in regards to the discussion at 5:09 about the speed of light and how the light we see on earth that is, for example 500 light years, would be 500 years in the past. How do we know that the recently discovered 'twin planet' Kepler-452b, which is 1,400 light years away, has not yet expired some how? Also, why is the zone that this planet was found in called the habital zone. Is it not impossible to ever travel that distance?",
        "A": "At this time it is not possible to go to that planet it is called the habitable zone simply because it is at a point in space that its not to hot or cold for humans."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "At 12:08 how did they get a picture of the milky way?",
        "A": "It s not a picture. It s a computer generated image, based on measurements and observations by astronomers."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "If as Sal said at 5:08 is correct, then if we somehow beat the speed of light and travel 10 lightyears away from Earth (in one second presumablly), then when we gaze at Earth, in a very powerfull telescope, will we see it as it was 10 years ago?",
        "A": "Yes, it is the illusion made by the Finite speed of light. If you were 80 light years away, you would see WWII. If you were 230 light years away, you would see the Revolutionary war. If you were 100 million light years away, you would see dinosaurs."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "At 5:05 is shows some sort of a galaxy. Does that spiral use the fibonacci sequence?",
        "A": "No. the formation of galaxies into spirals is caused by gravitational accretion, not by necessity or efficiency. All galaxies started out as amorphous globulars, and over billions of years of rotating, flattened out and formed arms."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "At 3:51 Sal talks about light years. What are light years?",
        "A": "A light year is a unit of length equal to just under 10 trillion kilometers. I found that out on Wikipedia. Hopefully that will help."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "at 9:11, How can we guess that there is a supermassive blackhole in the middle of most galaxies if we dont even know what that big bright thing is in the middle of our own galaxy?",
        "A": "We observe gravitational effects that would be explained by the presence of a black hole."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "At 7:11, what model was used to come up with the estimate of 200 - 400 billion stars? What factors cause the estimate to vary so greatly and how were they entered into the model?",
        "A": "We count the stars in one particular area of the sky with a cretain density. Then we apply that factor to all areas of the galaxy because we can determine the star density without actually counting the stars. This method is not exact and becaus of the density of the core, we can t see a third of our galaxy. That is why it varies so much."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "On 12:42, What is the bright object on he bottom left of the screen?",
        "A": "It could also be an average star that happens to be close, from our point of view."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "At 12:50 or so, what is the brownish/black stuff? Or is it just absence of stars?",
        "A": "They all are stars, but most of them are just brown dwarfs."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "at 5:28, it means that the changes we observe now being taking place at the \"Tabbi's Star\" would have been occurred years before today.",
        "A": "Absolutely! If the star is 1000 light-years away, the light you see today left that star 1,000 years ago. So if something changed on that star today, for instance it goes Nova, it would take 1000 years before we could detect that light here on Earth."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "1:13 what is the small dot near the boundary of the oort clowd?",
        "A": "what does that mean?"
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "At 12:20, what are the black, cloud-like objects in the galaxy?",
        "A": "It is likely a large dust cloud blocking some light."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "At 7:05 he said 200-400 billion stars... or solar systems, but star is much smaller than a solar system isnt it?",
        "A": "it is a ton smaller than a solar system"
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "11:20 i think he meant to say galaxy and not universe, as the universe is infinate, correct?",
        "A": "We don t know yet. As far as we understand it the universe could only possibly be about 30 billion light years in diameter. As the big bang happened around 15 billion years ago. As we understand physics nothing can travel faster then the speed of light. So nothing could be faster then the distance light could of traveled since the big bang."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "at 12:23 there is a large bright cluster in the bottom of the picture what is this cluster called",
        "A": "Yeah, I think that might be a close star. Probably Sirius."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "how do scientists figure out how many stars are in out galaxy because if you were to count every star it would be impossable and sal said something about one star or even a million stars at 12:46 in the video.",
        "A": "thx aegon targaryen"
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "At 8:46, Sal shows us a artist's depiction of the galaxy. What is at the center of it?",
        "A": "The center of the galaxy is filled with old and bright stars packed tightly together, that is why it is so bright. At the very center, there is a black hole at 4 million solar masses keeping a few dozen stars in very fast and close orbits around it."
    },
    {
        "video_name": "rcLnMe1ELPA",
        "Q": "At 6:02 Sal says that our galaxy is spiral , however he also mentioned that we cannot observe the Milky Way from above.Then how do we know that our galaxy is spiral?",
        "A": "Good question! After long and patient observation of the stars and other objects making up our galaxy, measuring their distance and relative velocity from us, scientists were able to reconstruct the shape of the Milky Way. By the way, the most recent consensus is that the shape is that of a barred spiral."
    },
    {
        "video_name": "T1p_tQZsPLg",
        "Q": "At 0:38, how could Sal use Ohm's Law to determine if the bulb is nonohmic? Isn't it a paradox?",
        "A": "You can use Ohms law at any point to find a value. Its just that you cant have the same resistance at multiple voltages/currents if its non ohmic."
    },
    {
        "video_name": "pBZ-RiT5nEE",
        "Q": "What about the two CH3 groups in the dipole-dipole interactions example (0:01 - 2:54). Was it cancelled out or something? If yes, then why? Also, is there any particular difference between hydrogen bonding and other dipole-dipole interactions, or is it just the same thing but about something specific?",
        "A": "Those are quite insignificant compared to the C=O one, also since they re arranged symmetrically they will cancel out. Hydrogen bonding is a particularly strong form of dipole-dipole interactions when hydrogen interacts with a small highly electronegative atom."
    },
    {
        "video_name": "pBZ-RiT5nEE",
        "Q": "At 7:50, why the left molecule is negative and the other is positive?",
        "A": "This is only an instantaneous picture of the charge distribution. In another microsecond, the picture might be reversed."
    },
    {
        "video_name": "pBZ-RiT5nEE",
        "Q": "at 8:16, what extra forces did you talk about?",
        "A": "He is saying that if you have two big molecules coming together side by side, the London dispersion forces will be exerted all along the sides where they are touching."
    },
    {
        "video_name": "pBZ-RiT5nEE",
        "Q": "I know that hydrogen bonding is supposed to be stronger than the dipole-dipole interaction but other than hydrogen bonding needing a hydrogen atom, I don't understand how the two are different in nature or just why hydrogen bonding is stronger. Jay explains this in the video (3:45-5:15) but there's still something I must be missing. Anyone?",
        "A": "H-bonding is also a dipole-dipole interaction. It is just so much stronger than other dipole-dipole interactions that it gets its own special name."
    },
    {
        "video_name": "pBZ-RiT5nEE",
        "Q": "Around 4:28 you stated that hydrogen bonding is the strongest intermolecular force. Is Ion-Ion bonding not stronger?",
        "A": "The key word here is intermolecular, or between discrete molecules. If the molecules interacted more strongly, in either a covalent or ionic bond, it would, by our definitions, then become an INTRAmolecular bond."
    },
    {
        "video_name": "pBZ-RiT5nEE",
        "Q": "In 3:10 you said that electronegative, what did you mean?",
        "A": "Electronegative is a word that describes how much pull an atom s nucleus has on the electrons in its atomic or molecular orbitals. Some atoms are more electronegative than others, and consequently will often induce polarity in simple molecules, or partial local charges in bigger more complicated molecules."
    },
    {
        "video_name": "pBZ-RiT5nEE",
        "Q": "at 2:30 minutes you explained to us that acetone's room temp. is 56*C, why does it evaporate at room temperature if it is left open?",
        "A": "Yes like other liquids it evaporates at room temperature"
    },
    {
        "video_name": "pBZ-RiT5nEE",
        "Q": "Is London Dispersion (5:51) the same as Induced-Dipole? Thanks!",
        "A": "It is not, however it can caused an induced dipole. London dispersion is when a temporary imbalance it charge causes a dipole. Induced-dipoles can appear when the other atom with a dipole causes it to form one."
    },
    {
        "video_name": "pBZ-RiT5nEE",
        "Q": "LDFs can only occur between non-polar molecules?\nAbout 6:40.",
        "A": "They occur in all types of molecules, but are most dominant in non-polar molecules as there are no other forces to dominate (due to symmetry in non-polar molecules). LDFs are the weakest type of force interactions."
    },
    {
        "video_name": "pBZ-RiT5nEE",
        "Q": "At 7:12, he says the electrons are moving in the opposite direction making that side partial positive.The direction of electrons won't change their sign, they will still remain negative.So what is he really trying to convey here?",
        "A": "Where the electrons are at any given moment is random. If there are more on one side than the other, that side will temporarily feel a negative charge and the other side will temporarily feel a positive charge. This temporary dipole can then cause a temporary dipole in a neighbouring atom or molecule. We know there has to be some form of attraction between noble gas atoms, as they have liquid states at low temperatures. These temporary dipoles are the answer to that."
    },
    {
        "video_name": "pBZ-RiT5nEE",
        "Q": "At 8:18, so the speaker means that the larger the molecules get, the greater attractive forces they have? But why?",
        "A": "This is true for London dispersion forces, simply because the more electrons there are in the molecule, the more these temporary dipoles can form."
    },
    {
        "video_name": "pBZ-RiT5nEE",
        "Q": "At 7:00, if the momentary arrangement of the electrons is completely random, shouldn't it be equally likely for the two molecules to repel? And in large groups, shouldn't these forces on average cancel out, thus giving the substance no \"stickiness\" for it to turn into a liquid?",
        "A": "no, because the momentary arrangement is irrelevant to the bonding."
    },
    {
        "video_name": "W9AVRhqEkek",
        "Q": "is it just me or does anyone else think he added WAY too much hot glue at 1:40\n?",
        "A": "It was to make it stick better i guess."
    },
    {
        "video_name": "dJhxphep_gY",
        "Q": "At 11:39 why does this example not concern stereochemistry but the previous example did? What makes them different?",
        "A": "It does involve stereochemistry. At 11:20 he days, We know that these are chirality centres."
    },
    {
        "video_name": "dJhxphep_gY",
        "Q": "At 1:20, why do the pi bond electrons attack the hydrogen atom in H3O+ instead of the oxygen? It seems like the oxygen would be a good electrophile with its positive charge.",
        "A": "The pi bond electrons attack the hydrogen rather than the oxygen because an acid-base reaction is taking place between the alkene and the H3O+. As he said, the pi bond is acting as a base and the H3O+ is acting as an acid in this case. Acids (H3O+) are proton donors, while bases are proton receptors, which is why the pi bond takes the hydrogen atom from the H3O+."
    },
    {
        "video_name": "dJhxphep_gY",
        "Q": "How comes at 2:07 a hydride shift doesn't occur on the secondary carbon to make a tertiary carbon, but at 6:15 one does occur? What are/how do you know the conditions for a hydride shift to occur.",
        "A": "Nevermind, worked it out... silly me"
    },
    {
        "video_name": "dJhxphep_gY",
        "Q": "At 7:04, how is hydride shift initiated? Does the molecule just automatically rearrange to provide stability or does it need to be catalyzed, or even yet is it just random?",
        "A": "From what I understand based on previous videos, hydride and methyl group shifts will occur when, in this case, the carbocation is at secondary and that achieving tertiary is possible. Same rules apply if the carbocation is primary, then rearrangement would occur in order for it to increase stability from primary to secondary or primary to tertiary."
    },
    {
        "video_name": "dJhxphep_gY",
        "Q": "@ 12:39 you talk about how you do not have too worry about the stereochemistry for the OH molecule. Why is that? Because it is on a tertiary or because it is not at a chiral center?",
        "A": "Because 2 identical substituents are attached to that carbon."
    },
    {
        "video_name": "4rsswT_Rv1M",
        "Q": "in 2:18 ,he says blown up version of resistor what that mean?",
        "A": "Blown up = enlarged"
    },
    {
        "video_name": "4rsswT_Rv1M",
        "Q": "Why, at 0:32, is current shown as \"I\"?",
        "A": "That s the standard symbol for current. It originated with the word for current in another language."
    },
    {
        "video_name": "4rsswT_Rv1M",
        "Q": "8:35 but in case of water the speed of water will increase while going through the constriction acc. to bernoulli's theorem",
        "A": "Yes. The analogy between water in pipes and electric current in wires only goes so far. You can certainly never prove anything about electrical circuits using the water pipe analogy. But it is ok for understanding some things, like why putting resistors in parallel reduces resistance, and why putting them in series increases resistance."
    },
    {
        "video_name": "HvYUKRMT0VI",
        "Q": "At 4:18. If something can be observed that requires at least light particles to move at that place (at the time that corresponds with the information of your observation) which implies that the temperature their is not an absolute zero (for the light-particles are moving). Hence a temperature of absolute zero cannot be observed. Does that make sense?",
        "A": "Intuitively yes, but light is funny and doesn t always act like a particle. Even at absolute zero mass-less things, or nearly mass-less like electrons, can move and interact with particles."
    },
    {
        "video_name": "HvYUKRMT0VI",
        "Q": "At 4:27, Sal said \"electrons collapse into the nucleus\" when reaching 0 Kelvin. So, what are the differences between electron behaviour at room temperature and nearly 0 Kelvin? Is electron has velocity or speed? If so, what is the velocity or speed of an electron? Did the electron slowed down at cold temperature? Can electricity be frozen? Can light also be frozen? Can time be frozen when reaching 0 Kelvin? I'm just curious.",
        "A": "Electrons are in constant motion around the nucleus."
    },
    {
        "video_name": "HvYUKRMT0VI",
        "Q": "At 0:18, why did he use 2 V1's?",
        "A": "That s wrong. He corrected that at 9:22"
    },
    {
        "video_name": "HvYUKRMT0VI",
        "Q": "4:27 Does that mean that there is no \"motion\" in our practical every-day sense? There is absolutely no change in physical position relative to anything else?",
        "A": "That s the idea. Although it s a different question whether absolute zero is achievable at all, due to quantum mechanical phenomena."
    },
    {
        "video_name": "HvYUKRMT0VI",
        "Q": "At 3:53 should the v1 next p2 be v2",
        "A": "Yes, a blue box says that next to the fraction"
    },
    {
        "video_name": "HvYUKRMT0VI",
        "Q": "it is v2 or v1 1:31",
        "A": "On the bottom of the video screen he corrects himself. It is v2"
    },
    {
        "video_name": "Z_GWBW_GVGA",
        "Q": "at 5:00 i didn't understand how the hydrogen can take electron from carbon ?",
        "A": "The H has a \u00ce\u00b4\u00e2\u0081\u00ba charge. The \u00cf\u0080 electrons in the double bond are weakly held and move to form a bond to the H. At the same time, the electrons on the other side of the H move onto the Br."
    },
    {
        "video_name": "Z_GWBW_GVGA",
        "Q": "On 7:50, the Bromine would be the nucleophile and the H is the Electrophile, right?",
        "A": "Well, at 7:50 (when the carbocation is formed) H is neither an Electrophile or a Nucleophile considering it has formed a bond with the 1st carbon and is now stable. The carbocation is now the Electrophile and the Bromide ion is the new Nucleophile :)"
    },
    {
        "video_name": "Z_GWBW_GVGA",
        "Q": "At 3:50, you circled the carbon on the right, but you applied what you said about the hydrogen on the carbon on the left. Does it matter if we put the hydrogen on the right carbon after breaking the double, and eventually put the Bromo group on the left carbon ?",
        "A": "I believe that question is answered in the next video when he explains Markovnikov s rule."
    },
    {
        "video_name": "Z_GWBW_GVGA",
        "Q": "I got lost at 4:51. Are the two equations close to each other when the carbon loses its electron?",
        "A": "As Sal says, the 1-pentene molecule and the HBr molecule must come in collision to start this reaction, so I suppose they must come very close. At that moment a electron from the secondary carbon goes to the H, and so forming a bond with the primary carbon."
    },
    {
        "video_name": "Z_GWBW_GVGA",
        "Q": "At 11:20, when Sal is naming the compound, why doesn't he put something in the name for the CH2-CH2-CH3 group? What would the new name be?",
        "A": "The chain is still 5 C atom s long so that s why it s Pentane, you can view the C atoms like 1 long chain and the length of the chain didn t change (no C-atoms were added or subtracted)."
    },
    {
        "video_name": "axB6uhEx628",
        "Q": "So, if Pangaea is the latest supercontinent(2:36 in the video) Then, can we(the human race) guess who, what, and when the supercontinent before Pangaea was?\nif so, how?!?",
        "A": "yes"
    },
    {
        "video_name": "axB6uhEx628",
        "Q": "On 1:56 it shows that India is near Antarctica. When they separated, was India an island?",
        "A": "Yes it was an island until it collided with Asia."
    },
    {
        "video_name": "axB6uhEx628",
        "Q": "At 3:06, Sal says that there were more supercontinents before pangaea. How do we know this and what did they look like if we know the specifics?",
        "A": "If we rewind time even further back, we can see the old supercontinents Hear are some examples Gondwana Laurasia Euramerica Oldredia Rodinia"
    },
    {
        "video_name": "FfNgm-w9Krw",
        "Q": "At 13:32, when you plug in the velocity for the equation, why do you not have to find the vertical velocity component? Previously you speak about only using vertical forces so why wouldn't we need to use trig. to find the y velocity component and use that in the equation when solving for Fn?",
        "A": "Because centripetal acceleration is v^2/r, and it is always toward the center."
    },
    {
        "video_name": "FfNgm-w9Krw",
        "Q": "With the bike going over the crest of the hill at 10:42, what makes the normal force less than if it was stationary? The ground doesnt know the bike is moving so why would it push up less?",
        "A": "If it were stationary it would not be accelerating. When it goes over the crest it has to accelerate downward. The force of gravity has to be larger than the normal force, otherwise the bike would be suspended in the air."
    },
    {
        "video_name": "FfNgm-w9Krw",
        "Q": "I don't understand.\nAt 11:30, what makes the weight greater than the normal force?\nI thought that Newton's third law states that for every action there is an equal and opposite reaction?",
        "A": "Weight and normal force are not a third law pair. Third law pairs are on different objects, not the same object. When you jump into a pool, your weight is greater than the normal force from the surface of the water, isn t it? When an elephant sits on a table, the weight is going to be greater than the normal force. What happens when that s the case?"
    },
    {
        "video_name": "ka8Yt4bTODs",
        "Q": "At 9:22 and two previous other times, how do you calculate the angles between the hydrogens?",
        "A": "So if the center has electrons that aren t bond to any other atom is also counted as a electron cloud."
    },
    {
        "video_name": "ka8Yt4bTODs",
        "Q": "At 2:40, why isn't the angle 90 degrees? Wouldn't all the electron clouds repel each other equally?",
        "A": "That s right. The electron clouds repel each other equally, but the repulsion is less if the angle between the clouds is 109.5\u00c2\u00b0 than if it is 90\u00c2\u00b0."
    },
    {
        "video_name": "ka8Yt4bTODs",
        "Q": "how can we know that NH3 molecule can also be written in the manner in which the instructor has written in the video ,time 4:53?",
        "A": "He s just redrawing it in 3 dimensions...molecules usually aren t flat"
    },
    {
        "video_name": "ka8Yt4bTODs",
        "Q": "Bond angles are given at 2;40,, at 7:00 and at 9:11 but there is no mention of how they are determined/calculated or their importance to a given molecule's structure. In fact, in over an hour of VSEPR there is no mention whatsoever of why a molecule's shape is important.... does anyone know?",
        "A": "Molecular shape is used in determining the stability of the molecule and how it can bond with other molecules; this becomes more relevant in the section on hybrid orbitals and hybridization."
    },
    {
        "video_name": "ka8Yt4bTODs",
        "Q": "At 8:31 , Why don't we consider all 4 lone pair of electrons as one whole single cloud on Oxygen? Why the ought to be in the pair of 2 and separated from each other?",
        "A": "An orbital can only contain two electrons. So if you had a single cloud , as you suggest, that would be four electrons in one orbital which isn t allowed. Consequently, there are two lone pairs of electrons, and since each of these orbitals contains negatively charged particles, it is natural for them to repel one another and move as far apart as they can."
    },
    {
        "video_name": "ka8Yt4bTODs",
        "Q": "at 9:22 ,why didn't we put the electron pairs on oxygen's up and down? why did we put em on up together?",
        "A": "Water has tetrahedral electron domain geometry so any which way you draw them they are in effect the same thing just rotated around. If you can get a molecular model set (or blu tack and toothpicks!) you might be able to prove this to yourself."
    },
    {
        "video_name": "ka8Yt4bTODs",
        "Q": "At 5:20, I understand how ammonia has a pyramidal shape, but only if you apply the lone electron pair's repulsion. If you ignore the lone electron pair or if they didn't have effect on the geometry, shouldn't ammonia be a flat triangle?",
        "A": "If it didn t have the lone pair yes it would be flat (planar). But we have done experiments and they tell us it is pyramidal not planar."
    },
    {
        "video_name": "ka8Yt4bTODs",
        "Q": "At 2:40, if the electrons are furthest away from each other as possible, why the bond angles are 109.5\u00c2\u00b0 apart but not 90\u00c2\u00b0??",
        "A": "Because the molecule is in 3 dimensional space not 2."
    },
    {
        "video_name": "ka8Yt4bTODs",
        "Q": "at 9:30, why all the four electron cloud in CH4 don't have equal amount of repulsive force",
        "A": "At 9:30 that is H2O not CH4...lone pairs repel more than bonding pairs which is why the angles are different"
    },
    {
        "video_name": "Ce4BGV1DVVg",
        "Q": "At 2:30, Although the atomic radius of iodide anion is greater than that of other halide ions, it also has more e- right? So .. even if we take that into account, iodide anion is more stable than others? Thanks.",
        "A": "Although I\u00e2\u0081\u00bb has more electrons than the other halides, it also has more protons. Each of the four conjugate bases has a 1- charge, so the number of electrons alone shouldn t have an effect."
    },
    {
        "video_name": "Ce4BGV1DVVg",
        "Q": "At 5:30 he talks about HI having the lowest bond energy, thus it's most likely to \"donate a proton\". Could someone please explain what he means by donating a proton? Thanks!",
        "A": "Proton and hydrogen ion are used to mean the same thing in acid/base chemistry. It means HI is the strongest acid of all the halogen acids."
    },
    {
        "video_name": "Ce4BGV1DVVg",
        "Q": "At 2:55, what does he mean exactly by being \"stable\"?",
        "A": "He means that their is room for them to be evenly spread out."
    },
    {
        "video_name": "w2QVg9mezcY",
        "Q": "why does Mr Verghese say, at 4:29 to 4:31 in the video that as soon as the doc lays hands on the patient, the symptoms give way, like in a stomach ache??(forgive me if there's some minor change, i may have heard wrong)",
        "A": "He says it gives weight not gives way . Essentially explaining that a patient will feel more understood and validated in their concerns if the doctor actually examines them physically rather than just ordering tests."
    },
    {
        "video_name": "jl_gQ-eL3xo",
        "Q": "at 2:55, is it 5m/s or 5 ft/sec?",
        "A": "What do you think? What would make sense, based on the units he is using, which he states?"
    },
    {
        "video_name": "jl_gQ-eL3xo",
        "Q": "2:39\nit is 5m/s shouldn't we convert it to feets/s",
        "A": "It was most likely a slip of the mind, he normally works in metric for these videos."
    },
    {
        "video_name": "jl_gQ-eL3xo",
        "Q": "At 8:48, why must 109t = 350?? Where does 350 come from?",
        "A": "I think that 350 is the horizontal displacement needed in order to be able to clear the wall (it s also the distance of the wall from the ball s origin)."
    },
    {
        "video_name": "FbP9nUrtKto",
        "Q": "About 5:00 , What's Carbon Excitation?",
        "A": "Carbon Fixation is the only thing around 5:00 that sounds like Carbon Excitation , so I ll assume that s what you are referring to. Carbon Fixation is a process by which certain organisms convert inorganic carbon from the environment into organic material. The most popular example is photosynthesis in plants in which the organic material is used in large part to form the physical structure of the plant. The other method of carbon fixation is chemosynthesis."
    },
    {
        "video_name": "FbP9nUrtKto",
        "Q": "1:03 shoudn't the electrons repel",
        "A": "Not really, if they repelled, there would be a lot less elements and all other stuffs"
    },
    {
        "video_name": "FbP9nUrtKto",
        "Q": "At 4:05 Sal says that silicon-silicon bonds are not as strong as carbon's. Why is that? Is it something to do with silicon's structure and hybridization?",
        "A": "Silicon has is larger and heavier, so its valence electrons are further from its nucleus and it is less electronegative, therefore its bonds are weaker and so easier to break and this prevents silicon from forming long chains."
    },
    {
        "video_name": "vnNlvnnBX0I",
        "Q": "At 0:02 negative sign with constant indicate to what?",
        "A": "your question doesn t seem complete. Well, as far as i understood lambda is an arbitrary constant which is unique to each element and it can be easily derived from it s half life as mentioned by Jay, (lambda=.693/half life) [ kindly refer to the video as what i wrote might be wrong. So yes, i believe you were referring to lambda which is a constant unique to each element which we can use in the equation, N=N(nought) * e^-(lambda)*t. Hope this helps."
    },
    {
        "video_name": "RQpBj8ebbNY",
        "Q": "At 8:13 he mentions tissue factor exactly what is that?",
        "A": "Tissue factor is part of the coagulation cascade. It s a molecule in the tissue that, onece exposed, acts as a catalyst on a clotting factor, that ultimately leads to the formation of the fibrin clot."
    },
    {
        "video_name": "Xw1CMTXQGPY",
        "Q": "Shouldn't the answer at 2:46 be re-alphabetized? Giving a final answer of:\n1,4 - cyclobutyl - 5,7 - bis(2 - methylpropyl) - 2 - cyclopropylcyclooctane\n\nAlphabetical order--> B, M, P",
        "A": "When we consider constructing the name alphabetically: * multipliers (bis, di, tri, etc.) are ignored * MOST prefixes (sec, neo, tert, etc.) are ignored * cyclo and iso are NOT ignored So cyclobutyl then cyclopropyl then bis(2-methylpropyl) is the correct order."
    },
    {
        "video_name": "BhQUW9s-R8M",
        "Q": "At 10:30 he tells about multiple nods. How can we tell how many nods it is in a tube?",
        "A": "You will have to compare the wave length to length of tube. Or no. of loops must be given."
    },
    {
        "video_name": "BhQUW9s-R8M",
        "Q": "Around 13:47 you refer to n for all possible wavelengths. Does n stand for the number of nodes?",
        "A": "No, n refers to the number of the harmonic. The lowest number of nodes gives you the fundamental, which is also called the first harmonic. The next possible standing wave gives you the second harmonic. The next one is the third, etc."
    },
    {
        "video_name": "49LcF9Zf9TI",
        "Q": "At 9:17 Please can you give some other examples for zero order?",
        "A": "Reactions catalysed by enzymes are often zero order with respect to substrate. The enzymes can only process so many substrate molecules per second so increasing the substrate concentration has no effect on rate."
    },
    {
        "video_name": "49LcF9Zf9TI",
        "Q": "At 8:30, we added more ammonia molecules but it did not affect the rate of reaction as the added molecules were not on ammonia surface.What if we added the extra molecules on the ammonia surface??Also, if we removed some molecules from the ammonia surface, will it decrease the rate of reaction??",
        "A": "If more molecules of ammonia were on the platinum surface this would increase the reaction rate. This could be achieved by increasing the surface area of the platinum by using more platinum and/or by grinding it to a smaller particle size. Conversely, decreasing the surface area of the platinum so that there were fewer ammonia molecules on the surface would decrease the reaction rate."
    },
    {
        "video_name": "49LcF9Zf9TI",
        "Q": "At 7:50, how do you know this equation is zero-order? Or it is just an assumption that it is zero-order?",
        "A": "The order of that reaction has been determined to be zero by experimental methods. On comparing experimental data(as shown in earlier videos),we could see that it is a zero order reaction."
    },
    {
        "video_name": "49LcF9Zf9TI",
        "Q": "At 3:39, is the integrated rate law used always the same for zero order reactions?",
        "A": "You are correct."
    },
    {
        "video_name": "Svvr5uF_FZ8",
        "Q": "I understand that having a larger space could increase the entropy because there would be more possible locations/configurations. How could increasing the temperature increase the entropy? Wouldn't that just make the molecules move faster, not have more options of location? The video mentions the concept around 5:31.",
        "A": "When heat is transferred into a system, the molecules move more. Not only do they have more translational kinetic energy allowing them to shoot around more, but they can also vibrate, bend and rotate more. All this increased movement adds more variety to what the molecules are doing - some molecules will be moving more than others, so there will be a greater range of available energy states. Hence the entropy of the system increases."
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "At 10:20, Sal, says that with NaCl breaking down, you get twice the moles of the solute that you have originally. His example being that two moles of NaCl being dissolved in water gives you four moles of the solute. SO my question is, does it always just double by two, or does it depend on the elements amu and/or how many elements are in the molecule?",
        "A": "It depends on the molecules involved. For example, when is dissolve 1 mol of MgCl2 in water, I get three moles of solute, as I have 1 mole of Mg and 2 moles of Cl, making 3 moles overall. With AlCl3, I d have 4 moles of solute for every 1 mole I dissolve."
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "At 13:38 we are shown a picture of what a Glucose molecule might look like but I don't understand how to read it. There isn't carbon anywhere and there are only five Hydrogen atoms. It appears, in my ignorance, that only Oxygen has the correct amount. I know I'm wrong (or sure hope so) and anyone who could help me understand this would be very much appreciated!",
        "A": "This is exactly what I needed. Thank you!"
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "At 8:36 does Sal mean molarity or molality? He says molality. Am I missing something? I've never heard molality before...",
        "A": "He is correct, it is molality. Here is how each term is defined: Molality is the number of moles of solute in one kg of solvent (note, solvent, not solution). Molarity is the number of moles of solute in 1 liter of solution (note, solution, not solvent). So, molality and molarity are different ways of measuring concentration. Boiling point elevation and freezing point depression require using molality instead of molarity."
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "At 7:19, what does the triangle thingy before Temperature change mean or represent?",
        "A": "That is the uppercase Greek letter Delta. The delta here means change in , so delta-T would be the change in Temperature for example. Similarly delta-H is change in enthalpy ."
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "At 3:30 Khan says adding a solute lowers the melting point.\n\nIs this only true for water? I don't see how this can be true for every chemical. Wouldn't making a solution have a high concentration of a chemical with a high melting point increase the melting point?",
        "A": "freezing point depression is a colligative property that means it does not depend on what the thing is. What happens is that the ions get in the way of the water molecules clumping together. So yes this works with any substance."
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "at 7:45 sal says Molality, did he mean molarity?",
        "A": "He said it correctly, the boiling point elevation and the freezing point depression are both a function of the molality, NOT the molarity."
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "At 6:12 what does it mean that the slute raises boiling",
        "A": "It means that the water will boil at a lower temperature without the solute than it will with it. It raises the lowest temperature that water will make water boil. Water normally boils at 100 degrees Celsius, but it will only boil at a higher temperature if a solute is added. If a solute is added to water it might raise the boiling point to 110 degrees Celsius."
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "At 7:30, what if you don't know what k is?",
        "A": "If you are thinking of exams, then you would have to be told what k was for the substance concerned. Outside of exams, it s something that can be determined experimentally by measuring temperature elevations at different molalities."
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "At 7:30, why is the molality used to determine the boiling point elevation/ freezing point depression?",
        "A": "Molarity is a measure of moles of solute per volume of solution (mol/L). The density (and therefore volume) of the solvent changes with temperature, meaning that molarity will change as the solution is heated or cooled. Molality is a measure of moles of solute per mass of solvent (mol/kg). We use this when temperature changes greatly because the mass of solvent remains constant through the temperature changes in the experiment."
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "At 3:25,shouldn't it be lower freezing point?",
        "A": "I believe so too! Adding a solute will decrease the freezing point"
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "6:47 so does that mean adding solute will lower surface tension too?",
        "A": "A look at the wikipedia article on surface tension reveals that the answer to this question is ... it s complicated! For sodium chloride the answer seems to be that surface tension increases ..."
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "At 3:10, Sal says that the solute molecules in the liquid make it harder to form a solid. But why does lowering the kinetic energy further than its regular melting point make it a solid?",
        "A": "It doesn t when the connecttic energy is lower they because solids more easily, think of it as a magnet if you wave 2 magnets back and forth next to each other there not going to come together. But if you do this slowly they will come together. Same thing with the molecules forming a solid. The lowering of the melting point is just a side affect of this."
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "Was that a mistake that he wrote solute lowers the boiling point at 3:30, it should be melting point right?",
        "A": "You re right - well spotted!"
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "Does the \"K\" stand for anything? I know it is a constant, but I mean does the \"K\" actually stand for a word starting with \"K\" ? 7:20",
        "A": "I strongly suspect it may stand for konstant, which is German for constant."
    },
    {
        "video_name": "z9LxdqYntlU",
        "Q": "At around 3:15, Sal says that the random particle will lower the melting point..I don't understand the intuition behind it. If it is harder to form into a lattice structure shouldn't the freezing point increase?",
        "A": "The logic is as you introduce a random particle, the lattice does not form properly as if it was pure, thus the strength of the lattice is weaker than the lattice of a pure sample. As the lattice is not as strong, then less energy in the form of heat is required to break the lattice, thus causing a lower than expected melting point. It may be harder to form into a lattice structure as you state, but the strength of this lattice is lower."
    },
    {
        "video_name": "slm6D2VEXYs",
        "Q": "At 04:47, does Sal mean to say Ribulose Bisphosphate and not biphosphate?",
        "A": "yes because i do not know what that means lawl"
    },
    {
        "video_name": "slm6D2VEXYs",
        "Q": "2:56 which one do people prefer to call it? PGAL? or G3P?",
        "A": "I prefer G3P, based on my AQA book, it s easier to understand from the reactions taking place."
    },
    {
        "video_name": "slm6D2VEXYs",
        "Q": "At 8:42, how is the 10 PGAL just get to the 6 ATP?",
        "A": "10 PGALs deliver 3x10= 30 carbon atoms. However, for every RuBP molecule, 5 carbon atoms are needed. From those 30 carbon atoms delivered by PGAL, (30:5=) 6 RuBP molecules can be formed. Every regeneration of PGAL into RuBP requires one ATP. Thus in total, 6 ATP is needed."
    },
    {
        "video_name": "slm6D2VEXYs",
        "Q": "At 9:30 when 10 PGAL's are used to make 6 RuBP's 2 more phosphate groups are needed to form the RuBP's, where do they come from?",
        "A": "The extra phosphate groups come from ATP."
    },
    {
        "video_name": "slm6D2VEXYs",
        "Q": "at 1:14 sal said called light reaction as a cycle but in the previous video he said that if NADP+ is being reduced to NADH than it is non-cyclic. Please help I am totally confused.",
        "A": "He corrected himself later and said that it should be light reaction instead of light cycle. Hope this helps :)"
    },
    {
        "video_name": "slm6D2VEXYs",
        "Q": "Could someone clarify what NADPH and NADP+ is? What is there function?\n00:47",
        "A": "NADP+ is a hydrogen acceptor. It accepts hydrogen at the end of the electron transport chain in the thylakoid membrane to form NADPH. NADPH can then be broken down to release energy which is needed in the Calvin Cycle."
    },
    {
        "video_name": "slm6D2VEXYs",
        "Q": "At 01:02 what were the two bi products?",
        "A": "The two bi-products are ATP and NADPH."
    },
    {
        "video_name": "slm6D2VEXYs",
        "Q": "So, just to confirm (as seen at 8:36), 18 ATPs are oxidized during the entire Calvin cycle?",
        "A": "A better way to remember it is that 3 ATP are converted to ADP per CO2 assimilated under Ideal conditions, but in this example. since he started with 6 CO2, yes it was 18 ATP converted."
    },
    {
        "video_name": "slm6D2VEXYs",
        "Q": "At 7:42 what happened to the H in NADPH? Wouldn't the product be NADP + H?",
        "A": "The electrons ( and then also the hydrogen) reduces an intermediate called 1,3 - bisphosphoglycerate (1,3- BPG) into G3P/ PGAL. Therefore, the H is now inside the G3P molecules and not released as a product."
    },
    {
        "video_name": "slm6D2VEXYs",
        "Q": "At 5:26, what is ribulose made of? 5 carbons?",
        "A": "Ribulose is a sugar, C5 H10 O5. The 5 carbon atoms form a 5-C-ring."
    },
    {
        "video_name": "slm6D2VEXYs",
        "Q": "At 5:06, RuBP, is that rubisco?",
        "A": "RuBP is Ribulose-1-5-biphosphate. RuBisCO is short for Ribulose1-5-biphosphateCarboxylase. RuBisCO is the enzyme that facilitates the Calvin Cycle, whereas RuBP is an element of the Calvin Cycle."
    },
    {
        "video_name": "J0gXdEAaSiA",
        "Q": "At 6:35 the OCH3 ion donates the extra electron to hydrogen.isn't that a co-ordinate bond?",
        "A": "not quite. methoxide anion isn t donating 2 electron groups to form a covalent bond with the hydrogen, therefore it can t by definition be considered a coordinate covalent bond."
    },
    {
        "video_name": "J0gXdEAaSiA",
        "Q": "At 3:18 how can methoxide ion give away electrons to that H just like that?? Shouldn't it give that electron to the carbon below Cl as it wud hv a partial positive character due to high electronegativity of Cl??",
        "A": "you can watch zaitsev s rule"
    },
    {
        "video_name": "J0gXdEAaSiA",
        "Q": "3:20 hydrogen will give its electron, but its neighboring carbon doesnt want it. why not? whats the issue?",
        "A": "Carbon wants it, but Cl wants it more. It forms a \u00cf\u0080 bond between the C atoms, and the Cl can then leave with a pair of electrons."
    },
    {
        "video_name": "J0gXdEAaSiA",
        "Q": "From 3:03 and forward, shouldn't he be using half arrows? (the \"fishhook arrow)",
        "A": "Yes we need to use half arrows because the electron is moving by itself and not as a pair."
    },
    {
        "video_name": "J0gXdEAaSiA",
        "Q": "At 5:01 why does the chlorine which was detached from butane have a negative charge even with 8 valence electrons around it? Doesn't it complete to it's noble gas configuration?(I know chlorine can only exist in it's molecular form on it's own but I don't get it)",
        "A": "A neutral chlorine atom has 7 valence electrons, but now it has 8, so it must have a -1 charge."
    },
    {
        "video_name": "J0gXdEAaSiA",
        "Q": "At 2:52 why didn't the electron from the O just go directly to the Cl? Why did it take the long excess route from one carbon to another and then finally to the Cl?",
        "A": "The O s electron doesn t go to the Cl in the end. The O s electron goes to the H, making methanol. The extra electrons from the bond to the H (left behind) make the double bond between Cs and then the Cl is pushed off to make chloride."
    },
    {
        "video_name": "J0gXdEAaSiA",
        "Q": "At 8:33 you specified that this reaction was E2 because of having two rate determining reagents. Does that mean all molecules above the arrow are rate determining reagents as well as the substrate to the left of the arrow?",
        "A": "Not at all. How you write the equation has nothing to do with the mechanism. You have to do experiments to determine the rate determining reactants."
    },
    {
        "video_name": "OIwp8m3S30c",
        "Q": "At 5:11, here speed of light seems to be half of the real value, and you said that it makes sense, but then you told that speed of light regardless of the frame of reference should be 3*10^8 .so i did not get it. Please help me.",
        "A": "He s telling you what the Galilean reasoning would suggest and then he s showing that it s wrong when you try to apply it to light, which is why the video has the title it has."
    },
    {
        "video_name": "OIwp8m3S30c",
        "Q": "At 6:14 it is said that sally's photon will go 3*10^(8) m in one second in sally\"s frame of reference .But sally is moving at 0.5c velocity.So the photon would go 1.5*10^(8)m in sally\"s frame of reference.\n(3-1.5)*10^(8)=1.5*10^(8)\nSo,at the end if I follow this line I will get that the velocity of light will be the same in my frame of reference.\nWhere am I wrong??",
        "A": "A better way to think about this situation is by taking sally s spaceship in motion when t<0, so at t=(-1), x=[-1.5x10^(8)]m and at t=0, it still has that vel, now v(Sally s photon)=u(1.5x10^(8))+c(3x10^(8)), thus at t=1 sec sally s photon is at 4.5x10^(8)m."
    },
    {
        "video_name": "rNJPNlgmhbk",
        "Q": "At 12:43, in the product formed by LiAlH4 the C atom in bottom left containing -OH has 2 valencies remaining. I mean that C atom has 2 bonds instead of 4.\nSorry if my question is wrong.",
        "A": "No, the C atom has four valencies. The missing two bonds are to H atoms. Remember that in line structures the H atoms are not shown explicitly, but they are still there in the molecule. If you draw the two C-H bonds, the four valencies become more obvious."
    },
    {
        "video_name": "rNJPNlgmhbk",
        "Q": "The compound formed at 3:00, why does LiAlH4 attack the carbon and not H2O attack the Oxygen?",
        "A": "At this point in the video, there is no water present. If there were, it would react vigourously with the LiAlH4 and destroy it. Water is added only after the reaction with LiAlH4 is complete."
    },
    {
        "video_name": "rNJPNlgmhbk",
        "Q": "at 4:27, is it possible to stop the reaction here, say if you wanted to make an aldehide.",
        "A": "No. LiAlH\u00e2\u0082\u0084 is such a powerful reducing agent that it goes all the way to a primary alcohol."
    },
    {
        "video_name": "rNJPNlgmhbk",
        "Q": "At 8:24 where did the hydrogen go when alcohol was formed from the aldehyde group at the top right?",
        "A": "It didn t go anywhere, it s still bonded to that carbon just implied not shown"
    },
    {
        "video_name": "rNJPNlgmhbk",
        "Q": "At 10:10, when we add the H2/Pd to substrate, why is the benzene ring untouched when it contains double bonds. It is said that it is not hydrogenated at normal conditions, so what conditions are then needed for that to happen?",
        "A": "Hydrogenation of benzene to cyclohexane is carried out commercially with specialized catalysts at pressures of 20 atm to 30 atm and temperatures of 300 \u00c2\u00b0C to 350 \u00c2\u00b0C. It IS possible to hydrogenate benzene in the lab in the presence of rhodium and palladium catalysts at room temperature and modest pressures, but it is much cheaper to buy the commercially produced cyclohexane."
    },
    {
        "video_name": "rNJPNlgmhbk",
        "Q": "At 3:28 when Jay says we have reformed our carbonyl, did he mean to say aldehyde?",
        "A": "Carbonyl is the general term for any functional group containing a C double bonded to an O (C=O). So Sal is correct in calling it a carbonyl, but you are also right that this particular carbonyl is an aldehyde!"
    },
    {
        "video_name": "rNJPNlgmhbk",
        "Q": "at 9:20 he says the O-H leaves as a \"leaving group\", does really leaves as OH or it first protonates and leaves as H2O",
        "A": "It first protonates, then leaves as H2O. As it stands, OH- is a terrible leaving group."
    },
    {
        "video_name": "rNJPNlgmhbk",
        "Q": "At 7:26 he explains that NaBH4 is selective for aldehydes and ketones. I don't really understand why though. Shouldn't the carbon on an ester be even more electron deficient than that of an aldehyde or ketone (since it's placed between 2 oxygens), and therefore more reactive towards the negatively charged NaBH4?",
        "A": "The carbonyl carbon in an ester is less reactive because the O attached to the alkyl group can donate electrons to it via resonance. This reduces the + charge on the carbon."
    },
    {
        "video_name": "rNJPNlgmhbk",
        "Q": "@6:20, are we only concerned with the C and not the -O-R group that left? I know its organic chemistry but is that normal to just forget it in the products list? I'm sure it would pick up a proton, that's not my question.",
        "A": "You are correct on all counts. If we were writing a balanced chemical equation, we would indeed have to account for the fate of the RO- group. In the laboratory work-up, the reaction mixture is usually treated with dilute acid, so the RO- ends up as ROH and is usually flushed down the drain."
    },
    {
        "video_name": "rNJPNlgmhbk",
        "Q": "at 11:05 isn't it H2 and Pt, not Pd? or what is the difference between the two?",
        "A": "it could be either Pd or Pt because both are good catalysts"
    },
    {
        "video_name": "rNJPNlgmhbk",
        "Q": "at 8:31 - I understand in excess that LiAlH4 will reduce aldehydes, ketones, carboxylic acids and esters - when not it excess, what order of those four functional groups does the LiAlH4 react with first, second, third, fourth?",
        "A": "LiAlH\u00e2\u0082\u0084 is such a powerful reducing agent that it reduces any functional group it collides with. If there is a limited amount of LiAlH4, it will react until it is used up, but it doesn t discriminate."
    },
    {
        "video_name": "rNJPNlgmhbk",
        "Q": "At 3:51, when the O-R' group leaves, can that oxygen take a proton and become HO-R' which is an alcohol? Which means you form 2 alcohols total from one ester",
        "A": "That s right. The O-R group also becomes HO-R ."
    },
    {
        "video_name": "TJeUOPc9pgs",
        "Q": "At 4:45, the closed captions say \"arterials\", but right after at 4:53 Mr. Punwani writes \"arterioles\". Which spelling is correct?",
        "A": "Arterioles is the correct spelling."
    },
    {
        "video_name": "f2BWsPVN7c4",
        "Q": "4:53 , If the asthenosphere of the upper mantle is solid, and so is the crust, how do you differentiate between the crust and the asthenosphere? What is the difference?",
        "A": "The asthenosphere is hotter than the crust, MUCH HOTTER."
    },
    {
        "video_name": "f2BWsPVN7c4",
        "Q": "At 7:41, how can you remember what is the lithosphere and the asthenosphere?",
        "A": "Lithosphere is above the asthenosphere and comprises of the lower crust and upper mantle."
    },
    {
        "video_name": "_8RohfPMklM",
        "Q": "At 3:43, David converts all of the Gravitational Potential Energy into Kinetic Energy. However, if I remember correctly, in previous articles, it said that some of the energy would be converted into thermal energy, since right at the moment the block hits the earth, some friction happened...\nIs it that the thermal energy is so small it is able to be neglected? Or am I getting confused here?\nHope my question is clear.",
        "A": "that conversion is after contact with the ground. we are concerned with what happens right up to the moment of contact"
    },
    {
        "video_name": "_8RohfPMklM",
        "Q": "At 12:19 when h=0 then how will be there gravitational potential energy?",
        "A": "It is referring to the potential energy at the top of the ramp. It started at the bottom of the ramp with a height of 0 and went up. So h does not = 0 when it has gone up the incline."
    },
    {
        "video_name": "3Gm4nAAc3zc",
        "Q": "At 2:10, is it okay to write \"-OH\" instead of \"OH-\"?",
        "A": "Yes its okay."
    },
    {
        "video_name": "3Gm4nAAc3zc",
        "Q": "At 1:44, Isn't NH3 a weak base, not an acid?",
        "A": "The equation was HA \u00e2\u0086\u0092 H\u00e2\u0081\u00ba + A\u00e2\u0081\u00bb, where A = NH\u00e2\u0082\u0083,F. The acid was HNH\u00e2\u0082\u0083 or NH\u00e2\u0082\u0084. NH\u00e2\u0082\u0084\u00e2\u0081\u00ba is the acid."
    },
    {
        "video_name": "3Gm4nAAc3zc",
        "Q": "At 5:41, Why is it useful/necessary to do the invers? Want to know the reason for doing it.",
        "A": "It s necessary to do the inverse so that A- is on the top, so that you could eventually end up setting Ka and Kb equal to each other."
    },
    {
        "video_name": "3Gm4nAAc3zc",
        "Q": "What? At 2:17, if you look at his weak base equation, it's not balanced. You have to include H2O in the equation, right? And if you do that, you get Ka = 1/Kb. How can he do this?",
        "A": "Correct. It is not balanced, but that does not affect the calculations, because the concentration of water is not included in the equilibrium constant expressions. He got Ka = [H+][[A-]/[HA] Kb = [HA][OH-]/[A-] If you multiply these together, you get KaKb = [H+][[A-]/[HA] \u00c3\u0097 [HA][OH-]/[A-] = [H+][OH-] This is exactly the result obtained in the video."
    },
    {
        "video_name": "3Gm4nAAc3zc",
        "Q": "At 13:35, should the second equilibrium reaction have a H2O in the reactants to balance it properly (so NH3 + H2O --> NH4+ + OH-)? And if so, why doesn't the first reaction (NH4+ --> H+ + NH3) need H2O?",
        "A": "This is because the NH4+ has the extra H+ (product) to offer, and doesn t need the hydrogen from the water. The reason why the first equation (NH3 + H2O --> NH4+ + OH-) needs water is because it needs an extra Hydrogen molecule to add to NH3 to make NH4+. The only real way to get that hydrogen (since one will most likely NOT break off from another NH3 molecule), is through the excess of water around it."
    },
    {
        "video_name": "3Gm4nAAc3zc",
        "Q": "At 2:15, is the equation balanced?",
        "A": "What he isn t writing is the how this is all in water(which is what aqueous means). So it s really (H2O) + (A- ) <=> (HA) + (OH-) Also if you wanted to get picky the first equation could also be (H2O) + (HA) <=> (H3O+) + (A-) The H3O+ is a hydronium ion."
    },
    {
        "video_name": "3Gm4nAAc3zc",
        "Q": "At 0:50, we write NH\u00e2\u0082\u0083\u00e2\u0081\u00ba using the \u00e2\u008a\u0095 sign because it only has a proton without its electron as I understand. Why do we when writing other acids such as HCL we don't use the \u00e2\u008a\u0095 sign because Cl is bonded to both the electron and the proton. What makes NH\u00e2\u0082\u0083\u00e2\u0081\u00ba only bond to a proton and HCL bond to both; proton and electron?",
        "A": "This is because hydrochloric acid has a covalent bond and so both hydrogen and chlorine share one electron each with each other. However, in ammonia, the nitrogen atom is already fully bonded with its original three hydrogen atoms. Nitrogen does have a lone pair though and so bonds datively to the fourth hydrogen proton without needing an electron from it."
    },
    {
        "video_name": "3Gm4nAAc3zc",
        "Q": "at 4:13 For what does the equilibrium constant stands for.",
        "A": "The equilibrium constant tells us the concentration of the reactants and products at equilibrium. In this case the equilibrium constant for the reaction HA = H+ + A- is called Ka, the acid dissociation constant. Is this what you needed to know?"
    },
    {
        "video_name": "3Gm4nAAc3zc",
        "Q": "Why did Sal choose to solve for A- at 5:25? Could he have solved for any of the variables in the Ka or Kb equation?",
        "A": "He could ve solved for HA but it would ve led to the same result, he just needed a variable that he could solve both equations for, so that he could set them equal to each other."
    },
    {
        "video_name": "YKKd5p4a5Lo",
        "Q": "at about 5:00 it is said that when the switch is closed, there is a potential difference of +3v across the inductor, isn't the main job of the inductor to \"resist change in current\" and since there is a change in current from 0 to i, then shouldn't the voltage across the inductor be so that VL = -L(di/dt). or in other words a induced potential difference that is opposite of the voltage of the circuit. its either iam utterly lost, misunderstanding something or iam right somehow loll!!",
        "A": "Hello Hussein, You have all the pieces you need - but you must ask what was the inductor current before the switch was closed? The answer is zero! When the switch is closed the inductor does resist change in current. To make the current zero the voltage on the inductor instantaneously jumps to 3 volts. Regards, APD"
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "At 8:35, what is a terminal atom, and why does the fact that Hydrogen is surrounded by 2 atoms mean that we have to assign the leftover atoms to Oxygen?",
        "A": "Terminal atoms are those on the outside of the structure, as opposed to central atoms which are in the middle. As far as we are concerned here, hydrogen only ever forms 1 chemical bond, so it only wants to have 2 electrons around it."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "At 12:22, I don't understand why Xe has 10 electrons in its outer shell. I though that elements could only bond with 4 other elements.",
        "A": "Elements can bond with more than 4 atoms. Xe is the central atom and is bonding with 5 pairs of valence electrons. So, 5 x 2 = 10. I hope that helped! ^^"
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "I didn't get how (@ about 09:45 of the class) if you replace a single bond with a double bond, using one more C electron, your dot diagram goes from 6 dots around it to 4. I was expecting 5. What am I missing.",
        "A": "Each bond is 2 electrons, one pair of electrons from oxygen has made the second bond to carbon. As per the steps introduced earlier we assign terminal atoms (the oxygen) their octet of electrons first. Then see if we need to make double bonds to complete octets on all atoms that need it."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "why at 4:24 does he say that each bond represents two valence electrons? how do we know that?",
        "A": "That s kind of just the definition of a covalent bond. 2 atoms sharing 2 electrons."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "hello sir, in third energy level you have s and p only . from 4 energy level d-block start. at time 2:28",
        "A": "The 3d orbitals are still part of the third energy level not the fourth, Jay is correct here. Elements like P, S and Cl (all in the third row) can accomodate more than valence 8 electrons around them. We usually attribute this to these elements bonding using the empty 3d orbitals (note that this isn t really correct, but that s going beyond this topic)"
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "Does he draw the CH2O incorrectly at 9:25 ( The single bonded) as he gave the Oxygen with seven valence electron like this -\n                                             .. \n                                            :O:\n                                             |\n                                        H - C - H",
        "A": "That was just his starting point. He knew that it was incorrect and drew it correctly a few seconds later."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "Wait, I'm confused, why does hydrogen have 2 valence electrons at 10:32? Aren't they only in group 1A on the Periodic Table?",
        "A": "There are two hydrogen atoms in formaldehyde, they each contribute 1 electron to this molecule. When they bond with carbon they now feel as though they have 2 valence electrons which is what hydrogen wants."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "How can Xe at 11:06 bond with anything when it is already octet? It dose not make any sense for Xe and F to bond.",
        "A": "The 5d and 5f orbitals of xenon are empty. By filling them it can become more stable.(see electronic configuration for reference)"
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "5:35 I don't understand why F already has 2 electrons? Doesn't it have 7 valence electrons. And they're adding 6 electrons from where? Why 6?",
        "A": "----F---- F--Si--F ----F--- Every F give one electron and every Si give one electron. So the Si has given its four valence electrons and F has given also four but still has {32(all valence electrons-8=24 e} or {28(four F valence electrons)-4)24} so divided 24 by 4 F and we get 6 electron. Which make sense as we know F has 7 valnce electron so 6 around it and one come from covalent bond."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "At 10:01,where should we draw the lone pair electrons?In the video the lone pair electrons are put on the two sides(right and left) of oxygen .Is it okay that we put the lone pair electrons on the top ?Or like the dot structure of O2,how do we know where to put the lone pair electrons or actually it doesn't.matter?",
        "A": "It technically is more correct if the lone pairs go on the left and right sides. The electrons are considered to be farther away from each other if they re put on the left and right, rather than on the top and bottom. My chemistry teacher explained this to us ; I don t know how it makes sense, but that s just the rule. I hope this helped! ^^"
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "At 11:40, why did he take away only one electron from 43 to 42? There are 5 F's and so wouldn't that have to end up taking away five electron instead of only one?",
        "A": "The positive charge was referring to the compound, not to the elements in particular, and if it was, then it would have a +5 charge."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "in 4:21 why does the bonds represent 2 valence electrons",
        "A": "Covalent bonds (which are usually how we think of the bonds in Lewis structures) are made when valence bonds are shared between atoms. Since we think of each atom donating one electron each in a covalent bond, each bond represents 2 valence electrons."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "At 12:09, he says that that Xenon puts on the central of the atom, why he puts Xenon on the atom?, and what i see is that Xenon is more electronegative than Fluorine, and the rule says that the least electronegative must put on the central of the atom.",
        "A": "Fluorine is the most electronegative element on the entire periodic table...not sure where you re getting that Xe is more electronegative from? F = 3.98 Xe = 2.6"
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "At 12:38, it is mentioned that Xenon is violating the octet rule, which is acceptable since \"Xenon is PAST the third period on the periodic table\". I was a bit confused, since I thought only the second period had to follow the octet rule. Past indicates that it is non inclusive for the third period, and thus indicates that the third period also has to follow the octet rule? Can anyone verify? Thanks!",
        "A": "What you say is correct in that d orbitals are available from Period 3 onwards, which is why sulfur can form more bonds than oxygen. However, if you are just looking at ground states, then the d orbitals don t come into use until Period 4, where you have the transition metals (but, of course, the Period 4 transition metals are using the Period 3 d orbitals)."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "At 8:56, why does the molecule have O and H connected to C? How do you know it's not H-C-H-O or something like that? Thanks.",
        "A": "Well (at this level) hydrogen only forms 1 covalent bond so it isn t likely to be in the middle like that. Any other arrangement would need charges in order for both the carbon and oxygen atoms to have octets, and there are no charges in the formula given."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "At 9:46, when Sal adds the 2 hydrogen atoms to the formaldehyde dot structure, why does he put them at an angle instead of putting them in a line?",
        "A": "In Lewis dot structures, they can be written in the same line or at an angle. They were written at an angle because that s closer to how the actual formaldehyde structure in real life, but it doesn t change the dot structure any."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "at 4:10 you put the Silicon at center because it is more electronegative but in second example you didn't put oxygen in center it is more electro negative then carbon",
        "A": "Actually, he put Si at the center because it is not more electronegative than Fluorine. The least electronegative atom (except for H) should be the central atom. In the second example, O is more electronegative than Carbon and hence Carbon is the central atom."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "At 10:50, Xe is a noble gas and it is bonding with fluorine. Noble gases can form bonds? They are very un-reactive and since they already have a stable electron configuration, they dont need to form bonds. Then how is Xenon bonding with Flourine??",
        "A": "Yes, in some relatively rare instances, some of the larger noble gas atoms can be coerced into bonding. The combination you mentioned is one of those."
    },
    {
        "video_name": "p7Fsb21B2Xg",
        "Q": "At 8:36 what did Sal mean when he said terminal atoms?",
        "A": "Terminal atoms are the atoms at the very edge/end"
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "At 9:59 Sal said that the protein chains within the E.R. will eventually butt out. Can the smooth endoplasmic reticulum shrink because of so many protein chains butting out?",
        "A": "it s the rugous ER that has vesicules going out...and between the smooth and the rugous ER they can swap places..so size never really interferes in anything"
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "At around 10:09, Sal said vesicle. What is the difference between a vesicle and a vacuole?",
        "A": "Vacuoles are large membrane-bounded organelles, while vesicles are a little like a mini vacuole. They can carry many things for the cell."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "At 3:37 Sal writes cytosol. Can someone please tell me what is cytosol?",
        "A": "Cytosol refers to the liquid part of the cytoplasm. Cytoplasm = cytosol + organelles."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "What is the \"lumen\" referred to at 7:04?",
        "A": "Lumen is just another word for the inside of the endoplasmic reticulum. It s a catch-all word that refers to the inner space of a tubular or cellular structure."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "what is lumen at 8:35 in the video?",
        "A": "At 8:35, Sal is referring to the lumen of the endoplasmic reticulum (ER). The word lumen just refers to the space inside of it, kind of like the lumen of our stomach (where all our food goes to be mashed up and digested). Inside of the ER (in the lumen), newly synthesized proteins get processed and modified. The lumen contains fluid that differs in composition from the rest of the fluid in the cell."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "at 2:07pm, what is the relationship between rough ER and smooth ER , the golgi apparatus and lysosomes?",
        "A": "Both ER and Golgi bodies are for cellular transportation while ER is for transportation inside the cell and Golgi bodies are for transportation outside the cell. ER has two types as smooth and rough. While smooth ER don t have ribosomes attached to ER rough ER have ribosomes attached to ER. Lysosomes are used for internal digestion of the cell. They are also called as the Suicide Bags. As a result of the lysosomes we have got fingers."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "3:43 what word is this? organals? probably a stupid question and what does it mean?",
        "A": "The word is Organelles. Organelles are small bodies within the cell such as the nucleus or mitochondria which have a specific function."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "what's the difference between protoplasm and cytosol? 3:56",
        "A": "Protoplasm is the colorless material comprising the living part of a cell, including the cytoplasm, nucleus, and other organelles. So basically, you could refer to it as the cytoplasm. As Sal Khan said, the cytoplasm (or protoplasm, if that s what you want to call it) is everything inside the cell, while cytosol is just the fluid between the organelles. And like the other answer says, cytosol makes up part of the cytoplasm."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "Starting at 1:10 and then throughout the rest of the video Sal said \"double bilipid membrane\", did he mean double lipid bilayer or am i mistaken?",
        "A": "Yes, that is what he meant."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "At 6:28 Sal mentions the rough E.R but not the Smooth E.R, I would like to know where the smooth E.R is located or distinguished from the rough E.R.\nThanks",
        "A": "Smooth ER lies within the cytoplasm of a cell, but not attached to the nucleus as the RER is. The way to tell the difference is that RER is covered in ribosomes, the smooth ER is not... it is smooth."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "At 9:00 - what happens to the mRNA that's been translated by the rough ER? The mRNA passes through the ribo. - does the endoplasmic reticulum \"consume\" the mRNA, or is the mRNA simply \"passing by\" to drop off the \"message\" then goes somewhere to \"die\"? Sorry - I'm trying to figure out if the mRNA is passing by or being \"consumed\".",
        "A": "The stability of an mRNA is highly variable, some last minutes, others days. Translation does not consume an mRNA, it can produce many copies of it s protein until it degrades over time by a variety of factors. Regulation at the level of translation is one of the many ways cells regulate gene expression."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "at 1:52, what are the fatty hydrophilic ends and the hydrophilic ends? I may have spelled these words wrong.",
        "A": "The fatty end is a chain of fatty acid, which is basically a string of carbons with hydrogens attached to them and a carboxyl group at the end. If you get two tails of fatty acid and a third glycerol molecule, this forms the hydrophobic end of the phospholipid. The hydrophilic end is a phosphate group attached to the top of the glycerol molecule."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "At 1:48 what is meant by hydrophilic and hydrophobic ends ?",
        "A": "Hydrophilic literally means water-loving. Hydrophobic is the opposite, meaning water-hating."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "At 1:09, the video talks about a \"double bi-lipid membrane\". What?",
        "A": "A double bilipid membrane is a membrane made of two lipid bilayers. Each lipid bilayer is made of two layers of lipids facing inward towards each other kind of like a zipper."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "Around 8:30, Sal names the inside fluid of the E.R \"The lumen\" and the outside \"The cytosol\".\nIs the inside fluid of the nucleus also called the lumen?\nAnd is the inside fluid of the mitocondrial matix also called the lumen?",
        "A": "lumen is the space"
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "At 10:31 Sal drew the Golgi Apparatus to be connected and continuous but shouldn't it be drawn to be in stacks and not continuous?",
        "A": "I will say that as a yes."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "HI!\nAt about 4:52-55, the person who's explaining (Sal?), said that the nucleus has two membranes, the inner and outer. He also said that the space between was called the prenuclear space. Now, what I've understood is that the membranes are made of bi-lipid layers or something...\nBut what is in between them?? Is the prenuclear space HOLLOW or FILLED with something?\nIf it is filled with something, what is it filled with??\nTQ,\nCookie!",
        "A": "it really depends on what you mean by filled. There are plenty of enter/ exit protein and little areas budding off with water and what not."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "This may sound like a silly question but at 4:55, is the endoplasmic reticulum attached to the nucleus? Thank you",
        "A": "yes. its inner surface"
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "At 10:10 Who discovered Golgi bodies?",
        "A": "Camillo Golgi. 1897 :)"
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "at 4:45 mRNA is a messenger RNA,then why does it convert into a protein?",
        "A": "Actually it does not convert into protein. mRNA is the messenger RNA which carries the message from the DNA to the protein making machinery inside the cells. This machinery is the ribosomes. Thus, mRNA is the blueprint of DNA is has all the instructions for making proteins, it feeds on these instructions to ribosomes and thus ribosomes attach amino acids corresponding to these messages incoded within the mRNA to make functional proteins."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "So in 9:53 when the little things (I don't remember what they're called) butt out wouldn't the E.R. just let it through instead of going with it? or would it actually go with it",
        "A": "the endoplasmatic membrane that goes with vesicles is needed for the proteins function and cell distribution"
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "At 3:06, what is a \"free\" ribosome?",
        "A": "A free ribosome is a ribosome that is not attached to the rough endoplasmic reticulum. Bound ribosomes (ribosomes attached to the rough endoplasmic reticulum) mostly produce proteins that are exported out of the cell to be used elsewhere whereas free ribosomes often produce proteins that are used within the cell."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "at 2:03 what is chromatin?",
        "A": "Chromatin is a combination of DNA and protein, and is what makes a Chromosome. It has many functions, but mainly a way to efficiently pack in as much DNA as possible, and improve the strength of the structure. DNA strand wraps around protein histones, to form a nucleosome, it looks like beads on a string . This in turn wraps around on itself to create a 30 nm structure , and wraps around on itself again. If you google image Chromatin you will see what I mean, there are some good diagrams illustrating it."
    },
    {
        "video_name": "6UqtgH_Zy1Y",
        "Q": "Quick Question, So at 8:43 we see a mRNA go through the ribosome that is connected with the er (to make rough er). So my question is: As the mRNA is being translated to protein, does it get used up? In other words, is it converted to a protein in the process?",
        "A": "It isn t exactly converted into a protein it is more like using google translate, the cell uses the original non-useful message, and turns it into something it can use. Once the process is over the mRNA detaches and eventually deteriorates."
    },
    {
        "video_name": "h_--qw-fv3k",
        "Q": "At 2:50 the speaker said that it is more likely that she is \"just a bank teller.\" The speaker should not have said \"just.\" If the stick figure is both a feminist and a bank teller, then she is necessarily a bank teller. Every single bank teller could be a feminist, and saying that she is a bank teller would still be true. Every square is a rectangle but not every rectangle is a square.",
        "A": "I think what she meant was since there are more bank tellers than feminist bank tellers. Then, the probability of her being a feminist bank teller is less than the probability of her being a bank teller. Therefore it is more likely that she is just a bank teller."
    },
    {
        "video_name": "ZD8THEz18gc",
        "Q": "at 7:35 how do we know it takes 26000 years to make a loop?",
        "A": "Astronomers make careful measurements of how fast the motion is."
    },
    {
        "video_name": "5kT7oUB0q6Q",
        "Q": "The gases the having high velocity should exert high pressure. But at 5:31 you said that gases have high velocity but low pressure. How is that possible?",
        "A": "Probably because he s talking of an Ideal gas, which does not really exist."
    },
    {
        "video_name": "mMHOM2qaRaE",
        "Q": "at 3:07 where does the NH3 come from to protonate the ring?",
        "A": "NH3 is the solvent - that is why Jay has written NH3 (liq) over the reaction arrow."
    },
    {
        "video_name": "RA7x81NI8C4",
        "Q": "At 17:40, he starts the problem using the formula for the fundamental -- how did he know that it was the first harmonic?",
        "A": "It doesn t matter what the initial harmonic was the closed tube will always resonate with a resonance frequency of F(open)/2."
    },
    {
        "video_name": "RA7x81NI8C4",
        "Q": "At 7:06, David notes that amplitude causes the sound to seem softer. Why is this?",
        "A": "It appears softer because there is less movement and therefore less displacement of the air around the wave so it is less loud. Also a higher amplitude makes it louder."
    },
    {
        "video_name": "LTuGQy4rmmo",
        "Q": "At 9:02, does the curl of the middle finger represent the direction of the magnetic field?",
        "A": "The middle finger not curling. It is pointing straight out of the plane of your screen, towards you. Only the ring finger and you pinky are curling."
    },
    {
        "video_name": "LTuGQy4rmmo",
        "Q": "Question on the circular motion created on the charge (@9:55):\n\nIf the instance the charge enters the magnetic field it starts to go in a circular patten, won't it exit the magnetic field before the circle is complete? Thanks",
        "A": "It depends on how big the field is. I think in this case, we are assuming that the field is big enough that we don t have to worry about the particle escaping the field before it starts arcing back."
    },
    {
        "video_name": "LTuGQy4rmmo",
        "Q": "At 10:00 Khan says the proton makes a circle because there is a downward force on the proton due to the magnetic field. 2 questions...\n1) Why is the proton going down initially\n2) If the proton is going down, wouldn't it just run into the magnetic field since the field is pointing straight up? Why does it make a circle versus how I'm picturing this?",
        "A": "Moving charges in magnetic fields experience a force that is perpendicular to the direction of motion. If you have a force that is always perpendicular to the direction of motion, that gives you a circle."
    },
    {
        "video_name": "4tctB1wZNiI",
        "Q": "Is crossing basically multiplication?\nI am confused because in the video, when Sal replaces the vectors with numbers at 9:10, he just makes the cross multiplication.",
        "A": "The cross product is a form of multiplication for vectors. The other form is the dot product. The cross product is maximized when the vectors are perpendicular to each other. The dot product is maximized when they are parallel ."
    },
    {
        "video_name": "4tctB1wZNiI",
        "Q": "At 1:15, Sal said that for the wire with current I2, the magnetic field on the right side is into the plane. I tried using the right hand thumb rule but for some reason I'm not getting it. Shouldn't it be out of the plane?",
        "A": "No, put your thumb in the direction of the current and grab the wire. Your fingers point into the screen on the right side and out on the left."
    },
    {
        "video_name": "4tctB1wZNiI",
        "Q": "at 1:38, how does length also is a vector?\nif he meant displacement, how can the displacement not be in the same direction as that o the flow of the current?",
        "A": "because we are taking the displacement of charged particles which can only occur in direction of current..although i dont know why we are using conventional current in this case"
    },
    {
        "video_name": "4tctB1wZNiI",
        "Q": "7:12 when Sal draws the arrow (in magenta around the top) for the Magnetic field on Current 2, shouldn't the arrow be facing the other direction?",
        "A": "Well, with your thumb pointing the direction of the current I2, which is downwards, imagine grasping the wire by forming a fist (a typical BOO sign). You will notice that your fingers wrap around the wire just as Sal described in the video...the arrow going into the screen on the left hand side of the wire and coming back out on the right hand side."
    },
    {
        "video_name": "4tctB1wZNiI",
        "Q": "why is the mag field of current two going into the page? isnt it going counter clockwise thus it should be coming out of the page? Time: 1:15ish",
        "A": "It s not. Watch again, and pay attention to what he is saying. The x field lines are from wire 1."
    },
    {
        "video_name": "zsdek-uH0NU",
        "Q": "In 9:20, was that your hair/?",
        "A": "Yes that was his hair because he needed to do something that required precision."
    },
    {
        "video_name": "zsdek-uH0NU",
        "Q": "at 2:53, Karl mentions that the switch will overheat and \"not work\". I'm curious as to why it ceases to function. Is it because the heats up and melts the spring or other metals?",
        "A": "The switch will make a short circuit and create a lot of heat and eventually make the batteries useless."
    },
    {
        "video_name": "DXc4pa9lIaA",
        "Q": "Table at 3:24 shows cyclobutane has more heat of combustion than cyclopropane why at 7:56 he said that cyclobutane has lower heat of combustion ?",
        "A": "At 3:24, cyclobutane has a higher heat of combustion than cyclopropane because it has one more carbon atom, but its heat of combustion per carbon atom is less than for cyclopropane. At 7:56, both compounds have the same number of carbon atoms. The lower heat of combustion from the four carbon atoms in the cyclobutane ring means that the methylcyclobutane has the lower heat of combustion."
    },
    {
        "video_name": "uqyLOuAzbvo",
        "Q": "At 4:00, Sal divides Force by area than multiplies by area. Why is he multiplying by area? Aren't we just looking for pressure which is simply Force/Area?",
        "A": "Because when F is divided by Area it also needs to be multiplied by it as it is being introduced by us and hence to maintain the equality in the equation ."
    },
    {
        "video_name": "uqyLOuAzbvo",
        "Q": "at \"3:38\" he says that Vi*t= D in the formula work=fd. Why is it this though? what does distance have to do with time?",
        "A": "Velocity * time = distance. E.g if you drove a car at speed of 100km/h for two hours, you ll travel 200km."
    },
    {
        "video_name": "uqyLOuAzbvo",
        "Q": "At 5:56 , Sal says\nv= m/p\nP A v t = Pm/p\nWhere did A and t go??",
        "A": "You can think of the pipe as a cylinder where A is the area and vt is the height. Area x height is volume (A x vt = V). Density (p) is mass (m)/ volume (V); or volume = mass/density. In conclusion, P A v t = P V = P m/p."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "2:00 says we only breathe in <1% CO2, 71% Nitrogen and 28% O2.\nHow do humans breathe in such certain percentage of gasses out of the air which is supposed to contain mixture of CO2, Nitrogen, O2,, every time ?\nwhat if i breathe in from a paper bag that contains the CO2 that i just breathed out?",
        "A": "Those numbers are the concentration of those gases in the earth s atmosphere. Of course, different places will have different concentrations of different gases. For example, in the forest one would expect a higher oxygen concentration than in urban areas."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "Is the reason you turn blue when youre being suffocated because there is no oxygen to make your blood appear red?\n\n1:28",
        "A": "jabezongzhenyu you speak words of wisdom, but yeah hallie nathan you are correct!(="
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "at 15:00 how could the body choose wether to have red blood cells with a nucleus or without? it does not make any sense.",
        "A": "Red blood cells probably used to have nuclei a very long time ago but they eventually evolved to lose it so that it could hold more oxygen."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 13:33 in the video you said that the \"Hemoglobin is sopping up all the Oxygen\", however is it the \"Heme Groups\" that attracts and sops up the Oxygen or the Iron?",
        "A": "The heme group contains Iron."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 13:16, is the 1.5 percent of oxygen not absorbed by the red blood cell diffused into the alveoli the next time that section of blood goes to the lungs again or just absorbed later when the oxygen-filled red blood cells give their oxygen to cells who need it?",
        "A": "the remaining 1.5% dissolves into the other liquids in the blood"
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 5:30, Sal said that the Carbon Dioxide in the blood diffuses into the lungs. I've learnt that diffusion evens out the concentration on either side of the membrane. If this is true, does this mean there is still some CO2 in the oxygenated blood, or does diffusion work in a different way?",
        "A": "Yes, there is still some CO2 left over in the oxygenated blood."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 1:50 how is it possible that while breathing in there is a minor trace of CO2 and while exhaling\nas people say major portion of exhaled air contains more CO2?",
        "A": "And actually the majority of what you breathe is nitrogen as that s the most common element in the atmosphere. What people mean is that your body uses oxygen and produces CO2, when you exhale there is a greater fraction of CO2 in that air than there was in the air you breathed in."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 13:40, what happens to the nitrogen that diffuses initially into bloodstream ? Are not they used in manufacturing proteins ?",
        "A": "No. Very, very few organisms can metabolize nitrogen from its elemental form into something else. No animal can do this. It is a process known as nitrogen fixation and it is mostly limited to a few types of bacteria and a few plants that are in symbiosis with bacteria that can fix nitrogen. The reason that so few organisms can fix nitrogen is that nitrogen gas is mostly inert. It is a rarity that any organism can evolve to make use of it."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "During the video at about 16:00, Sal said that red blood cells do have a very short live. He also said that red blood cells lack a nucleus after they are mature. Is it because that they dump their nucleus that they start to die or decay after that?",
        "A": "It might partly be because of their nucleus, but in the blood vessels, red blood cells are prone to constant wear and tear. This is caused by the cell squeezing through small capillaries. However, the fact that the red blood cell does not contain a nucleus contributes to the shorter life span, because the cell doesn t have the instructions to heal itself."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 10:33, Why is blood red",
        "A": "Each protein is made up bits called hemes, which are what gives blood the red color. More specifically, the hemes can bind iron molecules, and these iron molecules bind oxygen. Therefor blood cells are red because of the interaction between iron and oxygen..."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "at 4:45 why is your blood red",
        "A": "Blood is red because of an oxygen-binding protein called hemoglobin, which is essential for transporting oxygen throughout the body."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "Sal said around 12:45 that veins (like the ones we see on our wrists) look blue due to lack of oxygen. When we cut ourselves there, then, is the blood only red because it is reacting with oxygen outside out bodies and in the air?\nAnd, furthermore, if we could get that blood out of our system without exposing it to oxygen, would that blood remain blue?",
        "A": "to your first question, i guess thats the only way our blood changes colors is by oxagen so yes to your second maybe who knows unless you try but most likely not have you ever been given blood? its always red fro wherever it comes when you give blood its red so i guess not but near the end he sadi someting about the light"
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 3:54, how can something (an alveolus) be \"almost one cell thick\"?",
        "A": "The alveolar surface is very thin to allow the diffusion of oxygen from the inside of the alveolus sac to the red blood cells in the capillaries surrounding the alveolus. Sal is referring to squamous alveolar cells which are incredibly thin (on the scale of nanometers to allow this oxygen diffusion to take place)."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "1:24, I thought we breathed oxygen.But more than 70% is nitrogen and only 21% is oxygen.So , aren't we actually breathing nitrogen?",
        "A": "Yes, you are correct, our air is not only made up of Oxygen, we are also breathing in Nitrogen when we inhale. Hope this helped! :)"
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "Is there red blood cells in the alveolus membrane? Or does the oxygen diffuse from the red blood cells inside the capillary running next to it? I'm trying to understand 11:00",
        "A": "The oxygen diffuse through the alveoli into the capillaries and into the cells, so the red blood cells never leave the capillaries. The walls are quite thin here and there are concentration differences that drive the process :)"
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 13:13, if the oxygen and the hemoglobin is combining, then is it a chemical reaction?, and if it it, what compound does it make? Thanks, great video by the way.",
        "A": "It s not a chemical reaction. By combining it means they bind, but this isn t done by the formation of a new bond. It s achieved by intermolecular forces, so the two are attracted to each other but no reaction occurs."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "at 15:33 it's said that RBC don't have DNA then how does it get produced after every 120 days i.e old RBC dies and new RBC take their place?",
        "A": "RBCs get produced in the bone marrow by hematopoietic cells. Bone marrow constantly makes RBCs."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "at 13:40 Sal said that: \"You have nothing like that for nitrogen\" it meant that there is nothing to pick up nitrogen and to put into RBC (red blood cells), the question is : so where does all the nitrogen that we swallow by breathing go?",
        "A": "We breathe it back out again."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "From my observation, it looks as if the ratio between plants and animals (including humans) is not very small and the numbers are close together but if this is true, why does sal say that there is much more oxygen(what animals breathe) than carbon dioxide( what plants breathe) at 1:20",
        "A": "Plants are very abundant - the total mass of plants is about a thousand times the total mass of animals. Put differently, for every gram of every animal on Earth, there s about a kilogram of plants -- and those plants make just under a third of the free oxygen in the atmosphere. When you add in O2 from algae and cyanobacteria, oxygen production is (potentially) massively greater than oxygen consumption."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At around 05:06 you mention that CO2 isn't carried by red blood cells and is only in the blood plasma. I think this information is wrong, because from what I have learned, the CO2 is stored as bicarbonate inside the red blood cells.\n\nI'm not sure exactly how it works, but I think think it's worth looking up and posting some kind of updated video. Or have I misunderstood the whole thing?",
        "A": "Only 5% of carbon dioxide is carried by red blood cells. CO2 does travel in the blood as bicarbonate, and it is dissolved into our plasma."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 14:35, you said RBCs have no DNA, How to they replicate?",
        "A": "They don t replicate. They are continually produced by the bone marrow."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 2:13, Sal showed veins red and artery blue. Doesn't the vein carry the blood that is not oxygenated?",
        "A": "It does but the arteries carry blood toward to the area that needs it and the veins take blood away from the area. The pulmonary arteries carry not oxygenated blood toward the lungs then the pulmonary veins take the oxygenated blood from the lungs to the heart"
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 13:23 Sal says that the red blood cells take 98.5% of the oxygen. What happens to the remaining 1.5% of oxygen?",
        "A": "The remaining of you blood cells travels with the other percent , bound to molecules of hemoglobin."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 16:00 Sal mentions that a red blood cell dies. What will happen to the cell. Will it sort of like evaporate or just lay down in the vein/artery?",
        "A": "Red blood cells die on a regular basis, so it wouldn t be healthy to have them just pile up in the circulatory system. Dead blood cells are sent to the spleen where they are broken down/dissolved. Some parts of recycled by the body, others are eliminated from the body as waste products."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "What is a divet and a lounge he talked about at 9:25? sorry if im spelling it wrong",
        "A": "He is illustrating the physical appearance of the cell, slightly resembling a cough drop or lozenge. It is a rounded, flattened sphere with centralized indentations or divots on either side."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "at a little before 11:07 he talks about oxygen diffusing across the cell and bonding\ndo the bonds go through the membrane? also what does diffuse mean",
        "A": "Diffuse means to seep. Most gases can readily pass across the cell membrane. A cell membrane is made up mostly of phospholipids and the gases are small enough to slip between the phospholipid molecules. Oxygen passes across the membrane in its gaseous form of two oxygen molecules covalently bonded together (O2). So yes, oxygen passes across with its bonds intact. Outside the cell, oxygen gas may be encouraged to dissociate by changes in pressure and other factors."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 15:40, Sal discusses how the lack of a nucleus would be beneficial for red blood cells, and that the nucleus is pushed out of the cell after cell growth. What happens to the nucleus?",
        "A": "The nucleus is most likely broken down into its base molecules and recycled into the body."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "At 16:03 Sal wonders if cells are alive once they've lost their DNA, does that mean that they were alive in their primary form?",
        "A": "The tiniest living thing is a cell, and it contains most of the time a nucleus. If a cell loses it s nucleus we may wonder if it s alive or not"
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "11:46 So is our blood acidic in nature? Or are there other bases as well?",
        "A": "Yes, our blood is naturally acidic. The body is extremely, shall we say, picky about the pH of your blood. If it changes too much, you can die. Blood with high alkalinity would kill you."
    },
    {
        "video_name": "fLKOBQ6cZHA",
        "Q": "15:18 is the dent in the center of the red blood cell where the nucleus used to be?",
        "A": "Yes,that is true."
    },
    {
        "video_name": "tLOWcfeJ-74",
        "Q": "At 5:57, why is the 20cm placed at zero seconds and not at 1 second?",
        "A": "The 20cm is the amount that the spring has been pulled - once released, the spring begins to oscillate. So the 20cm amplitude is placed at the start of the oscillation, 0 seconds."
    },
    {
        "video_name": "tLOWcfeJ-74",
        "Q": "Beginning at 3:52, why does David graph the oscillation as a cosine wave, and not a sine wave? Although the weight at the end doesn't start bouncing until he pulls it to a certain distance and let go, the distance he pulls should count on the graph as a movement of the weight and half the amplitude. Therefore, should not the graph start at 0, rise in the first quadrant according to the distance the weight is first pulled, and then proceed with the negative movement of it bouncing back?",
        "A": "The body did not start oscillating at 0, it was pulled 0.2m (t = 0s) from equilibrium, then released. That s when it started oscillating."
    },
    {
        "video_name": "tLOWcfeJ-74",
        "Q": "At 0:54 - 0:57 IF amplitude is the madnitude of displacement why would it always be positive?",
        "A": "Because magnitude is always positive. It s like the vector version of absolute value."
    },
    {
        "video_name": "69V-60sga3M",
        "Q": "at 1:38 what are the kilopascals?",
        "A": "Pascal (unit) The pascal (symbol: Pa) is the SI derived unit of pressure, internal pressure, stress, Young s modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal."
    },
    {
        "video_name": "69V-60sga3M",
        "Q": "At 1:08 he talks about joules. Can someone explain that?",
        "A": "A Joule is a measurement of energy. So you can measure the energy output (in heat and light) in Joules and the energy consume by running to school is measured in Joules. The gas law is derived from energy in Joules but converted into the units shown in the video."
    },
    {
        "video_name": "69V-60sga3M",
        "Q": "At 4:04, which equation is it?",
        "A": "The rearranging of PV=nRT to give a value for n is given. n=PV/RT"
    },
    {
        "video_name": "69V-60sga3M",
        "Q": "At 1:10, what are joules",
        "A": "joules are the units used for energy!"
    },
    {
        "video_name": "69V-60sga3M",
        "Q": "At '1:09', I dont understand.... How is P*V in Joules ?",
        "A": "The joule is a measure of work - with units of N x m (Newtons x meters). Pressure is a measure of force/area - or N/m^2 (Newtons per square meter). So when you multiply pressure times volume you wind up with units that are equivalent to joules. N/m^2 x m^3 = N x m. Regardless of the units that you use for pressure and volume, their product always yields units of work."
    },
    {
        "video_name": "69V-60sga3M",
        "Q": "At 3:09, where there be a point where we have to memorize these R constant values?",
        "A": "Some particularly cruel teachers may require you to memorize the gas constant. Most will provide it to you."
    },
    {
        "video_name": "69V-60sga3M",
        "Q": "At 2:44, When you write 10 degrees Celsius I was wondering... I thought that in Chemistry we liked to measure temperature in Kelvin, or is Celsius the preferred unit of measurement",
        "A": "You can use either Kelvin or Celcuis, sometimes one is more convenient, sometimes the other is. Use whichever one makes your calculations easier to do, just make it clear which one you are using."
    },
    {
        "video_name": "69V-60sga3M",
        "Q": "At 2:45, what happens to the problem if you kept you volume in mL?",
        "A": "Then your answer would be 1000 times larger."
    },
    {
        "video_name": "69V-60sga3M",
        "Q": "At 1:08 he talks about joules. Can someone explain that?",
        "A": "Joules measure heat, energy, or work, depending on the context, but all of these require energy, so you can say that Joules are a measurement of heat."
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "If someone threw a ball 90 ft into the air , wouldn't it take way more than 5 seconds to come down ?? Is that because of air friction ?? Like what Sal says at 8:33 . :)",
        "A": "No, it would go up for about 2.5 seconds, and then it would come down for about 2.5 seconds. Air resistance on a ball is pretty small. It is complicated to bring air resistance into it because air resistance is not a linear function of velocity. But if you still want the ball to get to 90 feet, you have to throw it with higher initial velocity than you would have to throw it with no air, and it will come down with lower final velocity than it would have with air."
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "At 3:40 is the initial velocity zero or the final velocity? Earlier in the video he said that the initial velocity is zero.",
        "A": "I think what he means is that the initial velocity is at the moment where the ball leaves the hand and the stopwatch started. When you said that the initial velocity is zero, that is the moment you started the stopwatch when you were ABOUT to throw the ball. The purpose of this is that we can calculate the initial velocity by taking the velocity at the maximum point (which is stated as final velocity in the video) equals to zero. That is what I understand hopefully it s the right explainantion."
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "At 3:31,shouldn't the initial velocity be zero since the ball was at rest?",
        "A": "No, because the problem starts when the ball is released, not before the ball is thrown. Before the ball is released, you have force besides gravity applied to the ball. The point of this problem is to understand free fall. The Free in free fall means that the only force is gravity. That s after the ball is released."
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "At 8:32, Sal says that anyone who can throw a ball so that it stays in the air for 5 seconds has thrown it some 30.625m. what if the angle of launch is different? I mean, you could throw it at less of an angle, and it could still last just as long. Or in baseball, where it's almost parallel to the ground. If there was not one batting or catching and the ball was just left to fall down, it would take a long time, but wouldn't it stil be at a very low height?",
        "A": "Well the height it would attain would still be the same (30.625 m) but it would also get a certain range due to the horizontal component of the velocity . In baseball the ball stays for so long due the aerodynamics of the ball and air resistance and all other facts . In projectile motion we assume that there is no air resistance and dissipative forces are absent . (A vacuum like situation)."
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "at 3:40 seconds, Sal indicates that Vf = 0. But the given information states that Vi = 0. What am I missing?",
        "A": "At 2:30 he says we went from our initial velocity, whatever that is, to our final velocity, which is zero meters per second. His notation is basically saying from Vi to zero, not Vi is equal to zero. Just think of the arrow as symbolizing before and after, not an equal sign. It s definitely a confusing way to write that, but I hope this helps!"
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "At 6:18 why do we use the whole velocity instead of just its y-component? If we are trying to find the height of the object isn't that on the y-axis so wouldn't we need to use that value? Do we even need to find the y-component of the initial velocity? Thanks!",
        "A": "He says at 0:30 that we are throwing the ball straight up. So all the velocity is in the y direction, and there is no other component."
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "If someone threw a ball 90 ft into the air , wouldn't it take way more than 5 seconds to come down ?? Is that because of air resistance? Like what Sal says at 8:33 [:",
        "A": "Air resistance does not have a large effect on the air time of a ball thrown at those speeds."
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "Wasn't Sal supposed to consider significant figures in his answer, at 4:39? Shouldn't the answer be -25 m/s? Are there times when significant figures should not be considered in physics?",
        "A": "Yes, I agree with Andrew M. But if you do the same or not depends upon what your teachers s prefrances are."
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "At 8:05, the person says the displacement of the ball is 30.625 metres thrown in the air for five seconds. But before, he kept on saying that we are doing calculations for the first half as in for the first 2.5 seconds; so isn't he supposed to multiply 30.625 times 2 for the total displacement of a five second ball throw journey?",
        "A": "At the end of the journey the displacement is 0."
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "for 7:00, when sal is finding the displacement, should i be getting the same answer using the third kinematic equation?\nacceleration = 9.8\ninitial velocity= 24.5\ntime= 5\nI am getting a different answer using the kinematic equation...",
        "A": "You should get the same answer. Perhaps any mistake in the calculations ? Hope this helps :)"
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "At 6:45, why is it: ( initial velocity + final velocty ) / 2? Shouldn't it be: (final velocity - initial velocity) / 2?",
        "A": "How do you take the average of 10 and 8? Do you do (10-8)/2? Or (10+8)/2?"
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "3:39 confuses me. I need help understanding how 0-Vi is the same as vf-vi. Ex: if vf is 3 and vi is 2, change in v is 1. 0-vi of 2 is -2.",
        "A": "I think vf is 0 because when the ball hits the ground, there s no actual velocity"
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "At 2:22, why do you divide the time by 2 seconds? I mean, I understand that you do it to get the velocity when the ball is being thrown up, but why not take the 5 seconds?",
        "A": "Because the change in time when the ball is going up is half of 5 seconds. Since we said the time up = the time down , you divide the total change in time by 2."
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "At 7:16 why he mulitiplied by 2.5 ? instead of 5? Our change in time is 5s right? nope?",
        "A": "As of 7 minutes and 16 seconds into the video, we are still only considering the first half of the ball s journey, from it s initial position to its peak. We already figured out that since air resistance does not play a role, and the ball spends a total of 5 seconds in the air, the change in time between its initial point and peak is 2.5 seconds. Hope this helps!"
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "At 3:23 you said that the acceleretion of gravity is negative 9.8 meters per second squared. But I learned that the acceleration of gravity is positive 9.8 meters per second squared. So, what is the actual acceleration of gravity.",
        "A": "The acceleration is neither positive nor negative, it s downward. If you define up as positive, then you need to use -9.8. If you define down as positive, then you need to use +9.8 It doesn t matter which way you do it as long as you are consistent."
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "AT 3:20, Sal is saying a=-9.8m/s2.I am just confused on this, on the previous video you said going up is positive and going down is negative.So how are you saying a is negative when you are throwing the ball up?Please help me out.",
        "A": "when you re throwing up the ball its decelerating bcoz earth s gravitational force is pulling the ball down so the accleration is negetive OR you can think about it like this--since force on ball is downwards the acceleration will also be downwards [F=ma]{both are vectors}"
    },
    {
        "video_name": "IYS4Bd9F3LA",
        "Q": "What does he mean by acceleration of gravity in 3:12 ?? Isn't acceleration change in velocity over time?",
        "A": "Yes, and gravity provides an acceleration to something when you let it go, right?"
    },
    {
        "video_name": "aAfBSJObd6Y",
        "Q": "At 15:25 when we start adding pebbles to the system to move from state D to state A was it an adiabatic process?",
        "A": "The short answer is yes. B to C and D to A are both adiabatic processes. A to B and C to D are both isothermic processes."
    },
    {
        "video_name": "aAfBSJObd6Y",
        "Q": "At 7:00, why is there a change in temperature when there should be no change in temperature in an isothermic process?",
        "A": "I don t see a change in temperature. Is it possible you confused the labeling of the P-V diagram? The T1 written on the curve states that the temperature of the system is constantly T1 throughout the process from A to B. What changes is the volume of the system which increases, and the pressure of the system which decreases."
    },
    {
        "video_name": "aAfBSJObd6Y",
        "Q": "I couldn't exactly get what really happened in the case (not named, in the video) just after the 'C' case. When we have a gas at temp T2 put on a reservoir at T2, why and how will the heat be transferred to the reservoir(as the arrow depicts, at 12:59), when both are at same temp.??",
        "A": "This hypothetical reservoir is infinitely large and can take in any amount of heat without raising its temperature. When pebbles are added after the C point and the pressure increases, there is a tendency for the temperature (hence the total kinetic energy) to also rise. The reservoir allows the system to bleed off this excess heat into the cold reservoir, or sink, and thus allows the system to retain the temperature of T2 from C to D. The system is therefore isothermic between C and D."
    },
    {
        "video_name": "aAfBSJObd6Y",
        "Q": "@10:12 Khan says that his pressure would go down at T2 but his volume would go down also.\nI suppose that this is wrong and only a spoken mistake, because the move to the right o in the positive direction on the V axes can only mean more volume and lower pressure. Lower volume does not normally go hand in hand with lower pressure. So, although Khan probably meant more volume and lower pressure he said lower volume and lover pressure. Please confirm;)",
        "A": "i noticed the same. Yes, I m sure it was a small error"
    },
    {
        "video_name": "5-McqAO8_Qw",
        "Q": "at 8:36, If it is easier to send signals to the brain with only one connection, then isn't the tufted cell extra? Why can't the glomerulus send the information directly the brain?",
        "A": "It s more like condensing information from many neurons (think a circle full of neurons) into a smaller connection (think a dot s worth of neurons), rather than the number of connections the info has to pass through. Having multiple points of connection can be useful: it allows for greater regulation (e.g. increase the number of synapses to increase sensitivity) and organization (e.g. summarize odor info from many cells in the nose at the glomerulus, so the brain isn t overwhelmed trying to process everything.)"
    },
    {
        "video_name": "O9lL2KStW9s",
        "Q": "So which is healthier, margarine or butter? Margarine is made of hydrogenated vegetable oils, which creates transfats (6:34 starts the section on transfats). Butter is made of animal fat and contains saturated fats.",
        "A": "Margarines are often mistakenly associated with trans fats because they used to be the biggest source pre-90s. However it became known that they re not good for our health so manufacturers found the technology to make margarine without hydrogenation which is the process that creates trans fats. Now it s processed and take-away foods (chips, doughnuts, pies etc) that are the real problem."
    },
    {
        "video_name": "qZTeyhR1akA",
        "Q": "At about 4:50, isn't it pent-2-yne, not 2-pentyne?",
        "A": "Both names are acceptable, but pent-2-yne is preferred. IUPAC now prefers to put the number (the locant) immediately before the substituent it locates. It s difficult to get scientists to change old habits."
    },
    {
        "video_name": "qZTeyhR1akA",
        "Q": "why is it hept-2-yne instead of hepta-2-yne ? at 6:58",
        "A": "The root is actually hept, not hepta."
    },
    {
        "video_name": "qZTeyhR1akA",
        "Q": "I noticed in most other videos, molecules have their numbers inside the molecules as opposed to outside their molecules (ex: hept-2-yne vs. 2-heptyne). Does this mean that at 4:29 when Jay says the molecule is called 2-pentyne it can also be called pent-2-yne or is that different?",
        "A": "Both answers are valid but, the IUPAC way prefers hept-2-yne than 2-heptyne."
    },
    {
        "video_name": "qZTeyhR1akA",
        "Q": "At 5:20, when the molecule for 5,6-dimethyl-2-heptyne is being named, isn't the methyl on carbon five a chiral center and given a priority (i.e. shouldn't it be (5R) 5,6-dimethyl-2-heptyne)?",
        "A": "You re right, it is a chiral centre, but because it s drawn as a line (vs as a wedge or a dash) it is not defined as either R or S. When a chiral centre is drawn with a line instead of wedges/dashes we can usually assume it means we have a racemic mixture. You could specify this by writing R/S before the name or +/- or by not indicating any specific stereochemistry. Your suggested name would be correct if the stereochemistry was defined though - well spotted!"
    },
    {
        "video_name": "SHL_3JvfhXM",
        "Q": "At 1:58, the video mentioned that at the half-equivalence point, the [A-] and [HA] are equivalent. Is this only true when one mixes a weak base with a strong acid? (Thanks for these excellent videos, by the way.)",
        "A": "It is true when one mixes a weak base with a strong acid & also a weak acid with a strong base."
    },
    {
        "video_name": "l9-Qkl42-Tc",
        "Q": "At 1:02, is it a natural log or a common log?",
        "A": "It is absolutely log base 10. Depending on where you learned logarithms, you may refer to ln as either log base e or base 10 and log would refer to the remaining base- this often causes confusion between people from different universities at first. At any rate, we use log base 10 for the Henderson Hasselbalch equation because the base 10 is a bit more productive in our application since we are using a lot of scientific notation."
    },
    {
        "video_name": "l9-Qkl42-Tc",
        "Q": "At 7:20, What does \"buffer region\" mean?",
        "A": "The range of pH values (or component concentrations\u00c2\u00a7) for which a buffer system acts to stabilize the pH. This is usually said to be about 1 pH unit to either side of the pKa for the acid. \u00c2\u00a7note: concentrations of the buffer components: e.g. of a weak acid and its conjugate weak base"
    },
    {
        "video_name": "l9-Qkl42-Tc",
        "Q": "1:03 Shouldn't she have said conjugate base?",
        "A": "Saying conjugate base would also have been correct, but in a buffer both the acid and its conjugate base are weak (or vice versa)."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "01:45 Could you explain please why I has 7 electrons?",
        "A": "Iodine has atomic no. 53 {no. of electrons} So its main shell electronic configuration is 2,8,18,18,7 ( ll keep it easier to understand than going into the more complicated sub-shell configuration) Thus it has 7 electrons in its valence shell {outermost shell} So Sal says that I has 7 VALENCE electrons and these outermost electrons are the ones that cause bonding and the reaction... hope it helped! :)"
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "So increasing temperature will speed up reactions? 12:34",
        "A": "Yes, increasing the temperature will speed up reactions :)"
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "@ 6:30 how do you know that the final state H--I and H---I is the lowest state? I was told that it is possible that the final state can be higher than the initial state.",
        "A": "this is an exothermic reaction, where heat is released during the reaction. the energy released from the formation of H-I bonds is greater than the energy required to break the H-H and I-I bonds, and thus H-I is the lower energy state. in endothermic reactions, where heat is absorbed by the reactants to carry out the reaction, the reverse is true and thus the products become the higher energy state."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "At 3:13 when Sal mentions the electrons it hit me...\nSince Electrons are Negative, shouldn't they repeel the other electrons...",
        "A": "Electrons do repel other electrons."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "So, at 11:16 Sal said that \"everything\"( in this reaction is probabilistic ( so the \"bumps\" of the molecules have some chance of happening or not happening). I know it\u00c2\u00b4s difficult, but as the collisions are just a probability, is there any chance that during a reaction there are no \"perfect bumps\" and the reactants don\u00c2\u00b4t form products?",
        "A": "Keep in mind that even in a microgram of H2 and I2 that there are 10 million billion particles and they are moving extremely fast!"
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "At 7:20, why is the y-axis Potential Energy? Isn't it supposed to be kinetic energy? It's either that or I missed something in the video. Thank you so much...",
        "A": "Potential Energy is based on position so y-axis is position ,and x-axis is Kinetic Energy based on how it is direction"
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "At 14:45, why does the crushed salt have a larger surface area compared to its original state before it was crushed?",
        "A": "Try a thought experiment. What has more surface area: a cube, or the two pieces of a cube cut in half?"
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "in 6:29 it was mentioned that the products will go to a lowest energy state, what if the reaction was an endothermic reaction, wouldn't the products have higher energy state than the reactants??",
        "A": "I think Sal was specifically referring to the relative energies of the reactants and products in this reaction, which is exothermic. If the reaction is endothermic, you are right that the products have a higher energy state than the reactants. However, the products will still have a lower energy state compared to the activated complex."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "in what reactions, is activation energy not required? 8:03",
        "A": "Radioactive decay"
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "At 6:28 why would it change from a higher energy level to a lower energy level?",
        "A": "Nature always wants to get to the lowest energy level it can reach. It s like asking why does a ball roll downhill."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "At 0:25, Sal says you have two moles of hydrogen...but the subscript doesn't indicate the no.of moles....the coefficient does...H2 does not mean 2 moles of hydrogen; *2*H2 does. RIGHT? (Same logic applicable for iodine). BTW. This is not an accusation...I may be wrong...but I remember being told this in one of my classes and our teacher told us to remember this point carefully.",
        "A": "He probably should have said 2 moles of hydrogen ATOMS instead."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "At 0:20 sal says that it are 2 Moles of H and 2 Moles of I, but don\u00c2\u00b4t you in fact have 1 Mole of H2 and one Mole of I2? or is this just quibbling?",
        "A": "Yeah, you re right."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "at 8:20 he say you have to add energy?? how do you do that at more of whatever element your using?",
        "A": "You won t actually add energy its more of a lowering the activation energy when a catalyst is used. Heat speeds up reactions because it speeds particle movement."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "@9:34 would you include the catalyst in the equation or not and also is the catalyst part of the reaction like is it changing or is it just sitting there?",
        "A": "a catalyst is written above the arrow which means gives rise to it does not take part in the reaction as it is neither the reactant nor the product its not just sitting there it actually fastens the rate of the reaction."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "hi, in 13:59, i thought when you combine NaCl with H2O, you get:\nNaCl + H2O ---> NaOH + HCl ? i don't know, i may be just having a dumb moment...",
        "A": "NaCl is table salt. When you dissolve salt into water you don t get acid (HCl) and base(NaOH). The NaCl + H2O is a dissociation reaction. The main reason is because HCl is a strong acid and NaOH is a strong base. Strong acids and bases completely dissociate into their component ions in aqueous solution."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "at 2:29 you mentioned halogens are group 7 elements. aren't they in group 17?",
        "A": "Group 7 in Mendeleev s table and group 17 in the modern long table."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "at 14:25 sal said that with the increse in surface area the rte of reaction also increases ,how is it possible bcoz for the reacxtion to occur ,the molecules/atoms have to collide in a particular orientation.",
        "A": "That s correct, but he is talking about reactions with solids. Even in solids, a fraction of the surface molecules will have the correct orientation for reaction with the attacking molecules from the outside. If you break the solid into smaller particles, you increase the surface area and hence the number of molecules with the correct orientation, and thus the reaction will go faster."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "at 14:00 wouldn't the surface are decrease with smaller pieces? like isn't the surface area smaller because the pieces are smaller?",
        "A": "ohh okay thank you(:"
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "at 5:50 Sal gives the impression that the two Iodine atoms and the hydrogen combine into a 4 atom molecule. yet the product of the reaction is written as 2HI. how can this be?",
        "A": "This should be that 1 molecule of H2 (two atoms of H) and one molecule of I2 (again two atoms of I) so a total of 4 atoms that make two molecules of HI as a product."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "At 6:30, how do you know that the final state H--I and H---I is the lowest state?",
        "A": "it satisfies the octet rule"
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "At 8:02, Sal says that energy must be added to the system but the reaction H\u00e2\u0082\u0082+I\u00e2\u0082\u0082-->2HI is an exothermic reaction. Therefore, energy needn't added to the reaction, lest it is endothermic.\nPlease help...",
        "A": "The molecules must get over an energy barrier before they can react, even if the overall process is exothermic."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "Around 8:01 Sal says, \"you have to add energy to the system, at least not if it happens spontaneously\". How can a system gain energy spontaneously?",
        "A": "What he is saying is that in order for a reaction to occur, energy needs to be added, unless that reaction is spontaneous. A spontaneous reaction will just occur, with no added energy because it doesn t need it. It just does it by itself. I see where you are confused. He doesn t mean that energy is gained spontaneously, he means that the reaction occurs spontaneously. Hope this helps!"
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "In 4:07, when Sal says that the iodine molecule hits the hydrogen molecule, is this an exothermic reaction where heat is released during the reaction.",
        "A": "No. It is slightly endothermic. This is a formation reaction and the heat of formation of HI is +26 kJ/mol. The HI bonds are higher in energy than the H-H and the I-I bond. Sal makes a mistake at 6:35 because he says that the HI is the lowest energy state and in fact this is an endothermic reaction."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "At 2:35 how do electrons orbit in a covalent compound?",
        "A": "Electrons do not orbit (as a planet orbits the Sun), rather they are found in an orbital/region of space about the nucleus. Now for electrons in a covalent bond, they can be found most of the time in the overlap of the shells that have contributed them. This region of space is found between the two atoms that are sharing the pair of electrons."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "At 7:58, why is the potential energy lower for 2HI?",
        "A": "Well, all these bunch of atoms (or molecules.), they go through all these hardship of reacting with each other for a purpose * and that purpose is to settle down in a lower energy state. That s why sometimes we get products from reactants if we leave them for themselves and not the other way round. *the word purpose thing is not actually suitable to get the right feel but that s not the point .)"
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "At 2:39, do the atoms that share a bond \"fight\" over it?",
        "A": "no. iodine is extremely electronegative relative to hydrogen, this means it wants to draw the electron from hydrogen towards itself to form a polar covalent bond"
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "at 8:17 minutes in the video Sal drew a graph of the reaction going from a high energy state to a low energy state. Is this for an exothermic reaction and would a endothermic reaction graph look different.",
        "A": "It will be different graph because the situation just got reversed."
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "At 5:40, why is the activated complex at a high energy state? And also at 7:40, why is hydrogen iodide (2HI) at a lower potential energy? Thanks.",
        "A": "The Activated complex is at a higher potential energy as H2 and I2 are temporarily connected to each other in a Transition state .. Then as they separate as HI molecules their potential energy decreases. As they are unstable in a transition state they posses the maximum energy at that point and after formation of HI molecule there is a decrease in energy as they are more stable than reactants"
    },
    {
        "video_name": "__zy-oOLPug",
        "Q": "Polar dipole bonds? (14:26)",
        "A": "It s tautological but I think he just said polar dipole to stress that a dipole is polar!"
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "at 1:25 why do you want to do the most complicated first?",
        "A": "It makes balancing a lot easier at the end! If you do a more complicated balancing equation and balance single element molecules first, then you will end up going back and forth trying to re-balance the simple molecules again and again. If you do the most complicated molecules first then this problem doesn t arise. Hope that helps! :)"
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "At about 8:15, how does (1 mol of glucose/180 mol of glucose) equal 25/180. It would just cancel out to 1. And if it's 25g glucose TIMES (1 mol of glucose/180 mol of glucose) then I can't even begin to explain how much I don't understand that. Can someone please help fast cause I really need to get on with the chemistry course and it's just really awful knowing I'm stuck and I can't do ANYTHING (melodramatic but lkjrfhj;lka)",
        "A": "He is using what is called Dimensional Analysis to solve this problem. I cannot find any videos specifically about it here, but if you search on YouTube you will find many videos if you re still confused. Each bracket contains an equality. 1 mole of glucose has a mass of 180 grams, that s all (1 mol glucose / 180 grams glucose) means here. Go and plug 25 * (1 / 180) into your calculator, then plug 25 / 180 in. You ll find they re equal to the same thing. Multiplying by 1/x is the same as dividing by x."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "At 1:25, why is it suggested that you balance out the more complicated molecules first?",
        "A": "More complicated molecules tend to involve many elements. If you balance them out first, it makes it easier to then fine-tune the balance with the single-element molecules."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "at 7:52 how does multiplying the grams times the moles\\grams give us grams , i understand it mathematically but it doesn't make sense , why is it even ok for us to multiply them simply like that to get what we want ? shouldn't it be like an equation or something ? i try to think about it like adding the gram to itself several times and i don't get how that can give us moles ?",
        "A": "Multiplying grams times moles/grams gives you moles, not grams. It s just algebra. If you are going 60 miles/hour, and you multiply by hours, don t you find out how far you went? Same thing."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "He said at 14:25 for every mole of glucose there are 6 moles of CO2 and water, but why does the moles of O2 not determine the moles of the products?",
        "A": "In this problem glucose is the limiting reagent, that means the moles of glucose directly determines how many moles of product we can produce. If we limited the amount of oxygen then it could be the limiting reagent."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "at 12:51, you used every one mole of glucose we produce 6 mole of carbon dioxide\" and found the moles of co2 first, does it make any difference if i take the moles of glucose and find out the number of h2o moles first, or i cant do that because there is no carbon in h2o so i have to find the moles of carbons first?",
        "A": "Nope! You can work them out in any order :) As long as you know the mole ratios i.e 1 glucose + 6 O2 = 6 CO2 + 6 H20 there will be no problem in the order you work it out."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "When trying to find the moles of carbon dioxide and water at 13:00 , why did you use only the moles of glucose to find the moles of carbon dioxide? Shouldn't both the moles of glucose AND the moles of oxygen gas form the two products? In other words, shouldn't both the moles of the reactants become the ratio? Is it because the oxygen is unlimited and is therefore disregarded?\nPlease explain.\n\nThanks.",
        "A": "I asked the same question above, but I figured it out :) The equation is, simply to say, 1a + 6a-> 6a + 6a . a can be any number. 1x99 + 6x99 ->6x99 + 6x99 or 1x2 + 6x2 -> 6x2 + 6x2 etc. The reason he only used glucose to find out the moles of carbon dioxide is because glucose is the standard. It is given that there is 25g of glucose. Only matter of ratio. If we were given that there is 100g of Oxygen, he would have found the moles of carbon dioxide using oxygen."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "At 1:20, why do we have to make sure our equation is balanced?",
        "A": "Having a balanced equation is essential to figuring out these stoichiometry problems. It s very important because the balanced equation gives us correct ratios of reactants and products. We use these ratios a lot to convert between one type of element to another. If our equation is not balanced, our ratios will be incorrect, and so will our answers."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "At 9:24 Sal mentioned significant digits, and so I had a question: are significant digits really important? Could I still be technically right if, in a certain situation, I say \"0.1 is the product of 0.321 * 0.322\", or must I add all significant digits and say that \"0.103 is the product of 0.321 * 0.322\"?",
        "A": "As long as you don t say 0.100 is the product of 0.321 * .0322, you are ok. The best answer is 0.103. But you are allowed to round to 0.10, or 0.1, or even 0, depending on your needs. Usually you want to maintain all your significant figures until you reach the final answer of your problem, then decide what is the right degree of precision to report."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "At 12:00, when we wanted to find the grams of Oxygen required, we calculated that the atomic weight of oxygen was 16 and we multiplied that by 2, because it was two molecular oxygen. But when we balanced the equation in the beginning, we said that there is needed 6 times O2. We take that into account when calculating moles, how come we don't do for example, the atomic wight of molecular oxygen times 6?",
        "A": "There is nothing inherently wrong with multiplying the molar mass of 32 by 6, but it is not the standard way of doing it."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "How do you know in 0:49 that there are 2 oxygen as you say C6H12O6 reacts with oxygen in its molecular form and that there are 2 atoms when in the question it just states that oxygen is reacted .",
        "A": "On earth oxygen exists as O2 molecules not as lone O atoms. It s something you need to remember."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "It really starts to mess me up at 9:24, after Sal finished calculating the moles of glucose.I've got two questions.\n\nFirst, why if multiplying nC6H12O6 * M6O2 = n6O2?\nYou can check it out. 0.139 * 192 (which is 16 * 12) = 26.699, or 26.7, as Sal says.\nAnd what does Sal mean when he says \"for every mol of glucose, how many moles of oxygen?\"?\n\nI went through another route because I find it hard to understand the way it's done in this video. I would appreciate if someone could explain me the way is done here systematically.",
        "A": "When Sal says, For every mole of glucose, how many moles of oxygen? he is asking about the molar ratio of oxygen to glucose. C\u00e2\u0082\u0086H\u00e2\u0082\u0081\u00e2\u0082\u0082O\u00e2\u0082\u0086 + 6O\u00e2\u0082\u0082 \u00e2\u0086\u0092 6CO\u00e2\u0082\u0082 + 6H\u00e2\u0082\u0082O According to the equation, you need 6 mol of oxygen for 1 mol of glucose. That is, the molar ratio is 6 mol O\u00e2\u0082\u0082 to 1 mol glucose. That s the part that s missing in your calculation: 0.139 mol glucose \u00c3\u0097 (6 mol O\u00e2\u0082\u0082/1 mol glucose) \u00c3\u0097 (32.0 g O\u00e2\u0082\u0082/1 mol O\u00e2\u0082\u0082) = 26.7 g O\u00e2\u0082\u0082 The concept of molar ratio is central to all stoichiometric calculations."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "At 12:00 where did the 32 come from?",
        "A": "32 is the molar mass of molecular oxygen. Oxygen exists as the diatomic gas, O2. Each oxygen has a molar mass of 16 (from the periodic table), and there are two of them, so O2 s molar mass is 32 grams/mole."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "at 3:36, Sal wanted to convert grams into moles. isn't 1 mole equal to one gram when working w/ a chemical equation?",
        "A": "No. 1 mole is NOT equal to 1 gram. 1 mole is 6.022141\u00c3\u009710\u00c2\u00b2\u00c2\u00b3 particles of something, whatever mass that might happen to be. The way that you convert from grams to moles is to divide the grams by the molar mass."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "at 10:37 why is the moles of CO2 0.833......\nwhen i did 0.139*6 i got 0.834 moles\n?",
        "A": "You need to carry more decimal places in your computations. Also, note that Sal rounds of way too much (for example, you cannot get away with rounding the molar mass or molecular mass to the nearest whole number). You have to respect the number of significant digits."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "2:54 - should you always try to balance the smaller amount of elements last?",
        "A": "As a general rule it will make it easier if you balance less complicated molecules last :)"
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "At 12:16 : when he says 26.7 grams required does he mean 26.7 grams required to equal .833 moles?",
        "A": "Yep! 0.833 moles of Oxygen = 26.7 grams of Oxygen. Check a periodic table to find the Molar Mass of O2 And just remember the formula n x M = m OR (n) moles x (M) Molar Mass = (m) mass"
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "at 13:48 sal said that 1 mol of glucose is used to prepare 6 mol of Co2 but it prepares both Co2 as well as H2O also with help of O2 according to equation described above then how we can say that that 1 mol of glucose is used to prepare only CO2 or H2O in the above eg how we can do without taking the aid of O2? I know that I am not specific to describe this question but for now please understand",
        "A": "i checked what he precisely said. He related 1mol (0.139mol) of glucose to 6mol of carbon 0.139mol C6H12O6 -----> 6mol 6mol carbon hope it helps if not clear comment Rahul"
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "At 8:23 min. It says it's 1 mol glucose over 180g glucose. Why is that?",
        "A": "He had already calculated that the mass of 1 mol of glucose is 180 g. Now he is generating a conversion factor to convert 25.0 g of glucose to moles. The conversion factor is either (1 mol glucose/180 g glucose) or (180 g glucose/1 mol glucose). He chose the conversion factor with grams on the bottom to make g glucose cancel and put mol glucose in the answer."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "17:37 is their a shortcut when solving these equations?",
        "A": "Yes, when I took chemistry, I would always use Excel whenever I was allowed to not show my handwritten work. For this, you could just designate 2 cells, let s say A1 and B1, and then A2 would show our answer. Since it s just a simple multiplication problem, just enter into the A2 cell =A1*B1, and then you ll get instant answers for whatever you enter into A1 and B1. This is also useful in other chemistry applications like the Ideal Gas Law."
    },
    {
        "video_name": "eQf_EAYGo-k",
        "Q": "Throughout the video Sal calculates atomic weights of elements and compounds. What is the units of the calculations? ex. at 15:55 CO2 = 44 __? Is it amus? Is it grams? Thanks in advance!",
        "A": "The units are amu or grams per mol (g/mol). 1 amu = 1 g/mol."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "Aren't there an infinite amount of electric field lines? So how can he say that the electric field strength is stronger at the charge with the 3 lines then compared at like mid-way where there's 2 lines? Aren't electric field lines infinite in paths and quantities? 11:38",
        "A": "No, the whole point of field lines is to represent the strength and direction of the field. They way they represent strength is that stronger field is shown by more dense lines."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "At 8:58, aren't there infinite field lines? So theoretically the density should be sameat any point, right?",
        "A": "Yes, you are correct. For a given distance r from the point charge, the field strength is uniform over the surface of that sphere. There really isn t such a thing as the number of field lines (ie. the field lines are not quantized)."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "At 5:55 Sal writes that 9*10^3 / 4 is 2500 N/C. Am I not seeing something trivial here or is that math wrong?",
        "A": "He guesstimated the answer to 2500 because 9/4 = 2.25. Correct answer is 2250 N/C."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "at 7:00 sal said that if both charges are positive, they repel, agreed but why is the direction outwards from 1C( below the 1C) why not in the opposite direction",
        "A": "That s just the convention that positive charges have field pointing outwards."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "At 0:24, Sal mentioned something about \"sphere of influence\". How can I calculate the radius of that \"sphere\".",
        "A": "The electric field extends infinitely in all directions, although its strength declines with 1/r^2."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "@ 3:43 distance between Q and?",
        "A": "he started to introduce another charge of -1C, but as the example carried on, it turned into the general electric field at any point, 2m from the original charge, (Q)"
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "At 10:50 sal said rate of acceleration .Whats that?",
        "A": "Maybe he means the derivative of acceleration, which is jerk. Maybe he made a mistake and he meant to say rate of velocity. I don t know."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "Hey Sal, I think you meant to write 1 microcouloumb @ 6:28, you say -1 C but then in calculations use -1E-6 C.",
        "A": "A micro Coulomb is changed to coulombs when put into the formula so to convert it to coulomb s you have to add E-6"
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "In the example at 12:57, if E is 100 {meaning Q is positive) and we place an electron (meaning q is negative), why is F negative? Shouldnt it be positive cause positive and negative attract?",
        "A": "The direction of E is defined by the direction of the force experienced by a positive charge. So the force experienced by a negative charge is in the opposite direction of E."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "at 10:00 why do the vector lines get shorter and shorter as the distance increases? if the distance gets shorter and shorter as the distance from center increases, wouldn't that indicate that the force gets stronger as it gets further away (inconsistent with the electrostatic force equation: divided by distance squared)",
        "A": "Shorter arrows indicate weaker field (not stronger). The field gets weaker as you get further away from the charge."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "@8:23 Is there are an infinite number of electric field lines and at an infinite length, all going in different directions? Do the lines eventually seperate as they get farther away from the blue dot? What is the blue dot exactly and what does it do?",
        "A": "The field is continuous, it s not a bunch of lines. We just draw it as a bunch of lines."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "At 5:42, I keep getting E=2250 N/C. Is there something missing from my calculation or is that an error?",
        "A": "You re right. It is 2250N/C. It s a mistake in the video. I guess you missed the tiny grey box that appeared in the left bottom of the video showing the correction :)"
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "\"At 1:15 Sal comments as to whether an electric field \"really exists\". I would like it if someone could elaborate both sides of this argument a bit further. Is it a \"metaphysical matter\" or an objective debate?\"",
        "A": "electric field is just like air.. it can t be seen but can be felt therefore exists.."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "At 4:10, Sal says that electric field is a vector quantity. I don't understand. Where is the direction in an electric field. Doesn't the force vector require information form both particles (positive or negative) to determine its vector direction? With one charge removed, how can direction exist?",
        "A": "If you put a positive charge, say a proton, into an electric field, it is going to move one way, not the other. The direction it moves is, by definition, the direction of the electric field"
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "0:24 Isn't the sphere of influence of a charge theoretically the whole universe? Even if d is very big, the force is still greater than zero, isn't it?",
        "A": "sphere does not mean the end of electrical force at that exact, fine boundary, if that s confusing you. and we have to do the practical too so after going away from the charge a distance, the force becomes too small to affect the systems"
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "At 13:08 in the video near the end for -1.6E-19 C X 10E2 N/C shouldn't the answer be -16E-17 (1.6 multiplied by the 10 moves decimal one place to the right) then convert it back to scientific notation to only show one digit < 10 (16 is more than 10) to be -1.60E-18?",
        "A": "no?"
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "Just wondering, at 6:50 they will repel each other. But will they repel each other with the same acceleration, if they had the same mass (F = ma)? Or thus the charge play any part of this?",
        "A": "The force of repulsion is related to charge, not mass. Determine the force, then divide it by mass to get acceleration."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "At about 10:20 Sal is talking about field lines. He says he is talking about the path of a test charge, but the field lines should be indicative of acceleration or force rather than position or path. Would a test charge actually follow the field lines if it started out right next to the positive charge?",
        "A": "No. The field lines show the direction of the force at any point. If the charge started from rest, it would follow the field lines as it departed from that point. But once it has its own velocity, it will accelerate in the direction of the field line but that doesn t mean it will follow the field line"
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "At 10:48 he draws a line which decelerates and then accelerates. Why doesn't it maintain a constant speed, drawing from the pull of the electron at an equal rate to its losing of the push of the proton?",
        "A": "Speed and acceleration are different items, acceleration is the change in speed - Sal uses his words carefully stating its rate of acceleration will slow down however it may be a little confusing when he states things will speed up again - its really the acceleration will increase - the speed should always be increasing throughout the example @10:48"
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "at time 8:24 Sal uses the area method to depict the weakening force over a given distance but\ntechnically , a charge would have infinite field lines emitted in all directions, so wherever a box is drawn, u would have the same no. of field lines . how does this make sense with any thing we observe in reality?\nplease correct me if i made a mistake in my approach.Thank you",
        "A": "The charge lines from a point charge spread out. No matter how many you draw, they still get further apart as you move further away from the charge. Draw it and you will see. The number of lines per unit of area goes down and down. The lower density of lines corresponds to a weakening field."
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "How could we calculate the force between a charged particle such as Q from \"2:54\" to a separate particle, such as q, where q has no charge (for example, just an uncharged particle of iron, or a particle in a refrigerator door)? An uncharged piece of iron is attracted to both positive and negative charges (to both ends of a bar magnet), but using a value of 0 for q would not work in the equation. Is uncharged iron still somehow charged?",
        "A": "that is magnetism, not electric force. it is slightly different"
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "Are we assuming that Q is stationary, e.g. when Sal draws the force at 6:50, why does it only seem to be affecting q?",
        "A": "It is Because we are studying the effect of Q on any test charge in its electric field. If more charges were present Q would have affected them too"
    },
    {
        "video_name": "0YOGrTNgGhE",
        "Q": "@ 10:56 Are saying the path of the charge is going from the positive to negative? I know in DC the current flows from negative to positive and in AC the poles switch, thus Alternating Current.",
        "A": "In circuits we define conventional current as current made up of positive charges that flow from positive to negative."
    },
    {
        "video_name": "T2DaaGuKOTo",
        "Q": "What is the chemical notation for 6:51? I just wanted to know, because I'm pretty sure it's not H4C2O2N.",
        "A": "What they re showing you at 6:51 is a basic structure for an amino acid - There are 20 common ones and all of them have this piece plus some extra stuff bonded to the center carbon. The best way to write a linear formula would be: R-CH(NH2)-COOH (Where R is the side chain/the stuff that changes in each amino acid) because it shows the different functional groups that make up the molecule a carboxyl group -COOH an amine group -NH2 an atom of hydrogen -H a variable radical -R Hopefully that makes sense!"
    },
    {
        "video_name": "T2DaaGuKOTo",
        "Q": "At 5:52, What is it about an atom that causes it to want 8 electrons in it's outer shell?",
        "A": "It is still an issue in debate. Actually, atoms tend to be more stable when they have 8 electrons in their outermost shells. The noble gases at the extreme right of the periodic table are the most stable elements due to 8 electrons in their outermost orbits. All the other elements want to become like the noble gases. Hope this helps :)"
    },
    {
        "video_name": "T2DaaGuKOTo",
        "Q": "at 6:49 what is a Amino Group?",
        "A": "nitrogen atom attached by single bonds to hydrogen atoms, alkyl groups, aryl groups, or a combination of them"
    },
    {
        "video_name": "T2DaaGuKOTo",
        "Q": "at 1:20 he says that organic compound is more or less any chemical that contains carbon but there are so many chemicals like CO,CO2 etc which contain carbon but not called organic. So how can we identify an organic compound?",
        "A": "If you refer to organic when talking about chemistry it just means that there is carbon involved. You could rename organic chemistry to carbon chemistry. The organic is just a reference to the fact that living beings (that we know of) consist of compounds in which carbon plays an essential role and wouldn t be possible without it."
    },
    {
        "video_name": "T2DaaGuKOTo",
        "Q": "At 6:50 what is a amino acid and what does it do?",
        "A": "The ultimate value of a food protein or a protein supplement is in its amino acid composition. Amino acids are the building blocks of protein, and muscle tissue. Many physiological processes relating to bodybuilding from energy, recovery, muscle hypertrophy, fat loss, and strength gains are linked to amino acids."
    },
    {
        "video_name": "T2DaaGuKOTo",
        "Q": "At 7:53, why does he say chlorine needs one proton to fill its octlet, shouldn't it be one electron?",
        "A": "Yes, that it true. It should be electron , not proton ."
    },
    {
        "video_name": "T2DaaGuKOTo",
        "Q": "At 8:15, wouldn't sodium become negatively charged, because it had just lost an electron?",
        "A": "No, electrons are negatively charged, so losing an electron increases the positive charge on an atom."
    },
    {
        "video_name": "T2DaaGuKOTo",
        "Q": "At 3:42 Hank Green mentions \"S-Orbitals\" and \"P-Orbitals\", What are these, and what is the difference between them? Is there a specific reason they are called \"S\" and \"P\" Orbitals?",
        "A": "The S stands for sharp and the P stands for principal. These two characterizations were first formulated by the British in the 1870 s. Different spectral lines are placed into the S category or P category depending on the types of orbitals."
    },
    {
        "video_name": "T2DaaGuKOTo",
        "Q": "At around 7:06, he mentions things being positively charged. How is it determined what has a positive vs. a negative charge?",
        "A": "Things that are positively charged will be repelled by other positive charges and attracted to negative ones."
    },
    {
        "video_name": "T2DaaGuKOTo",
        "Q": "At 7:53, where he says that a Halogen is an element that only needs one proton to complete it's octet, does he mean that a Halogen is an element that only needs on electron to complete it's octet? The arrow does point to the outer electron shell......",
        "A": "Yes, he means one electron. The correction in the video at that time states so too and is also correct."
    },
    {
        "video_name": "pxKLeIjzxAk",
        "Q": "at 1:52 , what does he mean by the output voltage will be between +/- 12 volts , shouldn't it be 6 V ?",
        "A": "Hello Balraj, The +/- 12 volts is an assumption. Here Willy assumed the op amp is powered by a dual power supply. The upper rail is 12 VDC and the lower power rail is -12 VDC. The op-amp can operate from many voltage (with stipulation) but +/- 12 is a common value seen in the literature. The output of the op-amp may take on any value between the rails. In this example the output will be between +/- 12 volts. The 6 VDC is the calculated voltage for a given assumption. Regards, APD"
    },
    {
        "video_name": "SrrcBWllLbI",
        "Q": "At 3:02, the end product shows the stereochemistry of carboxylic acid as dashed- wouldn't it be a wedge since it is on the right side of the dienophile?",
        "A": "isn t there an oxygen missing on the other ring (the 5C ring)?"
    },
    {
        "video_name": "SrrcBWllLbI",
        "Q": "At 3:12 the carboxylic acid group is on the right of the dienophile. However, at 5:18 the carboxylic acid group is on the left side of the dienophile. I understand we switched the orientation, but why are we allowed to do this? How do we know how to look at it to determine what is right and what is left?",
        "A": "He was explaining that we could not obtain an endo product from the first reaction, and basically wanted us to work backwards as in, Which steriochemistry will give us the endo product? He rotated the dienophile portion of the molecule so that the carbonyl groups were inward in order to obtain the endo product. Otherwise, it would have been an exo product."
    },
    {
        "video_name": "ZA_D4O6l1lo",
        "Q": "At 5:12 why sal had subtracted 29.4 from 100 N?",
        "A": "Because the sum of all Forces F is F1 - F2 (in this situation) is used to find the total force enacted on the object so The bigger force would minus the smaller one"
    },
    {
        "video_name": "ZA_D4O6l1lo",
        "Q": "At 5:50 how can the object have any acceleration from the force of STATIC friction. i would think by its very nature there cannot be any acceleration because the block is not yet moving?",
        "A": "What if you put a block on a sheet of paper and then pull the paper slowly along a table. What force is moving the block?"
    },
    {
        "video_name": "ZA_D4O6l1lo",
        "Q": "At 5:20, Sal says that the friction force is 29.4 N, yet in previous videos he explained that the budging force is always 1 N more than the force of friction. However, in this video, he says that they are the same. Why is this so?",
        "A": "I am not exactly sure what you were asking but i will try to answer it. Its not always true that the bulging force has to be exactly 1 newton more than the force of friction, its only the amount of force that you need to just get the object in motion. It could me 0.45 newton more, or 0.000001 newton more! It depends on what type of object you are using(wood, ice, steel) and the surface over which the object is placed(ice on ice or wood on ice etc), Hope that helps, I can clarify more if you want. :)"
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "Previous video:\nThe sperm cells are created through meiosis, and the outcome of meiosis would be equal number of X-carrying and Y-carrying sperm cells (2 X and 2 Y).\n6:54 So why isn't there produced the same amount of X-carrying and Y-carrying sperm cells?",
        "A": "The amount of X carrying and Y carrying sperm cells should be nearly equal."
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "Around like 10:00 he starts talking about disorders or mutations in X chromosomes. Are there any mutations for Y? If so would this mean they can only happen in men?",
        "A": "There could very well be mutations to a Y chromosome. It s hard to say what effects they might have, though, because there are no known coding regions on the Y chromosome that are essential for life."
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "At 10:00 Sal says the chromosome for haemophilia is recessive.If so, then won't the Y chromosome in the male dominate hence no man should have haemophilia?",
        "A": "Because the gene is not present on the Y chromosome. It is not the presence of two recessive alleles that cause the disease, but the absence of one dominant one."
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "At 5:14, what do you mean by 'haploid genetic'?",
        "A": "genetics which deal with only one set of chromosomes is haploid genetics"
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "At 9:45 Sal says that red-green color blindness are a result of the X chromosome mutation.does that mean that certain chromosome can only have certain mutations, and if so does that mean that certain genders are less likely to have certain mutations?",
        "A": "In women if the gene is defective in one of the alleles, since it is on the X chromosome and since X chromosome inactivation occurs, why wouldn t the faulty allele be expressed when the X chromosome that carries it is expressed? How is it that there are carriers for X chromosome linked conditions if X chromosome inactivation randomly shuts down either x chromosome in females?"
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "At 12:18, what if the X chromosome without the hemophilia trait is inherited into the XY pair? Does the offspring still have hemophilia?",
        "A": "No.. The gene is only carried on the X-chromosome so a father cannot pass hemophilia on to his sons, only to his daughters."
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "At 6:17, Sal is talking about the difference between x sperms (girls), and y sperms (guys), but i had always learned that there was an Xx and an Xy. The mother can only give an x but the father determines by giving them an x or y. Is this true?",
        "A": "Yes, both of these are true. The mother has egg zygotes that can only carry an X chromosome. As the mother can only have XX, and zygotes only have half the information, the egg only carries X. Fathers produce sperm zygotes. Since fathers have XY, and zygotes only have half the info, the sperm either carries an X or a Y from the father. The sperm with X will produce girls when combined with the egg (Xsperm + Xegg = XXgirl). The sperm with Y will produce guys when combined (Ysperm + Xegg = XYboy)."
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "I would like to ask 2 questions.\n\n1. How can we determine B or b allele is dominant? (B stands for Blue eyes allele, b stands for brown eyes allele)\n\n2. in \"10:36\", assume X is dominant, why the offspring with X and Xh allele is no hemopheilia? If Xh is dominant, the offspring should have hemopheilia right?",
        "A": "1) The easiest way to determine whether an allele is dominant(at least, in the simple model) is to compare parents to children. If two brown-eyed parents have a blue-eyed kid, then there must be blue eyes genes lurking in there somewhere. If two blue-eyed parents never have a brown-eyed kid, then you know that there s no possible brown-eyes genes in blue-eyed parents. That means that blue eyes is always bb, whereas brown can be Bb, so brown is clearly dominant."
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "So there are about 142.8 hemophiliac women in the entire world? (with (as Sal says at 12:40) 1/49,000,000 and 7,000,000,000 people in the world)",
        "A": "That is a rough probability, at any given time there could be more or less women than that with the disease. Also you can t have a fraction of a person."
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "0:59 What are alleles?",
        "A": "One of two or more alternative forms of a gene that arise by mutation and are found at the same place on a chromosome. Hope this helps:) Also I think video seven defines alleles as well."
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "At 1:47, he said that there are 22 pairs, but I thought that there were 23?",
        "A": "There are 22 pairs of chromosomes excluding the sex-chromosomes. Including the sex-chromosomes, there are 23 pairs."
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "At 13:01, he mentions that if a male has a recessive sex-linked trait, he will show it, so it's not possible for him to be a carrier. If it's a dominant sex-linked trait, can a male be a carrier?",
        "A": "A carrier is someone who has the allele but doesn t show the phrnotype. Since dominant alleles always get expressed then they can never be present but not appear. Thus, there is no such thing as dominant carrier ."
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "At approx. 11:05 he said that only the X chromosome can carry hemophilia. If so, how could a man possibly obtain hemophilia?",
        "A": "A man has one X chromosome from his mother and one Y chromosome from his father. If his mother had the gene associated with hemophilia on one of her X chromosomes and that is the chromosome her son inherited from her, then he will have hemophilia. The Y chromosome in a male lacks the gene which would counteract the one he got from his mother."
    },
    {
        "video_name": "-ROhfKyxgCo",
        "Q": "At 7:00, Sal says that Henry the 8th was maybe producing more x-sperms than y-sperms, but I thought that the process of meiosis involved the division of a whole cell, meaning that a male must produce the same number of x-sperm and y-sperm. Isn't the case?",
        "A": "What Sal meant was that more sperm that carried an x chromosome reached the egg first than a sperm carrying a y chromosome."
    },
    {
        "video_name": "leudxqivIJI",
        "Q": "at 0:25 what to do with the angle?",
        "A": "We can think of the momentums as lengths of a triangle. We know from trig that angles can be found due to their relationship with side lengths."
    },
    {
        "video_name": "leudxqivIJI",
        "Q": "at 6:20 (basically when calculating the final velocity using the Pythagoras theorem), is it necessary to divide momentum by it's mass before finding the resultant velocity then re-multiply mass again? could you not just find the resultant vector for momentum itself?",
        "A": "Try it that way and see if you get the same answer. Good way to learn."
    },
    {
        "video_name": "leudxqivIJI",
        "Q": "at 0:30 how do you know momentum is conserved? does it tell you this in the problem??",
        "A": "Momentum is always conserved in any closed system. It s a law of physics."
    },
    {
        "video_name": "K2tIixiXGOM",
        "Q": "At 5:20 a Br atom gets added to only one C atom, but since both C's across the double bond are equivalent shouldn't it add to both of them to form a bromonium ion?",
        "A": "The bonds in benzene aren t true double bonds, they don t do normal addition reactions like alkenes do so the bromination of alkenes mechanism doesn t work here. What they do is this special electrophillic aromatic substitution reaction which is basically replacing one H with another group."
    },
    {
        "video_name": "K2tIixiXGOM",
        "Q": "At 03:28 why does the Fe 'nab' the electron from the Br in the first place? If afterwards the Br has a + charge ad the Fe has a - charge, surely this isn't a stable configuration?",
        "A": "because second Br is more crowded by other Br and FeBr3 and thus has steric hindrance"
    },
    {
        "video_name": "K2tIixiXGOM",
        "Q": "At 6:26 I don't understand why the carbon above where the Bromine is drawn only has three things bonded to it? Shouldn't it have two hydrogens as Carbon bonds to four things?",
        "A": "Well that is why it has a positive charge. It had 4 things bonded to it, but lost an electron when the bond broke so it becomes positive. You are right, it SHOULD have 4 things bonded to it, thats why it is less favored/stable than it wants to be...thats why the reaction continues until it gets its double bond back"
    },
    {
        "video_name": "K2tIixiXGOM",
        "Q": "at 5:30 the electron from bromine is bonded to iron and at same time Bromine is bonded to another bromine .\nWhy does not it breaks the Bromine Bromine bond and form FeBr4 - and a Br + molecule",
        "A": "That can happen, but the Br\u00e2\u0081\u00ba ion is not very stable. If the \u00cf\u0080 electrons of the ring attack the Br, the reaction doesn t have to go through that high-energy intermediate."
    },
    {
        "video_name": "K2tIixiXGOM",
        "Q": "At 11:23, as we use FeBr3 as a catalyst for bromination of an aromatic compound, what catalyst do we use for chlorination of an aromatic compound?",
        "A": "FeCl3 and it reacts in the same fashion."
    },
    {
        "video_name": "K2tIixiXGOM",
        "Q": "@1:33 isn't dipole movement 0 due to sp2 hybridization so then shouldn't Fe have o charge? then y does sal say it will have charge?",
        "A": "He is talking about electronegativity and oxidation numbers, not dipole moment or hybridization. Br is more electronegative than Fe, so Br will have a \u00ce\u00b4\u00e2\u0081\u00bb charge and Fe will have a \u00ce\u00b4+ charge. The oxidation numbers give the extreme case, where the oxidation number of Br is -1 and the oxidation number of Fe is +3. No matter how you look at it, Fe will have a partial positive charge and will act as a Lewis acid."
    },
    {
        "video_name": "K2tIixiXGOM",
        "Q": "At 01:29 how is iron bromide a covalent susbstance? Isn't it ionic?",
        "A": "The electronegativity difference between Fe and Br is 1.13. From this, we can calculate that the bond has about 27% ionic character. Thus, it is polar covalent. A bond must have more than 50% ionic character to be considered ionic."
    },
    {
        "video_name": "52wxpYnS64U",
        "Q": "At 2:20, he puts 196 in for the weight of the object. Why does he put 196 in instead of 20kg?",
        "A": "Kilograms is not a weight it is a mass. To get a weight in Newtons you need to multiply the mass of 20 kg by the acceleration of gravity which is 9.8 m/s^2 giving you 196 N."
    },
    {
        "video_name": "52wxpYnS64U",
        "Q": "At about 2:00, he's finding the tension in the hanging mass. He does Fnet=m.a However, he uses the acceleration of the system times mass... wouldn't you have to factor in the force of gravity into the total net force? Which would be an additional m.g?",
        "A": "Easier to understand is: mg-T=ma 196-T=85 T=196-85 =113 N"
    },
    {
        "video_name": "52wxpYnS64U",
        "Q": "At 6:00, why do the net forces have to be zero ? when we accelerate the pie, there is a net force, right ?",
        "A": "The vertical component of the net force must be zero, but the horizontal component will be positive to the right."
    },
    {
        "video_name": "52wxpYnS64U",
        "Q": "At 6:04, Sal says that the offsetting force of gravity is friction. When would the normal force be the offsetting force, and when would it be applicable? Would someone be able to explain what the normal force is? Thank you!",
        "A": "In Mathematical term normal refers to the force that is perpendicular to the contact object. So, the normal force in this particular problem would be the force that is counteracting force of the pie toward Sal s hand."
    },
    {
        "video_name": "52wxpYnS64U",
        "Q": "At 2:57, can someone explain the convention of using \"positive numbers for downwards and negative numbers for upwards\"?",
        "A": "You can define positive and negative however you want, as long as you are consistent throughout the problem."
    },
    {
        "video_name": "52wxpYnS64U",
        "Q": "at 6:34, why is force of friction 9.8m?",
        "A": "9.8 x mass of pie. The weight of an object is defined as its mass multiplied by the force of gravity (9.8N). Thus, 9.8m is the downwards force of gravity on pie. In the video, it is assumed that the pie is stationary. For the pie to be stationary, aka not falling, the upwards force (or resisting force of friction) must be equal to the downwards force. So it has to be 9.8m as well."
    },
    {
        "video_name": "3LiyCxCTrqo",
        "Q": "What is that centre is called where the carbon is situated at 5:10 ?",
        "A": "Are you referring to the Chiral centre? The Carbon that is attached to 4 other groups. This is the centre for the stereochemistry of the molecule."
    },
    {
        "video_name": "3LiyCxCTrqo",
        "Q": "8:33.... shouldn't Jay have drawn the OH group on the other side of the chiral carbon? Why did he draw it on the same side as what the bromine left from? (p.s. Hi Ernest!)",
        "A": "I think it dint really matter since the Br there was in front of the plane, whether u draw it going below or going on top, the OH is going to attack from behind the plane. Hope this helps."
    },
    {
        "video_name": "3LiyCxCTrqo",
        "Q": "At 6:00, NaOH is a strong base.. So wouldn't it be an E2 reaction instead of SN2 reaction? In some previous video we saw that presence of strong nucleophile favours SN2 reaction and presence of strong base favors E2 reaction.. Please correct me wherever I am wrong..",
        "A": "I think E2 mechanism takes place in the presence of an alchoholic medium. And for SN we are using aqueos medium And E1CB MECHANISM is followed in even stronger bases such as NANH2(SODAMIDE)"
    },
    {
        "video_name": "3LiyCxCTrqo",
        "Q": "8:00 - besides the presence of a strong base, what other clues are there that it will be an SN2 rxn?",
        "A": "If the substrate is primary or secondary, the reaction is usually SN2.. A tertiary substrate reacts via SN1."
    },
    {
        "video_name": "3LiyCxCTrqo",
        "Q": "At 7:18 shouldn't the molecule be 2-hydroxybutane as the alcohol is a side chain and not apart of the main chain?",
        "A": "You are right. The OH group is a substituent on the main chain. But the OH group takes priority in the name: buran-2-ol."
    },
    {
        "video_name": "649ZlWMp0LE",
        "Q": "At 7:07 it is told that that the IE of boron is lower than beryllium due to extra shielding effect. Why doesnt this shielding effect also affect the atomic size in similar way?\nAlso\nat 9:22 the video tells that electron added to the same orbital is repelled by the electron already present but if they have the opposite spin how can they repel eachother?",
        "A": "Although the two electrons may have opposite spin, they still both have negative charges. These negative charges will repel the two electrons, regardless of quantum numbers."
    },
    {
        "video_name": "649ZlWMp0LE",
        "Q": "Around 12:22 sal says that oxygen has lower ionization energy because the fourth 2p electron must go into an orbital that already has another electron in it and thus they repel each other. So, in that case wouldn't the ionization energy of Fluorine also be lower since the 4th and 5th 2p electron goes into another orbital thus causing repulsion between the orbitals",
        "A": "Apparently the increased repulsion is less than the increased attraction associated with increased nuclear charge ... I suspect we can t understand in detail without getting into some serious quantum mechanics ..."
    },
    {
        "video_name": "WLKEVfLFau4",
        "Q": "At 6:42 Sal says kinetic energy and temp. do not change from volume expansion unless work was done. What about in a refrigerator when the coolant passes through an expansion valve and gets cold as it boils in a low pressure environment? Is work being done here?",
        "A": "Pretty sure work is being done. This idea of a gas expanding into a vacuum is pretty theoretical."
    },
    {
        "video_name": "QQMZ1ljepWg",
        "Q": "At 8:57, when he explains that the axis of symmetry does not exist on 1,3 - Difluorocyclopentane, if you draw an axis of symmetry vertically directly down the middle of the molecule, it would show two mirror images. Can the axis of symmetry run vertically?",
        "A": "I don t think he said there was no axis of symmetry for the entire molecule. Instead, he was trying to see if any of the two carbon atoms attached to the Florine was a chiral carbon or not. I think he meant that a chiral carbon does not have a axis of symmetry and there is no axis of symmetry around either of the carbon atoms. That is why those axis went through those carbon atoms."
    },
    {
        "video_name": "QQMZ1ljepWg",
        "Q": "At 6:08, when Sal tries to figure out if the carbon is a chiral centre, how does he know he needs to break the ring at the CH (the one bonded to F)?",
        "A": "Sal isn t breaking the ring ... he is just looking along the ring starting at that carbon to see whether the ring is the same in both directions. When he says break it out @5:33 what he really means is something like draw it in more detail ."
    },
    {
        "video_name": "QQMZ1ljepWg",
        "Q": "At 9:19, you start talking about the mirror image of 1,3-Difluoro cyclcopentane, and you go on to say that although the molecule has chiral centers, it is not a chiral molecule. Is it a meso compound?",
        "A": "Yes, this is a meso compound. Well spotted!"
    },
    {
        "video_name": "QQMZ1ljepWg",
        "Q": "At 3:22, shouldn't the hydrogen and the butyl group switch? I believe that hydrogen should be in the back of the carbon.",
        "A": "putting the hydrogen in the back is simply for ease of identifying which stereoisomer it is. The drawing at 3:22 is entirely correct, it is just in a different orientation. rotating the molecule about the OH bond can the H in the back of the molecule, but that doesn t change the molecule in any way. Note that this rotation also would move the methyl group. If you were to to straight out swap the H and the butyl group, you d end up with a stereoisomer (the enantiomer in this case)"
    },
    {
        "video_name": "QQMZ1ljepWg",
        "Q": "at 2:03 how did he know that the methyl group is in front? how can i determine which group is in front and which one is in the back?",
        "A": "He just drew it that way to illustrate the stereochemistry. You can draw the structure any way you like unless you are instructed to draw a particular stereoisomer."
    },
    {
        "video_name": "QQMZ1ljepWg",
        "Q": "At 9:34, isn't there a possibility for the 1,3-difluorocyclopentane to be a chiral molecule given that it has a trans conformation (i.e, fluorine atoms opposite orientation in the plane.)? For the cis- isomer it would be identical, hence achiral but not for trans. Is that right?",
        "A": "That s right. The trans isomer is a chiral molecule. The cis isomer has two chiral centres but the molecule is achiral because it is identical with its mirror image."
    },
    {
        "video_name": "QQMZ1ljepWg",
        "Q": "At 2:20, how can a person know that the butyl group is in the back?",
        "A": "A person has no way of knowing, because a bond-line structure gives no stereochemical information. He arbitrarily put the butyl group in the back. He could just as easily put it somewhere else."
    },
    {
        "video_name": "QQMZ1ljepWg",
        "Q": "At 8:10, one can draw an axis of symmetry along vertically too, right, splitting the molecule equally?",
        "A": "No, that wouldn t work because a vertical split would give a chlorine on the right that is not balanced by a chlorine on the left. The axis of symmetry is as shown in the video."
    },
    {
        "video_name": "DpnUrVXSLaQ",
        "Q": "At 1:46, we can see that perchlorate has more oxygen than chlorate. Is there a CLO compound that has more oxygen atoms than perchlorate? If so, what is it called?",
        "A": "Not to my knowledge. Perchlorate already has chlorine in an oxidation state of +7 which is all of its valence electrons."
    },
    {
        "video_name": "DpnUrVXSLaQ",
        "Q": "1:35... How do you determine the name? Is that something you just have to memorize?",
        "A": "an acronym suggested by my school tutor. Nick ate Supper Clam in Phoenix nitrate, sulfate, Carbonate, phosphate"
    },
    {
        "video_name": "DpnUrVXSLaQ",
        "Q": "At 0:17 what the front door is he even talking about.",
        "A": "Jay explains it very clearly. At 0:17 he just refers to the ammonium cation."
    },
    {
        "video_name": "DpnUrVXSLaQ",
        "Q": "Why are are there so many more anions than cations? 0:19",
        "A": "It s because the list is looking only at polyatomic ions and the majority of cations are monatomic (eg, Na+, Mg2+, Fe3+, etc)."
    },
    {
        "video_name": "DpnUrVXSLaQ",
        "Q": "What about HO-? Would that be hypohydrite, following the pattern shown at 2:50?",
        "A": "Just hydroxide."
    },
    {
        "video_name": "DpnUrVXSLaQ",
        "Q": "In 0:39 =====> 1:02, it says that the suffix \u00c2\u00b4\u00c2\u00b4-ate\u00c2\u00b4\u00c2\u00b4 means 3 and \u00c2\u00b4\u00c2\u00b4-ite\u00c2\u00b4\u00c2\u00b4 means 2, while in\n1:11 ======> 1:31 , \u00c2\u00b4\u00c2\u00b4-ate\u00c2\u00b4\u00c2\u00b4becomes4 and \u00c2\u00b4\u00c2\u00b4-ite\u00c2\u00b4\u00c2\u00b4 means less.\nI don\u00c2\u00b4t understand. Please help",
        "A": "It can seem confusing! This is because ite just means one less oxygen that ate . So nitrite (NO2-) has one less oxygen that nitrate (NO3-). Similarly, chlorite (ClO2-) has one less oxygen that chlorate (ClO3-). And sulfite (SO3^2-) has one less oxygen that sulfate (SO4^2-)."
    },
    {
        "video_name": "DpnUrVXSLaQ",
        "Q": "At 1:24, what is the different between \"chloride\" & \"chlorite\"?",
        "A": "Chloride is Cl- Chlorite is ClO2- In chlorite the chlorine is covalently bonded to two oxygens."
    },
    {
        "video_name": "DpnUrVXSLaQ",
        "Q": "4:02 shouldn't it be called hydrogen carbonite",
        "A": "Why? The carbonite anion has the formula CO2^2-"
    },
    {
        "video_name": "DpnUrVXSLaQ",
        "Q": "At 5:11, We have O\u00e2\u0082\u0082^2\u00e2\u0081\u00bb, I thought O\u00e2\u0082\u0082 (oxygen) didn't have a charge. Isn't there a covalent bond? Not an ionic bond? And also, how do you write in superscript here? Thank you.",
        "A": "A molecule can be covalently bonded and also have a charge. Every molecule in this video is an example of this. Yes O2 (molecular oxygen or dioxygen) doesn t have a charge, but this molecule, O2^2- (peroxide), isn t the same thing. The 2- charge means that it has 2 more electrons than it has protons."
    },
    {
        "video_name": "D6mRPgvAEOc",
        "Q": "At time 3:00, how come when pyruvate loses a carboxyl group to form acetaldehyde , there is an extra hydrogen atom on the acetaldehyde? Where does this hydrogen come from?",
        "A": "Good question. This hydrogen most likely comes from the water in the cytosol, where glycolysis/fermentation is performed. Remember that there are always small amounts hydrogen protons and hydronium ions (hydrogen donators) in an aqueous (water-based) solution. :)"
    },
    {
        "video_name": "9IYTqlVk_ZI",
        "Q": "From 3:11 to 3:27 and 7:04 to 7:32, after finding out how the priority goes , the compound is named in as (2R, 3S) 3-methyl-2-pentanol. How did that happen? Please clarify this doubt ASAP.",
        "A": "What exactly aren t you sure about here? I just watched the video and it seemed pretty straight forward. We have the name. We just need to determine R or S for both the stereocentres."
    },
    {
        "video_name": "fYUwEAPejbY",
        "Q": "At 11:29 why do you add the cell potentials? I thought the formula for E cell was\nEcell= E cathode - E anode... So shouldn't you subtract?",
        "A": "Sal did this in two steps. First he reversed the reaction for the anode and rewrote the potential (for what is now an oxidation reaction) as the negative of the reduction potential, which gives us \u00e2\u0080\u0093Eanode. Second he added the reduction Ecathode and the oxidation \u00e2\u0080\u0093Eanode potentials to get Ecathode + \u00e2\u0080\u0093Eanode. So this works out to be the same as your equation. Note that the E values here are all for reductions ..."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "At approximately 5:50, Sal said the supernova can outshine a galaxy. So, if there was a supernova in our galaxy the Milky Way, would it turn night (if it was night during the supernova) into day for a brief moment?",
        "A": "Potentially, depending on how far away it was. In 1604 there was a supernova about 20000 light years away from earth. It was visible for over 3 weeks."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "at 9:18 Sal says that the super nova causing the crab nebula is estimated to have occurred 1000 years ago, but since the crab nebula is 65000 light years away, wouldn't the light take 65000 years to get to us?",
        "A": "Watch the next video. Then, he says we only OBSERVED the supernova 1000 years ago, and since it is traveling 65000 light years, it is actually estimated to have occurred 75000 years ago. P.S. Just watch the next video. It will make more sense. Trust me. P.P.S. I made sure that the number was 65000 because I thought it was 6500. It is 6500. To Elvin Blakaj and azitelberger."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "At around 10:20, I wondered, what is the Sun's core made of? This question has probably been answered, but I would still like to find out. Thanks!",
        "A": "hydrogen and some helium"
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "At 7:00, is that the only way a neutron star forms?",
        "A": "The only way to form a neutron star, that we know of, is with a supernova."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "@5:01 What causes the super nova/expansion? Before everything is trying to get to the center, causing pressure, but now the star doing the opposite.",
        "A": "Right before that happens, the star runs out of fuel in its core. With no fuel, there is no fusion. With no fusion, there is no outward core pressure and with no core pressure, gravity is unchallenged and everything is crushed into the core forming the supernova."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "At 3:02, what is a neutrino?",
        "A": "A neutrino is a product of nuclear fusion. It is a very elusive low mass particle that reacts very little with matter. A neutrino can fly through over a light year of lead without interacting with anything in the barrier."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "At 3:02, what is a \"nuetreno\".",
        "A": "A neutrino is a fundamental particle, classified as a lepton. It is similar to electrons except it has a smaller mass and no charge. Billions of neutrinos pass through every square cm of us every second and we never notice because they don t interact with other particles very easily."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "If the crab nebula is 6,500 light years away - how did the nebula occur only 1000 years ago? I am reference 9:20.",
        "A": "We saw the supernova 1000 years ago. As it is 6500 ly away, the actual supernova would have occurred 7500 years ago."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "At 4:32 you said that the star would explode outward at an unbelievable energetic explosion. I am wondering about what would happen if instead of the core collapsing it split in two?",
        "A": "Stars are made of gas. A core wouldn t be able to split in two because it s not solid. And you can t freeze it so it splits in two because, remember, we re in space. Space is SUPER cold, and if the stars haven t gone out from THAT kind of cold (not to mention the sheer size of these stars), then probably nothing could freeze a star."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "What is a Crab Nebula? (8:43)",
        "A": "About 7500 years ago, a massive star exploded as a supernova and it left behind what we call the crab nebula as a remnant."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "In the picture of the Crab Nebula at around 8:55, are those the real colours of the shockwave remnants, or a false colour photo? Also, it is indeed a photo and not an artist's depiction?",
        "A": "Always the color in those photos is added by artists (or some color-adding automated process). The telescopes that take those pictures can sense different wavelengths of light but the images do not look anything like the colorful depictions you see."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "At 9:18, How can we see something 6,500 lightyears away if the event only took place 1,000 years ago? We should still be 5,500 years from observing this event.",
        "A": "The actual event occurred 7500 years ago. The light reached Earth 1000 years ago."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "3:25 why don't the neutrons fuse into some sort of six-quark particle?",
        "A": "Because it would be unstable, the strong nuclear force is not strong enough to bind more than a few quarks together. Quarks have a charge called color. This color charge has a value of red, green and blue. These charges can not appear on their own. So you can have pairs of quarks that are a color and its anti-color or you can have a set of three, one of each color. With this and the range of the strong nuclear force we only have 2 or 3 quark particles."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "At 4:00, Sal says the great force of energy releases an outburst of neutrinos. How is this possible? Shouldn't the energy be pressing down the neutrinos too, trapping it like a prison, along with the neutrons?",
        "A": "Neutrinos are particles that very rarely interact and can travel through solid matter without much trouble. There are about 65 billion neutrinos passing through every square centimeter of you body every second."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "At 9:24, when talking about the crab nebula, Sal marks out approximately where the neutron star should be in the center of the Crab Nebula. My question is, how come the Crab Nebula is so irregular in shape. Wouldn't the remnants of a supernova basically show up as a giant perfectly spherical cloud?",
        "A": "Gravity of other stars, interactions with other parts of the cloud, or even the fact that the supernova rarely starts in the center of the star. It usually starts some distance away from the direct core of the star."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "at 5:32 how come sal talks about components of supernovae",
        "A": "Cause they are important he might not say anything that isnt important."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "What's exactly a \"White Dwarf\"? I heard it in 2:06.",
        "A": "Watch the red giant, or the white dwarf video to find out."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "In 10:05 how many light years does he mean about the supernova by relatively close to us?",
        "A": "He means a super nova that is so close that the explosion changed the disk of gas surond the sun in such a way that that the started to clump together until they became little bits of matirial that eventuly become larger and larger masses until we have planets."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "At 9:20, the video states that the supernova responsible for the Crab Nebula is believed to have happened 1,000 years ago, but it also states that the Crab Nebula is ~6,500 light years away. Wouldn't the supernova have to at least have occurred ~6,500 years ago to have allowed the light from that supernova to reach Earth?",
        "A": "From what I ve just read here in the comments, the Crab Nebula s star itself must have exploded 7,500 years ago. It s 6,500 light years away, so it s light has taken 7,500 years to reach us, thus predicting the time period it has exploded. Which was around 5500 BC. Chinese astronauts must have just observed it when it s light had finally reached us."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "At about 5:00 Sal introduces Supernova. How long does the SN material ejection process continue? How quickly does the intense light last for astronomers to observe? Is this one source of x-rays, gamma rays, etc?",
        "A": "1. The ejection continues for thousands of years as the nebula continues to expand. It will expand until it becomes just as dense as the interstellar medium. 2. The light can last one to two months. It is the most intense during the first five days and can sometimes be seen during the day during that time. 3. Supernovas produce all types of light. Even gamma rays and X-rays."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "If what Sal said at 10:30 is true, is our oldest ancestor a supernova?",
        "A": "Based on the elements present in our Solar System, we can say that our Sun is at least a second generation star, that is it was formed in the nebula of a previous supernova explosion. This because some chemical elements that are present on our planet can be formed only in the extreme energies of supernova explosions."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "3:40, what are Neutrinos?",
        "A": "a neutral subatomic particle with a mass close to zero and half-integral spin, rarely reacting with normal matter. Three kinds of neutrinos are known, associated with the electron, muon, and tau particle."
    },
    {
        "video_name": "qOwCpnQsDLM",
        "Q": "At 9:40, Sal describes the shape of the Crab Nebula as a \"sphere\". It makes sense that the shockwave would form a sphere, but it doesn't look spherical in the picture. Why is that?",
        "A": "it is due to the bending of time and space but logically it is a sphere"
    },
    {
        "video_name": "YdBXHm3edL8",
        "Q": "At 8:44, what happens if the viruses duplicate before a cytotoxic cell gets to it?",
        "A": "then when the other viruses get other cells they will all be killed by the cytotoxic cells. There are more than one throughout your whole body :)"
    },
    {
        "video_name": "LRmwUsxNSL0",
        "Q": "What now are you talking about at 3:16",
        "A": "He is talking about the Drake Equation. I think that is what you mean."
    },
    {
        "video_name": "LRmwUsxNSL0",
        "Q": "4:50 Isn't the life span of our sun 10 billion years? Sal says that's the average life span for a star, so that would place our sun at about average. Is the sun average in most other ways, too? Does a star need to be about average to have planets that could support life?",
        "A": "Our Sun is slightly larger than the average star but there are some limits to star size and life production. A very large star will only live a few million years, not enough time to develop. Small red stars may be subject to intense and seemingly random periods of increased activity that can double the stars luminosity in a matter of minutes. Not all red dwarfs do this but some do."
    },
    {
        "video_name": "yrxAKGGEWZY",
        "Q": "At 1:54, Green mentions aform of precipitaion. It sounded like \"graouple\" to me. What is it?",
        "A": "Grauple is really soft hail. It is pretty much super tiny snow balls falling from the sky."
    },
    {
        "video_name": "U9dGHwsewNk",
        "Q": "In 6:21, he draws a double headed arrow originating from a single electron. Double headed arrows represent the movement of a pair of electron. Although he is conceptually right, I think this was a mistake. He should have drawn the arrow going form a pair of electrons.",
        "A": "Agreed. The mechanism involves the movement of electron pairs."
    },
    {
        "video_name": "U9dGHwsewNk",
        "Q": "At 11:23, why is the reaction between t-butyl bromide and sodium methoxide an elimination reaction and not a sustitution reaction?",
        "A": "The video goes only to 9:21. The t-butyl bromide is sterically hindered to back-side attack by the base. The methoxide does the only thing it can and attacks an H atom to eliminate HBr."
    },
    {
        "video_name": "U9dGHwsewNk",
        "Q": "5:32. I thought Sal said that a tertiary carbocation is a somewhat stable one. Why therefore does this new ionic compound want to give away protons, in the form of one of its hydrogens? Doesn't this mean that it is not stable?",
        "A": "Tertiary Carbocation is SOMEWHAT stable and the alkene formed is much much more stable than the tertiary carbocation."
    },
    {
        "video_name": "U9dGHwsewNk",
        "Q": "At 5:59 you chose the Hydrogen to the right of the carbocation, was this an arbiturary choice? Could you have chosen the Hydrogen to the left of the carbocation just as easy ? Thanks",
        "A": "I would advice you to see the next video Zaitsev s rule . It would tell you all!"
    },
    {
        "video_name": "U9dGHwsewNk",
        "Q": "At 3:46-3:54, what would happen if the carbon was surrounded by 2 carbons or 1 carbon? would it still be stable enough to be on its own as a carboncation?",
        "A": "If it was surrounded by only one carbon, the carbocation would be very unstable so would not likely form. If it was surrounded by 2 carbons, the carbocation may form, but it would not be as stable as a tertiary carbocation. You could adjust the conditions (e.g. the solvent) to either help stabilize or destabilize the secondary carbocation to help control its formation."
    },
    {
        "video_name": "U9dGHwsewNk",
        "Q": "why at 5:35 or so, ethanol doesn't react with that carbon cation instead of that hydrogen?\none has partially + , the other one has - .",
        "A": "It does. If the ethanol attacks the carbocation, you get an ether (SN1). If it attacks the hydrogen, you get an alkene (E1). With tertiary alkyl halides in a relatively nucleophilic/basic solvent like ethanol, these two processes are usually in competition with each other, and you get significant amounts of each product. The video was explaining the mechanism if the E1 reaction, so it did not consider the competing process."
    },
    {
        "video_name": "aD6l5bNnJgE",
        "Q": "Virtual ground example at 5:54 you say \"Boom\" We did it - and I fail to see how U arrived at Uo=Uin\nThe non-invent diagram with no resistors is just the version of the diagram from Feedback video where R1=0, R2=infinity. For this diagram Uo=Uin*(R1+R2)/R2=infinity/infinity=uncertainty. How did U arrive at Uo=Uin?\nPS. this page shows minus instead equal and | insted of plus in my comments",
        "A": "It s tricky dealing with infinities. To derive the unity gain expression starting from the non-inverting amplifier equation, you start with R1 = 0 as you said. Instead of setting R2 directly to +infinity, write it as a limit, as in vo = Lim (as R2 goes to +infinity) of [ (0 + R2)/R2]. When you do it with limits, the ratio of R2/R2 stays gracefully equal to 1 all the way up to +infinity."
    },
    {
        "video_name": "xYoQQCO15GE",
        "Q": "At 2:35, we know that the first person to obtain a virus is known as the 'Index Case', but can we track the very first Index Case, to find out exactly how the Ebola virus was formed?",
        "A": "The very first index case of this outbreak was thought to be a 2-year-old child who died from Ebola in a remote village in Guinea."
    },
    {
        "video_name": "xYoQQCO15GE",
        "Q": "If you wash your hands thoroughly after you touch someone's blood like on 7:48, is it possible that, even if you touch parts of your face, you won't contract ebola?",
        "A": "Only if you wash your hands correctly. However, there s a low probability that says that you ll get ebola even if you washed your hands correctly."
    },
    {
        "video_name": "xYoQQCO15GE",
        "Q": "at 0:45, what does he mean about \"The geography is different\" ?",
        "A": "he means where the outbreaks started. one of them started on the coast of west Africa and another in central Africa."
    },
    {
        "video_name": "xYoQQCO15GE",
        "Q": "At 4:59, Dr. Rishi Desai spoke about blood pressure medication for infected individuals that experience Low Blood Pressure. Why does low blood pressure occur in Ebola patients?",
        "A": "Common symptoms of Ebola infection include vomiting, diarrhoea, and bleeding. All of these involve loss of fluid from the body, which will decrease the blood volume. With lower blood volume the patient will also have decreased venous return, decreasing preload on the heart chambers. The decreased preload will result in less contractility in the heart muscle and less volume of blood flow, which can be measured as lowered blood pressure."
    },
    {
        "video_name": "xHgPtFUbAeU",
        "Q": "At 1:39, I am curious, but why does 2.00 have 3 sf? I thought that would only be 1, because it is trailing zeros? I'm confused...",
        "A": "Trailing zeros are significant when they are after the decimal. It means that we know that the measurement taken was to the 1/100th of a unit and just happened to be 2.00 as opposed to just putting 2 which could be anything from 1.5 to 2.49."
    },
    {
        "video_name": "xHgPtFUbAeU",
        "Q": "Around the 2:50 mark of the video shouldnt the rounding have gone to 3.6? Because we are rounding to the tenths place in 3.56 so the 6 in the hundredths place should bump the 5 to a 6?",
        "A": "I m not sure if you can see it, but there is a clarification in the bottom right of the screen that corrects the mistake that you just pointed out. This correction was made ages ago :)"
    },
    {
        "video_name": "xHgPtFUbAeU",
        "Q": "at 2:36 i don't understand what you mean by a \"least precise\" number. Could you further explain this ?",
        "A": "because 1 meter is less precise than 1.2343434 meter. As you get more decimals it becomes more precise. However, you cannot be more precise than the least precise number, or 1 in this case."
    },
    {
        "video_name": "xHgPtFUbAeU",
        "Q": "At 6:13 on the tower measurement problem, if the blue tower had been 1.90*9* and you added to get 3.999 m...\n\na) Would you round to get 4m?\nb) In your answer would you write 4m or 4.00m?",
        "A": "In your answer if you rounded to 4 you would just write 4."
    },
    {
        "video_name": "xHgPtFUbAeU",
        "Q": "at 3:07 how could 3.56 be rounded up to 3.7??To the best of my knowledge it should have been 3.6??",
        "A": "You re right, there s even an annotation correcting the mistake."
    },
    {
        "video_name": "xHgPtFUbAeU",
        "Q": "how come at 3:07 3.56 was rounded to 3.7 isn't it supposed to be 3.6",
        "A": "You are correct, but you can see that a red pop up shows the correct answer."
    },
    {
        "video_name": "xHgPtFUbAeU",
        "Q": "he got it wrong on 2:57 wouldn't that be 3.6 instead of 3.7?",
        "A": "Yes. He did get it wrong. At 3:01 there s now a message saying 3.6 not 3.7! ."
    },
    {
        "video_name": "xHgPtFUbAeU",
        "Q": "4:38 - Me: This guy is color bling omg same highfiveeee",
        "A": "It looks like you re learning more than just Significant Numbers."
    },
    {
        "video_name": "xHgPtFUbAeU",
        "Q": "At 0:56 wouldn't 2.00 have 2 sig fig?",
        "A": "No, it is three; the two and both zeros are all significant figures."
    },
    {
        "video_name": "lVadjWOjvV8",
        "Q": "First of all what is the tangent? Also, at 4:34 you said that you know the distance is 90-theta. Is this how you would also get the perpendicular distance?",
        "A": "the tangent is the opposite of the angle over the ajasent side to the angle."
    },
    {
        "video_name": "lVadjWOjvV8",
        "Q": "In 5:23 what is Soh Cah Toa?",
        "A": "It s A Way To Remember The Three important trig Ratios... SOH-Sine is opposite over Hypotenuse CAH-Cosine is Adjacent over Hypotenuse TOA-Tangent is Opposite over Adjacent"
    },
    {
        "video_name": "lVadjWOjvV8",
        "Q": "In 6:58, arc sec is used. Why do we have to use arc sec? and what is the significance of arc sec?",
        "A": "An arc second is 1/3600 of a degree. I don t know why Khan bothered with them when he had to convert to degrees anyways, but they are used to give more precision than a degree without a bunch of numbers to the right of the decimal. I have no idea where the term comes from."
    },
    {
        "video_name": "lVadjWOjvV8",
        "Q": "At 6:56, Sal gives theta a value of 1.5374 arc seconds. Is this just an arbitrary number that he used for the purpose of this example or did I miss how he solved for theta? If so how do you solve for theta?",
        "A": "The whole point of the stellar parallax method is that we MEASURE the angle and then use that to determine distance."
    },
    {
        "video_name": "SlQDuTS6FR4",
        "Q": "At 4:03, why is the man is wearing a Khan Academy shirt?",
        "A": "It s on the khan academy website so they have to do that to represent khan academy."
    },
    {
        "video_name": "1FQPXtN7MeI",
        "Q": "From 9:10 to 9:30, why do nonbonding electrons have more energy than bonding electrons?",
        "A": "Bonding electrons are between two nuclei. This is a low-energy situation. Lone pairs are attracted to only one nucleus. This is a higher-energy situation."
    },
    {
        "video_name": "1FQPXtN7MeI",
        "Q": "at 9:42, why are there only two non-bonding electrons? Aren't there four?",
        "A": "Yes, there are two nonbonding orbitals, and each contains two electrons. He is just talking about an n\u00e2\u0086\u0092\u00cf\u0080* transition from one of these orbitals."
    },
    {
        "video_name": "1FQPXtN7MeI",
        "Q": "Min 10:30 >> What is the experimentally observed UV/Vis spectrum of ethanal, given that Jay mentions two wavelengths, ~180nm and ~290nm? Does the spectrum have two peaks? Or is it a mixture of the two? Thank you",
        "A": "There are two peaks. The intense peak at ~180 nm corresponds to the transition of a \u00cf\u0080 electron to an antibonding \u00cf\u0080 orbital. A much weaker peak at ~290 nm corresponds to the transition of a lone pair electron on O to an antibonding \u00cf\u0080 orbital."
    },
    {
        "video_name": "CafRuKs7EfE",
        "Q": "at 6:27 he explains that a base would take that proton. When writing out a reaction scheme, should the DCC be accompanied by a base like pyridine?",
        "A": "Both DCC and the amine are bases so I don t think you would need (or want) to add another base."
    },
    {
        "video_name": "J30zpvbmw7s",
        "Q": "At 14:25, where does the extra phosphate group come from to turn ADP's into ATP's? Thanks!",
        "A": "I believe it is from the soil that the plants absorb for the process, I m not sure about animals"
    },
    {
        "video_name": "J30zpvbmw7s",
        "Q": "At 1:58 where do the 2 electrons come from? How and from who does the NAD+ carrier get those in the first place?",
        "A": "Those 2 electrons and one hydrogen ion came from the substrate(like glucose)."
    },
    {
        "video_name": "b6VQv76BQDs",
        "Q": "I don't know if this is a theory, but wouldn't beyond the \"observable\", as Sal said in 0:32, maybe be more space and galaxies that their light hasn't reached our sight yet? Or is it really certain that the \"observable\" edge is actually the edge of space and the universe?",
        "A": "I do not believe there is any current theory that would accomodate the assertion that the edge of the observable universe is in fact the edge of the universe."
    },
    {
        "video_name": "o_puKe_lTKk",
        "Q": "what Sal meant by his word ''ambiguous'' 10:18",
        "A": "It means Unclear or vague. Sal likes that word. Haha ); Have a nice day."
    },
    {
        "video_name": "kxNaVzQ07yc",
        "Q": "at 6:16 isn't E*cell = Ecathode - Eanode ? why did he add those two?",
        "A": "He changed the sign of the standard reduction potential for the oxidation half-reaction - I suggest watching the rest of Jay s videos on the subject of voltaic cells."
    },
    {
        "video_name": "kxNaVzQ07yc",
        "Q": "At 2:09, I thought you weren't supposed to switch the sign of a standard potential, even if you flip the half-equation... Someone explain, please?",
        "A": "He flipped it because it was the oxidation reaction and standard potential or Ecell in volts is only in reduction reaction form. The one he flipped was originally oxidation but he had to write it in the reduction form of that makes sense"
    },
    {
        "video_name": "kxNaVzQ07yc",
        "Q": "2:56 do the coefficients in the original equation matter at all?",
        "A": "Not as far that the standard reduction potentials are concerned, no. What does matter, is how many moles of electrons are transferred."
    },
    {
        "video_name": "_yvdb_0NGT8",
        "Q": "What does the R at 2:20 stand for and where is it in the Periodic Table",
        "A": "The R is not an element for any element in the Periodic Table. It is a general symbol for an alkyl group such as methyl, ethyl, etc."
    },
    {
        "video_name": "_yvdb_0NGT8",
        "Q": "At 5:00, Jay says that CH2OH is cellulose molecule which is made up of glucose molecules. But what I know is that cellulose is a polymer made up of individual sugar/glucose molecules, so how does it contain only 1 carbon atom? Isn't it supposed to be a polymer? TIA and sorry for the silly doubt. :-)",
        "A": "He drew the braces around the CH\u00e2\u0082\u0082OH, but he meant them to apply to the whole glucose molecule. Cellulose is a polymer with the formula (C\u00e2\u0082\u0086H\u00e2\u0082\u0081\u00e2\u0082\u0082O\u00e2\u0082\u0085)\u00e2\u0082\u0099."
    },
    {
        "video_name": "_yvdb_0NGT8",
        "Q": "at 4:53 the nitrate ester that formed from the glycerol, i thought esters need the carbonyl functional group to be an ester, so how is this still an ester?",
        "A": "Technically, the functional group is not an ester , which, as you describe, has a general formula of RCOOR . The functional group here is a nitrate ester which is it s own distinct functional group with the general formula of RONO2."
    },
    {
        "video_name": "_yvdb_0NGT8",
        "Q": "3:01 i know there are exceptions to the rule, but i though oxygen could only bond twice, and if i remember correctly, how come oxygen bonds three times then?",
        "A": "Oxygen can t have more than 8 valence electrons, but it can have 3 bonds with a positive charge."
    },
    {
        "video_name": "_yvdb_0NGT8",
        "Q": "At 3:48, why does the R-group, when it binds to the notroniumion, get a formal charge of +1? Shouldn\u00c2\u00b4t it remain neutral?",
        "A": "That + charge is on the oxygen not the R group. It has 3 bonds and 1 lone pair so a formal charge of +1"
    },
    {
        "video_name": "DiBXxWBrV24",
        "Q": "4:30 anyone else think that the force be drawn perpendicular to the lever and not directly down? It may be an ok as an approximation, but my impression was Moment = (distance * force perpendicular)",
        "A": "Just wondering...does mechanical advantage affect energy?"
    },
    {
        "video_name": "DiBXxWBrV24",
        "Q": "At 3:14 Sal says that energy cannot be made out of thin air.\nBut what about if the machine is 101% efficient , say the finsurd's perpectum machine.\nPerpetual motion?",
        "A": "Energy cannot be created out of thin air, so you know that any machine that claims to be 101% efficient is a fraud or a trick."
    },
    {
        "video_name": "vh166DKxYiM",
        "Q": "6:20 it says -70 millivolts. Aren't measurements always positive :\\",
        "A": "No, they re not."
    },
    {
        "video_name": "EKKe7DBZVhI",
        "Q": "At 2:00, are you saying that Jesus started a new era, or that this is just what I believe?",
        "A": "To be historically accurate one must understand that this calendar system didn t originate with the supposed birth of anyone. It was adopted in 1582. If the birth was so important why wait 1582 years to make the change? It was politically motivated not divinely."
    },
    {
        "video_name": "EKKe7DBZVhI",
        "Q": "At 7:18, Why is there no year 0?",
        "A": "Zero is not a positive nor negative number so it is not really a number ."
    },
    {
        "video_name": "EKKe7DBZVhI",
        "Q": "At 8:15, the video said that there is no year 0. Why is this?",
        "A": "Because year zero would be before time existed!"
    },
    {
        "video_name": "3KrqQXsWP1M",
        "Q": "How is the first carbon less substituted at 1:12?",
        "A": "Jay did not draw the R group on the right carbon, presumably implying the right portion of the alkyne could have been attached to an even larger carbon chain such as (CH2CH2CH2CH3). This considered, the regiochemistry is correct."
    },
    {
        "video_name": "18HrvnXsgNs",
        "Q": "At1:00, what's the meaning of the different of two blocks is moving more mass? what's the difference bwtween the weight and the mass?",
        "A": "I will help you figure out the answer but you ll have to work with me too. Ready? Think of the situation when there was no block 3. How many external forces are acting on the system which includes block 1 + block 2 + the massless rope connecting the two blocks? Think about it and it doesn t matter whether your answer is wrong or right, just comment what you think. If it s wrong, you ll learn something new. If it s right, then there is one less thing to learn!"
    },
    {
        "video_name": "g_snytB7iQ0",
        "Q": "What is a dipole at 1:45?",
        "A": "A dipole is a molecule in which a concentration of positive electric charge is separated from a concentration of negative charge. Water is a dipole because it has a partially negative oxygen end, while the hydrogen atoms are at the positive end."
    },
    {
        "video_name": "g_snytB7iQ0",
        "Q": "At 6:15, Jay says the silver molecule is being surrounded by water molecules, but how is that when H2O is not included in the chemical equation?",
        "A": "Actually it is -- the little (aq) for aqueous after the dissolved species indicates that the solvent is water ..."
    },
    {
        "video_name": "g_snytB7iQ0",
        "Q": "from 4:23 , why does Na+ combine with NO3- when it is aldready combined with Cl- and why does Ag+ combine with Cl- when it is aldready combined with NO3- ?",
        "A": "The key here is that AgCl is insoluble and falls out of the solution. This removes both the Cl- and the Ag+ ions from the solution. So although Na+ is very happy to be in solution with Cl- and, likewise, NO3- is very happy to be in solution with Ag+, this is no longer possible because of the precipitation of the Cl- and Ag+ ions. The precipitation of the AgCl drives the reaction to completion."
    },
    {
        "video_name": "g_snytB7iQ0",
        "Q": "At 7:13, you said the electrostatic attraction of ionic crystal is much stronger than the force of hydration and therefore form AgCl precipitate. However, in the previous dissolution example of NaCl, the force of hydration was able to break the ionic bond and form sodium ions and chloride. Why is it? Which force is stronger, the electrostatic attraction or the force of hydration ? Thank you!",
        "A": "It will depend on the compound as to which force is the stronger. NaCl dissolves in water because the force of hydration is stronger than the electrostatic attraction, but for AgCl it is the other way around so AgCl is insoluble in water."
    },
    {
        "video_name": "g_snytB7iQ0",
        "Q": "At 6:10, why is there water molecules? We only have NaCl and AgNO3.",
        "A": "(aq) means they re dissolved in water"
    },
    {
        "video_name": "g_snytB7iQ0",
        "Q": "At 9:45, it is said that Na+ and NO3 - are spectators. But they form NaNO3, then why are they spectators?",
        "A": "Because the Na+ and NO3- will remain as ions in solution and not form NaNO3. This only occurs when you take them out of solution."
    },
    {
        "video_name": "g_snytB7iQ0",
        "Q": "At 0:56, what is the symbol that Jay writes to represent the partial positive and negative charge of the hydrogen and oxygen in the water?",
        "A": "It s a lower-case Greek letter delta (\u00ce\u00b4). It corresponds to the English letter d and looks like a figure 8 that hasn t been completed."
    },
    {
        "video_name": "g_snytB7iQ0",
        "Q": "At 1:27, they bring up the concept of water splitting Sodium Chloride into their individual parts. If this is the case, why does salt water still taste salty, if in fact you are tasting Chlorine and Sodium separately?",
        "A": "Because it s the sodium ions that our tongues detect as being salty."
    },
    {
        "video_name": "cKAF1v5hJoE",
        "Q": "At 4:50, if you look at the top you see an arrow with a W. what does the W stand for?",
        "A": "The W stands for work. The heat is doing work to lift something:)"
    },
    {
        "video_name": "omvNINaRdxg",
        "Q": "Does anyone else hear a beeping sound around 1:40?",
        "A": "I do - but it s really faint."
    },
    {
        "video_name": "omvNINaRdxg",
        "Q": "At 6:30-6:40ish, is he saying data or beta?",
        "A": "He is saying Theta. The symbol he drew, \u00ce\u00b8, is used in this case as the symbol to represent the angle of the plane. It is a letter used from the Greek alphabet to represent a lot of mathematical and scientific values. It is similar to using X or Y as variables in equations to represent a number or numbers on an axis."
    },
    {
        "video_name": "omvNINaRdxg",
        "Q": "Why are we calculating the tangent at 7:56",
        "A": "we are using the Tangent function to calculate the distance between our sun and that star."
    },
    {
        "video_name": "ImTwySWxA4U",
        "Q": "At 4:23-4:25 Sal mentions a tetrahedral shape. I really don't understand what it is. Please explain to me. What is a tetrahedral shape?",
        "A": "Whenever you encounter a tetrahedral molecular geometry, just visualize a tripod stand with its top towering up. This is the position in which 4 similar electron clouds come in the state of minimum repulsion"
    },
    {
        "video_name": "AvtS_IrlbYk",
        "Q": "Hi, at 6:20 we go over distributive shock. Would osmosis be a factor in thus type of shock?",
        "A": "Yes that is the mechanism of the fluid movement, it depends on the reason for the distributive shock. The are different factors that cause the fluid to move from the blood to the tissues."
    },
    {
        "video_name": "AvtS_IrlbYk",
        "Q": "At 5:53, what is a pulmonary embolism? I never understood that.",
        "A": "A pulmonary embolism is a blockage of an artery in the lungs, usually caused by a blood clot."
    },
    {
        "video_name": "pnPWdDOR9XE",
        "Q": "At 2:25 how do you get the value that [A]/[HA]=1.82?",
        "A": "in the previous video this process is explained. But [A]/[HA]= 10^pH-pKa. She gave an example PH=5, and then we were able to find the pKa from the Ka value she provided. Pka=4.75. So 10^5-4.75=1.82"
    },
    {
        "video_name": "pnPWdDOR9XE",
        "Q": "At 6:05 i don't get why the reaction only goes one way when CH3COO- is a weak acid. Wont its strong conjugate base dissociate?",
        "A": "The conjugate base of a weak acid is not a strong base. Acetate is quite a weak base. There is some reassociation with acetate and water but not much."
    },
    {
        "video_name": "pnPWdDOR9XE",
        "Q": "At 9:33, \"5.90M\" should just be \"5.90\".",
        "A": "Yeah.. you r correct. She just did a mistake.. Ignore it and go ahead.."
    },
    {
        "video_name": "eEUvRrhmcxM",
        "Q": "14:30 why are you starting with the big B?",
        "A": "because thats the dominant one"
    },
    {
        "video_name": "eEUvRrhmcxM",
        "Q": "At 11:17, are the two lower case b's (the individual that would have two alleles for blue eyes) be called homozygous recessive?",
        "A": "That is correct."
    },
    {
        "video_name": "eEUvRrhmcxM",
        "Q": "At 15:09 does the order in which you write the heterozygous trait matter? Like does the dominant allele always appear first?",
        "A": "No,it doesn t matter.But i think it s just convention to write the dominant allele firstly. Hope it helped :)"
    },
    {
        "video_name": "eEUvRrhmcxM",
        "Q": "At 2:50, Sal says that homo sapiens could only have blue or brown eyes. But how come ID's and Driver Licenses have a selection of blue, brown, and black eyes? Everybody in my family have black eyes.",
        "A": "He means to say (which is mentioned in the video) imagine a population where the brown and blue are the only eye color possibilities."
    },
    {
        "video_name": "eEUvRrhmcxM",
        "Q": "At 7:10, how do the scientists determine which allele is the dominating one for a trait?",
        "A": "They test the traits of offspring. Say you have a two homozygous parents with different phenotypes (what they show). Well, if they have a child, whichever trait that is dominant will show up in the child."
    },
    {
        "video_name": "bUCu7bPkZeI",
        "Q": "In the 1:42 mark, he has drawn the NO3 structure, but here all the 6 valence electrons of oxygen are shown, and there is a bond between N and O, but shouldn't he be showing 5 electrons because since there is a bond? I mean that bond is bonding to what on oxygen...an electron so instead of 6 dots there should be 5 right?",
        "A": "If I am understanding your question correctly you are asking why did he remove a lone pair of electrons from oxygen to create a double bond with nitrogen? Each bond requires 2 electrons to create (covalent bonding, shared electrons between two nuclei). The octet for nitrogen is satisfied, even though nitrogen can have exceptions to this rule; as well as sulfur, group 3a elements, and elements with higher d orbitals."
    },
    {
        "video_name": "bUCu7bPkZeI",
        "Q": "At 5:17 why are there 5 dots along the bonds??",
        "A": "Those dots represent partial double bond character across all three of the N-O bonds. The three resonance structures do not mean the molecule is rapidly shifting between them. It means that the double bond electrons are shared over all four of those atoms and that every bond is exactly the same."
    },
    {
        "video_name": "bUCu7bPkZeI",
        "Q": "@6:07, Jay says that \"we know the double bond would be shorter than a single bond.\" How do we know this and what is the reasoning behind it?",
        "A": "Double bond are formed by a sigma bond + a pi bond.. As there are two bonds the atoms are more attracted toward each other making them much more closer. But in a single bond only a single sigma bond is present, thus the attraction force is lesser than a double bond. So it s shorter than a single bond."
    },
    {
        "video_name": "bUCu7bPkZeI",
        "Q": "At 1:55, why does Nitrogen not have an octet? Since it has a valence number of 5, after forming three single bonds with the oxygens, wouldn't still have a lone pair of electrons that would complete it's octet?",
        "A": "The structure isn t finished at 1:55, be patient! When we draw dot structures we do not automatically assign each atom the valence electrons it normally has, because that can lead to the wrong structure. Nitrate is an example of why we do it this way, the nitrogen does not have a lone pair in nitrate, it has been oxidised."
    },
    {
        "video_name": "bUCu7bPkZeI",
        "Q": "At 5:57 , if structure is not applicable as double bond is shorter than other single bonds , how does other structures, say h2so4 , be correct(as it has a double bond)?",
        "A": "In H2SO4 experimentally we observe that there are 2 different S-O bond lengths. 2 S-O double bonds and 2 S-O single bonds could explain this. In the sulfate anion (SO4 2-) however all bond lengths are exactly the same and are shorter than a normal S-O single bond. So that would lead us to a similar conclusion as nitrate in this video, partial double bond character across all the bonds."
    },
    {
        "video_name": "bUCu7bPkZeI",
        "Q": "at 6:15, how is NO3 having a negative charge despite being a covalent compound ,where does the -ve charge come from?",
        "A": "The nitrogen in NO3 can be considered to have 6 valence electrons due to the covalent bonds and electron sharing with each oxygen atom. Since it only requires 5 valence electrons based on the periodic table and 6-5=1 extra electron, this results in a formal charge of -1 for the N atom. This is why this molecule will have an overall negative charge."
    },
    {
        "video_name": "bUCu7bPkZeI",
        "Q": "At 2:11 , why doesn't shifting the lone pair of electrons from the topmost oxygen atom give it a positive charge ?",
        "A": "Note that he hasn t even assigned formal charges then, he is finishing off the lewis structure. But: an oxygen with 1 bond and 3 lone pairs has a formal charge of -1 An oxygen with 2 bonds and 2 lone pairs has a formal charge of 0"
    },
    {
        "video_name": "bUCu7bPkZeI",
        "Q": "At 5:53, the hybrid resonance structure does not depict the other valence electrons on the oxygen atoms. Is this correct?",
        "A": "They should be there too"
    },
    {
        "video_name": "bUCu7bPkZeI",
        "Q": "At 2:50 he draws [ and ] to the structure to show the ion. Is that the way we show ion at the structure?",
        "A": "Square brackets [...] are used to show a resonance structure of an ion. The charge on the ion is placed outside the brackets."
    },
    {
        "video_name": "bUCu7bPkZeI",
        "Q": "At 0:52, Jay puts Nitrogen in the center. Isn't the most electronegative atom always in the center, not the least?",
        "A": "No the least electronegative element goes in the centre almost all of the time. Often the unique element goes in the centre too. Nitrogen is both the unique atom and the less electronegative in nitrate, so it s in the centre."
    },
    {
        "video_name": "bUCu7bPkZeI",
        "Q": "At 5:20, I get confused. When you de-localize the electrons, you added 4 electrons in a line connecting the first Oxygen to Nitrogen. Why does that Oxygen electron not need 8 total electrons anymore? Also, its seems that you added 14 de-localized electrons...why is it 18 total electrons to complete the octets? And doesn't adding those de-localized electrons between the Oxygens and Nitrogen give Nitrogen more than 8 electrons?",
        "A": "When he draws in the delocalized electrons, it s not literally showing that 14 of them are added \u00e2\u0080\u0094 those dots represent the idea of delocalized electrons generally, not individual electrons. No electrons are added, but rather some of them are no longer bound to or paired with any one atom. The octets of each atom are still satisfied \u00e2\u0080\u0094 you can think about it as if those electrons in the structure not associated with any one atom are spending enough time near each oxygen to keep all of them satisfied."
    },
    {
        "video_name": "bUCu7bPkZeI",
        "Q": "At 5:58, Jay says that an N=O double bond would be shorter than an N\u00e2\u0080\u0093O single bond. Why is this the case? Why do different bonds have different lengths?",
        "A": "More specifically, a bond length is determined by the size of the atoms. Let s take a look at O2 and S2, O2 would have a shorter bond length due to the atoms being able to get closer. If you meant why multiple bonds have different bond lengths that aren t equal, it s because bond length is also dependent on orbital overlap. A single bond which is sometimes called a sigma bond tends to have more electronic overlap than a double bond which is composed of a pi bond that tends to have less overlap."
    },
    {
        "video_name": "CHHu-iTwHjg",
        "Q": "At 6:32, why does the system have negative change in\nentropy?",
        "A": "System can have either positive or negative entropy. The universe has only positive entropy."
    },
    {
        "video_name": "CHHu-iTwHjg",
        "Q": "At 5:32, Sal says that if T is low enough delta G can be less then zero. I don't understand how you will get a negative product if your multiplying a negative (entropy) by a negative? ( - T)",
        "A": "T could NEVER be negative. If we refer to the formula, we could see T= 273.15+t (celsius ). There s law (I could not remember the name), T=0K or less can NEVER be reached."
    },
    {
        "video_name": "PvoigrzODdE",
        "Q": "There were originally 2n chromosomes, so where did the extra 2n chromosomes come from when the cell divides through mitosis at 1:35?",
        "A": "They split. So at the beginning the zygote is big. So when it differentiates it becomes smaller. When it is the size of the normal cell, it eats a lot to split at the same size. I hope this helps."
    },
    {
        "video_name": "PvoigrzODdE",
        "Q": "at 3:57 what is the name of the cell that is produced after mitosis? is it some thing like blood cell, nerve cell or anything please help",
        "A": "Mitosis (division of the nucleus) and cytokinesis together define the mitotic (M) phase of an animal cell cycle\u00e2\u0080\u0094the division of the mother cell into two daughter cells, genetically identical to each other and to their parent cell. mitosis happen in all kinds of cells (blood, nerve, body).. on the other hand there is also meiosis that happens in reproductive cells.. (excuse my english!!)"
    },
    {
        "video_name": "PvoigrzODdE",
        "Q": "At 8:06, I believe I am understanding this correctly: mutations in somatic cells do not affect your genetic makeup passed down to your offspring. Why are there people who \"pass down\" genes linked to cancer in the heart, brain, etc? Is it because the gametes contain mutations and the zygotes in the offspring then differentiate into somatic cells?",
        "A": "Somatic Cancers are not generally passed down. If this does happen, the cancer would be congenital (which is quite unlikely). Even if it true, the mutation should be in the GERM cells ONLY to be passed down."
    },
    {
        "video_name": "PvoigrzODdE",
        "Q": "in the video at 1:51 what does 2n represent",
        "A": "2n represents that diploid number of chromosomes .It means 46 chromosomes."
    },
    {
        "video_name": "PvoigrzODdE",
        "Q": "At 3:14 he goes into the different kinds of cells (brain cells, heart cells, etc.). Aren't these kinds of cells derived from germ layers? So for example, endodermal cells form the digestive system and ectodermal cells form nerves while mesodermal tissue forms bones and muscles...",
        "A": "Your intuition is correct."
    },
    {
        "video_name": "PvoigrzODdE",
        "Q": "@4:00 I am now curious about the process through which these cells specialize!",
        "A": "Cell-to-cell communication as well as the location of an individual cell within its tissue microenvironment defines what cell will become what."
    },
    {
        "video_name": "PvoigrzODdE",
        "Q": "At 3:15, Sal explains that the cells differentiate into specific types, like brain cells, heart cells, lung cells and so forth. How do those cells \"know\" which type they have to differentiate in ?",
        "A": "First, by the proteins they are made of, and second by communicating with the cells surrounding it."
    },
    {
        "video_name": "PvoigrzODdE",
        "Q": "at 7:27 Sal mentioned abt mutations. what are mutations?",
        "A": "A mutation is when there is a permanent change in the DNA sequence. It can be beneficial or harmful to the organism."
    },
    {
        "video_name": "uCLZyTliZj0",
        "Q": "@ 0:48 shouldn't f'(x)=0?cause when f(x)=k,f'(x)=0.Which means f'(x)=0*g(x).So when we multiply anything by zero is always zero right?",
        "A": "We have f(x)=k\u00e2\u0080\u00a2g(x). Take the derivative of both sides to get f (x)=d/dx (k\u00e2\u0080\u00a2g(x)). So what does the right hand side simplify to? The derivative of a product is not the product of the derivatives. That is, it s not the case that d/dx(f(x)g(x))=f (x)g (x). If that were the case, then every derivative would be 0, since g(x)=1\u00e2\u0080\u00a2g(x). That s not useful. Sal goes on to prove in the video why the constant gets moved outside the derivative."
    },
    {
        "video_name": "uCLZyTliZj0",
        "Q": "At 2:21 Sal Says \"And we know from our limit properties...\" to explain how he got from the expression on the left to the expression he began writing.\nI don't understand what limit properties he was referring to and how he got to the next step.",
        "A": "He is able to factor out the k from the limit...lim kf(x) = k lim f(x)."
    },
    {
        "video_name": "uCLZyTliZj0",
        "Q": "At 4:46, why did not sal write 2h in denominator? One from the subraction of g and one from j.",
        "A": "That isn t how division works - you see this for yourself by doing something like: Let a = 24, b = 8 and c = 4 (a - b)/2c = a/2c - b/2c vs. a/c - b/c (24 - 8)/(2*4) = 24/8 - 8/8 vs. 24/4 - 8/4 16/8 = 2 = 3 - 1 vs. 6 - 2 It might we worth brushing up on some of the earlier material to make sure you ve got a good grasp of the fundamentals ..."
    },
    {
        "video_name": "uCLZyTliZj0",
        "Q": "@1:35 where did you get the h from?",
        "A": "We get that h from the definition of a derivative: f (x) = lim(h->0) ( (f(x+h)-f(x)) / (x+h)-x) h basically represents a very little increase in x value. instead of h some book also have (delta x) [delta looks like a small triangle, its a greek symbol]"
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 6:22, on second hand, where did Theia come from? (P.S. please correct if I spelled Theia incorrectly. thanks!)",
        "A": "In the early solar system, there were many more planets than there are now. The majority of them were ejected from the system or flung into the Sun. A few of them collided and merged with Jupiter but one called Theia collided with earth."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "In 3:20, he is saying that the rocks have gravity and so collide and join each other. But actually, shouldn't they be breaking apart due to the force at which they collide each other?",
        "A": "Yes, but then the fragments get attracted back together. Eventually you get bits that can come together at speeds low enough that they don t shatter and disperse"
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "@ 3:31 ; like Pluto?",
        "A": "No. They are called planetoids. What basically happens is that two chunks of material in space stick together, creating a gravitational attraction that pulls in more chunks of material. That bigger chunk has a larger gravitational attraction, so it pulls in more and more material, and the cycle repeats. Pluto is a failed planet. A planetoid that did not have a large enough gravitational attraction to attract t he nearby chunks"
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 2:50, What does Sal mean by \"Early Sun\"",
        "A": "He means the Sun when it was younger"
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 4:55, Sal talks about theia hitting the earth. After this, What happened to Theia?",
        "A": "A part of Theia is now part of the Earth, the rest is part of the moon."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 10:02 it said that There was no possible life. Actually, there were tiny micro creatures who lived off off acids and carbon dioxide.",
        "A": "There was no possible life. Life did not come until there were stable oceans. During this period, Earth would have a cataclysmic collision every thousand years which would boil the oceans. Life can not develop in those conditions."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "On 5:30 he talks about theia and the asteroid colliding. Why did Theia not get stuck in the orbit of a planet that size?",
        "A": "It s actually quite difficult for an object to capture a passing object and force it into orbit. That s a very, very rare occurrence. Almost always, the passing object will either crash into the other one, or go off into space, but not enter a stable orbit."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "4:54 Is there a reason why Theia is called Theia? I understand some Greek so I am curious without googling it right away...",
        "A": "In Greek mythology, Theia (also known as Euryphaessa) is a Titaness and a goddess of the moon."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 0:26 Sal mentions a super nova. Where did it come from since our universe and the matter in it did not exist yet?",
        "A": "The universe had been in existence for about 9 billion years by the time our Solar System formed. Plenty of time for stars to form, grow old and explode many times over."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "on 4:30, when Sal said that Theia gave Earth a glancing blow, then what happened to Theia? Did it explode or what?",
        "A": "It was completely destroyed, with some of it becoming part of the Earth and some of it becoming part of the Moon."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 1:39 he mentions that he thinks that whats happened, but how could you actually think that an explosion of a star, which is a ball of gas, create a solar system and planets that can sustain life.",
        "A": "but how did the Big Bang happen if there where nothing. I already know everything else you both said I just want to know why you think that."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 05:28 thiea smashed into Earth at a angle. Did theias smash into Earth at an angle make Earths tilt?",
        "A": "That is possible but we can t be sure."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "4:25 Mr. Khan says that there was gas that would eventually form Earth. How could gas form a solid? Can the state of matter change like that in space? What would cause that?",
        "A": "In that gas, there were particles of silicon, iron, and other assorted heavy elements. Hydrogen is not heavy enough to start to collapse into small planets but the heavier ones are able to attract each other and form planets. Only after a rocky planet reaches a certain mass can it start to acreate gas and that is how the gas giants formed."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 8:50, Sal says human life would not have been able to breath during the Hadean Eon. Why is this so? How did Earth's atmosphere initially form? When did oxygen and nitrogen trap in the atmosphere?",
        "A": "It s because the atmosphere hasn t molded for humans to breath it. Earth s atmosphere initially formed when gasses, such as carbon dioxide and hydrogen, started orbiting the earth. This eventually formed the atmosphere. Oxygen and nitrogen made up the atmosphere near the end of the Hadean Eon. It was made when oxygen and nitrogen pockets in asteroids opened up, releasing it."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "My question is referring to what is said at about 0:32. Was the supernova in question brighter than the moon?",
        "A": "No, although it was brighter than any star in the sky and visible even in daylight. It got to about the same brightness as Mars, Jupiter and Venus are capable of getting."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 9:31, how did things fall if there was no gravity?",
        "A": "good q"
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "at 4:28, wouldn't earth be destroyed it thea hit full force?\n\nMichael",
        "A": "no, because it wasn t a direct hit to the center of earth. Maybe if it was a direct hit."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 9:08,Sal say that the Earth \"is still a molten ball\".So my question is do we really need this gaint molten ball ...some miles below the crust.Does it affect the Existence of life on Earth?",
        "A": "I think it is essential for life on Earth. Without that molten ball, the dynamo that powers Earth s magnetic field wouldn t work, which would expose Earth directly to the dangerous, radioactive solar wind. That would strip away most of Earth s atmosphere and sterilize most of the surface."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "So Theia basically skimmed across the Earth? 5:25",
        "A": "It didn t skim across earth but it also didn t collide right into Earth. If it skimmed Earth, then very little damage would have been done to both objects and Thea might have just kept flying away. If it collided head on, then both Earth and Thea might have been blasted to pieces. Thea collided at about a 45 degree angle."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 0:30, Sal says that Kepler's supernova happened 400 years ago. But the location of the supernova remnant is 20,000 light years away, so since it was first viewed in 1604, wouldn't that mean that it occured 20,000 years before then?",
        "A": "If the earth was still forming, it could still be short enough time from the Big Bang to throw rocks around, or a chain reaction starting from the Big Bang happened and happened to go across the solar system too."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "Regarding section 5:30 in the video of \"Earth Formation: How the Earth is a the byproduct of a local supernova\". It has been reported that the Moon rocks were dated older than then Earth. So the Moon could not had been of Earth, when it\u00e2\u0080\u0099s older than the Earth, right?",
        "A": "i agree with Geoff, but NASA said that part of Theia, became the moon as Theia broke and Earth survived."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 3:55 he said that the particles stick together to create clumps, but how did Jupiter and Saturn end up so big and everything else end up be so small?",
        "A": "Because Jupiter and Saturn are about the same size as Earth when it comes to rock. Its just the their atmosphere is more massive than the planet itself. In the area where the gas giants formed, there was an abundance of hydrogen. When the planets formed, they were around the same size as Earth but they just kept acreting gas until they swelled to their present size."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "6:30 Does \"thea\" mean anything? I typed it into google translate for Greek, and that says it means \"view\", but that seems like a coincidence.",
        "A": "Theia was a Greek titan who was the mother of Selene, the moon goddess. Essentially in mythology Theia is the creator of the moon goddess, such that in history Theia is the moon s creator."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "Didn't he just say that there was some uranium that dated back to about that time period mentioned on 8:28",
        "A": "Yes! Uranium has such a long half like and it is so difficult to decay and get rid of that it can live for a very long time indeed. In fact, the chernobyl disaster area will not be safe for another 20,000 years due to this phenomenon"
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "@ 5:33 ,if the planets were \"molten\" and the \"bits and pieces\" of our so called planet earth and Theia that was blown off from the resulting collision, if those pieces of rock were molten, wouldn't they have cooled off since space is like somewhere below zero. Plus since space is \"below zero\" how long did it take for the planets to cool off? I NEED SOME ANSWERS PLEASE",
        "A": "The planets are still cooling off Since space is a vacuum, the only way for the planets to give off heat is through radiation The sun is still shining on them, so that adds some heat, too. The core of the earth is very hot and is insulated from space by the rock around it."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 1:41, he says that a supernova exploded to cause Theia to run into Earth, which broke a piece off and formed the moon. Where did the supernova come from? Where did the star that exploded come from? Where did the universe come from?",
        "A": "Supernovas happen when large stars use up their nuclear fuel"
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "At 5:59 how would the would be Earth be able to collect some of the splash damage?",
        "A": "The debris from the impact would be caught in orbit of the earth and fall back on the earth."
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "around 3:30 planetesms are mini planets?",
        "A": "They are more like infant planets, as they exist still within the planetary nebulae,"
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "9:00 How did the rocks in the earth go molten?",
        "A": "because of intense preasure and heat due to the tremondose force above them"
    },
    {
        "video_name": "VbNXh0GaLYo",
        "Q": "t 0:37 how did the star explode?",
        "A": "ok thanks for telling me i id not know"
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "At, 3:50, why does he draw double bonds? what is the difference between the two? if the double bond isn't drawn will it change the structure of the atom?",
        "A": "That molecule is called benzene, if you don t draw the double bonds each carbon would only have 7 electrons around it not the 8 they want to have."
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "At 3:50, why are every other bonds \"double bonds?\"",
        "A": "That s the way they happen to be arranged in benzene, Any other arrangement would be impossibly strained."
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "At 3:46 he says that the six carbons in Benzene have double bonds. What are double bonds and how do they occur?",
        "A": "Two shared electrons form a bond, so a double bond consists of four shared electrons. A double bond is usually written as C=C or C=O, where each line represents a pair of electrons. Double bonds are usually formed when the atoms have more than enough electrons to form single bonds to each other, so they use the excess electrons to form double bonds."
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "At 6:08 can we say that for every oxygen, we have two hydrogen?",
        "A": "Yes, entirely correct. 2H per 1O, or otherwise 1O per 2H"
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "At 2:18 why did you write carbon 6 and hydrogen 6 only?\nit could be c 2 & h 2?",
        "A": "No, it couldn t be C2 and H2, because Benzene is actually composed of 6 atoms of carbon and 6 atoms of hydrogen."
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "at 5:07, Sal is talking about how Carbon atoms need four bonds, and how in organic chemistry, the vertices would be automatically assumed to be Hydrogen atoms. But couldn't any atom with a valency of one be a vertex?",
        "A": "Sal explains in another video that a Carbon is always at the vertex.I don t know why but they do it.Also Hydrogens are used so much that they thought they would make it too much work to draw Hydrogens."
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "At 3:46, how does a double bond in a molecule differ from a single bond?",
        "A": "A double bond has two pairs of electrons between the nuclei. A single bond has one pair of electrons between the two nuclei."
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "Around 2:40, Sal says that the empirical formula is a ratio of 1:1.\nIf the ratio of Carbon to Hydrogen were something like 2:3, how would you write it? Cv2Hv3? It would look exactly like a molecular formula!",
        "A": "Like Sal says later in the video using water as an example, we will often find that the molecular formula and empirical formula look the exact same. This is by pure coincidence. The empirical formula is not only a 1:1 ratio. **(New randomly selected example)*** CH3 is the empirical formula while the molecular formula is C2H6. All the empirical formula is is a reduced version of the molecular formula show the ratio of all the atoms in the molecule. I hope this helps!"
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "at 3:50, why are there double bonds, and what are they?",
        "A": "A single bond is one bond between two atoms, so a double bond is two bonds between two atoms. I don t like that he used benzene as the example here, because your question comes up all the time as bonding has barely been covered. But the explanation above should be acceptable for now at least."
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "At 3:54, Sal says double bonds. What are double bonds?",
        "A": "C-C is single bond(Saturated compounds),in simple terms, if the hyphen between C and C is 2 instead of 1,it is double bond,if 3 then it is triple bond. Double and Triple bonds are called unsaturated compounds!!"
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "At 3:48 when drawing the benzene in structural formula, why does every second one have a double bond? Why not just one bond?",
        "A": "You have to draw it that way to give every carbon an octet of valence electrons."
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "At 6:33, Sal says that each of the hydrogens are bonded to an oxygen, but what kind of bond is it? Is it ionic, or covalent?",
        "A": "It is covalent due to the sharing of electrons and the fact that oxygen and hydrogen are both non-metals. The bond is going to be weaker and longer."
    },
    {
        "video_name": "bmjg7lq4m4o",
        "Q": "At 6:13 Sal said that H2O is a molecule.\nBut isn't H2O a compound?\nMolecules are made of the same element & Compounds are made of different.\nWas this a mistake because Sal was also having \"problems\" in other parts of the video?",
        "A": "Molecules are a group of 2 or more atoms held together by covalent bonds. Water is both a molecule and a compound."
    },
    {
        "video_name": "uY2ZOsCnXIA",
        "Q": "At 14:43, are calcium ions the same calcium that you get in milk?",
        "A": "They are. In milk, they re usually bound to proteins and phosphate groups. Ca\u00c2\u00b2\u00e2\u0081\u00ba is also the form found in bone, as calcium phosphate. There s no calcium metal in living organisms, because the group 2 elements are all highly reactive."
    },
    {
        "video_name": "uY2ZOsCnXIA",
        "Q": "At 8:21 Sal mentions a \"sarcolemma\" what is that?",
        "A": "it s just the cell membrane of a muscle cell"
    },
    {
        "video_name": "uY2ZOsCnXIA",
        "Q": "Is the process of the contraction of a muscle cell similar to that of involuntary twitching? (16:19)",
        "A": "Involuntary twitching involves contractions of muscles in the same way that voluntary twitching does. The difference is the origin of the signal to contract."
    },
    {
        "video_name": "66_qW5UhYn0",
        "Q": "at 8:30 in the vedio there r 2 chiral centers so there r 4 enantiomers we draw 2 only",
        "A": "When there is 2 chiral centers then the max number of stereoisomers is 2^n which in this case is 2^2 which gives us 4 max stereoisomers. He drew only the 2 enantiomers because the other two are diastereoisomers."
    },
    {
        "video_name": "4Y2cY5YGxI4",
        "Q": "At 6:24, how did 0.4 become 40%?",
        "A": "They are the same thing. per cent means per 100 . 40% = 40/100 = 0.4 Need to review algebra."
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "at 4:53 ,in the figure about the copper wire,where does the negative charges go after the y reach the left end",
        "A": "You can imagine the copper wire continuing to the left as the blue line and bending around to touch the + terminal of the battery. Just inside the battery terminal there s a chemical reaction happening that absorbs (retires) electrons."
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "at 0:33 i understand how different and like charges work, but WHY do they work like that?",
        "A": "The reason charges exist and attract/repel each other is one of the mysteries of the natural world. I don t think anyone knows why charge exists, or even how they attract/repel. What I do know is that we can observe this natural thing happening, and best of all, as engineers, we can put it to use."
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "At 5:19, what is that q shaped symbol he draws? What's it meant to mean?",
        "A": "The letter q is often used as the variable representing an amount of charge. You see q in Coulomb s Law, representing the two charges in the numerator."
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "At 9:00, if the opposite signs add together to make 2 charges, why is current = 1 q/sec? Should it not be current = 2 q/sec?",
        "A": "I believe the presenter was just reiterating that current is the amount of charge moving across the boundary per second and expressed this as q/s. You are correct that the total amount of charge moving across the boundary (i.e. flux) would be a total charge of 2q/s for a Na cation and Cl anion traveling in opposite directions."
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "At approximately 7:40, in the example with water similar to a conductor, do the protons move toward the negative on the battery as the negatively charged Cl did in the water, and vice versa for the electrons and the positively charged Na?",
        "A": "In the salt water example, entire atoms are moving through the water. The whole Na+ ion bobs along towards the negative wire. The entire Cl- ion swims towards the positive wire."
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "At 1:09 , doesn't he mean \"orbitals\" instead of orbits, and that the electrons exist in approximate areas around the nucleus?",
        "A": "For the purposes of describing current flow, it is sufficient to think of an electron as a particle in orbit around the nucleus, and use the Bohr Model of the atom where the atom looks like a miniature solar system. It is not until you study solid state physics (how a transistor works) that you need a more sophisticated model of the atom."
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "5:13 why can't you just use C or CH for charge instead of Q?",
        "A": "Q or q is the traditional variable for charge. It stands for quantity of charge. It may have been the choice made by Coulomb himself in his publications. We use the letter C for capacitance, and the letter H stands for henry , the unit of inductance. While you are working on charge problems, you can pick whatever variable name you want, since your goal is to get the right answer. (It s not a spelling contest after all)."
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "At 6:49 when the salt is dissolved in the water does that mean that the bond between the Na and the Cl is broken? If so where, did the energy necessary to break the bond come from?",
        "A": "I think that may just have been a slip up. The bond of NaCl is ionic, meaning that, in order to bond, Na loses an electron to Cl. As a result, when the battery is turned on, the already + Na is attracted to the negative side. The battery is providing the energy necessary to break the bond."
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "At 0:38 did he mean a negative charge?",
        "A": "Hello Shreeram, Yes, negative would be aligned with the use of positive seconds before. BTW - I do this all the time. It is very hard to talk and write at the same time... Regards, APD"
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "At 2:15, it's said that the electron on the outermost shell is easiest to pull away. But doesn't that imbalance the copper atom in any way? It no longer has 29 electrons after it gives away one, so how can we be sure that a copper atom has 29 electrons, not 28 electrons?",
        "A": "There is no stopping it from having 28 electrons. This doesn t change the atom as the copper atom is defined by the number of protons it has. The copper atom will become a positive ion though."
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "At 7:54, would it be wrong to say that this movement of charge is similar/the same as alternate current seen in many electricals, because the charge is moving in both directions? If it is the same, what elements are used as wiring (it can't be saltwater, of course!)? And finally, is the copper used for direct current?",
        "A": "The movement of positive and negative charges in the saltwater experiment is not akin to AC (alternating current). To envision AC in both the copper wire and saltwater examples, flip the orientation of the battery 50 or 60 times per second. Copper wire (or gold or aluminum or other metals) are used to carry DC currents all the time. That s what is happening inside most electronic gizmo s you own, except for AC electric motors. Power generation and transmission (the electric power grid) uses AC exclusively, however."
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "So when in the video 7:52 we talk about the positive electrons moving one way and the negative moving the other - is this akin to Alternating Current (AC)?",
        "A": "Hello Dan, Nope, that is not an AC circuit. For simplicity let s assume current is flowing in a copper wire. Here the electrons are free to move but the atoms are stuck. In a DC circuit the electrons would on average move in one direction only. In an AC circuit the direction of the electrons changes. For example, the power system feeding my computer has a 60 Hertz (Hz) frequency. The electrons change direction 60 times a second. Regards, APD"
    },
    {
        "video_name": "ZRLXDiiUv8Q",
        "Q": "At 9:00, the presenter says, \"Because they have opposite signs, they add together and make two charges.\" Would you not get 2 charges from a positive current and another positive current? -Or maybe you would just get a current twice as strong? I would just like to know whether this matters or not. Thank you.",
        "A": "If two + charges are moving in the same direction through the imagined surface, you get a current of 2(charges). If you have 1 + charge and 1 - charge, they will be moving in opposite directions through the surface, and this also counts as a current of 2(charges). The - sign on the negative charge combined with the - sign on its direction of movement produces the same current as a + charge moving the other way."
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "At 3:18 ,the cell membrane is made mostly of a double layer of molecules called what?",
        "A": "the cell membrane is made out of a phospholipid bilayer. a phospholipid has a hydrophilic head and a hydrophobic tail also some organelles have a phospholipid bilayer."
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "0:39 - 0:47 When Hank says \"Everything you have eaten is plants or has come from plants.\" ; Well, what about mushrooms or a fungi? Are mushrooms a plant?",
        "A": "No, they are not. They are their own kingdom. They still come from plants because they decompose dead plants, since it s a cycle. Learn more in the ecology series."
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "at 8:26 he talks about turgor pressure...what exactly is that?",
        "A": "Turgor pressure is the pressure that builds up in a plant cell. This pressure is usually due to the movement of water into the cell. In this case, the cell would have to be in hypotonic solution to allow water to move into the cell. This pressure is created in the vacuole as it is filled up. It is also the optimal pressure in which a plant can thrive."
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "At approx. 5:56 Hank mentions that beavers eat wood. I know that they gnaw on wood to fell trees but I don't think that they eat wood. Am I wrong? Maybe Hank should have mentioned termites instead.",
        "A": "Hank did mention it. Animals that eat plant cells like grass and wood have a special bacterium in their stomachs that digest the cellulose that they eat. Some of these bacteria are Fibrobacter succinogenes, Ruminococcus flavifaciens, and Lactobacillus ruminis."
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "at 0:20, what about CO2 needed for plant cells",
        "A": "CO2 is the source of the carbon in the glucose made through the process of photosynthesis. The six carbons of glucose come from six molecules of CO2. Without CO2, plants could not make sugar. Without sugar, neither plants nor animals could burn that sugar in the process of cellular respiration to make energy."
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "At 9:20, Hank mentions the fact that nucleus is the headquarters at the eukaryotic cells, isn't is even so at the Porkaryotes? why does he have to mention eukaryotes?",
        "A": "prokaryotic cells have no nucleus, and are much simpler than the eukaryotic cells in the kingdoms of animals, plants, protists, fungi"
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "At 6:00, it is specified that some animals have a species of bacteria which actually breaks down cellulose for them. What's the name of this (or these) bacteria and also, what is keeping us from adding it to our gut flora artificially so we can actually digest cellulose, which might be useful in cases of great famine?",
        "A": "The problem is not that we don t have that bacteria, but that our intestines aren t long enough for the food to be in our intestines for long enough for the bacteria to break the material down. There are a variety of bacteria that can digest cellulose, so different animals have symbiosis with different species of bacteria. In some animals there are several species of bacteria that digest the cellulose for them."
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "at 2:28, what does he mean when he says \"not nucleus?\" is there a different type of nucleus?",
        "A": "no there is only one type of nucleus the nucleus is a part of the cell like the mitochondria but it is formed around the DNA or RNA and contains genetic information"
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "at 2:27 he says \"the nucleus\" and then says \"not nucleus\"\nwhat does he mean.",
        "A": "I think he says nucleus \u00e2\u0080\u0093 not nucilus \u00e2\u0080\u00a6"
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "at 4:58 why are plant cells different from animal cells?",
        "A": "Plant cells have chloroplasts which give plants their green coloring and helps the plant make food from sunlight and water. Animal cells don t have that, if our cells were plant cells a) we will be a plant b) We can make our own food c) we would be green. Plant cells also have a larger vacuole for storing water."
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "at 0:47, Hank says that everything you've ever eaten was either made from plants of from something that ate plants. what about when we eat carnivores? shouldn't he include carnivorous animals too?",
        "A": "Well carnivorous animals eat animals that are mainly herbivores which obviously eat plants, for example a lion eating a wilder-beast. I think he meant that everything can link back to the digestion of plants."
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "At 2:49, Hank mentions that prokaryotic means pro kernel, but then what is the difference from pro kernel, and good kernel (eukaryotic).",
        "A": "The root pro has multiple meanings, but in this case it means before . Karyo means nucleus, so the word prokaryote means before nucleus . In other words, prokaryotes are the more simple types of cells that formed before the nucleus had evolved. Since eu means good, eukaryote means good nucleus , as in eukaryotes contain a good nucleus."
    },
    {
        "video_name": "d9GkH4vpK3w",
        "Q": "At 1:02 Hank mentions plants having fossils. Where exactly do you call \"fossil\" in a plant?",
        "A": "Rarely, if the plant is strong enough, the fossil is the plant itself, but like Brandon said, its usually just the imprint in the stone."
    },
    {
        "video_name": "eCJ76hz7jPM",
        "Q": "At 3:30, Sal said that 370. has 3 significant figures, but since 370. is the same as 370.0, wouldn't this number also have 4 significant figures? This is also applied to 370.00 having five sig figs, and so on?",
        "A": "Good question! If there are zeros after the decimal points continuosly and no number other than 0 after that, its not counted. Ex: 23.4040 has 5 significant figures 21.0000 has 2 significant figures 21.0001 has 6 significant figures"
    },
    {
        "video_name": "eCJ76hz7jPM",
        "Q": "In Sal's last example at 4:12, he says that only the 3 and 7 are significant figures because it is not specified how precise the measurement was.\nI understand that part.\nSo let me get something straight- if 370. was written as 370 without the decimal point at the end, only the 3 and the 7 would be significant?\nBecause isn't 370 ambiguous about how precise the measurement is?",
        "A": "You are correct. 370 only has two significant figures. But if it were reported as 370.0, you d have 4, because even though those numbers are 0, they could have been something else. They are only 0 because of a consequence of a measurement or what-have-you."
    },
    {
        "video_name": "eCJ76hz7jPM",
        "Q": "I dont get how he determined the # of significant figures at 0:53",
        "A": "There s only 3 rules for significant figures. 1) any non-0 numbers. 2) trailing 0 s. and 3) numbers between sig figs. So, in 0.00700, the number 7 is significant because it is non-0. And the last 0 is significant because it is trailing. And also, the 0 in the middle is significant because it lies between significant figures. So, the last 3 digits in 0.00700 are the only significant figures."
    },
    {
        "video_name": "eCJ76hz7jPM",
        "Q": "at 2:31 0.052km=520m not 52m",
        "A": "Uhm - wrong. 0.52 km would equal 520m (think about it - half a km or 0.5 km would be 500 meters...). 0.052 km does indeed equal 52 meters."
    },
    {
        "video_name": "eCJ76hz7jPM",
        "Q": "At 2:28, how does he make the kilometers into meters?",
        "A": "Ahh OK. That makes a lot more sense. Thanks."
    },
    {
        "video_name": "eCJ76hz7jPM",
        "Q": "Wait a minute, at 0:55 Sal said that the number 0.00700 has three significant digits. I understand that the three zeroes beforehand are not counted as significant figures, but aren't the two zeroes on the right not sig. fig either? Because I thought any zeroes ending a number is not sig. fig. Can someone please explain my error?",
        "A": "your error is that any zeroes ending a number ARE significant as long as it s in a decimal and after a nonzero number"
    },
    {
        "video_name": "eCJ76hz7jPM",
        "Q": "At 4:15, Sal said that \"37,000\" had 2 significant figures while \"37,000.\" had 5 significant figures.\n\nHow would you write 37,000 if you wanted to convey that there were 3 (or 4) significant figures? Is the only way via scientific notation? (i.e. 3.70 x 10^4)",
        "A": "scientific notation is one way. The other possible way, is to write out the entire number, and add a decimal point at the end, as Sal did with the second 37000, indicating that all the digits are significant"
    },
    {
        "video_name": "eCJ76hz7jPM",
        "Q": "Is there a way to specify precision of measurement if a person measured to the 100s or 10s in the number 37,000? From 4:14 to 5:03 of the video.",
        "A": "You could use scientific notation. 3.700*10^4, or 3.70*10^4, depending on precision."
    },
    {
        "video_name": "eCJ76hz7jPM",
        "Q": "At 4:58, he says \"If you don't see the decimal point, I would go with two\" Why is the decimal point so important?",
        "A": "if there is a decimal point, than the zeros at the end of the number count as signifiant figures. if there is no decimal point, they don t count. ex. 47.00 four significant figures 4700 two significant figures"
    },
    {
        "video_name": "RdFfIEDxo18",
        "Q": "At 1:30, how do you know what to react it with?",
        "A": "At this point, you have to memorize the conditions. If you want to reduce an alkyne to a trans alkene, you use Na in liquid ammonia. If you want a cis alkene, you use H2 and a poisoned metal catalyst. It s all in the mechanisms of he two reactions."
    },
    {
        "video_name": "RdFfIEDxo18",
        "Q": "At 1:40, Jay introduces liquid ammonia as a reagent. If i remember correctly, ammonia is a gas at room temperature and at normal pressure. Does that mean that this reaction needs to take place at either a low temperature or a high pressure (or both)?\nWill this reaction occur with ammonia as a gas? Perhaps the ammonia is aqueous or in some other solution? ... or am I missing something?",
        "A": "That is correct. The boiling point of ammonia is -33\u00c2\u00b0C. If the flask is immersed in a mixture of dry ice and acetone (-78\u00c2\u00b0C), the ammonia will condense to a liquid. High pressure is not needed. There must not be any water present."
    },
    {
        "video_name": "RdFfIEDxo18",
        "Q": "at about 6:25, why doesn't the Na bond to the C? Why instead does the C get an H from the NH3?",
        "A": "It does bond to the C, but the bond is ionic rather than covalent. The electronegative difference between C and Na is great enough so that the bond is essentially ionic. The carbanion is a very strong base. It can easily remove a proton from NH3 to form the much weaker base NH2-."
    },
    {
        "video_name": "RdFfIEDxo18",
        "Q": "In case of free radical mechanism, I think no rearrangement occurs ? So why at 4:18, the free radicals move to the other side? I assume that this movement is also some kind of rearrangement.",
        "A": "That free electron must be in a \u00cf\u0080* orbital, so it can easily get to the other side of the double bond."
    },
    {
        "video_name": "pILGRZ0nT4o",
        "Q": "@6:07, is there any way to know for sure that pentane is a liquid at room temperature as opposed to a solid?",
        "A": "Not really no. This is a property that needs to be measured."
    },
    {
        "video_name": "pILGRZ0nT4o",
        "Q": "Does the number of bonds on a single atom lower it's boiling point? See at 4:00 how neopentane has 4 bonds on the central atom while pentane only has 2 per atom.",
        "A": "Pentane has 4 bonds per carbon atom too. Any bonds that aren t shown are to hydrogen atoms. The video explains why they have different boiling points."
    },
    {
        "video_name": "pILGRZ0nT4o",
        "Q": "@8:45, exactly why are dipole-dipole forces stronger than London dispersion forces?",
        "A": "Dipole-dipole forces are between molecules that always have a positive end and a negative end. London dispersion forces are between molecules that for a short time have a very small dipole in one direction, and in the next moment it may be in a different direction. Thus, they are weaker than dipole-dipole forces."
    },
    {
        "video_name": "pILGRZ0nT4o",
        "Q": "at 8:50 hexanone has a dipole dipole bond. shouldn't it be a hydrogen bond?\nis it because the hydrogen is bonded to carbon?",
        "A": "remember hydrogen bonding is simply a stronger type of dipole- dipole interaction. 3-Hexanone is not hydrogen bonded as the bond exist between a carbon and oxygen atom. for hydrogen bonding you need hydrogen to be bonded with flourine, oxygen, or nitrogen."
    },
    {
        "video_name": "8i0j3j16yFk",
        "Q": "6:18 so in theory, the force I feel in the soles of my feet is 1,000N? 500N from the Earth pulling me plus 500N from me pulling the Earth?",
        "A": "No, if you weigh 1000 N, the earth pulls on you with 1000 N and you pull on the earth with 1000 N. What you feel is the 1000N PUSH back from the ground on your feet, in response to your feet pushing on the ground."
    },
    {
        "video_name": "jEeJkkMXt6c",
        "Q": "If the oort cloud is hard to reach and it takes forever how do we get to all those other galaxies so quickly with a satelite? (look at 5:48)",
        "A": "When we look at the universe we have to remember, we are looking at how the universe was, not it s current state. For example the closest start is Alpha Centauri system. It take light about 4 years to reach us. That means we are looking at the system as it was 4 years ago. So when a satellite is taking picture of the universe it is taking snapshots of the past."
    },
    {
        "video_name": "jEeJkkMXt6c",
        "Q": "in 7:52 i see so many names how do ppl get those names",
        "A": "those closer stars all belong to constellations."
    },
    {
        "video_name": "jEeJkkMXt6c",
        "Q": "What was Sal talking about at 1:40 when he mentioned the dark side of the Moon and particles issuing from the Sun?",
        "A": "Basically, since the moon isn t protected by a magnetic field like Earth is, when you re on the light side (the side facing the Sun), you are in the direct path of the charged particles emitted from the Sun that we are normally protected from on Earth."
    },
    {
        "video_name": "jEeJkkMXt6c",
        "Q": "At 2:48, Sal says that the orbit of Sedna goes beyond the Heliopause. However, if the Heliopause is the edge of the solar system, then is Sedna part of the solar system? Same question for the Oort Cloud at 2:57.",
        "A": "Well that s the orbit. It s still part of it. Depends on what its orbiting"
    },
    {
        "video_name": "jEeJkkMXt6c",
        "Q": "At 2:08 Sal talked about the Solar \"wind\". Isn't the solar wind responsible for creating C.M.E. (Corona Mass Ejection)?",
        "A": "No, CME s emit charged particles from the surface of the Sun, which the solar wind can then carry off into the solar system."
    },
    {
        "video_name": "jEeJkkMXt6c",
        "Q": "7:04, how do you now it is 4.2 light years .",
        "A": "Astronomers have various tools, such as parallax, to determine how far something is from us. Another well known tools is standard candles."
    },
    {
        "video_name": "jEeJkkMXt6c",
        "Q": "Sal Khan said at 1:35-1:36 that we're protected by highly energetic particles because of Earth's magnetic field. Theoretically, if we somehow created a magnetic field just like the Earth's with the same volume, etc., then could we explore the sun slightly closer? What is it exactly that prevents us from examining the sun closer?",
        "A": "Although it would be theoretically possible for us to approach the Sun with no danger of radiation, the extreme temperature of the Sun would prevent us from getting anywhere near it."
    },
    {
        "video_name": "jEeJkkMXt6c",
        "Q": "At 3:20 are there other galaxies within the oort cloud?",
        "A": "No, the orort cloud is around our solar system not the galaxy."
    },
    {
        "video_name": "jEeJkkMXt6c",
        "Q": "Sal. you mentioned at 6:20 that we can't alpha proximus centauri with naked eye. does that mean we can see alpha centauri A and B with our eye? If so, you said even the Earth Cannot be seen if looked at from the Sun. Im rather confused :S how can i look at a star which is so far away?",
        "A": "except from certain vantages and within certain aspects of earth s orbit >> at which times the light from the sun will reflect off the earth and the earth would appear as do other planets which we see as stars in the night sky."
    },
    {
        "video_name": "36Rym2q4H94",
        "Q": "I'm still convinced the brick would still, albeit imperceptably, \"hit the surface of the moon\" (the question posed in 0:42) before the feather (even assuming that the moon is a true vacuum). F_f and F_b aren't only the forces of the moon pulling on the objects, but also the same force that the object pulls on the moon, so the moon is accelerating at G*m_f and G*m_b respectively, where the latter is larger and thus the brick collides first. My question is: am I wrong?",
        "A": "Technically, you re correct. When you drop the brick, it accelerates towards the moon at the same rate as a feather would, but the moon also accelerates towards the brick a tiny amount. This tiny amount is a slightly bigger tiny amount than the moon would accelerate toward a feather. Of course, if you drop the brick and the feather at the same time, the moon will accelerate towards them both, so they will collide at the same time."
    },
    {
        "video_name": "36Rym2q4H94",
        "Q": "At 2:20, why does it have to be the distance between the brick and the moon's center? Can't it just be the distance between them?",
        "A": "No, Newton s law of gravitation assumes that masses are point masses. It would not make sense to measure the distance from the surface of the earth since all the mass of the earth is involved in the gravitational field"
    },
    {
        "video_name": "36Rym2q4H94",
        "Q": "Can you please explain the comments at 9:38, \"the air does two things for constant pressure\" and \"if you have two object with the same shape, the object that is heavier that has more weight will fall faster because it will provide more net force against the air pressure\" ?",
        "A": "So, basically what Sal says at 9:38 is that if you have 2 objects which have the same shape, and different masses, 1) They will experience the same amount of air resistance i.e. upward force. 2) Since we know that the one with a greater mass will have a greater force, hence the net force acting on the object will be greater."
    },
    {
        "video_name": "36Rym2q4H94",
        "Q": "At 2:04, Sal used G in the formula f = G x mm x mb / (r)^2. If g is equal to the gravitational field on the moon, what is G?",
        "A": "G is the Universal Gravitational constant. The acceleration of gravity on the surface of a planet (or moon) of mass M, if the radius of that planet is r, is related to G by the equation g=G*M/(r^2)."
    },
    {
        "video_name": "jA1NyCE4M2g",
        "Q": "At 10:45, the lecture mentioned salivary glands. Where are salivary glands located?",
        "A": "Salivary glands are the glands in your mouth which produce saliva which assists in breaking down your food (they re not really IN your mouth more like above and below your mouth are where the glands are but their ducts release into your mouth)"
    },
    {
        "video_name": "jA1NyCE4M2g",
        "Q": "When he's writing out the components of the ANS at 0:23, why doesn't he include afferent neurons and efferent neurons? He only writes out the efferent neurons.",
        "A": "that s because the Autonomic nervous system deals with response involving only the efferent neurons. Information is carried from the brain/spinal cord to the target cells to provide response."
    },
    {
        "video_name": "jA1NyCE4M2g",
        "Q": "IN 00:05 and 00:12, what is the difference between a functional division and a structural division of the nervous system. Instructor mentioned that Automatic Nervous System is a functional division, are there any other functional divisions?",
        "A": "The functional divisions are the autonomic and somatic. The autonomic can be further subdivided into the parasympathetic, sympathetic and enteric nervous systems. The functional divisions are based around the various jobs of the nervous system and how it accomplishes them, and the structural divisions are based upon the structure of the nervous system."
    },
    {
        "video_name": "1Oopq5U0b08",
        "Q": "At 2:50, I think the current direction is only correct when the capacitor is charging, when it is discharging, the current direction turns the opposite. Am I right?",
        "A": "Hello Wcyi, Correct - the flow of current changes between charging and discharging. Regards, APD"
    },
    {
        "video_name": "9Ng6Zv9oLzk",
        "Q": "At 5:29, how is it that the diene can simply be rotated to complete the reaction? I thought an example such as this was stuck in the trans conformation due to the presence of the double bond.",
        "A": "Generally,as a rule of thumb, everything can be rotated about a single bond!"
    },
    {
        "video_name": "9Ng6Zv9oLzk",
        "Q": "At 1:26, Does the cyclohexene exhibit resonance? Why or why not?",
        "A": "No, because there is only one pi bond and nowhere else for those electrons to go."
    },
    {
        "video_name": "P-1mWGeJjmg",
        "Q": "In 6:36, could we have started figuring out the equivalent capacitance with the 3F and 27F capacitors, which are in series, instead of with the parallel ones? Or would that be incorrect, and if so, why? Thank you!",
        "A": "no because the 27 is not in series with the 3, it s in series with the combination of the 3 and the 6. Try it and see if you get the same answer. Good way to learn."
    },
    {
        "video_name": "P-1mWGeJjmg",
        "Q": "When combining the 3 capacitors into 1 capacitor in the second example (beginning at 6:30), could you have combined the two capacitors in series and then combined that capacitor with the capacitor in parallel and gotten the same answer? In other words, does it matter which order you get you equivalent capacitance?",
        "A": "Try it and see if you get the same answer. Good way to learn."
    },
    {
        "video_name": "P-1mWGeJjmg",
        "Q": "in the diagram, @9:38 we see that the equivalent capacitors connected in parallel have a voltage of 6 volts, doesn't that mean that each the capacitor in that arrangement should have 3v + 3v = 6 v to make sure when the voltages r added the total system should be equal to 8v? plz help!",
        "A": "No, because remember that across individual capacitors connected in parallel, the voltage is the same as the voltage for their equivalent capacitor."
    },
    {
        "video_name": "P-1mWGeJjmg",
        "Q": "@9:28 Why don't the voltages of the parallel capacitors + the voltage of the 27F capacitor add up to 8V?",
        "A": "They do add up to 8V. The video shows the parallel capacitors have an equal amount of voltage(6V) and the 27F capacitor with a voltage of (2V). You do not add voltage in a parallel circuit; add charge. In a series circuit you do the opposite."
    },
    {
        "video_name": "3G_Q6AggQF8",
        "Q": "At 6:01, Sal states that depending on where the two patterns of interference shine, either proves or disproves Ether wind, but how does a different interference pattern mean different velocities?",
        "A": "The interferometer works by comparing travel time in the x direction to travel time in the y direction. The difference in travel time determines the pattern. The travel time is different depending on whether you are facing into or away from the ether wind. Therefore the interference patter will differ."
    },
    {
        "video_name": "3G_Q6AggQF8",
        "Q": "At 4:40, sal said \"first light speeds up then slows down\" maybe, that's the reason we get same interference pattern in all direction because both lights will have same speed at the time of interference?",
        "A": "They might have the same speed, but they d be at different places in their wave phase. Think of what the graph of sin(t)+sin(t+1) would look like - when one wave is sightly behind the other; compared to 2sin(t) - when they re perfectly in sync. For the record, I m not saying light behaves exactly like sine waves, just that the analogy illustrates the difference."
    },
    {
        "video_name": "3G_Q6AggQF8",
        "Q": "At 5:30 why does Sal say longitudinal wave, isn't that a transverse wave? Plus, light is a transverse wave in general",
        "A": "Light is a transverse wave. If he said longitudinal, he misspoke."
    },
    {
        "video_name": "3G_Q6AggQF8",
        "Q": "At 1:11, How exactly is the Galaxy \"Moving\"?",
        "A": "Since every object in the Milky Way is bound together by gravity, you can think of our galaxy as a very large object. This object is just like any other and can have a velocity relative to other galaxies. In fact, the Milky Way and our nearest neighbor Andromeda are moving towards each other and will meet in some billions of years."
    },
    {
        "video_name": "c547u94QRFU",
        "Q": "10:27 Simon Says (lol Jay) one way to determine if a substituent is meta directing is by it's positive formal and partial charge, but in EAS (electrophilic aromatic substitution) the electrophile being attacked by the pi bond is positive?",
        "A": "Jay is talking about a positively charged substituent on the benzene -- i.e. it is already bonded to the ring. In EAS the positive charge is on a separate molecule that the \u00cf\u0080 electrons are going to attack. These are different situations ..."
    },
    {
        "video_name": "EPvd-3712U8",
        "Q": "At 1:21, does it matter which isotopes of an element are used to find the atomic weight as long as you know how common the isotope is?",
        "A": "you have to use all the isotopes"
    },
    {
        "video_name": "EPvd-3712U8",
        "Q": "At 3:45, is a carbon neutron and proton about 1.0034 amu?",
        "A": "The mass of a neutron is 1.00866491588 u and the mass of a proton is 1.007276466879 u. The neutrons and protons in carbon are identical to those in any other atom. Note that the unit of amu is now obsolete and the correct unit for atomic mass unit is u . You will find amu is still used but amu and u have different definitions and u is the one to use."
    },
    {
        "video_name": "EPvd-3712U8",
        "Q": "At 1:17, Sal refers to the isotope, Carbon-12, as having the atomic weight of 12.01. Doesn't that number represent the atomic mass? In one of Sal's earlier videos, he explained the difference between atomic mass and atomic weight, although I still do not understand the use of that terminology.",
        "A": "Atomic mass is the actual mass of a specific isotope, for carbon-12 this is defined as being exactly 12 u, for carbon-13 this is 13.0034 u. Atomic weight is the weighted average mass of all isotopes of an element and is the number found on the periodic table, eg. the 12.011 for carbon."
    },
    {
        "video_name": "EPvd-3712U8",
        "Q": "At 1:33, why did you convert the percentage as a decimal in both isotopes but not the atomic mass ?",
        "A": "Why would you want to do that? That doesn t make sense. We are trying to find the average mass of an atom. We are weighting the masses of each isotope by how much there is on Earth."
    },
    {
        "video_name": "EPvd-3712U8",
        "Q": "In general, don't the electrons also influence the atomic mass 4:20 . Also, what does the percentage represent?",
        "A": "They do but they only have about 1/1800 of the mass of a proton or neutron. The percentage is how much of each isotope is on earth. Because some isotopes are significantly more common than others so we can t just take the arithmetic mean here."
    },
    {
        "video_name": "E1P79uFLCMc",
        "Q": "At about 9:00-9:20 Sal says Earth goes into \"Snowball Earth\" and ices over. Is that like an Ice Age, and if not, how are Snowball Earth and an Ice Age different?",
        "A": "Snowball Earth is much colder than an ice age. All of the water on Earth basically freezes and stays frozen because ice has a way of reflecting heat back into space."
    },
    {
        "video_name": "E1P79uFLCMc",
        "Q": "If water is composed of hydrogen and oxygen, why did the oxygen catastrophe (3:39) occur? Wouldn't the organisms be used to at least some oxygen?",
        "A": "For example, the chlorine in your table salt (NaCl) is very different from the highly toxic chlorine gas (Cl2)."
    },
    {
        "video_name": "E1P79uFLCMc",
        "Q": "At 7:11 I'm confused what are ultra violet lights and what do they do?",
        "A": "Ultraviolet lights are in the electromagnetic spectrum. They have smaller wavelengths and higher frequencies than the visible spectrum we see. However, over-exposure to ultraviolet lights can cause skin cancer."
    },
    {
        "video_name": "E1P79uFLCMc",
        "Q": "3:35 What were all of the extinctions that Earth experienced?",
        "A": "Scientist believe that 99% of species are extinct."
    },
    {
        "video_name": "E1P79uFLCMc",
        "Q": "If the entire ozone layer were to somehow disappear for about 10 minutes, what would happen? Would the UV rays harm us as it would to life before (Ex: 7:09)? What would happen if it disappeared for 24 hours?",
        "A": "Without the ozone layer\u00e2\u0080\u0099s protection from the sun, people, animals and plant life would be destroyed. Even underwater life would not be safe since UV rays can penetrate clear water to a certain depth before being absorbed. Of course, the actual effect on mankind of less ozone depends on the extent to which it is depleted. Experts believe that for every 1% drop in ozone protection, there is an increase of around 2% in UV-B rays which get through to the planet\u00e2\u0080\u0099s surface."
    },
    {
        "video_name": "E1P79uFLCMc",
        "Q": "during the time of huronic glaciation (9:03) did the polar caps had more ice as compared to the equator........",
        "A": "I believe yes"
    },
    {
        "video_name": "E1P79uFLCMc",
        "Q": "At 7:58. It is stated that the ozone layer by protecting Earth from UV rays was important in the formation of life. Do any of the other planets or moons in the solar system have an ozone layer?",
        "A": "Some do have ozone layers yet some of the planets ozone are still a whole.... pretty good they didn t have humans such as always destroying our ozone layer... :)"
    },
    {
        "video_name": "E1P79uFLCMc",
        "Q": "at 9:19 when whole world iced over than what happened to the water in the water bodies",
        "A": "They iced over ..."
    },
    {
        "video_name": "E1P79uFLCMc",
        "Q": "at 3:09 sal said that the atmosphe became oxygen rich? then how much oxygen content (in %) was present in that time in that atmosphere?",
        "A": "i think 40 percent"
    },
    {
        "video_name": "E1P79uFLCMc",
        "Q": "At 7:14, the video said that the organisims creating oxygen had to be in the ocean for this (the oxygen to me made) to happen, Yet, there was no oxygen to make the water to house this event. How did it happen?",
        "A": "Well, technically there was. The water hitched a ride on comets crashing into earth, thus why there was water here. Also, there just wasn t any airborne oxygen, but plenty in the water which had been bonded to 2 hydrogen atoms, thus not allowing them to be available to breathe. Hope this helped"
    },
    {
        "video_name": "E1P79uFLCMc",
        "Q": "at 9:00 to 9:20 he says that the snowball earth occuredthen how did the earth become of normal temperature again?",
        "A": "During the Ice Age, when there was very little carbon dioxide, the Earth was very cold, leading up to Snowball Earth . Eventually, the carbon dioxide built up and warmed our planet through the greenhouse effect. I m afraid I don t know how the carbon dioxide built up when there was little to start with, but I hope I helped! :)"
    },
    {
        "video_name": "E1P79uFLCMc",
        "Q": "At 1:23 sal mentions greek. He has also mentioned greek in some other videos. Why does science use greek to name things?",
        "A": "The Ancient Greeks were one of the earliest civilisations to study Science. Many made important contributions and the roots of modern science lies on many of their ideas. Most of the things they named are still called so in Greek."
    },
    {
        "video_name": "aKeDUvpfOfI",
        "Q": "Are all electrophilic structures electron deficient based on what you said about Sulfur being electron deficient at 04:28 or does that happen to sulfur only?",
        "A": "Yes. Electrophile means electron loving. An electrophilic atom is electron loving because there is another or several other atoms pulling harder on the shared electrons in the bond. Since it is then electron deficient, it will readily accept electrons in order to fix the imbalance. Nucleophiles have too much electron density and will therefore want to share that extra density with another atom. Hence the simple meaning of nucleus loving."
    },
    {
        "video_name": "_0nDUXO0k7o",
        "Q": "At 3:33 how come the Fext would have to subtract mgsin(30)(4*9.8*1/2)?",
        "A": "Because that mass is in an incline, so... some of the gravity affects that mass, how much? well the component parallel to the incline of mg, that is the mgsin(30)."
    },
    {
        "video_name": "_0nDUXO0k7o",
        "Q": "At 4:00, how come the force of friction is opposing the system? I thought that the force of friction is always upwards on an incline to prevent it from sliding down. In class for example, the professor said that friction acts upward on a truck driving up an incline. Doesn't friction only oppose motion when an object is sliding, but the 9 kg box is pulling the 4 kg uphill, so wouldn't the friction of the 4 kg box be positive then?",
        "A": "Dynamic friction is always in the opposite direction of motion and static friction is opposite to the net forces (excluding friction) on an object."
    },
    {
        "video_name": "_0nDUXO0k7o",
        "Q": "Are the two tension forces equal? I know at 6:25 he said that the internal forces cancel, but is that the same thing as saying they are equal in separate directions?",
        "A": "A pulley is a rotating piece that is meant to convert horizontal tension force into vertical tension force. In these videos, we are assuming there s no resistance from the pulley, so the tension of one string is converted into the tension of the other string with no force being subtracted. In short, yes they are equal, but in different directions."
    },
    {
        "video_name": "_0nDUXO0k7o",
        "Q": "At around 8:50, why is the force of friction neglected? Shouldn't it also be included as a vertical force acting upon the 9kg mass?",
        "A": "There is no friction force acting upon the 9 kg mass because it is suspended and is not sliding on any surface."
    },
    {
        "video_name": "_0nDUXO0k7o",
        "Q": "For solving the normal force, how did he get mgcos(30)? Since the cos(3) would equal mg/(Fn) if you drew it out. so Fn would equal mg/cos(30). At around 4:40.",
        "A": "Cos30 is not equal to mg/Fn. It is equal to Fn/mg. The component of the 4kg mass force which resists the motion is parallel to the incline. So the second leg of the triangle which is equal to the normal force is mg*cos30."
    },
    {
        "video_name": "_0nDUXO0k7o",
        "Q": "When SantoPietro was calculating the two components of gravity on the 4kg block, he did an interesting thing: he extended the parallel line (and the floor line) and assumed it takes a 30 deg angle, is that valid? around 5:40 Thank you, in advance.",
        "A": "Yes it is. You need a little geometry to prove it."
    },
    {
        "video_name": "_0nDUXO0k7o",
        "Q": "1:37 How exactly do we determine which body is more massive? Do we compare the vertical components of the gravitational forces on the two bodies or something?",
        "A": "In the video, the masses are given to us: The 9 kg mass is falling vertically, while the 4 kg mass is on the incline. The forces of gravity, or Weight, is directly proportional to mass, and both be positioned vertically."
    },
    {
        "video_name": "_0nDUXO0k7o",
        "Q": "At 4:41, why is the normal force MGcos(theta)?",
        "A": "he explains it. Listen carefully. You need to understand about using trig to break vectors into components to understand what he is saying. If you don t know that, you need to go back to some other videos that explain it."
    },
    {
        "video_name": "gFFvaLzhYew",
        "Q": "2:53, How do they etch away the copper? I see how they could do it, but how could they do it efficiently?",
        "A": "They Use Chemicals"
    },
    {
        "video_name": "gFFvaLzhYew",
        "Q": "At 4:20, the pin on the solenoid pops to stop electricity flowing, how does relying on a mechnical mechanism make something safe electrically?\n\nSurely there's enough time for electric current to do something bad, as it moves a lot quicker than a switch.",
        "A": "It is an electromagnet, and the pin is metal. So the electromagnet triggers the pin to pop out."
    },
    {
        "video_name": "AiGGaJfoQ1Y",
        "Q": "Around 2:40 he says that it is sp2 hydridized, yet he's drawn 3 H's around the C. If the C is sp2 hydridized that means it is double-bonded to another C, and should therefore only have 2 H's around it. Right?",
        "A": "If a C is sp2 hybridized, that doesn\u00e2\u0080\u0099t mean that the C has to be double bonded to another C, sp2 hybridization requires only that there be three other atoms directly attached to the C. This is the case with the methyl cation, CH3+."
    },
    {
        "video_name": "FssCS9RhFt0",
        "Q": "Around 6:30, couldn't temperature also be detected? Because the superhero's body would still be warmer to the touch than the outside air, wouldn't temperature receptors still fire more in the area that he hits the man than the surrounding areas of his body?",
        "A": "Good point, maybe some of the energy (in the form of heat) could also be transferred/released as the result of the collision"
    },
    {
        "video_name": "aaZ-isZs4ko",
        "Q": "At 2:48 wouldn't it be hexane instead of pentane? Isn't the Bromine attached to a Carbon?",
        "A": "Yes, the Br is attached to C1, the first blue carbon of the 5-carbon (pentane) chain."
    },
    {
        "video_name": "aaZ-isZs4ko",
        "Q": "At 0:50, Jay says that the name of the compound is chloroethane. Why isn't it 2-chloroethane, because the chlorine atom is bonded to the second carbon atom?",
        "A": "There is no difference between 2-chloroethane and 1-chloroethane, they are the same molecule flipped around. The number is redundant here."
    },
    {
        "video_name": "aaZ-isZs4ko",
        "Q": "At 7:08, isn't carbon 3 a chiral carbon? So shouldn't that give the whole molecule another name?",
        "A": "why do you have 5,5 methyl heptane than 3,3 2 which gives the lowesr position"
    },
    {
        "video_name": "aaZ-isZs4ko",
        "Q": "At about 6:30, when applying R and S to 2-iodo-3-methylpentane, why isn't the number 3 carbon chiral as well? Isn't it also attached to four different groups?",
        "A": "Carbon 3 is chiral, but you can t determine its stereochemistry because there isn t a dash or wedge to denote which group (methyl or hydrogen) is in front of the other."
    },
    {
        "video_name": "XjFNmfLv9_Q",
        "Q": "At 1:33, why are there two protons to nabb off of them?",
        "A": "Because oxalic acid has 2 acidic hydrogens , one for each of the carboxylic acid functional groups."
    },
    {
        "video_name": "XjFNmfLv9_Q",
        "Q": "Is there a difference between moles and molecules? 3:30",
        "A": "A mole to a chemist is just a very, very large number: 6.02 times 10 to the 23rd power. A collection of a mole of molecules, with each molecule having an atomic weight of N will weigh N grams."
    },
    {
        "video_name": "XjFNmfLv9_Q",
        "Q": "At 0:19, what's the difference between impure and pure oxalic acid?",
        "A": "Pure oxalic acid contains only oxalic acid, while impure oxalic acid can contain other substances as well."
    },
    {
        "video_name": "XjFNmfLv9_Q",
        "Q": "3:46-4:02 ?\nwhere do the hydrogens come from? Oxalic acids or Oxalic acid molecule? whaat?",
        "A": "Well there are 2(H2O) s. So that basically means (H2O)+(H2O). And both (H2O)s are made up of an (OH-) and an (H+). So that means there are 2(H+) and 2(OH-). The 2(OH-) came from the 2(NaOH) since it ionizes. The 2(H+) came from the (H2C2O4) since as the empirical formula shows it has 2(H+) which will ionize."
    },
    {
        "video_name": "XjFNmfLv9_Q",
        "Q": "At 2:42, why will the oxalic acid have lost both of its hydrogen's?",
        "A": "Because it has been completely neutralised through the addition of hydroxide ions to the solution."
    },
    {
        "video_name": "XjFNmfLv9_Q",
        "Q": "at 5:24, so M is moles per liter?",
        "A": "Yes. It is moles per litre. Thus, if it is 3M, it means there are three moles per litre of the solution. Hope this helps :)"
    },
    {
        "video_name": "Y4HzGldIAss",
        "Q": "sal explain a reaction at 4:21 is it a reversible reaction?",
        "A": "HCl+H2O=H3O+Cl is an equilibrium reaction and can go backwards. There is a double set of acids and bases in this reaction .......HCl is an acid and H2O is a rxn base.........H3O is a reverse acid and Cl is a rxn base"
    },
    {
        "video_name": "Y4HzGldIAss",
        "Q": "0:37 what is concentration??",
        "A": "Concentration is the number of molecules present of a particular liquid in 100 molecules of its liquid solvent. It is expressed in %"
    },
    {
        "video_name": "Y4HzGldIAss",
        "Q": "Why did he say that oxygen has nabbed an electron at 5:39 ?",
        "A": "I assume you are referring to what he is saying at around 5:40. Basically, all he s saying is that the hydroxide ion has a charge of negative 1 on account of the oxygen in this ion having an extra electron. Although he implies it s a mystery where this extra electron came from, of course it came from sodium, because he s talking at this point about sodium hydroxide (NaOH). Sodium hydroxide is Na+ OH-, where sodium has lost an electron, and the oxygen has gained an electron."
    },
    {
        "video_name": "Y4HzGldIAss",
        "Q": "At around 4:00, why does the Cl anion not bond with the positive side of water to make H20Cl with a 1- charge? Why does only the hydronium cation bond with water?",
        "A": "The hydrogens of a water molecule have a partial positive charge but they cannot bond to chlorine. A proton bonds with a water molecule to form the hydronium cation - the hydronium cation does not bond to water."
    },
    {
        "video_name": "Y4HzGldIAss",
        "Q": "At 1:55 Sal says that the chlorine atom strips off the two electrons from the hydrogen atom. How can this be if the hydrogen atom has only one electron?",
        "A": "There are two electrons in the bond between the H and the Cl. The Cl takes both of them, leaving the H with no valence electrons."
    },
    {
        "video_name": "Y4HzGldIAss",
        "Q": "at 3:15, what is the difference between aqueous solution (aq) and liquid form (l)?",
        "A": "A liquid is a substance in the liquid state (e.g. NaCl heated up until it is molten/liquid). An aqueous solution is a substance dissolved in water (e.g. NaCl dissolved in water). The key difference is whether you have a pure substance in liquid state versus a substance dissolved in water."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "So it's been a while sense I had organic chemistry. Can someone tell me why the 5th carbon at 4:05 is not a stereocenter?",
        "A": "It is a stereocentre. For that matter, so are C2 and C3."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "At 10:40, is the example called trans-1,4-cyclohexanediol because the two alcohol substituents are opposite from each other or simply because one is coming out and the other is going into the page? For instance, can a molecule be referred to as \"trans-1,3-cyclohexanediol?",
        "A": "It s called trans-1,4-cyclohexanediol because one OH is coming out and the other is going into the page. Bear in mind that while the molecule has been drawn flat, the cyclohexane ring will actually be puckered into a chair conformation where both OH groups can be either equatorial or axial, with the equatorial conformation favoured. As you say, you can also have trans-1,3-cyclohexanediol, and, for that matter, also trans-1,2-cyclohexanediol, as well as the corresponding cis isomers."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "Why is the compound at 4:45 heptanol and not octanol? The longer chain starts on the end of that \"propyl\" group. I'm not sure how to tell when to start the base chain or consider it an attached group.",
        "A": "When there is an alcohol group,preference is given to the chain containing alcohol and not the longest chain group"
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "At 3:35 Jay uses the path which has 7 carbon atoms,but I found a path which has 8 carbon atoms.How can this Happen ??",
        "A": "You need to include the OH group. Like when naming alkenes the parent chain must include the double bond even though there is a longer single-carboned chain."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "At 4:04 Jay says it is An alkyl group. How? It just has single bonds.",
        "A": "Alkyl group is what we call any carbon chain that is coming off the main parent chain. It is not meaning alkynes or anything like that."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "When do you know to use the cis/trans notation instead of R/S? E.g. at 10:31, the lower molecule. I know R/S and cis/trans mean different things, but how do you know when to use what?",
        "A": "The cis/trans notation is used only for disubstituted alkenes that have one substituent on each carbon atom, such as RCH=CHR (You may also use the E/Z notation). The E/Z notation must be used for trisubstituted alkenes (RR C=CHR ) and for tetrasubstituted alkenes (RR C=CR R ). Thus, the cis/trans notation is a subset of the E/Z system. You may always use E/Z, but cis/trans only in special cases."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "At 6:48, sal mentions that the carbon-3 is a chiral carbon but the carbon has two methylene group ,one hydroxy and one hydrogen.since a chiral carbon should be surrounded by four different functional group how can this carbon be a chiral center if it has two methylene bonded to it?",
        "A": "It does not have two methylene groups attahed to it. It has one ethyl and one hexyl-group attached to it, as well as the hydroxy and hydrogen. A chiral atom does not have to have four different FUNCTIONAL groups attached to it. Four different groups will do, so an ethyl and hexyl are two different chains, which means they are two different groups."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "At 3:50, We should make the longest isn't it we have longest available is from that above the above",
        "A": "The longest chain has to include the highest priority functional group (the OH) This was mentioned in the video."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "At 4:46 Shouldn't the molecule be 3-chloro 3-methyl 5-(ethan-1-ol) octane?",
        "A": "No, the OH group is the highest priority group here so the main carbon chain HAS to include it, even though there is a longer carbon chain possible. The name Jay gives is correct."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "Hi there,\nAt 4:45, is it correct to right 5,5 chloromethyl, instead of 5-chloro-5-methyl?\n\nThanks",
        "A": "No because chloromethyl means a -CH2Cl group, not what you intend it to mean. 5-chloro-5-methyl-... is correct here."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "When naming the last phenol at 11:25 would it be correct to write the name as ortho-Bromo phenol instead of 2-Bromo phenol?",
        "A": "Yes, that name could also describe 2-bromophenol, but it is not the IUPAC name."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "At about 10:40, why didn't we assign R/S to the 1 and 4 carbons?",
        "A": "Because carbon atoms 1 and 4 are not chiral. The molecule has an internal plane of symmetry: The 1,2,3,4 side of he ring is identical to the 1,6,5,4 side."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "In 2:24 is it fine to write it as propan-2-ol for 2-propanol....",
        "A": "yes of course you can."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "At 3:28 shouldn't the numbering be at 8 because of the 3 -propyl group at the end? If so how would you then name it because the alcohol is on a branch",
        "A": "When you find the main carbon chain you have to include the highest priority functional group which in this case is the OH. So the main chain is what it is in the video."
    },
    {
        "video_name": "kFpLDQfEg1E",
        "Q": "Doesn't the example compound at 3:45 have 8 carbons in longest chain? I haven't watched other videos, so I could be wrong here.",
        "A": "No, it can t have 8 carbons because you have to include the alcohol group in the chain...."
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "At 8:10,why can't the charge be movable?",
        "A": "Sal mentioned that an ionic compound isn t a good conductor of electricity. Because there s no free e- to flow. e- is negatively charged. So he said, in that state charges aren t movable..."
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "The way he draws the metallic bond of iron in 5:02, wouldn't the electrons all repel each other?",
        "A": "It s not drawn to scale so the electrons are further apart than they look."
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "At 1:59 , Sal said that,\" that type of bond is the strongest\" , I suppose that it is the covalent bond as he was talking about covalent network so can I say that the covalent bond is stronger than the ionic or the metallic bonds ??",
        "A": "Giant covalent structures are stronger than ionic, which stronger than metallic and finally simple covalent structures."
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "at 6:30 is it hydrofluoric acid not hydrofluoride?",
        "A": "HF is both. When it is not in aqueous solution, but is a gas, it is hydrogen fluoride. When it is dissolved in aqueous solution, it is hydrofluoric acid."
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "6:34.... i thought hydrogen fluoride had hydrogen bonds which is the strongest intermolecular force",
        "A": "Hydrogen bonds are the strongest of the van der Waals forces (which are intermolecular), but metallic, ionic and covalent bonds are basically _intra_molecular bonds, and play in another league when it comes to bond strength."
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "1:56, Sal said covalent were the strongest. Aren't ionic bonds stronger?",
        "A": "He said they were the strongest of all molecular bonds. Ionic bonds are not molecular bonds."
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "At 6:38 Sal says that HF doesn't form crystals. Why is that? I looked at the electronegativity difference between H and F and it is almost identical to that between Na and Cl.",
        "A": "HF doesn t form crystals because they don t form ionic bonds. The reason they do not form ionic bond is that hydrogen is not fully willing to give away its electron.This is why sal say that partially negative and partially positive."
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "By looking at the structure on 3:57 , is there a possibility to end up with more Na or Cl ions on one side when cut?",
        "A": "Yes, there is, but the crystal usually cleaves along a plane. In any case, we wouldn t be able to detect the imbalance of a few ions on a macroscopic scale."
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "He says at 7:24 that NaCl dissolves in water because the partial positive and negative charges of the water molecule attract the sodium and chloride ions. Why the, does sugar dissolve, since it is not ionic?",
        "A": "Sucrose, or table sugar, is a heterodimer made up of two simpler sugar molecules, one glucose and one fructose. This makes a molecule with 8 polar OH groups around the outside, and all of these interact nicely with the partial chages on the H2O molecules. Interestingly, sucrose is easier to dissolve than either glucose or fructose, because the structure of sucrose molecules means they can t actually bond to each other."
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "at 3:46 when he draws the sodium chloride, if the Na atoms have lost one electron and the Cl atoms have gained one electron giving them +1 and -1 charges, how are they able to bond with more than one ion?",
        "A": "As they now have a static charge, ions of opposite charge will bond with them. In the case of NaCl, for each Na+1 ion, it will be surrounded by 6 Cl-1 ions (top, bottom, front, back, left and right). Similarly, the Cl-1 ion will be surrounded by 6 Na+1 ions. The NaCl is the simplest ratio between positive cations and negative anions. In this way they form a giant ionic lattice of alternating ions."
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "At 2:00 you say that a diamond is so strong because it has so many covalent bonds and covalent bonds are the strongest type of bonds. But in your Ionic, covalent, and metallic bonds video, you say that ionic bonds are the strongest and covalent bonds are not very strong. Which one is it?",
        "A": "Didn t he say/meant the covalent network or even the covalent network of diamonds was the strongest structure, and not covalent bonds in general?"
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "At @5:05 How do the Iron (or any element who forms metallic bonding) atoms allow their electrons to roam? Do they just give up their mobile electrons?",
        "A": "Metals give there valence electrons up because they are more stable with out them (they have low electron negativity). When metals lose electrons they get positively charge and hence the negative electrons roam around them. This is called the Electrons Sea Model."
    },
    {
        "video_name": "Rw_pDVbnfQk",
        "Q": "At around 8:04 , Sal says that NaCl by itself, as an ionic crystal, is a not good conductor of electricity, whereas dissolved NaCl is. Can someone please explain why this happens?",
        "A": "To conduct a current, there must be a flow of charged particles. In a wire, it is electrons that move from the negative electrode to the positive electrode. In solution, it is the motion of ions that carries the current. The positive Na+ ions move towards the negative electrode, and the negative Cl- electrons move toward the positive electrode. In solid NaCl, the Na+ and Cl- ions are locked in place in the crystal lattice. They are unable to move, so the solid NaCl cannot conduct electricity."
    },
    {
        "video_name": "QXYbGZ3T3_k",
        "Q": "At 2:07, why does a lot of the material spiral off into space",
        "A": "The force of the collision causes much material/gases to be thrown into space. Earth could possibly be part of that mess in the future."
    },
    {
        "video_name": "QXYbGZ3T3_k",
        "Q": "Sal mentions at 1:26 the probability of stars colliding is very low. I did notice however, that since the number density of stars increases as one approaches the galactic center, wouldn't this increase the probability of stars colliding?",
        "A": "Stars are a lot and they re big, but the distances between them are even bigger, even if they re in the galactic center."
    },
    {
        "video_name": "QXYbGZ3T3_k",
        "Q": "At 2:16-2:19, there is a 'burst' of stars or dust shooting out from the Andromeda galaxy, after the first pass. What causes this?",
        "A": "It s material/stars/gas being thrown out from the Andromeda due to the collision s strong force."
    },
    {
        "video_name": "QXYbGZ3T3_k",
        "Q": "Around 1:40 Sal said the time it takes for a star to orbit our galaxy's core. Is that near the core itself or at the edge of the galaxy?",
        "A": "That,s the time it takes for our Sun, which is half way between the core and edge to orbit the galaxy."
    },
    {
        "video_name": "QXYbGZ3T3_k",
        "Q": "Sal says that our galaxy would crash with the Andromeda galaxy in about 3 to 5 billions of years in 0:14. that is about the same time the sun will die. would our millions of years of grandsons or granddaughters see the collision before the sun dies or would they be dead already because the sun died?",
        "A": "I would assume that by 3 to 5 billion years from now, humanity will have at least gotten to other star systems to escape the whole sun thing if they aren t extinct by that time. The galaxies will likely collide before the Sun starts to expand but by 1 to 2 billion years from now, the Sun will have warmed enough to do considerable harm to life on Earth."
    },
    {
        "video_name": "QXYbGZ3T3_k",
        "Q": "WHAT will happen to earth i'm so woried the time is 2:12",
        "A": "nothing because we are so close to the sun that we are permanent orbit some of the smaller planets that are far out like Pluto or Neptune or Uranus but they would float to another star and we might get some extra plantes too"
    },
    {
        "video_name": "QXYbGZ3T3_k",
        "Q": "At 3:06 the \"new\" galaxy is orbiting one center. Does that mean the the black holes have merged?",
        "A": "They might not have joined yet but they may be orbiting each other at close distance."
    },
    {
        "video_name": "QXYbGZ3T3_k",
        "Q": "2:45, Sal says that the Milky Way and the Andromeda galaxies will merge, but is it into a new galaxy, or something else? I mean, would they destroy each other?",
        "A": "Their collision will fling countless stars out of both galaxies but they will merge into a large elliptical galaxy."
    },
    {
        "video_name": "QXYbGZ3T3_k",
        "Q": "What happens when the 2 black holes at the center of each galaxy collide at 2:37? do they both collide to make a big black hole, or does one black hole eat the other?",
        "A": "The two blackholes will orbit each other until their gravity draws one into the other, forming a gigantic blackhole."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "8:30pm. In this video, it is said that in order to obtain the number of neutrons, we use the formula A = Z + N, and we get the A from the superscript of the element symbol. For example, the superscript of C (carbon) is 12, which equals to the number of protons plus the number of neutrons. So, my question is: how do you know how many neutrons are there in the superscript (A or mass number, if you will) to begin with?",
        "A": "You subtract Carbon s atomic number, 6, from its atomic mass number, 12, and you get 6 neutrons. The atomic number tells you the number of protons. The mass number tells you the number of protons plus neutrons."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "At about 1:20 to 5:30 protium, deuterium, and tritium are mentioned. Does anyone know how these are used?",
        "A": "I think that they are used as a malleable substance for experiments in chemistry, so students are able to understand properties of chemistry, but maybe I am wrong."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "2:15 - 2:18\nhow do we calculate the mass?",
        "A": "Protium only has one proton in the nucleus. Deuterium has one proton and one neutron. You don t need to calculate it here, but if you wanted to, you look up the individual masses of the nucleons (typically given on some sheet with constants in textbooks) and add them together."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "3:26 into video we have gone over protium, deuterium, and tritium. My question is do all elements have these isotopes or just Hydrogen?",
        "A": "All elements have isotopes, but only H has isotopes with those names."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "First, at 1:14, we see the names of the isotopes of carbon. Did they get their names from how many protons and neutrons they have in their nucleus? (does pro stand for \"one\", deu stand for \"two\", and tri \"stand for 3\"?). Second, at 3:21, does the number of electrons always equal the number of protons?",
        "A": "Pro stands for first but the deu and tri stand for two and three. In a neutral atom the number of electrons always equals the number of protons."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "at 6:23, he mentions that it must be a neutral atom of carbon, so the number of protons is equal to the number of neutrons. but what if it wasn't neutral? what would happen then? how would we find the number of electrons then? i hope this question makes sense and is sensible...",
        "A": "In a neutral atom the number of protons is equal to the number of ELECTRONS. Not neutrons. If it isn t neutral, the charge will tell you how many extra electrons are there or how many are missing. +2 charge means 2 missing electrons. -1 means 1 extra."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "At 3:24, why would the subscript be the atomic number, which is the number of protons in the nucleus, if the H for hydrogen already tells you the number of protons in the nucleus.",
        "A": "Not everyone will know off the top of their head what the atomic number (z) of a given element would be. Hydrogen is generally an easy one because it s commonly used and referenced. However, there are many other elements whose atomic mass you won t readily know, so it s good to have it written down for reference!"
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "At 2:13, he said that protium and deuterium have different masses, so which isotope is the atomic weight calculated from?",
        "A": "The atomic weight is the weighted average of the masses of all three isotopes. Protium comprises 99.98% of all naturally occurring hydrogen. Deuterium comprises about 0.02%, and tritium comprises about 10\u00e2\u0081\u00bb\u00c2\u00b9\u00e2\u0081\u00b8 percent. You can safely ignore the contribution of tritium, unless you need an enormous number of significant figures."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "How can one tell if an atom is a neutral atom with just looking at the formula (thought of at 6:40 in the video)?",
        "A": "If it were an ion rather than a neutral atom there would be a positive or negative charge to the top right of the atom s symbol"
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "at around 1:55 is my question of: Is water made up of 2 hydrogen atoms plus 1 Oxygen atom, or a Deuterium atom plus 1 Oxygen atom... or am i talking about the same thing?",
        "A": "Water is 2 hydrogens and an oxygen Deuterium is a very rare version of hydrogen, so some water molecules might have deuterium instead of normal hydrogen."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "In 0:33, is he saying that when there is an equal amount of protons and electrons, there is no charge?",
        "A": "Right- because protons have positive charge and electrons have negative charge, when there is an equal number of each, the charge on the atom is neutral (or equal to zero)."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "wait i thought that carbon-13 could not happen?\n7:39",
        "A": "Carbon-13 is the second most common isotope of carbon, with carbon-12 being far more common. The other naturally occurring isotope, carbon-14, is quite rare, existing only in trace amounts."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "At 6:23\nhe says that the atom is neutral\nhow do we know that that atom is neutral ?",
        "A": "because atom contains proton which is positive and electrons which is negative and are equal which when combined give neutral form"
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "At 0:02 the video shows a picture of the writing of the element hydrogen as if it were on the periodic table. However, my periodic table on my wall shows that hydrogen's average mass number is 1.0079, not 1.01 like in the video. Are just some periodic table rounded differently?",
        "A": "Yes some tables round differently. In practice it doesn t affect school type calculations very much."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "But how do we know that it is a neutral atom? 7:41",
        "A": "A neutral atom is an atom which has no net charge. The protons each contribute a charge of +1 and the electrons each contribute a charge of -1, so an atom which has no overall charge will be an atom which has an equal number of protons and electrons."
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "At 1:13, How is protium named?",
        "A": "This specific isotope of hydrogen is named protium because it distinctly has 1 proton and 1 neutron, along with only 1 neutron. Since the isotopes of hydrogen have very different properties, they are given different names, instead of just Hydrogen-1 or Hydrogen-2"
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "9:28 Does the number of electrons include valence electrons?",
        "A": "Yes they are included"
    },
    {
        "video_name": "I-Or4bUAIfo",
        "Q": "Wait. At, 2:52, he says that tritium has TWO neutrons. I thought it would be three because of the tri in the word. So, does tritium have 3 or does it have 2?",
        "A": "It has two neutrons but it has three nucleons, a proton and two neutrons. It s named for its mass number, not the number of neutrons."
    },
    {
        "video_name": "Ezp8F7XJHWE",
        "Q": "at 0:25, what is the meaning of polyunsaturated?",
        "A": "Poly means many (ie, more than one) and unsaturated means that there are carbon-carbon double bonds (ie, C=C bonds). So polyunsaturated means that the fat contains two or more carbon-carbon double bonds."
    },
    {
        "video_name": "Ezp8F7XJHWE",
        "Q": "At 10:44 in the video, why do the hydrophobic tails go below the head in the top row, and why do they go above the hydrophobic head in the bottom row?",
        "A": "By having the hydrophilic heads on the outside and the hydrophobic tails on the inside allows the tails to stay away from water by putting the head next to it."
    },
    {
        "video_name": "Ezp8F7XJHWE",
        "Q": "What's the molecule being talked about with the beeswax at around 3:50? As in, the name. Thanks!",
        "A": "Triacontanyl palmitate is the a wax ester that is a major component of beeswax and that s the one being discussed. Don t worry: it s unimportant :P"
    },
    {
        "video_name": "KzpRivhr6HI",
        "Q": "1. At 5:03 is there also a pheophytin presesnt in ps1 ?\n2. 5:51 where does the phosphate jammed with ADP come from ?",
        "A": "1. No, there is not. PSI has different primary electron acceptor called an A0 chlorophyll , which acts the same as pheophytin in PSII. 2. The phosphate group which is added to ADP comes from the stroma. It was released during the Calvin cycle from an ATP molecule, and floats around in the chloroplast until it is synthesized with ADP to make ATP again. The ADP is also a product of the Calvin cycle."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "At 7:24, I think the non-bonding electrons of the top and bottom oxygen would be 5 electrons because if it was 6 non-bonding electrons, it would make each oxygen has 7 valent electrons instead of 6 valent electron like in the periodic table",
        "A": "Okay. But in dot structures we do not assign each element the valence electrons it normally has, because it causes issues with certain structures (eg nitrate) Regardless, in the structure at that time stamp, that top and bottom oxygens do have 6 non bonding electrons. Count them up again. Fast forward to 9:45 for what is the whole point of this video. The preferred dot structure is the one that minimises formal charge."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "7:23 what's up with those operations, why is subtracting if they are electrons how can the charge be positive?",
        "A": "What exactly confuses you? He introduced the formula for calculating formal charge near the start of the video. Remember atoms have a positively charged nucleus and a neutral atom will have the number of electrons equal to the number of protons. So when there is less electrons than protons (based on this completely arbitrary method of calculating formal charge) there will be a positive charge."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "In the second example, H2SO4 when the final dot structure was complete, why the sulphur is sharing 12 electrons? I mean it can allow maximum 8 of them, right?\n\nAlso a bond means 2 electrons so if you see a bond between atom A and B you naturally assume that one electron from atom A and one electron from atom B are together, at 8:57 mark takes out the 2 electrons from oxygen and made a double bond between O and S?",
        "A": "Sulfur has an exception to the octet rule. Look up other exceptions, especially group 3a. I answered your other question in a previous comment."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "At 9:20 to 9:25 you drew sulphur with double bonds, 6 bonds in total. My question is why to give so many electrons to sulphur atoms when its electronic configuration is [Ne] 3s2 3p4 which is just two electrons short of Argon's configuration. I need a little clarification in that aspect i'm little confused...",
        "A": "Sulfur can expand its octet when it has to do so. It has an octet with four single bonds, but it also has two + charges and two - charges. It takes energy to separate + and - charges from each other. The molecule can become even more stable by creating two double bonds and thereby reducing the charges to zero."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "At 3:22\nQ: Is there a different method to calculate the formal charges?\nI think my teacher has a different method.\n( V electrons) - (the bonds)- (the number of free electrons).\nIs it correct or did I make a mistake ?",
        "A": "Both methods are correct. The method your teacher used is best when you know the electron-dot structure of the molecule ( e.g, H-O-SO\u00e2\u0082\u0082-OH). The methods used in the video are best for when you don t know the electron dot structure (e.g. H\u00e2\u0082\u0082SO\u00e2\u0082\u0084). They are just summaries of the results obtained from your teacher s formula."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "at 10:25, why did you move 2 electrons to sulfur when the formal charge of oxygen is -1?",
        "A": "To make the formal charge 0 on each atom. lower formal charges are preferred in deciding which dot structure is the best."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "At 7:10, can someone explain why we are giving one electron to oxygen and other to sulphur?",
        "A": "That s the way we calculate formal charge. The formal charge is the charge that an atom would have if the shared electrons were shared equally, i.e. if we give one electron to S and one electron to O."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "Where do we write the formal charge in the final dot structure? Do we write it right next to the atom it is associated with (like in the video at 7:41 with the -1 next to the oxygen atom)?",
        "A": "It s usually written next to the atom that has the formal charge and in brackets"
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "At 7:26, formal charge is no. of valence electrons - no. of bonded atoms. So formal charge of oxygen should be 6-1=+5. Why is it -1? (For oxygen at the top).",
        "A": "Your definition of formal charge is incorrect. FC = no. of electrons in an isolated atom - no. of electrons in the bound atom. Isolated atom has 6 valence electrons. Bound atom has 3 lone pairs + half of the bonding pair = 6 + 1 = 7 electrons. FC = 6 - 7 = -1"
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "At 4:40, how do you know where to put the Hydrogen, and where the bonds go out? Why would they not attach to the Sulphur? I understand that they don't, but I'm not sure how to tell that for other molecules as well.",
        "A": "As H2SO4 is an acid, in acids H goes with O ...."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "When you say Oxygen donates those two electrons to create a double bond with Sulphur at 8:59, wouldnt that give oxygen a -2 charge?",
        "A": "why does it move 2 electrons down instead of just 1 to be shared?"
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "At 5:01, why do the hydrogen atoms bond to different oxygen molecules rather than just one?",
        "A": "Its really not possible, if you are thinking of taking that rightmost hydrogen and bonding it to the leftmost oxygen, the hydrogen octet will be completed but oxygen will be surrounded by more than 8 electrons which is not possible. The arrangement at 5:01 mark is the most stable formation of h2so4, how do you know if it s the most stable? well, if you see the formal charge of all the elements in the compound you will see they have the lowest formal charge possible."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "On the video at minute 8:55, how do you know two oxygen must share 2 of its electron with Sulfur? Why not just 1 oxygen or 3 oxygen that forms a double bond with the Sulfur?",
        "A": "All of the arrangements are possible, but the most stable one has the smallest possible formal charges. That s the one with two double bonds."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "At 5:50, why did you know to add more electrons to the non-terminal oxygens? I know you said because oxygen follows the octet rule but would that apply to all molecules wherever there's an oxygen, you just know to make sure there are 8 around it (as long as it isn't charged)?",
        "A": "At this point in the video you have some oxygens that have a single bond to sulfur; the bond gives them two electrons, so they need 3 lone pairs (6 more) to complete the octet and have 8 overall. Other oxygens have two bonds, one to sulfur and one to hydrogen, so they have 4 electrons (2 from each bond) and only need two more pairs (4 electrons) to complete the octet."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "At 0:49, he says you put the least electronegative atom in the centre, but isn't nitrogen the most electronegative atom?",
        "A": "H is never a central atom though. It only forms 1 bond."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "at 5:49 ,when oxygen originally has only 6 electrons in its valence shell,how does it have 7 electrons after bonding?",
        "A": "We don t really care about how many electrons each element brought to the molecule (other than the initial count) when we use this method. We create single bonds, then assign electrons to the more electronegative elements first, then to the central atom if any are left over. You could assume each oxygen only brought 6 electrons (and the sulfur also brought 6) and still arrive at the same structure in this case, but this method does not work for every Lewis dot structure."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "At 4:20: Why is the least electronegative atom (excluding hydrogen) always in the middle?",
        "A": "because atoms in the middle share most of the electrons, atoms around typically have lone electron pairs so electronegative atom likes electrons more so he would want to have as much unshared electrons as possible."
    },
    {
        "video_name": "dNPs-cr_6Bk",
        "Q": "At 9:15, specifically, Why don't elements in the 3rd period follow the Octet Rule? I am struggling to understand this concept.",
        "A": "All it really boils down to is that they are larger than the second period elements so can fit more groups around them. The common answer is that the empty d orbitals are used but this isn t backed up by the evidence."
    },
    {
        "video_name": "DGMs81-Rp1o",
        "Q": "At 6:55, where did he get the value of Kw?",
        "A": "It s a known value and should be in any chemistry textbook."
    },
    {
        "video_name": "hA5jddDYcyg",
        "Q": "In the part 10:45, high looks like aigh.",
        "A": "Agreed. The tips of his H came together to make it look like an H. Let s forgive him. Is your handwriting always perfect?"
    },
    {
        "video_name": "hA5jddDYcyg",
        "Q": "At 10:44 when Sal mentions iron, does melted iron evaporate just like every other liquid?",
        "A": "Yes, it s boiling point is 2862\u00c2\u00b0C"
    },
    {
        "video_name": "hA5jddDYcyg",
        "Q": "I don't understand from \"12:04\" and beyond. Can you explain it further? The comparisons with temperature and pressure confuse me.",
        "A": "I get it now, after watching the video again. Thanks Mr.Sal! -Sruveera"
    },
    {
        "video_name": "hA5jddDYcyg",
        "Q": "at 1:50 what do u mean \"temp. is the avg. k.E\" energy is energy and temp. is just temperature isnt it? pardon me if this doesnt make any sense at all:-P",
        "A": "What we experience as temperature is the effect of the particles in a substance randomly bouncing around. As they gain more energy and move faster, they seem to become hotter. This is why 0K (-273.15\u00c2\u00b0C) is the coldest temperature possible - at this point, there s no kinetic energy left and all movement stops."
    },
    {
        "video_name": "hA5jddDYcyg",
        "Q": "@14:50 whats boiling? when VP=ATM Pressure liquid boils but what is boiling? whats its significance? what happens inside a liquid while it boils??",
        "A": "The atoms, in a gas state (boiling) isnt at all bonded to each other, they have been given so much energy that they break their bonds and are completely free. Opposite to Solid where the atoms are bonded and structured"
    },
    {
        "video_name": "hA5jddDYcyg",
        "Q": "around 12:10 when Sal mentions the atmosphiric pressure, does this mean that water would vaporize faster in a less pressurized place, say outer space?",
        "A": "Precisely. Fun fact, liquid water will start evaporating in space because there is so little pressure AND it will also freeze at the same time because it is so cold."
    },
    {
        "video_name": "hA5jddDYcyg",
        "Q": "At 9:58 Why does a lot of its molecules have to enter into the vapor state in order for the equilibrium to be reached?",
        "A": "Because for each molecule that leaves the liquid state one molecule from the vapor state has to enter the liquid state. The more that leaves, more has to enter."
    },
    {
        "video_name": "hA5jddDYcyg",
        "Q": "At 4:35, they talk about how the water molecules with higher KE evaporate then the wind blows them away. Wouldn't this drain the system of KE until the water molecules still in the pan/dish \"run out\" of KE since the majority of the molecules with high KE that reach the surface escape and take their KE with them?",
        "A": "Evaporation has a cooling effect, exactly for this reason. The molecules left behind have lower average KE than the molecules that run off."
    },
    {
        "video_name": "hA5jddDYcyg",
        "Q": "at 0:40, should'nt it be the heat of vaporisation that the liquid molecules have to overcome?",
        "A": "Correct. The molecules have to overcome the heat of vaporization to move from the liquid to the gas phase."
    },
    {
        "video_name": "hA5jddDYcyg",
        "Q": "At 14:15 , why sal said SEA LEVEL ATMOSPHERIC PRESSURE ?\nWhat does it mean ?",
        "A": "The air pressure is different depending on your altitude. Sea level is a common reference point for air pressure. Sea level is considered 0 altitude."
    },
    {
        "video_name": "hA5jddDYcyg",
        "Q": "at 9:22 what are hydrocarbons?",
        "A": "A hydrocarbon is any substance made up from only hydrogen and carbon atoms. Many of the fuels in crude oil are hydrocarbons."
    },
    {
        "video_name": "hA5jddDYcyg",
        "Q": "at 10:46, vapor pressure is the pressure exerted by the vapor till the equilibrium state is reached, right?",
        "A": "Correct. Vapor pressure is the pressure exerted by a vapor that is in equilibrium with its liquid form."
    },
    {
        "video_name": "MODnIkQvyz0",
        "Q": "at 8:35 Sal says that any structure that has a single polypeptide will have primary, secondary and tertiary structure. But in secondary structure we interact the backbones of TWO polypeptides init?",
        "A": "Maybe the polypeptide interact with himself."
    },
    {
        "video_name": "MODnIkQvyz0",
        "Q": "When talking about the interactions at the secondary level, (5:10) he refers to the second carbon as \"the carbonyl carbon,\" but this is the carbon of the carboxyl group of the A.A. Is the name of the functional group changed once the Amino Acid has bonded? Is this just a mistake?",
        "A": "Yes, the formation of an amino acid chain gets rid of the OH from the carboxyl turning it into a carbonyl CO, and the other H to form water comes from the amino group on the other amino acid."
    },
    {
        "video_name": "l-g2xEV-z7o",
        "Q": "(8:35) why wouldn't the other 2 methyl groups connected to the alpha carbons be labeled as beta carbons",
        "A": "They are also beta carbons, they just haven t been labelled"
    },
    {
        "video_name": "l-g2xEV-z7o",
        "Q": "At 6:47, the protons from the acid are responsible for the -OH leaving the cyclohexanol ring. How is it an E1 reaction if the concentration of both cyclohexanol and sulfuric acid determine the rate of this step?",
        "A": "The concentration of H2S04 is not a factor in this case because it is your solvent. You can assume that there is an endless supply available."
    },
    {
        "video_name": "l-g2xEV-z7o",
        "Q": "At 7:23 how is the 1 carbon a carbocation once the hydrogen is taken off? Does it not have two bonds and two lone pairs giving it 4-4= 0 formal charge?",
        "A": "It doesn t have any lone pairs. It has 2 bonds to other carbons and 1 bond to an implied hydrogen. 4 - 0 - 6/2 = +1"
    },
    {
        "video_name": "l-g2xEV-z7o",
        "Q": "For the alkyl halide example (starting at 8:17), why wouldn't that undergo an Sn1 reaction? It's a tertiary carbocation with a weak nucleophile, isn't it?",
        "A": "You re right, SN1 is possible. In reality, both E1 and SN1 will often happen at the same time and you will get both products. You can, however, adjust the conditions to help favour one mechanism over the other. In this case your hint is that Sal drew a triangle under the reaction arrow indicating that heat has been added. Heat helps drive elimination reactions, so whenever a substitution or an elimination reaction is possible and you add heat, you can usually assume that the elimination mechanism dominates."
    },
    {
        "video_name": "l-g2xEV-z7o",
        "Q": "at 0:21, he says the first step is loss of leaving group. Cant we view the mechanism to start with the base plucking a proton from Beta Carbon? the electrons of C-H bond can then move to C-C bond to make C = C. Since we made a new bond, we must break an old one, so the weakest bond i.e. C-X (due to polarization) should be broken. So we get alkene as result. I think this is more probable since an acid-base reaction is faster than leaving of a leaving group. so that should happen first and then loss of halogen",
        "A": "What you are describing is an E2 mechanism The 1 in E1 just means the kinetic order of the reaction only depends on the concentration of the molecule not the nucleophile. The 2 in E2 means it depends on both. Same is true with SN1 and SN2. In SN1 and E1 reactions the leaving group has to leave first and then the nucleophile attacks the carbocation. In SN2 and E2 reactions the nucleophile or base attacks and the leaving group leaves at the same time."
    },
    {
        "video_name": "l-g2xEV-z7o",
        "Q": "8:33 - The delta represents heat?",
        "A": "The upward facing triangle used to represent the element fire in Medieval times. It has been carried over in modern chemistry to mean either heat, or a change in a quantity (from mathematics)."
    },
    {
        "video_name": "l-g2xEV-z7o",
        "Q": "10:25 how come the ch3 not on the ring not concidered as beta?",
        "A": "You are correct. There are three \u00c3\u009f carbons: C-2 and C-6 of the ring and the C of the methyl group. The \u00c3\u009f hydrogen atoms on the methyl group can also be removed to give an alkene, but the product is not as stable as the one with the double bond inside the ring. That is probably why they were ignored in the video."
    },
    {
        "video_name": "l-g2xEV-z7o",
        "Q": "At 4:14 how the electrons are gonna make a double bond ?",
        "A": "The sp\u00c2\u00b2 C-H orbital reverts to a p orbital, which overlaps with the vacant p orbital on the carbocation to form a \u00cf\u0080 bond."
    },
    {
        "video_name": "l-g2xEV-z7o",
        "Q": "where did the H+ come from at 12:30?",
        "A": "from the cyclohexane. Unlike charges attract. So the oxygen with valence electrons takes the H+ and the electron that it was sharing in its bond with the carbon goes to form a double bond."
    },
    {
        "video_name": "t_f8bB1kf6M",
        "Q": "At 2:42, does the \"groups\" refer to the vertical columns or the types of elements such as \"alkali metals\"?",
        "A": "both. groups are vertical columns in the periodic table that contain elements with similar chemical properties. some have special names (alkali metal, alkaline earth metals, halogens, halkogens, noble gases) and some have trivial names (boron group, carbon group, nitrogen group)"
    },
    {
        "video_name": "t_f8bB1kf6M",
        "Q": "At 7:39 antimony has been classified as a metalloid. but isn't it a metal?\nplease clarify.",
        "A": "It s regularly classed as a metalloid in literature"
    },
    {
        "video_name": "t_f8bB1kf6M",
        "Q": "At 1:09, why are we ignoring the group between 3 and 12 is this because they are metal ?",
        "A": "That s simply how some old US chemists decided to number the groups. And that system is not consistent with another that was used in Europe during the same time period."
    },
    {
        "video_name": "OYerdzZPSI0",
        "Q": "At 5:27, does the expression for Kirchhoff's current law equal zero?",
        "A": "KCL can be written different ways. The yellow box on the left at time 5:27 says sum of currents going in = sum of currents going out . A few seconds later when I put yellow boxes around other forms of KCL, the box in the center says sum of all currents going in = 0 , and on the right, the box says sum of all currents going out = 0 . These are all different ways of saying the same thing."
    },
    {
        "video_name": "OYerdzZPSI0",
        "Q": "At 5:04\nHow is the current entering the circut?\nShouldn\u00c2\u00b4t it be +4A if the other nodes are leaving the circut\nor vice versa?\n\nThanks in advance!",
        "A": "This is a tricky part of Kirchhoff s Current Law. Pay close attention to the numbers and the sign of the numbers and the arrow directions. I drew all the arrows going out of the node. I wanted to do this so I could show you the form of KCL that says SUM(currents going out) = 0. The answer came out -4 flowing out, which is exactly the same as +4 flowing in."
    },
    {
        "video_name": "OYerdzZPSI0",
        "Q": "Ok I am confused around 3:14, where that one current equals -3. I get how current in general can be negative, but I am confused on how the current going in is equal to zero and if the current going on/out is always equal to zero.",
        "A": "Another way of putting it is all the current going in is equal to all the current going out. So if you have 3A of current going into a node, then there must be 3A of current going out of the node. The reason Sal says that i = -3A is because he defined the direction of current i to be going in, so negative would mean the direction is going in the opposite direction as it was defined."
    },
    {
        "video_name": "OYerdzZPSI0",
        "Q": "At 1:41 i3 is assumed going downward. How can that assumption be made? Isn't it possible for i1 +i3 = i2 if i3 was upward?",
        "A": "We just start out assuming it s going down. If it turns out it is going up, we will get a negative number for an answer."
    },
    {
        "video_name": "QspmZf_yWyU",
        "Q": "At 1:41 he wrote \"food vacoule\" but why didn't he write a food vesicle instead ? , can someone explain the deffirence please",
        "A": "You cant drink food, and since pinocytosis is the cell s way of drinking, then its isn t food. But i would say that a vacuole is a type of vesicle, just a larger one used for storage."
    },
    {
        "video_name": "-aGYytZ7K7M",
        "Q": "At 0:36 Sal talks about facilitated diffusion. Why is facilitated diffusion called uniporters?",
        "A": "Facilitated diffusion is the process that uniporters like channel proteins follow. Channel proteins are called uniporters because they transport one molecule or atom at a time to the substance s lower concentration gradient."
    },
    {
        "video_name": "FOkQszg1-j8",
        "Q": "At 1:38, Sal indicated that acceleration was a vector quantity. Is there an equal value of acceleration for scalars, for example, can you calculate the \"acceleration\" of an object if you are given a speed, not a velocity?",
        "A": "that would be the magnitude of acceleration. We don t have a separate name for it."
    },
    {
        "video_name": "FOkQszg1-j8",
        "Q": "at 4:36 why does it matter that the car is going east?",
        "A": "It doesn t matter here; Sal just said that particular direction to illustrate an example. However, in some situations, stating the direction of a moving object would be needed."
    },
    {
        "video_name": "FOkQszg1-j8",
        "Q": "At 7:11 Sal says you might want to change it into feet. How would I change 1/180 m/s^2 to feet?",
        "A": "m = meter mi = mile Just to clarify a bit :) To convert it into feet, simply take into account how many feet there are in a mile (5280 feet per mile) and swap the two terms. 1/180 mi/s^2 = 5280/180 ft/s^2 There s a more formal way to do it, but for a simple problem like this, a swap should suffice. Simplifying the final answer would also be good: 5280/180 ft/s^2 is approximately equal to 29 ft/s^2"
    },
    {
        "video_name": "FOkQszg1-j8",
        "Q": "At 2:07, sal has written an a with a arrow above it. What does that mean?",
        "A": "The arrow indicates that the variable is a vector."
    },
    {
        "video_name": "FOkQszg1-j8",
        "Q": "At 8:08, why do you say it as 1/180 miles per second per second? Isn't it the same as 1/180 miles per second?",
        "A": "miles per second is a unit of speed. Acceleration measures how much speed changes per second, which is why it s measured in (miles per second) per second."
    },
    {
        "video_name": "FOkQszg1-j8",
        "Q": "In 3:01 Mr.Khan said that 60 mph was the velocity. Isn't supposed to be speed?",
        "A": "At times, for covenience we don t give the directions. As you ll progress further and solve numericals, you will notice that directions are not given and the terms velocity and speed are used interchangeably."
    },
    {
        "video_name": "FOkQszg1-j8",
        "Q": "When Sal is giving a visual representation with a speedometer at 8:21, doesn't the 20 mph/s represent an average acceleration of 20 mph/s, instead of an exact amount each second?",
        "A": "Yes. The speedometer is just helpful for visualizing it if we assume that the average acceleration is the actual acceleration the whole time."
    },
    {
        "video_name": "FOkQszg1-j8",
        "Q": "At 0:22, Sal says acceleration is change in velocity over time. Would it be correct to say acceleration is change in speed over time? I know that velocity is a vector and speed is a scalar but what difference does it make?",
        "A": "There s nothing wrong with dividing a scalar by a scalar, but the result of that is a scalar. Acceleration is a vector. So it has to be velocity / time, not speed/time. If you do speed/time, you will get the magnitude of acceleration, and you will have lost any information about direction."
    },
    {
        "video_name": "FOkQszg1-j8",
        "Q": "At 7:02, why would it be second squared? Thanks",
        "A": "if it is 4 meters per second squared it means that the object s speed is going up by 4 m/s every second. Second 1: 4 m/s Second 2: 8 m/s"
    },
    {
        "video_name": "FOkQszg1-j8",
        "Q": "At 7:00, when Sal starts talking about the astronaut propelling himself by throwing something, why would the mass of the object be considered? When observing Newton's Second Law in conjunction with the Third, it would seem instead that the action of throwing should exert similar force on any object, in turn reacting that force onto him. Therefore, wouldn't the man accelerate based solely on his own mass and the force he could generate, independent of how his force affected the object?",
        "A": "I ll give you a mathematical answer: According to Newton s second law F=m*g According to the third law, every action has an equal and opposite reaction. So we get F1 = m1*a1 = m2*a2 = F2 We know that F1 = F2 (third law) so this is the same thing as m1*a1 = m2*a2 finally we get a1 = m2*a2/m1 (a1 is the acceleration of the man throwing the object). You can see now that it does depend on his mass. The heavier the man is the less he will accelerate for a given force of a throw. Hope that helps :)"
    },
    {
        "video_name": "BWs-ONRDDG4",
        "Q": "At 7:18 you state that if you know the absolute luminosity of a cepheid variable star, then you know the absolute luminosity of any other cepheid variable star. How can that be? Isn't that an assumption? Don't they vary in their absolute luminosity from one star to another?",
        "A": "By measuring the distance to cepheid variable using other methods like parallax and then using this to compute the max absolute luminosity and comparing it to the variance rate we were able to see that there is a relationship."
    },
    {
        "video_name": "M_5KYncYNyc",
        "Q": "At 13:12 Sal says that the efficiency equation 1- Q_2/Q_1 applies to ANY engine, but this equation was derived using the assumption that all net heat added is converted to Work. This would not be true in a non-Carnot engine in which some of the heat added is lost to friction. Please explain.",
        "A": "What I think Sal meant is that 1 - Q_2/Q_1 is the maximum efficiency any engine can obtain. The Carnot cycle is an (ideal, unobtainable) engine that attains that efficiency, and the best any other engine can manage is to be as good as the Carnot cycle. It is very easy being worse."
    },
    {
        "video_name": "tvO0358YUYM",
        "Q": "What equation is Sal using to solve for Joules at 4:00? Is it PV=nRT since PV=joules?\nWhat equation is Sal using at 6:54 to solve for the number of joules as well?\nIt seems as though he is taking whatever numbers/units given and multiplying them to get \"joules\" isolated... Sorry for any inconvenience but if someone could clarify it would be much appreciated.",
        "A": "I believe at 4:00, he uses the equation E= (m)x(C)x(delta T) m= mass in grams, C= specific heat, and delta T = change in temperature"
    },
    {
        "video_name": "tvO0358YUYM",
        "Q": "At 2:11, how come you can change the units of temperature freely( went from58.29 degrees Celsius to 58.29K)?",
        "A": "Its because the difference between two numbers in the Celsius scale as well as the Kelvin scale is equal. They change at the same measure. For example, 90 degree Celcius - 20 degrees Celcius= 70 degress Celsius. Converting into Kelvin by adding 273, 363 Kelvin - 293 Kelvin = 70 Kelvin. Thus the magnitude between the differences remains equal even though the individual values are different. Hope this helps :)"
    },
    {
        "video_name": "tvO0358YUYM",
        "Q": "At 2:07 what is the kelvin scale exactly?",
        "A": "The Kelvin scale is a temperature scale that puts the zero at absolute zero, unlike Celsius which puts zero at the freezing point of water and Fahrenheit which puts it at the freezing point of brine. The degrees on the Kelvin scale are the same size as on the Celsius scale."
    },
    {
        "video_name": "tvO0358YUYM",
        "Q": "at 1:23, how do you find the significant figure in the problem sal was working on?",
        "A": "Take the first decimals."
    },
    {
        "video_name": "tvO0358YUYM",
        "Q": "At 0:19, what is the meaning of Ethanol and is that the formula that's after the word ethanol? can there we a more specific what to the heat capacity?",
        "A": "Ethanol is an organic substance and its formula is C2H5OH. It is in a class of substances known as alcohols and, in fact, it is what non-scientists call alcohol , being present in beer, wine, whisky, vodka, etc."
    },
    {
        "video_name": "tvO0358YUYM",
        "Q": "At 2:10, why is there a kelvin scale? Why can't we have f. and c.",
        "A": "Kelvin is necessary because C and F are both relative scales. That means that 0 in either C or F does not mean no heat, likewise doubling degrees in C or F does not mean that you have doubled the heat. In Kelvin, which is an absolute scale, 0 K really is no heat and doubling the number of Kelvins really is doubling the amount of heat. So, for computations, we must use an absolute, not a relative, scale. So, that is why we need the Kelvin scale."
    },
    {
        "video_name": "2UTr46btzaY",
        "Q": "At 2:24 onward the polar coordinates for the top two hydrogen atoms are designated as (sqrt 2, 1, 0) and (-sqrt 2,1,0). Why is that? If they were equidistant but with different values for x and y I don't think the theta will be equal to 35.26 degrees. Is there a reason for this designation?",
        "A": "The C atom is at the centre of a cube with side = 2. So the distance from the C atom to the top of the cube (y) is 1. The H atoms are at the diagonally opposite corners of the top face of the cube. According to Pythagoras, the diagonal of a cube with side 2 is d = \u00e2\u0088\u009a(2\u00c2\u00b2 + 2\u00c2\u00b2) = \u00e2\u0088\u009a8 = 2\u00e2\u0088\u009a2. So, half the distance between the two H atoms (x) is \u00e2\u0088\u009a2. The Cartesian coordinates of the H atoms are therefore (\u00e2\u0088\u009a2, 1, 0) and (-\u00e2\u0088\u009a2, 1, 0). There is no other possibility for a regular tetrahedron."
    },
    {
        "video_name": "CqN-XIPhMpo",
        "Q": "5:45 Where does this molecule comes from ?",
        "A": "There are genes for this protein in the DNA. It is expressed in all photoreceptor cells of the retina."
    },
    {
        "video_name": "CqN-XIPhMpo",
        "Q": "At 9:05 Ronald describes the differences between on-center and off-center bipolar cells. Can you please clarify this difference? Is it that on-center bipolar cells are turned on when the rods are turned off? And off-center bipolar cells are turned off when rods are turned off? Thanks.",
        "A": "So, On-center bipolar cells turn on when rods and cones are turned off, Off-center bipolar cells only turn off when cones are turned off, right?"
    },
    {
        "video_name": "aASSUcUSBro",
        "Q": "its best to mention after 4:15 the difference between contralateral and ipsilateral in terms of crossing over of cortical fibers.\n\nipsilateral- situated or appearing on or affecting the same side of the body (affecting the same side of the body)\n\ncontralateral- occurring on, affecting, or acting in conjunction with a part on the opposite side of the body. EX: motor cortex controls contralateral muscles. =)",
        "A": "Its something that the user wants the maker of the video to fix, not a question."
    },
    {
        "video_name": "-_xZZt99MzY",
        "Q": "I'm a bit confused. At 0:33 the horizontal line is said to represent time but doesn't it represent average density or pressure? And the area above it as high pressure or density (of the air molecules) and the area below it as low pressure or density? Also I couldn't get the difference between the 2 kinds of graphs shown in the end.",
        "A": "No, the horizontal line is time. Watch the video again, more slowly, and pause it when you need to in order to make sure you are following along. Look closely at the axes of the two graphs."
    },
    {
        "video_name": "-_xZZt99MzY",
        "Q": "At 3:04, the definition of wavelength has been explained. I didn't get it properly. Can you please explain the difference between period and the wavelength of a sound wave using an example from our daily lives?",
        "A": "Think about waves on the sea, because they re quite similar to other types of waves. The period is that time it takes between each peak of the wave to reach the shore, while the wavelength is the distance between two consecutive peaks."
    },
    {
        "video_name": "-_xZZt99MzY",
        "Q": "in 2:00 we can see that frequency = number of oscillation per second. and it is also written that f = 1/T, where T = second. So does the numerator represents the number of oscilliation? will it keep on changing?",
        "A": "f=cycles/second i.e cycles per one second and... T=second/cycle i.e T is the time taken to complete one cycle f=1/T means that they are reciprocals of each other.."
    },
    {
        "video_name": "-_xZZt99MzY",
        "Q": "At 0:30 ,this graph doesn't represent compressions and rarefactions , but sound waves travels in the form of compressions and rarefactions.Confused!",
        "A": "The graph represents displacement of molecules, and the displacement is what causes compressions and rarefactions."
    },
    {
        "video_name": "VqAa_cmZ7OY",
        "Q": "at 12:16, how is L/Mol the same as Mol/L?",
        "A": "Its NOT the same. He is dividing a number by L/Mol which essentially means multiplying it by Mol/L. You have to flip the fraction. Hope this helps :)"
    },
    {
        "video_name": "VqAa_cmZ7OY",
        "Q": "at 3:23 what is potassium permanganate?",
        "A": "It is a chemical compound."
    },
    {
        "video_name": "VqAa_cmZ7OY",
        "Q": "At 3:09, do you have to use the graph to answer it?",
        "A": "No. You can use the other process that Sal explains"
    },
    {
        "video_name": "VqAa_cmZ7OY",
        "Q": "At 3:14 what values go on the y-axis of the graph?",
        "A": "Absorbance goes on the y-axis of the graph."
    },
    {
        "video_name": "VqAa_cmZ7OY",
        "Q": "At 5:07, will it always be a line?",
        "A": "According to the Beer-Lambert Law, this equation will always be linear and therefor, a straight line."
    },
    {
        "video_name": "VqAa_cmZ7OY",
        "Q": "At 1:15, what is difference between spectrometer and spectrophotometer?",
        "A": "A spectrometer is an apparatus used for recording and measuring spectra, esp. as a method of analysis. A spectrophotometer is an apparatus for measuring the intensity of light in a part of the spectrum, esp. as transmitted or emitted by particular substances. So, the spectrometer determines the spectra (lines of transmittance/absorbance), while the spectrophotometer measures the light strength/intensity of the particular light of one or more bands in the given spectra."
    },
    {
        "video_name": "VqAa_cmZ7OY",
        "Q": "at 11:39 will you always end with concentration?",
        "A": "If you work the equation this way then yes. There is a way to work the equation to find each of the variables. If you work it a different way you would be able to find length or absorbance, etc. It works like any algebra problem"
    },
    {
        "video_name": "VqAa_cmZ7OY",
        "Q": "At 3:14 what values go on the y-axis of the graph?",
        "A": "Absorbance goes on the y-axis of the graph."
    },
    {
        "video_name": "VqAa_cmZ7OY",
        "Q": "At 8:45, what is an epilon?",
        "A": "Epsilon is the fifth letter of the Greek alphabet. \u00ce\u00b5 In the Beer-Lambert Law Epsilon represent a constant, called the molar absorbtivity. Sal writes this law at 6:55"
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "At 1:48, they said that the Milky Way was 2.5 million light years away from the nearest galaxy, the Andromeda Galaxy so the Local Group isn't really the Local Group, it's the Unimaginably Huge Group right?",
        "A": "Pretty much, but Andromeda is not the closest galaxy, it is however the closest large one. The closest galaxy to the Milky way is that Sagittarius dwarf galaxy which happens to be orbiting the Milky Way."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "At about 7:26 Sal has just about finished talking about sizes of galaxies and clusters of galaxies, there is a diagram on the right that has many clusters of galaxies of different forms. How do we know this much about sizes of galaxies, or even clusters of galaxies? Do we really have telescopes, or satellites that look/go this far into space?",
        "A": "In fact our hubble telescope has the potential to look much farther than it does now but it can t due to the limits of the observable universe."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "So when the observable universe was mentioned at 7:07 of this video, I am wondering- So by observable universe the dude means only what our telescopes can see and observe? I guess that means our telescopes are limited to seeing only the observable universe? Is there a limit to size of the universe, does the universe ever end? Does it endlessly keep expanding? So is the universe unlimited in size? God, a unlimited universe is hard to contemplate.",
        "A": "Ha! We likely will never explore the whole universe then. 0_o Expanding forever. Hum."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "At about 8:50 into the video, Khan says that the point on the edge of the visible universe that he is referencing is now...\"40 to 45 Billion Light Years Away\" Is that possible? I thought the universe was estimated to be only 13 to 14 Billion years old. I know that a light year is distance, not time/age -- but can something be 40 Billion light years away in a universe that is \"only\" 14 Billion years old? How would it get so far away in that time if it can't travel faster than the speed of light?",
        "A": "I watched Radius of Observable Universe after this one. It does a nice job of answering the question."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "At 8:50 in \"Scale of the galaxy\" video, Sal said that the center of our galaxy is too bright to see what is on the other side. How are we able to know and see what is 13 billion light years away if we can't see through the center of our galaxy? Thanks for any replies!",
        "A": "We re not looking through the center of our galaxy when we see objects that far. We re looking out of the top and bottom of the galactic plane in our current location, where there isn t much galactic debris to interfere."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "At around 7:07, he said the light left some point about 13 billion light years ago and is just reaching us right now. Isn't light year a measure of distance, not time? or did he just slip?",
        "A": "You are correct. If that s what he said, it was a slip of the tongue."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "what and how big is an astronomical unit at 2:19 in the video?",
        "A": "Or the distance between the Earth and Sun."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "At 3:11 Andromeda is blue. Is that related to the doppler effect?",
        "A": "The shift in Andromeda is to small to be seen by the human eye, the blue comes from an array of bright young stars, if our galaxy was seen from a distance, it would likely have the same color."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "At 1:09, Sal talks about the \"local group\" of galaxies. Andromeda and the Milky Way are in that group, and they are on a collision course. Are other galaxies in that \"local group\" also on a collision course, or are they drifting away from each other as the universe expands?",
        "A": "The local group comprises of approximately 30 galaxies with the two largest galaxies being the Milky Way and the Andromeda Galaxy. Many of the other galaxies are gravitationally bound to the Milky Way and Andromeda Galaxy and are called the Milky Way satellites and Andromeda Satellites respectively. Several of these including the Magellanic Cloud are expected to merge with the Milky Way and Andromeda galaxy eventually."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "At 8:36, how do we know where all of those galaxy's are? It would take to long to send camera's out there.",
        "A": "The light comes to us and then we can use telescopes to magnify the light."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "at 1:49 he said it would take 2.5 million years. so can we make a ship that can go tripled the speed of light we will be able todothat",
        "A": "Even if we could generate the infinite energy necessary or somehow fold space to go triple the speed of light, it would still take nearly a million years. If we used a multi generational ship people could eventually get there, provided nothing failed, but by the time they got there the ship s people would have evolved a little bit. If we could develop some sort of stasis pod, maybe that would be doable."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "Will those small satellite galaxies Sal talked about at 1:28 someday collide with the Milky Way?",
        "A": "probably not, if they re in orbit around the galaxy"
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "At 1:54 the dude says the distance to the nearest galaxy is 2.5 million light years. What is 9,460,730,472,580 kilometres (1 light year) x 2.5 million? In other words how many kilometres would that be?",
        "A": "0_0 WOW! I do not see getting to the nearest galaxy anytime soon"
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "at 7:18 sal keeps talking about how big and vast things are, and I had a question is the universe infinite? My brain is kind of having a hard time comprehending every thing.",
        "A": "well first off there are two parts to the universe first is the observable universe which is how far we can see with our current technology the other part is unobserved due to the fact the the light has not yet reached earth so we cannot not see it, however we do know that things are expanding or floating away from each other therefore until we hit a cosmic wall or something the universe is infinite."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "On 7:07, it says that the visible universe is 13 billion light years. So after 13 billion years, we can see that object on the corner of the visible universe right? But does that mean after 13 billion years, we can see the object forever (if the object didn't move or expand)?",
        "A": "Yes, once the light reaches us we can see the object as it was 13 billion years ago."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "At 3:49, Sal mentioned the Virgo Supercluster. What is the Virgo Supercluster?",
        "A": "It is a cluster of galaxies that are gravitationally bound to each other. It includes our own galaxy as well as the Andromeda and various other local galaxies."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "At 1:35 Sal talks a little bit about the Andromeda galaxy. Will we (our Milky Way Galaxy), ever crash into andromeda?",
        "A": "Watch the video on quasars. We will eventually collide and he discusses it further there."
    },
    {
        "video_name": "JiE_kNk3ucI",
        "Q": "at 7:00 sal said that that light left 13 billion light years ago. Isn't light years distance?",
        "A": "Yes, light years are a measure of distance. If he said light years ago he misspoke, and probably he just meant years."
    },
    {
        "video_name": "Tbq-KZaXiL4",
        "Q": "At 12:05, Sal explains the process of the the NT's exiting the post synaptic neuron and entering the synaptic cleft. He describes the process as Exocytosis, but, correct me if I am wrong, I had thought this process was receptor-mediated exocytosis?",
        "A": "There is no receptor in the plasma membrane here causing excytosis, so I would no categorize it as receptor-mediated. Calcium influx triggers conformation changes in proteins that link the vesicles with the presynaptic membrane, causing the two to merge and thereby spill the contents into the cleft."
    },
    {
        "video_name": "Tbq-KZaXiL4",
        "Q": "At 14:54, Sal mentioned that if the neurotransmitters stimulated the potassium gates, it will inhibit the signal since the potential gradient goes out. But my question is, is it purely arbitrary whether the transmitters will affect the sodium or potassium gates? So some signals travel by luck? I did not understand that last part...",
        "A": "The sodium and potassium travel through the same gate which is actually called a ligand gated ion channel but for the purposes of this it could be called a neurotransmitter gated ion channel. So there aren t independent sodium and potassium gates only one ion channel which allows all sorts of ions to pass through potassium and sodium included."
    },
    {
        "video_name": "Tbq-KZaXiL4",
        "Q": "at 9:36, it introduces the neurotransmitters, but where do the neuron transmitters come from? I know they are molecules and they have different classes of them, but what \"trains\" them to get into the vesicles?",
        "A": "The neurotransmitters are manufactured in the Rough Endoplasmic Reticulum (which in neurons are called Nissl bodies ). They travel along a microtubule highway (created by the golgi aparatus) from the cell body to the axon bulb. If you aren t familiar with that terminology, check out the videos on generalized cell structure. :)"
    },
    {
        "video_name": "Tbq-KZaXiL4",
        "Q": "At 0:20, what type of ions are released into the cell?",
        "A": "Typically Na+ or Ca2+ for excitatory postsynaptic potentials (or EPSPs) as they bring the membrane potential closer to threshold, and Cl- for inhibitory postsynaptic potentials (or IPSPs) as it hyperpolarises the membrane to prevent action potentials."
    },
    {
        "video_name": "Tbq-KZaXiL4",
        "Q": "At 13:50, Sal mentions the possibility of a potassium channel being triggered instead of a sodium channel. What is required to open a potassium channel that's different from opening a sodium channel?",
        "A": "Sometimes it needs a different ligand, or the dendrite may have more K+ channels than Na+."
    },
    {
        "video_name": "Tbq-KZaXiL4",
        "Q": "How does a neuron know which neurotransmitter to release? 12:00",
        "A": "Every neuron has its own special neurotransmitters when the body is in a different state. Acetylcholine is a neurotransmitter that is released when a muscle is moving. It helps with muscle activation. Epinephrine is a neurotransmitter that is nicknamed adrenaline. It is found in the sympathetic nervous system which is activated at a time of stress. It helps with the brain s fight or flight response."
    },
    {
        "video_name": "Tbq-KZaXiL4",
        "Q": "At 12:81 you refer to \"serotonin,\" pronounced a short 'o' sound,\n\"sa-rotten-in\". I've heard of a neurotransmitter pronounced with a long 'o' sound, like \"sarah-tone-in.\" Are these different transmitters? If not, are both pronunciations correct?",
        "A": "Based on MerriamWebster.com, your pronunciation, sarah-tone-in is accurate."
    },
    {
        "video_name": "Tbq-KZaXiL4",
        "Q": "@ 12:15 you mention the names of many neurotransmitters like epinephrine, norepinephrine, dopamine, seratonin, etc... my question is this - epinephrine is made in the adrenal medulla on the kidneys, dopamine and seratonin are made in the brain, but you show these neurotransmitters in presynaptic vesicles. How is it possible that a specific neurotransmitter like epinephrine can be released when needed? are there many different neurotransmitters in one presynaptic neuron?",
        "A": "no, usually neurons only release 1 type of neurotransmitter. so to release dopamine you would stimulate dopaminergic neurons"
    },
    {
        "video_name": "Tbq-KZaXiL4",
        "Q": "At 1:10, Sal talks about the signal moving down the axon, so does the wave (the depolarization/repolarization graph) just keep going and repeating itself until the signal reaches the end of the axon?",
        "A": "It transmit into the terminal, if I am right. It may go to the brain too, but I m not so sure."
    },
    {
        "video_name": "Tbq-KZaXiL4",
        "Q": "9:36 Why do these vesicles have two membranes? Usually vesicles only have one membrane, if I am correct?",
        "A": "The vesicles only have one membrane, he is merely drawing the phospholipid bilayer. The membrane is made up of mostly fats one end of which is hydrophilic(attracted to water) and the other being hydrophobic(repulsed to water), so the double layer is the hydrophilic side of the fats being on the outsides and the hydrophobic being on the inside."
    },
    {
        "video_name": "Tbq-KZaXiL4",
        "Q": "If synapses can be inhibitory (15:53), why do we even have synapses? why isn't everything one big neuron?",
        "A": "Because then if we send out one signal, we can only get one response. Like if you want to pick up an apple, you send signals to your arm to extend, if you had one BIG neuron all over your body, then every body part would get the same signal as well. Hope that helps!!"
    },
    {
        "video_name": "YddDLLDhPvM",
        "Q": "at 4:12, I don't quite understand why the oxygen has a plus 1 formal charge?",
        "A": "Formal charge = valence electrons - lone pair electrons - bonds From this formula we get: An oxygen with 2 bonds and 2 lone pairs will have a formal charge of 0 An oxygen with 3 bonds and 1 lone pair will have a formal charge of +1 An oxygen with 1 bond and 3 lone pairs will have a formal charge of -1"
    },
    {
        "video_name": "YddDLLDhPvM",
        "Q": "At 3:45, why is hydroxide a bad leaving group. Is it because you're in a polar aprotic solvent and hydroxide is a very strong base?",
        "A": "Hydroxide is a strong base and a bad leaving group. Once we protonate it, it will become water which is a weaker base and hence good leaving group"
    },
    {
        "video_name": "XUT7zEffaTY",
        "Q": "At 7:35 , Sal said that a pi bond and an electronegative element should be present in the substituent to a benzene ring , its a moderate deactivator . Then why NO2 is a strong deactivator?",
        "A": "The difference is that the N atom has a formal positive charge as well as a \u00cf\u0080 bond."
    },
    {
        "video_name": "XUT7zEffaTY",
        "Q": "at 4:36 could someone explain why the electrons on F are too far away to take part in resonance? I'm just thinking F is the most electronegative atom, so electrons would be close to F and far from C. How are the electrons far?",
        "A": "The electrons are too far from the benzene ring (close to the F atoms) and are thus not able to participate in resonance."
    },
    {
        "video_name": "wEQxtUwiV1E",
        "Q": "At 9:41,, shouldn't the root yield a positive as well as a negative value for velocity indicating the opposite direction of the 2 charges?",
        "A": "When dealing with square roots in physics problems you often have to look at the physical system you are dealing with to understand if you really need both roots or have to choose one of them. In the video you are dealing with 2 positive charges so you need to choose the root(s) that are constant with the charges being pushed apart."
    },
    {
        "video_name": "wEQxtUwiV1E",
        "Q": "At 8:58, What is a charge anyway?",
        "A": "charge is a fundamental property, like mass is. Just as mass determines how an object will respond to a gravitational field, charge determines how it will respond to an electric field"
    },
    {
        "video_name": "iF-f2-KSw6E",
        "Q": "At 1:50. Is it possible to get 1,3-dichloro-2-methyl-5-nitrobenzene as a side product?",
        "A": "It is possible, but I would expect 2,6-dichloro-4-nitrotoluene to be a very minor product for two reasons: 1. Once the first Cl has been added, it acts as a deactivating group, so you would have to use more vigorous conditions to get the second Cl atom to add to the ring. 2. Because of steric hindrance, it is not easy to get three substituents adjacent to each other in a benzene ring. This would require still more vigorous conditions."
    },
    {
        "video_name": "iF-f2-KSw6E",
        "Q": "At 6:25 which of the two products is major?",
        "A": "The major product will be the one in which the steric hindrance is lesser. If groups attached to the benzene ring are closer, then there will be steric hindrance due to the non-bonding electrons present of the groups."
    },
    {
        "video_name": "yE3eQ6q39f4",
        "Q": "Hey, sorry I'm really new to physics, I'm just learning casually. Quick question though: why would the amps be the same on either side of R3? I thought resistors reduced the current flow meaning that the amps on the input side would be greater than the output? 1:52",
        "A": "Resistors reduce the current everywhere in the circuit, not just at one spot. Think of the electrons in a series circuit like the cars of a train. They move together, at the same speed. If the train is going uphill, it faces more resistance, so the cars go slower - together."
    },
    {
        "video_name": "yE3eQ6q39f4",
        "Q": "At 6:47, David says the current would completely skip the voltmeter and the resistor, but if you think about it, he's connecting it in parallel to the other 2 resistors at the bottom, so some current should flow through the circuit, but the voltmeter would end up measuring the difference in electrical potential across R_1 and R_2.",
        "A": "An ideal voltmeter has infinite internal resistance, so no current at all goes through it. Real voltmeters have very very high internal resistance."
    },
    {
        "video_name": "yE3eQ6q39f4",
        "Q": "At 6:50, how does the current \"know\" which path has less resistance without passing through there?",
        "A": "The same way the cars at the back of a traffic jam know that the cars way up in front have encountered a reason to slow down. The same way the back car of a train knows to slow down when the front car does. The same way the water in your sink knows that there is a clog in the pipe underneath."
    },
    {
        "video_name": "esJ5MbAHswc",
        "Q": "What is \"R prime?\" He said it around 3:24. Thanks in advance.",
        "A": "It s how you pronounce R ."
    },
    {
        "video_name": "QymONNa5C6s",
        "Q": "At 0:20 Hank mentions that water can exist in three forms of matter. Can it occur naturally as a plasma, or even as a plasma?",
        "A": "No, the bonds between the H and the O will break long before you reach plasma temperatures."
    },
    {
        "video_name": "QymONNa5C6s",
        "Q": "9:10 Given that a floating block of ice displaces the same amount of surrounding liquid water as it does when melted, surely if the ice at the North Pole sank, it would have to be denser than liquid water, thus having a smaller volume and (the immediate effect) would be for sea levels to fall?",
        "A": "The ice at the north pole is all floating ice. It can t sink. If it melts, sea level will not change. The ice at Antarctica is on land. If it melts, sea level will rise. If you are contemplating climate change, the biggest contributor to a rise in sea level is going to be the fact that warm water is less dense than cold water. As the water warms, it will expand, and this will raise sea level all over the world."
    },
    {
        "video_name": "QymONNa5C6s",
        "Q": "At 4:29 he said that water is a polar molecule. What is a polar molecule?",
        "A": "When i think of a polar molecule, i think of the word bipolar. A polar molecule has a slightly negative charge on one side; however, there is a slightly positive charge on the other."
    },
    {
        "video_name": "QymONNa5C6s",
        "Q": "at 6:46, what's aire?",
        "A": "Air (or aire) is the invisible gaseous substance surrounding the earth, a mixture mainly of oxygen and nitrogen."
    },
    {
        "video_name": "QymONNa5C6s",
        "Q": "Which lizard can walk on water? 3:25 I know that there are bugs and spiders that can (mosquitoes water bugs June bugs) but I did not know that lizards could too.",
        "A": "Basiliscus lizards, also known as Jesus Lizards can sprint across short stretches of water."
    },
    {
        "video_name": "QymONNa5C6s",
        "Q": "At 5:39, what liquid did he pour into the water?\n\nLorenzo, age 9",
        "A": "It looks to me like it is some kind of vegetable oil, perhaps olive oil. Certainly vegetable oils behave as shown when you pour them into water...but some other things do too."
    },
    {
        "video_name": "QymONNa5C6s",
        "Q": "Is heat capacity, as mentioned at 9:20, another term for specific heat or is there a difference between the two?",
        "A": "Usually heat capacity (thermal capacity) refers to the TOTAL thermal capacity of a particular mass of material, whereas the specific heat capacity refers to thermal capacity PER UNIT of mass. So, thermal capacity = c * mass."
    },
    {
        "video_name": "QymONNa5C6s",
        "Q": "Why is the bonding between O and H called hydrogen bonding, although even oxygen is involved? And at 5:05 , even when the bond between O and H breaks, and they bond to other substances, why is it still called HYDROGEN bond?",
        "A": "Hydrogen bonds are not only the bond between Hydrogen and Oxygen. Hydrogen bonds form when Hydrogen is bound to Nitrogen or Fluorine as well. It is not only Hydrogen and Oxygen. Since Hydrogen bonds are really quite weak, they can in their most basic form be considered an attraction between Hydrogen and a highly electronegative atom."
    },
    {
        "video_name": "QymONNa5C6s",
        "Q": "Why is water necessary for life? Why are scientists trying so hard to find water on other planets? At 1:35 he asks this question but doesn't really answer it. Does anyone know the answer?",
        "A": "all life on earth needs water to live and most if not all scientists agree that life originated in water"
    },
    {
        "video_name": "QymONNa5C6s",
        "Q": "At 6:33, there was something said about water being an \"element\". Soon, what are elements and Is there really such thing as an \"element\" and Is there a list of elements?(if it is a thing)",
        "A": "Water isn t an element; it is a compound. Elements are the building blocks , one could say, of everything there is. Elements are composed of atoms, which are the smallest units that still retain the properties of their respective elements. The Periodic Table of the Elements lists all the known elements, and some list the properties of those elements."
    },
    {
        "video_name": "QymONNa5C6s",
        "Q": "At 4:41, what is the strongest acid that we have ever created that Hank spoke of?",
        "A": "I don t know what acid Hank meant, but the strongest known acid is generally considered to be Fluoroantimonic acid, though there are some competitors. The ability to dissolve other substances is irrelevant to how strong an acid might be."
    },
    {
        "video_name": "QymONNa5C6s",
        "Q": "At 0:20 he says that water is the only substance that occurs naturally in all three forms.\nWhat does this mean?",
        "A": "He pretty much says that water is the only substance, it can occur as water(liquid), ice (solid), or fog, or steam(gas)"
    },
    {
        "video_name": "TGwXMpWvcYU",
        "Q": "What does EEG stand for?\n\nsaid at 0:33",
        "A": "EEG stands for electroencephalogram, which is using a machine to map the electrical activity in the brain. It was invented by a German doctor, Hans Berger, in 1924."
    },
    {
        "video_name": "8-ccnvn9DxI",
        "Q": "at 2:36, wouldn't carbonyl oxygen more likely to be protonated by acid instead of grabbing hydrogen from protonated ethanol?",
        "A": "@dohyh88: yes you re correct. It would most likely be protonated by the H2SO4 in this case, but this does not dismiss the possibility of it being protonated by the protonated ethanol instead. But it is much more likely for it to be protonated by the H2SO4"
    },
    {
        "video_name": "8-ccnvn9DxI",
        "Q": "At 11:06, how do you know that the reaction will happen twice? How do you know the butanol ( minus the Hydrogen, I don't know what the nomenclature for that would be) attaches twice to the the ethanal? This part has me confused.",
        "A": "You can t know in advance. But many chemists before us have done the reaction, so we know that it happens. We build on the results of those who have gone before us."
    },
    {
        "video_name": "8-ccnvn9DxI",
        "Q": "At 12:00, how is it \"ethyl(ene) glycol\" ? Where is the double bond? Isn't ene only supposed to be used in case of double bond and epoxides?",
        "A": "That is a trivial name. It means the glycol made from ethylene. The correct name is ethane-1,2-diol."
    },
    {
        "video_name": "8-ccnvn9DxI",
        "Q": "at 6:55, why is water an excellent leaving group?",
        "A": "Because there is +1 Formal Charge on the Oxygen atom along with two Hydrogen atoms... thus its ability to leave from the intermediate favors the furthering of reaction without any disturbances."
    },
    {
        "video_name": "8-ccnvn9DxI",
        "Q": "At 13:40, Jay meant a cyclic ketal, not a cyclic acetal, right? Because the starting molecule was ketone, not aldehyde.",
        "A": "Right. At one time, both acetals and ketals were called acetals, but they now have separate names."
    },
    {
        "video_name": "8-ccnvn9DxI",
        "Q": "What is a hemiketal (at 5:39) ?",
        "A": "A hemiketal is a compound that has an alcohol and an ether group attached to the same carbon. Its general formula is R\u00e2\u0082\u0082C(OH)OR."
    },
    {
        "video_name": "8-ccnvn9DxI",
        "Q": "After step 3, around 5:30, the molecule has an OH attached. Why does the molecule have to go through the dehydration step? Isn't the molecule stable as it is after step 3?",
        "A": "The formation of a hemiketal is highly reversible: a hemiketal easily converts back to a ketone plus alcohol. The presence of acid improves the leaving capacity of the hydroxyl group and enables its substitution with an alkoxy group (-OR). The ketal (a diether) is much more stable."
    },
    {
        "video_name": "8-ccnvn9DxI",
        "Q": "at 11:17 , why is it that in the first problem, its acetal product's carbon is lacking a hydrogen? all it has are 2 OR\" and a R' substituent?",
        "A": "The H is there. Remember that in bond-line structures you don t show the H atoms on C. Any unfilled valences on C are assumed to consist of H atoms."
    },
    {
        "video_name": "15zliAL4llE",
        "Q": "At 5:09, why did Sal write acceleration x time?",
        "A": "He must have wanted to know a change in velocity"
    },
    {
        "video_name": "15zliAL4llE",
        "Q": "At 1:36, I hear velocity is a \"scalar quantity.\" Isn't velocity a vector quantity?",
        "A": "velocity is a vector quantity, it has both a magnitude (the number), and a direction. Speed is a scalar, and equals the magnitude of the velocity."
    },
    {
        "video_name": "15zliAL4llE",
        "Q": "At 2:02, would gravity be positive because its going with gravity not against it?",
        "A": "No. Gravity is always directed towards the centre of the (body) Planet. If this is the opposite direction to velocity or displacement then it will be negative."
    },
    {
        "video_name": "15zliAL4llE",
        "Q": "at 2:41 sal says the formula is d=vt but should not it be d=ut-1/2(gt)^2 because it has an acceleration in y direction?",
        "A": "There s no acceleration in the x direction, so the horizontal displacement is = v*t."
    },
    {
        "video_name": "15zliAL4llE",
        "Q": "At 3:13, why did Sal use distance with velocity?",
        "A": "In this problem, distance traveled = displacement covered by the object."
    },
    {
        "video_name": "15zliAL4llE",
        "Q": "At 6:13, if the acceleration due to gravity is -10 m/s squared, how does it take 10 seconds to reach a final velocity of 100 m/s? Shouldn't it only take between 4 and 5 seconds? Sal's falling faster the whole time he's in the air.",
        "A": "He s talking about the velocity, not the the vertical distance. The velocity increases by 10 meters every second, so the velocity after 10 seconds is 10x10 = 100 m/s. If it was the distance though, it would indeed be far greater than 100 meters."
    },
    {
        "video_name": "15zliAL4llE",
        "Q": "At 2:56, why can't we just use (Vf)^2=(Vi)^2 +2gs? Even the acceleration is constant.",
        "A": "Does it give you the same answer? Then you can use it. There is often more than one way to solve these problems."
    },
    {
        "video_name": "15zliAL4llE",
        "Q": "Hi Sal, at the point of 5:28, you wrote -100 . 0. I dnt know how I can get the right dot in there, but you got what i'm saying. is the dot addition or multiplication because you wrote after that that = -100m/s. it seems like you said you kinda jump from there. can you please explain that part. thanks you and God bless bro",
        "A": "At 5:28, I assume that you are referring to the point where Sal substitutes in the values for the final velocity minus the initial velocity. After substitution, Sal is not adding or multiplying. He is actually subtracting. What he actually wrote is -100-0."
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "At 7:09 isn't the momentum of the \"cartruck\" equal to the initial momentum (of the car and the truck) only when the momentum is conserved? what happens when the momentum is not conserved?",
        "A": "Momentum is always conserved. That s the point."
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "At 7:54 in the video, how does sal make 3000V = 9000? And then why does he divide everything by 3000?",
        "A": "M*V = P , he is trying to get the velocity for the new mass (which is the mass combined -> 3000kg) and like no force is acting on the sistem, the momentum P stays the same (9000 kgm/s). He only solved for V"
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "At 8:07, Sal says that there is no change in momentum from before to after the collision because there are no net forces acting on the system. Does that mean there are no external forces at all, and if so, how is that determined? What would be an example of net forces? Thank you!",
        "A": "He really means no external forces. You could of course have two, perfectly countervailing external forces that would add up to no net external force."
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "doesn't change in velocity occur over time? So shouldn't it just be (delta)V rather than (delta)v/(delta)T? 2:06",
        "A": "velocity does change with time, but the thing is... how much time does it take? that is what is represented by delta v. the difference in time intervals- to know how much time it took from the period of the initial velocity to that of the final. Besides, that is how we calculate acceleration. its important to indicate whether it took 5 seconds for a body to reach 10m/s from rest or whether it took 10."
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "At 8:24 how did he get from 3000 to 3 meters per second? What units was the 3000 in?",
        "A": "Divide both sides by 3000 to solve for v The 3000 is kg. v is in m/s 9000/3000 = 3"
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "in the video during 2:05 to 2:18, the equation was re-written as Force = change in (m * v)/change in time. When the mass was assumed to be constaint ( change in mass = 0), then force will equal 0 assuming a change in time that doesn't equal 0. But in the equation above, Force doesn't equal 0 assuming that mass or acceleration doesn't equal 0. I don't understand how that step is possible.",
        "A": "No, you are misunderstanding the math of derivatives here. F = d(mv)/dt. If mass is constant, then this becomes m*dv/dt. Which is m*a. F = ma is familiar, right? If you want to know why d(mv)/dt becomes m*dv/dt then review the product rule for derivatives. Sal has a vid about it."
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "at 3:41 how does Impulse have units kg*m/s AND N*s as the N*s= mass x acceleration x time ?\nand kg*m/s=mass x velocity",
        "A": "Because a Newton, N, is defined as kg*m/s^2. When multiplying by a second, you get kg*m*s/s^2, which cancels out into kg*m/s. Also, acceleration is defined as velocity over time, so (m/s^2)/s, giving m/s."
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "At 4:35, Sal mentions that the car is moving at 9 m/s. He later on says that the velocity is zero. I assumed that the 9 m/s was the velocity. I'm a bit confused.",
        "A": "9m/s is the velocity it is the truck s velocity is zero."
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "at 1:56 what does the triangle mean",
        "A": "The triangle known as delta is a greek letter that is used to represent variation."
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "At 8:09, what is a net force ?",
        "A": "the net force is the vector sum of all forces acting on the body."
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "At 7:40, why does he say that momentum has to be conserved in relation to the 3000kg and 9000kg m/s. I do not understand that part/",
        "A": "Momentum is a conserved quantity. Meaning: the momentum of any closed system will not change. There s nothing much to understand here. It s just the way, nature works."
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "at 3:53, what is meant by consevation of momentum?",
        "A": "It s a basic principle of physics. It s the idea that for a closed system, the total amount of momentum in that system is unchanged at the end of an interaction from the beginning of the interaction."
    },
    {
        "video_name": "XFhntPxow0U",
        "Q": "What if you need to find what the velocity of the car only, if they are both going at each other, unlike 4:54?",
        "A": "Well mass and weight are very different things including density. density is how much mass there is in a certain amount of space. And mass is the amount of stuff in something. Unlike Weight where that is how strong you are pulled down onto the earths surface. Good Question!"
    },
    {
        "video_name": "jQ2nD8ZGeEw",
        "Q": "at 1:00 he says distance vector, isnt distance vector and displacement the same??",
        "A": "Precisely, Distance is Scalar Quantity(Without Direction). So, Distance Vector(With Direction) gives you the Displacement. He Could have said said Displacement instead of Distance Vector."
    },
    {
        "video_name": "jQ2nD8ZGeEw",
        "Q": "At 3:24 and at 5:25 How does the current have direction if it's a scalar quantity?",
        "A": "Scalars can be negative or positive. The current can flow in only two directions. It s not a vector because it can t go in any direction except along the wire, either way. In other contexts, current can be a vector field but not when it s just a simple situation like a current assumed to flow straight through a wire."
    },
    {
        "video_name": "jQ2nD8ZGeEw",
        "Q": "@ 7:00 \"Holes..Well I don't wanna get into that.\" What did he mean by that. I am very very curious now. perhaps a mathematical abstraction?",
        "A": "Actually it is visualization exercise. Have you ever seen a long fluorescent light that is partially lit up where it looks like dark rings are traveling from one end toward the other. If so, concentrate on looking at the lighter rings between the dark ones. They will appear to be moving in the opposite direction. The holes will appear to move in the opposite direction of the electron flow in your circuit."
    },
    {
        "video_name": "Bc9bhLk_AhI",
        "Q": "at 1:00 mins Brown eyes are assumed to be dominant. Is that just an example? Or can Blue eyes be dominant and Brown eyes be recessive?",
        "A": "Brown eyes are dominant and blue eyes are recessive. In world population, 55% are brown eyes and only 8% of blue eyes."
    },
    {
        "video_name": "Bc9bhLk_AhI",
        "Q": "At 6:07 Sal says that p is the frequency of the dominant allele and q is the frequency of the recessive allele. Is this always the case/convention? Or can p denote the frequency of the recessive and q the frequency of the dominant?",
        "A": "No that s always the case. p denotes dominant alleles and q denotes recessive alleles. There is a thing unrelated to this question, but it is helpful at times. Always remember that things in biology exist in a particular way and you study that way . It is never the other way round that you are studying the way and that s why it is in that way . Similarly the above concept of p and q is the exactly the way you should read and understand it. I hope that s clear.:)"
    },
    {
        "video_name": "Bc9bhLk_AhI",
        "Q": "at 5:24 mins, are there four GENES or four ALLELES?",
        "A": "There are 4 genes. Genes are the actual pieces of DNA we are considering. Genes come in different types, called alleles. There were two types of allies, Capital B, and lower-case b. So for the first gene, the allele type was B. For the other 3 genes, the allele type was b."
    },
    {
        "video_name": "12lYK1U4dyY",
        "Q": "At 1:35 Sal mentions a torr. What is this?",
        "A": "You will be better off using atm as a unit for pressure. Roughly one atm is 760 torr (760 mm of mercury)"
    },
    {
        "video_name": "b1QFKLZC11U",
        "Q": "At 6:00, Sal says that the radius is 1.25 m. But since the proton is moving at 1/5th the speed of light, shouldn't we take relativity into account? The mass of the proton actually increases as per the formula\nM=Mp*(1-(v^2/c^2))^-0.5\nAnd the corrected radius is actually 1.28m. Isn't it??",
        "A": "I didn t check your math but it looks right. He probably should have used a lower velocity. As you probably know, relativistic answers are always more correct than Newtonian ones, but the difference is usually too small to worry about and it makes it a lot harder to teach and learn basic physics. All those problems where we ignore friction are also wrong , right?"
    },
    {
        "video_name": "b1QFKLZC11U",
        "Q": "at 10:20 an electron will follow the opposite path , against the right hand rule?\ni mean for proton thumb directed downwards , it will do the same for electron right? but electron goes the other way around!\nhow do i tell my right hand that i'm talking about electron not the proton?",
        "A": "Even for the electron the force using the right hand rule is downwards but since electron is negative in charge, the force ends up being negative. So the downward force according to right hand rule is negative of the force experienced by the electron so the positive direction is opposite to it hence for electron the direction is reversed and it ends up going other way around."
    },
    {
        "video_name": "b1QFKLZC11U",
        "Q": "Sal did an awesome job on this video! However I do understand the centripetal acceleration formula but I do not understand why at 2:42 he put an M in front of the MA formula? I do know that he may have broken up the force in MA but where did the A go? Thank you! Also please explain why he made it equal the force after denoting the MA, thank you!",
        "A": "because centripetal force=ma and a=v^2/r so F centripetal =(mv^2)/r F magnetic=F centripetal therefore (mv^2)/r=QvB simplifying (mv)/r=QB"
    },
    {
        "video_name": "b1QFKLZC11U",
        "Q": "Durin Magentism4 Sal talks about at 5:14, About centipetal force pls explain that ?",
        "A": "it is a force that makes a body follow a curved path: its direction is always orthogonal to the velocity of the body, toward the fixed point of the instantaneous center of curvature of the path. Centripetal force is generally the cause of circular motion. In simple terms, centripetal force is defined as a force which keeps a body moving with a uniform speed along a circular path and is directed along the radius towards the centre."
    },
    {
        "video_name": "b1QFKLZC11U",
        "Q": "At 9:15, how can the magnitude of the force be a negative number? I thought that magnitudes lacked direction (similar to an absolute value, neither positive nor negative). Of course, I know we can give them direction, but shouldn't the notation for the force be without the absolute value signs if we assign it a direction?",
        "A": "Negative just tells you that the force is in the opposite direction from whatever direction you have defined as positive."
    },
    {
        "video_name": "xJf6pHqLzs0",
        "Q": "at 3:16 it is little bit confusing.\nWhen the colder one is gaining heat and hotter one is losing heat, T isn't constant.( i mean it is variable)\nIn this case, I think we should use integration instead of just deviding them with their own temperatures.\nIs the T is very specific temperature? So can it be treated as constant?",
        "A": "That is a great question! In this example, we are assuming they are both infinite reservoirs (which is impossible in real life, but maybe they are just really really big). Thus, the temperature of the cold reservoir and the temperature of the hot reservoir are constant (or have changed very very very little) even after the heat transfer."
    },
    {
        "video_name": "xJf6pHqLzs0",
        "Q": "At 1:49, how is delta S >= 0 when the second law says that entropy is always increasing and therefore delta S>0 instead of ever being equal to 0? This would imply a perfect natural process.",
        "A": "Entropy generated is zero for any reversible process, because S is the function of state. (so you can theoretically change volume and pressure, then go back to initial state and the change in entropy will be 0) For the whole system (universe)\u00e2\u0096\u00b3S =0 if the transformations are quasi static. I think Sal didn t make this video very clear."
    },
    {
        "video_name": "xJf6pHqLzs0",
        "Q": "at 4:05, calculation of entropy change of this irreversible process is done by the Q/T formula which only works for reversible processes?",
        "A": "Each infinitesimal step of heat transfer gives a new pseudo equilibrium state. Theoretically, the reaction can also go in the reverse direction at each state, (although the probability is extremely small that this will happen). Thus, the use of the definition (dS = Q/T) is ok in this case."
    },
    {
        "video_name": "qzNokxFaAAg",
        "Q": "At 6:20 you show that the for faster speeds the ct' axis converges with the photon. I was wondering, do they actually meet at full light speed? And if so, since the path of the photon would cover the entire axis, does that mean a photon is really everywhere at once?",
        "A": "Yes, from the photons perspective, it is already everywhere that it ever was or ever will be. The universe has no depth for a photon."
    },
    {
        "video_name": "qzNokxFaAAg",
        "Q": "I don't understand what Sal says at 5:50 - \"In one light meter the thing travels a meter\" How does this prove that light travels at light speed in Sally's frame of reference?",
        "A": "From experimental results we have been able to show that regardless the relative velocity of two inertial frames of reference the speed of light is always measured as the same. So since Sally is moving at a constant speed she is in a inertial frame of reference and will measure the speed of light to be the same as Sal does."
    },
    {
        "video_name": "tk-SNvCPLCE",
        "Q": "Out of curiosity, what is the name of the molecule he's drawing at 4:10?",
        "A": "1-bromo-1-fluoroethane"
    },
    {
        "video_name": "o52BW3B4efc",
        "Q": "At 5:32 the video states that- V1/T1 = V2/T2 (assuming that the provided pressure remains constant throughout the experiment) But during the experiment, while applying heat, don't you think that as the volume of the gas increases, the pressure will decrease? In that case how does the pressure remain constant?",
        "A": "Mahera, keeping the pressure constant. Means that the pressure that we apply or the environment should remain the constant initially and finally...the pressure you re talking about is the pressure exerted by the gas particles"
    },
    {
        "video_name": "o52BW3B4efc",
        "Q": "at 34:22 0 K is given with a symbol is it that we should write in that symbol?",
        "A": "That symbol is Phi...it is often used for denoting zero...so it really is just 0K or Absolute Zero:)"
    },
    {
        "video_name": "o52BW3B4efc",
        "Q": "6:30 - Why is the conversion necessary to go from C to K?",
        "A": "The temp cannot be a negative value which is crucial for the calculations in ideal gas law. So you can always convert C to K by adding 273."
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "at 7:02, why are field vectors the same size no matter how far away they are from the field? doesn't the field become weaker farther out?",
        "A": "He specifically says just for the simplicity of it, let s assume a constant electric field"
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "At 7:05, What did you mean when you said that the electric field points upwards?",
        "A": "A small positive charge placed at that point will experience an upward force."
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "At 7:00 Sal said that we the size of the vectors stay the same no matter how far we get from the source. How is this possible?",
        "A": "If you have a large charged plate and you are not too far from the surface and not too close to the edge, the field is close to uniform. It s an approximation. It s similar to how we assume g is constant, even though it s not either."
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "in 3:56, it say potential energy needs a reference point . if the object is on earth , why dont we always take the core of the earth as reference point ?",
        "A": "It makes it very inconvenient to calculate the potential difference of two points a small distance apart when the center of the earth is 6,371,000 meters away."
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "12:00 why doesn't the electric p.e get converted to gravitational p.e?",
        "A": "Electric potential energy is not converted to gravitational p.e but it is converted to kinetic energy by repulsion between the two identical charges."
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "\"at around 12:00 Sal said force of 5N will be needed to move the 2C charge up\". should it be 10N force instead of 5N",
        "A": "it will be 5 N as the value of electric field is specified as 5N/C"
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "Around 3:28, Sal states W=Fg*h=PEg. So, as work increases, so does the potential energy, right? However, can work can be done without a gain in potential energy? i.e. if i push a cart along a flat surface. My main question is, does an increase in work only result in an increase in PE if that work is done against the direction of a field? i.e. against earth's gravitational field, or a proton moving towards a positively charged plate.\nI greatly appreciate any help with this matter.",
        "A": "Yes, when you do work against a field, you get PE."
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "At 11:13, wouldn't the electric field continue to move the 2C of charge away?\nIt doesn't make sense to me why the 2C of charge would move back 3 meters and stay there. If you're placing a positive 2C of charge in an electric field, wouldn't it just keep moving away? How could 2C of charge just stand still in an electric field?",
        "A": "Ooops, I just read the comment below. I think my question is answered."
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "at 3:40,sal says that the work done is equal to force of gravity into h,but shouldn't the force be a little bit more if we want to lift it upward against the force of gravity?",
        "A": "No, because whatever little extra you put in to get it moving, you get back when you stop it, so that work cancels out. Alternatively, imagine that the object is already in motion when it passes your starting point."
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "At 12:00 Sal shows that the work required to shift the charge of 2 coulomb from position p1 to p2 would be 30joules. Do we ignore the PE due to gravitation when we talk about electricity, or do we take that in account also while solving problems?",
        "A": "when you were doing gravity problems, did you ignore the electric PE? Yes, right? Same now. You would need to know the mass of the charge to do anything with the gravitational force."
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "At 7:49 sec Sal had chose a positively charged particle. What if we kept a negatively charged particle in the same electric field. Would it simply move in the opposite direction or does something else happen",
        "A": "It would move in the opposite direction."
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "hello !\nat 7:00 it want be of same size\nm i right.\nas we go far the vector sign goes on decreasing.\nrefer next Sal video , it is in my support .\njust explain me i am bit confused.",
        "A": "This is an infinite uniformly positively charged plate. As explained in the previous videos , it is a uniform electric field (which is hypothetical), so the vectors from the field have equal magnitude even at longer distances. Refer 6:12 and previous videos. Cheers! ;-)"
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "Positive charge diverges field in every direction, but at 6:53, the direction of electric field is directed upwards only....Why?",
        "A": "Because this is not just a single point charge, but a plate full of charges."
    },
    {
        "video_name": "elJUghWSVh4",
        "Q": "at 6:12 , what is an ' infinitely charged uniform plain'?",
        "A": "This is answered in the previous video. Imagine a flat 2 dimensional plate that stretches in every direction forever. Ie A square with a width of infinity and a height of infinity. Then imagine that this entire plate has a charge, lets say +1. And this charge is uniform across the plate such that if you were to choose any individual point on the plate, its charge would be +1."
    },
    {
        "video_name": "Akpl6wmAfZQ",
        "Q": "he says at 3:31 that we don't know how life happened, but i thought that the Stanley Miller experiment proved how life arose. Is this true?",
        "A": "The Miller experiment shows how amino acids can be created. Amino acids are essential to life on earth, but making an amino acid is not the same as creating life. We do not know how the first living organism came into existence."
    },
    {
        "video_name": "Akpl6wmAfZQ",
        "Q": "At 3:12; As Hank says... this probably can't be answered, but HOW do organic molecules form from chemicals? Could we re-create this scenario in a lab to form new organic molecules/life?",
        "A": "Scientists have tried to recreate this scenario. You can look up the Miller-Urey experiment, it has some amazing results. However, they don t know all the details about how life first formed."
    },
    {
        "video_name": "Akpl6wmAfZQ",
        "Q": "At 3:41 what is a bio membrane?? Also, at 4:05 what is a DNA nucleotides?",
        "A": "He was referring to a phospholipid bilayer and the answer to your second question is that nucleotides are the monomers that make up the macromolecule DNA. A nucleotide is made up of a nitrogenous base that is accompanied with a phosphate and a 5 carbon sugar."
    },
    {
        "video_name": "DD58B2siDv0",
        "Q": "At 7:08, what makes the velocity at time 0 1m/s?",
        "A": "He just says that the velocity at t=0 is 1m/s as and example so he could get the velocity wherever on the graph."
    },
    {
        "video_name": "DD58B2siDv0",
        "Q": "At 10:53, David mentions that the blue triangle will have negative area. I understand that we are dealing with negative acceleration values, but is it actually possible to have negative area? Thanks!",
        "A": "Usually negative area would be totally meaningless (try to imagine what a square with negative area would look like!). But it makes a little sense if you treat the negative sign as a direction, like with velocity. Here the fact that the area is negative means that the area is below the x-axis, so saying it s negative is like saying its direction is below the x-axis."
    },
    {
        "video_name": "DD58B2siDv0",
        "Q": "@7:30 , at time t=0 how daisy had v0 =1 m/s ?",
        "A": "she was already moving at the time the stopwatch was started."
    },
    {
        "video_name": "DD58B2siDv0",
        "Q": "At 5:25 , can you explain what does he mean\nthanks",
        "A": "He is just trying to tell that when we add up the area of all those small rectangles, their sum gives the area of the triangle. Therefore we need to find the area of the triangle. Thus we must apply the area of triangle formula. Which is why we divide it by 2. Hope that helps :)"
    },
    {
        "video_name": "DD58B2siDv0",
        "Q": "at 8:17 where did the -1 m/s come from? i would think it was 0, but i had no idea where the -1 came from. and why is the acceleration negative? what does that mean, why does it go under the horizontal axes?",
        "A": "Actually, this is an arbitrary number. In this case, David is assuming we start on 1 m/s, to show how we can add acceleration to velocity."
    },
    {
        "video_name": "DD58B2siDv0",
        "Q": "At 1:24 in the video, David says that we can't tell if Daisy is speeding up or slowing down. But if an object is accelerating in the positive direction at 2 m/s, then isn't it automatically speeding up?",
        "A": "No, what if the velocity is negative, and the the acceleration is positive? Then the object is slowing down until it turns around and has positive velocity."
    },
    {
        "video_name": "5B1i26dUwME",
        "Q": "At 8:35 , Sal told that we got a relation between p,v,R,t,n and he told that in the relation n is no.of molecules but he clearly wrote it is no.of moles , so which one is correct the no.of molecules or moles?",
        "A": "a mole is a number, so n is the number of molecules, measured in units of moles."
    },
    {
        "video_name": "5B1i26dUwME",
        "Q": "At 9:06, how does one get joules multiplying pascals with meter cubed?\npa*m3\n=newton/m2 * m3\n=newton*meter\nHow?",
        "A": "You just showed how."
    },
    {
        "video_name": "5B1i26dUwME",
        "Q": "At 8:55 we have a mole as a unit while it is also a number. Isn't this problematic? Thanks!",
        "A": "I guess I just shouldn t interpret it as a unit but as a number although it is at the right ."
    },
    {
        "video_name": "5B1i26dUwME",
        "Q": "at 1:45, how is a mole useful for counting atoms? I thought we couldn't see atoms. And how was this number calculated?",
        "A": "Atoms are really small, so if you want to count them, you need a unit that is really big. How is a dozen useful for counting donuts? Same idea. When you do chemistry, you need to know how many atoms are reacting with how many other atoms. It does not matter whether you can see them or not."
    },
    {
        "video_name": "5B1i26dUwME",
        "Q": "At 1:39 what is a mole",
        "A": "a mole is a unit of measurement in chemistry which is used to find number of atoms or molecules, this is done using the avagardo number i.e. 6.023 * 10^23"
    },
    {
        "video_name": "5B1i26dUwME",
        "Q": "at 5:33, Sal says \"If we had a mole of niobium -- if we had 6.022 times 10 to the 23 of niobium...\" what does he mean by that? does he mean 6.022*10^23 atoms of niobium or 6.022*10^23 molecules of niobium?? what is it?",
        "A": "A mole is 6.02*10^23 of anything. So he probably meant atoms of niobium."
    },
    {
        "video_name": "okjt-OZY1JU",
        "Q": "At 4:30, Sal says that the area of the trapezium or trapezoid is average height times the base, but from I recall that the area of a trapezium or trapezoid is (upper bound+lower bound)*height*0.5? Where does the area formula for the trapezium Sal had stated come from?",
        "A": "Sal chose to divide the trapezoid into a square and a triangle - both easy to compute the area in your head. He could have used your formula, but then we would have to imagine the trapezoid laying on its side since what you call the upper and lower bound need to be parallel to each other. In that case the upper bound is 20, the lower bound is 10 and the height is 40 giving (20+10)(40)(1/2) = (30)(40)(1/2)=(30)(20)=600, the same result."
    },
    {
        "video_name": "ygXkdSKXQoA",
        "Q": "I couldn't understand how are the groups are being arranged in alphabetical order and in the last in 7:29 on what basis they are arranged!How can dicyclobutyl be arranged before cyclopropyl!If b and p are considered then what can be said in case of diisobutyl!As diisobutyl consists b then it should have been placed before cyclopropyl!",
        "A": "When putting the substituent groups in alphabetical order, ignore the prefixes like mono, di, and tri etc. So we are looking at cyclobutyl, vs cyclopropyl, vs isobutyl. The alphabetical order of these is: cyclobutyl, cyclopropyl, isobutyl. Now add back in the prefixes, so that it s dicyclobutyl, cyclopropyl, diisobutyl."
    },
    {
        "video_name": "ygXkdSKXQoA",
        "Q": "At 4:10 when he numbers the carbons starting from the cyclobutyl on the right and then at 5:02 again when he numbers the carbons starting from the cyclobutyl on the left, why are they numbered clockwise? Could you have numbered them counterclockwise? Would numbering counterclockwise give you the same outcome (left cyclobutyl being more preferable)?",
        "A": "You could. The instructions as per IUPAC is that we use set of numbers that has the lowest number at the first point of difference. That could be clockwise or anticlockwise depending on the molecule. With regards to this one, there is another numbering that also uses 1 2 4 5 7 that Sal missed, but when we have such a tie, we break the tie by assigning the alphabetically first group as being on carbon #1 So that\u00e2\u0080\u0099s why he chooses that cyclobutyl group to start numbering."
    },
    {
        "video_name": "s38l6nmTrvM",
        "Q": "At 00:16, Sal says that T=r X F. I don't understand how radial distance is a vector; I've always seen it treated as a scalar. How does distance have direction?",
        "A": "how is radial distance a vector?"
    },
    {
        "video_name": "s38l6nmTrvM",
        "Q": "in 6:23, how do we now that it is r cross F, and not, F cross r? Surely it doesn't come for the equation for torque ( T=r F ). Because T= F r , is also valid. Is it just whatever vector (r or F) is closest to the pivot point?",
        "A": "Cross product is not commutative, so r cross F is different from F cross r. The torque is defined as T = r x F, so the vector from the pivot to the force should always be the first term."
    },
    {
        "video_name": "s38l6nmTrvM",
        "Q": "at 0:38 how is the radius a vector?",
        "A": "A vector is something given magnitude (length) and direction. A radius meets both of these criteria."
    },
    {
        "video_name": "jzcB3faNdq0",
        "Q": "At 2:28, why don't we split up the sorbic acid into its cation and anion parts. Isn't that what we do to aqueous compounds?",
        "A": "Yes and no. Potassium Sorbate breaks up int K+ and C6H7O2- ions in solution. Then the H+ ions from HCl attach to the C6H7O2- to make sorbic acid. Since Sorbic Acid is a weak acid a tiny little amount will ionize but it will mostly stay together in water."
    },
    {
        "video_name": "jzcB3faNdq0",
        "Q": "This question is a general question: Once we have separated all the ions, as in 2:18, how do we know which one will go together with which? How do we know that H+ rather goes with C6H7O2- than with Cl-? Does it have something to do with electronegativity, or with the fact that HCl is a strong acid? Would it be different with a weak acid?\n\nThank you so much!",
        "A": "It s that HCl is a strong acid, So even if it reformed it would split up again immediately. Whereas HC6H7O2 is a weak acid so it will not split up immediately."
    },
    {
        "video_name": "jzcB3faNdq0",
        "Q": "At 3:53 wouldn't the equilibrium arrow be used in the net ionic equation since sorbic acid is a weak acid?",
        "A": "Yes it technically should have been but the fact is that since a strong acid is being mixed with a weak base, virtually all will react."
    },
    {
        "video_name": "jzcB3faNdq0",
        "Q": "at time 2:40, how do you know that sorbic acid is aqueous? It isn't a strong acid",
        "A": "(aq) means dissolved in water. It doesn t have to be a strong acid to be dissolved in water. Dissolve and dissociate do not mean the same thing when you re talking about acids. Dissociate means splitting up into H+ and the conjugate base, dissolve means that it forms a solution when put in water rather than staying as a solid at the bottom of the beaker or whatever."
    },
    {
        "video_name": "jzcB3faNdq0",
        "Q": "At 2:35, why didn't the K react with Cl to form Potassium Chloride?",
        "A": "Yousef Sheikh, Potassium Chloride is a salt that readily dissociates into its respective ions. If it were to hypothetically react and form Potassium Chloride, it would break back down into K and Cl."
    },
    {
        "video_name": "jzcB3faNdq0",
        "Q": "At 2:27, Isn't Sorbic Acid soluble in water? Why was it not broken into spectator ions to be canceled out like the K and Cl?",
        "A": "It is soluble in water, but as it is a weak acid it does not dissocate very much. Weak electrolytes do not get separated in ionic equations."
    },
    {
        "video_name": "jzcB3faNdq0",
        "Q": "At 3:14, what is a net-ionic equation?",
        "A": "A net ionic equation only shows ions that take part in the chemical reaction, any so called spectator ions are left out."
    },
    {
        "video_name": "q--2WP8wXtk",
        "Q": "Why did Jay not specifically tell which atom was the largest or has the largest radius as he told the smallest atom was helium? (video 9:20-9:24 approx.)",
        "A": "Some of the Largest atoms are Radio-active, so it is quite difficult to actually measure their radius, and it cannot be determined which one is the largest. But according to the Periodic Table trends, Francium is predicted to be the largest."
    },
    {
        "video_name": "q--2WP8wXtk",
        "Q": "(at 4:22) could u tell me about that Vander walls force?",
        "A": "Van der Waals forces is another name for all intermolecular forces. There will be a video on this."
    },
    {
        "video_name": "q--2WP8wXtk",
        "Q": "At 8:11, Sal says there Is more of a Coulomb attraction, what is that?",
        "A": "Coulomb attraction, as dictated by Coulomb s Law, is the attraction between positively-charged particles and negatively-charged particles. This often refers to the force that pulls electrons towards an atom s nucleus."
    },
    {
        "video_name": "q--2WP8wXtk",
        "Q": "at 3:24 what is the Van der Waals radius?",
        "A": "It is a measure of the size of an atom. It refers to half of the distance between the nuclei of two unbonded atoms when the atoms are as close together as possible. If they try to get any closer than this then repulsion between electrons pushes them apart again."
    },
    {
        "video_name": "q--2WP8wXtk",
        "Q": "At 1:30 How are Scientists able to calculate the number of electrons if they exist in a probability space?",
        "A": "We can detect their charges, even if we cannot isolate them to an exact location."
    },
    {
        "video_name": "q--2WP8wXtk",
        "Q": "Around 8:20 Sal said helium is the smallest atom because of the +2 and the 2 electrons and the electrons are being pulled in because of the +2. Isn't that the sane for hydrogen because there is +1 and 1 electron. So the size would be the same because they are pulling in their same amount?",
        "A": "Atomic radius, or atomic size decreases across a period. This is because the electrons added are still on the same electron shell, so the added protons can easily hold the atoms closer. If you were comparing hydrogen and Lithium, Lithium would be larger because its 3rd electron is placed in another shell and therefore farther from the proton s attraction."
    },
    {
        "video_name": "q--2WP8wXtk",
        "Q": "At 2:44, Sal calls two atoms are not yet part of the same MOLECULE. I have also heard people saying that two atoms in the same molecule are called DIATOMS. Which are two atoms of the same element that are bonded called: molecules, or diatoms? Please verify this for me. I AM CONFUZZLED!",
        "A": "Molecules are any particle that contains two or more atoms. Diatoms are a type of molecule that contain two atoms. Hope that helps."
    },
    {
        "video_name": "q--2WP8wXtk",
        "Q": "At 6:11 Sal mentions Coulom force, what is that?",
        "A": "Coulombic force also known as electrostatic force, is the attraction between oppositely charge particles such as electrons, protons and ions. Named after the scientist Charles Ausgustin de Coulomb, which is also used for the unit for the fundamental unit of electrical charge. Thus, from Coulomb s law or Coulomb s inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles."
    },
    {
        "video_name": "q--2WP8wXtk",
        "Q": "At 2:30 two ways to calculate the atomic radius are listed. Which one should I normally use?",
        "A": "It really depends on why you are trying to determine the radius. For example, if you were trying to find the size of a gaseous atom in a very dilute concentration, then you d probably use the van der waal s radius because you d assume that the atom wasn t interacting with other atoms."
    },
    {
        "video_name": "q--2WP8wXtk",
        "Q": "At 6:10, Sal mentions the Coulomb force. I'm not entirely clear on this. Is Coulomb force merely magnetic attraction? Or is it something more abstract or complicated in understanding?\nCould someone please help?",
        "A": "Coulomb force is the electrical attraction between positive and negative charges"
    },
    {
        "video_name": "pMoA65Dj-zk",
        "Q": "At 6:46 min in the video Sal shows the simplest way to write a chain carbon.\nHow is that the simplest way ?",
        "A": "Do you know of a simpler way? Just confirming, the simplest way to represent the structure of propane is the line diagram he drew at 5:35 that looks basically like a ^"
    },
    {
        "video_name": "pMoA65Dj-zk",
        "Q": "@5:31 with the condensed structure isn't it bad to say CH3-CH2, doesn't that imply that a hydrogen is bonded to the first carbon and the second? would it be better to just put CH3CH2?",
        "A": "The condensed structure is only a way of writing a formula on a single typewriter line. You have to use your knowledge of structural formulas to figure out which atoms are bonded to which. However, many chemists write the formula as H\u00e2\u0082\u0083C-CH\u00e2\u0082\u0082 or H\u00e2\u0082\u0083CCH\u00e2\u0082\u0082 for that very reason."
    },
    {
        "video_name": "pMoA65Dj-zk",
        "Q": "at 5:30 how do you know what atom bonds to the ends of the carbon atoms",
        "A": "Unless there is another molecule drawn at the end of the line, assume that the carbon is bonded to hydrogen atoms."
    },
    {
        "video_name": "pMoA65Dj-zk",
        "Q": "the line diagram mentioned in 7:24 is also known as skeletal formula.",
        "A": "yes."
    },
    {
        "video_name": "pMoA65Dj-zk",
        "Q": "At 4:50 , Sal mentioned that we could write propane as CH3-CH2-CH3. So would it be wrong if I just write it CH3-C2H5 or C2H5-CH3?",
        "A": "It d be unusual. C3H8 is the overall molecular formula, while CH3CH2CH3 gives you a good idea of the structure. Sometimes parts of molecules are abbreviated while others are expanded, but you wouldn t do it for something as simple as propane."
    },
    {
        "video_name": "pMoA65Dj-zk",
        "Q": "I couldn't help but notice at 4:25 he said he said we would see the reason behind the similar names, and yet he never told us. Is this just because they are both only C and H molecules?",
        "A": "generally compounds which follow the same basic structure sound similar ... for example structures of the form C(n)H(2n+2) are called alkanes .. you can substitute the n for any integer 1,2,3,4 etc this just shows they are of the same general group"
    },
    {
        "video_name": "pMoA65Dj-zk",
        "Q": "this is not a question but at 5:49 the diagram is mistakenly labelled it is in fact a skeletal diagram",
        "A": "The two terms are interchangeable."
    },
    {
        "video_name": "pMoA65Dj-zk",
        "Q": "5:43 to 5:47 -\n1. What are the other uses of line or line-angle diagrams?\n2. What are the other names for line or line-angle diagrams for the purposes other than in organic chemistry?",
        "A": "They are just short-hand ways for quickly writing a formula. They are also called bond-line structures, bond-line formulas, skeletal structures, and skeletal formulas."
    },
    {
        "video_name": "pMoA65Dj-zk",
        "Q": "Around 1:24, Sal addresses the Octet Rule, which does not apply to Hydrogen and Helium because they stabilize by filling their 1s orbitals. However, does this mean that atoms close to H and He like Lithium and Beryllium prefer to give up their electrons to enter the stable 1s configuration? Or do they try to pair up with other atoms to fill the 2p orbital?",
        "A": "The tend to give up their electrons to get the stable 1s\u00c2\u00b2 configuration."
    },
    {
        "video_name": "pMoA65Dj-zk",
        "Q": "Is the molecular structure drawn at 2:00 a electron dot structure ? If not what is the electron dot structure of methane ?",
        "A": "the very first structure in the video is an electron dot structure"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 6:26, why is the velocity -5 m/s? shouldn't it be 0?",
        "A": "The object is going 5 m/s when it lands. We don t care that after it hits the ground, it stays on the ground. That s obvious."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "@8:42 Why does Sal get 1.02? Can't you only have two significant figures?",
        "A": "I d wager a guess that it was a small oversight on his part."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At around 4:00, Sal said that you may have remembered that the sine of 30 degrees is 1/2. What are some other \"should remember\" angles for sine, cosine, tangent, and cotangent?",
        "A": "0, 30, 45, 60, 90 you really only need to learn two triangles though a 30-60-90 and an equilateral,which will be 90-45-45"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 1:13 , Sal says that it decelerates, isn't it and then reaccelerate, but isn't it just supposed to be a constat acceleration on -9.8 m/s^2. So techincally it is still accelerating and not decelerating right?",
        "A": "We don t use the term decelerate in physics. The object is under constant acceleration as long as it is in free fall. That may lead to slowing down before speeding up, but that s just constant negative acceleration from an initial positive velocity."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "5:52 why is the initial velocity 5m/s wouldn't it be 10 because the object is launched at 10m/s?",
        "A": "The velocity IN THE VERTICAL DIRECTION is 5 m/s."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "at 9:15 sal says our change in time is 1.02 SECONDS how do we know that the unit SECONDS and not MINUTES OR HAOURS",
        "A": "Because the velocity was in m/s and the distance was in m."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 7:35 I noticed something weird. When the acceleration is contrary to the direction of movement (rising object) it should be negative and when it is in the same movement direction (falling object) it should be positive, isn't ? If yes, then why at 7:35 it was considered the acceleration as only negative (-9.8m/s^2) for the whole movement (up and down) of the object? Someone could explain me this?",
        "A": "No, your understanding of the relationship of the sign is wrong. You define positive as a certain direction. Motion in that direction is positive. Acceleration in that direction is positive. If the motion is in the opposite direction, the motion is negative, by you can still have positive acceleration. The acceleration is the rate of change of the velocity. Moving in the negative direction but at a slowing pace is positive acceleration."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 7:400, how do we know accelerations is -9.8?",
        "A": "The acceleration due to gravity (g) is an average value that is obtained after: 1) Repeated measurements (you can do this by dropping an object from different known heights and tracking how long it takes to hit the ground at each height). 2) Getting it from Newton s Universal Law of Gravitation. You just need to know the mass of the Earth (M), its radius (r) and the gravitational constant (G): g = G*M/ (r^2) The negative sign is due to the direction chosen to solve the problem."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 6:55 what is delta T",
        "A": "delta T stands for time or how much time it takes an object to do something within that amount of time"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "3:10 I didn't understand Soh cah toa",
        "A": "soh...sin = opposite/ hypotenuse cah....cos = ajacent / hyp toa....tan = opp / adj"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "0:00, how does it change vertically and why would we need to know that if we're just looking for the displacement?",
        "A": "You need to know how long it is in the air if you want to know the displacement"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 6:16 how does Sal determine that v_f=- -5?\nI would think that one has to find the height the rocket reaches and then multiply that by g(9.8) to get v_f",
        "A": "if air resistance is negligible you can assume that the vertical component of the velocity is unchanged by this factor upon its parabolic flight. Therefore if gravity is constant at 9.81 then the maximum height achieved by the initial projection of 5ms (when total vertical velocity =0) in the vertical direction (only acted upon by gravity) is mirrored in the opposing downward direction and due to velocity being a vector quantity this results in the velocity being -5ms."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 2:50, is it absolutely necessary to use trigonometry to solve for the unknown sides? Couldn't we just use the rule of 30-60-90 triangles, which would be faster?",
        "A": "How do we know the rule of the 30-60-90 triangle? Trigonometry."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "how do we know at 6:40 that the initial velocity and final velocity are the same for the vertical?",
        "A": "Because the acceleration is constant and the distance up is the same as the distance down."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "at 11:41, why is the average velocity in the horizontal direction is 5 square roots of 3 metres per second? I know Sal said it is because it doesn't change, but why does it not change?",
        "A": "Because there is no force acting on it in the horizontal direction. In case of the vertical direction there was gravity to change it but no such force in the horizontal direction. (Sal mentioned that he is ignoring air resistance and explained it somehow at 01:30)"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "at around the 10:00 mark, we have found our change in time. Why wouldn't multiplying our time by its velocity work to find distance? What would this be calculating?",
        "A": "Which velocity? The rest of the calculation in the video is multiplying the time by the horizontal velocity to find the horizontal distance travelled. You can t multiply the time by the vertical velocity because that s changing all the time and s=vt only works for when there is no acceleration."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "at 6:15, why is the final magnitude of velocity is the same as the initial?",
        "A": "Conservation of energy"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 6:12, Sal says that we will have the same magnitude of velocity (expect negative) while coming down. Well, due to gravity, wouldn't the downwards velocity be greater in number? He says that air resistance is negligible, but gravity isn't air resistance, is it?",
        "A": "gravity slows you down on the way up and speeds you up on the way down. The effect is equal but opposite in both directions. That means you slow down on the way up exactly as much as you speed up on the way down. That means you land with the same speed you had taking off"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "11:42 In a long jump which factors determines the span of the jump?",
        "A": "The span of the jump is determined by the initial velocity and the angle of elevation."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "The value of g was -ve when he plugged it at 9:12.Suppose I want to do some calculations at the point just when the projectile hits the ground,suppose i want to find the impact velocity,will I be plugging -ve g or +ve g value in my calculations?",
        "A": "g is always down. If you define down as positive, it s +9.8. If you define up as positive, it s -9.8"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "at 8:05 i thought the force of gravity was postive 9.8 m/s^2 not negative 9.8 m/s^2?",
        "A": "The acceleration due to gravity is DOWN 9.8 m/s^2. If you want to call down negative, you can. If you want to call down positive, you can do that, too. Gravity doesn t know about plus and minus signs. Just be consistent throughout the problem you are working."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 5:03 the magnitude of Vy = 5m/s. But why you take initial velocity on vertical as 5m/s.\nBut Why?\n\nWhy you have not taken Vx as Initial and final ?",
        "A": "Vx does not change. Vy is what determines how much time you have to travel in the x direction. If Vx changed, then you would have to do an additional set of calculations for Vx initial and Vx final."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 6:30, wouldn't the final velocity be 0 m/s, because when it's gonna stop when it hits the ground?",
        "A": "at the instant just as a projectile hits the ground, it has the velocity V but as soon as it hits the ground, it is due to the opposing force of the ground due to which it comes to rest(V becomes 0.) Obviously, the projectile would not hit the ground with 0 m/s velocity."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 1:44 , how much would air resistance affect the object?",
        "A": "The object would not travel as far, and could possibly have a slower velocity when going down than when it was going up."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 7:26, How did you get the vertical acceleration as -9.8 m/s? Is it because of gravity?",
        "A": "Yes. The - 9.8 m/s^2 is the force of the gravity of the earth."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "How is cos of 30 degrees = sqrt 3/2 ?? I don't have a calculator how do I know that its sqrt 3/2. @10:54",
        "A": "Hello Arman, There are several special angles that you should commit to memory. You would be well served to memorize SIN, COS, and TAN for 0\u00c2\u00b0, 30\u00c2\u00b0, 45\u00c2\u00b0, 60\u00c2\u00b0 and 90\u00c2\u00b0. You will see these angles again and again as you study trig, calc, and physics. Also, once you know these angles you can apply them in the other quadrants for angles such as 120\u00c2\u00b0, 135\u00c2\u00b0, 150\u00c2\u00b0, and 180\u00c2\u00b0 as so on into the 3rd 4th quadrants. Regards, APD"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "What does the -9.8 m/s*s mean? I don't really understand. (8:20)",
        "A": "-9.8 m/s^2 is about the acceleration due to gravity on earth."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 6:41 why change in velocity is = -5 m/s - 5m/s? isn't it suppose to be the sum of the period which is = (5 m/s + -5 m/s)/2? because that what we did with the horizontal velocity!",
        "A": "The formula you are suggesting is that of average velocity. Average Velocity, v-bar = (Final Velocity+Initial Velocity)/2 This gives us as idea (or the exact value) of the mean velocity during the flight of the object. On the other hand, change in velocity, as calculated here, gives us the idea of how much the velocity changed from the beginning of the flight to the end. It is calculated thus: Change in Velocity, delta~v = Final Velocity-Initial Velocity"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "6:00 How do we know final velocity is also 5 m/s in the opposite direction (-5)?? doesn't acceleration due to gravity have an effect on an object going downwards differently than an object going upwards in terms of their velocity?",
        "A": "it does have some effect on it but we solve these problems based on assumptions"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 11:35 Sal says the formula is s = (v init)(time). But isnt the formula supposed to be s = (v init)(time) + (1/2)(acc)(time^2)",
        "A": "Acceleration in the horizontal direction is 0."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At approximately 7:15 why do we say that change in velocity equals acceleration times change in time??..",
        "A": "The acceleration is what is actually causing the velocity to change, so if you multiply the time by the acceleration, the answer will be how much the acceleration caused the velocity to change (change in velocity)"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 4:05, do I have to memorize the values of the sine, cosine, tangent?",
        "A": "If it helps, use the acronym Soh, Cah, Toa. All this means is: sin(angle) = opposite (typically the y component) over hypotenuse cos(angle) = adjacent (typically the x component) over hypotenuse tan(angle) = opposite (typically the y component) over adjacent (typically the x component)"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 6:31, it is said that the final velocity is -5m/s. But due to acceleration caused by gravity, the velocity would increase. So, how could the initial velocity be equal to final velocity?",
        "A": "It s not. It s in the opposite direction. That s not equal."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 6:41 why is the final velocity negative?\nIs the negative the direction and the direction is down?",
        "A": "Because it is in the opposite direction of the initial velocity, which was defined as the positive direction"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 3:18, why did Sal put 2 line on both sides of Vy ?",
        "A": "That s the vector version of absolute value. It means Take the magnitude of what s between these bars"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 6:25 why does sal say that final velocity is -5m/s? isn't the final velocity going to be different from the initial velocity ? because as far as i know that when i throw an object with with some initial velocity the magnitude of the final velocity is going to be much greater than the magnitude of the initial velocity.",
        "A": "As far as you know is wrong. When you throw an object straight up in the air, it comes back to your hand with the same speed it left. That has to be true, because the acceleration is constant during the entire flight, and the distance up is equal to the distance down."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 7:41 how did he find the acceleration -9.8 m/s^2?",
        "A": "This is the acceleration due to gravity at Earth s surface. It s a known quantity which you will see often in physics. You would do well to memorize this bit of information."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "at 6:26 why final velocity is in the opposite direction and why it is negative?",
        "A": "The projectile started out going up, then it is going down. Down is opposite of up, right?"
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "at 7:39 , why did he use the acceleration by negative value , I know that way it will get the time positive as there is no negative time , but the body first goes upward and then downward so why he used negative value not positive !",
        "A": "If you define up as the positive direction, then the acceleration due to gravity is negative."
    },
    {
        "video_name": "ZZ39o1rAZWY",
        "Q": "At 1:48 Sal said that doing this experiment on the Moon would remove air resistance. Would that not effect the vertical direction, since there is less gravity on the Moon? I would think that the object would go higher when thrown on the Moon.",
        "A": "It would also affect the vertical direction since the moon would cause a lesser gravitational force on the projectile. In his example, he was just pointing out a method of removing the force of air resistance."
    },
    {
        "video_name": "0XSSPow5oAc",
        "Q": "In your final example, (10:21), why won't a rotation of 180 degrees along the horizontal axis of the molecule give a superimposable image?",
        "A": "Since it is a 2-dimensional representation you can t rotate the bonds or move the molecule in 3-dimensions."
    },
    {
        "video_name": "0XSSPow5oAc",
        "Q": "In the first example, Sal seems to be rotating the molecule in a 2D way to check for chirality (at 3:40). I imagine chirality from a 3D perspective (i.e. rotating counterclockwise around an imaginary axis running through the center or topmost CH2 straight down). Is it wrong to think of it that way?\nThank you!",
        "A": "It isn t wrong to think of it that way. But, if you re thinking it 3d, then perhaps you should try looking at it from a fischer projection way; where the primary functional groups are closer to you, and the other substituents are farther. In this case, if you want to turn it around, then, the primary functional groups will be away from you and vice versa. This way, you get the concept right, and you re looking at it the considered right way."
    },
    {
        "video_name": "0XSSPow5oAc",
        "Q": "At 9:30 , why that carbon is a chiral molecules as it is bonded to a carbon itself ?\nPlease someone clear my doubt. Thank you.",
        "A": "It is not a problem that it is bonded to a carbon. For the carbon to be chiral here, it just has to be bonded to 4 different groups; carbon is one of those groups, just like H, Br, and Cl are each one of those groups."
    },
    {
        "video_name": "0XSSPow5oAc",
        "Q": "In the first example at around 4:44, if you rotate from the CH-Cl 180 degrees to the left, wouldn't the two bottom CH2's then be on the top and the one CH2 on the bottom? So how is it supper impossible?",
        "A": "That s what it looks like, doesn t it?This is one of the problems caused by drawing things free-hand. You have to remember that the ring is a regular pentagon, and he is really rotating the ring by only \u00c2\u00b2/\u00e2\u0082\u0085 of a circle (144\u00c2\u00b0), not half a circle (180\u00c2\u00b0). Try it with a regular pentagon, and you will see that they really are superimposable."
    },
    {
        "video_name": "0XSSPow5oAc",
        "Q": "At 01:51 why does Sal assume that we go around in two different directions? Does it have to do with the molecule's chirality? Don't we have to know only whether the 4 neighbors of an atom are different?",
        "A": "He is hunting for chirality. He is trying to see if the two ring carbons attached to C-1 are different. He finds that they are identical (both CH2 groups). So he goes one more C atom out to see if he can find a difference. Again they are both CH2 groups. He continues this process in both directions around the ring and never finds a difference, so he concludes that C-1 is not a chiral atom. Hope this helps."
    },
    {
        "video_name": "0XSSPow5oAc",
        "Q": "at 3:30, we rotate on the plane of the paper. why aren't we allowed to rotate it about any other axis(ie.) other than the plane of the paper??",
        "A": "You are allowed to rotate in any axis. However, here we are trying to compare the molecule on the left with the molecule on the right. Rotating in the plane of the paper is the simplest way to put these two molecules in the same orientation to demonstrate that they are identical."
    },
    {
        "video_name": "1k6MUeM-pEo",
        "Q": "at 8:40 ish, if one were to make a grignard reagent with the dibromide molecule shown, which Br would bond with Mg?",
        "A": "Grignard reagents are made from alkyl halides and magnesium, not diatomic halogen molecules. even if you did make a Grignard reagant from diatomic bromine, which Br bonds to Mg is irrelevant because the two Br atoms are indistinguishable (unless you want to talk about different isotopes)"
    },
    {
        "video_name": "1k6MUeM-pEo",
        "Q": "At 6:32 you show adding a bromine group to a tertiary carbon and again at 6:58. Are these carbons not sterically hindered? Is this hindrance different than the hindrance in the next example with benzene ring which prohibits the Sn2 mechanism?",
        "A": "Those carbons appear to be sterically hindered but I think the mechanism in this case is SN1 with the leaving group leaving first (the C - O bond is broken) and then the Br attacks the resulting carbocation"
    },
    {
        "video_name": "1k6MUeM-pEo",
        "Q": "at 5:05, if we were to do the mechanism, would we show an SN1 or SN2 mechanism and why?",
        "A": "SN2 - Strong acid (HBr) protonates the ether oxygen, which turns it into a better leaving group. Next, the Br ion attacks the methyl carbon in an SN2 reaction to give the alcohol and methyl bromide. OH protonated by second mole HBr, Br- attacks alpha carbon, releasing H2O."
    },
    {
        "video_name": "1k6MUeM-pEo",
        "Q": "1:46 Why does the nucleophile attack the carbon (in R') bonded to the oxygen rather than the oxygen itself? After all, the latter carries a +1 positive charge.",
        "A": "It probably does, but the attack is not productive from the oxygen s point of view. The oxygen atom would still have three bonds and a positive charge. It is much better energetically for the nucleophile to attack the oxonium ion and form two uncharged products."
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "7:07 sal said its distance travelled it should be displacement right?",
        "A": "It is actually displacement, but here we are considering only the magnitude of displacement, which is the same as distance in this case. Hope this answers your question. Cheers!"
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "At 0:46, is speed the magnitude of velocity ? if it is, can we sketch speed vs time graph ? ( please i need the answer and why . i want to understand every thing of this point. )",
        "A": "Speed is indeed the magnitude of velocity."
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "At around 8:30 Sal discusses the area under the curve is how far you have traveled while using the formula for displacement: s = vt. What is the difference between how far one has traveled and displacement. I was under the impression that distance traveled and displacement are two separate things.",
        "A": "If you go around in a circle back to where you started, your displacement is zero, but you traveled a non-zero distance."
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "I understand the logic behind the answer in 8:18. However, I was wondering why the answer could not be 15 (which was what I originally thought) because in the first second, you travel 1 meter, then in the second, you accelerate to 2, so 1 + 2 + 3 + 4 + 5 = 15?",
        "A": "Because you are traveling 1 m/s at the END of the first second. You didn t go 1 meter in the first second. You went 1/2 meter. Etc."
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "At 8:44, Sal puts magnitude marks around the acceleration. I understand why he would say the magnitude of values like velocity and displacement, but why did he put \"the magnitude of acceleration\"? Isn't it possible to have the acceleration being negative? And if you do put the magnitude marks around acceleration, doesn't that say that the acceleration can never be negative and that the slope will always be zero or above? But that is not true...",
        "A": "acceleration is a vector, just like velocity and displacement. The magnitude of any vector cannot be negative.That doesn t mean the vector can t be negative"
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "At around 7:54 you make a calculation of the total Displacment or distance when the a Vector is 1 m/s*s, the initial speed was 0, and the time was 5 seconds.\nWhat i didnt understand, was why cant you do the following calculation:\nsince i started in 0 Velocity, after 1 second i was 1 mile away. then after the next second i was another 2 miles away. so this adds up to 3. and if you go on, you'l get 15 meters, and not 12.5 as you calculated. why is my calculation incorrect?\n\nThanks :)",
        "A": "the graph drawn was velocity vs time graph. the velocity is uniformly increasing wrt time. hence it will keep on increasing. the best way to calculate the displacement here is area under the curve."
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "At 7:15, it says distance traveled. He later corrects himself because distance traveled does not include a direction, but this does not address whether it is distance or distance traveled. In the title for the video, it says distance. The letter s is used in the picture also. I'm confused. Is the area under the line distance or distance traveled? (I am assuming it cannot be displacement because area does not have a direction)",
        "A": "The distance Sal is talking about is where the object started at to when it stopped. Thus, this is the same as the distance traveled. He also talks about the area under the curve to be the magnitude of the displacement. This is the same as distance since magnitude is the size of the vector, not the direction. Hope this helps!"
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "Doubt:-\nAt 1:51 and that's also the basic formula too.\nV = s/delta(t)\nso can we also write it as\nv = delta(s)/delta(t)?",
        "A": "V=Delta (s)/Delta (t) is the actual equation used in physics."
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "I see this everywhere used in physics what does the ||v|| mean? Sal uses them at 0:35 and I also see this used around other variables like force ||F|| too. Does this mean magnitude?",
        "A": "Yes, magnitude."
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "At 2:30 Sal says that distance is the area of the rectangle? Shouldn't it be a square as both the sides have a numerical value of 5 units.",
        "A": "Geometrically speaking, all squares are rectangles, yet not all rectangles are squares. Therefore, it is perfectly acceptable to refer to what you know to be a square, as a rectangle. Hope this clarifies. =]"
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "At about 6:51, Sal says, \"The smaller the rectangles you try to make here, the closer you're going to get to the area under this curve.\" I'm beginning to sense a problem. To get a perfect answer that tells you the area under a curve, wouldn't you need rectangles that are infinitely small?",
        "A": "yes, and you would have infinitely many of them, and that s what integral calculus is all about"
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "at 8:43 Sal says that the slope of the line is 0 because it has a constant velocity but wouldn't the slope of the line be undefined?",
        "A": "No, the slope of the line is the change along the vertical axes divided by the change along the horizontal so that would be the change in velocity divided by change in time. So in 1 second you have 0 change in velocity which would be 0/1 = 0."
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "at 6:55,why did sal took area under the graph as distance whereas in V-T graph area under the graph is displacement?please simplify\nthanks.",
        "A": "You can t fully represent displacement by finding the area, since as a vector quantity, displacement also requires a direction. Finding the area only gives an amount, no direction. The area under the curve is the magnitude of the displacement, which is equal to the distance traveled (only for constant acceleration). So in this case, they are interchangeable although it was probably a mistake by Sal to use both."
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "in SAL example of at 2:20 v=5m/s and \u00ce\u0094t=5sec the displacement becomes 25m by the formula v=s/ \u00ce\u0094t and by area of graph too but in second graph when velocity is increasing why formula( v=s/ \u00ce\u0094t ) is not satisfying..it should come 25 m by the formula( v=s/ \u00ce\u0094t ) but 12.5m (at 8:17) by area of triangle.",
        "A": "You need to use average velocity when you use v = s/t. If velocity is constant, then v is always the same as the average v, but if velocity is changing, you need to take that into account and calculate the average. If acceleration is constant, then the average velocity will be (Vf + Vi)/2, which is the height of the triangle."
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "At 4:58 when he was writing change in velocity wasn't he supposed to put an arrow over the V?",
        "A": "No. An arrow over a number or variable means that it is a vector. Change in velocity is delta (v). Delta means change in"
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "at 8:39 Sal says the slope of the line is completely flat because the velocity is unchanging. Shouldn't it be the acceleration that is unchanging? The slope tells you your acceleration, so if the slope is flat the acceleration is constant which means that velocity is changing correct?",
        "A": "also it is velocity on the axis not acceleration"
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "2:10-2:24\nSal states that the area in the velocity per time graph is equal to the velocity we get by multiplying velocity by time to get or = displacement which we get as 25 meters and is similar to the area covered in the the rectangle in the velocity per time graph's rectangle formed.\nHow is the the area covered expressed in meters and not meters square?\nPlease give a response Sal or anybody.",
        "A": "because you are multiplying meters/sec times sec so the seconds cancel out and you get meters"
    },
    {
        "video_name": "d-_eqgj5-K8",
        "Q": "At 0:31, the symbol of magnitude of velocity was ||v||. I learnt it as |v| and <v>.\nPlease explain the correctness of the proper symbol.",
        "A": "All are correct. As long as you use your symbols consistently, it doesn t matter which you choose to express magnitude (or anything for that matter) with."
    },
    {
        "video_name": "UQHT5RktlVk",
        "Q": "At 3:30 why is the total pressure 2.5atm, where did that number come from and how did you calculate it?",
        "A": "The number 2.5atm was just made up as a starting value for the question, and wasn t calculated from anything. The part that was calculated was what is the partial pressure of the oxygen in the container, since the pressure changed from 2.0atm to 2.5 atm when we added oxygen in with the nitrogen? Hope that helps."
    },
    {
        "video_name": "pQWwP7YYH6o",
        "Q": "at 4:06, you say adiabatic contraction, dont you mean adiabatic compression?\nor is contraction the same as compression?",
        "A": "If something is pushing on the system, it s a compression. If the system is doing the pulling, it s a contraction."
    },
    {
        "video_name": "pQWwP7YYH6o",
        "Q": "Can someone explain why the engine, when run in reverse (refridgerator), takes heat from the cold reservoir? Sal says this at 11:02. He says that the engine takes a smaller amount of heat from the cold reservoir. I thought that in the reverse process work is done ONTO the system and as a result the heat generated by this is expelled into the cold reservoir. Can someone please explain?",
        "A": "I think it s because if you think of the reverse Carnot cycle, although it expells the heat Q1 to the surroundings when isothermically contracting, the engine also takes in the heat Q2 during the isothermic expansion. Here I think Sal is talking about the heat Q2 whereas you are thinking of the heat Q1."
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "AT 8:15 he mention something about 10 million Kelvin. What do Sal mean by 'Kelvin'",
        "A": "Kelvin is a temperature scale like Celsius except that 0 is at absolute zero, the coldest temperature possible, instead of the freezing point of water."
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "At 3:10, why does the helium core become denser than the hydrogen core? Wouldn't they both have the same amount of mass? Unless some of the mass was released in energy form, but then how would the two cores have the same gravitational pull?",
        "A": "Welcome to nuclear science. Firstly, Helium is greater in mass to Hydrogen. While not much it is enough to become the core. Gravity always pulls the heaviest thing into the center in any case. The Earth, for example, has a huge iron core because gravity pulled it in that way when the Earth formed. So as the Helium is fusing in the core Hydrogen is still fussing on the outside of the core. It s just not massive enough to become the core itself. I hoped this helped. I wouldn t mind clearing things up."
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "at 4:25,the core get under a lot of pressure and H is fusing in a shell which is also under enormous pressure, but dosen't the He start fusing into C under that much pressure?",
        "A": "No, He needs much more pressure to start fusing into Carbon."
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "At 9:40, I don't understand why does the helium fuse into another core?",
        "A": "At 9:40, you can see that the hydrogen coreturns into helium as the core is hotter and there is faster fusion. The density also causes it to move in to the core."
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "@ 3:32, what limits the hydrogen to fuse only inside the\"shell\"? How does the shell enclose the hydrogen?",
        "A": "The shell does not enclose the hydrogen; the shell is made of hydrogen, and it encloses a sphere of helium. The hydrogen making up the shell is undergoing fusion because it s the only region in the star currently at the correct pressure and temperature to be capable of it."
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "At about 10:40, Sal says three heliums fused together makes a carbon and four heliums fused together makes an oxygen. Can you fuse two heliums together? If so, what is that get you?",
        "A": "You would get beryllium but due to the low fusing point of beryllium, a helium atom would instantly fuse with it becoming carbon."
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "At 3:31 , Sal says that Hydrogen will start to fuse around the helium core. What will happen, if by chance, there is no hydrogen outside the helium core?",
        "A": "When the hydrogen runs out, the star s life will come to an end for medium stars. It will eventually explode and become a white dwarf. This, in 5 billion years, will happen to our sun. Larger stars will fuse carbon and other elements all the way up to iron. Once iron is done fusing, the star will become a supernova or black hole, and in that explosion is where all elements heavier than iron are formed."
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "how does pressure and temperature get high enough in space? 00:13 and how hot does it have to be?",
        "A": "10,000,000 K is the required temperature to fuse hydrogen."
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "At 2:56, it shows that the core is completely helium, but then goes to say (at 3:29) that there's hydrogen diffusing into helium in a shell. Where is all the hydrogen coming from? Do stars just produce tons of hydrogen, or are the stars getting it from somewhere?",
        "A": "Stars begin composed mainly of hydrogen. It will start fusing it at its core into helium, and then work its way outward from there."
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "I don't understand Sal's point at 3:10 about how there's now a hydrogen fusing shell around the core because of stronger gravity due to being closer to the denser core. How is the existence of a hydrogen fusing shell related to stronger pull of gravity to the core?",
        "A": "To fuse hydrogen you need to put it under a lot of pressure. The stronger pull of gravity provides that pressure."
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "At 0:46, What does it mean for the star to collapse in itself?",
        "A": "When a star s gravity overcomes it s internal pressure, it s gravity compresses itself thus collapsing into itself. Eventually, the gravitational force is balanced out by thermal pressure causing the star to stop collapsing into itself."
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "At around 10:17 he said that Helium mainly forms into Carbon and Oxygen. Why does it not mainly form into Beryllium as it has an atomic number of 4",
        "A": "because of its different physical properties"
    },
    {
        "video_name": "kJSOqlcFpJw",
        "Q": "At 5:00 how long does this gradual process take?",
        "A": "a LOOOONG time. millions of years."
    },
    {
        "video_name": "3gUE_P9T-Wk",
        "Q": "at 1:06 what does A stand for",
        "A": "Hello, I m not sure what you mean. If you refer to when he writes 1.2 Ga, it means 1.2 billion years ago or 1.2 Giga (that s 1 billion) years ago. We don t writhe this as Gy, because Gy stands for Gray, which is a radiometric unit. Hope that helps!"
    },
    {
        "video_name": "3gUE_P9T-Wk",
        "Q": "At 3:00 sal says that we only have fossils of spores not plants. Why is that?",
        "A": "Spores have something inside them that can t be decomposed and can be turned to rock when compressed by layers and layers of dirt (that s how a fossil forms). But plants are made mostly of tissues, and everything is degradable. Plants can be preserved though, with the help of amber."
    },
    {
        "video_name": "3gUE_P9T-Wk",
        "Q": "at about 3:37 Sal said that the oldest fossil of one that was found of an animal that spent it's entire life on land. My question is that how do they know it actually lived it's whole life on the land? I understand that the fossil was on land but what if there was water in that region of where it was found. Or, maybe the animal lived under water but-let's say because of food problems it came to land but couldn't evolve correctly to survive all conditions or didn't evolve at all but died.",
        "A": "They know it was a land animal because it had to breathe air and it had legs. We know it breathes air because it was found with spiracles on its segments, which are what millipedes use for respiration."
    },
    {
        "video_name": "3gUE_P9T-Wk",
        "Q": "4:30 What is the thing in that picture? A dinosaur",
        "A": "It s actually a myriapod fossil. Scientists think that myriapods appeared on earth about 428 million years ago. If you start watching it from 3:34 Sal will sort of explain about it."
    },
    {
        "video_name": "BgTpPM9BMuU",
        "Q": "At 1:48 ,would the formula have equal charges or not? Both sides have a charge of 0, but the different charges are different. One side has \"+,-,+,-\" but the other is just \"+,-\". Can someone help me, I'm confused.",
        "A": "As you point out, both sides have a net charge of zero and this is the important bit when balancing ionic equations. In ionic equations, precipitates, which are solids, are not written as ions. Hence, AgCl, which is a precipitate, is not written as Ag+ and Cl-. In general, solids, gases and liquids (eg, H2O) are not written as ions when writing ionic equations."
    },
    {
        "video_name": "5RXRr8PKunk",
        "Q": "what's obease? in 5:9",
        "A": "Obesity is a BMI of 30.0 or over."
    },
    {
        "video_name": "GuaozMpFS3w",
        "Q": "At around 4:00, Jay said that the triple bond had two pi bonds and one sigma. I thought that a triple bond always had one pi and two sigma,",
        "A": "Sigma bonds are the FIRST bonds to be made between two atoms. They are made from hybridized orbitals. Pi bonds are the SECOND and THIRD bonds to be made. They are made from leftover p orbitals."
    },
    {
        "video_name": "GuaozMpFS3w",
        "Q": "At 4:59 what is the name of that molecule?",
        "A": "Can be named diethyl ether or ethoxyethane. Normally in a lab, when you say ether , you are not referring to the functional group, you are referring to diethyl ether."
    },
    {
        "video_name": "GuaozMpFS3w",
        "Q": "At 7:39 you started with last molecule. What is the hybridization states of oxygen atom? I have a similar question in an exam paper.",
        "A": "O can be sp\u00c2\u00b3 (as in H\u00e2\u0082\u0082O) or sp\u00c2\u00b2 [as in (CH\u00e2\u0082\u0083)\u00e2\u0082\u0082CO]."
    },
    {
        "video_name": "GuaozMpFS3w",
        "Q": "At 7:30, does the oxygen atom have a sigma and a pi bond?",
        "A": "Yes, in that double bond one is sigma and one is pi."
    },
    {
        "video_name": "GuaozMpFS3w",
        "Q": "At 6:45 what does the bond angle here? Is it \u00e2\u0088\u00bd105 ?",
        "A": "Experimentally it s found to be around 110\u00c2\u00b0 The best rationale for this vs the much lower bond angle seen in water is due to increased steric repulsion from the bulkier ethyl groups."
    },
    {
        "video_name": "7p2qfyqiXHc",
        "Q": "at 6:46 I saw a lone pair of electrons , can they be connected in a way?",
        "A": "they can t bound together, they can however bond to another atom"
    },
    {
        "video_name": "7p2qfyqiXHc",
        "Q": "At 7:20, Carbon is shown as having 5 valence electrons. Why is that? Doesn't carbon only have 4 valence electrons?",
        "A": "Let s say hypothetically that some super strong base reacted with methane by taking off a hydrogen, the electrons that were in the C-H bond are now going to be a lone pair of the carbanion. This is the sort of situation Jay is describing. Carbon brought 4 electrons to the molecule, but one of the bonded atoms has been removed and the electrons in the old bond went to carbon giving it one more electron that it normally owns and a negative formal charge."
    },
    {
        "video_name": "7p2qfyqiXHc",
        "Q": "At 4:02 you say that the carbon is sp2 hybridized, why isn't it considered sp3?",
        "A": "The carbon is directly bonded to only three other atoms. Hence, it is sp\u00c2\u00b2 hybridized."
    },
    {
        "video_name": "7p2qfyqiXHc",
        "Q": "At 3:10,you see CH3 having no lone pairs but while explaining the carbanion at 6:15,Carbon has a lone pair.Could you please tell how?",
        "A": "These are both well-known species, although they are highly unstable. You don t see them in ordinary chemical reactions. CH\u00e2\u0082\u0083\u00e2\u0081\u00ba has only 3 C-H bonds (6 valence electrons). CH\u00e2\u0082\u0083:\u00e2\u0081\u00bb has 3 C-H bonds and 1 lone pair of electrons (8 valence electrons)."
    },
    {
        "video_name": "7p2qfyqiXHc",
        "Q": "at 2:48 why does carbon not have an octet .... pls help merci",
        "A": "As mentioned in the video, the carbon you refer to is a carbocation which is a kind of intermediate compound mostly found in organic reaction.This carbon has given one of its shared pair of electrons to a more electronegative element bonded to it like oxygen so it is left with a positive charge hence it doesn t have an octet."
    },
    {
        "video_name": "7p2qfyqiXHc",
        "Q": "At 5:43, i wanna know how do you know exactly that carbon is bonded with hidrogen atom? Can that be another atom?",
        "A": "The only atom you don t have to draw is hydrogen, so if you re missing one bound in the structure it must by hydrogen, assuming the drawing is correct."
    },
    {
        "video_name": "7p2qfyqiXHc",
        "Q": "At 7:05 mark, we know the formula for calculating the formal charge is (no. of valence electrons) - (no. of bonded electrons). Then why for the Carbanion he has also included lone pair of carbon? They are not bonded electrons in any way.",
        "A": "The formal charge of any atom in a molecule can be calculated by the following equation: FC = V - (N + B/2) where V is the number of valence electrons of the atom in isolation (atom in ground state); N is the number of non-bonding valence electrons on this atom in the molecule; and B is the total number of electrons shared in covalent bonds with other atoms in the molecule. There are two electrons shared per single covalent bond."
    },
    {
        "video_name": "7p2qfyqiXHc",
        "Q": "At 2:53, Jay says carbon can never exceed an octet, but can't phosphorus achieve expanded octet? Can phosphorus sometimes exceed an octet, but carbon can't?",
        "A": "A phosphorus atom is larger than a carbon atom (or other second period atoms eg. N and O) so it is better able accomodate extra electron pairs around it, compared with the smaller atoms in the second period."
    },
    {
        "video_name": "7p2qfyqiXHc",
        "Q": "at 6:49 where did that lone 'pair' suddenly appear from. carbon has only four valence electrons!",
        "A": "One electron in the lone pair is one of carbon s four valence electrons. The other electron came from some other atom."
    },
    {
        "video_name": "7p2qfyqiXHc",
        "Q": "At 6:39, carbon is having a lonepair electron. How is this possible without having a bond with any atom or groups.",
        "A": "Essentially it has taken an electron from some other atom or molecule."
    },
    {
        "video_name": "mDttV5UmJ1Q",
        "Q": "At 0:08 how long should the paper clip be?\nHow long will it take to make the bot?\nIs there a whole kit with all the parts I need?",
        "A": "The length doesn t matter. The reason there s a tail is simply to lift the back of the robot up & off the floor. Whatever length you want, you just need to lift the robot to reduce the friction of it dragging around."
    },
    {
        "video_name": "5MOn8X-tyFw",
        "Q": "What chemical do they put in the tube to prevent clotting at 1:28?",
        "A": "some of those chemical are heparin, or EDTA , or sometimes both depending on the tube."
    },
    {
        "video_name": "5MOn8X-tyFw",
        "Q": "At 2:19 Dr.Rishi mentions that a centrifuge spins around to split the blood. How fast does the centrifuge spin? At least 100 mph.",
        "A": "Centrifugation speed will depend largely on the laboratory performing the testing and what their preferences are. The centrifuges at the laboratory I work in are capable of spinning anywhere from 500 to 5,000 RPM, and most of the blood that comes through the laboratory is spun at 5,000 RPM."
    },
    {
        "video_name": "5MOn8X-tyFw",
        "Q": "At 6:10 Sal refers to platelets. What are they and what do they do?",
        "A": "Platelets are small fragments of cells that circulate around in your blood. When one of your blood vessels becomes damaged, the platelets will clump together and form a plug that blocks off the hole in the blood vessel from the inside so that the bleeding will stop. Platelets also play many different roles in the general functioning of your immune system."
    },
    {
        "video_name": "5MOn8X-tyFw",
        "Q": "At 6:59 Rishi says that hemoglobin is a protein. Isn't it actually the pigment that gives RBCs and blood its redness? Thanks",
        "A": "hemoglobin is nothing but the pigment which transport the oxygen to various parts of the body using the red blood cells. Yes you are right. it give the blood and the RBC s its redness"
    },
    {
        "video_name": "5MOn8X-tyFw",
        "Q": "At 2:00, does it matter which way the tubes are facing in the centrifuge?",
        "A": "No. The tubes are usually arranged how he drew them and to balance the centrifuge. Sometimes they are closer to vertical. They need to be upright (with the rubber stopper on top)."
    },
    {
        "video_name": "Biua3BgRf54",
        "Q": "At 3:58, why does A's velocity have to be V in the formula? I know that if we place it as U, we would get -2V on the numerator vs. getting 2V, so it is different. But how do you choose which velocity goes where properly?",
        "A": "I believe, if I understand you correctly, that you are asking if there would be any change, or if this proof would be made different if the axis used to define the motion of the objects were changed. To that, the answer is actually no, for any inertial reference frame, and any set of coordinates in which you define the motion of the bodies handled in this video, you will consistently get the same result for the dilation that occurs between the two."
    },
    {
        "video_name": "Biua3BgRf54",
        "Q": "At 2:19 why the velocities of both A and C have same magnitude?\nShouldn't we use Einstein's velocity addition formula to determine velocity of B in C's frame?",
        "A": "He s finding a velocity V in reference frame A that will lead C to see A and B moving away with the same magnitude."
    },
    {
        "video_name": "1S4DtuMY88I",
        "Q": "So at 7:45 he calls it the 3rd harmonic, but is this also the 1st overtone?",
        "A": "We don t use the term overtone in physics"
    },
    {
        "video_name": "1S4DtuMY88I",
        "Q": "At 9:01,sound can pass through water , isn't it?",
        "A": "Yes, but the sound you hear is that which rebounds from the water. To hear the sound waves passed through water, you would have to basically put your ear to the water, or in the water."
    },
    {
        "video_name": "1S4DtuMY88I",
        "Q": "At 1:36 in the video, David mentions that molecules at the closed end of the tube do not have any displacement because they keep bumping into the closed end. Isn't that only on one side? What about the other side of the tube, the open end? Is SHM motion the only type considered?",
        "A": "At the open end, the molecules do move, and that s why it is an anti-node rather than a node."
    },
    {
        "video_name": "1S4DtuMY88I",
        "Q": "Hello! At 9:45- 9:47, it was mentioned that note would only get lower and lower, not louder and louder. I wanted to know if the intensity (or volume) of the sound would change (in addition to the tone) since the area is changing. Thanks so much!",
        "A": "I m not entirely sure, but I think the reason that the volume doesn t change is because the change in area is relatively small (and therefore, negligible) in a regular bottle/pipe: the area in the intensity formula is measured in meters, and I haven t seen a bottle that is a meter tall or wide. And since the amplitude of the sound wave isn t changing in this case, neither does the volume. Hope this helps!"
    },
    {
        "video_name": "1S4DtuMY88I",
        "Q": "At 4:38 the teacher just placed at not in the tube. I don't understand how practically that would work? Why would a section of the air in the middle of the tube not oscillate?",
        "A": "Because that s a spot where the incoming wave and the reflected wave are always interfering destructively. That s what a standing wave is - a wave that has points that don t move (called nodes) and points that move a lot (called anti-nodes)."
    },
    {
        "video_name": "PX_XSnVWlNc",
        "Q": "At 1:14, what do you mean with the higher frequencies being red shifted?",
        "A": "There is a video on red shifts on here. So at 1:14 he is just saying it was red shifted."
    },
    {
        "video_name": "i6gz9VFyYks",
        "Q": "At 9:40 he said a Newton is a (Kg*m^2)/s and he also said that the units for pressure are N/m^3 when they are N/m^2?",
        "A": "Good catch. He made a mistake with saying Newton is kg*m^2/s when it should be kg*m/s^2. He actually put in everything into that small space so not only it confuses someone who watches it, I think he ended up confusing himself. Here is how the final calculation would look like: h=(103000 kg/m*s^2) / ((13600kg/m^3)*(9.8m/s^2) h=0.77m Sorry this is hard to write it in here with nice formatting. Just to clear it up. Pressure = kg/m*s^2, Density=kg/m^3, Gravity=m/s^2"
    },
    {
        "video_name": "i6gz9VFyYks",
        "Q": "4:24 - Am I right in saying that specific gravity is the same as relative density?",
        "A": "Yes. It is the same. Hope this helps :)"
    },
    {
        "video_name": "i6gz9VFyYks",
        "Q": "At 9:38 you first say that newton is equal to kilograms times meter squared per second then at 9:42 you say newton is equal to kilograms times meters squared per seconds squared! Either way the units cancel out all the way or seconds is all that is left.",
        "A": "F = m*a m is in kg a is in m/s^2 F is in newtons. So 1 newton must be 1 kg*m/s^2"
    },
    {
        "video_name": "i6gz9VFyYks",
        "Q": "At 6:20, He says the pressure is being created by \"This column of water\". Shouldn't it be \"column of mercury\"?",
        "A": "Yeah ur right He just made an unknowing mistake. Its easy to get muddled......we r human afterall."
    },
    {
        "video_name": "i6gz9VFyYks",
        "Q": "I have one question... When you look video at 2:45 does it pressure in every point of \"horizontal green line\" (inside and outside of inverted tube) equal 1 atm?",
        "A": "only the inside yeah....not the outside tht wouldnt make any sence"
    },
    {
        "video_name": "i6gz9VFyYks",
        "Q": "it is actually around 5:00 the Fluids part(3), Sal started using the P to represent density, as far as know D is used, couldn't he just use the D?",
        "A": "He is using p, but really he means rho which looks sort of like a p. It is the most widely accepted symbol for density."
    },
    {
        "video_name": "i6gz9VFyYks",
        "Q": "At 8:52, why did we take 13,600 times height times 9.8 equal to 1atm or 10300 Pa or 10300 n/m^2?",
        "A": "From the previous video, the pressure at a given depth equals the density of a liquid multiplied by the depth (height) of the liquid multiplied by the acceleration due to gravity (P=(rho)gh) The pressure was given as one atmosphere, which is 101300 Pa, the density was given as 13600 kg/m^3, and the acceleration due to gravity was 9.8 m/s^2."
    },
    {
        "video_name": "i6gz9VFyYks",
        "Q": "There is a warning at 08:18 which says it is meter squared not cubed. But I can't understand, density is mass/ Volume and mass is kg, volume is meter cubed. So Sal did said right when he said meter cubed right?",
        "A": "ya he said it right i mean the notation"
    },
    {
        "video_name": "6G1evL7ELwE",
        "Q": "3:25 if h20 is extremely electronegative then why dose it transmit electricity so well",
        "A": "Water itself doesn t conduct electricity very well. Usually water s conductivity is a result of ions and dissolved elements present in most water. Completely de-ionized water is a poor conductor of electricity."
    },
    {
        "video_name": "6G1evL7ELwE",
        "Q": "At 4:16, he mentions that the symbol that he uses for charge is the Greek letter delta. What does delta mean?",
        "A": "In this context it means partial , as in a partial positive or negative charge."
    },
    {
        "video_name": "6G1evL7ELwE",
        "Q": "In 4:26 how does it make a half positive force?Isn't hydrogen and oxygen ametals?And ametals are negative forced",
        "A": "because oxygen takes all the electrons. oxygen is electronegative. Hydrogen and Oxygen are nonmetals. when NONMETALS are combined with NONMETALS they donate electrons so they acquire positive charges and thats what hydrogen its doing, gives electros to oxygen"
    },
    {
        "video_name": "6G1evL7ELwE",
        "Q": "What is the scientific word for 'hogging the electrons'? (3:30- -3:16)",
        "A": "I think it would be the polarity of the atom :)"
    },
    {
        "video_name": "6G1evL7ELwE",
        "Q": "At 4:15, what does the delta really mean?",
        "A": "In chemistry, \u00ce\u00b4 (delta) means a small amount. It s the standard symbol used to show partial electronic charges in atoms or molecules."
    },
    {
        "video_name": "6G1evL7ELwE",
        "Q": "Does waters polarity every neutrlaize because sal says at 3:53 that oxygen says its wats the elctrons to spend a little more time around me. So that led me to wonder water becomes a temporary covalent bond that is not polar? Like a covalent bond where the oxyegen and hydrogens actually share electrons?",
        "A": "Hydrogen bonding is far weaker than a covalent bond. No electrons are transferred or shared in a hydrogen bond."
    },
    {
        "video_name": "ob5U8zPbAX4",
        "Q": "at about 1:33, Sal mentions dendrites. What is the origin of that word and what does it mean?",
        "A": "Gk dendrit\u00c4\u0093s tree-like"
    },
    {
        "video_name": "ob5U8zPbAX4",
        "Q": "At 2:12 ,what is the purpose of the soma?",
        "A": "The soma mostly gives the cell nucleus and organelles somewhere to be, because dendrites and axons are quite narrow structures. The soma is where proteins are made, for example. The soma also has a role in action potentials and post-synaptic potentials in some types of neurons."
    },
    {
        "video_name": "ob5U8zPbAX4",
        "Q": "Does this video end prematurely at 6:12. It seems like an unatural point to cut off.",
        "A": "I thought that too...this is a good comment to put in Tips & Thanks next time, by the way. I think it s because the next video in the series, Sodium potassium pump , picks up where this one left off. Kind of a good strategy for Sal, leaving you hanging like that!"
    },
    {
        "video_name": "ob5U8zPbAX4",
        "Q": "At 3:17, he says that at the end of the Axon is the axon terminal. What is that?",
        "A": "axon terminal refers to nerve endings where there is a phase that is synapse where a chemical secretion is done to let message go in one direction by enzymes like norepinephrine and acetylcholine"
    },
    {
        "video_name": "ob5U8zPbAX4",
        "Q": "At 5:26, what does Sal mean by \"large enough\"? Is there a certain level of electricity that needs to be crossed before the neuron will transmit a signal?",
        "A": "Yup, there is a minimum amount of potential difference required to initiate a signal."
    },
    {
        "video_name": "ob5U8zPbAX4",
        "Q": "At 5:40, Sal mentions synapses. What are synapses?",
        "A": "Synapses are basically junctions over which communication can occur. Think of pass the parcel, the space between the people is the synapse the parcel is the message they want to pass to each other."
    },
    {
        "video_name": "ob5U8zPbAX4",
        "Q": "At 4:56 Salman talks about Ranvier. Who was Ranvier? And if he was not a person then why are the nodes called that?",
        "A": "Louis-Antoine Ranvier was a French medical scientist in the 1800s who discovered the nodes of Ranvier."
    },
    {
        "video_name": "ob5U8zPbAX4",
        "Q": "What do the Myelin Sheaths do? At 4:57 he says something about them but what is it exactly?",
        "A": "The short answer is that the myelin sheaths increase neuron conductance of the action potential. Instead of having to travel down every inch of the neuron, it jumps from space to space that the myelin coating does not cover."
    },
    {
        "video_name": "ob5U8zPbAX4",
        "Q": "At about 3:25 Sal mentions something called and Axon, i was wondering how Axon got its name",
        "A": "It is axis in Greek. In 1898, von K\u00c3\u00b6lliker (he discovered axons) proposed that these fibers, which form the long axis of the neuron and had come to be called axis-cylinders , be referred to as axons instead. Around the same time, some began using axon as an alternative name for the notocord, but as von K\u00c3\u00b6lliker s axon gained in popularity, axon-as-notocord faded from use."
    },
    {
        "video_name": "lKq-10ysDb4",
        "Q": "At 12:00, Sal says that there is no loss of energy (in a frictionless world). But how is this possible? The gas in the container applies a force in the direction of the displacement of the piston (up), thus it must be doing positive work. Shouldn't the gas then have less energy?",
        "A": "The gas has less energy, but whatever it gave work to now has an equal amount of energy, so none was dissipated as heat (which is really what we usually mean when we say lost )"
    },
    {
        "video_name": "lKq-10ysDb4",
        "Q": "at around 1:51 he says there would wpould be lower temperature how is that possible\nwhen v is proportional to temp",
        "A": "v is proportional to temp only at constant pressure, not all the times"
    },
    {
        "video_name": "lKq-10ysDb4",
        "Q": "At 11:08 ,Sal says there are few quasi-static processes which aren't reversible processes.\nWhat are the few examples?",
        "A": "An example of a quasi-static process that is not reversible is a compression against a system with a piston subject to friction \u00e2\u0080\u0094 although the system is always in thermal equilibrium, the friction ensures the generation of dissipative entropy, which directly goes against the definition of reversible."
    },
    {
        "video_name": "lKq-10ysDb4",
        "Q": "at 9:36, what if I used something that is in theory not discreet, such as water as the weight?",
        "A": "But water is made up of molecules, so you could remove 1 molecule at a time couldn t you?"
    },
    {
        "video_name": "lKq-10ysDb4",
        "Q": "At 07:49 , i pondered that every process HAS to be quasi static , equilibrium can never be obtained and the true path cannot be obtained ! take the process of removing sand for example , even if u take out infinitesimally small pieces of sand grains there will be some disturbance in the system ! am i right ?? sorry if my question is nt very clear but i dunno hw to frame it !",
        "A": "Only infinitesimally small disturbance, I guess. Quasi-static process is much like a square circle.: disrurbance = no disturbance. I marvel at the fact that such not just physically impossible but illogical notions are helpful in physics. Leibniz must be right and there must be some incomprehensible principle of continuity in Nature..."
    },
    {
        "video_name": "lKq-10ysDb4",
        "Q": "At 11:01 sal said that most quasi-static processes are reversible but some are not. Is there an example of that?",
        "A": "I think if you slowly released gas into the surroundings. It would be impossible to get the gas back, but the changes in pressure, volume etc could be graphed in a quasi static way"
    },
    {
        "video_name": "h-31O7CaF2o",
        "Q": "how will the water be less cold at 8:00 ?? Why does water radiate less heat?",
        "A": "The water does not radiate less heat, it radiates the same amount of heat as the land. However, since the Specific Heat Capacity of water is higher than that of the land, the water s temperature decreases less per unit of heat lost. In other words, both the water and land lost the same number of calories, but the water needs to lose more calories than the land to have the same temperature decrease."
    },
    {
        "video_name": "h-31O7CaF2o",
        "Q": "At 8:07, is there any differences on night and day? At day, there is sun but at night there is no sun.",
        "A": "The main difference is that during the day the sun is radiating heat, at night there is no sun to radiate heat."
    },
    {
        "video_name": "h-31O7CaF2o",
        "Q": "At 8:00, why is the water colder than in the air? Is it because it is more dense than air?",
        "A": "As temperature decreases, the density increases, so water is colder than air. at 8:01ish."
    },
    {
        "video_name": "h-31O7CaF2o",
        "Q": "0:16, isn't it the amount of energy required to raise 1 kg of a substance by 1 degree Celsius.",
        "A": "Yes, that s correct because heat is a form of energy. Therefore, technically the definition of specific heat could be the amount of energy (heat) required to raise 1 g of substance 1 degree Celsius."
    },
    {
        "video_name": "TC23wD34C7k",
        "Q": "I don't understand at 10:42 when he says multiply \"boat sides\" what is boat sides?",
        "A": "both sides.. :|"
    },
    {
        "video_name": "TC23wD34C7k",
        "Q": "At 3:23 Sal said something about a theta degree incline. What exactly is a theta degree angle?",
        "A": "Theta is a variable. It is a greek letter. So it is an angle of theta degrees."
    },
    {
        "video_name": "TC23wD34C7k",
        "Q": "On 1:03 Sal said that 'g' meant the gravitational field near the surface of the earth. I thought g meant the acceleration due to gravity. Are these the same thing or are they different?",
        "A": "They are two ways of talking about the same thing. The acceleration due to gravity at the surface of the earth is because of the strength of the gravitational field of the earth near its surface. The units of the strength of a gravitational field is acceleration."
    },
    {
        "video_name": "TC23wD34C7k",
        "Q": "in 8:40, how did the hypotenuse become mass times gravity?",
        "A": "When he draws the vector that will become hypotenuse at 0:53, he states that it s the force due to gravity. The whole point of the exercise is to decompose that vector into component vectors that are perpendicular and parallel to the inclined plane. Those vectors are the other two sides of the triangle. Since the component vectors form a right angle, the gravitational force vector is the hypotenuse of a right triangle."
    },
    {
        "video_name": "TC23wD34C7k",
        "Q": "At 7:31 , Sal just picks up the vector (the one parallel to the inclined plane) and shifts it to complete the right triangle. I am very confused about this part. How can you just shift a vector like that?",
        "A": "When you are trying to add or subtract vectors, their location has nothing to do with the answer, so you can move them around however you like, as long as you don t change the size or the direction."
    },
    {
        "video_name": "TC23wD34C7k",
        "Q": "At 5:17, what do you mean by \"A the'da (sorry if I spelt it wrong) degree angle\"?",
        "A": "Theta is a variable frequently used in trigonometry and physics to represent an unknown angle. It comes from the Greek alphabet, and looks like this: \u00ce\u00b8"
    },
    {
        "video_name": "TC23wD34C7k",
        "Q": "I'm still rather confused about some of the notation Sal uses throughout the physics videos, such as ||x||, used at 8:08. If someone could explain that to me, I'd be very grateful. Thank you.",
        "A": "Everybody: |x| means absolute value of x, (so if x = -1, then |x| = 1 ) ||x|| means magnitude of x, (length, as opposed to/excluding direction )"
    },
    {
        "video_name": "TC23wD34C7k",
        "Q": "at 2:37 Is the force due to gravity that is perpendicular to the ramp the normal force? In that case shouldn't the arrow be pointing the other way?",
        "A": "The force due to gravity is down. A normal force is always perpendicular to the surface that applies it."
    },
    {
        "video_name": "TC23wD34C7k",
        "Q": "When Sal talks from 11:20 to the end of the video about internalizing which one is cosine and which one is sine, it makes sense. But when doing vector addition, the magnitude of the horizontal component is (using v for the vector) ||v||cos(\u00ce\u00b8\u00c2\u00b0) and the vertical component is ||v||sin(\u00ce\u00b8\u00c2\u00b0). So in other words, when doing 2D vector addition and given a vector, sine is vertical, cosine is horizontal. In the diagram, the yellow component is cosine and is vertical, and the blue sine is vertical?",
        "A": "Yeah, it s counter-intuitive if you re used to normal vector addition. Bear in mind, though, that we rotated our triangle to decompose the gravity vector, so sine and cosine end up switched around in this case."
    },
    {
        "video_name": "TC23wD34C7k",
        "Q": "7:20 Will \u00ce\u00b8 of the ramp always be equal to the inner angle where the force gravity and the force gravity perpendicular meet or are there situations where this isn't true?",
        "A": "it is always true because of the property of angles of parallel lines"
    },
    {
        "video_name": "TC23wD34C7k",
        "Q": "How can something be perpendicular to the surface of the earth which is curved.5:20",
        "A": "on a small scale it can be neglected. if you stand on ground, you dont see it curved, you see it plane and straight. Thats because you are tiny comparatively"
    },
    {
        "video_name": "TC23wD34C7k",
        "Q": "9:30 What is the component of a vector?",
        "A": "there are normally 2 components to a vector, x and y. Imagine the vector was on a graph. The x component is how far the vector goes across and the y component is how far the vector goes up. Sometimes it helps if you imagine the vector makes a right triangle with the base on the x- axis. the base would be your x component and your y component would be the height of the triangle."
    },
    {
        "video_name": "_Ut-nQ535iE",
        "Q": "@11:46, (R1 + R2) / R2 is not 2 if R1 is not close to being equal to R2; right?",
        "A": "Hi Julio, Correct, the op-amp circuit s gain is determined by the resistors as : Vout = [(R1 + R2) / R2] x Vin There is nothing special about the gain of 2. It simply occurs when R1 = R2: (A + B)/B = 2 when A = B I m not sure if I answered your question. Please leave a comment below if you would like to continue the conversation. Regards, APD"
    },
    {
        "video_name": "_Ut-nQ535iE",
        "Q": "I don't understand. At 9:10 he says the gain is really large. We used this assumption to derive the equation Vo = f(Vin) but then he says if the R1 and R2 are equal, the gain is only two. So is it two or is it really large? Isn't this contradictory?",
        "A": "The gain (A) of the opamp (the triangle symbol) is huge. When you surround the opamp with resistors you get an amplifier where the overall gain is nicely determined by the resistors."
    },
    {
        "video_name": "_Ut-nQ535iE",
        "Q": "At 7:33, how can you say V+ is Vin ? Wasn't Vin=V+-V-?",
        "A": "Hello Kunal, Assuming the gain is infinite then feedback will make V+ = V-. This is a good assumption as long as the signal has a low frequency e.g. audio. Things start to fall apart when the signals get higher. Let us know when you reach these more interesting levels with op-amp. Quick question for you - have you had the chance to build and experiment with an op-amp? Regards, APD"
    },
    {
        "video_name": "NGpJPz44JYc",
        "Q": "What is zero height mentioned in this video at 4:51.\nPlease help.\nThank you.",
        "A": "I believe he is just saying that the height at which the hole is, is zero"
    },
    {
        "video_name": "NGpJPz44JYc",
        "Q": "Shouldn't you multiply all the terms by two at 6:01 -> ?",
        "A": "If you notice, he did multiply all the terms at 6:01 by 2. The terms that already had a 2 in their denominators cancelled out with the 2 being multiplied to their numerators."
    },
    {
        "video_name": "NGpJPz44JYc",
        "Q": "at 3:55 how can we say that pressure is zero.\nwhile explaining the equation of continuity sal said that this pressure is d pressure exerted BY the liquid and not ON the liquid??",
        "A": "the pressure P2 that is external is zero. That s because it doesn t work in terms of preventing the fluid flowing out of the hole. The liquid flows out without any obstacles that could be created by pressure. If say that the hole is closed the pressure does work by blocking the water flow and not letting the plug fall down. Hope it helps."
    },
    {
        "video_name": "NGpJPz44JYc",
        "Q": "I think sal made a mistake when he multiplied everything by 2 at 6:30, he forgot to multiply v2^2/1x10^6 term by 2",
        "A": "He did that one OK - before multiplying, that term was (1/2)(v2^2/1,000,000) so multiplying by 2 just cleared our the 1/2 term. At 6:33 you ll see the 1/2 term in the middle equation. He cancelled out the rho factors by dividing the equation by rho so all those terms became 1"
    },
    {
        "video_name": "NGpJPz44JYc",
        "Q": "Starting at 5:50, Sal cancels out two rho (p) on the left side of the equation, but only one rho (p) on the right side. shouldn't there still be a rho (p) on the left side of the equation?",
        "A": "Sal cancels out one rho from each term on both sides, so that s ok. Another way to think about it is to rearrange the left side first. You start with: rho*g*h + rho*(v1^2)/2 = rho*(v2^2) There is a rho in each term on the left, so we can write: rho*(g*h + (v1^2)/2) = rho*(v2^2)/2 Now, we divide both sides by rho: g*h + (v1^2)/2 = (v2^2)/2 Does that help?"
    },
    {
        "video_name": "gMKtGulgGH8",
        "Q": "At 4:15, doesn't the amine need to get deprotonated first and then the Cl get kicked off? Wouldn't the protonated amine be a better leaving group than the Cl?",
        "A": "How would you know? You have to look at a pKa table. Then you can compare the strengths of the conjugate bases. The pKa of NH4+ (ammonium) is 10. the pka of HCl is -6 or -7. Therefore, Cl- is 1x10^17 times better as a leaving group than a protonated amine. Basically, there s no comparison."
    },
    {
        "video_name": "gMKtGulgGH8",
        "Q": "6:20 why wouldn't the Cl anion take the hydrogen. why was is necessary that the dimethyl amine take it?",
        "A": "Chloride is not a very good base, which makes sense as it is the conjugate base of a strong acid (HCl)"
    },
    {
        "video_name": "lxnAwd8OXaA",
        "Q": "So with the last reaction, was that a redox reaction followed by an SN1 reaction? 5:27",
        "A": "The first part is an acid-base reaction, but it is not a redox reaction. You can know this by observing that no electrons are being transferred. You could also calculate the oxidation state for the oxygen before and after protonation and see that it is -2 in each case. You may have been confused by the fact that the oxygen acquires a positive formal charge, but this does not indicate that a redox reaction has occurred."
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "At 3:11, why do we use the 1.00 M from the CH3COONa if we are just working with the Ka and therefore pH for the 1.00 M of CH3COOH?",
        "A": "That part of the question says we have a solution that is both 1.00M CH3COONa and 1.00M CH3COOH It makes sense that we would need to include both in the calculation if they re both in solution right? So I m not really sure exactly what your question is."
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "At 1:30, why does the pH go up when the molarity of the acid goes down?",
        "A": "pH is inversely related to the concentration of hydrogen ions, which come from acids. More acid = more H+ = lower pH"
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "At 9:38, doesn't the nitrate ion affect the situation at all? Couldn't it take H+ away from H2O itself and create more OH-?",
        "A": "No, because when Nitrate does bond with an H+, it will immediately dissociate because it is a strong acid, and all strong acids go completely to their ions when in water."
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "At 1:21, How does decreasing the concentration of the hydronium ion increase the pH?",
        "A": "The less acidic a solution is, the higher the pH will be. Hydronium is an acid, so as less of it is in the products, the pH rises."
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "At 9:05, why is the concentration of NH4+ 0.35? Isn't that the concentration of ammonium AND nitrate? Is it ok to just ignore the nitrate here?",
        "A": "1 litre of a 0.35 M solution of NH4NO3 has 0.35 moles of NH4+ ions and 0.35 moles of NO3- ions. So the concentration of NH4+ is the same as the concentration of NH4NO3."
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "at 0:40 why did he put +ve and _ve sign in CH3COO- Na+",
        "A": "Because it is an ionic compound, made up of the sodium cation and the acetate anion."
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "At 7:40 How you know to set up the second equation as such?",
        "A": "Since ammonia is a weak base, it receives a proton from H2O, or water. The resulting right side of the equation is NH4+ + OH-, since NH3 gained a proton and water lost a proton."
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "at 3:17, what is the dissociation constant of CH3COONa? How do we know that it completely dissolves and there is 1M CH3COO- and 1M Na+?",
        "A": "When an ionic bond breaks down, it dissociates completely. When a compound breaks down in an acid/base reaction, it may not be complete because the water is needed to pull off the H+ to form hydronium, but in the case of Na+ it will completely fall off"
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "At 3:38 why is the mol value assigned to [H30+] a zero? I think I do understand why it is 1.00 mol for [ch3cooh] and [ch3oo-] though...",
        "A": "I in ICE stands for initial so it represents the hypothetical situation before the equilibrium is established. Hence all that is there is the 1M acetic acid and 1M acetate. The next line, C shows the changes that happens as equilibrium is reached. This is where the hydronium ion is formed as the change is +x."
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "at 6:37, your using the calculator to take a -log of 1.8 x 10^-5 to get your ph? The mcat is not a calculator baed test so, how can this done without a calculator?",
        "A": "one way is to rearrange so that pH/pOH= 5 - log(1.8). Essentially, you bring the exponent down (drop the negative, of course) and then subtract the log of the coefficient. Since the coefficients in these problems are usually between 1 -> 10, its log will be between 0 -> 1 (closer the coefficient is to 1, the closer the log will be to 0)."
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "At 11:35 , why is the x in the numerator (outside the bracket) left while the other x's are neglected taking into consideration that all the values of the x are very small and are neglected everywhere else?",
        "A": "Because you re multiplying the x outside the bracket (see 11:10), not adding or subtracting. Multiplying or dividing by 0.35 is considerable compared to adding or subtracting a tiny value from 0.35."
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "At 5:40, why do we always say that X is very very small?",
        "A": "With a weak acid or a weak base, x is usually small. But the presence of a common ion, as in buffers, forces the position of equilibrium so far to the left that x is always very, very small and it can be ignored in the buffer calculations."
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "At roughly 10:00 he crossed out H2O, why is this? Guessing it's because water is the solution that all of these wonderful things react in? Am I correct?",
        "A": "H2O is the solvent, and is therefore ignored because its concentration is essentially 1. H2O is everywhere, meaning its concentration has no bearing on the reaction."
    },
    {
        "video_name": "223KLPnJCBI",
        "Q": "At 5:17 why is x assumed to be much smaller than 1? How would I know to make this assumption in other problems?",
        "A": "Chemists are lazy creatures. They assume that x\u00e2\u0089\u00aa1 in order to avoid having to solve a quadratic equation. But they always have to check whether the assumption is valid. A common rule of thumb is that x is negligible if the initial concentration of HA divided by Ka is greater than 400."
    },
    {
        "video_name": "iLFcSfzrlMk",
        "Q": "1:17\nHow do we know for sure this t constant (half life time for potassium 40) 1.25 billion years?",
        "A": "We ve measured it."
    },
    {
        "video_name": "xuQcB-oo-4U",
        "Q": "@2:45 how do they tint the plastic",
        "A": "this paint isn t conductive?"
    },
    {
        "video_name": "xuQcB-oo-4U",
        "Q": "what is the use of jumper circuits? as specified @8:04",
        "A": "You mean a jumper wire? Jumper wires transfer power to alter the functionality of a component."
    },
    {
        "video_name": "xuQcB-oo-4U",
        "Q": "6:18 ....... Is that why a lot of wristwatches say QUARTZ on them?",
        "A": "Yes, when QUARTZ is written on a watch, it means that it uses an electronic oscillator with a piece of quartz to divide time into seconds, minutes and hours. However, there are other systems to create divisions of time such as automatic movement watches or mechanical watches."
    },
    {
        "video_name": "xuQcB-oo-4U",
        "Q": "At 0:57 how many times is the wire wound around the transformer on average?",
        "A": "well it depends how big is the transformer, the one in the video had around 6-10 coils"
    },
    {
        "video_name": "xuQcB-oo-4U",
        "Q": "at 1:55 , karl mentions about injection moulded plastic , what is it?",
        "A": "Basically, plastic injection molding is when you have a mold and you inject hot plastics into the mold and allow it to cool down to form a certain shape. Injection molding also uses other materials such as glass and metal. It is a very common process that is used in manufacturing. Hope that helps out!"
    },
    {
        "video_name": "xuQcB-oo-4U",
        "Q": "What is a diode? (At 5:29)",
        "A": "A diode is a thingamabob that goes on a circuit. It keeps electricity from going the wrong way."
    },
    {
        "video_name": "F8RCR_1jIAk",
        "Q": "At the 02:18, Jay draws bond line structure for propane.\nHow to draw methane with this shorthand (bond-line structure) method?\n\nA single point would look like a bit unclear. So i guess they apply bond-line structure to structures consisting of at least ethane-like structure. Am i right?",
        "A": "You are. CH4 is as simple as a hydrocarbon can possibly be, so it s easier to write the formula instead of trying to draw the structure."
    },
    {
        "video_name": "F8RCR_1jIAk",
        "Q": "At Jay 12:16 says Cyclohexanes are important, why?",
        "A": "They are the most common cyclo- structure encountered in organic chemistry."
    },
    {
        "video_name": "F8RCR_1jIAk",
        "Q": "At 11:25 Jay said the formula of alkanes as CnHn is it correct? Because as far as I know the formula of alkanes is CnH2n+2 ?",
        "A": "He said the formula for CYCLOalkanes is CnH2n which is correct."
    },
    {
        "video_name": "F8RCR_1jIAk",
        "Q": "At 10:15 which one of the top two's numbering is correct?",
        "A": "There is no difference. Both numberings are correct."
    },
    {
        "video_name": "F8RCR_1jIAk",
        "Q": "At 11:00 when dealing with the smaller cycloalkanes the nomenclature makes sense, but the bond angles not so much. Given that a SP3 hybridized Carbon has a bond angle of about 109.5\u00c2\u00b0 how are these smaller rings possible?",
        "A": "With quite some ring strain, and as such they are generally quite reactive. With regard to hybridisation, our model does not work here very well, it needs to be adapted somewhat. Cyclopropane can be described as being sp^5 hybridised with the 5 not literally meaning five p orbitals but referring to a certain % of p character in the hybrid orbitals instead. It gets quite mathy from there on so might be better to ignore for now. By all means look up Bent bonds on Wiki if you want to have a read though."
    },
    {
        "video_name": "3858MaULDdI",
        "Q": "At 1:01, What does Stenosis mean?",
        "A": "by plaque and cells"
    },
    {
        "video_name": "3858MaULDdI",
        "Q": "From 1:07 to 1:10 Sal says that stenosis is the narrowing of blood vessels. Isn't the formation of an atheroma also the narrowing of blood vessels? What's the difference between atheroma and stenosis?",
        "A": "I would say that the formation of an atheroma results in stenosis. The atheroma is the plaque, and stenosis is what happens in the vessel that has the plaque."
    },
    {
        "video_name": "BeHOvYchtBg",
        "Q": "At 8:41 how can you tell that acetic acid is \"not good\" at donating a proton?",
        "A": "It has a Ka value of 1.76E-5, that tells you it is a weak acid so not good at donating its ionisable proton. Strong acids have Ka values > 1 All Ka really does is tell us where exactly does the equilibrium lie for an acid reacting with water. For strong acids it s effectively 100% products, for weak acids it s more like 99% reactants and 1% products, maybe even less. It depends on the weak acid s Ka."
    },
    {
        "video_name": "BeHOvYchtBg",
        "Q": "At 8:31, the teacher left the water out, why?",
        "A": "We ignore H2O in Ka calculations"
    },
    {
        "video_name": "BeHOvYchtBg",
        "Q": "at 4:25 cl has -1 charge it's normal charge or formal charge?",
        "A": "The -1 is both its normal charge and its formal charge."
    },
    {
        "video_name": "BeHOvYchtBg",
        "Q": "at 7:00 he explains how aceticacid leaves its hydrogen to water. in the process the hydrogen leaves an electrone behind. know if you look at the picture, oxygen hasa total of eight electrons after the protolyse reaction and it is claimed that this oxygen has 1- charge. but wouldn't it be 2- since oxygen has 6 valence electrones? afterall, two extra electrons is equal to a charge of 2-. pls help me i'm confused",
        "A": "Before that hydrogen ion is lost, that oxygen has 8 electrons around it but only 6 of them belong to oxygen, 1 belongs to carbon and 1 belongs to hydrogen. After the hydrogen is lost, the oxygen still has 8 electrons around it and 1 of them still belongs to the carbon, but now that leaves 7 electrons that belong to oxygen. 6-7=-1"
    },
    {
        "video_name": "BeHOvYchtBg",
        "Q": "@6:00 You mentioned that the stronger the acid the weaker the conjugate base. Is there a way to say the reverse (the weaker the acid the stronger the conjugate base)? And if so, can the conjugate base of a weak acid ever be a true strong base?",
        "A": "For what we would call weak acids, no, they still have weak conjugate bases. For something to have a conjugate base that would be a strong base it needs to be beyond what we would call a weak acid. Ethanol for example has a strong conjugate base (ethoxide), but we would not generally call ethanol an acid as it has such a high pKa which means it hardly gives up an H+ at all."
    },
    {
        "video_name": "BeHOvYchtBg",
        "Q": "at 7:29 what do you mean by acidic proton?",
        "A": "It means a hydrogen that bases are able to remove easily"
    },
    {
        "video_name": "BeHOvYchtBg",
        "Q": "at 7:54, how did he know that acetic acid was a weak acid just by looking at it. He said it doesn't donate protons very well, but how can you tell whether or not an acid donates protons well?",
        "A": "HCl, HBr, HI, HNO3, H2SO4, HNO3, HClO4 are common strong acids, there are others but that list is enough for now. It s fine to assume anything else is a weak acid unless you re told otherwise. If you re given the Ka and its less than about 1 it s a weak acid."
    },
    {
        "video_name": "BeHOvYchtBg",
        "Q": "Starting from 7:53, the proctor stated that acetic acid is a weak acid due to it's poor H+ donation ability. So that makes acetate a strong base. My question is, why is the acetate ion a strong base when the two oxygen atoms and carbon atom can participate in resonance, or electron de-localization to stabilize the negative charge? Shouldn't resonance make the ion more stable, or am I missing a point here?",
        "A": "Acetate (CHCOO-) isn t a strong base. The strong bases by definition are those compounds with a kb >> 1 and are LiOH, KOH, NaOH, RbOH and Ca(OH)2, Ba(OH)2, and Sr(OH)2. Acetate ion is a weak base, but it s a better base than its conjugate acid (acetic acid) is. I think the point is the molecule s ability to either donate OH- or accept H+ because either of these will increase the pH of the solution and is less about resonance."
    },
    {
        "video_name": "BeHOvYchtBg",
        "Q": "at 6:22 why chlorine anion is not good at accepting protons from hcl??",
        "A": "We know HCl in water is a strong acid which must mean H+ + Cl- -> HCl is an extremely unfavourable reaction. Cl- is quite happy by itself, it has a full octet."
    },
    {
        "video_name": "gDWy3GEEcx8",
        "Q": "At 6:30, why did you name it cyclohexene and not cyclohex-1-ene",
        "A": "Both names are correct! It s just that since you have just one double bond in this molecule, it is assumed that it is located between C1 and C2. Therefore, just cyclohexene would be an easier name for the molecule."
    },
    {
        "video_name": "gDWy3GEEcx8",
        "Q": "At 8:19 can that molecule also be named 1 vinyl hex-1-ene?\nThank you!",
        "A": "The correct name is 1-vinylcyclohexene."
    },
    {
        "video_name": "gDWy3GEEcx8",
        "Q": "At around 2:30, the video counts the carbons. However, I thought that the double bond should be between #1 and #2 which would make this a pentene. Can you please explain?",
        "A": "Your primary objective is to find the LONGEST parent carbon chain. That is more important than having the double bond be between 1 and 2."
    },
    {
        "video_name": "gDWy3GEEcx8",
        "Q": "At 6:40 when the molecule is named 3-methylcyclohexene, why isn't it written as 3-methyl-1-cyclohexene? Is it just automatically assumed that the double bond will always start at 1 in a ring?",
        "A": "it should be written as- 3-methylcyclohex-1-ene. here it is assumed that the double bond is at first carbon if we dont write that -1-"
    },
    {
        "video_name": "gDWy3GEEcx8",
        "Q": "at 6:32 its is said that it should have been 3-methyl-1-cyclohexene but will it be wrong to write 3-methylcyclohexene?",
        "A": "The preferred name is 3-methylcyclohexene. You don\u00e2\u0080\u0099t need the 1 because the double bond carbons are automatically 1 and 2."
    },
    {
        "video_name": "gDWy3GEEcx8",
        "Q": "when you are trying to identify the numbering. at 7:52 the substituent take priority. However in the molecule before the methyl group does not take the priority. What determines whether or not the double bond takes the priority in numbering or not?",
        "A": "At 7:52 the substituent is not taking priority; the double bond is, as usual. The substituent just happens to be on that same carbon this time."
    },
    {
        "video_name": "gDWy3GEEcx8",
        "Q": "At 10:29 instead of calling the molecule '2-chloro-1,3-butadiene', we could also call it '2-chloro-but-1,3-ene' or 2-chloro-di-but-1,3-ene'.\n\nPlease correct me if I'm wrong. What might be all the possible names we could give to it?",
        "A": "actually....his answer is approaching coherence....many people like to shove locants into names to make their referent transparent. However, for a butadiene there is no place to insert the locants into the name without making it incoherent or unpalatable. however...2-chlorobuta-1,3-diene is also correct as you implied. you are just arranging the chunks in hard-to-pronounce-ways. which will never give you a good answer by scrambling -di- -ene- & -but- together in random ways.."
    },
    {
        "video_name": "gDWy3GEEcx8",
        "Q": "At 4:25 couldn't this molecule also be called 4,5-dimethyl-2E-hexene? As the substituted R groups are on opposite sides of the Carbon-Carbon double bond?",
        "A": "Yes, it could, At this stage, he is just interested in naming constitutional isomers. He hasn t yet gotten to the naming of stereoisomers."
    },
    {
        "video_name": "Ad0aaYixFJg",
        "Q": "Where did the 1280 degrees at 10:02 in the video come from?",
        "A": "If you go way back to the start of the video you ll see it was given to us in the question."
    },
    {
        "video_name": "l-h72j2-X0o",
        "Q": "At 3:38, if Ohm's law says that Voltage is the current times the resistance, why can you not use a resistor to lower 9V to 5V with a resistor? I have only seen it done with a 5V regulator. Shouldn't a resistor work as well?",
        "A": "Hello Addison, Correct! You could do this but, what if the input voltage changes. Also, what if the current demand of the load changed? In both cases the 5 VDC will no longer be 5VDC. You could think of a regulator such as the 7805 as a variable resistor. Electronic circuitry constantly adjusts the pass element (transistor) to maintain 5VDC on the output. Please leave a comment below if you would like to continue the conversation. Regards, APD"
    },
    {
        "video_name": "l-h72j2-X0o",
        "Q": "So I was looking through my EE notes from last semester and I see an inconsistency:\n\n@7:26 when you start labeling the Voltage sources direction for current you show that the direction of the current is into the source as opposed to out of the source! I verified with the textbook and for a voltage source the current points out from the positive terminal.",
        "A": "Depends how you want to look at the system. Conventional current (which I have been taught in college/high school classwork) does have current flow from the positive terminal, but the actual electron current flows from the negative to positive. I admit the conventional current is easier to work with in circuit analysis, but when I do electron flow models, I do have to use current flow coming out of the negative terminal."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "7:23 Before Sal said it was 333.55, but here he changed it to 335. Which one is it supposed to be?",
        "A": "333.5. He made the mistake later in the video. You get a brownie point if you can write the true value of the total energy calculation."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "I noticed that at 7:15 , it says that the heat of fusion is 333.55j/g however Sal writes 335.55j/g in his calculation which would make the whole answer wrong. Was it just his mistake or was there something that i've missed?",
        "A": "Looks like just a mistake--a transcription error."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "In \"6:30\", I would like to ask why we can change the unit of the formula from K to C?\n\"2.05 j/gk was changed to 2.05 j/gc\"",
        "A": "explica50, that is actually wrong. 1k/273c is not a conversion factor. The correlation between the two is that degrees K = degrees C + 273. i.e., 10C = 283K. To answer the original question, raywilliammiller did a great job. The only concern is the /difference/. If there is a ten degree difference on the Celsius scale, there is a ten degree difference on the Kelvin scale."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "at 1:12 , on the graph does the temperature always start at a negative number then increase to a positive number?",
        "A": "Only if the temperature scale is Celsius and the starting temperature is below 0 C."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "At 2:35, where did Sal get these figures? I thought liquid water's specific heat capacity was 4.184 J/gK?",
        "A": "4178 is pretty close to 4184. I wouldn t worry about it. I believe the difference has to do with the temperature at which the measurement is taken."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "How come Sal said he had to change 100 degree water to 100 degree ice at 2:57, when ice doesn't exist at 100 degrees? And on the graph he marks off the zero degree horizontal, where water freezes and ice melts.",
        "A": "This was a slip of the tongue by Sal. He meant to say 0 C not, 100 C."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "I think we're skipping a step at 9:45. Shouldn't we then go from 100 degree water to 100 degree vapor THEN onto 100 degree vapor to 110 degree vapor?",
        "A": "Should ve watched the rest of the video before I asked this question. Sal realizes the mistake. Very helpful, thank you! I ve been studying for my DAT coming up and Sal has been infinitely helpful refreshing my memory."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "at 00:24 sal talks about celcius and kelvin i know they are both tempretures but what is the difference",
        "A": "celsius is orientated at the changing of state of water. so when water freezes we have 0\u00c2\u00b0C, when it boils 100\u00c2\u00b0C. Kelvin starts at the absolute zero where the molecules nearly don t move annymore. so 0 K = -273\u00c2\u00b0C 0\u00c2\u00b0C = 273 K you normally use Kelvin for specifieing the change of temperatures. I hope you understand it and that there are no spelling mistakes. I m no native English speaker."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "At 5:20 , Why doesn't Sal just calculate the entire difference in temperature instead of going from -10 degrees to 0 degrees?",
        "A": "Because at 0 you have to account for a phase change, and then when you are warming liquid water you are using a different c than you use for warming ice."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "At 1:06 Sal mentions a lattice structure. What is that?",
        "A": "Its is the arrangement of the molecules in a pattern when they are in the solid state ."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "At 0:00, why is specific heat and phase change important in life. What do they use these calculations for?",
        "A": "for any different purposes , for example to figure out how much heat energy to be supplied to melt metals to pour them int molds , to calculate how much energy is need to react certain substances (chemistry ) and many more ."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "at 5:47, Sal said that to convert -10 degrees to 0 degrees, the difference is 10 degrees C.\nif i were to convert from 0 to -10 degrees C, would the difference be -10?\nif so, do i multiply -10 to find the amount of energy required??",
        "A": "Yes, the difference being that one you will get a positive sign for the answer, meaning heat is being added (from -10 to 0), while the other heat is being removed (going from 0 to -10)."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "at 6:20 pm NOW,what is fusion?!",
        "A": "Fusion is the process of joining two or more atoms together to form a larger atom. The Sun and stars do this initially with their primary fuel of Hydrogen to make all larger atoms up to and including Iron. The nuclei merge and form heavier atoms with more protons and neutrons than the original atoms that formed them."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "At 5:53 How did you get from -10 ice to 110 vapor?",
        "A": "Warm the ice using the specific heat of ice. Melt the ice using the latent heat of fusion. Warm the melted water using specific heat of water. Vaporize the water using latent heat of vaporization. Heat the vapor 10 more degrees. Add up all that energy."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "at 3:56 he says assuming the pressure is the same. would a difference in pressure effect the formula or just the melting and vaporizing points?",
        "A": "Changing the pressure could change the specific heat, heat of fusion, and/or the heat of vaporization as well as change the boiling point and maybe even the melting point. It has the greatest effect on the boiling point, but the others could be affected as well. You would use the same math to calculate how much energy was involved in the phase transition, but the values for the heat and temperature might be different."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "At 10:48 what does irrevacable mean?",
        "A": "It means incapable of being revoked, or in other words, there s no way to fix the problem. Hope that helps you."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "Why not just round the numbers instead of using 333.55 just use 334, it's easier, and less confusing, because i almost got confused at 2:44",
        "A": "The 333.55 is the number you get when you look it up in a text or online. At about 3:10 he does round it off to 333 (it should have been 334). It is perfectly acceptable to round off a tabulated value. You wouldn t want to write the atomic mass of H as 1.007 825 0 u all the time, would you? But you should keep one more significant figure than your answer requires. If you need only 2 significant figures, you can safely round to 334. If you need 3 significant figures, you can use 333.6, etc."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "why is it p.e and not k.e in 1:21?",
        "A": "potential energy you mean? probably because it has the potential to reach there but does not always."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "@9:48 Don't you have to calculate the heat of vaporization from 100 degree liquid water to 100 degree water vapor and THEN use the specific heat of vapor?",
        "A": "Never mind I m sorry I didn t watch the other part until after posting."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "At 1:51 sal says that that the heat energy is not used to increase the temperature but it its used to break the lattice structure. why does this happens so?",
        "A": "Once something has reached its boiling point, it takes even more energy to change its state of matter. For example, say you are heating a pot of water over a stove. At 100 degrees Celsius, the water begins to boil. Until all of the water boils away into vapor, the temperature will remain at a constant 100 degrees. Because the heat isn t increasing in kinetic energy but rather increasing in potential energy (through a phase change), the temperature remains the same."
    },
    {
        "video_name": "zz4KbvF_X-0",
        "Q": "Why does he always round down? I mean, like @14:40, 609.95 is way closer to 610.",
        "A": "I m not sure, although I think a lot of us disregard rounding numbers when saying them out loud."
    },
    {
        "video_name": "NRsM8kXtT5s",
        "Q": "@5:42, why did the ethanol get a plus charge after it attacked the epoxide? Is it because it gave an electron away to the epoxide?",
        "A": "The Oxygen on the ethanol molecule has a plus-1 charge on it because it is bound to 3 other atoms: the Hydrogen, the Carbon belonging to the Benzene ring, and the remainder of the ethanol molecule. Hope that clears things up!"
    },
    {
        "video_name": "Df2escG-Vu0",
        "Q": "At 1:42, what does f x prime mean? Then at 2:03, what is f a prime mean?",
        "A": "f prime of x (denoted as f (x)) refers to the derivative of a curve f(x). The long version of f (x) is lim ( f(x+h) - f(x) ) / h h>0 This could work for any variable, whether its f (x) or f (a) or f (?), just substitute a or ? in for x in the equation"
    },
    {
        "video_name": "ZrMw7P6P2Gw",
        "Q": "at 3:10 when he says \"when the currents i1 and i2 merge they combine to become i1\" did he mean when i2 and i3 merge?",
        "A": "Absolutely.. The net current is equal to the sum of individual currents though the resistors"
    },
    {
        "video_name": "ZrMw7P6P2Gw",
        "Q": "At 11:59 Sal says that 3.2 amps added with 8 amps becomes 4 amps how is that possible.",
        "A": "That s not possible, so either you misunderstood or he misspoke."
    },
    {
        "video_name": "ZrMw7P6P2Gw",
        "Q": "At 9:22, Sal mentioned that the resistance is 4 ohms, before he got to that calculation he worked it out and said it was R=1/4 which changed to R=4, why?",
        "A": "Ah, see, if you go back an re-watch the video, you ll find that it was actually 1/R = 1/4, not R=1/4. If it is 1/R=1/4, you can just flip them to get R=4."
    },
    {
        "video_name": "ZrMw7P6P2Gw",
        "Q": "In Circuits (part 2) at 8:00, Sal said that the electrons were more 'reluctant' to flow after running through a resistor. Higher resistances create larger voltage drops in series circuits, so why are the voltage drops caused by resistors in parallel circuits unaffected by the actual value of the resistor?",
        "A": "Because the voltage has to be the same because all the resistors are connected to the same two points. So instead the current adjusts."
    },
    {
        "video_name": "ZrMw7P6P2Gw",
        "Q": "I'm confused at 3:08.. i1 = i1 + i2.. does he mean i1 = i2 + i3? and then he says i1 + i2 = i1 on the other side.. (the parallel currents are labelled i2 and i3 right? which do add up to i1 right) thanks in advance",
        "A": "its just a mistake. Basically what he is trying to say is that the sum of the current in those 2 pathways is equivalent to the current of the wire, i1"
    },
    {
        "video_name": "ZrMw7P6P2Gw",
        "Q": "At 0:45, can someone explain why there is less resistance when there are more resistors without using math? Thank you!",
        "A": "More resistors in parallel means more different pathways for the current to flow. It s like when you have more lanes on a highway. More lanes gives less resistance, so more traffic flows."
    },
    {
        "video_name": "ZrMw7P6P2Gw",
        "Q": "At 5:15 of the video, Sal said that the voltage between the + and the - is the same as the voltage between the two ends of the resistors. Should it be different?",
        "A": "No, because the resistors are directly connected to the + and -. It s like the battery has really long terminals."
    },
    {
        "video_name": "ZrMw7P6P2Gw",
        "Q": "Starting at 8:44, I was confused that total resistance= 1/r, I thought total resistance =r1+r2?",
        "A": "1/r applies to parallel circuits r1+r2 applies for series circuits"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "Could you theoretically melt a rock? 12:50 :O",
        "A": "Yes, as lava (as it is called above the surface) or magma (as it is called in the Earth s core - the mantle) is as you ask --> molten rock. Molten being a hot liquid substance. Then when this lava or magma cools it forms igneous rocks such as basalt or granite as examples."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "2:33 What's that? \"S\" ?",
        "A": "s is the state description of a solid."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 4:09, i couldn't get the name of the bond",
        "A": "Polar Bonds. A polar bond is a covalent bond in which the electron pair is unequally shared."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At :19 what are the three main states of matter?\nAt 4:15 what are polar bonds?",
        "A": "1)Solid 2)Liquid 3)Gas a polar bond is a type of bond between two atoms in which electrons are shared unequally. Because of this, one end of the bond has a fractional negative charge and the other a positive charge."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "at 9:02 i was wondering why does q represent heat?",
        "A": "Q represents The Quantity of heat either gained or lost in an object"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 2:15, Sal mentions electronegativity. What would make a molecule electronegative?",
        "A": "Atoms can be electronegative, but not molecules. Electronegativity is the power of an atom in a molecule to attract electrons to itself. There are videos on the Khan Academy on electronegativity if you are not familiar with the concept."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "What could be the 4th state of matter? 0:07",
        "A": "Scientists have now proven that there are five states of matter. - Solids - Liquids - Gases - Plasmas - Bose-Einstein Condensate"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 3:21, what does he mean by multiple gases?",
        "A": "actually it was multiple oxygen. he meant that there are many oxygen atoms in a molecule. you can understand all this by learning the mole concept"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At about 18:00 you said the \"Heat of Fusion\" on the x-axis of the graph, but later on the same axis you said it would change into the \"Heat of Vapor\". I thought that the Heat of Fusion was for every molecule of every element. Is the Heat of Vapor like that too or are they both just for the water molecules?",
        "A": "heat of fusion refers to the point where the substance is liquid and solid, as when it is freezing or melting. Heat of vaporization refers to the point where the substance is liquid and gas, as when it is boiling or condensing"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 15:07, the property of matter only depends on the temperature?",
        "A": "Yes like when its cold, water would become ice and when its hot evaporation accurs"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 0:39, Sal mentioned solids being colder. The ground is a solid. When you lay no the ground in the afternoon it is really hot, but when you jump in the pool, it is all of a sudden really cold for a while. Why is this?",
        "A": "What makes you feel hot or cold is how fast heat is entering or leaving your body. The water is colder than your body, and water conducts head away from your body very well, so you get cold very fast."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 13:05, the phase diagram is drawn at temperature versus heat. At school we use temperature versus ATM(pressure). I barely understand the one at school. Would ATM be equivalent to heat?",
        "A": "No, ATM cannot be equivalent to heat. ATM stands for atmospheric pressure, it is a unit for measuring pressure. Heat is the form of energy that flows between two samples of matter due to their difference in temperature. You see, the two are very different."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "What is polar and non polar bonds that Sal mentions in 4:12?",
        "A": "polar means one atom is quite a bit more electronegative than the other. nonpolar means that there isn t a big electronegativity difference"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "What is the fourth state of matter that he mentioned at 0:08?",
        "A": "The fourth state of matter is plasma."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 13:25, why doesn't it change temperature?",
        "A": "Because the energy is going into breaking bonds rather than increasing kinetic energy of molecules."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "1:16 How does water vaporize when it is in a pot, at what temperature does it vaporize, is there chemical reaction's in vaporization, why does vaporization ocur",
        "A": "Vaporization occurs at any temperature. It is simply when the molecules on the surface receive enough energy to boil off, and is a physical change - not a chemical one (vapor is still H2O)"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "at 7:51, it says that gases just bounces around randomly because it has a lot of kinetic energy. and before stated in the video that solids have less kinetic energy thus molecules stay close to each other. So, my question is that- if they move around so much then definitely the energy would have been used ,so there will be a time when all the kinetic energy will be used up in you know moving around so why don't the gases convert into liquid after losing kinetic energy?",
        "A": "How is energy getting used up ? Where do you think it goes?"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "When it was at 0:23 i was wondering how many atoms does liquid have?",
        "A": "Depends on the amount and type of liquid."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 2:39 Sal makes \"S\" like symbols with positive and negative signs next to each atom in the water molecule to indicate that each one has a partial charge (making it polar to my understanding). Is this actually an \"S\" he is writing, or is it a special symbol for partial charge?",
        "A": "It is not a normal S , but as you said a special symbol for partial charge. Actually it is a Greek lowercase letter that is called delta, and looks like this \u00ce\u00b4. If it is not lowercased then it looks like this \u00ce\u0094, when used as symbols they have two different meanings."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "why does the graph at 15:20 flatten. please can you explain using particle theory?",
        "A": "During this flat section of the graph, all the energy that is put into the system is used to break H-bonds between the water molecules in ice, rather than to increase the kinetic energy of the individual particles (which would equate to an increase in temperature). Thus, the temperature stays constant while ice converts to liquid water."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "It is very confusing around 15:00 when Sal is making his graph. He says that you heat up the water, but the temperature doesn't get higher. What's that supposed to mean?",
        "A": "The energy is used to change the state of matter rather than increase the temperature."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 10:05, Sal says \"the change in enthalpy is the change in heat content\"\nIsn't 'enthalpy': the energy that is stored in bonds?\nLike in the 'enthalpy of fusion' or the 'enthalpy of vapourisation' it is just the extra energy needed to break the bonds, but the heat doesn't really change!\nSo why would enthalpy be refered to as 'heat content'?",
        "A": "You can take change in heat content to mean quantity of heat transferred."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "at 4:00, why does he add the oxygen and hydrogens?",
        "A": "he tried to make a state of water structure"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "at 19:15 did Sal mean 0 celsius water or 0 celsius ice?",
        "A": "Ice and liquid water coexist at 0 C. The temperature doesn t move from 0 C until either all the water has frozen or all the ice has melted."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "When Sal draws the water molecule at 1:41, shouldn't both loose pairs of electrons be on top of the O?\nIf he draws one pair on top, and one pair on the bottom, that would essentially make the water molecule a linear molecule and not a bent one like it's supposed to be.",
        "A": "Well, when he draws it on a 2 dimensional surface, he may not be showing the three dimensional nature of the 2 lone pairs and the 2 bonded pairs. The 3D shape would be tetrahedral, with angles of 109.5 degrees from one another in 3D. Usually it is drawn with both lone pairs above or both below the bonded pairs, but not always."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "at 12:17 what is kelvin?",
        "A": "Kelvin is a unit used to measure temperature. It used to be called degrees kelvin, but then they changed the name to just kelvin. A kelvin is the same size as a degree celcius. But zero kelvin is not the freezing point of water. Zero kelvin is absolute zero, the coldest something could theoretically be. Absolute zero is at minus 273.15 degrees celcius, or 0 kelvin."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 0:23, Sal shows 3 states of matter, solid,liquid,and gas. But if you search it up at science websites, and many other places such as google and bing, they say there is a 4th state of matter (PLASMA).So is plasma really a state of matter?",
        "A": "Yes, plasma is considered the 4th state of matter, but note there are other states of matter that do occur in extreme conditions (e.g. Bose\u00e2\u0080\u0093Einstein condensates and neutron-degenerate matter)."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 13:07 he was talking about how the temperature stays the same for a short amount of time... Can someone explain that to me?",
        "A": "Adding heat (energy) to the system normally means an increase in temperature. However when you change phase (solid to liquid, or liquid to gas) you need energy to break the bonds between molecules. This bond-breaking energy is absorbed so the temperature won t increase until all the bonds are broken. Once the bonds are broken the temperature will increase again. Sal does explain this later in the video."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "0:01, how many states of matter are there?",
        "A": "Scientists have now proven that there are five states of matter. - Solids - Liquids - Gases - Plasmas - Bose-Einstein Condensate"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "at about 9:22, Sal said the word enthalpy. I still don't understand it's meaning. Please help me",
        "A": "Enthalpy is defined as the internal energy of a system plus the product of its pressure and volume. This cannot be directly measured, so we usually speak of changes in the enthalpy. Enthalpy is also called heat ; however, heat is too vague of a term and is not preferred."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At around 11:13, I am confused by the graph that shows temperature versus heat. Isn't heat and temperature the same thing?",
        "A": "Heat is a form of energy measured in Joules, where temperature is the intensity of that energy in a given body/system while using a comparative scale (degrees fahrenheit, celsius, etc.)."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "at 9:02 i was wondering why does q represent heat?",
        "A": "Because in the future you will probably use the equation Q= M*C*(Change in Temperature) Where Q is heat, M is Mass and C is Specific heat"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "at 2:41 what is the symbol Sal draws?",
        "A": "S for slightly :p"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At about 15:20, what are the periods in which the temperature does not increase called?",
        "A": "The lower plateau represents melting if you are gradually adding heat and freezing or fusion if you are gradually moving heat. The upper plateau represents vaporization if you are gradually adding heat and condensation if you are gradually moving heat."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "8:25 Sal said the liquid form of water is more dense then the gas. Isn't density an intensive property? So, it should have the same density right??",
        "A": "as a liquid turns into a gas, its molecules begin to spread apart very very far from eachother. this is why a gas is less dense than a liquid. think of a hand full of water vs. a hand full of steam, all gaseous forms of matter are less dense than their liquid state."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 4:04 the bonds between the molecules are described as polar but what does polar mean?",
        "A": "A polar bond is one where electrons are NOT shared equally. One end of the bond is slightly positively charged, while the other end of the bond is slightly negatively charged."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 8:30, how is liquid more dense than gas?",
        "A": "Liquid is more dense than gas because the molecules in a liquid are more densely packed then the molecules in a gas, so liquid would be more dense than gas."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "At 0:04 Sal mentions a fourth state of matter, what is it?",
        "A": "Plasma is the 4th state of matter"
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "He drew the H2O molecule at around 1:45, but since there is an unshared pair of electrons on the top and an unshared pair on the bottom, wouldn't this be a linear molecule? It would have the same force pressing upwards as downwards. Is it just that he didn't draw it as a linear molecule?",
        "A": "Actually, the molecular geometry would be bent. So even though it doesn t really matter where the electrons are drawn, its geometry would be represented with the two pairs of electrons next to each other, pushing the shared pairs out through forces of repulsion, forming a bent form."
    },
    {
        "video_name": "pKvo0XWZtjo",
        "Q": "17:30 Falling = work?",
        "A": "work=force *displacement so anything that on application of force gets displaced can be said to have done work"
    },
    {
        "video_name": "2o-Sef6wllg",
        "Q": "At about 2:00 in this lesson is Sal saying that a year is not based on the time it takes for the Earth to orbit the Sun, but the time it takes the for one precession orbit (the period from one perihelion to the next) to occur?",
        "A": "I have always been taught that a year is based on the time it takes to orbit the Sun. I have heard that a year for other planets would be different since their orbits take differing amounts of time. Based on this a Mercury year is shorter that an Earth year and a Neptune year is longer. Based on the description of a year given in this video how long is a year on a planet like Mercury where there is no change in seasons?"
    },
    {
        "video_name": "2o-Sef6wllg",
        "Q": "At 6:00, is the plane of our orbit perfectly flat? Or is there some deviation up or down along the way (from the reference of looking at it from the side)?",
        "A": "every orbit is flat in the sense that it is on one plane. but the orbits of other planets are not on the same plane as Earth - they are tilted a few degrees from the plane of Earth s orbit."
    },
    {
        "video_name": "BrLbtCnrPaA",
        "Q": "at 2:20 why make the Fg of 5kg positive? shouldnt it be negative since its downward and down is usually negative? I understand that we decide which direction is + or - but still am I missing something?",
        "A": "the body is moving in that direction so they take Fg of 5kg positive and the forces opposing the motion of the 5kg body will be negative."
    },
    {
        "video_name": "BrLbtCnrPaA",
        "Q": "At 5:15 wouldn't acceleration be positive because the system is accelerating down indicated by curved line with +ve sign.",
        "A": "Hi Raju. The curved line indicates the direction of motion of the system of the two masses . For the system, down is +ve. But if we view it normally, down is treated as -ve. Sure, you could change every other sign and use down as positive and up as negative, but it s still the same thing. Hope this helps!"
    },
    {
        "video_name": "Yiy84xYQ3es",
        "Q": "at 3:40, I don't understand fully why the halogen attack from the bottom instead of the top? Shouldn't it also attach themselves together because one halogen is positive and the other is negative thus attracting, or if not attracted, helps each other to be closer thus being beside each other on the ethene?",
        "A": "While there s a formal positive charge on the halogen it isn t electron deficient. Sterics stop the halide from attacking the top."
    },
    {
        "video_name": "Yiy84xYQ3es",
        "Q": "At 7:56, why aren't they Diastereomers?",
        "A": "They aren t diastereomers because they are enantiomers. Diastereomers are optical isomers that aren t enantiomers."
    },
    {
        "video_name": "Yiy84xYQ3es",
        "Q": "At 2:51 why is the carbon with the purple bond on the right have a positive charge? It is bonded just like the carbon with the magenta bond on the left. Is it completely arbitrary? Or could the magenta carbon the left also form the positive dipole while the purple carbon to the right remain neutral?",
        "A": "Correct. Both carbons have a partial positive charge. The video just picks one of the carbons as having a partial positive charge in order to explain the mechanism."
    },
    {
        "video_name": "VFtOcdXeP0Y",
        "Q": "At 2:11 he mentions the 200,000 years old patch of sea gras. How do they know how old it is?",
        "A": "Scientists tested its radioactive materials and using a very complex equation, and volia! They figured it out."
    },
    {
        "video_name": "00qYQahwuSQ",
        "Q": "At around 2:00, he says the right carbon gets a partial positive charge. Why isn't it a full carbocation?",
        "A": "Cuz the double bond still exits. So the pi-electrons belongs to both 2 carbons. :)"
    },
    {
        "video_name": "00qYQahwuSQ",
        "Q": "At 6:20, why does the hydro peroxide ion attack the trialkylborane if the boron doesn't have a positive charge or anything? Thank you!",
        "A": "Remember that the HOO\u00e2\u0081\u00bb is in solution and colliding with billions of other molecules every second. Most of the collisions result in no reaction. The HOO\u00e2\u0081\u00bb isn t actively looking for something to react with. But the B atom in the trialkylborane as an incomplete octet (only 6 valence electrons!). If the HOO\u00e2\u0081\u00bb happens to hit a B atom, it can form a fourth bond to the B and give the B an octet."
    },
    {
        "video_name": "00qYQahwuSQ",
        "Q": "At 7:05, why are the electrons in the alkyl group attracted to the oxygen? I thought the oxygen is nucleophilic, not electrophilic.",
        "A": "oxygen being highly electro negative attracts the cloud of electrons towards itself. (even with complete octate) think of it to be a greedy man who has acquired the wealth needed by him but desires for more."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "at 1:33, what is the force?",
        "A": "London Dispersion Forces. Sal explain it further at 3:13"
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "at 9:18 Sal says that flouride is one of the most electronegative elements.\nIsn't it actually the most electronegative element?\nCould somebody confirm or clarify?",
        "A": "Fluorine is the most electronegative element, be careful as fluoride is the ion of fluorine (fluoride --> F^-1)."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "I'm familiar with London forces and am happy with the ideas behind them, but there has always been one thing that has not been clear. If they are caused by random movement of electrons (4:26). Surely electrons will produce - / - or + /+ repulsion forces as often as they produce +/- attraction and this would even itself out over time. Why is it that London forces are an attractive force?",
        "A": "The electron movements are correlated. That is, a + in one molecule induces a - in the other molecule, so + / - interactions are much more common than + / + interactions, and they do not average out over time. Also, the force is inversely proportional to the sixth power of the distance between the two molecules, so they have to be quite close together for the force to be significant."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "Sal said at \" 6:39 \" that oxygen, nitrogen and fluorine are 'super-duper' electron hogs. But then, how come there is a triple covalent bond in an N2 molecule ?",
        "A": "A covalent bond in a molecule occurs when two atoms share an electron. In a triple covalent bond, the nitrogen atoms are sharing three electrons. Since nitrogen is a super duper electron hog, the nitrogen atoms are hogging electrons by sharing each others electrons."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "At 4:26 Sal says that for a small amount of time one side of the Neon atom is positive and the other negative. How long is this time?",
        "A": "It is extremely short. I was unable to find a specific time for this but, since the electron is a wave, I would guess that it is on the order of nanoseconds or picoseconds."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "Around 1:00 neon is illustrated as having essentially no spare electrons. How does this translate into neon lights working? Don't they need to conduct, thus have spare electrons?",
        "A": "Neon lights work because when an electric current is run on a wire through them, the neon electrons get excited and give of energy as light."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "at 6:55pm,what is dipole induced dipole force?",
        "A": "Induced, put very simply, means created. Just like how a magnetic has opposite poles which attract each other, a dipole has opposite charges that attract each other, creating an electric field that exerts a force on any nearby charged object. Dipole, consists of two oppositely charged poles at either of its end -just like a bar magnet having a north and a south pole."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "At 9:25 Why are N , O , and F the most electronegative atoms on the periodic table?",
        "A": "The effective nuclear charge is a lot greater as compared to the atomic radius. And also the factor of needing just 3,2 or 1 electron to become completely stable helps."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "At 11:09, Sal said that hydrogen bond is a kind of van der Waals' forces, is that really true? My teacher said van der Waals' forces are much weaker then other types of bonds like hydrogen bond, covalent, ionic etc. It is quite confusing.",
        "A": "At about 5:30, Sal says that van der Waal forces consist of all types of molecular bonds that are not ionic or covalent (so hydrogen bonds are a type of van der Waal forces). Your teacher may have been referring to the london dispersion bonds that Sal was talking about. London dispersion bonds are the weakest of the van der Waal forces. So they are weaker than hydrogen bonds."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "At 9:51 Sal said that Flourine has a partial positive charge.But that's not possible as Flourine is more elctronegative than hydrogen,so it would attract the electrons towards itself.\n\nCan somebody explain this?",
        "A": "In the annotations it does state that this was a mistake on his part it is meant to be negative but for some reason the annotations or notes don t show up when the video is on full screen."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "At 3:20, Sal said the Neon atom \"just happens\" to have the electrons weighted over to one side (or there abouts). Does this just happen randomly when electrons orbit around? If so, am I correct in thinking that every atom has this happen occasionally (where electrons are 'weighted' to one side')?\n\nThanks.",
        "A": "Yes, you re right, this does just happen randomly and can happen in any atom."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "At 9:41 shouldn't Fluorine have the negative charge since electrons are attracted towards it? And isn't the symbol of Fluorine F, and not Fl?",
        "A": "@Janice But Fluorine is more electronegative. Hence electrons should be attracted towards Fluorine, not Hydrogen."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "sal has mentioned fluorine partially positive and hydrogen partially negativeat 10:35\nIsn't that wrong?",
        "A": "Yes, that\u00e2\u0080\u0099s wrong. And the symbol for fluorine is F, not Fl (for flerovium)."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "at 9:40 Sal says the hydrogen has a negative charge and F has positive charge I don't understand doesn't the fluorine want to hog more electrons?",
        "A": "Sal made a mistake. The charges should be reversed."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "It seems to me that the fluorine partial charge (9:35) should be negative instead of positive.\nIs this correct?\n***",
        "A": "You are correct. There was a correction in the annotations. Also, the symbol for Fluorine is F -- the symbol Fl is for flerovium"
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "at 5:47 Sal said that the noble gases only experience london dispersion forces , if i heard right. but doesnt Xenon form compounds with flourine ?",
        "A": "Sure, and in some of those molecules there will also be dipole-dipole forces as long as the molecule isn t symmetrical. But between two noble gas atoms there is only LDF."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "At 9:30, Sal says that the H-F attraction is an example of a super strong dipole-dipole interaction. Isn't it an example of a polar bond? I'm really confused. Isn't dipole-dipole interaction between 2 molecules?",
        "A": "It is. He must ve meant H of one molecule and the F of another. Within the molecule, it is an example of polarity."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "With London Dispersion Forces ( 4:16 ) - wouldn't the atoms repel each other due to having electron heavy or electron light sides facing each other as often as they attract, resulting in no net force at all?",
        "A": "No, electrons have virtually no mass (they do have mass and I don t know the constant off the top of my head). So when the electrons are zipping around, one side becomes more positive (only slightly) and the other side of the atom becomes more negative. This slight charge causes a very weak bond to be created called London Dispersion Forces."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "in 1:22 sal said that neon has a liquid form, is it dangerous?",
        "A": "no, it isn t. it is useful"
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "At 10:11 isn't Fluorine supposed to be partial negative and hydrogen be partial positive?",
        "A": "Yes it should be that way..."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "At 9:40, Sal says that in an HF molecule, the F would have a partial positive charge and the H would have a partial negative charge, but if the F is the one hogging the electrons, wouldn't F be negative and H be positive?",
        "A": "Yes, you are correct. There was a correction that appeared onscreen, but perhaps it is not showing on your device. Compounds of F usually have a partial negative charge and hydrogen often has a partial positive charge (though not in hydrides). BTW, Fl is the symbol not for fluorine but for flerovium, which is element 114 and the most recent addition to the periodic table (though we should be getting two more elements fairly soon)."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "At around 10:45, shouldn't the F be partially negative and the H be partially positive due to the electron spending more time around the F?",
        "A": "Yes, Sal got that backward. Remember that Sal is not a scientist."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "7:15 what is the curly \"s\" like symbol Sal uses drawing the charges?",
        "A": "That is the lower case Greek letter delta ( \u00f0\u009d\u009c\u00b9 ). It means, in this case, partial. So \u00f0\u009d\u009c\u00b9 \u00e2\u0081\u00bb means a partial negative charge and \u00f0\u009d\u009c\u00b9 \u00e2\u0081\u00ba means a partial positive charge."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "In 9:40 sal says flourine is partly+ then why chlorine is partly- ?",
        "A": "He made a correction- Fluorine is more electronegative than Oxygen so Fluorine is partly negative. Always refer to the periodic table when figuring out electronegativity."
    },
    {
        "video_name": "8qfzpJvsp04",
        "Q": "At 09:45 isn't flourine meant to be delta negative instead of delta positive? Since it's more electronegative than the hydrogen",
        "A": "Yes indeed you are correct. The Fluorine should be delta negative. This is also shown in the video as a notification around 09:43."
    },
    {
        "video_name": "q3g3jsmCOEQ",
        "Q": "At 8:35, why does he draw a vector for the c-h bond if c-h is a nonpolar bond? Do you draw vectors for any bond as long as there is an electronegativity difference between them?",
        "A": "Because there is still a bond dipole there, it is just very negligible. Yes any difference in electronegativity means there is a dipole. But if the difference is low we usually ignore it."
    },
    {
        "video_name": "q3g3jsmCOEQ",
        "Q": "What is \"Debyes\"? 1:10",
        "A": "Im not sure but I think it is a non-SI metric unit of electric dipole moment (that is a measure of the separation of positive and negative electrical charges in a system of electric charges, that is, a measure of the charge system s overall polarity). The unit is named after the creator, Debye. The debye gives a convenient unit for molecular dipole moments."
    },
    {
        "video_name": "q3g3jsmCOEQ",
        "Q": "At around 5:50, why don't the two O-H dipoles cancel each other out, just like in CO2?",
        "A": "Because CO2 is a linear molecule while water is not because of oxygen s two lone pairs of electrons."
    },
    {
        "video_name": "q3g3jsmCOEQ",
        "Q": "Around 5:39, why are the dipole moments of the two unbonded electron regions pointed away from the O atom? I feel like they would be pointed towards the O atom because the electron density is in the O atom, which would mean that the net dipole moment for the water molecule is zero.",
        "A": "Because electrons will repel each other as like charges repel, so electrons near the lone pair of electrons of oxygen will shoot away from the oxygen atom"
    },
    {
        "video_name": "q3g3jsmCOEQ",
        "Q": "@ 1:25 Jay calls HCl a covalent bond, but wouldn't it be ionic?",
        "A": "It s a covalent bond as the electronegativity between Hydrogen and Chlorine is less than 1.7"
    },
    {
        "video_name": "q3g3jsmCOEQ",
        "Q": "Why is Hydrogen not the most electronegative element? It needs one electron to complete its shell and has a very small radius. This confused me because It was mentioned that Hydrogen is less Electronegative than Carbon at @8:20",
        "A": "Being small isn t the only factor though. Fluorine is also a small atom but it has 8 more protons than hydrogen so its has a significantly higher nuclear charge."
    },
    {
        "video_name": "q3g3jsmCOEQ",
        "Q": "At 5:23 how do the lone pairs experience a dipole?",
        "A": "Lone pairs have a dipole because they are negative, with the nucleus being the positive end."
    },
    {
        "video_name": "q3g3jsmCOEQ",
        "Q": "At 5:19, why do the lone pairs go upwards instead of towards the central atom? In my chemistry book, all the lone pairs (vectors) point to the central atom instead of away from it..",
        "A": "They are repelled by the other electrons on the other bonded atoms."
    },
    {
        "video_name": "q3g3jsmCOEQ",
        "Q": "At 5:21 to 5:29 - I could not understand why the lone pairs of Oxygen will be pulled in opposite direction. I mean, Oxygen is more electro-negative than H. So Electrons in bond with H will be attracted towards Oxygen. But I could not understand the theory with the lone pairs of Oxygen. Why they have positive sign near Oxygen and point in opposite directions?",
        "A": "A lone pair even by itself contributes to the dipole moment of a molecule. The nucleus is the positive end, and the electrons re the negative end. In water, then, there are four dipoles: the two O-H dipoles and the two lone pair dipoles. We often talk about just the bond dipoles, but the lone pair dipoles are still there."
    },
    {
        "video_name": "d4bqNf37mBY",
        "Q": "why does sal write 30.48% of O2 as .3048 4:04",
        "A": "30.48% is the same as 0.3048, just like 50% is 0.5. % means per cent, or per 100. 30.48/100 = 0.3048. Watch sal s vids on percentages if you need to."
    },
    {
        "video_name": "d4bqNf37mBY",
        "Q": "at 02:28 can someone please explain how did sal calculate the no. of moles of nitrogen?",
        "A": "We have : # of grams Nitrogen (2.1 kg * percentage by weight of N2). Weight of N2 is 28 amu, which means that 1 mole of N2 molecules will weigh 28 g. So if we divide weight of nitrogen in the problem by the weight of 1 mole of N2 molecules (28g), we get the number of moles."
    },
    {
        "video_name": "d4bqNf37mBY",
        "Q": "hey a question just striked my mind when sal said about hydrogen at 6:00 ..\nas we know that in most of the cases a hydrogen atom consists of an electron revolving around the proton(as there are no neutrons).. so my question is that.. IF WE TOOK AN ELECTRON NEAR A ISOLATED PROTON THEN WILL I BE ABLE TO CREATE HYDROGEN ATOM?",
        "A": "This happens all the time. A proton is also a positive hydrogen ion."
    },
    {
        "video_name": "d4bqNf37mBY",
        "Q": "at 5:46 why doesn't it have a neutron?",
        "A": "Neutrons are important to the nucleus. Most importantly, neutrons hold the nucleus together."
    },
    {
        "video_name": "d4bqNf37mBY",
        "Q": "where did the different Rs come from?? on 8:00",
        "A": "The constants are mainly used so that you can cancel out units and translate all the different units to the unit that you are interested to solve for. Hope this helps :)"
    },
    {
        "video_name": "d4bqNf37mBY",
        "Q": "at 02:28 can someone please explain how did sal calculate the no. of moles of nitrogen?",
        "A": "I recommend you go watch the video called The mole and avogadro s number . It explains conversion between grams and moles."
    },
    {
        "video_name": "d4bqNf37mBY",
        "Q": "At 8:55, Sal says to divide both sides of the equation by 4, but only divides the 100 moles by 4. Shouldn't the R rate (8.3145) and 273K also be divided by 4?",
        "A": "I had the same question, thanks for the clear answers!"
    },
    {
        "video_name": "d4bqNf37mBY",
        "Q": "at 2:22\nshould we convert 66.67% to deceimall first (.6667)(2100g) to get 1400 cus the other wway around would give us (66.67)(2100) = 140,007. Whats the correct way of thinking about this?\nThanks again.",
        "A": "Yes, convert to decimal first. Just note though, that 66.67% IS 0.667. There is no real conversion. In fact, some calculators have a percent button which make the conversion for you."
    },
    {
        "video_name": "d4bqNf37mBY",
        "Q": "at 7:52, where do you get these R values from?",
        "A": "R is a constant called the Universal Gas Constant. You should be given its value in a test or exam situation but, for a homework assignment, you may have to look it up in your text or on line."
    },
    {
        "video_name": "d4bqNf37mBY",
        "Q": "Hey guys, at 9:03, Sal divides 100 by 4 to isolate P on the left side, my question is: Why didn't he distributively divide 4 amongst the other values on the right side?",
        "A": "If you were to multiply everything out first on the left then divide by 4 you would get the same answer. So, I think he just divided 100 by 4 since it was quicker."
    },
    {
        "video_name": "d4bqNf37mBY",
        "Q": "At 7:56, Sal scrolls down to show us the three types of \"R\"s. My high school offers this Chemistry course in high school so I am doing a little prep work by watching Chemistry videos before starting the course. My question is, when tests come up regarding this topic, do we generally have to memorize these \"R\"s or are they given to us. I am asking since these look very daunting if we have to memorize these. I know I will be going to a different school but I want to know in your case what happened. Thanks.",
        "A": "It is not as rigorous as it seems. Unless you have a very strict teacher, you will probably be given formula sheets. If you aren t given formula sheets, you tend to pick up alot of the more difficult topics naturally as you hear about them in class, rather than brute force memorization."
    },
    {
        "video_name": "UhIwMAhZpCo",
        "Q": "at 5:10 is told that it takes energie to form heavier elements then iron, why is that?",
        "A": "It is primarily because of the size of the atomic nucleus. The range of the strong nuclear force binds the atoms smaller than iron together strong enough to produce an energy of bonding that helps overcome the proton s repulsion. The larger atoms need more energy to force the pieces together to make them. The fact iron is the pivot point is just because of the relative strengths of the strong nuclear and electromagnetic forces."
    },
    {
        "video_name": "UhIwMAhZpCo",
        "Q": "5:50.. Isn't it that Li, Be, and B are formed from heavier elements breaking down from cosmic rays? Not in the stars themselves?",
        "A": "Most of those elements that we come across on Earth are produced in such a manner. But they can also be fused in stars. The difference is that the conditions inside stars that produce these elements also burns them up just as quickly, so no lasting concentration of them is really produced."
    },
    {
        "video_name": "UhIwMAhZpCo",
        "Q": "At 3:58-4:03 Sal says \"you have a carbon core and around that is a helium core\". Does he mean to say around that is the helium layer or is the statement \"a helium core\" correct.",
        "A": "yes he means that"
    },
    {
        "video_name": "UhIwMAhZpCo",
        "Q": "Around 5:24 Sal mentions endothermic process, what is an endothermic process?",
        "A": "Endothermic means it makes it s own heat. Exothermic means it takes heat from outside."
    },
    {
        "video_name": "UhIwMAhZpCo",
        "Q": "At 1:56 Sal says that our Sun's lifespan is going to be 10 or 11 billion years, but how can we know how long will it take for a star to fuse all of its elements?",
        "A": "We make estimates by studying other stars we see in the sky, as well as by modeling what we think is happening inside the star"
    },
    {
        "video_name": "UhIwMAhZpCo",
        "Q": "At 6:35 mention is made of the fact that the star with an iron (FE) core cannot fuse anymore; is it that we have not observed that reality of iron fusing or we know definitively that iron will never fuse?",
        "A": "That is a slight misstatement. Iron can fuse further in a star. But there is no energy gain from that fusion so a star cannot sustain itself from these reactions."
    },
    {
        "video_name": "UhIwMAhZpCo",
        "Q": "Why does helium just fuse directly into carbon? Doesn't carbon have six protons and helium only have two? Shouldn't it fuse into beryllium, which have four protons? I was kind of confused when Sal mentioned that at 3:55. Another question: Why is the oxygen layer under the neon layer? Doesn't the lighter elements end outside and the heavier inside (5:41)?",
        "A": "Only one isotope of beryllium is stable, Be-9. Be-8, the product of two alpha particles fusing, is extremely unstable and will decay right back into two alpha particles. All other likely produced isotopes of beryllium are also unstable and will have a decay chain that leads right back to helium. So, the next likely and stable product in stellar fusion is via the triple alpha process, where three alpha particles fuse into carbon-12."
    },
    {
        "video_name": "UhIwMAhZpCo",
        "Q": "At 5:45, how does the other elements, such as lithium, are formed? in the exact same process as carbon, for example? is there a reason why these elements are not noted in the diagram?",
        "A": "Lithium, Beryllium and Boron can be produced in stars. But they are easily burned at temperatures less than stars burn at (brown dwarfs are even capable of lithium burning), so they are quickly burned up after being generated. Most of the elemental occurrences of these elements aren t generated in stars but via other nucleosynthesis process, primarily cosmic ray spallation."
    },
    {
        "video_name": "UhIwMAhZpCo",
        "Q": "at 1:24, how do we know that at this stage it realeses more energy than the sun? we did not see it happen, did we?",
        "A": "Actually, we (or at least the astronomers) can observe stars in various stages of their life cycles, and of various masses. They compare these stars absolute magnitude to the sun s to get these comparisons, too, so distance won t mess up the results."
    },
    {
        "video_name": "udgMh3Y-dTk",
        "Q": "At 4:42, what if the velocity was not constant when it moved 2m up (but even after 2m it was still moving) what would be the work then?",
        "A": "Well then the work would be calculated like the example given before using KE= 1/2 (mv^2). We would need to know the velocity to calculate the KE (correct me if I m wrong)"
    },
    {
        "video_name": "udgMh3Y-dTk",
        "Q": "At 3:37,\nIf mg=39.2N Downwards You have to pull it with more than 39.2N in the opposite direction to move it!\nIsn't it?",
        "A": "Yeah but it is moving with a constant velocity that means that the extra energy you used to lift it is converted into the kinetic energy (thus constant velocity) and now the force you re exerting on it equals it s weight."
    },
    {
        "video_name": "udgMh3Y-dTk",
        "Q": "At 4:33 he says no net work due to no change in kinetic energy. Is there not a change in potential energy that would account for work being done?",
        "A": "I also don t understand this very well. Hopefully someone can correct my possibly wrong answer. I think potential and kinetic energy are different. The trashcan has potential energy at 2m above the ground, but no kinetic energy. When you would drop the traschan, then you d have potential energy being converted into kinetic energy and work would be done by gravity? Would appreciate if someone could back this up or correct it! :)"
    },
    {
        "video_name": "udgMh3Y-dTk",
        "Q": "at 3:33 and 3:35 y do we need to push with force equal to the weight can i didnt understood plz explain this clearly to me",
        "A": "we need to apply the force at least equal to or more than weight to produce displacement. if we don t push it by enough force it won t displace. hope that helps!"
    },
    {
        "video_name": "udgMh3Y-dTk",
        "Q": "At 4:06, I don't understand why the force lifting the trash can up is 39.2N. If it's 39.2N up and 39.2N down, would the forces balance on both sides and the trash can would not be lifted up? Doesn't the upward force have to be greater in some way for it to move in that direction?",
        "A": "He says that the trashcan is lifted with constant velocity. It would take some work to begin the movement initially but to continue its movement with constantly velocity requires no work. Think of Newton s first and second laws."
    },
    {
        "video_name": "udgMh3Y-dTk",
        "Q": "Can anyone explain what the Normal Forces is at 0:46?",
        "A": "the normal force acts perpendicular to the ground in this case the ground is flat so it would cancel out with the weight force"
    },
    {
        "video_name": "udgMh3Y-dTk",
        "Q": "At 1:50, how can the work done be a negative value when work is a scalar quantity? I understand that cosine of 180 is -1, but is the negative value still relevant?",
        "A": "Yes. It means that the force and displacement vectors are opposite. So, lifting a crate up means gravity is doing negative work and you do positive work. If you lower a crate, the opposite is true."
    },
    {
        "video_name": "udgMh3Y-dTk",
        "Q": "At 4:34, since the kinetic energy of the trashcan doesn't change, is it implied that the trashcan was not lifted from rest?",
        "A": "yes."
    },
    {
        "video_name": "udgMh3Y-dTk",
        "Q": "At 2:15 why is the angle listed at 90 degrees and not 270 degrees? I'm watching all of the Khan Academy videos on physics, and in an earlier physics video on vectors, they said that vector angles are measured in a counter clockwise direction starting from the east.",
        "A": "It does not matter. There certainly is no law of physics that says angles must be measured that way."
    },
    {
        "video_name": "aobWzlXIooQ",
        "Q": "So why is the first ray of light at around 1:10 bent \"AWAY from the normal\" (as if the eye's lens' refractive index was LESS than the refractive index of the air), and therefore angles AWAY from the back of the eye, but then the second ray of light bends \"TOWARD the normal\" (as if the eye's lens' refractive index was GREATER than the refractive index of the air), and therefore angles TOWARD back of the eye?",
        "A": "Maybe it s a mistake in the drawing? Either way, the end result is the same. The right visual field will hit the nasal side of the right eye and the temporal side of the left eye, and same for the left visual field."
    },
    {
        "video_name": "aobWzlXIooQ",
        "Q": "At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?",
        "A": "When the sun is blocked out in an eclipse, the moon may seem to be huge, but that s just our perception from Earth. From where we stand, the moon is far closer than the sun, so it appears larger and seems to cover the sun. Things like this happen in everyday life, too. Next time you re in a busy place, look at someone who s far away. If you hold your hand up, your hand may seem to cover the faraway person, even though you know your hand is much smaller. It s all in the way you perceive things."
    },
    {
        "video_name": "HahjsBApxLE",
        "Q": "At 0:10, is the snell's law applicable to all types of light rays like uv light, etc",
        "A": "Yes. It is applicable to all types of electromagnetic radiation."
    },
    {
        "video_name": "HahjsBApxLE",
        "Q": "At 8:52 can we keep it as 3.00x10^8 m/s instead of 300, 000, 000?",
        "A": "They are two different ways of writing the same thing."
    },
    {
        "video_name": "HahjsBApxLE",
        "Q": "at 7:11 wouldn't the angle of light exiting the 'denser' (without a use of a better word) material be less than that of the angle of light in the denser material? Seeing as (using the car analogy) the front left tire would exit the dense medium first wouldn't this make the car head westerly as it exits?",
        "A": "Make sure you are measuring the angle the right way. We measure it against a line that is perpendicular to the boundary between the materials. We call this the normal line. Light going from air to water bends towards the normal. Light exiting water bends away."
    },
    {
        "video_name": "RpOHZc6cDIw",
        "Q": "\"5:28\" no make sense",
        "A": "Is it easier to look at a bunch of bright lights spread out or a bunch of bright lights put together? Mass produces gravity like a light bulb produces light. The intensity of the gravity functions the same as the brightness of the light. If you put the same number of lightbulbs in a smaller area, that area will get brighter, just the same as if you put the same amount of mass in a smaller area, it will have a larger gravitational force."
    },
    {
        "video_name": "RpOHZc6cDIw",
        "Q": "Ok, if you were in the middle of a MASSIVE object like 10,000 times bigger than the sun, then would you be completely torn apart? or do i have this backwards? here's an example: 7:23",
        "A": "If it had the same mass yes But other stars do exist which are much bigger than the sun"
    },
    {
        "video_name": "oIZV-ixRTcY",
        "Q": "At 6:30 , Sal says hat those astronauts are in continuous free-fall. If is it so, the should always be stuck to that face of the shuttle that faces earth. But they appear to float around like there is no gravity up there. Why is t so ?",
        "A": "They space shuttle is also in free fall. They all fall together."
    },
    {
        "video_name": "oIZV-ixRTcY",
        "Q": "Can you explain what you mean by keep missing the earth at 3:34 and if velocity is what keeping the satellite in the air, how does the satellite maintain that speed?",
        "A": "The satellite maintains that speed because there is nothing to make it slow down. Objects in motion tend to stay in motion; that s Newton s first law. An orbiting satellite is constantly accelerating toward the center of the earth. In common terms, we describe that as falling. It doesn t hit the earth because it is moving sideways so fast that it misses the ground."
    },
    {
        "video_name": "oIZV-ixRTcY",
        "Q": "4:31 \"This thing can keep on going for a substantial amount of time\". So even if there were no drag, would all orbits eventually decay?",
        "A": "They would only decay if there were external forces. Such as a gravitational field from nearby bodies, or even unknown forces like dark matter."
    },
    {
        "video_name": "oIZV-ixRTcY",
        "Q": "At 1:53, I was not able to understand the answer to the respective question properly. In space, the astronauts are away from the magnetic field of the earth, then whose gravity is working upon the astronauts?",
        "A": "magnetic field has nothing to do with gravity earth s gravity pulls on the astronauts"
    },
    {
        "video_name": "oIZV-ixRTcY",
        "Q": "At 3:45 does the space shuttle come back to earth?",
        "A": "Yes, that is why space shuttle is a great invention, because they can be reused"
    },
    {
        "video_name": "oIZV-ixRTcY",
        "Q": "at 3:18, (around that time), sal says that the object keeps missing the earth as it is moving fast enough. So why doesnt gravity act on it immediately and cause it to come down?",
        "A": "It does, that s the whole point of what he is saying. The object IS falling, but it is also moving sideways, and it is moving sideways so fast that even though it is also accelerating toward the center of the earth, it never hits the earth."
    },
    {
        "video_name": "oIZV-ixRTcY",
        "Q": "At 0:15 if you pushed off an object in space, would you eventually slow down or keep going for eternity?",
        "A": "Newton s first law says unless there s a force to stop you, you will keep going."
    },
    {
        "video_name": "oIZV-ixRTcY",
        "Q": "At 4:20 he says that there is almost no air resistance. So theoretically, if there was no air resistance would this sort of free fall be considered as perpetual motion?",
        "A": "Yes, but note that it is not a violation of any of the laws of physics because what the laws say is that you can t get more energy out of a machine than you put in."
    },
    {
        "video_name": "jtPvDycHVQw",
        "Q": "At 0:27 you said to try it out on your own, and I did........ The answer that I got was she did not do the negative and minus sign. Is that correct?",
        "A": "Yes. She didn t put those in, and therefore only got one answer. To get the correct answer, the plus/minus sign must be there."
    },
    {
        "video_name": "jtPvDycHVQw",
        "Q": "At 1:33 in the lower left corner, what does \"Sal said \"square root of 11\" but meant \"11\" \" mean?",
        "A": "It is letting you know that when he said square root of 11 it was a mistake and what he really meant is 11 . Hope this helps."
    },
    {
        "video_name": "LpY8Qa3IP1w",
        "Q": "at 9:10 when Sal begins to evaluate the boundaries, would it be equivilent to substitute x for cos(t) and y for sin(t) and then evaluate [((cos(t))^3)/3]-[(cos(t))((sin(t))^2)] from 0 to pi? I'm assuming this is simply a different method of substitution he used, but I feel more comfortable actually subing the parameterized functions of t. Thanks in advance.",
        "A": "Absolutely. It s fine to rewrite F as a pure function of t. Sal s way just involves a little less writing so it looks nicer on the screen, but both are equally valid."
    },
    {
        "video_name": "LpY8Qa3IP1w",
        "Q": "At 2:20, why can we write dr=dxi+dyj? I was actually stuck on this since the last video. How do we know when to write dr=dxi+dyj or dr=(dx/dt i + dy/dt j)dt?",
        "A": "dr is defined to be dx \u00c3\u00ae + dy \u00c4\u00b5. You can, if you want, multiply this expression by dt/dt, which is just 1, so that you get dr = (dx/dt \u00c3\u00ae + dy/dt \u00c4\u00b5) dt, but that is just if you want to. Normally, you will do that if you want to get dr in terms of just one variable, t."
    },
    {
        "video_name": "LpY8Qa3IP1w",
        "Q": "At 1:50, it looks like you could solve this by:\n1) substituting x = cos(t), y = sin(t), dx = -sin(t)\u00e2\u0080\u00a2dt, dy = cos(t)\u00e2\u0080\u00a2dt\n2) integrate from t=0 to t=pi\n\nAnyone know of a reason why that wouldn't be valid?",
        "A": "Yes, that is correct. It s the same method used in earlier videos. I suggest starting with the Line Integrals and vector fields video here and viewing all of them. He is really just showing an alternate method of solving the integral here, a method which in some cases may actually be quicker."
    },
    {
        "video_name": "_k3aWF6_b4w",
        "Q": "Is there a faster way of doing this? One problem goes up to 1:47!",
        "A": "Sure. Just use a calculator. ha"
    },
    {
        "video_name": "_k3aWF6_b4w",
        "Q": "At 3:51, Sal said, \"I think it'll hit the point home.\"What did \"hit the point home\" mean? Alternatively, could it be expressed as \"I think it'll make you understand the point (the knowledge I'm teaching you) completely?\"",
        "A": "Basically it s so that you are confirmed to understand the technique."
    },
    {
        "video_name": "_k3aWF6_b4w",
        "Q": "At 0:10, Sal said, \"I'm going to use two different colors here, so bear with me a second.\" What did he mean by \"bear with me\"? Does he mean that he wanted to ask us to be patient 'cause it would take a short time to change the color?",
        "A": "yes bear with me means be patient."
    },
    {
        "video_name": "dstNU7It-Ro",
        "Q": "At 6:34 Sal says: \"...And the simplified version of this expression is x+4/x+6\" What I don't understand is why that is the simplified version. Why can't the two \"x\"s cancel each other out? Is it because the part in brackets is addition?",
        "A": "yes, you can t divide the x s with each other without also dividing the 4 and 6. You could try and get rid of the x but then you would be left with (1+4/x)/(1+6/x) which is even worse than what you started with"
    },
    {
        "video_name": "dstNU7It-Ro",
        "Q": "At 6:40, Sal said that the simplified fraction is x+4/x+6. Wouldn't that simplify to 2/3?",
        "A": "if x=0 then the expression is equal to 2/3 but if y=0 then x=-4 by multiplying (x+6) at both sides of the equation."
    },
    {
        "video_name": "dstNU7It-Ro",
        "Q": "At 3:10, I do not understand how Sal does a+b and gets 17. From my knowledge with quadratic equations, the a is 2 and the b is 17. So when you add together a and b won't you get 19?",
        "A": "It is confusing the way Sal showed it in the video. When he wrote that a+b=17, he did not say that a = 2 and b = 17 like you assumed. He is using these as variables to solve for the constants in his factored expression. To solve for a and b , you need to use the other expression that Sal wrote, which was of the form a*b=k , where k is some constant."
    },
    {
        "video_name": "dstNU7It-Ro",
        "Q": "At 5:40, Mr. Khan starts to explain the domain and says that \"the domain is all real numbers except x=-6 and x=5/2.\" Does that mean that any number that is a real number can be x?",
        "A": "Yes, except for the restriction, x=-6 and x=5/2, of course."
    },
    {
        "video_name": "dstNU7It-Ro",
        "Q": "At 1:06, why did he multiply 2x times 20? Shouldn't it be 2x times 13? plz answer",
        "A": "It s a little hard to explain. When you factor x^2 + bx + c, do you see that you want 2 numbers that add up to b and their product is c? Because: (x+d)(x+e) = x^2 + (d+e)x + de. When you have ax^2 + bx + c, with a != 1, it gets a lot more complicated. I don t have a good explanation, but it makes some sense and turns out to be true that the numbers you want have a sum = b and a product = ac."
    },
    {
        "video_name": "dstNU7It-Ro",
        "Q": "2:58 I'm not sure how he factored 2x(x+4)+5(x+4) into (x+4)(2x+5). Can anyone clarify?",
        "A": "In the most generic case, if we have an expression at+bt, we can factor out the t so that we ll get t(a+b). Hopefully, that s not a question. In the example you are working on, we can substitute t instead of (x+4) just to see the analogy more clearly. And then we ll get 2x*t+5*t. Factoring out the t again, we get t(2x+5) or, recalling that t=x+4, (x+4)(2x+5). I hope this will help."
    },
    {
        "video_name": "dstNU7It-Ro",
        "Q": "I don't understand why at 1:13 and 3:26 he splits up the 13x and 7x. Couldn't you just factor the problem without splitting the coefficients? Do we have to do that for every factoring problem?",
        "A": "Did you actually factor this without grouping? If you can, great! If you re like me and grouping is faster, then use grouping."
    },
    {
        "video_name": "dstNU7It-Ro",
        "Q": "At 3:09, why is a*b=60?",
        "A": "Because: For Ax^2 +Bx +C a*b = AC a + b = B Thus a*b = AC = 2*30 =60"
    },
    {
        "video_name": "dstNU7It-Ro",
        "Q": "at 2:51 why didn't sal just combine both (x+4)s into (x+4)^2 because he did that with (2x+5)",
        "A": "He could ve, but I guess he wanted to show you both ways to write the simplified form of an expression."
    },
    {
        "video_name": "I3vIAzMcm4Y",
        "Q": "At 1:25, did he mean first derivative?",
        "A": "Yes. Sal made a mistake by saying second instead of first."
    },
    {
        "video_name": "I3vIAzMcm4Y",
        "Q": "It seems like Sal's method of finding the particular solution is to just make a guess and then solve for the constant coefficients (0:44). What happens when the right hand side of the equation is really complicated and you can't guess?",
        "A": "I am a second-year student at UCD with only 1 class in Differential Equations (ODE), but I would imagine the next step would be to apply a Laplace Transform. Although, many differential equations have g(t) [y +p(t)y +q(t)y=g(t)] which may be guessed for."
    },
    {
        "video_name": "I3vIAzMcm4Y",
        "Q": "At 4:04, what is the technique Sal is using to solve the problem?",
        "A": "I dont know the name of it but he is adding the like terms/ powers on the left hand side and setting them equal to the right hand side."
    },
    {
        "video_name": "DRpdoZQtvOM",
        "Q": "In 6:00, Sal subtracts 5yx - yx to get 4yx. My question: why doesn't Sal switch the variables around to make it xy?",
        "A": "He could have. They usually are in alphabetical order, but perhaps he wanted to make the point that they can be in any order."
    },
    {
        "video_name": "DRpdoZQtvOM",
        "Q": "At 1:37, why did you use the top one? Also, at 0:48, how did -1 get involved?",
        "A": "The goal when working to solve a quadratic equation such as the one in the example is to find the two factors of the the y-intercept (last term) which can add (or subtract) to come up with the value of the middle term. In other words, since the y-intercept was -5, the factors of -5 were listed. The top set of factors was chosen because -1+5=4, which is the value of the middle coefficient. Negative one got involved since it is a factor of -5. (This is because -1 times 5 equals negative 5.) I hope that helps!!"
    },
    {
        "video_name": "DRpdoZQtvOM",
        "Q": "Wait, how is he able to ignore the x that 4 had and just focus on the y? Everything worked out, but I don't understand the reasoning. He did it again in the next video, and I still don't get it. It's around 2:54.",
        "A": "It is because the x, y, and 4 are being multiplied with each other. I believe this is called the associative property of multiplication, where 4x \u00c3\u0097 y is the same thing as 4y \u00c3\u0097 x, which is the same as xy \u00c3\u0097 4, et cetera."
    },
    {
        "video_name": "PDReqvXfkBA",
        "Q": "So, just to confirm, at 0:09 if in a Maths Test the question was 1/0. Would the answer be undefined or would you say, +infinity. Do they mean the same? Thank you.",
        "A": "I think if you put +\u00e2\u0088\u009e it might be wrong. You could put undefined. Maybe in the future anything divided by 0 and 0/0 could be defined. And maybe \u00e2\u0088\u009e (infinity) could be a number!"
    },
    {
        "video_name": "PDReqvXfkBA",
        "Q": "At 2:20 isn't 0/0 indeterminate?\nI have proof...The next video says so and, the internet says so, too.\n\"Why x/0 is undefined and 0/0 is indeterminate\"",
        "A": "I agree with you because there is a slight difference in the definitions of the two words. I don t know why Sal made the first video undefined and the second indeterminate. Try asking the site and help center, or perhaps just report a mistake."
    },
    {
        "video_name": "PDReqvXfkBA",
        "Q": "At 1:37 in the video Sal says that 0/0.1 = 0. And I've heard that 0 divided by any number is 0, but I don't know why? Could someone please explain. Thank You in advance",
        "A": "Let s look at the problem 0/5 = 0. Why is this true? Well, let s also look at the problem 8/4 = 2. We know we can switch these numbers around and make an equally true multiplication problem. So, 2 x 4 = 8. Our answer times the denominator, will equal our numerator. We can do the same thing to our 0/5 = 0 problem. So, 0 x 5 = 0. Our answer times our denominator will equal our numerator. That s why 0 divided by anything = 0."
    },
    {
        "video_name": "PDReqvXfkBA",
        "Q": "At 1:36, Sal says that 0 goes into 0.01 0 times. But isn't that be kind of impossible? Wouldn't 0 go into a number bigger than 0 infinity times, because 0 is equal to nothing?",
        "A": "Well, reverse it. 0.01*0=0. So it goes in 0 times. However, what you said is also true because an infinite number of 0s fit into .01. Now you have two answers that must be equal, so \u00e2\u0088\u009e=0, which is not true and part of why the whole thing is undefined."
    },
    {
        "video_name": "3aDV3L8aZtY",
        "Q": "At 0:52 Sal said\nso a reflection over the line y equals x minus 2.\nWhat does this mean?",
        "A": "He s naming the line: y=x-2"
    },
    {
        "video_name": "3aDV3L8aZtY",
        "Q": "I dont get this. why did sal translate to (3,-9) in 3:43",
        "A": "(3,-9) is how far away from the reflection the image needs to be. so each point is moved 3 units to the right and 9 units down, to find the image of the figure."
    },
    {
        "video_name": "3aDV3L8aZtY",
        "Q": "1:18 - 1:47, how did he know to use those numbers? For both of the coordinate pairs? Mostly this whole video was hard to understand.",
        "A": "They re just two points on the line y=x-2. Since it s such a simple function, he probably just did it in his head. The y-coordinate is obviously -2, so one point is (0, -2). And 2-2=0, so another point is (2,0)."
    },
    {
        "video_name": "MUQfl385Yug",
        "Q": "why/how is the derivative of x=1? ( @ 2:13) ?",
        "A": "f(x) = x is is linear function, whose slope (i.e derivative) is a constant, 1."
    },
    {
        "video_name": "MUQfl385Yug",
        "Q": "When applying the chain rule at 2:30, why did he multiply by the derivative of the exponent? isn't he supposed to multiply by the derivative of whats inside the parenthesis, which can be re-written as (e)^(-2x^2), which is this case was the d/dx of the \"e\". Appreciate your answers!!",
        "A": "e is a constant, so its derivative would be zero."
    },
    {
        "video_name": "MUQfl385Yug",
        "Q": "At 3:10, why is it that when applying the chain rule, the coefficient of -2x^2 is not multiplied to the e^-2x^2?",
        "A": "You appear to be focusing on the part of the problem in which we find the derivative of e^(-2x\u00c2\u00b2). The chain rule says we get the derivative of the outer function (e^u, where u = -2x\u00c2\u00b2) multiplied by the derivative of the inner function (-2x\u00c2\u00b2). The derivative of e^u is just e^u, so that part gives us e^(-2x\u00c2\u00b2). The other part is the derivative of -2x\u00c2\u00b2, which is -4x."
    },
    {
        "video_name": "-TpiL4J_yUA",
        "Q": "So at 0:16, he says that when you multiply exponents with the same base, you basically add them. So does that mean that if you had z^5 times z^11, that simplified is z^16?",
        "A": "Yes, that is correct."
    },
    {
        "video_name": "MarqSlyz-lU",
        "Q": "At 1:07, Sal says there are a couple of ways you can think about the problem.\nHe only gives one way.\nWhat are the other ways ?",
        "A": "Well he asked a question, so it can go here. Anyways, you can always create a venn diagram out of it"
    },
    {
        "video_name": "NSSoMafbBqQ",
        "Q": "2:53 Wouldn't the life insurance company be drowning in money?",
        "A": "usually, any insurance company will give insurance to people after considering their age, sex and health condition and this usually ensures that they ultimately make money and not go into loss."
    },
    {
        "video_name": "NSSoMafbBqQ",
        "Q": "I think for the calculations to be correct, the insurance company would pay out the insured amount minus the premium not yet paid, in case of an early death. In case the company paid out the whole amount, shouldn't the premium in the numerator at 2:37 be adjusted for the probability of Sal's death in the next 20 years?",
        "A": "Life insurance pays the entire death benefit at the time of death."
    },
    {
        "video_name": "eBzQdDgBB_8",
        "Q": "At 0:58, shouldn't the two circles (each having three parts) add up to six parts? Shouldn't the answer be 5/6 as opposed to 5/3?",
        "A": "The denominator of a fraction always tells you the number of parts that will create 1 whole unit. Each circle is a single unit. To make up one circle (or one unit) it take 3 parts. If you have 5/6, you are saying that 5 parts of an object that has been split into 6 parts. Or, one circle split into 6 parts with 5 shaded. Hope this helps."
    },
    {
        "video_name": "Ld7Vxb5XV6A",
        "Q": "At 2:45 ,why is the number -6?",
        "A": "At 2:45 the number is -6 because Sal divided -90 by 15"
    },
    {
        "video_name": "Ld7Vxb5XV6A",
        "Q": "Sal said at 2:29 that 15x is -90. Why? I am very confused at what Sal is doing at around from 2:29 to 2:49. I am very confused can someone please explain.",
        "A": "He said that 15x= -90 because you only need to find the value of x at the moment, so you need to cancel out the 270. In order to do that, you need to keep both sides equal, so you subtract 270 from both sides of the = sign. If 15x+270=180, you subtract 270 from 270. Then you subtract 270 from 180, which leaves you with 15x= -90. After that (to find the simplified value of x), you divide both sides by 15 (again, to keep both sides equal) and you come out with x= -6. Make sense?"
    },
    {
        "video_name": "Ld7Vxb5XV6A",
        "Q": "At 2:20 What does he mean by subtracting 270 on both sides.\nDo you subtract 270 by 180 or 15x?",
        "A": "15x is a variable, so u can only subtract it by another variable x. So, Sal was stuck at 15x+270=180. He wanted to solve for x. So he had to get x on one side and the constants on the other. He subtracted 270 on both sides, which means on the left u have 270-270, which is 0 and on the right u have 180-270, which is -90. You arrive at 15x=-90."
    },
    {
        "video_name": "Ld7Vxb5XV6A",
        "Q": "at 1:08 what does the x stand for",
        "A": "marinabell, x = -6 hope that helps!"
    },
    {
        "video_name": "Ld7Vxb5XV6A",
        "Q": "At around 1:51, he says it will be equal to 180 degrees, so what would I do if the two angles didn't equal 180 degrees. Like, in this case, there is a bigger angle and a smaller angle that add up to 180 degrees, but what if there were 2 of the bigger angles that obviously don't add up to 180. What number would I use instead? And, how would I solve it?",
        "A": "If you have two parallel lines and a transversal, you only have two choices when you are talking about two different angles - Either they are supplementary (add to be 180) or they are congruent (same measure) So if you are talking about two of the obtuse angles, they must be congruent"
    },
    {
        "video_name": "Ld7Vxb5XV6A",
        "Q": "I don't really understand how Sal got -90 from 15x at minute 2:28. PLZ HELP!",
        "A": "Sal had the equation 15x + 270 = 180. He wanted to unlock the variable, so first of all he added - 270 to both sides of the equation, getting 15x + 270 - 270 = 180 - 270, which simplified is 15x = - 90"
    },
    {
        "video_name": "Ld7Vxb5XV6A",
        "Q": "At 2:18 how did he get -90?",
        "A": "He was simplifying the equation. To isolate the variable, he subtracted 270 from both sides. 180-270= negative 90"
    },
    {
        "video_name": "TqRReFvbpXA",
        "Q": "at 0:25 why would you put the seven under the three",
        "A": "Because 7 is only in the ones place. If you put it under any of the other numbers that would mean it was 70, or 700."
    },
    {
        "video_name": "TqRReFvbpXA",
        "Q": "At 0:33, Sal uses the word compute. Math-wise what does compute mean?",
        "A": "Computing and calculating are almost synonyms, but sometimes one word is better suited than the other. Calculating refers to doing simple arithmetics while computing implies more complex and long-winded mathematical operations than that. You would calculate your daily budget and compute the inverse of 2X2 matrix for example."
    },
    {
        "video_name": "LrxZMdQ6tiw",
        "Q": "what is dropping an altitude\n4:51\nI'm a sixth grader",
        "A": "Dropping an altitude means that you draw a line segment from one vertex of a triangle to the opposite side so that the line segment is perpendicular (makes a right angle) with opposite side."
    },
    {
        "video_name": "LrxZMdQ6tiw",
        "Q": "Hmmm... At 1:16, I think he said that the apothem is squared, but I don't think that's right!",
        "A": "Is he talking about squaring the apothem to use the pythagorean theorem to find another side length? Hope this helped!"
    },
    {
        "video_name": "LrxZMdQ6tiw",
        "Q": "At 6:18 ish, can you use the Pythagorean therom to find the altitude of the triangle?",
        "A": "Yes, you could. Good job on figuring that out!"
    },
    {
        "video_name": "LrxZMdQ6tiw",
        "Q": "thing is at 4:00 in the video you say that all triangles are 60,60,60 but earlier you said they were icocelies (sorry for spelling) if all angles are 60 degrees that would make it equilateral yes?",
        "A": "yes, the triangle would be equilateral all equilateral triangles are considered to be isosceles"
    },
    {
        "video_name": "LrxZMdQ6tiw",
        "Q": "At 7:04, isn't the area of an equilateral triangle (sqrt(3)*s^2)/4? I still get 3*sqrt(3), so I guess it's not as important as I thought...",
        "A": "Well, you are actually right. That would be the special formula that gives you the area of equilateral triangles. However the general area formula for triangles used in the video (A = 1/2*h*b), works for all triangles, including equilateral ones. Using the special formula as suggested by you would have been quicker though, as you only need to know the side measurement of the equilateral, while the general formula requires the height and the base measurement."
    },
    {
        "video_name": "GvgJ2_7P9SA",
        "Q": "At 1:16, Sal writes a little line through a fraction. What does that line mean?",
        "A": "watch the fractions introduction video It ll help you find out the answer :)"
    },
    {
        "video_name": "bml74_PsfwA",
        "Q": "At 0:48 Sal says that x cannot be -2 because it would make both terms undefined.\nBut, I still don't get it:\n\n(-2)^2/(-2+2) = 4/(-2+2)\n4/0 = 4/0\n\nWhy it isn't a solution ? Yeah, we don't know what is the quotient of 4/0, but, isn't it logical to assume that whatever the quotient is, it is going to be equal to 4/0? I mean, they're both the same thing. (4/0 = 4/0)",
        "A": "4/0 is undefined (in a way, its not even a number), so it can t be a solution and if you try to solve it with 4/0 your equation will just be messed up. There are contexts where it is defined, but they don t usually appear until much later in your math studies."
    },
    {
        "video_name": "bml74_PsfwA",
        "Q": "at 2:13 in this video, I'm a little confused on where he went to x^2-4=0 and then went and did (x+2)(x-2)=0. Where did the x+2 and x-2 come from?",
        "A": "The x^2-4 is a difference of squares. When factored out, a difference of squares will always be in the form (x+a) (x-a). These are called conjugates. Since the square root of 4 is 2 and the square root of x^2 is x, the factored form is (x+2) (x-2). You can prove this by FOILing out the conjugates; you will get x^2+4."
    },
    {
        "video_name": "bml74_PsfwA",
        "Q": "At 1:36, couldn't you just use square roots to solve it instead of factoring a quadratic equation?",
        "A": "when you do sq in the calculator you only get the positive half of the root, you need to remember it could also equal the negative number. bottom line: you could. :)"
    },
    {
        "video_name": "bml74_PsfwA",
        "Q": "At 1:10, how do you know to multiply both sides by the denominator rather than dividing one side by x",
        "A": "Well, it is always a mess to be working with fractions if you don t have to, so getting rid of fractions by multiplying by their common denominator should be really high on your solving checklist! Also, dividing by x just makes things messier. You will now have an x on both sides of the equation, and even worse, the fractions will now have two different denominators. Nope, getting rid of the fractions is what you want to do!"
    },
    {
        "video_name": "Q9t1LghwdGc",
        "Q": "At 9:03 couldn't g(y) be equal to any constant, not just zero? Surely g(y)=k will give the same result when you take the partial derivatives?",
        "A": "Yes, the overall answer should have a constant I think. If you take the gradient of F(x,y) = x^3/3 + xy^2 + C, you get the correct vector field."
    },
    {
        "video_name": "Q9t1LghwdGc",
        "Q": "At 2:17 the video says that normally you can treat differentials (such as dt) as numbers and multiply them. What are some examples where you can't?",
        "A": "I can think of one example \u00e2\u0080\u0093 although I can t think of a specific problem in which this is the case off the top of my head \u00e2\u0080\u0093 and that would be if your differential (such as dt) just so happens to have the value of 0. Then you would be dividing/multiplying by zero, which does not work."
    },
    {
        "video_name": "dAvosUEUH6I",
        "Q": "at 2:15 how is 2^0 equals to 1?",
        "A": "At first glance, it seems logical that something to the 0 power is equal to 0, but it s not. And there s a reason why. Say you had 2^3 / 2^3. This is equal to 1, because anything over itself is equal to 1. 2^3 / 2^3 is also equal to 2^0, after subtracting the exponents. So, if 2^3 / 2^3 equals 1, and also equals 2^0, then 2^0 equals 1 as well."
    },
    {
        "video_name": "EDlZAyhWxhk",
        "Q": "His reflect tool at 1:12 has these balls he can move it with but when i do it mine does not help!",
        "A": "I went back and looked through all the old exercises. It appears they changed their user interface a little, so now there s just some double arrows that you click and drag to spin the reflect line, instead of those dots. It still works functionally the same."
    },
    {
        "video_name": "1ul8g55dYA4",
        "Q": "at 1:59, Sal says he doesn't know what the plural of hypotenuse is.\nWhat is it?",
        "A": "Hypotenuses, which sounds very weird and awkward so it feels wrong but is actually correct."
    },
    {
        "video_name": "1ul8g55dYA4",
        "Q": "Around 2:30, is he saying \"Beta\" or \"Thata\"?",
        "A": "Sal is saying Theta which is a Greek alphabet. It is used as a variable like we commonly use x or y , etc.. It is usually used to denote an angle."
    },
    {
        "video_name": "1ul8g55dYA4",
        "Q": "At 1:59 Sal says he doesn't know the accepted plural version of \"hypotenuse\". After some research, I found that the acceptable plural version is \"hypotenuses\".",
        "A": "Thank you for that information! Good stuff."
    },
    {
        "video_name": "1ul8g55dYA4",
        "Q": "at 2:25, is theta just a fancy placeholder? or is a number like tau, pi, or aleph null?",
        "A": "It s just a fancy variable. Theta is often used for an unknown angle in geometry."
    },
    {
        "video_name": "1ul8g55dYA4",
        "Q": "at 3:08 what does sal mean by theta and 90 - theta, what is theta?",
        "A": "To add to Kubleeka, if x and y are complementary x+y=90 x+90-x=90 y+90-y=90"
    },
    {
        "video_name": "1ul8g55dYA4",
        "Q": "At 2:10, Sal says that if we can prove that all corresponding angles are the same, it means that the triangles are congruent. Is he implying that AAA might also be used to prove congruence under certain circumstances?",
        "A": "AAA plus knowledge that any corresponding sides are congruent proves the triangles are congruent. However, AAA by itself proves only similarity, not congruency."
    },
    {
        "video_name": "1ul8g55dYA4",
        "Q": "How exactly did Sal cut the square into the 4 triangles? At time1:13.",
        "A": "He just picked an angle, then drew a line from each vertex across into the square at that angle. His angle choice was arbitrary. because as he shows later, he ends up with 4 identical right triangles. By just picking a random angle he shows that it works for any right triangle."
    },
    {
        "video_name": "JKvmAexeMgY",
        "Q": "At 2:39, wouldn't it also be possible to factor X out to further simplify? or would that be unnecessary.",
        "A": "It would be possible but not really necessary. Usually it is considered a polynomial s simplest for as he left it, with just a series of terms added (or subtracted) together. Unless you were asked specifically to factor it, I would leave it as is."
    },
    {
        "video_name": "JKvmAexeMgY",
        "Q": "At~1:30 why are we supposed to multiply 7x-5 by x^3 and then by 4x but not do the perenthathys first and then multiply?",
        "A": "In the parentheses, there are two terms (7x-5), which are in a deferent degree and separates them a minus sign - not multiplication sign x or *. So you supposed to not do what is in the parentheses."
    },
    {
        "video_name": "JKvmAexeMgY",
        "Q": "at 2:30 he multiplys 7x times 4x why does it equal 28x squared?",
        "A": "7x * 4x is the same as 7 * 4 * x * x 28x^2"
    },
    {
        "video_name": "JKvmAexeMgY",
        "Q": "At 1:48, why is Sal putting x^3?\nI tried to solve this my way and it went like this :\n(7x - 5)(x^3 + 4)\n7x^4 + 28x -5x^3 - 20\nafter rearranging I got : 7x^4 - 5x^3 + 28 - 20",
        "A": "thanks a lot! it really helped :D"
    },
    {
        "video_name": "JKvmAexeMgY",
        "Q": "At ~ 1:30 he mentions \"FOIL\"... What is that?",
        "A": "Holly, FOIL is a pneumonic or memory jogger to remember how to multiply two binomials together. For example, if you want to do (x+4)(x-2) you could remember how to do that by multiplying: The FIRST terms in each binomial: x*x=x^2 The OUTER two terms: x*-2=-2x The INNER two terms: 4*x=4x And the LAST two terms: -2*4=-8 Then add everything together: x^2-2x+4x-8= x^2+2x-8. First, Outer, Inner, Last or the F.O.I.L. method. Hope that helps!"
    },
    {
        "video_name": "JKvmAexeMgY",
        "Q": "At 2:43, Sal mentions fourth, third, second, and first degree terms. I just started working on functions today and have a pretty basic understanding of how they work. What is the difference between the 4th, 3rd, and 2nd degree terms in the video, and why can't you just simplify them? They all consist of a constant followed by a variable.",
        "A": "The degree of a term is whatever the exponent of its variable (not its constant) happens to be. You can only add or subtract terms whose variables (but not constants) are absolutely identical -- those are called like terms . Since this problem did not have like terms you cannot add them. You may need to review earlier lessons because knowing when you can and cannot combine terms is a skill you need to have learned before covering this topic."
    },
    {
        "video_name": "JKvmAexeMgY",
        "Q": "at 2:46, why could he not simplify the 7x^4 -5x^3 +28x^2 -20x to 7x^3 -5x^2 +28x -20?",
        "A": "You could simplify it to x(7x^3 -5x^2 +28x -20) You could not just drop the extra factor of x. It is debatable which form is most simplified, and is a matter of preference."
    },
    {
        "video_name": "muC7HMIEYDU",
        "Q": "This might be really splitting hairs here, but at around 2:51 the bank floor is measured at 4 meters. Just noticing that the floor falls a little short of the 4 meter mark's rightmost edge on that ruler... which brings up my question: WHERE, exactly, are you supposed to begin measuring on a ruler? I've always started from the leftmost edge of a ruler mark to the rightmost edge of another mark. (Sorry if this doesn't make sense...)",
        "A": "You should start your measurement on the 0 mark (tick) instead of the very edge. Otherwise, your measurements will be offset by about 1/16 of an inch."
    },
    {
        "video_name": "muC7HMIEYDU",
        "Q": "What he means by longest dimension? 4:00",
        "A": "According to the Webster s Ninth Collegiate Dictionary, length means the longer or longest dimension of an object. So the length of a rectangle is the longest side whether it is vertical or horizontal."
    },
    {
        "video_name": "ENKH97PYssg",
        "Q": "I notice you use a circle for the multiplication sign at 0:35, does that signify a difference in the problem?",
        "A": "It s a dot. This is different notation for the same operator. Early in school we use x, but as we progress in math we often use the letter x to represent numbers, so it can get confusing with x. More often you ll be seeing * or the dot."
    },
    {
        "video_name": "ENKH97PYssg",
        "Q": "At 1:06 doesn't 34 have more digits than 2??",
        "A": "Yes, but Sal is saying that you CAN put the number with more digits on the bottom and multiply."
    },
    {
        "video_name": "rfyq32mHcYs",
        "Q": "I'm not familiar with the notation at 1:04. What is the meaning of s\u00c2\u00b2Y(s)...?",
        "A": "s\u00c2\u00b2Y(s) is equivalent to s^2 times the Laplace transform of y. As Sal mentions, Y(s) is the Laplace transform of y (Y as a function of s). It s equivalent to L[y], if you prefer that notation. I d assume that if you re this far into the differential equations content, the variable s is familiar by now. If not, it s just the typical variable we use after applying the Laplace transform to a function. I d go back and check out the earlier videos on Laplace transforms. Hope that helps."
    },
    {
        "video_name": "JbfVrwxuPxM",
        "Q": "At 3:45, why is b on top and not a?",
        "A": "In blue, he s taking the integral from c to b, where b > c. In green, the integral covers from x=a to x=b, with b > a. The x-value at the top of the integral sign is the upper bound."
    },
    {
        "video_name": "JbfVrwxuPxM",
        "Q": "4:31 I'm confused about the use of the word \"anti-derivative\", isn't it the same as saying \"integral\"?",
        "A": "An antiderivative is an indefinite integral. There are other kinds of integrals. The main one you ll be dealing with is the definite integral. In more advanced mathematics there are some subtleties of definitions of various kinds of integrals. I will leave that to a professional mathematician because I am not fully-versed in the nuances."
    },
    {
        "video_name": "JbfVrwxuPxM",
        "Q": "At 0:54, it is stated that if f(t) is continuous at a given interval, then it differentiable at every point in it's domain.......\nBut in the differential calculus course, we learned that a continuous function is not necessarily differentiable...eg:- f(x) = |x| is not differentiable at x=0, even though it is continuous.....\n\nPlease help",
        "A": "It is only claimed that the continuity of \u00c6\u0092 implies the differentiability of F."
    },
    {
        "video_name": "JlOyb_RTejo",
        "Q": "At 2:08, does a limit ever tend to be infinity?",
        "A": "You ll sometimes see people say a limit is equal to infinity or negative infinity, but others say in that case it just doesn t exist."
    },
    {
        "video_name": "JlOyb_RTejo",
        "Q": "At 1:01, how does Sal just know all this stuff about the natural log? How does he know that the natural log isn't defined for all x's, and how does he know that ln(x) IS continuous at x=1?",
        "A": "Some basic properties of the logarithmic function should be known before learning limits. In particular, the natural log function is continuous at \u00f0\u009d\u0091\u00a5 = 1 as the inverse (exponential function) is continuous at \u00f0\u009d\u0091\u00a5 = 0. Furthermore, there are domain restrictions to the natural log function as there are range restrictions in its inverse. If you are unsure about these properties, I would suggest watching the KA videos on inverse functions, logarithms, and exponential functions. Comment if you have questions!"
    },
    {
        "video_name": "7QMoNY6FzvM",
        "Q": "At 1:19, how did he get that 2?",
        "A": "The vertex of the parabola (when written in vertex form) is when the squared part is equal to zero, since this maximises or minimises the expression. In this case the squared part is (x-2), so Sal simply solved for the x that made this zero. x-2 = 0 \u00e2\u0087\u0092 x = 2"
    },
    {
        "video_name": "7QMoNY6FzvM",
        "Q": "What are those swirly things at 1:08? I've seen my teacher use them before, but I have no idea what they are or used for?",
        "A": "Do you mean the purple, curly bracket beneath -2(x-2)? It s called a brace. It s another type of grouping symbol, just like (parenthesis) and [brackets]."
    },
    {
        "video_name": "7QMoNY6FzvM",
        "Q": "I have a really important question, I always wanted to ask it;\n\nAt 2:14, based on what do we put \"x\" values? And, is the \"x\" point in the vertex coordination (2,5) which is 2, should always be in the middle between other points that you assigned in the video?",
        "A": "1, 2, and 3 are all small numbers meaning they are convenient to work with. Also since a parabola is symmetric it would be nice to have the vertex point somewhere in the middle of the \u00f0\u009d\u0091\u00a5-coordinates you choose so you can graph both halves of the parabola (otherwise, you would end up drawing only one side of the parabola)."
    },
    {
        "video_name": "7QMoNY6FzvM",
        "Q": "How do you know that X is equal to 2 at 1:15?",
        "A": "He is picking 2 because that value zeroes out the expression (-2(2-2)^2 = 0) leaving y=5 which is the vertex point we want, (2,5). This equation is in vertex form, A(x - B) + C. The vertex can be found at (B, C)."
    },
    {
        "video_name": "7QMoNY6FzvM",
        "Q": "at 2:16 why do you use numbers 1,2 and 3?",
        "A": "He started by noticing that x=2 would find the vertex of the parabola. Then he just chose two numbers that were equidistant to 2. 1 and 3 are the same distance from 2 and are easy enough to plug into the equation, so that is why he is also interested in x=1 and x=3."
    },
    {
        "video_name": "7QMoNY6FzvM",
        "Q": "1:52 why is ( 2,5 ) the maximum point or vertex?",
        "A": "Actually, he explained the maximum earlier in the video around the 1:22 time. The principal is that when a quadratic equation in in vertex form a(x-h)^2 + k then the vertex is at (h,k) - one thing to notice is that if (x-3)^2, then h=3 and if (x+3)^2, then h = - 3"
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "At 9:55, Sal drew another graph. In this graph, it is an odd function, but what is y when x is 0. If it is a function, the same value shouldn't produce two values.",
        "A": "If you go back and look at the graph he drew, the function is actually supposed to be on either side of the y axis (x=0). This is because the function jumps certain values on the coordinate plane. Such a function is entirely plausible as long as no two points line up vertically (which they don t on the function Sal drew)."
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "Doesn't the discussion around 8:23 of odd functions have an error. Sal states that f(x) = -f(x) but goes on to state that it follows that f(x) = f(-x) which defines an even function. Great video but this seems like an error.",
        "A": "Ok. that makes sense. thanks for the feedback"
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "What is the even function that Sal starts drawing at 4:00 that is negative near the origin but then jumps up into a diagonal line and a curve (it almost resembles an angry emoji face)? He uses the example f(3)=5...",
        "A": "I don t think Sal drew it with the equation in mind. You could probably find it, but it would be pretty wacky. All Sal was trying to do was give an extreme example of an even function."
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "Wouldn't he have to say f(2) squared= f(-2) squared and explain that the exponent has to be positive? 3:15 - 3:40\nI know he did have the definition, but some people may get confused due to the lack (very little) of details.",
        "A": "It s not f(2) squared, but 2 squared. It s the function that is either even or odd. The definition is the important part."
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "At 4:14, is what Sal draws a function? I mean, doesn't a function only have one y value for every x value? It looks like in Sal's example of an even function, he has more than one, or does it just look that way?",
        "A": "The part of the function where there is a break (or where the lines are not connected) seem to leave and pick up in the same spot, maybe around (-1)-ish on the x-axis. So, since the lines don t overlap on the x-axis, it is a function."
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "at 2:58 the relation between even function and even number is given by the power of x -_- ?",
        "A": "No, there are other even functions, e.g. f(x) = |x| and f(x) = cos(x)."
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "At 6:40, instead of reflecting over the y-axis and then over the x-axis, could you also think of it as taking that portion of the graph and rotating it 180\u00c2\u00b0 around the origin? Would that work for every odd function?",
        "A": "Yes. The result of reflecting any point over the y-axis and then over the x-axis is indeed the same as rotating the point 180\u00c2\u00b0 around the origin. For any point (a, b), reflecting over the y-axis gives the point (-a, b), then reflecting (-a, b) over the x-axis gives the final point (-a, -b). Rotating the point (a, b), 180\u00c2\u00b0 around the origin, also gives the point (-a, -b). Have a blessed, wonderful day!"
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "At 2:40 sal talks about even functions and even numbers being different ideas. if thats the case why are they called even and odd instead of something more specific\\less repetitive? Where do the terms odd and even come from?",
        "A": "Even-powered polynomials, when summed or alone consistently produce even-functions. Odd-powered-polynomials similarly, consistently produce odd functions. This is why the names match and are repetitive. A mix of even- and odd-powered polynomials will produce neither an even nor an odd function. Non-polynomial functions are often neither even nor odd."
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "At 9:42, I don't understand how on the second type of odd function Sal has described that if f(0) has two corresponding values that the -f(x)=f(-x) rule works",
        "A": "I guess Sal assumed this function isn t defined at 0, otherwise we won t be able to call it a function , since it would give 2 different outputs for a single input."
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "In 2:40 you say that there is no connection between even number and function but if you have an x to any odd number power there is no way this function is even. So way can't I say that if a function is powered by a even number it's automatically even?",
        "A": "A function could have an even power but not be an even function - if it were shifted so that it does not reflect over the y-axis. So, perhaps and even function must have an even exponent, but an even exponent is not necessarily an even function. In the same way that a square is parallelogram, but not all parallelograms are squares. This would mean that there is a connection, but they are not the same thing."
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "At 11:20 Sal said parabola, what does that mean?",
        "A": "A parabola is basically a line that curves in a U-shape and is symmetrical down the middle of the U (vertically). That s how I think of it anyway."
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "wouldn't the graph that Sal is drawing at 9:29 not a function? Aren't there two values of zero on that graph, one positive and one negative, making it not a\nfunction?",
        "A": "I think Sal was just sketching the shape of the graph without that level of detail; but you are correct that if there were two values at zero then it wouldn t be the graph of a function."
    },
    {
        "video_name": "8VgmBe3ulb8",
        "Q": "At 6:53, Sal says that you do two reflections (one over the y-axis and one over the x-axis) to map an odd function to itself. Would it be correct to say that you could rotate an odd function 180 degrees to map it to itself?",
        "A": "Not 100% certain that holds in all cases, but intuitively it does seem to be true so long as the rotation is around the origin."
    },
    {
        "video_name": "RFesGHsuFZw",
        "Q": "At 4:07, -to At 4:18 He says \"angle\" ( < )but defines \"triangle\" ( uses delta ). I'm assuming he is just confused. Is he confused ?",
        "A": "He states that the angles BEA and CED are congruent (which they are); thus triangles BEA and CED are proven to be congruent by the AAS postulate (which is why he used the triangle notation). However, that s not the answer we are looking for. So, he then went on to explain how this indicates that sides BE and CE are equal, making E the midpoint of line BC."
    },
    {
        "video_name": "RFesGHsuFZw",
        "Q": "I thought Sal said \"green angles\" at 5:00. To me they look yellow.",
        "A": "They re green for him, but they look yellow to me too."
    },
    {
        "video_name": "RFesGHsuFZw",
        "Q": "What does he mean when he says Alternate Interior Angle (3:42)",
        "A": "If two parallel lines have a diagonal line through them, then you get four angles that are between the parallel lines. The lower left and upper right angles are congruent as are the upper left and lower right. Thus they are alternate interior angles"
    },
    {
        "video_name": "PhfbEr2btGQ",
        "Q": "At 4:31, He talks about transformation and basis, what are those?\nCould I get a link?",
        "A": "Check out the linear algebra video list (by successive clicking on the titles at the upper left, etc., and find the video where you need to start working. If you find other things that you don t understand and can t learn there, keep going further back until you know what he s talking about, then work forward asking any questions you want answered."
    },
    {
        "video_name": "PhfbEr2btGQ",
        "Q": "Maybe this is a stupid question, but at 7:15 it is said that eigenvectors just get scaled up or reversed but that their direction doesn't change. But the direction of v2 does change, so is it correct to say that this is an eigenvector?",
        "A": "You said ...or reversed... . Is v2 reversed?"
    },
    {
        "video_name": "i1i2_9wg6N8",
        "Q": "i dont understand what he did at 1:18. can someone tell me?",
        "A": "What he is doing is that he is adding each of the separate fractions together. The 3/3 fractions are equal to one so he adds up those two to equal two and then adds the 1/3 fraction to have the number equal 2 1/3"
    },
    {
        "video_name": "0gzSreH8nUI",
        "Q": "At 0:07, I have trouble visualizing word problems. How can I sketch the problem accurately?",
        "A": "If the word problem says you are dealing with a triangle I would say, go ahead and draw a triangle. Then fill in the angles as it says. If one angle (as in Sal s example) is 10 degrees, write that in. I think one of the best ways to learn how to visualize word problems is to watch several videos in which Sal does it, like this one. Then pause the video and try to draw one of the word problems he has. When you re done, turn the video back on and see if your drawing matches his."
    },
    {
        "video_name": "0gzSreH8nUI",
        "Q": "At 0:20.What does an arbitrary triangle mean?",
        "A": "Arbitrary just means any triangle, not right, not special, just drawn."
    },
    {
        "video_name": "0gzSreH8nUI",
        "Q": "how do you find x in 1:26 in the vidio",
        "A": "Video* no Vidio"
    },
    {
        "video_name": "An46SYAxhtc",
        "Q": "At 0:23, the video could have been even better if the Voice-over didn't go on I-guess-ing or whatever-ish-ing like \"pinkish\".",
        "A": "uhm... there s a human being explaining it and not a computer? maybe that s why the [ I-guess-ing or whatever-ish-ing like pinkish ] !"
    },
    {
        "video_name": "An46SYAxhtc",
        "Q": "At 1:13, what is the answer to the equation?",
        "A": "6+4=10 and 10-6=4"
    },
    {
        "video_name": "cnCQGKx6LhE",
        "Q": "Hi, at 06:13 please tell me how did he get (sqrt phi-1)?",
        "A": "He factored out a 6371: (6371\u00e2\u0088\u009a\u00cf\u0086) - 6371 = 6371([\u00e2\u0088\u009a\u00cf\u0086] - 1) Note that when you distribute the 6371 on the right side of the equation, you get back to the left side of the equation. Therefore, both are equal."
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "In 1:07, Sal mentions non-negative square roots. Does that mean a root that is always positive? I don't understand.",
        "A": "Square roots can be both negative and positive. The square root of 25 is bot 5 and -5. Upvote if you understood!"
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "At 5:42 he says You can't put a negative number in a square root, But you can put it in a cube root. Why?",
        "A": "The simple answer is this: Two negatives make a positive. The only way that you can get a negative number is to multiply a positive number times a negative number. For example:__sqrt(100)=10 because 10*10=100. How would you get the sqrt(-100)? It is not going to be -10*(-10) because -10*(-10)=100. Taking the square root of a negative number is asking what number times itself equals a positive square root number....."
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "At 5:47 Sal mentions that eventually we will learn about something called imaginary numbers. But can someone please explain what they are?",
        "A": "Imaginary numbers make up a subset of all numbers called complex numbers. They have a real component (which is the numbers you are familiar with) and an imaginary component, which is denoted by the letter i . This imaginary component, i, is equivalent to the square root of -1."
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "at about 6:40, I don't get how a a number cubed can be negative, but not a number squared? Help!",
        "A": "Consider : (-2)^2 = (-2)(-2) = +4. We get positive 4 because a negative * a negative = a positive. Now consider: (-2)^3 = (-2)(-2)(-2) = 4(-2) = -8 Why does this become negative? ... because there are an odd number of minus signs. The 1st 2 negatives create a positive. Then, you have a positive * a negative = a negative Hope this helps."
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "At 5:34, \"Can write something like...Can we pick a new color?\" I don't believe I understand what you said. Is it part of the problem? The Color I mean,",
        "A": "he was just trying to make a half joke but he actually meant it to color code the thing to make it more easier to visualize"
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "3:23 If the sides of the cube is two to the third power, how does it equal eight?",
        "A": "One side = 2 If you multiply all 3 sides (the 3 dimensions), you get 2^3, or 2 to the third power. 2 to the third power = 2*2*2 = 4*2 = 8"
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "5:33 So, if you have an odd root of a negative number - like in Sal's example - will the answer always be negative?",
        "A": "Yes, any cube root of a negative number is negative. for example -27 cube root answer is -3 because -3*-3 is 9 and 9*-3 is -27"
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "I did not get 0:19 to 0:49 because hows is x^2 x?",
        "A": "Remember, it is not x^2. It is (radical of x)^2 = x Example: radical of 9 = 3 Now (radical of 9)^2 = 3^2 = 9 That s it."
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "7:22 in the video. With the 125 cubed could you just take a calculator and divide by three?",
        "A": "Cube roots are not the same as dividing by 3. Cube root 125 = 5 125 / 3 = 41 2/3 As you can see... they aren t the same answer. A cube root is reversing an exponent of 3. So, when you get a problem like cube root 125, you need to find the number raised to the power of 3 that will create 125. The answer = 5 because 5^3 =125 Hope this helps."
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "4:28 If there is a 3, and a 2, for square root and cube root, is there a 1 for the square root (or something before it?) Just wondering.",
        "A": "The 1st root is the square root. The is no one root because a number raised to the power of 1 is just itself. 5^1 = 5. Since the power of 1 changes nothing, there is no need to have a radical to reverse it."
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "Sal,at 1:07 you say non-negative.What is the difference between positive and non-negative?",
        "A": "Positive excludes zero, non-negative includes zero. In other words if we say x is positive then x > 0 But if we say x is non-negative then x \u00e2\u0089\u00a5 0"
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "At 4:31, how can people figure out that the cube root of 8 is 2? Can you just make a factoring tree?",
        "A": "yes u could. Also 2 times 2 times 2 is equal to 8"
    },
    {
        "video_name": "87_qIofPwhg",
        "Q": "@3:02 i dont understand why theres 3 numbers if a cube has 6 sides?",
        "A": "This is because we are finding the volume of the cube. The volume being the measure of the space inside the cube. To find the volume of the cube, you multiply Length by Width by Height. In the case of the cube however, because all of the side lengths in a cube are equal to each other, to find the volume of a cube, all you have to do is cube one of the side lengths (multiply it by itself 3 times)."
    },
    {
        "video_name": "8RasCV_Lggg",
        "Q": "At 3:50, where does the quantity 'r' come from?",
        "A": "r is how far the complex number is from the origin. Using just sine and cosine only allows us to represent numbers exactly 1 unit away from the origin. Once we introduce the r variable, we can now represent any number we want in polar form."
    },
    {
        "video_name": "8RasCV_Lggg",
        "Q": "i dont clearly get what happened at 3:20\nplease help",
        "A": "At that point in the video, Sal is setting us up for finding the radius by mentioning the unit circle (from trigonometry). Using the unit circle, the horizontal coordinate will be cos \u00ce\u00b8 and the vertical coordinate will be sin \u00ce\u00b8. Then he says we need to scale by the actual magnitude of r in order to reach the coordinates we are given in this problem. y = r sin \u00ce\u00b8 and x = r cos \u00ce\u00b8. Using the unit circle also helps determine the angle \u00ce\u00b8, because now we can use tan of the ratio between y and x to find \u00ce\u00b8."
    },
    {
        "video_name": "rufFQZDDXCE",
        "Q": "At 2:41, Sal said that 'sec' is the inverse of 'cos' when it is the reciprocal. I just wanted to clarify whether I am right.",
        "A": "You re spot on!"
    },
    {
        "video_name": "rufFQZDDXCE",
        "Q": "At 2:03, I thought a radical is never suppose to be left in the denominator?",
        "A": "Usually your not supposed to, but sometimes if you are not told to change the fraction (radical), you don t have to. At least, in my opinion"
    },
    {
        "video_name": "rufFQZDDXCE",
        "Q": "At 0:55 does Sin= 5 over the sqrt of 41 simplify to 5 sqrt of 41 over 41? or is that only on certain types of triangles? because i saw in one video you did that but i was not sure if thats only 30 60 90 triangles.",
        "A": "the simplification is useful on all triangles"
    },
    {
        "video_name": "2t0003_sxtU",
        "Q": "why is at 0:12 he just lists mXn why not just list how",
        "A": "A visual example helps in remembering how to do something. In this particular example, seeing the matrix helps make the connection between doing it and simply comprehending it."
    },
    {
        "video_name": "ourH3ueWNmA",
        "Q": "At 0:59 why does he say Lurie and what does it mean?",
        "A": "Sal is referring to the mathematicician Jacob Lurie, panelist who talks about that concept."
    },
    {
        "video_name": "ourH3ueWNmA",
        "Q": "What are 2 digit numbers? I heard sal said it at 0:17.",
        "A": "Two digit numbers are numbers like 17,25,36,54,68 and 99."
    },
    {
        "video_name": "xgoqcc2-RS4",
        "Q": "At 0:30 does it matter what order you put it in?",
        "A": "To Olivia- Yeah the highest exponent is also known as the degree of the polynomial."
    },
    {
        "video_name": "xgoqcc2-RS4",
        "Q": "at 3:04 why dose this apply to only subtraction and not any others?",
        "A": "I am not sure exactly what you mean, but it would also apply to addition and multiplication of polynomials. Division is a question mark that I do not think always applies."
    },
    {
        "video_name": "xgoqcc2-RS4",
        "Q": "At 3:03, Sal says that if you add or subtract two polynomials, you will always get another polynomial, but sometimes they terms will cancel each other out, making either a monomial or nothing! Is this a mistake?",
        "A": "No, it isn t a mistake. A monomial is a polynomial with one term. And, 0 is a monomial."
    },
    {
        "video_name": "xgoqcc2-RS4",
        "Q": "What is the circle thing beginning at 3:08?",
        "A": "That my friend, is the set of all polynomials (or at least the ones taken into consideration). Any point outside is not a part of our circle ,which means, the point or expression would not be a polynomial.And anything inside the circle, you might have guessed this one,is a polynomial."
    },
    {
        "video_name": "vl1uMOYFyf4",
        "Q": "At 2:37 I am not so sure how you added the dimensions that is what always messes me up in school i am always confused on how to add the dimensions.This video showed me how to write the Net but everything else is a Blur.",
        "A": "I have dimensions 2(36)+2(8)+2(3)=x to simplify first: 72+16+6 then to 88+6 And finally I have 94. Hope this helps BTW, i had 2 times each number because i found the dimensions for the sides i could see, then i knew it was identical on the back, so i did 2 times."
    },
    {
        "video_name": "RGbA2IyJILY",
        "Q": "At 4:45, could Sal have added 12 to the equation instead of 84?",
        "A": "The way to solve an equation is to move things away from the variable, so while it is mathematically possible to move the 12, it is not very mathematically expedient because you would have to move it back at some point to finish the solution."
    },
    {
        "video_name": "RGbA2IyJILY",
        "Q": "At 5:42 how does the equation tell us that Ben's age now is 24? it should only tell us that Ben was 24 12 years ago, now he should be 36, and William should be 108, I'm i missing something?",
        "A": "you must remember that b=ben s age NOW. look at the table above the equation."
    },
    {
        "video_name": "RGbA2IyJILY",
        "Q": "At 3:33, why does Sal use 7(b-12 )=4b-12? Shouldn't it be the other way around? = 7(4b-12 )=b-12. How does he know which one to use? It doesn't make sense to me.",
        "A": "Because Ben isn t 7 times as old as William. William is 7 times as old as Ben. So Ben is 1/7 of William s age and 7(4b-12 )=b-12 doesn t work in that scenario."
    },
    {
        "video_name": "RGbA2IyJILY",
        "Q": "At 2:32; I don't understand why for William (12 years ago column) it's 4b-12 and not just 7b?",
        "A": "12 years ago Willliam was 7 times as old as Ben. That means 7 times as old as Ben s at that time, which was 12 year ago. That s why we set 4b-12 equal to 7 times Ben s age 12 years ago or 7(b-12). Hope this helps."
    },
    {
        "video_name": "fWi-i_mdUOI",
        "Q": "at 2:22, what did the /100 come from when there was already /10",
        "A": "/10 times /10"
    },
    {
        "video_name": "gKywkLHV6Ko",
        "Q": "at 0:55, when he tries to get the variable to one side, why is it that he divides, when some times you can subtract to balance out the scale? please help",
        "A": "Because to simplify .30x=15 you need to remove the coefficient .30x. To do that, subtraction would not work. The .30x would jump over to the other side, which does not accomplish anything. Division of both sides by .30 simplifies the equation while keeping it balanced."
    },
    {
        "video_name": "gKywkLHV6Ko",
        "Q": "At 1:27 why didn't sal simplify his work to 3/15?",
        "A": "he wanted to show that there are different ways to do it than just 3/15."
    },
    {
        "video_name": "gKywkLHV6Ko",
        "Q": "At 0:54 he divides both sides of the equation by 0.30, what is the reason for him doing that?",
        "A": "This is for equality. He wanted to get the variable x by itself so he divided the co-efficient 0.30 from x. However, when he did that he had to divide both sides by 0.30 to keep it equal. Imagine the equation x - 1 = 5, when you are solving for x you add 1 to both sides, if you only added 1 to one side it would be x = 5 which is incorrect."
    },
    {
        "video_name": "gKywkLHV6Ko",
        "Q": "At 1:51, wouldn't it be easier to multiply .3 and 15 by 10 instead of 100?",
        "A": "no it WOULDENT be"
    },
    {
        "video_name": "gKywkLHV6Ko",
        "Q": "at 1:24 why didn't sal just do this?\n150/3\n\ninstead of this:\n1500/30",
        "A": "When dividing by a integer percentage, such as 31% you can always multiply through by 100 to turn .31 into 31 etc. In this case of 30% the second digit is zero, so you only need to multiply through by 10. I think Sal was trying to be generic in his method and just get past that quickly, but really it would be nice to explain than in the case of 30% you only need to multiply through by 10."
    },
    {
        "video_name": "gKywkLHV6Ko",
        "Q": "I figured out the answer at 1:06,",
        "A": "Good job! You should try to do problems ahead."
    },
    {
        "video_name": "14-NdQwKz9w",
        "Q": "at 1:30 when Lucas and Fibonacci numbers are being talked about the one thing i noticed is that a number is added to the previous to get the next(ex , 1+1= 2+1= 3+2= 5 +3=8 cont.) is there a special name for such a sequence of is it confided to Fibonacci\\Lucas",
        "A": "Yes, they are called Lucas sequences."
    },
    {
        "video_name": "14-NdQwKz9w",
        "Q": "Does this rule work for all spirals? At 2:52",
        "A": "yes you can use the rule on all spiral"
    },
    {
        "video_name": "14-NdQwKz9w",
        "Q": "Another question: At 4:50, there is the sentence: \"An accidental fourish beginning might stabalizing into Lucas.\" Anyone who could explain that to me?",
        "A": "Yes. Vi means that a slight variation on the regular pattern could still settle into the regular pattern."
    },
    {
        "video_name": "14-NdQwKz9w",
        "Q": "0:54",
        "A": "the flower at 0:54 is very cute!"
    },
    {
        "video_name": "14-NdQwKz9w",
        "Q": "Could someone please explain to me what she meant at 3:32-4:06? I am really confused.",
        "A": "Plants send out a chemical called a hormone, and leaves grow out from the center, called the meristem, and eat the hormone. Leaves will grow away from each other because there is the most hormone where there aren t other leaves."
    },
    {
        "video_name": "ku4KOFQ-bB4",
        "Q": "At 2:30, Sal talks about the base-10 system that uses 10 symbols. In view of that, how could we classify the Roman numeral system? It has no 0, but a 10 (X). Also it doesn't have 10 different symbols (I-V-X-L-C-D-M) to express numbers. Is it still considered to be base-10 since it uses multiple of 10s? Thank you for your answers.",
        "A": "Most sources (but not all) I ve seen classify it as base-10, at least in the sense you mention that powers of ten (10, 100, and 1000) are named (X, C, and M); but since the system has no zero and no concept of place value , it operates differently than positional systems such as decimal and binary."
    },
    {
        "video_name": "ku4KOFQ-bB4",
        "Q": "at 8:27 when representing 231, why is there no 16s or 8s?",
        "A": "Work through the problem (Do you know how?) and find out."
    },
    {
        "video_name": "FSpSrZQzzVs",
        "Q": "At 00:45 Why convert to decimals? Why not convert all to fractions, or all to percentages?",
        "A": "Decimals are easier to work with while when you take a fraction or percentages, you have to do a lot of simplifications and conversations."
    },
    {
        "video_name": "FSpSrZQzzVs",
        "Q": "At 1:41, Sal factors out a negative sign. Can someone explain to me what that it is and why he did it? Also, can you explain how they became positive? Thanks!",
        "A": "Sal is using the distributive property to factor out a -1. This essentially causes all the numbers to be divided by -1, thus changing their signs. (Note: he didn t need to do this. I think it would have been clearer if he hadn t). If you haven t learned about the distributive property and factoring, here is the problem done without the factoring: 0.79 - 79.1 - 58.1 = -78.31 - 58.1 = -136.41"
    },
    {
        "video_name": "FSpSrZQzzVs",
        "Q": "At 3:28, there is a box that says \"Sal said 'numerator', but meant 'minuend'. What is minuend?",
        "A": "Minuend. The first number in a subtraction. The number from which another number (the Subtrahend) is to be subtracted. Example: in 8 \u00e2\u0088\u0092 3 = 5, 8 is the minuend."
    },
    {
        "video_name": "NYd6wzYkQIM",
        "Q": "@4:30 actually, if you pulled a 9 and a 6, you would get 7 and a half.",
        "A": "You re right! They should have one of those little correction pop-ups."
    },
    {
        "video_name": "NYd6wzYkQIM",
        "Q": "Is there a difference between 1000 times taking samples of 10 (as at 7:54) and 10 times taking samples of 1000?\nWhat about taking 1 time taking a sample of 10000?\n...Or 10000 times taking a sample of 1?",
        "A": "Yes. There is the difference. The bigger sample size you have - the narrow normal distribution you will get. For example, Sample size = 25, number of iteration = 5 || Sample size = 5, number of iteration = 25 mean ___________ 13.74 __________ || ____________ 14.32 ____________ median __________ 13.00 __________ || ____________ 15.00 ____________ SD ______________ 1.19 ___________ || ____________ 2.99 ____________"
    },
    {
        "video_name": "oEgeUk_Ix2c",
        "Q": "At 0:55, I feel that it's kind of irrelevant to multiply the 1. Why does Sal mention that as another strategy? It's confusing me.",
        "A": "It helps when things are to the power of 0."
    },
    {
        "video_name": "oEgeUk_Ix2c",
        "Q": "At 1:27: Is (4/9)^2 the same as (4^2/9^2)?",
        "A": "Yes, it is. 16/81 is the product of the second expression you mentioned, and when converted to a fraction is equal to the first."
    },
    {
        "video_name": "W-5liMGKgHA",
        "Q": "At 4:18, why did he divide by 2?",
        "A": "The y was multiplied by 2 and he just wanted y, so he had to divide by 2, and to balance it he did that to the other side as well."
    },
    {
        "video_name": "W-5liMGKgHA",
        "Q": "At 1:45 why was he able to subtract y and get 18? He had 18y and when he did this it should have left him with 17y.",
        "A": "He had 18+y, not 18y, this is why subtracting y leaves just 18. Hope this helps."
    },
    {
        "video_name": "W-5liMGKgHA",
        "Q": "At 1:45 if y = years when he says y years isn't he saying years years? Plz explain!",
        "A": "y = the NUMBER of years, so it could be 10 years or 24 years or 3 years. So, when he says y years, it is like saying a number and then saying the word years . Saying y years is like saying 12 years. We just don t know what number y is yet."
    },
    {
        "video_name": "W-5liMGKgHA",
        "Q": "In 3:50, Sal subtracted y from 18y and that is 17y, but he wrote 18. And on the other side, he subtracted y from 3y and he got 2y, that is correct, but the first one is incorrect.",
        "A": "Actually, the equation on the left, which you assumed was 18y, was really 18 + y. The equation would therefore be correct. Hope this helps!"
    },
    {
        "video_name": "W-5liMGKgHA",
        "Q": "What 3:45 in the video does he take -3y to 2 but not 18 y to 17?",
        "A": "He is able to do that because there was a y on the left side and he is removing the y from the left by subtracting y from 3y. Also there isn t a -3y at that time."
    },
    {
        "video_name": "W-5liMGKgHA",
        "Q": "At 0:45, Sal asked \"how many years will it take Arman to be 3 times as old as Diya?\" he is 9 times is old as Diya. What is he asking?",
        "A": "He s nine times as old now. So you can set up equations to work out their age difference in years. Then set up equations to work out when one will be three times as old."
    },
    {
        "video_name": "B0R3MJOrST0",
        "Q": "at 5:01 the pizza could have been an obtuse",
        "A": "How do you think it could have been an obtuse angle?"
    },
    {
        "video_name": "CxcP4ylUP5w",
        "Q": "at 1:07 why didnt he just leave the 8, I mean why did he simplify?? that just makes the problem and process harder to comprehend IF that is not needed. I mean the number 8 was already cubed. it was 8c^3 so why make it 2c^3 and what happened to the other cube in that same number? that confused me a lot. and also, do we have to simplify numbers even if the a^3-b^3 expression is already there?",
        "A": "Sal is factoring the polynomial. Factoring is the opposite of multiplication. It s like taking the number 15 and changing it into 3*5 (two factors of 15). There are many instances where we need factors, especially when working with fractions and solving polynomial equations. Sal is trying to teach you how to factor a polynomial that is a difference of 2 cubes."
    },
    {
        "video_name": "CxcP4ylUP5w",
        "Q": "Actually I think there is a mistake because like at the end around 4:08, he says 5(2c = d)(4c^2...) but if it is a^2, and if a= 2c, isn't the 4c^2 supposed to be JUST 4c? Because we did square that 2c so it should be 2c, instead of 4c^2... or am I wrong?",
        "A": "Sal did it correctly. Your correct that a is 2c. So (a)^2 is (2c)^2 = 2^2 * c^2. = 4c^2 Does that help at all?"
    },
    {
        "video_name": "CxcP4ylUP5w",
        "Q": "At 2:34 Sal said that the value of b was d. Is my understanding correct in that you only take the number and ignore the sign that comes in front of it? In this case, Sal ignored the negative that came in front of d.",
        "A": "I don t think so, after he is factoring there is a specific sign come with the rule which is minus if it s difference (a-b) (a^2 + ab + b^2 ) and it s plus if it s sum of cube as he wrote it in the top right side. (a+b) (a^2 - ab + b^2 )"
    },
    {
        "video_name": "vWXMDIazHjA",
        "Q": "At 4:30, I used the Pathagorean Theorem, but it does not give the correct answer. Any ideas? Thank you.",
        "A": "The Pythagorean theorem finds the length of the hypotenuse. In this problem they were trying to find area, so while you probably found the correct length of the hypotenuse, it is not the same as finding the area."
    },
    {
        "video_name": "vWXMDIazHjA",
        "Q": "AT 2:01 , does it count six sides too",
        "A": "It is only counting the 5 inside line segments."
    },
    {
        "video_name": "vWXMDIazHjA",
        "Q": "I didn't understand the 5:22 onwards part",
        "A": "The third problem starts at 5:22. He shifts the sides to make it clearer to solve"
    },
    {
        "video_name": "vWXMDIazHjA",
        "Q": "At 2:35, what shape is that? Isn't that 1/4 of a octagon?",
        "A": "that is a trapezoid"
    },
    {
        "video_name": "vWXMDIazHjA",
        "Q": "At 3:50, why is the distance AB = 6?\nDidn't get this point really..",
        "A": "We were given AB=6 in the original drawing. Since Sal dropped an altitude (a perpendicular line from point A to side CD, he split the trapezoid ABCD into a rectangle ABCE and a triangle AED. AE = 6 because opposite sides of a rectangle are always equal in length. That left us with DE = 9-6 = 3."
    },
    {
        "video_name": "vWXMDIazHjA",
        "Q": "Pardon me if I'm overseeing something, but around 7:25, Sal says that the lengths of the two circled sides are both 2. How does he know that? Couldn't the white side be a bit longer, and then the purple side would be shifted down and the circled side on the left would be shorter?",
        "A": "First of all, it s because all of the angles are right angles. Now, if the white side would have been longer, then it s true that the purple side would have been shorter. but it would be shorter by the same amount that the white would ve been longer! So that it doesn t really matter."
    },
    {
        "video_name": "vWXMDIazHjA",
        "Q": "At 4:30, to find the perimeter wouldn't Sal have to include the 9 for the perimeter of the square?",
        "A": "The part you were at was the AREA. Plus, 9 was the length of one side of the square AND a side of a triangle. (Both make a line of 9)."
    },
    {
        "video_name": "vWXMDIazHjA",
        "Q": "at the question around 6:05 could you also chop it into sections? or will that be harder?",
        "A": "you could chop it into sections but that will be harder to do that moving the sides around. it is much simpler to just shift the sides like Sal says"
    },
    {
        "video_name": "Iur13MNO0Ro",
        "Q": "At 3:57 - It would be helpful to have more explanation regarding \"pi squared, but that's just a number\". Why doesn't the last term become 2pi*-2/3x^-5/3?",
        "A": "When taking derivatives, pi is not taken as a variable. It is instead taken as a constant, so the constant multiple rule can be applied to pi. Hope this helps."
    },
    {
        "video_name": "uvgru6FcyxU",
        "Q": "at 11:40 when you move in the v2 direction aren't you also moving in the v1 direction?",
        "A": "Technically, yes. You re partially moving in the same direction of both vectors.That s why the graph paper is skewed. If the vectors were orthogonal you would move according to the standard basis / normal, unskewed graph paper."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "at 5:50,if we add 3a+(-2b),the resulting vector is the green arrow or the blue one?I'm confused",
        "A": "hi, the resulting vector is the blue vector."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "at ~17:00 in the video Sal multiplies the first equation by -2 ... sadly it's been 20+ years since I did any high school math, so I was struggling to understand why he picked -2, was it just based on the value of c1 in the second equation so it could eliminate the variable?",
        "A": "Yes, he wanted to eliminate c1 from the two equations so he could solve for c2."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "5:53-- Drawing the sum vector.. Why do we draw from 3a and not from -2b? Can we draw from both (or choose whichever one we want?)\nThanks for breaking it down.\n\n13:20 ...\"just so youre satisfied... \"... you are the best. talking down to us or your up to your boss, i love it.\n\n... and you sound like obama and I am going to miss him no matter who wins.",
        "A": "We can draw from both... after all -2b+3a is the same as 3a-2b by the axioms of arithmetic (though notice a and b are vectors here)."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "in 17:33, why did khan add the equations of x1 and x2 together?",
        "A": "He first multiplied x1 by -2 and then added both eqs of x1 and x2 just to eliminate c1 and get value of c2. However substitution of c1 value from x1 into x2 is also possible but khan prefered elimination method.Maybe because he didnt noticed that substition is possible or may be because he likes elimination method more than substitution and is better at elimination."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "11:18 why can't you talk about span (vector a) as linear combinations..?\nthanks, please reply asap",
        "A": "is it bc c1 (vect a) + c2 (vect a) = (some constant k) * (vect a) ?"
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "At 12:39 when he is describing the i and j vector, he writes them as [1, 0] and [0,1] respectively yet on drawing them he draws them to a scale of [2,0] and [0,2]. Is this an honest mistake or is it just a property of unit vectors having no fixed dimension?",
        "A": "No, that looks like a mistake, he must of been thinking that each square was of unit one and not the unit 2 marker as stated on the scale."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "1:36 what is a \"constant\"?",
        "A": "In this case, Sal means a scalar."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "10:46 Sal says the span of zero vector is a zero vector, i.e. span( 0 ) = 0. Shouldn't it be a set which only member is zero vector? i.e. span( 0 ) = { 0 }",
        "A": "Thats very astute of you. Yes, the span is by definition a set. Thumbs up!"
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "What is the difference between Rn and Rm in the footnote at 0:42?",
        "A": "At that point, Sal is saying that he has n vectors in the set. Each vector is a member of Rm. The reason it s m and not n is because the number of vectors doesn t have to be the same as the number of components in each vector."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "In the video at 0:32, Sal says we are in R^n, but then the correction says we are in R^m. Why does it have to be R^m? Is it because the number of vectors doesn't have to be the same as the size of the space?",
        "A": "Correct. The number of vectors don t have to be the same as the dimension you re working within."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "At 14:10 you give the mathematical definition for span. I am studying in South Africa and want to check if the notation used here is acceptable. Instead of c we use the letter k to represent a constant that is also a real number so if I were to write k1v1+k2v2+ ... +knvn is that acceptable?\n\nThanks for the help :)\nTammy",
        "A": "Yeah, it is perfectly acceptable. :) c and k are both used as pronumerals and can both represent the constants."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "on 17:25 why did Mr Khan multiply X1 equation by -2?",
        "A": "Multiplying the equation by -2 cancels (-2Csub1 + 2Csub1) when you add them then you re just left with one unknown in the equation which is much easier to solve. 1Csub1 + 0Csub2 = Xsub1 *Multiply by -2 -2Csub1 + 0 = -2Xsub1 *If you add that to 2Csub1 + 3Csub2 = Xsub2, -2Csub1 and 2Csub1 cancels out."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "At 17:58, it looks like Sal forgot the 2 in front of x1. Am I right or am I not looking at the problem correctly?",
        "A": "never mind. He caught it at 19:20."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "I am so very confused and frustrated. This video was all over the place. So why does on span of a, b is all of R2 but then the other is not? What? @9:35",
        "A": "I m sure there are rules for this. But in his first example, the vectors were not parallel, so any 2-dimensional vector could be made out of linear combinations of a and b. In his second example, the vectors were parallel so the span was restricted to the line. He did cite a 3rd example with only one vector (0,0), which doesn t really exist in 2 dimensions, and is why the span is restricted to a single point."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "@0:34, why does it matter if it's in R^m or R^n ? What's difference/significance and why do we choose one over the other?",
        "A": "Sal already used n for the number of vectors. The number of vectors doesn t necessarily have to be the dimension of the space. So, n shouldn t be used twice."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "At about 2:10 Sal denoted\n|0|\n|0|\n\nas a bold O. Can you do that for others? Like, is\n|2|\n|2|\n|2|\n\na bold 2?",
        "A": "No, that s not standard. For one, the zero vector represents the origin in any context, which might not be true for other numbers."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "at 6:15 he have joined 3a + (-2b) vector ..shouldn't it be joined graphically from arrow of 3a vector to arrow of -2b ..just asking as i've read that graphically on adding and subtracting we join those two vectors on which we are working on...\nanother question is what are 2-tuple..i've seen that in previous sal's vdo too..",
        "A": "No, what Sal did was correct. When you add vectors, you draw the first vector from the origin, then you draw the next vector starting at the head of the first. A 2-tuple is simply a 2 numbers collected in order. Vectors can be thought of as tuples."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "At about 5:30, why didn't he extend 3a altheway to (3,6)?",
        "A": "Watch carefully, Sal did so."
    },
    {
        "video_name": "Qm_OS-8COwU",
        "Q": "Can't i be < 1 as it is defined at 13:50?",
        "A": "No, the coefficients C\u00e1\u00b5\u00a2 can be any number (positive, negative, etc), but the index i represents the base vector that coefficient is associated with, that is why 1 \u00e2\u0089\u00a4 i \u00e2\u0089\u00a4 n (that menas that i should be a number between 1 and n)."
    },
    {
        "video_name": "tC1SfxRvtmM",
        "Q": "In around 0:47, Sal tells us to \"divide the eventual product by 5\", but I don't get why he does this. Is there a video on this trick, and why, for the sake of convenience, does he divide the 25/1's numerator and then divide 9/5's denominator by 5 and make it work? How does that trick preserve the essential value of both fractions?",
        "A": "So, 9/5 x 25/1 . If you multiply it out, it s 9 x 25 / 5 because you multiply both numerators to get (9 x 25) . You get 5 because you multiply both denominators ( 5 and 1 ) to get 5. Now you can divide the 25 by 5 ----> ( 25 / 5 ) . And you get 5. Now you have ( 9 x 5 ). 9 x 5 = 45. If you have anymore questions, ask me. :D"
    },
    {
        "video_name": "tC1SfxRvtmM",
        "Q": "I don't get it. How did you get the 5 and then the 1 at 00:50 to 00:57?",
        "A": "There are videos that will explain Greatest Common Factors and simplification better on the site. But, to get the gist of it: In order to simplify the fractions, you need to divide the numerator(25) and the denominator(5) by the Greatest Common Factor, which is 5 in this case. Mr. Khan did this before multiplying, but it s not necessary. 25 x 9 is 225. 225/5 is 45. Simplifying it made it a lot easier to get to the answer (45). Ask for clarification if you need it, okay? :)"
    },
    {
        "video_name": "9T3AAn-Cw3g",
        "Q": "At 1:01, why does Sal use -600 - 50 -9",
        "A": "He is splitting the whole problem into 3 separate problems, one for each place value."
    },
    {
        "video_name": "Oe1PKI_6-38",
        "Q": "When Sal simplifies -7x+ 13x into 2x^2+6x at 0:50, why does he add a plus sign inbetween 2x^2 and 6x? Doesn't it default to multiplication when there are no signs between numbers?",
        "A": "It is because 6x is positive 6x, and the sign for positive is +. Therefore, it wouldnt be multiplication"
    },
    {
        "video_name": "pwh1dK3vTkM",
        "Q": "At 2:40 in the video, how does one tell if a turn is too sharp?",
        "A": "Formally, if taking the limit of the derivative up to a certain value from both the right and left side results in different values, then the turn is too sharp. The turn not being too sharp simply means that the rate of change from both sides of a certain point should converge at the same value, i.e. for some input value a: lim f(h)-f(a) lim f(h)-f(a) h -> a^- ----------- = h -> a^+ ----------- h - a h - a"
    },
    {
        "video_name": "pwh1dK3vTkM",
        "Q": "at 4:23 why is f not differentiable on 1 ?",
        "A": "Starting at 4:06 Sal said the situation where it is not differentiable. - Vertical tangent (which isn t present in this example) - Not continuous (discontinuity) which happens at x=-3, and x=1 - Sharp point, which happens at x=3 So because at x=1, it is not continuous, it s not differentiable."
    },
    {
        "video_name": "pwh1dK3vTkM",
        "Q": "At 5:04 Sal said the derivative of f(3) is not defined because the slope in the negative direction is different from the slope in the positive direction. Would the derivative at f(2.9999999999) and f(3.00000001) still be defined?",
        "A": "I don t think they ll be very large. f (2.9999999999) looks to be about 3/2 and f (3.00000001) seems to be about -2/3, but it s hard to tell for sure."
    },
    {
        "video_name": "pwh1dK3vTkM",
        "Q": "Hey,\n4:12 , isn't the vertical asymptote at -3 a vertical tangent too? I thought that it is a vertical tangent because it is approaching infinity, which would mean it is not differentiable!\nThanks",
        "A": "No, they are not the same. Since f(-3) doesn t exist or undefined, there are no tangents at x=-3. But because it is undefined, created a discontinuity which made it not differentiable."
    },
    {
        "video_name": "pwh1dK3vTkM",
        "Q": "Isn't the explanation at 1:10 is wrong? I.e. I see that vertical tangent is not differentiable because, as it is in a \"sharp point\", its left derivative \u00e2\u0089\u00a0 right derivative. But it have nothing to do with infinity! Where did he took it from?",
        "A": "It s correct. Moreover your reasoning is a bit misleading. When x moves close to 3 from negative side, the rate of change is growing up to infinity (I imagine it like 1, 10, 100, ..., infinity). Then, when x moves from 3 rightward, the rate of change is decreasing from infinity to zero at x=6. So, the difference is that in sharp point the limits are different (like -2 != 2), and at vertical tangent the limit equals infinity. it was obscure for me as well, but some googling helped =)"
    },
    {
        "video_name": "a7wYAtMjORQ",
        "Q": "At 07:08 how we obtained psi (x,y) = C? did he integrate?",
        "A": "The solution you are looking for is always C. The derivative of a constant is zero, so when he has the derivative of psi = 0, this means Psi = C. So, yes, he did integrate."
    },
    {
        "video_name": "a7wYAtMjORQ",
        "Q": "at 9:50, why is ( My = Nx ) strong enough to imply an exact equation. It seems possible to me that My might equal Nx butthat no function Psi (P) exists such that M=Px and N=Py. (not withstanding that Fxy=Fyx for well behaved functons) Thanks",
        "A": "This is under the premise that M is in fact Px in the first place. So if M=Px, then My=Pxy, and similar logic applies to N, which is Py (again may or may not be in the real world). So if My=Pxy and Nx=Pyx, and Pxy=Pyx, then My=Nx."
    },
    {
        "video_name": "aDCXPdzyS0s",
        "Q": "@2:43 in the video. What was the postulate that says we are allowed to say that all three angle measures are equal to 180 degrees.",
        "A": "Sal does a video with a proof of that. It can be found at Geometry->Angles&Intersections-> Angles with triangles and polygons -> Proof: Sum of a measure of angles in a triangle are 180\u00c2\u00b0"
    },
    {
        "video_name": "JNm3M9cqWyc",
        "Q": "At 8:00 it is said that if we take a sample size of 10 or 20 from the population we would get a norm,al distribution. But how can we take 10 or 20 sample when the popution is itsels six{1,2,3,4,5,6}.",
        "A": "You can think of it as 10 or 20 rolls of a die. Coin flip has only two outcomes, but this in no way limits the number of time you can flip it."
    },
    {
        "video_name": "JNm3M9cqWyc",
        "Q": "At 2:50, what is that bar that Sal draws on top of x?\nWhat is it called and where is it used?",
        "A": "That s incorrect. X bar is used to symbolize the sample mean."
    },
    {
        "video_name": "Ec-BKdC8vOo",
        "Q": "At 11:44, how did he turn S = 2/(sqrt(3/28)) into S = 2 x sqrt(28/3) ? I checked on a calculator, and those values are not equal!",
        "A": "Did you check it again? :) Because i got the same values for both of them. Or using mathematics you can rewrite it as 2*(3/28)^(-1/2) = 2*(28/3)^(-1)^(-1/2) = 2*(28/3)^(-1 *-1/2) = 2*(28/3)^(1/2) note that (x)^(1/2) = sqrt(x)"
    },
    {
        "video_name": "Ec-BKdC8vOo",
        "Q": "at 9:58 couldn't you just use the inverse tan of radical three over 5?",
        "A": "You could, there are a lot of ways of solving trig identities, boils down to what you are comfortable with."
    },
    {
        "video_name": "Ec-BKdC8vOo",
        "Q": "At 4:18, how can you be certain that point E will take you back to the Y-Axis? Why CAN'T E have a positive x value (that is to the right of the Y-Axis)? Is there some sort of proof to that? Because I could imagine a Hexagon that fits all the criteria that have vertex E not on the Y-Axis. Please explain.",
        "A": "We know that vertex E has to have a y-coordinate of 8. Sal already used up the y-coordinates 0, 2, 4, and 6, and if Sal wanted to get a y-coordinate of 10, he would need a side length of at least 6, but the side length is only 5. So, that means that the change in y has to be 4. Since the total side length is 5, the change in x has to be 3, by the Pythagorean theorem. The change in x also has to be rightward, since if it were leftward A, F and E, would be collinear, and that can t happen."
    },
    {
        "video_name": "Ec-BKdC8vOo",
        "Q": "at around 9:19, how can he get the sin(60-\u00ce\u00b8)=sin60*cos\u00ce\u00b8-cos60*sin\u00ce\u00b8",
        "A": "That is a trigonometric Identity sin (x-y) = sin x cos y - cos x sin y and sin (x+y) = sin x cos y + cos x sin y there are many trig identities..."
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "At about 6:20 in the video, Sal wrote log(Base B)A - log(Base B) C=log(Base B)(A/C). Wouldn't it be log(Base B)(A-C)? Also, around 2:15 of the video he does a similar thing with adding two logarithms. Why does it work this way?",
        "A": "It s an interesting property of logarithms. log_a(b) + log_a(c) = log_a(bc) log_a(b) - log_a(c) = log_a(b/c) A proof is as follows: Let a = log_n(x), so n^a = x Let b = log_n(y), so n^b= y n^a * n^b = xy n^(a + b) = xy a + b = log_n(xy) log_n(x) + log_n(y) = log_n(xy)"
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "In 7:15 why did you flip 81 to its reciprocal on the right side?",
        "A": "Because given Log3(1/9) - Log3(81) = Log3( (1/9)/ 81 ) In order to divide a fraction by an integer, you just have to multiply said integer by the denominator of that fraction. That s why Sal represented it in this way"
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "At 7:11, why does he write \"1/81\" instead of \"81\"? I thought he would just multiply 1/9 and 81.",
        "A": "thanks"
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "sooo At 6:25 Sal mentioned the logarithms being subtracted and divided. Will that same property apply when Sal divides the two logarithms or multiply them?",
        "A": "when you have logA+logB it is same as log(AxB) and vice versa and when you have logA-logB it is same as log(A/B) and vice versa"
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "At about 5:20, I'm getting a little confused by the explanation of the log property: in exponential form, the numbers are being multiplied, but in logarithmic form, they're being added.\nI understand that this is how logs simply work, but is there a particular reason why?",
        "A": "In both multiplying exponents and adding logarithms, you are really just adding the value of the exponents in both forms."
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "At 7:15, wouldn't the fraction form of \"81\" be \"81/1\"?",
        "A": "81=81/1, yes, but he is using the identity log(a)-log(b)=log(a/b), so that log(1/9)-log(81)=log((1/9)/81)=log(1/729)."
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "in 7:23, sal said 9 times 8 is 720!!!\n\nIsn't 9 times 8 is 72?",
        "A": "Yea. he meant to say 9 by 80...but the way he said it, it s the same"
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "At 4:52 what Khan is trying is say is the law of indices, eg. a^m x a^n = a^m+n. Am i right?",
        "A": "Yes, exactly right. From Wolfram (which I use as my highest authority on maths) . . . The exponent laws, also called the laws of indices (Higgens 1998) or power rules (Derbyshire 2004, p. 65), are the rules governing the combination of exponents (powers). x^m * x^n = x^(m+n)"
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "So, (-2)-(+4)=(-6), and 3^-6=1/729. So why does Sal (@ 8:42) flip the 1/729 into 729? Is he just checking? Should the log in the solution read: log(base 3)1/729=(-6)?",
        "A": "proven that log(base 3)1/9 - log(base 3)81 = log(base 3)1/729"
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "At 2:29, how can I add two very big powers using logarithms?",
        "A": "You add large ones the same way that small ones are added."
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "At 3:20 I didn't understand what he meant.I understood that the equations are equal but in the first case he adds 2 logs and in the second when he wrote 2^3 * 2^5.\nWhat I mean is that in the first case he shows an example of addition and in the second we know the rules applies only to multiplication , where does he get the multiplication if it was an addition?",
        "A": "Try this: In (2^3)*(2^5) = 2^n what is n? It s 8, right? How did you get n? You added 3 and 5. So in the multiplication (2^3)*(2^5) = 2^8, the exponents (logs) are added: 3+5 = 8 So the multiplication and addition go together. To get the log of the product you add the logs of the factors."
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "at 1:59 he says the bases have to be the same. what if they aren't?",
        "A": "The bases have to be the same. otherwise, they aren t logarithms"
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "At about 8:05 it sorta stops making sense, the function relationships don't seem right, \"Suddenly, a wild square appeared.\" Whats going on there?",
        "A": "lol sounds like pokemon: a wild square appeared! Go Algebra! but anyway, that part is fine. what do you not understand?"
    },
    {
        "video_name": "PupNgv49_WY",
        "Q": "Around 8:00, why is that it isn't 81/1? and it's 1/81?\nI would assume that it is 3^4?\n\nCan someone clarify? :(",
        "A": "Great question. Since he is dividing 1/9 by 81, to him it just seemed easier to write is as 1/9 * 1/81. I hope I found the correct place you were asking about."
    },
    {
        "video_name": "wV59XWLmv0c",
        "Q": "at 3:45, during the calculation of X^2+Y^2=S^2, are we using the implicit differentiation? since there are multiple variables, I wondered if we should use either partial derivative or implicit derivative techniques...",
        "A": "Implicit differentiation. This is single-variable calculus and doesn t require you to use any partial derivatives. I hope I have helped you."
    },
    {
        "video_name": "wV59XWLmv0c",
        "Q": "at 2:01: Why is dy/dt decreasing?",
        "A": "Because y is the distance from the car to the intersection which is decreasing as the car approaches it."
    },
    {
        "video_name": "XtJwkIacpi8",
        "Q": "when Sal pulls up a calculator at 1:29 and other time, how does he get it because it looks exactly like a real caculator not the one you have on the computer",
        "A": "It is an emulator of a Texas Instruments graphing calculator."
    },
    {
        "video_name": "XtJwkIacpi8",
        "Q": "At 8:23 , Why does Sal use 5280 ?.",
        "A": "Because there are 5,280 feet in a mile. 1 mile = 1,760 yards and 1 yard = 3 feet. Then apply unit analysis."
    },
    {
        "video_name": "XtJwkIacpi8",
        "Q": "At 2:32 you got it wrong its 23",
        "A": "Please elaborate"
    },
    {
        "video_name": "lBscLuttQq0",
        "Q": "at 0:52 don't you mean (-3 + 3 = 3)",
        "A": "-3 + 3 is 0. Imagine a number line. Go to 3 below 0 and go forward by three. It should end up at 0."
    },
    {
        "video_name": "lBscLuttQq0",
        "Q": "At 0:43, why did you place the money, the three dollars from his uncle, in the parenthesis? I thought that was only in the order of operations?",
        "A": "Notice, the 3 has 2 minus signs in front of it. There is a convention in math that 2 operations are never written side by side. We use parentheses to separate them. Thus, we don t write: - - 3, we write - (- 3)"
    },
    {
        "video_name": "lBscLuttQq0",
        "Q": "at 0:15 he says net worth can someone please tell me what a net worth is .thank you",
        "A": "its like what gets caught in the net after all other considerations have been taken into account....after all additions and subtractions are finished, you get the net worth"
    },
    {
        "video_name": "lBscLuttQq0",
        "Q": "at 0:50 i have a question if you minus a negetive from a negetive is it a positive",
        "A": "Depends on the values and the problem. If I minus a negative from a negative and the beginning negative is larger no. Example: -9-(-5) would give us -4 since we are taking away -5 from the -9 and -4 is left over. However if I did -9-(-15) I would get +6. It s easier to write the problem out and just change the double negative to a + and work from there."
    },
    {
        "video_name": "3_DxJwDTbyQ",
        "Q": "When he says 1/10x @2:15 shouldnt it be x(1/10) because you multiply 1/10(x) because it is the reciprocal?",
        "A": "you re right. It s a mistake."
    },
    {
        "video_name": "3_DxJwDTbyQ",
        "Q": "At 6:28, I don't understand how he got 0 as a b value. I thought that he would have to multiply -2/3 by 1/3, like he did with the a value, and that gets -2/9. Could someone explain where he gets this number?",
        "A": "because if a is 1/3 and the equation is b=1/3-a b=1/3-1/3 b=0"
    },
    {
        "video_name": "3_DxJwDTbyQ",
        "Q": "At 1:37 Sal takes -3/a and separates it into -3*(1/a)\n\nWhat's the trick called and is there an exercise for it?",
        "A": "Do you mean how he pulled the -3 out of that fraction? Just think of how you multiply a whole number by a fraction... you multiply it by the numerator, and you are done. So if you were to multiply -3 times 1/a, that would equal -3/a. The parentheses are not significant, except to divide and clarify the problem. :)"
    },
    {
        "video_name": "3_DxJwDTbyQ",
        "Q": "At 4:28 Sal says \"when n is 7 m is going to be 1\".How did he get this?Can anyone please help me?",
        "A": "Sal has the equation: m = 1/7 * n If n = 7 , substitute into the equation and calculate m m = 1/7 * 7/1 = 1. Hope this helps."
    },
    {
        "video_name": "3_DxJwDTbyQ",
        "Q": "At 1:27, does it matter which variable you divide both sides by?",
        "A": "No. The point is that you need to recognize the formula. If you have a=1/b or b=1/a, it doesn t matter. They both have the y=k/x structure, which tells you that its an inverse formula. it s not until you start solving for those problems, that you need to be more specific on the variable you re doing things with..."
    },
    {
        "video_name": "3_DxJwDTbyQ",
        "Q": "At 5:31, Sal says that 1/3 of -3 is -1. Is that right? Is it possible for part of something to be larger than the actual something?",
        "A": "((-3)(1))/3 = (-3)/3 = (-1)/1 = -1 What s 3*(-1)? 3*(-1) = -3 (1/3)*(-3) = (1/3)*(3*(-1)) = ((1/3)(3))(-1) = 1(-1) = -1"
    },
    {
        "video_name": "3_DxJwDTbyQ",
        "Q": "5:37 Isn't when -3 multiply by 1 over 3 is increasing? Isn't that should be direct variation?",
        "A": "direct variation is when both sides increase or decrese by the same amount. indirect varation is when one side increase the same as the other side decrease. the latter is the example he s showing."
    },
    {
        "video_name": "lOIP_Z_-0Hs",
        "Q": "If plants don't grow well if their leaves cover other leaves, why do people always draw flowers like the flower at 1:16. What happens if the leaves do cover each other? Will the plant just die?",
        "A": "If the leaves cover each other, then that would be a waste of leaves. The leaf that is covered would not be generating anything for the plant yet at the same time the plant would still be supplying the leaf with nutrients to live. So the leaf would be a drain on the plant s resources."
    },
    {
        "video_name": "lOIP_Z_-0Hs",
        "Q": "I don't really understand at 2:07 what she was saying about 'PHI'",
        "A": "phi is a greek letter used to denote the golden ratio"
    },
    {
        "video_name": "lOIP_Z_-0Hs",
        "Q": "At 1:50, what is ratio?",
        "A": "A ratio is the quantity of one thing compared to the quantity of another thing. It also applies to fractions. For example if I have the improper fraction 2/2, the ratio would be 1:1 because both quantities are equal."
    },
    {
        "video_name": "lOIP_Z_-0Hs",
        "Q": "at 2:13 ish what is phi?",
        "A": "Phi is a Greek Letter and an Irrational number that is approximately 1.6180339887..."
    },
    {
        "video_name": "lOIP_Z_-0Hs",
        "Q": "At 5:19, what is a postulate?",
        "A": "A postulate is a thing suggested or assumed as true as the basis for reasoning, discussion, or belief. So, in that instance, mathematicians made a few simple assumptions which led to all of geometry."
    },
    {
        "video_name": "lOIP_Z_-0Hs",
        "Q": "At 3:56 she said angle-a-tron. Is that a real thing? does it work with other angles",
        "A": "There is a video all about angle-a-trons. It s in the same place as the video you watched. #Midnight Luna"
    },
    {
        "video_name": "lOIP_Z_-0Hs",
        "Q": "Why do some spiral based plants like artichokes, pineapples, and pine cones, not fit the pattern we learned before? At 5:40",
        "A": "It s because sometimes the growing cycle In a plant begins differently sometimes. This causes non-Fibonacci spirals to form."
    },
    {
        "video_name": "lOIP_Z_-0Hs",
        "Q": "Can the combinations of Fibonacci be something other that 2 and 5, 8 and 13 etc.? (Time in video, 4:44) Can it be like 2 and 8, 2 and 13, 5 and 8, 5 and 13? Also, What are the other combinations, and are there patterns in the 2 numbers in each pair? I don't get it.",
        "A": "No. They will always be two adjacent numbers in the series."
    },
    {
        "video_name": "lOIP_Z_-0Hs",
        "Q": "At 2:16, what is Phi?\nAnd what is - lcm(phi, pi)?\nWhat is - gcf(phi, pi)?",
        "A": "Phi is an irrational number equal to (1+sqrt(5))/2. The lcm and gcf of phi and pi are undefined because both pi and phi are irrational, while the lcm and gcf only work on whole numbers."
    },
    {
        "video_name": "kaxfCiP9d0w",
        "Q": "Hi! At 3:15 Sal said that 1/2 can be called common ratio. But the \"step\" for the common ratio should be 1. Here t goes from 0 to 5730. Shouldn't the common ration be (1/2)^(1/5730)? As always thanks for the grate job. Greeting from Greece!",
        "A": "Yes. He doesn t stress this, in the interest, maybe, of making this easy to learn. Every year the C14 is reduced by a factor of the 5,730th root of 1/2 . But a step of 5730 yrs is just as valid."
    },
    {
        "video_name": "kaxfCiP9d0w",
        "Q": "At 1:29 why didn't you multiply 741 by 1.5 instead of 0.5?",
        "A": "Read the problem carefully. It tells you that the carbon-14 loses half its mass. Your method of multiplying by 1.5 would mean that its mass is increasing by 1/2. Hope this helps."
    },
    {
        "video_name": "wB3QCk0MGuw",
        "Q": "at 2:22 why doesn't he just add 17.5 and 6.5 to get 24 then divide by 6 to get 4?",
        "A": "This is what Sal does, he just shows every step of the problem."
    },
    {
        "video_name": "wB3QCk0MGuw",
        "Q": "so you put what y ='s in were it has a y? (1:20)",
        "A": "oh ok so i was right thank you!"
    },
    {
        "video_name": "L5ErlC0COxI",
        "Q": "at 2:46 sal left a space and added and subtracted something to get in the required form but what about the real things which we get after multiplying {f(x+h)-f(x)/h}X {g(x+h)-g(x)/h}",
        "A": "This is just very difficult for us to do; as Sal said, at this point, there would be no easy way to work with that expression. The method he used was a little trick to get around that and make the expression a little more manageable. He didn t change the original expression, however, because he added AND subtracted the new part."
    },
    {
        "video_name": "L5ErlC0COxI",
        "Q": "At 1:45 why doesnt he do FOIL?",
        "A": "No, that s not right. You could do that with (x + h) (x + h) but he has f(x + h) * g(x + h) Which is entirely different."
    },
    {
        "video_name": "C3MI_mspN3o",
        "Q": "At 0:38, Sal says it's a safe assumption that both x and y are functions of t. Does it mean it isn't 100% sure to say that both x and y are functions of t? I'm a little confused to see the word 'assumption'..Can't anyone help me please?",
        "A": "It is just an english idiom that means the assumption we are about to make is certain."
    },
    {
        "video_name": "e7vA_S7abSY",
        "Q": "At 1:05, Sal wrote down \"5(x^2+4yx+4y^2)\", should it be 5(x^2+4xy+4y^2)? Please explain this to me.",
        "A": "The expressions are equivalent. xy=yx for any real numbers x and y."
    },
    {
        "video_name": "e7vA_S7abSY",
        "Q": "At 4:35 why does Kal say \"Once again, two y times two y is four y squared\"?",
        "A": "because 2*y*2*y=2*2*y*y=4*y^2=4y^2"
    },
    {
        "video_name": "mzULbezO6iA",
        "Q": "Aren't there infintly many solutions for a and b in this equation? All you have to do to get another correct solution if you choose an a other than 2 (2:32) is adjusting the value of b, right?",
        "A": "You can t have multiple solutions for the equation in the video. There is only one solution for a and b, but you can have different form for the solution. a is only 2 for a+4=6, however, you can write it as 4/2, 8/4 and so on. b is only -8 for 3b= -24, however, you can do as I did with a solution. Mathematics is so order."
    },
    {
        "video_name": "mzULbezO6iA",
        "Q": "At,0:31 how they get\n3x to the a power and bx to the fourth power ?\nHow they did it?",
        "A": "It was part of the original problem, so I m not sure what you mean."
    },
    {
        "video_name": "bIFdW0NZ9W4",
        "Q": "Shouldn't it be (5/3)^2 at 0:45 ?",
        "A": "Yes it should be... Sal s version only squares the 5. Good catch."
    },
    {
        "video_name": "bIFdW0NZ9W4",
        "Q": "Hi.\n\nIn 2:19 can we figure out what the nummer would be by by using the calculator:\ndon't know how to pronounce this so iam going to tell how I used the calculator.\nUsing ti-83 I press the number 4 then I go to MATH select 5th option. And write (256) and get 4\n\n4 times its self 4 times gives 256. Can I do this?",
        "A": "Yeah that sounds fine! It s good brain practice to do it manually, but if you ve already mastered deriving roots with prime factorization and it s just computational labor for you, I don t see why not."
    },
    {
        "video_name": "bIFdW0NZ9W4",
        "Q": "At 1:38 Sal does the reciprocal to make the fraction positive, is this the same as moving the entire fraction to the denominator and going from there? Thanks!",
        "A": "It does end up with the same result to take the fraction and put it as a denominator under a one as the numerator, however it is faster and less confusing to do it the way Sal does in the example."
    },
    {
        "video_name": "bIFdW0NZ9W4",
        "Q": "At 0:48 shouldn't the fraction be in parenthesis, because otherwise if we follow the order of operations it would give us 25/3",
        "A": "Good observation! Yes... the 5/3 should be in parentheses to square the entire fraction."
    },
    {
        "video_name": "bIFdW0NZ9W4",
        "Q": "At 1:15 when he said to pause and try to solve it for yourself, i solved but did it different from the way he explained it. What i did was square 81 and 256 than simplified it again to get 3/4 but since the exponent was negative i just did the reciprocal and got 4/3. Is the way i solved it a legal math move?",
        "A": "Your way worked because the exponent of 1/4 is the same as 1/2 * 1/2. Each exponent of 1/2 would tells you to take the square root (which is what you did). A word of caution... try to deal with the negative exponent first. I ve seen a lot of instances there people deal with the exponent of 1/4 and by the time they finish that work, they have completely forgotten that the exponent was negative. Thus, do the reciprocal 1st then, do the 1/4 and the negative doesn t get lost."
    },
    {
        "video_name": "GluohfOedQE",
        "Q": "At 7:00, what property is he refering to?",
        "A": "Supplementary angles."
    },
    {
        "video_name": "QS1LMomm0Gk",
        "Q": "at 4:04, I didn't understand why Sal subtracted -2 1/6 when it says to add them all.",
        "A": "thx"
    },
    {
        "video_name": "QS1LMomm0Gk",
        "Q": "At 4:41, why is 2 1/6 a positive?",
        "A": "Because when you add a negative integer to a positive integer. It is the same as subtracting a positive integer."
    },
    {
        "video_name": "QS1LMomm0Gk",
        "Q": "At 1:47, Sal said integers but there was a little box that said he meant numbers. So, that brings me to this question: Why aren't fractions integers?",
        "A": "Integers can only be whole numbers, fractions can also be in between \u00f0\u009f\u0098\u008a"
    },
    {
        "video_name": "QS1LMomm0Gk",
        "Q": "What is the point of the orange arrow at 4:04? It just doesn't seem to serve much of a purpose to me.",
        "A": "the orange arrow is showing the positive value of the problem because it is pointing to the right which means it is a positive the left is negative like the purple arrow"
    },
    {
        "video_name": "QS1LMomm0Gk",
        "Q": "At 5:49, can't you just switch the 93 here as in 31(3)=93 so 13(3) is 39?",
        "A": "Having the digits switched is really just a coincidence; you wouldn t use that to simplify the fractional expression. Also, if done this way, what would do with the denominator? However, the digits switching and having a product that follows the same properties does happen for many numbers. For example, 11*12 = 132, and 11*21 = 231. This is a concept that would probably be addressed in a class on number theory."
    },
    {
        "video_name": "QS1LMomm0Gk",
        "Q": "whoever put \"sal said 'integers' but meant 'numbers'\" at 1:48, wouldn't integers mean the same as numbers?",
        "A": "No, integers are whole numbers, numbers that are not fractions or decimals. Numbers are any number. So basically, integers are a kind of number."
    },
    {
        "video_name": "QS1LMomm0Gk",
        "Q": "there was a time in this video around 3:00 or somewhere it said Sai said Intager aand i didnt get a chance to read the rest",
        "A": "Integers are whole numbers. Like 1 or 2, but 1.5 or 2.5 is not an integer."
    },
    {
        "video_name": "QS1LMomm0Gk",
        "Q": "At 1:57 Why do we need to find a common denominator? Can't we just add 25/8 plus 3/4 directly?",
        "A": "Cause a common denominators solves all of life s problems"
    },
    {
        "video_name": "F5RyVWI4Onk",
        "Q": "At 1:27 to 1:42 doesn't it matter how you blow it up or does it always happen?",
        "A": "how about if you put a hexagonal brick inside it when blowing it up?"
    },
    {
        "video_name": "F5RyVWI4Onk",
        "Q": "@ 2:10 wouldn't vi make the biggest and deepest potato cylinder for maximum gravy?",
        "A": "Because the question was to find out the optimal gravy dish for a given amount of s*math*ed potatoes."
    },
    {
        "video_name": "1m9p9iubMLU",
        "Q": "At 8:52 , Sal says y/x is equal to tan theta . So y/x is also equal to slope,right? So can we call this tan theta as a slope ? If yes then slope of what? Terminal side of theta?",
        "A": "Yes to all your questions! Tan theta would be the slope of the line that creates the hypotenuse of the triangle, or as you put it, the terminal side of theta."
    },
    {
        "video_name": "1m9p9iubMLU",
        "Q": "i didnt understood from 7:45 plz help me",
        "A": "Positive and negative number cos sin tan"
    },
    {
        "video_name": "1m9p9iubMLU",
        "Q": "At about 7:00 Sal says soh cah toa doesn't work for triangles with a 0 degree or negative angle. How is it possible to have these angles in a triangle?",
        "A": "You re right! If you were to try and draw the triangle it would be impossible (unless you assume the negative angle was just a reverse positive angle) ! What Sal is trying to point out is that you can t just blindly use equations, you have to make sure it s possible too."
    },
    {
        "video_name": "1m9p9iubMLU",
        "Q": "6:25 ... why does soh cah toa have a problem? why doesn't it work for everything if it is just a use of proportions? Like isn't it just opposite over adjacent, and isn't opposite over adjacent the same no matter the angle?",
        "A": "The standard definitions of sin, cosine, and tangent only help us for angles that are greater than 0 degrees and less than 90 degrees. Defining them with the unit circle works for any arbitrary angle, as Sal explains in the video."
    },
    {
        "video_name": "1m9p9iubMLU",
        "Q": "At 0:59 He says We haven't moved in the xy direction......\nWhat does that mean?",
        "A": "I think he means that we haven t moved in the positive xy direction. It was poorly worded."
    },
    {
        "video_name": "1m9p9iubMLU",
        "Q": "At 3:40, the length of the opposite side (from theta) of the triangle is equal to the y-coordinate of the point that the triangle intersects the unit circle. But isn't that exactly what the length of the adjacent side is? If so, the triangle is isosceles, and it sure doesn't seem to me like it is. So does o = a or not?",
        "A": "No, they are different. The adjacent side is the horizontal side, and the opposite side is the vertical one, they are clearly not the same. In the drawing a<b just by looking at it, so the adjacent side is smaller than the opposite side."
    },
    {
        "video_name": "1m9p9iubMLU",
        "Q": "At 1:24, does the positive x axis always have to be the starting side?",
        "A": "As you define cos(p) to be the part within the x-axis and sin(p) to be the part within the y-axis, the point (cos(0), sin(0)) = (1,0) has to be your starting point."
    },
    {
        "video_name": "1m9p9iubMLU",
        "Q": "And also, at 8:40, what is the terminal side?",
        "A": "The terminal side of the angle is the point where the two lines forming the angle meet at an endpoint. The endpoint is where both lines end or terminate making this the terminal side of the angle."
    },
    {
        "video_name": "1m9p9iubMLU",
        "Q": "Is this the same as circular functions? At 2:33 Sal says it's an extension of basic trig functions. Is that what circular functions are? I have a math test tomorrow and my teacher didn't teach is this... Just playing it safe!",
        "A": "The basic principles are similar between circular function and the unit circle of trigonometry. Circular function to study about the circle on a graph paper. Unit circle of trigonometry is to study the trigonometry of angles, including angles that higher than 180 or 360 degrees."
    },
    {
        "video_name": "1m9p9iubMLU",
        "Q": "At 4:51, what is a hypotenuse?",
        "A": "The hypotenuse is the longest side of a right triangle or the side across from the right angle."
    },
    {
        "video_name": "1m9p9iubMLU",
        "Q": "What is \"theta\"? 2:17",
        "A": "Theta is a representation for angle . Wherever the symbol theta is used , it means that we are talking about any unknown angle . Hope this helps !"
    },
    {
        "video_name": "1m9p9iubMLU",
        "Q": "What is the \" symbol he uses at 8:25?",
        "A": "When he uses the symbols, what he means is to copy the statement he just wrote above it. IF I wrote: The answer is 2. 3. You can interpret the second line as The answer is 3."
    },
    {
        "video_name": "ZN2PfqZ4ihM",
        "Q": "At 11:11, if we want the area between z = 2 and 2x + 3z + y = 6, why do we integrate from x = 3 - y/2 to x= 0? Wouldn't we integrate from x = 3 to x = 3 - y/2 since x = 3 - y/2 is the lower bound?",
        "A": "Very good observation. I followed in the footsteps of those who solved this hideous problem, without realizing this, and now I know my solution is meaningless since the limits are mixed up. I would recommend mentioning this as an error in the video so that others may be able to avoid similar meaningless work. Thank you for your attention to detail and precision!"
    },
    {
        "video_name": "ZN2PfqZ4ihM",
        "Q": "At 9:21, i don't understand why you used the x-y plane instead of the x-z plane. I am not very good at visualizing 3D diagrams, so could you please show me why?",
        "A": "Because if your integration order takes care of Z first, i.e. dzdxdy, then once you find your Z limits in the first integral, then you are done with Z altogether, the next step to solve the triple integral is to project into the remaining variables plane, in other words, project into the x-y plane. You would only project your solid into the x-z plane if you integrated with the order dydxdz or dydzdx, in other words, integrating y first."
    },
    {
        "video_name": "ZN2PfqZ4ihM",
        "Q": "My question is at 10:10 of the video, when setting the bounds for the x and then y planes. I understand the bottom bounds being 0. What I don't understand is why the upper bound of x is the line x=3-y/2 and why the upper bound of y IS NOT also that line, only in terms of y (y=6-2x). To me it seems like both upper boundaries would either be the line or their simpler numbers, like we gave y at the end.\nThanks for your help, I always get mixed up using boundaries!",
        "A": "Sal drew a little triangle in the bottom right of the screen which is the projection of the volume on to the x-y plane. Sal also drew a horizontal rectangular bar with height dy. The area of the triangle is the sum of all of the little bars stacked on top of each other. Each bar runs from 0 to the end of the line in the x direction, so these are the x boundaries. And the bars are stacked from 0 to 6 in the y direction, so these are the y boundaries. Hope this helps."
    },
    {
        "video_name": "ZN2PfqZ4ihM",
        "Q": "At 7:18, why is lower bound of the dz integral equal to 2-(2x/3)-(y/3)? isn't this for the upper bound? and lower bound to be zero?",
        "A": "In the problem he s taken the triangle and made that the bottom of the region he s trying to integrate \u00e2\u0080\u0093 which explains why the lower bound is 2-(2x/3) - (y/3) and the upper bound is 2."
    },
    {
        "video_name": "QGfdhqbilY8",
        "Q": "At 1:45 Sal said to \"restrict the domain\". What does it mean, and how do you do it?",
        "A": "That means that you specify that certain values of x are not allowed. Typically this is done either when you are only interested in a particular portion of the domain for some reason or when the function fails to exist at some point or points."
    },
    {
        "video_name": "QGfdhqbilY8",
        "Q": "at 1:00 how can he just swap the x with g^-1(x)?",
        "A": "same question here"
    },
    {
        "video_name": "QGfdhqbilY8",
        "Q": "At around 6:00, how does Sal determine that the range is -pi/2 to pi/2?",
        "A": "From around 6:11 to around 7:10, Sal explains that the domain of the function tan must be restricted in order to make it invertible (so that no value in the domain of the inverse will have more than one image in the range). There are many ways to do this, but the easiest and most logical choice is to restrict the domain to the interval -pi/2 to pi/2, so this is the choice we agree (by convention) to make. This will be the range of arctan. Can you see why the interval 0 - pi will not work, for example?"
    },
    {
        "video_name": "QGfdhqbilY8",
        "Q": "At 8:10, Sal said that the domain was all real numbers. Here, the domain refers to the domain of (x-6) or x? If it refers to (x-6), then what is the domain for x? Why it is not the domain for x? I thought the domain referred to the domain of x.",
        "A": "Sal is speaking of the domain of the arctan function. You can input any real number into the arctan function, with the output, or range, of -pi/2 to pi/2."
    },
    {
        "video_name": "QGfdhqbilY8",
        "Q": "At 8:36, he said the domain is negative infinity to infinity, but earlier in the video (5:28)) he says the domain is all reals except for pi/2 + pi(k). If the domain isn't ALL values, how can it go from negative infinity to infinity? Also, the range is all real numbers (6:00), but at 8:51, he writes down the range is -pi/2 to pi/2\n\nCould you explain to me why? I'm not understanding how this works",
        "A": "From 3:50-6:12, Sal was talking about the normal tan(theta) function. Later, he talks about the inverse of tan and the restrictions."
    },
    {
        "video_name": "QGfdhqbilY8",
        "Q": "At around 9:15 Sal says that you add 3pi/2 to The low end of our range restriction. Why are we using the Low end and not the high end (-pi/2 instead of pi/2) ?",
        "A": "tan\u00e2\u0081\u00bb\u00c2\u00b9 has a range of (-\u00cf\u0080/2, \u00cf\u0080/2), but the equation is g\u00e2\u0081\u00bb\u00c2\u00b9(x) = tan\u00e2\u0081\u00bb\u00c2\u00b9(x-6) + 3\u00cf\u0080/2. So, in order to find the range of the whole thing, we need to add 3\u00cf\u0080/2 to both ends of the range of tan\u00e2\u0081\u00bb\u00c2\u00b9, giving us (-\u00cf\u0080/2+3\u00cf\u0080/2, \u00cf\u0080/2+3\u00cf\u0080/2) or (\u00cf\u0080, 2\u00cf\u0080)."
    },
    {
        "video_name": "QGfdhqbilY8",
        "Q": "At 7:53, Sal identifies the range of g^-1(x) as (-pi/2,pi.2). Why is the negative listed first? Is this simply a matter of convention or a matter of logic?",
        "A": "The convention is to list the numbers in numeric order... smallest to largest. That s why the negative is listed 1st."
    },
    {
        "video_name": "QGfdhqbilY8",
        "Q": "At 5:34 Sal wrote pi*k, but shouldn't it be k*pi?",
        "A": "It can be either way. No matter the way its put youll get the same thing"
    },
    {
        "video_name": "QGfdhqbilY8",
        "Q": "i didn't understand what was done between 0:50 to 0:59.\nplease explain!",
        "A": "All he did here was replace the value of x with g\u00e2\u0081\u00bb\u00c2\u00b9(x) and g(x) with x. This is the first step in finding the inverse of any function. So if you have g(x) = 2x - 1, the first step of finding the inverse is to replace x and g(x) with g\u00e2\u0081\u00bb\u00c2\u00b9(x) and x respectively, giving you x = 2g\u00e2\u0081\u00bb\u00c2\u00b9(x) - 1."
    },
    {
        "video_name": "QGfdhqbilY8",
        "Q": "At about 7:56, why is the restriction placed on the range, and not the domain? Wouldn't the restriction be affecting the x-axis, and should therefore be on the domain?",
        "A": "Hopefully this makes more sense with a calculator. Try inputing any real number x for arctan(x) and you will always get a solution, meaning the domain is all real numbers. However, the answer will never be 90 or above or -90 and less, always between those two, therefore the range is {-90,90} or {-(pi/2),+(pi/2)}"
    },
    {
        "video_name": "QGfdhqbilY8",
        "Q": "what does he mean at 6:35?\nBcz as you can see on of the slope is negative while the other is positive.",
        "A": "Not exactly. Slope is essentially change in y (\u00ce\u0094y) divided by the change in x (\u00ce\u0094x). For the line in Quadrant I, if you consider it moving upwards, then the slope is: \u00ce\u0094y/\u00ce\u0094x For the line in Quadrant III, if you consider it moving downwards, then the slope is: -\u00ce\u0094y/-\u00ce\u0094x The minus signs cancel out, giving you the same expression: \u00ce\u0094y/\u00ce\u0094x. Hence, whichever direction you consider a line to be moving, if it is the same line, then the slope is the same."
    },
    {
        "video_name": "1pHhMX0_4Bw",
        "Q": "2:15 a square is always a rhombus",
        "A": "Of course it really is. A square has the same identities of a rhombus."
    },
    {
        "video_name": "1pHhMX0_4Bw",
        "Q": "at 1:53,i couldn't understand why a rhombus isn't a quadrilateral",
        "A": "No, all rhombuses are a quadrilateral. You misheard Sal. he said that not all quadrilaterals are rhombuses."
    },
    {
        "video_name": "zqwVKhQV_2w",
        "Q": "at 2:27 and 2:31 in the video, why did sal count 1-8 twice",
        "A": "Sal probably counted 1-8 twice to focus on that part of the problem and help you focus to understand the problem. :)"
    },
    {
        "video_name": "9uxvm-USEYE",
        "Q": "So why, at about 3:43, does the dy/dx on the left side turn into -1(dy/dx)? Additionally, how does that turn (2y-2x) into (2y-2x-1)(dy/dx)?",
        "A": "The left side didn t turn into -(dy/dx), Sal just moved the dy/dx from the right side to the left. (2x - 2y) + (2y-2x)(dy/dx) = 1 + dy/dx -Subtract dy/dx from both sides So given that, the left side would be equal to: (2y - 2x)(dy/dx) - (dy/dx) = 1 - (2x - 2y) If we simplify things a little bit on both sides: 2y(dy/dx) - 2x(dy/dx) - (dy/dx) = 1 - 2x + 2y You can factor out dy/dx on the left side: (2y - 2x - 1)(dy/dx) = 1 - 2x + 2y"
    },
    {
        "video_name": "9uxvm-USEYE",
        "Q": "at 2:06 why are you distributing the 2x-2y as a whole rather than a part like when foiling?",
        "A": "It is to simplify the later operations, since implicit differentiation normally involves separating or isolating the dy/dx term . He did it his way to easier simplify :D"
    },
    {
        "video_name": "9uxvm-USEYE",
        "Q": "At 3:40 i don't understand the process of subtracting the dy/dx",
        "A": "The notation: dy/dx literally stands for the derivative of the function. Since it is an unknown you can treat it just as if it were a variable, which in a way, it is. I hope this helps."
    },
    {
        "video_name": "9uxvm-USEYE",
        "Q": "At 1:10 why is it the derivative of (x-y)^2 with respect to (x-y), and not with respect to x?",
        "A": "Because the Chain Rule states that if you are taking the derivative of f(g(x)), then it equals to f (g(x)) (with respect of g(x)), multiplied by g (x). If you are still not sure what I mean, you can go to the Calculus video called Chain Rule Introduction."
    },
    {
        "video_name": "9uxvm-USEYE",
        "Q": "at 1:23, what rule is Sal using and how. I understand its chain rule but i dont see how its used here",
        "A": "When taking the derivative of (x-y)^2 with respect to x, we can think of it as a composite function. It is in the form w(u(x)), where: w(x) = x^2 u(x) = x-y We can then take the derivative by using chain rule. d/dx[w(u(x))] = w (u(x)) * u (x) d/dx[(x-y)^2] = 2(x-y) * (1-dy/dx)"
    },
    {
        "video_name": "9uxvm-USEYE",
        "Q": "At 2:08 Sal decides to distribute (2x - 2y) and at 2:26 he distributes as -> (2x - 2y) + (2x - 2y)(1 - dx/dy).\nMy concern revolves around the distribution, that it should be -> (2x - 2y) - (2x - 2y)(1 - dx/dy) because (2x - 2y) distributes itself into (1 - dx/dy).\nWhy has Sal put a + ?",
        "A": "Sal s work here was correct but used a little sleight of hand that s easy to overlook. For the second part of the expression, instead of writing what you were expecting - (2x - 2y)(1 - dy/dx) he wrote + (2y - 2x)(1 - dy/dx) Notice that he reversed the order of the terms in the first parentheses: instead of 2x - 2y he wrote 2y - 2x. That reversal of order is equivalent to multiplying the expression by -1, and that allows him to change the minus sign to a +."
    },
    {
        "video_name": "9uxvm-USEYE",
        "Q": "At 2:24, did Sal accidentally forget the - when distributing (2x - 2y) into (1 - (dy/dx))?",
        "A": "No. (2x-2y)(1-(dy/dx)) Part 1: ((2x-2y)*1) = (2x-2y) Part 2: ((2x-2y)*(-(dy/dx))) = (2y-2x)(dy/dx) Notice that, in part 2, X and Y were switched, because (-(2x-2y)) = (2y-2x)"
    },
    {
        "video_name": "vAlazPPFlyY",
        "Q": "At 1:25 Sal lists the adjacent angles wouldn't angle AGE also work?",
        "A": "No, because it doesn t share a common side with angle BGD, andgle AGE can t be an adjacent angle. By definition, djacent angles have to have a common side and a common vertex (but can t overlap each other)."
    },
    {
        "video_name": "vAlazPPFlyY",
        "Q": "so at 2:44 , an angle that forms a linear pair with another angle means that the two angles hen added together make a 180 degree line ?",
        "A": "Well, that s a strange way to say it. 2 linear angles add up to 180 degrees because 180 degrees is the angle swept by either side of a line."
    },
    {
        "video_name": "vAlazPPFlyY",
        "Q": "AT 3:25 would DGC be a linear pair ?",
        "A": "Yes. DGC and DGF are linear pairs or supplementary angles."
    },
    {
        "video_name": "vAlazPPFlyY",
        "Q": "At 1:26 would it be incorrect for the question \"Name an angle adjacent to angle BGD\" to say angle BGD? Even though it shares two lines with BGD?",
        "A": "the definition of adjacent angle is an angle that shares one side, not two. Ihope this helped,"
    },
    {
        "video_name": "vAlazPPFlyY",
        "Q": "at 3:35, what's a linear pair?",
        "A": "A linear pair is a pair of angles where the two angles form a line (180 degrees), hence the name line -ar pair."
    },
    {
        "video_name": "vAlazPPFlyY",
        "Q": "Would angle AGD and angle BGE also be adjacent to angle BGD at 1:36 ?",
        "A": "Yes, because the angles are located next to each other and share a common vertex."
    },
    {
        "video_name": "vAlazPPFlyY",
        "Q": "at 2:06 does it have to have 2 rays in common to be an adjacent?",
        "A": "Adjacent angles have a corner point in common, and share one ray."
    },
    {
        "video_name": "vAlazPPFlyY",
        "Q": "FROM 0:08 to 1:28, where Sal listed all the answers to the first question, why isn't something like \"angle DGA\" and \"angle EGB\" listed above? Is that an answer? If not, please explain.\n\nThanks, FUZZI",
        "A": "In that diagram, angle DGA and angle EGB are also straight lines (straight angles). I m not sure if they count. Someone else please chime in on this."
    },
    {
        "video_name": "vAlazPPFlyY",
        "Q": "0:42 <ABG can also be called <BAG? ignore the fact that spells Bag.",
        "A": "Nope, not really. The vertex has to be the middle letter, so for 3-letter notation there are 2 possible ways of naming the same angle: \u00e2\u0088\u00a0BGA and \u00e2\u0088\u00a0AGB Angle \u00e2\u0088\u00a0BAG is not on the picture. It would look like an angle with sides (rays) AB and AG and vertex A. By the way, \u00e2\u0088\u00a0ABG is not on this picture either. Edit: fixed."
    },
    {
        "video_name": "vAlazPPFlyY",
        "Q": "At 0:53, couldn't angle <AGD be adjacent to angle <BGD? Doesn't <AGD have <GD in it?",
        "A": "The term adjacent means that the angle must be next to the designated angle (in this case, <BGD). The adjacent angle cannot contain (in part or in whole) the original angle in which you are comparing to (In short, no overlapping). Because <AGD contains the whole <BGD, <AGD cannot be adjacent to <BGD. As said in the video, some examples include <AGB and <EGD."
    },
    {
        "video_name": "vAlazPPFlyY",
        "Q": "At 8:34, What does linear pair mean?",
        "A": "Basically, it is a supplementary angle. That s another name for it."
    },
    {
        "video_name": "G4r9PpYQiL8",
        "Q": "at 2:24 why did you change the equation on top?",
        "A": "He is changing the equation to slope-intercept form so that it is easier to find the slope. Hope this helps!"
    },
    {
        "video_name": "s2_NTiISZl4",
        "Q": "I wanted to ask why at about 2:45 b + acoss didn't change. Why it didn't get raised to the 2nd power like the rest of the factors of the vectors. But then I thought maybe it did, but got cancelled out with the 1/2 power over the entire thing. I don't know if I'm right for sure though so could someone tell me for sure? Thanks in advance.",
        "A": "You are correct, he just simplified it immediately!"
    },
    {
        "video_name": "P0ZgqB44Do4",
        "Q": "(Referring to 6:37) Why can't the numerator be equal to 0? Also, what do you mean when you use the terms \"defined\" and \"undefined?\"",
        "A": "I may be wrong here, but I believe that is called a removable discontinuity. It is when a factor from both the numberator and denominator cancel out. When both the numerator and denominator equal zero, there is not an asymptote; instead there is a hole. This means that the graph is continuous (looks like one line or piece) but one point is undefined."
    },
    {
        "video_name": "P0ZgqB44Do4",
        "Q": "At 2:37, why does Sal say a horizontal asymptote at y=1/2? I am sorry I am so confused but don't we always refer to x when we say horizontal?",
        "A": "If y always equals a fixed value, then you ll get a horizontal line on that fixed value, which will be parallel to to x axis. So y = 1/2 gives you a horizontal line at 1/2. x = 1/2 would mean your y value is 1/2 for every single x value, which would produce a vertical line. The x coordinates are indeed the horizontal ones, and that s why the equations of lines work the way they do."
    },
    {
        "video_name": "P0ZgqB44Do4",
        "Q": "I am confused, at around 8:31 where Sal said that x doesn't equal -3 isn't that also true for positive 3? Since you look at the equation before to find when the equation is undefined?",
        "A": "Hi Nikhil, In this case, we are trying to find the vertical asymptote. So, the answer is whatever makes the denominator zero. So, only 3 can make (x -3) = 0. If -3 was used, it would be equal to -6 and that does not help us. Hope that helps! - JK"
    },
    {
        "video_name": "BQMyeQOLvpg",
        "Q": "At 8:46, shouldnt the range be {2}, instead of 2. Since the range is a set of numbers, not just a number?",
        "A": "Yes, I think it should be {2}."
    },
    {
        "video_name": "BQMyeQOLvpg",
        "Q": "at 15:00 Sal said scalar value function or real value function. Which one does he refer to as scalar or real value function R or R2,R.\nsilly question yes but can't grasp that exactly in the video tried watching again\nthanks for your help",
        "A": "Scalars are the normal numbers we worked with before the introduction to vectors. So all of the real numbers are scalars, mathematically, the real numbers are represented as \u00e2\u0084\u009d. Vectors are higher dimension entities, and thus they are commonly represented as \u00e2\u0084\u009d\u00e2\u0081\u00bf."
    },
    {
        "video_name": "BQMyeQOLvpg",
        "Q": "at the beginning, at around 3:28, he says that y=X^2 maps to all real numbers.. but that isnt true since x squared only leads to positive numbers?",
        "A": "For this particular function the codomain is the set of real numbers while the range (subset of codomain) is the set of positive numbers."
    },
    {
        "video_name": "BQMyeQOLvpg",
        "Q": "At 7:00 Sal says \"two tuples\" or something along that line. What does that mean and what do tuples have to with the domain?\nThanks.",
        "A": "A tuple is a set of numbers, so a 2-tuple is a set of 2 numbers. Every vector in Rn can be represented by an ordered n-tuple. So when he says that he s mapping 2-tuples to R, he means he s taking a 2D vector and mapping it to a 1D vector."
    },
    {
        "video_name": "BQMyeQOLvpg",
        "Q": "at 14:20 he says that (5, 1, 0) is not in the rage but I don't think that is correct. I believe that since I can find two numbers for x1 and x2 (being 5 and 0) that will satisfy the condition to produce(5, 1, 0), that in and of itself makes it in the range. I'm I correct?",
        "A": "no since u cant get the difference of 1. like he explained in the vid at 13:47 (5,0) will not give u 1. x2-x1(0-5) will give u -5"
    },
    {
        "video_name": "BQMyeQOLvpg",
        "Q": "Is a ratio a function? Like with 4:7, 4 would map to 7.",
        "A": "The ratio 4:7 is just the fraction 4/7. But say we had the set {4}, and the set {7}. The only possible function f: {4} -> {7} is f(4)=7. In such a sense like this, I suppose you could describe every rational number like this with two singleton sets like {4} and {7} for example. Now, if you have f: R -> R defined by: f(x)=4/7 for all x in R, then that is a function as well."
    },
    {
        "video_name": "6phoVfGKKec",
        "Q": "At 3:56, Sal says that the number has to be less than 10. What if I move the decimal point 12 places over to the right, and make the number 5.7? Wouldn't the notation, in this case be 5.7 x 10^-12 ? Does it always have to be the first non-zero digit that has a decimal point after it? Why?",
        "A": "But then you have a different number. 5.7 x 10^-12 = 0.0000000000057. The number in the video was 0.0000000003457. That is indeed the same amount of digits, but it s a very different number: the one from the video is much larger, just like 3457 is much larger than 57."
    },
    {
        "video_name": "6phoVfGKKec",
        "Q": "At 0:24 Sal says that \"a\" is equal to or greater than 1 and less than 10. If your answer is 35.6x10^2 is this still scientific notation?",
        "A": "no because the number is not in correct scientific notation. It should be greater than 1 and less than 10"
    },
    {
        "video_name": "6phoVfGKKec",
        "Q": "At 2:33, Sal says that 1x10 to the -2 power is basically 1/10 squared or 1/100.\n\nI didn't really seem to understand that. Can someone explain that to me?\n\nThanks =D",
        "A": "To see a decimal you got to see the fractions too so basically it is the recipricol of 100"
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "in 7:34 how do u get 1/2 for the square root of 1/4?",
        "A": "sqrt(1/4) = 1/2, because the square root of 1 is 1 (as 1 * 1 = 1), and the square root of 4 is 2 (as 2 * 2 = 4)."
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "why \u00e2\u0088\u009a10 x \u00e2\u0088\u009a10 makes 10? 9:34",
        "A": "That s what a square root of a number of is: a number that you multiply by itself to get the first number. Even if you don t know what the radical of 10 is, you know that when it s multiplied by another radical 10 it will equal 10."
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "At around 2:30, Sal says that the product of two irrational numbers, the square root of 2 and 3, respectively, in this case, is also irrational. However, isn't 1/pie times pie equal to 1 and therefore this statement is not correct?",
        "A": "1/pi * pi = pi/pi It is division. Any number divided by itself will = 1."
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "at 9:20 Sal talks about getting the \"radical denominator\" what does he mean. and how does he do what he does for simplifying that question?",
        "A": "The denominator of the fraction is expressed as a square root. The symbol used for a square root is called the radical sign. Sal wanted to make his denominator nicer by getting rid of the square root. This puts a square root in the numerator, but that is usually easier to work with."
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "THEN WHAT IS THE SQUARE ROOT OF 10? (9:00)",
        "A": "That would be 3.16227766"
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "Just a question, at 0:22 why does Sal do the square root of 5*5 ??",
        "A": "So, he his not finding the square root, he is factoring and finding the square root of the factorization?"
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "at 6:42 you said that the square root of 4*4 would be 4 why wouldn't it be 2? doesn't 2*2=4 and would be a perfect square?",
        "A": "\u00e2\u0088\u009a4*4 is the same as \u00e2\u0088\u009a16 which is 4. 4 to the second power is the same as 4*4. The square root of 16 is the same as 4*4. So everytime you take a square root of the product of two of the same integers. The answer is the first or second integer: \u00e2\u0088\u009a125*125 = 125 \u00e2\u0088\u009a82394*82394 = 82394 \u00e2\u0088\u009a11*11 = 11 \u00e2\u0088\u009a25*25 = 25"
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "@ 2:45 What about Pi? It is Circum./diameter. Do the numbers in the ratio have to be defined?",
        "A": "Oh I see. So the numbers have to be integers. Thanks for the explanation :D"
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "I'm sorry but is this dumb in 2:41 why is 2 times the square root of six not both rational and irrational. Because I thought that a rational number can have decimals that terminate or repeat , So why is it that 2 times the square root of 6 not both rational and irrational?",
        "A": "The answer is a single number 2Ssqrt(6). a single number can only be rational or irrational. That is an irrational number."
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "at 7:24 i thought it should be the \"square root of 1 over 2, why its only \"1\" and not the \"square root of 1\"?\"",
        "A": "The implied question is what s the sq root of \u00c2\u00bc not \u00c2\u00bd Sq root of .25 is .50 check with your calculator. You have to apply the sq root function to both the enumerator and denominator. Sq root of 1 = 1 Sq root of 4 = 2"
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "9:40, I don't understand why 1/sqrt 10 * sqrt10 over sqrt10 = sqrt10 over 10 ?",
        "A": "At the numerator, 1* sqrt 10 = sqrt 10. Anything multiplied by one is itself. At the denominator, sqrt 10 * sqrt 10 = 10. It s the same way the sqrt 4 * sqrt 4 = 4. sqrt 4 = 2. 2*2 = 4. Basically, you are re-squaring the 10."
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "At 8:00 is the answer to g rational?",
        "A": "3/2 is rational. It can even be written as 1.5 that is why it is rational. I hope this helps!"
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "around 6:50 solving the square root of 2000. why cant it be broken down into 2x1000 and then 1000 being broken down to 10x100 and 100 broken into 10x10. Because isn't 100 a square root? so why all the extra work. I mean you still get 10 times the square root of 20 which has to be broken down further, but its just a little quicker.",
        "A": "He did that so all the twos would be together so it is easier to see that they are the same."
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "At 5:45, he hit a perfect square-- 250. Why didn't he just stop there? (50)50 = 250!",
        "A": "50(50) is not 250."
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "At 7:32, why did he take the one out of the radical?",
        "A": "I am assuming you meant why is the one the numerator? 1/2 x 1/2 = 1/4 Multiply fractions by multiplying the numerators together and the denominators together with each fraction. So to find out 1/2 x 1/2, you multiply the numerators 1 x 1 and the denominators 2 x 2 What you are left with is 1/4, because 1 x 1 = 1"
    },
    {
        "video_name": "BpBh8gvMifs",
        "Q": "At 2:32, Sal says that 2\u00e2\u0088\u009a6 cannot be expressed as a fraction.\n\nNot to be difficult, but can't I just write 2\u00e2\u0088\u009a6/1, the same way as I can write 5/1.?",
        "A": "I see. So, its irrationality is determined by the fact that it can not be expressed as a ratio of two integers then?"
    },
    {
        "video_name": "ccrYD6iX_SY",
        "Q": "At 6:20, Sal says that (32*31*30*29*28)/(5!) is the number of hands with four ones.\nBut isn't he only really finding the number of positions the other numbers can be in when in the pattern 1 1 1 1 _ _ _ _ _? (see the video at 6:20 for clarification)\nDoesn't that only give us the number of possible ways to arrange the numbers in the five _ _ _ _ _ spaces when four ones are used already?\nOr are we just looking for that... I thought that 1 _ 1 _ 1 _ 1 _ _ is considered different from\n1 1 1 1 _ _ _ _ _?\nPlease help!",
        "A": "The value we re looking for here is a combination, not a permutation. Order does not matter in this example as the question says where the players can arrange the cards in any order so as each hand can be shuffled around, position doesn t matter. So in that case we are only looking for the number of combinations of 5 cards you can have from 32. Hope this helped!"
    },
    {
        "video_name": "ccrYD6iX_SY",
        "Q": "At 6:30 Sal divides the no. of hands with 4 1's with 5!, why does he assume that the 1's will only be the first four cards, shouldn't he divide the whole hand by 9!, as that's how one arranges 9 cards in 9 spaces?",
        "A": "Let call 1 as ace. It doesn t matter which slot you want to put the aces in as the 5th slot will have 32 choices, then 31 choices and so on. I believe in one of the videos he did on heads and tails he explained this. So arranging them in the first 4 slots will make the math neater. As to why he divided by 5!, he explained that in the video itself."
    },
    {
        "video_name": "ccrYD6iX_SY",
        "Q": "Sal, what if you would enter the ones at 5:18 in the last slots like 36 35 34 33 32 1 1 1 1? Would the probability still be the same or am I mistaken?\nThanks.",
        "A": "It says that hands can be assorted however the player wants, so the place of the 1s does not matter for the probability. You can also see it like this. In Sall last fraction on the bottom of the page the top fraction is (1 x 1 x 1 x 1 x 32 x 31 x 30 x 29 x 28). No matter where you place the ones, the top part of the fraction is always the same."
    },
    {
        "video_name": "ccrYD6iX_SY",
        "Q": "At 4:04 Sal got 9!: how does it become a factorial instead of 362,880?",
        "A": "While the two are equivalent, we often want to leave in factorial form because parts of them may cancel out later as Sal ended up doing in the video. Why work with such large numbers if we can cancel out before we get to the large numbers and work with smaller numbers?"
    },
    {
        "video_name": "ccrYD6iX_SY",
        "Q": "At 3:34, Sal wrote 9*8*7*6*3*2*1. Why did he do that?",
        "A": "He meant to write 9*8*7*6*5*4*3*2*1. He corrected it later in the video."
    },
    {
        "video_name": "ccrYD6iX_SY",
        "Q": "3:33 9 x 8 x 7 x 6 x 3 x 2 x 1? isnt it 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1?",
        "A": "@zootlewurdle: yeh, Khan works hard at what he does and sometimes makes mistakes, forgive the guy, and try to understand the video."
    },
    {
        "video_name": "ccrYD6iX_SY",
        "Q": "I'm so confused. What's the difference between in order and not in order?? He mentioned that at 3:06.",
        "A": "If order matters in the problem, I m pretty sure it s classified as a permutation. If it doesn t, it s combinations. The difference is whether or not the numbers must be in order, like if I have three heads, Head A, B, and C, and their places in the problem are not interchangeable."
    },
    {
        "video_name": "ccrYD6iX_SY",
        "Q": "At 5:41, shouldn't it be (32....28/9!) instead of (32....28/5!) ? I assume that the four 1's can be rearranged too.",
        "A": "Hi, Nop. You can view this as placing the four 1 s that you want in a fixed position and permuting the remaining 5 with any of the remaining 32 cards that were left. Remember that as long as the four 1 s appear in a hand, you don t care on their order either, and so the fixed position you chose at the beginning is enough."
    },
    {
        "video_name": "TErJ-Yr67BI",
        "Q": "At 4:10, could you also use opp. sides of congruent angles are congruent?",
        "A": "There is no such thing, just because angles are congruent, that does not speak to the sides opposite them, this can only be true if the sides forming both angles are congruent, then there is no use because SAS would suffice."
    },
    {
        "video_name": "TErJ-Yr67BI",
        "Q": "@ 1:13 Sal says DEC is congruent to BAE. I don't understand how he got there though.",
        "A": "With a parallelogram, opposite sides are equal, and with parallel lines, alternate interior angles are congruent, so the triangles are congruent by ASA."
    },
    {
        "video_name": "TErJ-Yr67BI",
        "Q": "At 4:34, Sal labels the midpoint, E. What should I do when I must prove the shape to be a parallelogram, and the midpoint is not labeled? Will I be allowed to label the midpoint, or Is there another way to complete this proof without labeling the midpoint?",
        "A": "There is no other way to prove that this is a parallelogram. Even if you say your auxiliary point is the midpoint of both diagonals, it is no solid proof that your polygon is a parallelogram."
    },
    {
        "video_name": "TErJ-Yr67BI",
        "Q": "At 1:35, he says that DEC is congruent to BAE but wait a minute, how can they be congruent? I don't get this... Because Angle E should be > than Angle BAE because DEC is obtuse and BAE is acute right? Or am I missing something?",
        "A": ":v sometimes look at down right corner too. Well, humans do make mistakes. No offence."
    },
    {
        "video_name": "TErJ-Yr67BI",
        "Q": "Isn't <AEC = <BED at 5:01?",
        "A": "yes because of the vertical angles conjecture. It states that vertical angles are congruent"
    },
    {
        "video_name": "TErJ-Yr67BI",
        "Q": "At 2:23, when u say angle-side-angle congruency, is that the same thing as angle-side-angle postulate?",
        "A": "Yes. The postulate proves ASA congruency."
    },
    {
        "video_name": "TErJ-Yr67BI",
        "Q": "Does the order of the points when naming angles matter? For example, at 2:33, did it matter if the triangle was named triangle ABE or could it have been named triangle AEB?",
        "A": "the order of the points on the triangle doesn t matter. however, when naming angles, the middle letter must be the vertex."
    },
    {
        "video_name": "TErJ-Yr67BI",
        "Q": "at 1:35, how are DEC and BAE congruent, I mean look at them they don't look the same",
        "A": "It s actually DCE and BAE. And the picture doesn t matter to the proof."
    },
    {
        "video_name": "9zreFD3T0qY",
        "Q": "At 6:25 why is the answer not -1.71 i? I thought if you take the the root of a negative number you have to put i after the answer to specify that it is imaginary.",
        "A": "He is taking the cube root of -5. If you take an odd powered root like 3, of a negative number your solution is real. For example the cube root of -8 is -2 because -2*-2*-2=-8."
    },
    {
        "video_name": "9zreFD3T0qY",
        "Q": "at 4:50, why does (3x-8)^3=0 go to 3x-8=0 ?",
        "A": "When solving a polynomial equation using factors, 1) We factor the polynomial. This is already factored into: (3x-8)(3x-8)(3x-8)=0. Notice all the factors are the same. 2) We then use the Zero product rule to split the factors and set each individually = to 0. 3x-8 = 0 and 3x-8 = 0 and 3x-8 = 0 3) We then solve each mini equation. But, since all of these are the same, we just solve one instance. Hope this helps."
    },
    {
        "video_name": "9zreFD3T0qY",
        "Q": "at 4:48, why did he not take into account the ^3 on 3x-8 when he was trying to figure out why it was equal to zero",
        "A": "It is because (3x-8)\u00c2\u00b3=0 is the same as (3x-8)(3x-8)(3x-8)=0. You can set all 3 of them equal to 0 if you want, but you ll end up getting the same answer for all 3."
    },
    {
        "video_name": "ho87DN9wO70",
        "Q": "I don't get what he does at 2:11..... Can somebody explain?",
        "A": "What he did was manipulating it algebraically. f(x)/g(x) = f(x) \u00e2\u0080\u00a2 1/g(x) = f(x) \u00e2\u0080\u00a2 (g(x))\u00e2\u0080\u00be\u00c2\u00b9 by using properties of exponent a\u00e2\u0080\u00be\u00c2\u00b9 = 1/a. Then he took d/dx of f(x) \u00e2\u0080\u00a2 (g(x))\u00e2\u0080\u00be\u00c2\u00b9 using product rule and chain rule on (g(x))\u00e2\u0080\u00be\u00c2\u00b9"
    },
    {
        "video_name": "ho87DN9wO70",
        "Q": "1:40 How does g(x) becomes [g(x)]^-1?",
        "A": "Because we originally had f(x)/g(x) which is =f(x)*1/g(x)=f(x)*[g(x)]^-1, that is why 1/g(x)=g(x)^-1, and actually this is true for any number x except zero. So 1/x=x^-1 and similarly we can substitute in any value for x except zero. 1/2=2^-1 1/3=3^-1 1/777=777^-1 If you don t believe me, try it out using a calculator ;-) Hope this helps :-)"
    },
    {
        "video_name": "ho87DN9wO70",
        "Q": "At 2:29. Using the chain rule to solve (g'(x)^-1) Sal gets (-1) (g(x)^-2)(g'(x). How is it that the outer term is g instead of g' ? Obviously it has to be g for quotient rule proof to work. Could you take me step by step through the chain rule for g'(x)^-1 ?",
        "A": "I think that you misread the problem. He took the derivative of (g(x))^-1, not (g (x))^-1, so the inner term is g(x), not g (x)."
    },
    {
        "video_name": "ho87DN9wO70",
        "Q": "At 2:40 why Sal multiplies (g(x)^-1) derivative, by g`(x) again? please someone can explain me whats wrong with my thinking?",
        "A": "He is using the chain rule, first he takes the derivative of g(x))^-1 with respect to g(x), which is -g(x)^-2;;; and then it has to be multiplied by the derivative of the inner function with respect to x, which is g (x)"
    },
    {
        "video_name": "d8qtbGMB2gI",
        "Q": "from 1:26 to 1:35 he tells that 0/0 has some chance of having limit but non-zero number by 0 will not have limit. why so?",
        "A": "\u00f0\u009d\u0090\u00bf = \u00f0\u009d\u0091\u008e \u00e2\u0088\u0095 0 Algebraically, this would mean that 0\u00f0\u009d\u0090\u00bf = \u00f0\u009d\u0091\u008e. If \u00f0\u009d\u0091\u008e \u00e2\u0089\u00a0 0, then \u00f0\u009d\u0090\u00bf has to be infinity (i.e. the limit definitely doesn t exist) If \u00f0\u009d\u0091\u008e = 0, then we have 0\u00f0\u009d\u0090\u00bf = 0, which is fulfilled for any \u00f0\u009d\u0090\u00bf (including infinity), so in this case the limit might exist."
    },
    {
        "video_name": "d8qtbGMB2gI",
        "Q": "I'm still having confusion understanding why Sal changes the function from f(x) to g(x) at 5:10. Why does he do that?",
        "A": "The function f(x) is undefined at x is zero. He creates g(x), which is a function similar to f(x), but where the discontinuity at zero is removed. Once the discontinuity is removed, he can just substitute x for zero an evaluate it. He is mostly making the point that you still shouldn t evaluate f(x) at zero, because it is undefined there. By removing that discontinuity, we actually can. And, by the way, throughout the video het is using theta instead of x, so it should in fact be f(theta)."
    },
    {
        "video_name": "d8qtbGMB2gI",
        "Q": "At 5:36 Why is limit of f(x)=limit of g(x)?\nCould anyone explain",
        "A": "Because f(x)=g(x). They re both 1/(2+2cos\u00ce\u00b8)."
    },
    {
        "video_name": "-UagBvxCReA",
        "Q": "2:25 what is the seven for?\n\nand I WANT TO REGISTER even though i'm only in the fourth grade and 9 1/2 years old",
        "A": "Register for what? If you think you can do an AMC 10 or AMC 12, ask your school teacher(prob in middle school, since those comps are INSANE). If you mean Aops, that just sign up using aops.com and friend me.. My name is GeneralCobra19."
    },
    {
        "video_name": "-UagBvxCReA",
        "Q": "at 3:21 he simplyfide the fraction, how did he do that",
        "A": "11 x (2 to the power of 5) x 7 11 x 25 x 7 275 x 7 = 1925"
    },
    {
        "video_name": "-UagBvxCReA",
        "Q": "At 2:30, how did he simplify 11 ! to 11\u00c3\u00975^2\u00c3\u00977 , when 11! is not equal to 11\u00c3\u00975^2\u00c3\u00977",
        "A": "When he simplify 11! to that value he was actually factoring out all factors of 2 and 3 from 11 factorial. So nine was able to factor out 2 threes, 8 was three 2 s, but the 5 in ten and 5 could not be factored along with the 7. It was those factors of 2 s and 3 s that he removed from the factors of 12^11"
    },
    {
        "video_name": "-UagBvxCReA",
        "Q": "At 5:02 why did he switch from 11 to 55 ?",
        "A": "it s the second pirate, who takes 2/12 of what is left - so what is remaining is 10/12=5/6. the denominator ( /6) chips a 2 and a 3, while the numerator adds a 5. since he already had 11 from the first pirate, he does the multiplication and gets 55."
    },
    {
        "video_name": "-UagBvxCReA",
        "Q": "At 1:40 he uses !. I don't really get what ! means. Can someone please explain that to me? Thanks.",
        "A": "! is factorial. Factorial just means for a number n to multiply all integers [n,0). Here s an example: 5! 5 * 4 * 3 * 2 * 1 20 * 3 * 2 * 1 60 * 2 * 1 120 * 1 120"
    },
    {
        "video_name": "BFW2lHobO4E",
        "Q": "How come at 3:20, Sal doesn't include the 2abx in his final answer of x(4x+3)^2",
        "A": "The 2abx is created when you multiply the 2 factors. Your question is like asking where is the 36 if you factor 36 into 2*12 The 36 is still there, but it is in factored form. Squaring 2 binomials creates a trinomial. (a+b)^2 = (a+b)(a+b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2 The middle term exists after multiplying. It won t visibly exist in factored form. Hope this makes sense."
    },
    {
        "video_name": "IW4Reburjpc",
        "Q": "At 8:46, instead of substituting for sin\u00cf\u0084, I multiplied and divided the equation with 2 to get (1/2)2(sin\u00cf\u0084)*(cos\u00cf\u0084) which equals (1/2)*(sin2\u00cf\u0084). On integrating the whole thing, I ended up with the final answer being (1/2)*t*sint + (1/4)*cost\nCan anyone explain where I've gone wrong?",
        "A": "You forgot to subtract the \u00c2\u00bc cos(t) which comes from plugging 0 in for \u00cf\u0084."
    },
    {
        "video_name": "IW4Reburjpc",
        "Q": "at 15:56, why is 2sin(t)cos(t) multiplied onto the function instead of added as above. If you have sin(2t) = 2sin(t)cos(t) then (1/4)*sin(t) + sin(2t) = (1/4)*sin(t) + 2*sin(t)cos(t), not\n(1/4)*sin(t)*(2*sin(t)cos(t))",
        "A": "Sal s t looks like a + at times. He meant a t instead of a + ."
    },
    {
        "video_name": "IW4Reburjpc",
        "Q": "At 10:33, why did Sal not use the trig identity Sin2a = 2SinACosA ? Instead of substituting U",
        "A": "I did it myself and you it just requires a little extra work because you have to use a trig identity to get back to sal s expression. Of course though, you can do it your way if you find its easier :)"
    },
    {
        "video_name": "ZS157czfx4E",
        "Q": "At 3:40 Sal says \"P*5=J\", but translating the first sentence on my own I thought J*5=P. Why does this work?",
        "A": "It doesn t. Let s exchange P and J for numbers, like P=8 and J=40. Now fill in the equation: 8*5=40, which is correct, but 40*5 isn t 8, it s 200, so that s incorrect."
    },
    {
        "video_name": "ZS157czfx4E",
        "Q": "at 0:11 i believe it is eleven",
        "A": "cool idc what u think"
    },
    {
        "video_name": "zid7J4EhZN8",
        "Q": "I dont understand at all from 1:05, what does he mean \"has to be true for all x's\" im very confused at what he means by that.?",
        "A": "he means has to be true for all values of x"
    },
    {
        "video_name": "zid7J4EhZN8",
        "Q": "The ratio of number of goats to the number of cows to the number of chicken in a form is 2:5:7.The number of cows is 35. What is the number of goats and the number of chicken",
        "A": "To solve problems like this, I like to consider all of the values in terms of another variable, call it x. So 2x is the number of goats, 5x is the number of cows, and 7x is the number of chickens. This keeps the ratio intact. Because we know there are 35 cows, 5x=35. This means that x=7. We can then use that to find that 2x=14 and 7x=49. So there are 14 goats, 35 cows, and 49 chickens. I hope this helps."
    },
    {
        "video_name": "zid7J4EhZN8",
        "Q": "The starting point in solving this problem was knowing that the solution would be linear, of the form y=mx+b (0:20 in the video). How do we know that the (or one) solution would be linear, i.e. why is a linear solution assumed?",
        "A": "A linear solution is assumed because that is how Sal set out the question, perhaps to make it easier to solve."
    },
    {
        "video_name": "zid7J4EhZN8",
        "Q": "at @6:18 sal asks us to verify it , can anyone explain how to do so ?",
        "A": "Yes. You can take the solution: y = 2/3 x + 17/9 and plug it back into the original equation. From the solution then dy/dx = 2/3 plugging back into the original: 2/3 = -2x + 3(2/3 x + 17/9) - 5 2/3 = 17/3 - 5 2/3 = 17/3 - 15/3 2/3 = 2/3 So you verified the solution"
    },
    {
        "video_name": "zid7J4EhZN8",
        "Q": "I didn't understand what Sal did at 4:30?",
        "A": "I did it a slightly different way, I took m across to the other side to make the whole thing equal to 0. And then because the statement has to be true for all values of x, the only way the whole equation can equal 0 is if the x term (3m-2)x doesn t vary, which it would unless it s equal to 0, that means that the 3m-2 must equal 0. Same logic, I just found it slightly easier to see. :)"
    },
    {
        "video_name": "zid7J4EhZN8",
        "Q": "6:14 is that the only solution or it is just one? i need to know if in this way (linear solution) we can find all the solutions or just one of them (if they are many)?",
        "A": "Initial value problems have only 1 solution. Otherwise, differential equations in general have infinitely many solutions, because there is a generally a constant of integration obtained when solving the differential equation. This constant of integration has infinitely many choices. What Sal has here is a particular solution, but the general solution to this DE is: y=c1e^(3x) +2x/3 - 13/9, where c1 is an arbitrary number."
    },
    {
        "video_name": "zid7J4EhZN8",
        "Q": "I don't understand at 4:20. Why would the x term have to equal zero? We don't know the value of m or 3b-5. So how can we say that it's correlary to ax+5=5? We don't know that the constants are both 5. We have them written as m and 3b-5. m = (3m-2)x + 3b-5 could easily be different than ax+5=5. For example couldn't it be ax+4=5. Then there is no need for the x term to be zero.",
        "A": "Actually, your example is the opposite. For ax+4=5, neither a nor x can ever be zero. Think of it this way: subtract m to the other side. What do you have? 0 = (3m-2)x + (3b-5) - m . When is this statement always true (not just when x=0, m=0, b=5/3)? It s true for any x, when (3m-2) = 0 AND (3b-5) = m."
    },
    {
        "video_name": "zid7J4EhZN8",
        "Q": "1:08 what does sal mean by \"True for all x's\" , i have no idea what he means by this statement?",
        "A": "Sal means true for all values of x ."
    },
    {
        "video_name": "leDYnoNSvD4",
        "Q": "at 0:58 arnt they all the same face, like is it one face?",
        "A": "Yes, they are all of Marilyn Monroe."
    },
    {
        "video_name": "APNkWrD-U1k",
        "Q": "At 1:22 , why does he cut it inhalf?",
        "A": "he didn t, he found the altitude, so therefore created two right triangles"
    },
    {
        "video_name": "APNkWrD-U1k",
        "Q": "At 1:00, he labeled the angle alpha and beta. Are those just unknown variables, like theta? Or do they represent a specific value as well??\nThanks. Sylvia.",
        "A": "The triangle that Sal used to derive the law of sines was fixed and arbitrary. In other words, the letters represent specific values for angles and sides which could be anything as long as the figure is a triangle. This assumption was made for the purpose of derivation. In practice, usually you are given three of the values(two sides and an angle, or two angles and a side) and are expected to use the law of sines to solve for the unknown value."
    },
    {
        "video_name": "APNkWrD-U1k",
        "Q": "At 2:36, we can also use cosec beta and rename the law as law of cosecants Then why don't we use so??",
        "A": "With so many equivalent values in trigonometry from reciprocal ratios or complimentary angles, you COULD rewrite almost all of trig. The version we teach is based on primary ratios and soh cah toa because it is a good place to start. Once you are a master, you may choose to remember the laws or ratios however you like."
    },
    {
        "video_name": "APNkWrD-U1k",
        "Q": "at 5:08 you skipped a step. When you divide both sides by B, you don't get SinA/A and SinB/B, I could be mistaken but I think you get AsinB/B and SinAlpha/B, not SinA/A and SinB/B.",
        "A": "No, you are mistaken. He s right. sin(b)=Bsin(a) is same the as(sin(b))/B=(sin(a))/A"
    },
    {
        "video_name": "APNkWrD-U1k",
        "Q": "0:07, What is an arbitrary triangle?",
        "A": "Because it just has random values and measures. Arbitrary means based on random choice or personal whim, rather than any reason or system."
    },
    {
        "video_name": "ZtI94pI4Uzc",
        "Q": "At 3:04 why does he multiply 2y by y prime?",
        "A": "Because he is using the product rule to differentiate x*y^2"
    },
    {
        "video_name": "ZtI94pI4Uzc",
        "Q": "At 5:05, when Sal is rewriting the right-hand side, wouldn't the two negative signs be simplified as 1 + y^2e^(xy^2)? Instead he writes 1 - y^2e^(xy^2). Or is it the case that 0 remains when the y' cancels out, leaving just 1?",
        "A": "consider this example: a - 1 = b - 1 //now lets add to both sides 1 and subtract a -a + 1 | -a + 1 now what we get is: a - 1 - a + 1 = b - 1 - a + 1 because it s adding/subtraction two negative signs don t turn into plus sign and in the end we have: 0 = b - a On the video there s similar thing, maybe the notation is a bit misleading."
    },
    {
        "video_name": "ZtI94pI4Uzc",
        "Q": "At 1:51 , Which rule was that ? I mean Derative of (x-y)",
        "A": "It s the sum/difference rule. It states that d ____[f(x)\u00c2\u00b1g(x)]=f (x)\u00c2\u00b1g (x) dx"
    },
    {
        "video_name": "ZtI94pI4Uzc",
        "Q": "at 4:10, when we add y' to each side, we do not have 2y'. The only difference is that the y' on the left just disappears. Why is this?",
        "A": "Think of it like this: blah*y + 1*y = (blah + 1)y this is the same thing, distributive property and Sal added +1y to both sides of the equation to get rid of the y on the right side of the equation and you can do that because doing the same thing to both sides of the equation doesn t change the equality of the equation."
    },
    {
        "video_name": "f6OnoxctvUk",
        "Q": "Proof Part 2 Minimizing Squared Error To Line:How is it that you are taking the mean of the parenthetical values (@ 4:22/9:53) in the video? It appears that this operation to simplify is being performed on only one side of the equation. How is this algebraically correct to not perform this on both sides of the equation?",
        "A": "I believe he is using substitution. Therefore he is not changing any values, but simply rewriting the equation."
    },
    {
        "video_name": "f6OnoxctvUk",
        "Q": "At 8:42, the way the local axes are drawn it seems like the surface (and the minimum) goes below the SE=0 plane, which isn't possible for a sum of squares. Or am I getting this wrong?",
        "A": "You are correct, it is not possible for the SE < 0. But it will be relatively close to 0. It is difficult to draw these XD"
    },
    {
        "video_name": "qMZnsj4ZO9g",
        "Q": "In 1:00-1:12, wouldn't the 9 be a negative, or do you make it non negative when doing this?",
        "A": "The 9 would not be negative because you are squaring the -3 for example squaring is the same thing as multiplying by the same number, in this case it is (-3)*(-3) and this equals to 9. The negatives cancel each other out - + - equal a positive"
    },
    {
        "video_name": "NCwwfvPEVn8",
        "Q": "at 6:00 when Sal starts explaining that dr_u is just a 'differential' and not a 'partial derivative,' I am very confused by his explanation.",
        "A": "The derivative is the ratio of the change in r (dr) with the change in u (du). The differential, r (dr) is simply the change in r. The derivative or partial, as is the case here, would be like Miles per Hour. The differential is just Miles traveled without without the concerned for the time it took to travel that distance."
    },
    {
        "video_name": "NCwwfvPEVn8",
        "Q": "At 8:21 Sal is talking about dr_u and dr_v but in the integral its just the vectors r_u and r_v.... are these equivalent?",
        "A": "No, they are NOT equivalent. r_u = \u00e2\u0088\u0082r/\u00e2\u0088\u0082u, and r_v = \u00e2\u0088\u0082r/\u00e2\u0088\u0082v in the integral, these are the derivatives with respect to parameters u or v. At 4:30 in the video, the \u00e2\u0088\u0082r_u and \u00e2\u0088\u0082r_v came from the expression: (\u00e2\u0088\u0082r/\u00e2\u0088\u0082u)du \u00c3\u0097 (\u00e2\u0088\u0082r/\u00e2\u0088\u0082v)dv. He simplifies it by saying that du cancels with \u00e2\u0088\u0082u, and dv cancels with \u00e2\u0088\u0082v. But if you do that, then you end up with something that looks like: \u00e2\u0088\u0082r\u00c3\u0097\u00e2\u0088\u0082r. He just tacked on _u and _v so that you could tell them apart."
    },
    {
        "video_name": "NCwwfvPEVn8",
        "Q": "At 8:54, shouldn't it be over the region R? Also, at 1:55, what if the magnitude of the cross product is 0? How can we cancel them out?",
        "A": "If the magnitude was zero then you wouldn t have a normal. It was be zero giving us an integral of zero and the integral of zero is zero."
    },
    {
        "video_name": "kLLsJXc1E1E",
        "Q": "At 0:56, how does he relate vector v to vector w there?\n\nWhy is it \"3x the magnitude of vector v\" ? I don't understand that part.\nThanks.",
        "A": "the correlation relies on the assumption that the (x,y) values for the vector v are the same as the (x, y) values for vector w v = (x, y) w = (3x, 3y) and then you can write 3v = 3(x, y) = (3x, 3y) = w"
    },
    {
        "video_name": "vhfjEpQWWeE",
        "Q": "At 8:23, I don't agree with your explanation of bringing the constant outside the integral bc u have to evaluate the integral at all points. From 0 to infinity, e^(-st)*f(t) does have values. It is misleading and I hope you or some1 else can come up with a better explanation.",
        "A": "He pulls e^(-sc) * f(c) outside of the integral not e^(-st)*f(t). Because we have a c in place of t the entire expression is a constant and can be pulled out of the integral"
    },
    {
        "video_name": "vhfjEpQWWeE",
        "Q": "at 11:27 how is it that f(t)=1 at c=0\n?",
        "A": "he does the general case for the laplace transform of f(t) * dirac(t-c). f(t) here is an arbitrary function so when he wants to find the transform of just dirac(t) he takes c = 0 and f(t) = 1"
    },
    {
        "video_name": "vhfjEpQWWeE",
        "Q": "At 7:39, shouldn't it be d(0), and not d(t-c), since he is evaluating at t=c?",
        "A": "No, he is giving the function for the entire graph, not just the function evaluated at t=c."
    },
    {
        "video_name": "vhfjEpQWWeE",
        "Q": "At the end of the video (10:50 - 11:50) it is stated that:\n\nL{d(t)} = 1\nL{d(t-c)} = e^(-cs)\n\nBoth d(t) and d(t-c) represent a line of the same height, one line is on 0 and the other is on c. Since both d(t) and d(t-c) are lines of the same height, howcome their area (and therefore their result to the laplace-transform) are different?\n\nThanks a lot for your help and another great thanks to Khan. Very helpful indeed.",
        "A": "if c=0, e^(-cs)=1, so those two equations you wrote are the same Its not that the delta functions have different areas, just that the laplace transforms look different"
    },
    {
        "video_name": "vhfjEpQWWeE",
        "Q": "\"infinity\" at 00:40 in the video can be either positive or negative?",
        "A": "Sometimes, it does represent minus infinity but usually its positive infinity."
    },
    {
        "video_name": "vhfjEpQWWeE",
        "Q": "at 11:20 why is f(c)=1? shouldn't it be just f(0) since c=0? have we defined what f(x) is?",
        "A": "integrate f(t-c) from infinity to zero, which includes the value of ZERO. If you recall the last video (Dirac delta function), it did mention the value of \u00cf\u0084 could be infinite small to make infinite high in y-axis. In this case, the integration range is mainly dependent on -\u00cf\u0084 and \u00cf\u0084. Assumed \u00cf\u0084 infinitely close to 0, then the range can be assumed as zero. Back to your question. when c=0, 0 is still in the integration domain and therefore the calculated result is still the unit."
    },
    {
        "video_name": "tnoPnyvVjyc",
        "Q": "At 3:33 we try to derive the integral evaluated from 0 to x. Would we not want to subtract the lower limit of f(0) from f(x)? Why is g'(x)=2+f(x) and not g'(x)=2+(f(x)-f(0))? This changes the answer since f(0) has a value.",
        "A": "When you integrate, you get something like F(x) - F(0). F(x) is a function of x, while F(0) is an evaluation of F(x) at x = 0, which is some constant number. When you derive F(x) - F(0), d[F(x)]/dx = f(x), while d[F(0)]/dx = 0, since the derivative of a constant is zero. This equates to f(x) + 0 = f(x). Therefore, g (x) = 2 + [f(x) - 0] = 2 + f(x)."
    },
    {
        "video_name": "wdAgEzdkFZI",
        "Q": "At 0:49, how do we know that the acceleration is the derivative of the velocity function?",
        "A": "By definition: Velocity is the rate of change in position, and therefore is directly computed from the derivative of position ( given that the rate of change of a function is by definition a derivative of that function ); then, by extension, since the definition of acceleration is the rate of change in velocity, it is therefore computed from the derivative of velocity."
    },
    {
        "video_name": "wdAgEzdkFZI",
        "Q": "Around the Time 4:11 you stated that to take an average of interval [1, 2] its [-4 * 1/8 - -4]\nwhat if the interval was [1, 3]?",
        "A": "At that point it would be [-4*(1/81)--4}"
    },
    {
        "video_name": "wdAgEzdkFZI",
        "Q": "At 1:51 shouldn't you have differentiated with the quotient rule?",
        "A": "S(t) was simplified to the sum of two terms : t + 2/ t^2 ; Derivative of the two terms can be taken individually. It comes to the same answer even if one did not simplify and chose to go with the quotient rule route."
    },
    {
        "video_name": "Vtcmyr5IGYY",
        "Q": "At 0:41, why did he write the denominator as \"15x-5\" instead of \"15x+5\"?",
        "A": "Sal made a transcription error. After that the problem was correct."
    },
    {
        "video_name": "Vtcmyr5IGYY",
        "Q": "At 6:50, instead of writing 6/(x^4), could he have written 6(x^-4)? Or would that have made the problem a lot harder to finish?",
        "A": "Many people find 6/(x^4) easier to read than 6 * x^(-4)."
    },
    {
        "video_name": "Vtcmyr5IGYY",
        "Q": "Why did Sal multiply both the numerator and denominator by 1/x at 1:25?",
        "A": "As he stated, he simplified it to lower degree of x to help to see what happens when x get very negative."
    },
    {
        "video_name": "01c12NaUQDw",
        "Q": "At 1:11 Sal says it is in a \"typical algebra II class\" then why is the topic in Pre Calculus",
        "A": "Khan academy has videos and exercises for matrices in Algebra II as well, just less."
    },
    {
        "video_name": "01c12NaUQDw",
        "Q": "2:35 do you multiply by 1/determinant?",
        "A": "Yes. You are exactly right"
    },
    {
        "video_name": "qfQv8GzyjB4",
        "Q": "At 0:04 The line above a digit indicates that the number(s) repeat, in a video about significant digits, Sal mentions that this same convention is sometimes used to indicate a significant digit. Is there a way to avoid this apparent confusion?",
        "A": "Unfortunately not. The rule of thumb that I use is that if it s on the left side of the decimal place they are referring to a significant digit (I run across these fairly often in calibration), and if it s on the right side of the decimal it is most likely a repeating decimal. You really don t see it very often to designate a significant digit and I don t think I ve ever seen someone use it to refer to a significant digit in a decimal place (that would be very dumb of them for exactly that reason)."
    },
    {
        "video_name": "qfQv8GzyjB4",
        "Q": "Wow, many questions. Approx. at 3:57 Sal set ups the manipulation to get the number to be rational. But he only eliminates the first set of repeating digits. What about the extension of 28 bar to the right of what is being operated on? Aren't there infinite repeating 28s? And won't we have a leftover at some point? Any conceptual assistance is appreciated.",
        "A": ".28888...... is the same as 28/99. Hope that helps!"
    },
    {
        "video_name": "qfQv8GzyjB4",
        "Q": "Why did he subtract the left side with 100x and the right side 340.28 (@3:43)? I thought we add/subtract/multiply/divide by the same number for both sides when dealing with equalities?",
        "A": "Yes, we should always do the same operation for both sides. However, remember that 100x = 340.28 is an equation. So we can use this to subtract from both sides and it would not break any mathematical rule. Just in case you are still confused: you can also subtract 100x from both sides. Then on the right side, substitute 100x with 340.28 since 100x = 340.28. I hope that clears things up!"
    },
    {
        "video_name": "KkaU2ur3Ymw",
        "Q": "At 5:15, it still seems misleading to name these s_n and s_{n-1}, since the numerator sums up to n in both. Any insight in this naming convention?",
        "A": "The n and the n-1 refer to the amount you divide by. Since that is what differs the biased from the unbiased sample variance, it s the most useful subscript for s."
    },
    {
        "video_name": "KkaU2ur3Ymw",
        "Q": "At 8:11 When you take a sample, your sample mean is going to sit within your sample.\nAnd so there is a possibility that when you take your sample, your mean could even be outside of the sample.\n\nIf the sample mean is going to sit within my sample, when the video also says that the mean could be outside of the sample?\n\nIs the second mean the population mean? Thanks I am quit confusing since English is not my native language.",
        "A": "Yes, the second mean mentioned at 8:11 is referencing the population mean. What it means when it says the pop. mean can sit outside of the sample is that the range of the sample doesn t even include the value of the population mean. Hope this helps :)"
    },
    {
        "video_name": "vsgrWDLEzcQ",
        "Q": "At 4:50, why would it not be a parallelogram or trapezoid?",
        "A": "It s not a trapezoid or parallelogram because there is no pair of opposite sides that are parallel to each other."
    },
    {
        "video_name": "WZPpbkVESVk",
        "Q": "@3:08 shouldn't there be an x after ln4",
        "A": "for v(x) it would, but this is v(u(x)). The x in v(x) is replaced with u(x)"
    },
    {
        "video_name": "zrqzG6xKa1A",
        "Q": "Did sal say in 1:35 \"let me suis colors for the sake of reidy\"?????",
        "A": "but like i have it showing it in words ..english(transcribed)"
    },
    {
        "video_name": "zrqzG6xKa1A",
        "Q": "what does supplemetary mean? At I think 1:20 he said it but i still don't get it. please tell me.",
        "A": "Supplementary angles are 2 angles that add up to 180 degrees."
    },
    {
        "video_name": "zrqzG6xKa1A",
        "Q": "1:51 What does \"perpendicular\" mean?",
        "A": "Perpendicular lines meet in a + sign. Like in a graph, where the x axis and y axis are perpendicular to each other."
    },
    {
        "video_name": "zrqzG6xKa1A",
        "Q": "at 1:10 he says the angles are supplementary how does he know that without measuring the angles",
        "A": "where sal is more experianced he sorta knows or he could be estimating, hope this helped"
    },
    {
        "video_name": "zrqzG6xKa1A",
        "Q": "at 4:55, wouldn't Sal be completely wrong? since x is a numerical variable and wouldn't y be a completely different variable?",
        "A": "Two variables that have different names can still end up having the same numerical value. In this case they definitely have the same value because, as demonstrated, vertical angles are always equal so x=y."
    },
    {
        "video_name": "Iqws-qzyZwc",
        "Q": "At 7:53 why is it called rise over run? Why can't it just be called y over x?",
        "A": "Rise over Run is a slang definition for slope. The alliteration makes it easy for students to remember. Slope is more precisely defined as the change in y over the change in x You would not want to say that slope is y over x because x and y are the values of one of the coordinates, and slope is ratio of the differences between two of those values."
    },
    {
        "video_name": "Iqws-qzyZwc",
        "Q": "At 00:27, Sal explains that slope is the change in y with the with in x. What is the triangle doing next to each of the variables?",
        "A": "The triangle is the greek letter delta . In math it means the change in ___ . For example, slope is the change in y divided by the change in x, which would be written as \u00ce\u0094y/\u00ce\u0094x (read as: delta y over delta x ) Please let me know if I can explain this better for you"
    },
    {
        "video_name": "Iqws-qzyZwc",
        "Q": "In 0:15,is change in y divided by change in x important? I was taught using rise/run instead",
        "A": "They are very important! However, the rise is really just the change in y and the run is really just the change in x. Let s do an example. We have two data points (0,0) and (1,1). The change in y = y2 - y1 (1 - 0 = 1) and the change in x = x2 - x1 (again 1 - 0 = 1). If you take the change in y over the change in x you are calculating the slope of a line (m = (y2 - y1) / (x2 - x1), in our case m = 1 / 1 = 1). The slope of our line is rising 1 for every run of 1 it moves to the right."
    },
    {
        "video_name": "Iqws-qzyZwc",
        "Q": "I'm confused. for 13:02 cant the answer be both ways? Ex. -3/-9 = correct, 3/9 = correct?",
        "A": "I think that s right."
    },
    {
        "video_name": "Iqws-qzyZwc",
        "Q": "11:51\n\nWhy is a fraction with 0 at the bottom undefined?\n\n5/0 is 0, so why is 5 over 0 undefined?",
        "A": "5/0 is not 0. Something times 0 is 0. What times 0 is some number other than 0?"
    },
    {
        "video_name": "Iqws-qzyZwc",
        "Q": "I still do not understand how you can flip what numbers you subtract first. It starts at 5:30. H e talks about how you can do it in any order and i do not understand it. Sorry",
        "A": "He says that because you can pick which coordinates you want to label as 1 or 2. For example, say the the groups of coordinates you had were (4, 7) and (9, 3) and you labeled (4, 7) as 1 and (9, 3) as 2. Then, you would subtract 2-1. (3-7/9-4). However, if you labeled (4, 7) as 2 and (9, 3) as 1, you would still do 2-1, only with 7-3/4-9. You just need to keep which is group one and which is group two straight in your head. If you put the y in group 2 first, put the x in group two first."
    },
    {
        "video_name": "Iqws-qzyZwc",
        "Q": "1:47 how is it +6 if it was just negative 6?",
        "A": "if you subtract a negative from something then its the same as adding a positive of the same value for example 9--3=9+3"
    },
    {
        "video_name": "Iqws-qzyZwc",
        "Q": "At 11:51 in the video, Khan says a vertical slope would be undefined. Why is that?",
        "A": "All vertical lines have undefined slopes. If you try to calculate their slope, you ll notice that you get a fraction with 0 in the denominator. Like in other parts of mathematics, fractions with 0 in the denominator are undefined."
    },
    {
        "video_name": "Iqws-qzyZwc",
        "Q": "At about 11:30 we see the undefined slope of a vertical line. Said line only crosses two quadrants, Aside from undefined and 0 slopes, is it possible to have a line that only crosses two quadrants? Is it possible or does the infinite coordinate plane explicitly make it impossible?",
        "A": "Yes, but the line would have to pass through the origin. The line would pass either through I and III or II and IV. Let k be a positive real number: y = kx passes through only quadrants I and III. And y = -kx passes through only II and IV. All lines other than this form and the ones you mentioned must pass through exactly three of the quadrants. No line can pass through all four quadrants."
    },
    {
        "video_name": "Iqws-qzyZwc",
        "Q": "Why can you put the x and y #s in any order? About 5:10 to about 6:00",
        "A": "So we can Divdiing 6:30 by 2 is 6:30/2=315. But we chang 6:30 to 630.Also we can Dividiing 5:10/2=255.But we can change 5:10 ot 510."
    },
    {
        "video_name": "Iqws-qzyZwc",
        "Q": "at 12:02 why does Sal say vanilla?",
        "A": "vanilla , or plain vanilla (as in vanilla ice cream) is slang for plain (or simple , with nothing added , etc.). He also says straight up there, which means open, honest, with no disguise or deception . Academics use the word straightforward a lot for this."
    },
    {
        "video_name": "Iqws-qzyZwc",
        "Q": "what is an delta?, Sal mentions this at 4:05",
        "A": "Delta just means Change in ."
    },
    {
        "video_name": "J43CIbKpdWc",
        "Q": "4:48 So I guess I'll assume when given both the coordinates for an X and Y for a vertical line the slope is the X point. As well as it not mattering what that values are on a horizontal line because a straight one will always match-up to undefined.",
        "A": "The slope of a vertical line = undefined. You said it would be the x point , which is incorrect. The reason the slope is undefined is because there is no change in X, there is only a change in Y. Since the change in X (denominator of the slope) = 0, we re dividing by zero which is undefined. The slope of a horizontal line = 0 because there is no change in Y, there is only a change in X."
    },
    {
        "video_name": "J43CIbKpdWc",
        "Q": "At 3:10, Sal talks about the slope of the vertical line. He says it's undefined because the change in y could be viewed by some people as infinity or negative infinity. But isn't it really undefined because you would have a change in x of 0, and therefore a 0 in the denominator of the fraction, which is by definition undefined?",
        "A": "Both are valid approaches. In the video where Sal explains why division by zero is not allowed, he uses the concepts of the denominator getting very close to zero from both a positive and negative perspective. This shows that as the denominator gets smaller it will trend toward infinity or negative infinity. But, your assessment is good as well and simpler to understand."
    },
    {
        "video_name": "KXDOOmquZag",
        "Q": "At 2:15, why does Sal say \"53.2 seconds\" twice when he was calculating the mean of the final round? Did he mean 53.0 or 53.3 seconds instead?",
        "A": "I m pretty sure he s just clarifying/conveying his first answer again to maybe emphasize it..."
    },
    {
        "video_name": "9eSPhvhuInw",
        "Q": "Why is it that at 2:44 minutes into the video Sal states that - 5/-b is the same as 5/b, whereas later in the video at 4:08 when we look at - -e/-f this equates to just e/f? Surely if we apply the same logic, then the first example is wrong because it would equate to -5/b?",
        "A": "Multiply and divide rules are: Odd number of negative signs give negative answer, even number of negative signs give positive answer. Addition rule is two negatives in a row are positive, So first has two negatives which give a positive answer, second has two negatives in a row, so that is positive, then there is only one negative which gives a negative answer (it sort of has three negatives)."
    },
    {
        "video_name": "2RbKfRfzD-M",
        "Q": "AT 1:50, how do you know that angle is Pi/12?",
        "A": "The angle we start with is 7pi/12, which is slightly larger than a right angle. The angle Sal is looking for (in the triangle he is constructing) is the small one that remains when we subtract the right angle from our original angle. Since a right angle has a measure of pi/2, the small angle we are looking for is 7pi/12 - pi/2, and that is: pi/12. (Thanks again for the timestamp! :-)"
    },
    {
        "video_name": "2RbKfRfzD-M",
        "Q": "At 2:22 why is the sin value (y axis) referred to as cosine as well?",
        "A": "If you are asking why the side he highlighted in purple is called cosine of pi over 12 , it is because of the way the triangle is drawn. Instead of drawing down to the x-axis, he drew across to the y-axis. If you look at it sideways, you ll see that the side he called cosine can be called cosine at that perspective. He s taking the cosine of the compliment of the angle you are thinking of. Although as commonly used as drawing down to the x-axis, this is a legitimate way to draw the triangle."
    },
    {
        "video_name": "2RbKfRfzD-M",
        "Q": "1:42 how does he know that it's another pi/12?",
        "A": "Because if you have 7pi/12 as your angle, the angle in quadrant 1 up to the Y-axis is pi/2 (which is the same as 6pi/12) which then leaves 1pi/12 to go into quadrant 2."
    },
    {
        "video_name": "2RbKfRfzD-M",
        "Q": "At 4:00, he says he arrives at the other two sides directly from the Pythagorean theorem. Can someone show me how to arrive at sqrt(2)/2 as the length of a side, only knowing that the hypotenuse is 1, and only using the Pythagorean theorem?",
        "A": "In that example, Sal has drawn an isosceles right triangle, so we know the 2 sides have the same length (let s call it X), and he told us the hypotenuse = 1. Using the Pythagorean theorem, we get: X^2 + X^2 = 1^2 Simplify 2X^2 = 1 Divide by 2 X^2 = 1/2 Take the square root of both sides X = sqrt(1/2) = 1 / sqrt(2) Rationalize the denominator X = sqrt(2) / 2 Hope that helps."
    },
    {
        "video_name": "OjzfQDFf7Uk",
        "Q": "At um 1:12 what does he mean by data Points?",
        "A": "Points in the data."
    },
    {
        "video_name": "OjzfQDFf7Uk",
        "Q": "at 0:29 is does statistictics mean others things too, other than data!",
        "A": "a nice question indeed. though you must take Khan Academy s grammar classes. Statistics in mathematical terms, is all that is referred to data, however its applications go beyond data, like, analysing a situation or inferring an outcome, etc, although it s all based on data. So yeah, Statistics is pretty much DATA. And we must heed and hail statistics."
    },
    {
        "video_name": "OjzfQDFf7Uk",
        "Q": "At 1:01, Sal mentions variability but doesn't describe it clearly. What is variability?",
        "A": "the difference one data point to the other data point, say you asked: what is the average age of students in your class? student1: 10 yr, student2: 11 yr, student3: 12 yr, etc so you see that there is a difference between one data point to the other data point. That differences is variablity"
    },
    {
        "video_name": "OjzfQDFf7Uk",
        "Q": "at 7:12, with the mother's teeth question, isn't it possible for that to vary?As in if one tooth falls or something?",
        "A": "Regardless of the fallen teeth, there will be only one data point to collect."
    },
    {
        "video_name": "oQpKtm5TtxU",
        "Q": "at 7:14, I became a bit unsure about what he meant when he said n!/(n-k)!k!\nDoes he mean (using the card example given):\n36!/(27!)(9!)\nPlus, is (27*9)! the same as (27!)(9!) the same as (243!)\nsorry if my question is a bit unclear",
        "A": "7:14? This video is only 3:45 minutes long. n!/((n-k)!k!) means, (Total number) factorial/((Total-number picked) factorial * (Number picked) factorial). 27!*9! is not the same as 243!. An easier way to see why is to look at 2!*3!. 2!*3!=2*1*3*2*1. That equals 12. If you do 6!, that equals 6*5*4*3*2*1=720. Much, much bigger."
    },
    {
        "video_name": "oQpKtm5TtxU",
        "Q": "At 2:16 to 2:54, wouldn't the second slot be 2?",
        "A": "That s because after choosing 1 (out of 6 colours) for slot 1, there are 5 more colours remaining to choose from, for the 2nd slot."
    },
    {
        "video_name": "byjmR7JBXKc",
        "Q": "At 3:48 why did he change it to a decimal point?",
        "A": "10,000 cannot go into 128,000 evenly, so there will be a remainder: 10,000 can go into 128,000 12 times with 8,000 left over so we write it as 12.8. Hopefully that helps. :)"
    },
    {
        "video_name": "byjmR7JBXKc",
        "Q": "This is the first video in this series, (Scale Drawings) and it operates through a ratio e.g. 1:80 But Sal never defines what either of these numbers are?? Which is which? I'm finding this very hard to understand despite the already present clarification on the video which states that the ration should be 1:80 not 80:1.\n\nSal still uses 80 in his multiplication and I am confused as heck as to what is what. I hope some good comes from this comment, cheers.",
        "A": "It would be very inconvenient to draw things the same size they are in real life, so they are drawn smaller and the ratio is given. 1:80 just means that, for every unit in the drawing, there are 80 units in the real thing. Sal still multiplies by 80 because he interpreted the ratio the other way around, he read it as: for every 80 units in the house, there is 1 unit in the drawing. But, conventionally, the first number refers to the scaled version and the second number, to the actual thing."
    },
    {
        "video_name": "byjmR7JBXKc",
        "Q": "Wait but it says the plan to the house ratio is 80:1. That means that the plan is 80 times larger than the house which makes no sense!",
        "A": "I don t think it says that anywhere - The video starts with a 80 : 1 scale floor plan of their new house . This does not in any way imply that the 80 is the plan and the 1 the house."
    },
    {
        "video_name": "byjmR7JBXKc",
        "Q": "I don't understand why on the screen it said 1:80 should be 80:1.",
        "A": "no it should be 1:80 because it is the unit rate"
    },
    {
        "video_name": "byjmR7JBXKc",
        "Q": "i confused why it must be 1:80 can it be 80:1",
        "A": "1:80 must be the ratio because the first number of the ratio is the scale drawing, and the second number is the actual object. If it was 80:1, then the scale drawing would be way larger then the actual object. I hope that helps you, cause I m terrible at explaining. :)"
    },
    {
        "video_name": "byjmR7JBXKc",
        "Q": "I don't understand what he means at 1:14 that \"we can represent that for every 1 floor plan their is 80 cm in the house\" How does every 1 floor plan their is 80 cm in the house",
        "A": "He means for every 1 cm on the floor plan (the drawing) there are 80 cm in the real world (the house). That is because the scale is 1:80 If the scale is 1 :20 , every 1 cm on the floor plan are 20 cm in the real world. The scale is just a ratio telling you how much bigger a distance is in the real world than it is on the drawing."
    },
    {
        "video_name": "byjmR7JBXKc",
        "Q": "At 3:21, Sal said that 10,000 square centimeters equal 1 square meter, but wouldn't 100 square centimeter equal 1 square meter, since 100 centimeters equal one meter?\n\n(I'm confused)",
        "A": "2 square = 2 * 2 = 4 similarly 1 square metre = ( 1 metre )* (1 metre) or (1 metre)^2 = (100 centimetre ) * ( 100 centimetre ) = 10000centimeter *centimetre = 10000 ( centimetre )^2 = 10000 square centimeter"
    },
    {
        "video_name": "byjmR7JBXKc",
        "Q": "Isnt it suppose to be 1:80 instead of 80:1?",
        "A": "there is a note in the bottom right corner saying that its supposed to be 1:80"
    },
    {
        "video_name": "qYjiVWwefto",
        "Q": "The ratio of venison to beef is 2:3. If he has 8 pounds of venison and wants to make 1/3 hpound beef burgers, how many burgers can he make?",
        "A": "2:3 = 8:? 2x4= 8 3x4 = 12 8 :12 = 2:3 8 venison, 12 beef."
    },
    {
        "video_name": "qYjiVWwefto",
        "Q": "At 1:26, why do the relationships have to be the same?",
        "A": "I know this is going to sound weird but let s say your dad is 2 and your mom is 5. They are married or in a relationship , so no matter how many dads and moms you have, they will still represent 2 and 5. If you have 3 dads and 3 moms, as long as the number of moms is equivalent to the number of dads, when you simplify the relationship, you ll always get 2 and 5. So 3 moms and 3 dads is 6/15, which can be simplified to 2/5. Hope that satisfies you question!"
    },
    {
        "video_name": "jpKjXtywTlQ",
        "Q": "At 12:25 Sal divides by 5a. we know diving by a is legal since a is not equal to zero. but how can we know it doesn't make us lose solutions or something similar? how do we know it's a totally legit action?",
        "A": "Precisely because a is not equal to zero. That s the only algebraic solution you d lose here by making that move."
    },
    {
        "video_name": "jpKjXtywTlQ",
        "Q": "At 12:16, Sal makes (2a)^2, 4a^2... why is this?",
        "A": "(2a)^2 = (2a)(2a) = 2*2*a*a = 4a^2. Doing this lets him add the like terms on that side of the equation."
    },
    {
        "video_name": "jpKjXtywTlQ",
        "Q": "At 12:16, how does 2a squared become 4a squared?",
        "A": "Because 2a * 2a = 4a squared"
    },
    {
        "video_name": "jpKjXtywTlQ",
        "Q": "At 3:00 how do we know that AC would pass through the center of the first circle",
        "A": "The thirst one will be the same"
    },
    {
        "video_name": "jpKjXtywTlQ",
        "Q": "at 1:30 Sal says that the diameter of the circles circumscribed around the triangles abd and acd are 12.5 and 25 but the text says radii",
        "A": "Yeah, he did the wording wrong, but he still did indeed solve the problem correctly."
    },
    {
        "video_name": "6YRGEsQWZzY",
        "Q": "At 7:55, since c is an arbitrary constant, couldn't Sal make it +cx\u00c2\u00b3, for looks?",
        "A": "Gabriel is right. The arbitrary constant is an arbitrary constant, so x^2+y^2-cx^3=0 is the same as x^2+y^2-cx^3=0. When you evaluate for c, If given initial conditions, it the sign will become significant."
    },
    {
        "video_name": "6YRGEsQWZzY",
        "Q": "at around 5:55, when you integrate, where is 2v?",
        "A": "Let u = 1+v^2. Then du = 2v dv."
    },
    {
        "video_name": "6YRGEsQWZzY",
        "Q": "At around 7:10, when you integrate, why did you put +ln(c) instead of just +c?",
        "A": "Whenever you re solving differential equation by separating into both sides, the constant you get is an arbitrary constant. Which mean you can change it to any form in order to simplify the equation. Maybe it would be clearer if it s written in the following way: ln(1+v^2) = ln(x) + c Let C = ln(d), ln(1+v^2) = ln(x) + ln(d) ...... Both c and d are arbitrary constant. Hope that this helps a little."
    },
    {
        "video_name": "6YRGEsQWZzY",
        "Q": "At 5:50 Sal takes the anti-derivative of 2v/(1+v^2)... And gets ln(1+v^2). But isn't also correct to use arctan? So that the anti-derivative becomes arctan(v)",
        "A": "No, the integral of 2v/(1+v^2) does not equal arctan(v). You could however use the product rule setting w=2v, w =2, u =1/(1+v^2), u=arctan(v) which would lead you to the same answer as Sal."
    },
    {
        "video_name": "6YRGEsQWZzY",
        "Q": "At 7:07 how come we can just drop the absolute value function from the cx term?",
        "A": "ln |1+v\u00c2\u00b2| = ln |cx| \u00e2\u0087\u0092 ln (1+v\u00c2\u00b2) = ln |cx| \u00e2\u0087\u0092 1+v\u00c2\u00b2= \u00c2\u00b1cx \u00e2\u0087\u0092 1+v\u00c2\u00b2 = Cx"
    },
    {
        "video_name": "6YRGEsQWZzY",
        "Q": "In \"5:40\", he says that 1+u^2 is always positive but that doesn't work out when u is a complex number.",
        "A": "We are considering only real numbers in this example... Same as when we said that the square root had to be positive a few vids back!"
    },
    {
        "video_name": "6YRGEsQWZzY",
        "Q": "The step at 7:00 : Is this actually allowed to write ln( |c*x| ) = c*x ?\nx can be negative so c*x can be negative so this isnt equal right? Doesn't it equal to | c*x | ?",
        "A": "If x is negative, then we can just say define c to be negative as well."
    },
    {
        "video_name": "6YRGEsQWZzY",
        "Q": "6:00 wouldn't it be right to see 2v/(1+v^2) as sin(v)==>-cos(y/x) after substituting and taking the integral? Or would it get harder to solve?",
        "A": "2v is the derivative of the expression (1 + v^2). So to use u substitution you get F(du/u) which is ln(abs(u)) or ln|1+v^2| + C. Hope that helps"
    },
    {
        "video_name": "H-HfmyUzPw8",
        "Q": "How come at around 2:08 he made the line diagonally and not any other way?",
        "A": "If you go back to pre-algebra, you see information about the slope of the equations. (slope is how slanty the line is on the graph.) In this case both lines have a negative slope so the graph in lines that point downward when moving left to right."
    },
    {
        "video_name": "H-HfmyUzPw8",
        "Q": "i dont understand the table of t and f can some body explain? 2:23",
        "A": "He just puts there the numbers of the equation 5f+10t=5500 when it crosses the f axes(t=0) and the t axes(f=0). from this table he plots the two coordinates in the graph so you can draw the line 5f+10t=5500"
    },
    {
        "video_name": "H-HfmyUzPw8",
        "Q": "At 2:41 how did he come up with 1100?",
        "A": "if there are no $10 dollar bills, then there are only $5 dollar bills, right? We are told that the total value of the bills is $5500. To find out how many five dollar bills that is divide 5500 by 5 like so: 5500/5 = 1100."
    },
    {
        "video_name": "H-HfmyUzPw8",
        "Q": "How did he come up with the two numbers: 500, and 1,000 on the graph on 1:39? Is there a specific thing you look for to find those two numbers?",
        "A": "There sure is! We are told that the total number of bills the Troll has is 900. So to make the graph, Sal chose the maximum value as 1000. For clarity he also marked the middle value between 1000 and 0, which is 500. It the number of bills was 700, Sal probably would have made a graph and marked 800 and 400. It depends on the range of values under consideration."
    },
    {
        "video_name": "H-HfmyUzPw8",
        "Q": "At 2:55, how did Sal get f=1100 and t =550?",
        "A": "Those are the maximums. If t=0, then f=1100. So, f cannot possibly be more than 1100. If f=0, then t=550. So, t cannot possibly be more than 550."
    },
    {
        "video_name": "H-HfmyUzPw8",
        "Q": "At 5:22, Sal says \"I know how many $5 and $10 bills you.\" Is that grammatically correct because it doesn't sound like it is.",
        "A": "He said, I know how many $5 and$10 bills you have."
    },
    {
        "video_name": "H-HfmyUzPw8",
        "Q": "At moment 2:32, you point out that the value of f (while the value for t is zero) would be 1,100. On the other hand, if f is zero, there are 550. I expected that the two equations to have parallel lines on the graph. Is the reason for this the fact that the value of f is different to the value for t, causing f to divide by five and t to divide by ten?\nIn other words, is the reason for the two equations not having a parallel line because f has a different value than t has?\nThanks.",
        "A": "For 2 lines to be parallel, they must have the same slope and different y-intercepts (in this case f-intercepts). Having different x and y intercepts is not sufficient to make lines be parallel. The only time 2 lines would have the same x and y - intercepts would be if they were the same line. Hope this helps."
    },
    {
        "video_name": "JYQqml4-4q4",
        "Q": "at 2:49, why does the approximation not alternate between + and - ?",
        "A": "When you take the derivative of e^x, the derivative is always e^x. For trig functions, the derivative oscillates between positive and negative. That s why sinx and cosx had negative terms."
    },
    {
        "video_name": "JYQqml4-4q4",
        "Q": "At 6:08, why wouldn't abs(cos (1)) + abs (sin(1)) = e? It seems that the absolute value of the poly approx. of cos(x) is one half of the approx. of e, and that the absolute value of the poly approx. of sin(x) is the other half. Added together, that should equal approximately e, but it doesn't?",
        "A": "Because abs(1-x^2/2!) is not the same as 1+x^2/2!. The absolute value sign doesn t distribute. If you want to get rid of the negative signs, you d have to calculate cos(ix)=1+x^2/2!+... sin(ix)/i=x+x^3/3!+... So then yes, cos(i)+sin(i)/i=e. (cos(i)=cosh(1), sin(i)/i=sinh(1), just FYI ) this is related to euler s formula e^(ix)=cos(x)+i sin(x)"
    },
    {
        "video_name": "JYQqml4-4q4",
        "Q": "at 0:25, what if x is negative?would the derivative still equal e^x or would it equal -e^-x?",
        "A": "The first derivative would equal -e^-x. However, the sign is going to alternate as the second derivative would then be e^-x. Then when you evaluate these derivatives at x=0 they will alternate 1 and -1. All the odd nth derivatives will be -1 and all the even nth derivatives will be 1. So you end up with the same Maclaurin series as e^x, except the signs alternate between + and -."
    },
    {
        "video_name": "JYQqml4-4q4",
        "Q": "Around 3:15 Sal starts to approximate e by using the Maclaurin series at x=1. Is this really correct? Shouldn't a Taylor series at x=1 be used instead? For how big x:s can we assume that the Maclaurin series work well enough instead of using Taylor?\nThank you very much! :)",
        "A": "All sal is doing at 3:15 is writing out e^1 as a Maclaurin series. The right-hand side, when a lot of terms are added up, will represent the same number as e^1. I think you are confused with the differences between Taylor and Maclaurin series. The difference between Maclaurin and Taylor series is where they are centered. That is, the a in (x-a). for Maclaurin, a = 0. For Taylor series, a = a number. Sometimes they are used interchangeably."
    },
    {
        "video_name": "GA_yxxeFYBU",
        "Q": "at 1:14, i don't think Sal meant \"keep subtracting negative 7\" because you'd just be adding seven instead of subtracting 7",
        "A": "Yes you re right, he meant decreasing by 7."
    },
    {
        "video_name": "GA_yxxeFYBU",
        "Q": "At 5:56, the subtitle says subtract 0.1 thee times. Shouldn't it be Subtract 0.1 three times?",
        "A": "the subtitles sometimes are a little off, but hopefully that doesn t affect your understanding"
    },
    {
        "video_name": "tyVCA_8MUV4",
        "Q": "in about 1:30. why the order of the cross product is Rs *Rt. But not the reversed order? I dont know how to tell which order to use? Thank you very much.",
        "A": "The order doesn t matter because either way the magnitude will be the same. Essentially Rs*Rt = -Rt*Rs, if you reverse the order you get a minus sign, but when you take the magnitude the minus sign is irrelevant."
    },
    {
        "video_name": "tyVCA_8MUV4",
        "Q": "At 10:45 when he does the square root, can you take out the cos(t) and use it as a scalar? He did something similar to that with the torus example. Does leaving it in just make it simpler to, well simplify.\nIt would look like:\nCos(t) * sqrt(magnitude)",
        "A": "I think it s too late, but, yes you can."
    },
    {
        "video_name": "tyVCA_8MUV4",
        "Q": "At about 10:25, Sal tries to convince us that cos^2(t)cos(s) + cos^2(t)sin(s) + cos(t)sin(t) = (cos^4(t)cos^2(s) + cos^4(t)sin^2(s) + cos^2(t)sin^2(t))^(1/2). I think he may be wrong. For example, 3 + 4 = 7. (3^2 + 4^2)^(1/2) = (9 + 16)^(1/2) = 25^(1/2)=5. These aren't equal.",
        "A": "cos and sin are interrelated in a way that 3 and 4 are not. Instead of just using numbers like you did, why don t you pick values for t and s and try the equation again, as it is."
    },
    {
        "video_name": "oIslHyWJHEY",
        "Q": "At 2:00, can someone explain how d is turned into 5/2? Also explain how 2d equals 5, and not 2dx.",
        "A": "Hopefully, you can see that the 2 middle terms must equal. You can use: 2dx = 5x or you can just use Sal s version: 2d = 5 . If you solve either of these for d , you will get d = 5/2 . -- if you start with: 2dx = 5x , you need to divide by 2x to solve for d 2dx / (2x) = 5x / (2x) d = 5x / (2x) Reduce d = 5/2 -- if you start with 2d = 5 , just divide both sides by 2 and you get d = 5/2 Hope this helps."
    },
    {
        "video_name": "3E8BO7VRMEA",
        "Q": "at 3:20 how did you get the area as 68 %?",
        "A": "hope this helps: For datasets that have a normal distribution the standard deviation can be used to determine the proportion of values that lie within a particular range of the mean value. For such distributions it is always the case that 68% of values are less than one standard deviation (1SD) away from the mean value, that 95% of values are less than two standard deviations (2SD) away from the mean and that 99% of values are less than three standard deviations (3SD) away from the mean."
    },
    {
        "video_name": "3E8BO7VRMEA",
        "Q": "Shouldn't the answer be 60% at the 1:26 mark? There is a possibility x could be 5, right?",
        "A": "There is no probability that x is less than 1. X can be from 1 to 5 so the chance x is from 1 to 4 is equal to (4-1) /(5-1) =3/4*100% = 75%"
    },
    {
        "video_name": "TOobWKx1xe0",
        "Q": "At 0:54, how can vertical angels be equal?",
        "A": "Since they re congruent they re equal. Really it has to do with the basic way a vertical angle is created. When two lines cross they create two sets of vertical angles with each set being congruent. Try it out IRL to see!"
    },
    {
        "video_name": "qW-Ce44ll0Q",
        "Q": "at 0:45 could you move by one space on the number line?",
        "A": "Yes, just move by -4"
    },
    {
        "video_name": "qW-Ce44ll0Q",
        "Q": "0:44-0:47 did not understand\ncan you please help me?",
        "A": "its like tempiture think of negative numbers as as the tempiture....so is a number is -456 think of it as -456 degrese. :3"
    },
    {
        "video_name": "qW-Ce44ll0Q",
        "Q": "how do you get from negative to positive at \"2:18\"",
        "A": "The number line was counting by 3 s so you keep adding 3 to -5 until you get to the blue dot. Hope this helped you."
    },
    {
        "video_name": "TpIBLnRAslI",
        "Q": "At 6:02, Sal said something about an angle angle postulate? What is a postulate? Is it some kind of formula or theory?",
        "A": "This is the dictionary s definition: Postulate (noun): 1. something taken as self-evident or assumed without proof as a basis for reasoning. 2. a proposition that requires no proof, being self-evident, or that is for a specific purpose assumed true, and that is used in the proof of other propositions; axiom. 3. a fundamental principle. 4. a necessary condition; prerequisite."
    },
    {
        "video_name": "TpIBLnRAslI",
        "Q": "0:09 What is an arbitrary triangle?",
        "A": "An arbitrary triangle is one with random side lengths and random angles (as long as it works). Hope this helps! \u00e2\u0080\u0094CT-2/002-24"
    },
    {
        "video_name": "TpIBLnRAslI",
        "Q": "At 7:02, what is AA Similarity? I've never heard of it or learned it before....",
        "A": "If two angles of one triangle are congruent to two angles of a second triangle then the triangles have to be similar. Meaning all corresponding angles are congruent and the corresponding sides are proportional."
    },
    {
        "video_name": "TpIBLnRAslI",
        "Q": "So technically the angle bisector forms a line that intersects the base and forms 2 right triangles? shown at 0:36?",
        "A": "That could happen only if the triangle is isosceles (and you draw the bisector of the third angle, not one of the two equal angles) or equilateral (in this case every bisector is also a height). Otherwise the bisector won t form two right angles. In this case, if you look closely you will see that CDB>ADB."
    },
    {
        "video_name": "TpIBLnRAslI",
        "Q": "at 1:22, in this case when you are writing the formula with the letters AB and AD, etc.\ndo you need to write a line over the letters?",
        "A": "You don t write the line because the line is the actual segment. Without the line it is the measure of the segment."
    },
    {
        "video_name": "TpIBLnRAslI",
        "Q": "At 7:26 , How is AD same as CD?",
        "A": "so triangle BAD and angle BCD is the same, so B is equal to B and D is equal to D and because of that A must equal to C. Or you can Use a compass to find the bisector and measure it and since a bisector cut though the middle then A is the same to C, so then AD is the same to CD."
    },
    {
        "video_name": "TpIBLnRAslI",
        "Q": "At 0:25 to 0:35, how do we know those two angles are equal?\nAnd also what would we do if they were not equal?",
        "A": "You mean angle ABD and angle DBC? We know the re equal because they have been bisected. An angle bisector bisects an angle into two equal angles by definition."
    },
    {
        "video_name": "63mWxNXQQAk",
        "Q": "At 0:58, you start to say equidistant, what does equidistant mean?",
        "A": "Equidistant means equally distant on both sides. Hope this helps :)"
    },
    {
        "video_name": "63mWxNXQQAk",
        "Q": "At 0:59, Sal says \"equadistant\" or something like that. How is that word properly spelled, and what is its definition?\nThank you in advanced!",
        "A": "It is spelled equidistant , and it simply means at equal distances . In other words, if two points are equidistant from something, then they are both the same distance from it. So if I m standing 10 feet from a building, and my friend is standing 10 feet from the building, then that means we are equidistant from that building. I hope this helps."
    },
    {
        "video_name": "_sIuZHYrdWM",
        "Q": "At 2:45, why does he divide by 4x^3 y?",
        "A": "Because 4 is the greatest common factor between the numbers in each term, the two terms share the variable x and you always have to take the smallest exponent so you get x^3, and the two terms also share the variable y (like before you take the smallest exponent) so you get just y. I hope this makes sense."
    },
    {
        "video_name": "_sIuZHYrdWM",
        "Q": "@2:39 i dont get what he did",
        "A": "Explain more your issue."
    },
    {
        "video_name": "_sIuZHYrdWM",
        "Q": "At 5:09 he says it as if he is subtracting, how does taking a 4 out of an 8 result in a 2? Is he saying we are dividing? Then at 4:57 why is there nothing resulted instead of a 1?",
        "A": "You divide the coefficient to divide (which is what the lesson is, basically), and you subtract exponents to divide, so Sal did want that two, he just misspoke when talking about his operations"
    },
    {
        "video_name": "_sIuZHYrdWM",
        "Q": "I don't understand how he gets x to the third at 4:40. ??",
        "A": "x^3 is considered to be a factor of both x^4 and x^3 (x * x^3 = x^4 and 1 * x^3 = x^3). Not only is it a factor of them, but it is the largest of their factors. Just like 4 was the largest shared factor of 4 and 8, x^3 is the largest factor of the x variables. So Sal factors out x^3 like he did for 4 (and y)."
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "There are 350 pupils in school.the ratio of girls to boys is 3:2.how many boys and girls are there in school",
        "A": "350 boys. 350:2 700:3"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "At 1:47 was the ratio simplifyed?\nFrom 6:9\nTo 2:3",
        "A": "Yes. He divided by 3 to simplify them."
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "At 1:54 why do you put this sign between the numbers(:)",
        "A": "At 1:54 the : -sign shows there is a ratio. 6 to 9 or 6:9 is exactly the same. 47%* is *47 :100. In the second case : looks more like a division but it is still a ratio tho. When you have double the first number, 12 (2x6) you got twice the second number, 18 (2x9)"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "what is the relationship of 6:9 and 2:3",
        "A": "If you simplify 6:9, you d get 2:3 because the greatest common factor of 6:9 is 3, therefore, you d divide 6 by 3 and 9 by 3 to get 2:3. Got it?"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "So if you have 5 apples and lets say Bob has 3 bananas... The ratio would be 5:3?",
        "A": "Yeah, if it was asking the ratio of apples to bananas. You can t simplify that, so it would be 5:3."
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "so the thing that sal is saying is that 2 apples to 3 oranges, that would mean 2 apples for every 3 oranges? if there was a group of 4 peaches and a group of 2 onions, the ratio would be 4:2 or 2:1?",
        "A": "Yes, it does mean 2 apples for every 3 oranges. As for your peaches and onions example, both 4:2 and 2:1 are equivalent ratios, but 2:1 is considered the more correct term since it is fully simplified."
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "at 5:34\nhow is the moon big?",
        "A": "? Are you trolling people, ask a question related to the video"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "so what is the actual ratio 9:6 or 3:2",
        "A": "both of them are correct but 3:2 is the simplified version"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "at 3:23 do we have to switch the order of the question?",
        "A": "Yes you will becaus of the way the question is worded"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "two number are the ratio 4:5\none of the number is 60\ntheir are two possible values for the other number\nwhat are the two values for the other two number",
        "A": "48 and 75....are the answers"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "Can you write the ratio of three numbers to each other Like maybe 3:5:6?",
        "A": "What you re describing is similar to the concept of similar triangles, which you ll learn all about in geometry. We don t usually write it like that, though."
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "The ratio 6 to 9 is written 6:9. Could it be written as 6 parts to 9 parts, or can it be written as 6 to 9. Does it really depend on how you write the ratio?",
        "A": "no, it does t, as long as you and your teacher or mentor understand what it means, for example 6:9 is the same as 6/9 or 6 parts to 9 parts. i hope this answers your question"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "My homework is finding out what 90,60,30,and45 are In their ratios,(30:45,60:90)Can you help me find out how to do this?",
        "A": "30/45 =2/3 60/90 =2/3 you can factor them out and cancel common factors or simplifying."
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "How do you work out this question:\nPat and Julie share some money in the ratio 2:5.\nJulie gets \u00c2\u00a345 more than Pat.\nHow much did Pat get?",
        "A": "45 is the 3 between 2 and 5, 3 is 2/3 larger than 2, so you caculate 45/3, then take that answer and times it by 2"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "@3:47 So you can have a ratio of 3 oranges to 2 apples or 3:2, so is 3:2 = 450:300?\n900:600? 9:6?",
        "A": "3:2 = 450:300"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "isnt there 3 ways to write ration 1-6 to 9\n6 to 9\n6:9\n6/9",
        "A": "yes and you did it right they all mean the same as 6 to 9 but all are different forms"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "Is there a direct correlation between fractions and ratios? I noticed the ratio and fraction of the total was 6:9 and 6/9. (Even thiugh a fraction is a portion of a whole, or total)",
        "A": "yes I have watched all the ratio videos 12 times sooo I know how to do it u r correct"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "At 3:13 you can switch the numbers(Oranges to Apples or Apples to Oranges)",
        "A": "can you do 3 to 6 or 6 to 3 does it matter?"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "So would the ratio 6:9 be equal to the ratio 9:6?",
        "A": "No. For ratios, order matters. Consider a bag filled with balls of either red or blue. A 6:9 would mean there are 6 red balls and 9 blue balls. A ratio of 9:6 would mean there are 9 red balls and 6 blue balls."
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "So do you do... for example, 3:9 equals that there is 3 bananas and 9 cucumbers?",
        "A": "3 bananas for every 9 cucumbers is more appropriate."
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "I don't understand ratios. Are they ways of fractions being shown? (I.E 5/6 =5:6)",
        "A": "no it cant be"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "so its like fractions but there is four ways you could put it?\nLike 9;10, 9/10, 9 to 10 9:10?",
        "A": "Basically it is a fraction because something is part of a whole"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "Is 1:3 the same as 1/3??",
        "A": "Yes. Both the fraction 1/3(one third) and the ratio 1:3(one to three) are equal. 1:3 and 1/3 are also equal to 33%. They are all the same amount."
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "At 3:13 Why did you swap them?",
        "A": "The sequence of the words in the description of the ratio tell you which number goes where. -- Sal started by created the ratio of apples to oranges = 2 : 3. The 2 refers to apples and the 3 to oranges. There are 2 apples to every 3 oranges. At 3:13, he is creating the ratio of oranges to apples = 3 : 2. Notice: the description now wants oranges to be first. This is why Sal swapped the numbers. There are 3 oranges to every 2 apples. Hope this helps."
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "Wouldn't it be 2:3 because for every 2 apples, there are 3 oranges?",
        "A": "no, because the question asked the ratio of apples to FRUIT, not oranges. so it would be for every 5 pieces of fruit, there are 2 apples. 2:5. it would only be 2:3 if it was apples to ORANGES"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "can a ratio give us a relationship of more than 2 quantities?\nIn other words can there be a ratio like 2:3:4?",
        "A": "Yes there can but it isn t really used as much"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "If you have 5 apples and 6 oranges could you write like this 5:6 and 6:5",
        "A": "If you write this as 5:6 then you are comparing the ratio of apples to oranges and if you write this as 6:5 then you will be comparing oranges to apples . You can write it both the ways , but make sure you do it according to the information you have . If you compare apples to oranges and write it as 6:5 , then you are wrong , the arguments should match with each other . Hope this helps :)"
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "what does percent meanin 3:15",
        "A": "Percent is kind of like a decimal or fraction, you can change these 3 into each other. pretend you have the decimal 93.0 it equals 93%."
    },
    {
        "video_name": "bIKmw0aTmYc",
        "Q": "The sum of 3 numbers is 192. If the ratio of the first number to the second number is 3 :5 and that of the second to the third number is 7:8, then find the first number",
        "A": "42+70+80=192 42:70 = 3:5 70:80=7:8"
    },
    {
        "video_name": "UqyN7-tRS00",
        "Q": "From 9:00 I could't understand how he represented vectors as a linear combination. Also how did he represent those vectors graphically? Could someone explain that to me? Thanks in advance.",
        "A": "By definition a vectors are written in linear algebra as an ordered n tuple of the form (a, b, . . . , c) or as a matrix that is a m x 1 matrix. In linear algebra you are using row reduction algorithm to find how to write the vector you are looking at in terms of the matrix you have."
    },
    {
        "video_name": "UqyN7-tRS00",
        "Q": "At 5:32 is it f/d instead of f/y",
        "A": "This is a known error and, as such, a pop-up box appears in the lower right-hand corner to say so."
    },
    {
        "video_name": "UqyN7-tRS00",
        "Q": "A vocabulary question: at 11:35, Sal shows two vectors, [a c] et [b d], that have the same direction, although, maybe, a different magnitude. Is it what is meant by two vectors being \u00e2\u0080\u009ccolinear?\u00e2\u0080\u009d",
        "A": "Yes, two vectors are colinear if they have the same direction."
    },
    {
        "video_name": "UqyN7-tRS00",
        "Q": "Dear Sal, Dear All,\n\ncould there be a small error at 6:26 min. where \"y\" should be: y= -cx/d + f/d ?\n\nAt this occasion, I would like to express my deep gratitude to you for all what you do and what we may discover through you !!! It's just fantastic !!!\n\nTruely (t)Ankhfully, Wilhelm",
        "A": "He did corrected that by a notation ( you might be disabling these your youtube )"
    },
    {
        "video_name": "UqyN7-tRS00",
        "Q": "At 1:55, what does Sal mean by a \"singular matrix\"? I read it in my linear algebra book but they didn't define it properly.",
        "A": "singular matrix is matrix that cannot be inverted as it has no solution or infinitely many solutions"
    },
    {
        "video_name": "PwDnpb_ZJvc",
        "Q": "at 0:38 he said 0^1 is 1x1 which is confusing",
        "A": "no, he said 0^1= 1*0"
    },
    {
        "video_name": "PwDnpb_ZJvc",
        "Q": "At 2:48, 0 is not a nonzero number so 0 to the 0th power should be 0. Am i right?",
        "A": "Watch the video! 0^0 is undefined."
    },
    {
        "video_name": "PwDnpb_ZJvc",
        "Q": "At 2:53, Sal says that any nonzero number to the zeroth power is one. How does that work if zero times ANYTHING is zero? So shouldn't it be something like this:\nIf the nonzero number is, for instance, 6, then 6 times itself 0 times ought to be 0.",
        "A": "That is correct that 0 * any number = 0 but we are talking about exponents, and Mr Khan adds a one into the calculation and then you get: 1*2^2= 1*2*2= 4 1*2^0= 1= 1!"
    },
    {
        "video_name": "PwDnpb_ZJvc",
        "Q": "Hold on a second.\n\nSal says, \"0 to the trillionth power is going to be 0. Even NEGATIVE or fractional exponents, which we haven't talked about yet, as long as they're non-zero, this is just going to be equal to 0...\" (1:40-1:51).\n\nBut I don't understand. If you had 0 with a NEGATIVE exponent, you'd end up dividing by 0 which is undefined.\n\nMy question is, how can you have 0^n if n is NEGATIVE like Sal said? Shouldn't 0 to a NEGATIVE exponent be undefined?",
        "A": "A message box pops up at 1:22 that says Sal said nonzero but meant positive And later at 1:48 a box pops up saying 0 to a negative exponent is undefined"
    },
    {
        "video_name": "PwDnpb_ZJvc",
        "Q": "At 1:37 every nonzero number equals 0?",
        "A": "0 TO every nonzero number equals 0. Powers can be a little tricky but think of it in terms of multiplication, 0^2 for example is simply (1) x 0 x 0, which of course equals 0. 0^ (every nonzero number) is simply (1) x 0 x 0 x 0 x 0 x 0 x 0 ... for what ever the nonzero number is, and of course will always equal 0."
    },
    {
        "video_name": "PwDnpb_ZJvc",
        "Q": "At 4:00, Sal said that there are only two possibilities for 0 to the power of 0 ,1 or 0: but how about infinity?",
        "A": "Infinity is not a number; it s just a concept. Since infinity is represented by the infinity symbol, and not by a number, a number cannot really be raised to the infinite power."
    },
    {
        "video_name": "PwDnpb_ZJvc",
        "Q": "@ 3:50, can 0^0 be defined? like using some magic math definition?",
        "A": "Because of what i know, x^0 is 1, so if x=0, i guess it would still be one."
    },
    {
        "video_name": "PwDnpb_ZJvc",
        "Q": "At 2:10, how do you know what 0 to the 0 power is?",
        "A": "Anything to the 0 power is always 1"
    },
    {
        "video_name": "5wUJLMWZ5Fw",
        "Q": "At 0:35 He Lost Me. Can Someone Explain What He Is Talking About?",
        "A": "well if you multiply (x+2)(x-2) you will get x^2-2x+2x-4. the -2x and the 2x cancel out so you are left with x^2-4 Hope that helps :)"
    },
    {
        "video_name": "5wUJLMWZ5Fw",
        "Q": "why did he not add 2 to both sides at 1:20?",
        "A": "Because that would get you to x\u00c2\u00b2-x = 6, which isn t the usual method of solving a quadratic equation. He wanted to get everything over to one side so that he could factor the polynomial."
    },
    {
        "video_name": "5wUJLMWZ5Fw",
        "Q": "2:15 has me a little confused, I might forget this on a test. Is there any way to do this in an easier manner rather than taking the factors of 6? I can not remember what I was taught at school but it did not involve factors.",
        "A": "I m not aware of an easier way to do it. The factoring part is really quite simple. All you do is find 2 numbers that multiply to six, and add to one. In another example quadratic equation: x^2 + 8x + 15 Find two numbers that multiply to 15... 3 and 5. 3 and 5 also add to 8, so youre answer will be (x + 3) (x + 5) or x = -3 or x = -5 Hope this helped"
    },
    {
        "video_name": "5wUJLMWZ5Fw",
        "Q": "I don't understand what happened at @2:34... What happened with -x? I understand that (x-3)(x+2)=0 is equal to x squared minus x, but I don't understand what happened to the -x",
        "A": "He was factoring as a trinomial, so he began with (x )(x ) from x^2. Then he was doing trial and error (x ?)(x ?) to find out what would equal to -6 as a multiplication, and he found that -3*2 would work, but your also suppose to see if those numbers would equal to a sum of -1 or the -x from the middle term of the trinomial, which is true because -3+2= -1 or -x. If you still need help just look up the trinomial videos."
    },
    {
        "video_name": "5wUJLMWZ5Fw",
        "Q": "At 0:48, why did Sal write (x+2)(x-2)?",
        "A": "Sal had x^2 - 4 in the numerator and he knows that this is equal to (x+2)(x-2) -- try multiplying out the brackets yourself and you ll see this is true. He has spotted that there is an x-2 in the denominator and sees that if he writes (x+2)(x-2) instead of x^2 - 4 then he will be able to cancel the x-2 that appears on both top and bottom of the fraction. This helps simplify the fraction."
    },
    {
        "video_name": "5wUJLMWZ5Fw",
        "Q": "At 0:47, why did you took (x-2)(x+2)\nPlease need fast answere, have math test tomorrow",
        "A": "x\u00c2\u00b2-4=x\u00c2\u00b2-2\u00c2\u00b2=(x-2)(x+2) This is a difference of squares, by definition, a\u00c2\u00b2-b\u00c2\u00b2=(a-b)(a+b)."
    },
    {
        "video_name": "5wUJLMWZ5Fw",
        "Q": "At 0:18 I opted to add the rational expression to both sides, leaving:\nx^2 = 4 + (x^2 - 4) / (x-2)\nThen subtracted the 4 from both:\nx^2 - 4 = (x^2 - 4) / (x-2)\nThen multiplied both sides by (x-2)\n(x^2 - 4)(x-2) = (x^2 - 4)\nDivided both sides by (x^2 - 4)\n(x -2) = 1\nLeaving one correct solution, x=3.\n\nWhy do I lose one solution using this method? Where do I lose information?",
        "A": "You lost solutions when you divided both sides by (x^2 - 4). To avoid losing solutions, never divide the equation by any value containing the variable."
    },
    {
        "video_name": "x-GV2v0dRNE",
        "Q": "At 9:42, the determinant of the transposed matrix is being calculated. Why would he be calculating it with a11, a12 (going down along the 1st column) rather than a11, a21 (going along the 1st row)?? Can you calculate a matrix like this?",
        "A": "You can calculate the determinant by going down the rows, or going down the columns. It doesn t even have to be the first row, or first column, (as long as you put the negative sign for even rows or columns)."
    },
    {
        "video_name": "3nQejB-XPoY",
        "Q": "At about 5:50, Sal brings '1/x' outside of the limit. His claim is that he is allowed to do this because the limit varies with n (instead of x), so x is essentially acting as a constant.\n\nBut n is defined in terms of x! If you change x, then n changes to! (?)\n\nPlease answer this conundrum in a more general way, as this is a problem I have been having in various areas of calculus.",
        "A": "The limit is only concerned with driving n to infinity, not x! If we re only varying n and not x, then we can pull the 1/x out of the limit. It s like if you had a 1/2 or 1/3 instead of 1/x; the n isn t concerned with constants it s only varying n."
    },
    {
        "video_name": "3nQejB-XPoY",
        "Q": "At 3:24, how is delta x/x = 1/n ? He makes the substitution, but I don't understand where he got that substitution from.",
        "A": "Pretty late, but may help someone with the same question. I think he does that for convenience of the proof (so he can get to the limit that is e). Every number and fraction can be written in 1/n form by changing the n value, so there is no problem in assuming that (for example, 2 can be written as 1/ (1/2) and so on)."
    },
    {
        "video_name": "3nQejB-XPoY",
        "Q": "at 5:47 can we take 1/x outside limit. I think x is defined in terms of n as x=deltax * n.\nso is it correct to assume x as constant whereas x is depending on variable n?.",
        "A": "No, he is right there - x is just some number that you pick, \\delta x is your small number -> to 0 , and if he wants to, he can make n to be 1/delta x."
    },
    {
        "video_name": "3nQejB-XPoY",
        "Q": "How does sal do the lim--->0 to lim--->inf. at 3:45 ?",
        "A": "Consider this. Let U=1/N. If N goes to INF where U should go? 1/INF = 0 so U goes to zero. In reverse, if U goes to zero, then N should go to INF. Sal has also some X in the mix, but it is just a finite number and does not influence the limit. HTH"
    },
    {
        "video_name": "3nQejB-XPoY",
        "Q": "When using the chain rule in the proof that derivative os e^x=e^x, in 9:29, before proving that the statement is correct, I can't say that the derivativo od ln(e^x) = (e^x)(1/e^x). I'm assuming that de derivative o g(x) in the chain rule, in this case, e^x, is equal to e^x, that is just what I'm still trying to prove... I'm not 100% convinced. Is there any other way to prove that?",
        "A": "He didn t assert that d/dx [ln(e^x)] = (e^x)(1/e^x). He asserted that d/dx [ln(e^x)] = (d/dx [e^x]) (1/e^x). Do you see the difference?"
    },
    {
        "video_name": "3nQejB-XPoY",
        "Q": "At approx. 8:30, why does Sal use d/dx ln e^x? Can't you just use implicit differentiation? i.e:\n\ny = e^x\nln y = ln e^x\nln y = x ln e\nln y = x\nd/dx ln y = d/dx x\n1/y dy/dx = 1\ndy/dx = y = e^x",
        "A": "The trouble with that derivative is that it uses the derivative of the natural log, which you would need to prove. I personally prefer to show this proof from the definition of a derivative and the definition of e. This relies only on basic definitions and a few properties of limits and exponents. As such it does not require proving or assuming the derivative of anything."
    },
    {
        "video_name": "3nQejB-XPoY",
        "Q": "I don't understand why the substitution at 3:20 is true. DeltaX/x = 1/n ; Where did this 1/n come from and why are they equivalent??",
        "A": "Understood, thanks."
    },
    {
        "video_name": "3nQejB-XPoY",
        "Q": "At approximately 4:03, Sal substitutes the lim as n -> infinity for lim as delta x -> 0. If infinity is a quantity without bound, beyond any real number, how can we substitute it for 0? Is the abstraction of infinity equivalent to 0?",
        "A": "This is valid because Sal defines \u00ce\u0094x = x/n. When he takes the limit as n approaches infinity on the right side, the limit will equal 0 because x will be a constant at any point. The limit as \u00ce\u0094x approaches 0 of the left side is also 0 (which is fairly clear). Thus the two limits are equal. Note that Sal in no way substitutes infinity for 0. He takes two different limits on two different expressions using two different limiting variables. Hope that helps."
    },
    {
        "video_name": "3nQejB-XPoY",
        "Q": "Right around 4:00, doesn't he mean \"as the denominator approaches infiniti?",
        "A": "Yes - I think he just misspoke."
    },
    {
        "video_name": "3nQejB-XPoY",
        "Q": "I feel like there is a leap of understanding happening at 8:18 when Sal starts his proof of the derivative of e^x. It is not obvious to me why he chose to use the derivative of ln(e^x) to prove the derivative of e^x. I understand the proof in the end, but how would someone that had not already known it would work know to use this?",
        "A": "e^h= lim h\u00e2\u0086\u00920 (1+h) I feel that this part is a little handwavey, as on the right the variable h is involved in a limit, but not on the left. Does it need to be justified?"
    },
    {
        "video_name": "gdE46YSedvE",
        "Q": "At 2:30, I was wondering if you could use a frequency table to make a histogram. Would you have enough data to make a histogram from it?",
        "A": "possibly, I am not very sure though, you could search it up."
    },
    {
        "video_name": "RQPC_rUvHas",
        "Q": "Around 3:27, Sal says that taking two petri dishes with similar conditions would be a better way to ascertain for sure if the antibiotic is effective or not. Wouldn't it be better if we take the two petri dishes under different conditions rather than similar conditions?",
        "A": "No, because you would prefer to have everything to be exactly the same except for the condition being tested, namely the antibiotic. If the petri dishes, were in different conditions (for example: one in the dark and one in the sun) we don t know if it was the antibiotic or some other difference that lead to the result."
    },
    {
        "video_name": "RQPC_rUvHas",
        "Q": "At 2:43 Sal talks about the margin of error. When can I difference if there is a possible margin of error in any statistical study?",
        "A": "You are correct: there is always a margin of error. In the section on hypothesis testing you will learn about when the margin of error should lead you to conclude that the study can t support the claim that there is a real difference between the things you are comparing, and when it should reassure you that the differences you see in your sample most likely reflect real differences in the population you are interested in."
    },
    {
        "video_name": "Obch1OP5QyA",
        "Q": "How does Sal get a -3 @1:35?",
        "A": "As you know, logarithm in the form log b(a)=c can also be written in the (exponential) form, a=b^c. Taking the example in the video,log 5(1/125)=-3 can also be written in the (exponential) form 5^-3=1/125. Note: a^-n is equal to 1/a^n, therefore in the above mentioned example, 5^-3 it equal to 1/5^3 which is in turn equal to 1/125"
    },
    {
        "video_name": "Obch1OP5QyA",
        "Q": "At 0:26, how did Sal insert both the brackets at a time?",
        "A": "The Khan Academy website does it automatically."
    },
    {
        "video_name": "Obch1OP5QyA",
        "Q": "In the video @ 1:25 Sy's answer is 5^(-3) = 1/125\nWould it be wrong to answer 1/125 = 5^(-3)?",
        "A": "No, it wouldn t be incorrect. As both sides are equal, they can be switched either way, as long as each side is still equal. :P"
    },
    {
        "video_name": "9SOSfRNCQZQ",
        "Q": "At 1:08 Sal said that 5^0= 1. Why does it equal 1",
        "A": "a^0 = a^(x-x) = (a^x) / (a^x) = 1; for any a and any x. Andrew M had a good explanation here 3 years ago (it works of course for 5, or any other number): Watch: 2^3 = 8 2^2=4. We divided by two. 2^1 = 2. we divided by two again. 2^0 = 1. Divided by two again. 2^-1 =? Sure seems like we should divide by two again, doesn t it? That would make 2^-1 = 1/2 Keep going."
    },
    {
        "video_name": "9SOSfRNCQZQ",
        "Q": "At 0:40, why does 5^-2 becomes 1/25 and not -25 like how 5^2 becomes 25? Can't 5^-2 become -25 or is it the fact that because 2 is negative that you have to follow the law of indices?",
        "A": "Think of it this way. 5\u00c2\u00b3 = 125. 5\u00c2\u00b2 = 25. 5^1 = 5. Notice how as the exponential value decreases by one, the overall value is divided by 5. So, 5^0 = 1 (anything ^ 0 will always be 1). Now, if it is 5^-1, it would not be -5, because 1 divided by 5 is not -5, it is 1/5. So, 5^-2 will be 1/25. As the exponential value gets higher and higher, the overall value will get closer and closer to 0, but it will never reach 0."
    },
    {
        "video_name": "9SOSfRNCQZQ",
        "Q": "At 0:00,\nIf it's y = 5^x, is the initial value like 1 or something?",
        "A": "Mathematically, you can make it any real number, A, you want, right? So the equation would be y = A*5^x. When x=0, A*5^x = A, and when x=1, it s 5*A, and so on. It increases by a factor of 5 for every unit of increase of x. When you use it to describe how something changes, for example how bacteria increase, A depends on how much there is at the start."
    },
    {
        "video_name": "9SOSfRNCQZQ",
        "Q": "at 4:12, Sal talks about how as the exponents increase towards the negative side, the number gets really close to zero, but not exactly zero. Wouldn't that be considered a limit?",
        "A": "Yes, horizontal asymptotes are the limit of the function at infinity."
    },
    {
        "video_name": "sXP7VhU1gYE",
        "Q": "Around 2:25 Sal puts one of the numbers in as 0. I still don't understand why he did that. Could someone please help me?",
        "A": "You were asked to find the range (not the domain) of the function f(x). The range is the vertical value while the domain is horizontal value. Naturally you want to find the lowest point and highest point of the graph vertically. 0 is the lowest point on that function while 8 is as highest as it goes."
    },
    {
        "video_name": "hC6zx9WAiC4",
        "Q": "At 4:46, why did he square root that?",
        "A": "The radius was squared so he had to cancel that out by square rooting the radius and then, to balance the equation, he square rooted the other side too."
    },
    {
        "video_name": "hC6zx9WAiC4",
        "Q": "Why for 1/3 do you need to divide it by itself? Sal does this in this video at 4:14. Is there any logic behind this, and if so, what is it?",
        "A": "You just need to isolate the variable to solve for the volume of the cone. You could also multiply both sides by 3, since 3 is the reciprocal of 1/3."
    },
    {
        "video_name": "hC6zx9WAiC4",
        "Q": "at 5:00 cm^3-cm should be cm^2, why is it become a cm only??",
        "A": "Well, he decided that the measurment was going to be centimeters. But he could have done kilometers or decameters or meters. He just decided that he would use centimeters."
    },
    {
        "video_name": "hC6zx9WAiC4",
        "Q": "At 4:46, why did he square root that?",
        "A": "they reason he did that was because he had to get ruin of the r squared"
    },
    {
        "video_name": "hC6zx9WAiC4",
        "Q": "At 4:36 can't you leave it in terms of pi?",
        "A": "yes because it does t change anything"
    },
    {
        "video_name": "hC6zx9WAiC4",
        "Q": "at 2:23, so one third of a cylinder is a cone",
        "A": "Yes, one third the volume of a cylinder will give you the volume of a cone."
    },
    {
        "video_name": "t3cJYNdQLYg",
        "Q": "If you were to plot f(x,y) at 7:20 on the z axis would you also get a surface like you would if you had a scalar field? I can't find the connection between line integrals for vector fields and line integrals for scalar fields.. Are you still finding the area of the so called \"curtain\" when you take the line integral?",
        "A": "I think a line integral through a vector field is not related to the area of a curtain with the scalar field, other than the fact that they re both integrals and they are both over a path."
    },
    {
        "video_name": "t3cJYNdQLYg",
        "Q": "At 11:33, why would we call it a differential of work? And why would the infinitely small change on the curve be called the differential vector/dr?",
        "A": "Work is defined to be force dotted with displacement. We performed f \u00c2\u00b7 dr, where f is the force and dr is the displacement. So, that means that it is work. Because dr is a differential, that must mean that the product is a differential as well, so we have a differential of work."
    },
    {
        "video_name": "sRVGcYGjUk8",
        "Q": "around 4:25 where did 1 come from, next to the Sum?",
        "A": "Firstly, it s grabbed from the \u00e2\u0088\u0091( \u00e2\u0080\u00a6 \u00ce\u00bc\u00c2\u00b2) above. Salman moved \u00ce\u00bc\u00c2\u00b2 to the left of \u00e2\u0088\u0091 by dividing it away (to multiply it onto the left side). \u00e2\u0088\u0091(\u00ce\u00bc\u00c2\u00b2) = \u00ce\u00bc\u00c2\u00b2\u00e2\u0088\u0091(1), Because \u00ce\u00bc\u00c2\u00b2 \u00c3\u00b7 \u00ce\u00bc\u00c2\u00b2 = 1, and it s only the x\u00e1\u00b5\u00a2 stuff that can t be divided away to the left side. Got that now? =)"
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "at the near 6:00 you address the angles by one lowercase letter, shouldn't it be two letters like <la or <lb (I use < for the angle symbol)",
        "A": "The convention is to use one letter, such as <a, unless there is some ambiguity. It is always appropriate (if the points are known) to express an angle as <DEF where E is the vertex of the angle which is formed by the two rays ED and EF."
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "At 5:25, what is a corresponding angle?",
        "A": "Corresponding angles are angles that are congruent to each other, meaning equal to each other. In this video they would be the angles that he circled. Hopefully this helped! :)"
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "I have a very important question: At about 4:45 it is stated that certain angles formed in parallel lines by a transversal are equal; it is further stated that this needs no demonstratiin, but is intuitively obvious\u00e2\u0080\u0094it is not. There has got to be a proof in analytic geometry that those angles are congruent. If it does exist, Could someone\u00e2\u0080\u0094anyone\u00e2\u0080\u0094please post it?\n\nThank You Very Much",
        "A": "There is no proof. It s just intuition. I don t know how to explain it, but those angles are just translations of each other, so they have the same measure."
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "At 2:14, Sal mentions a transversal. Is that just a line that crosses two or more parallel lines?",
        "A": "A transversal line is a line that passes through two or more other coplanar lines at different points.."
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "If at 6:27 in the video -\nb = c = f = g then will angle b... be equal to angles g and f??",
        "A": "Yep! That s what the equal sign means. They are all equal."
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "At 2:53 where d have the circles at the meeting point, the 2 angles together make 180 degrees right?",
        "A": "Supplementary angles definitely add to 180 degrees...vertical angles do not HAVE to equal 180 degrees (not sure which circle meeting point you are referring to)."
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "At 2:07, Sal mentions transversal line... can someone explain to me what that means?",
        "A": "A transversal is a line that intersects two or more lines (PARALLEL or NON PARALLEL) at different points. if two lines AB and CD are drawn and a third line DE intersects AB at P and CD at Q then DE is a transversal"
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "At 1:50, Sal started to talk about Transversal lines. Do they only refer to lines that cross over parallel lines or any line at all? Thanks. (in advance)",
        "A": "a transversal is a line crossing over any two lines \u00e2\u0097\u0089\u00e2\u0097\u00a1\u00e2\u0097\u0089"
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "At 0:14, what does Sal mean when he says it goes through point C and D?",
        "A": "connect point C and D with a same line. In other word, literally construct a line going through point C and D"
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "At 0:36 can the parallel lines work with rays and a line segment.",
        "A": "Well... yes... because if two rays go on infinitely forever in one direction and don t touch each other, it would be parallel."
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "At 1:27 can triple arrows be used to show parallel lines?",
        "A": "Yes, I am certain that if there are more than one set of parallel lines, triples arrows can be used to denote that a set of lines are parallel. For clarity, you ll likely only see one or two arrows to denote parallel lines, at least that this stage in Geometry."
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "At 3:26 How does Sal know the angles are similar?\nJust asking.",
        "A": "because they are parallel to each other"
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "3:28 How are those two angles equal?? What makes them equal? Is there an exact word or way to describe this?? What I mean is how would I tell that these two angles are equal?",
        "A": "These two angles are called corresponding angles (I talk with my class about being on the same corner of the two roads - in this case the northeast corner). The two lines are defined as being parallel, so the lines are parallel if and only if the corresponding angles are congruent."
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "At 2:41, why are they called vertical angles?",
        "A": "At 2:41, they are called vertical angles because they are symmetrical to each other. Say you put a vertical line between them they d both be a perfect reflection of the other. I hope this helps."
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "At 6:18, b=c=f=g. So, does that mean b=g?",
        "A": "That is correct. Because of the transitive property of equality, a statement involving multiple equals signs implies that any two of the quantities involved are equal to each other."
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "what does intersect mean at 0:40?",
        "A": "When two lines intersect on a coordinate plane, they cross one another. Parallel lines never touch, so they never cross one another."
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "At 4:15, Sal says that there is no proof for the angels of parallel lines being the same, and that it is \"intuitively obvious\". Is there really no proof? Would a teacher accept this?",
        "A": "Yes, the Corresponding Angles POSTULATE allows for that. Postulates are those rules that we accept without proof because they are generally pretty obvious. Consider collapsing the two parallel lines onto one another. The corresponding angles would then be the same angle and therefore obviously the same measure. But if you tried collapsing two non-parallel lines to a common point on their transversal then the corresponding angles would not be the same and therefore not congruent."
    },
    {
        "video_name": "H-E5rlpCVu4",
        "Q": "What was ~ sign when Sal wrote it next to the angle? It is 6:00 seconds.",
        "A": "It looked like he was just marking the angle, but then changed his mind and labeled them."
    },
    {
        "video_name": "lSwsAFgWqR8",
        "Q": "At 3:21, what does \"k\" mean?",
        "A": "It should be understood to be any fixed number."
    },
    {
        "video_name": "lSwsAFgWqR8",
        "Q": "I understand limits, but at about 2:15, wouldn't that be the product rule? Also, at 3:35, I am a little confused about that not being the quotient rule.",
        "A": "Well, it s basically the same idea as the quotient/product rule, but relating to functions, not variables."
    },
    {
        "video_name": "lSwsAFgWqR8",
        "Q": "2:26 So what if you had something like: lim x ---> a ( f(x) * g(x) +1 )?\n\nWould that be: L * M + 1\n\nAnd what if I had something like: lim x ---> a ( ( f(x) + 1 ) * g(x) )\n\nWould that be: ( L + 1 ) * M",
        "A": "Yes that s right."
    },
    {
        "video_name": "lSwsAFgWqR8",
        "Q": "Concerning 2:20, If you get the question \"what is the product of the limit g(x) when x approaches c and the limit f(x) when x approaches c\" given that f(x) is discontinuous at c. Would the answer be zero or just that the limit does not exist?",
        "A": "No, if one of the limit is undefined the product of the limits can t be defined"
    },
    {
        "video_name": "lSwsAFgWqR8",
        "Q": "Regarding the limit of the product of f(x) and g(x) at 2:15, does this have a special name?",
        "A": "well it does have a name multiplication rule"
    },
    {
        "video_name": "lSwsAFgWqR8",
        "Q": "At 2:21 you wrote:\nlim f(x)g(x) = lim f(x) * g(x)\nThis leads me to understand that lim ab = lim a * lim b\n\nHowever, at 3:22, you wrote:\nlim kf(x) = k * lim f(x)\n\nIf lim ab = lim a * lim b, shouldn't lim kf(x) = lim k * lim f(x)?",
        "A": "k is a constant, not a function. Hence, it doesn t approach anything - it s value is always the same."
    },
    {
        "video_name": "y-O_B-wWivQ",
        "Q": "At 1:05, Sal had shown line segment AD with points B and C in them, (reference from Language and notation of basic geometry video) so does it means the line segment is collinear even though there are two points in them?",
        "A": "I Think it does"
    },
    {
        "video_name": "UquFdMg6Z_U",
        "Q": "At 4:22, could I write -x/2 instead of -1/2 x? Is it the same?",
        "A": "they are same"
    },
    {
        "video_name": "UquFdMg6Z_U",
        "Q": "at 2:54, Sal said \"anything divided by anything is 1,\" but I think he meant to say \"anything divided by itself is 1\" ?",
        "A": "Yes, I agree with you, but would also add except zero ."
    },
    {
        "video_name": "UquFdMg6Z_U",
        "Q": "at 0:30, where did he get all the 6xs from?",
        "A": "the equation was 18x^4 - 3x^2 + 6x - 4 ------------------------------- 6x so that basically is 18^4/6x - 3x^2/6x + 6x/6x - 4/6x So that s where they came from"
    },
    {
        "video_name": "1BGlbx67B6s",
        "Q": "Does anyone know a proof for the property Sal uses at 6:40 where he just replaces x with x^2?",
        "A": "It does not need a proof, it is a very simple algebraic manipulation. If f(x)=x, then f(x^2)=x^2. This isn t special, we can say the same thing with f(x+y)=x+y."
    },
    {
        "video_name": "NfU7cmajLAY",
        "Q": "at 2:26 sal said hash mark what is the meaning of that",
        "A": "The hashmarks are the little lines dividing up the number line."
    },
    {
        "video_name": "NfU7cmajLAY",
        "Q": "On 2:57 it should be 27 hundredths",
        "A": "27/100 = 27 hundredths = 0.27"
    },
    {
        "video_name": "V6Xynlqc_tc",
        "Q": "At 2:22, if the slope is a whole number can you just convert it to the whole number over 1?",
        "A": "yes. integer a = a/1"
    },
    {
        "video_name": "V6Xynlqc_tc",
        "Q": "This is a stupid question but at 1:05 if its (-8) - 4x why does he put it down as -4x - -(8) and when should you switch it in this fashion ?",
        "A": "Sal writes it in this form because it is similar to the equation y=mx+b with the slope*x going first before the y-intercept."
    },
    {
        "video_name": "V6Xynlqc_tc",
        "Q": "doesn't it make a difference if you do -8 -4x or -4x -8?\nif not how?\n1:11",
        "A": "Krazy, -8 -4x could be written as (-8)+(-4x) and the commutative property of additon allows you to reverse this to (-4x)+(-8) which is -4x-8 As long as you keep the negative signs with the numbers they go with you can change the order. I hope that helps."
    },
    {
        "video_name": "V6Xynlqc_tc",
        "Q": "At time 3:30you said that you can't make it into slope interval form. Then you made a line at -2x. Why where you able to do that. Also what is the x mean in \"y=mx+b\".",
        "A": "when your m is -2 why did you go over one and down two?"
    },
    {
        "video_name": "V6Xynlqc_tc",
        "Q": "Just to make sure, when he is doing the slope @ 2:28 he is going over 1 in the x direction for every time he goes down 2 in the y direction, because of the slope in change in Y/ change in X",
        "A": "Yes, you are right"
    },
    {
        "video_name": "V6Xynlqc_tc",
        "Q": "ok....i understood everything till like 1:50 . i dont get how to do the slope thing. what do you do after y=-2x-4?",
        "A": "y = -2x - 4 is the equations in slope-intercept form. There is nothing more to do."
    },
    {
        "video_name": "V6Xynlqc_tc",
        "Q": "(2:34) I don't understand the relationship between the y intercept and the subsequent points on the slope. How is that derived?",
        "A": "The coefficient of x is the slope. In the video, the slope was -2. The slope is also the (change in y)/(change in x). So if x changes by 1, when the slope is -2, y changes by -2. change in y = -2 change in x = 1, so slope is -2/1 = -2. So, if you take a point you know is on the line, which was the y-incercept, when you go right by 1 (change the x by 1), you then go down 2 (change y by -2). I hope that helps."
    },
    {
        "video_name": "V6Xynlqc_tc",
        "Q": "At 1:15 why he just subtract the -4 from the -8?",
        "A": "Like the person before me said, he wasn t exactly subtracting it. [Even though he did...Let me explain.] He had to do that in order to get 4X to the other side of the equation, in order to make it like y = mx + b form. :) So he had to move 4x to the other side to get it in that form. So he subtracted 4x from itself so that it would cancel out there, but then to get it to the other side, that meant he had to subtract it from -8 too.. Hope this helps!! :)"
    },
    {
        "video_name": "V6Xynlqc_tc",
        "Q": "Back to the beginning of the video (1:04)....I though you were supposed to subtract 4x and 2y not 4x and -8?\n(Its how I was taught to do multi-step equations) I'm going to the 9th grade by the way so I'm a little lost lol.",
        "A": "im not sure what you have been taught but no matter what you need y on one side and x and a constant on the other"
    },
    {
        "video_name": "V6Xynlqc_tc",
        "Q": "At around 2:40 or so, I don't understand what Sal meant by the 1 and the -2x. How did he determine that he had to go over one first?",
        "A": "Hi Melly, He is referring to the slope of the line which says that every time you change X then Y changes. This line s slope is M=y2-y1/x2-x1 = -2 This means that every 1 step that x changes, y changes 2 steps. So if you start at (0,-4) you go to (-1,2)"
    },
    {
        "video_name": "V6Xynlqc_tc",
        "Q": "At 1:06, why did he change it to -4x-8? Wasn't it -8-4x?",
        "A": "because in an equation you re aloud to change the order of it as long as it s on the same side of the equal sign."
    },
    {
        "video_name": "V6Xynlqc_tc",
        "Q": "For line B, at 3:20, why does he say that you cannot convert it into slope intercept? Couldn't the y value be 0 and written as 0 = 4x - 8? In line C, he automatically puts the slope as a 0, so what is the difference?",
        "A": "The slope is not 0, it s undefined. Remember that slope is rise over run, in this case the run is 0 and dividing by 0 is undefined. The only time the slope is undefined is when the line is perpendicular to the x-axis and parallel or equal to the y-axis. So in this special case the y is insignificant as it can be any number, anytime you are given a linear equation and there is no y involved you can know that you are dealing with a vertical line and should just solve for x."
    },
    {
        "video_name": "7GEUgRcnfVE",
        "Q": "If f(x)=f(a) =/=> x=a\n\ncos(0) = cos(2pi) ... 0 =/= 2pi\n\nAt 17:30 Sal alludes to this being true",
        "A": "The pre-condition is the uniqueness of the solution."
    },
    {
        "video_name": "7GEUgRcnfVE",
        "Q": "At 21:09, I think Sal meant to say f(x) = y instead of f(x) = x. Please correct me if I am wrong.",
        "A": "Yes, I think you re right."
    },
    {
        "video_name": "zXyQI4lD4wI",
        "Q": "at 1:38 Sal says that point (-1,7) is a relative max. Isn't that point (-1,7) the Global or absolute max, or can it be both?",
        "A": "A global maxima is always a relative maxima."
    },
    {
        "video_name": "zXyQI4lD4wI",
        "Q": "At 0:57, he says it is lower than the points around it, but then, why did he make all those other points orange? I'm not sure I get it.",
        "A": "A minimum is a point where the direction of the graph changes. For instance, two of the three points he marked as relative minimums are at -3. There is another point at -3, but it is not a minimum, because the graph is still going down there (in fact it goes down to the absolute minimum visible, -8). A minimum is a U or V shape, where a maximum is upside down of those shapes."
    },
    {
        "video_name": "XD-FDGdWnR8",
        "Q": "At 2:55 I noticed that basically you are reversing the question like instead of k - 8 = 11.8 you practically have k= 11.8-8. Right?",
        "A": "No, actually it is k=11.8+8. Because k - 8 = 11.8, if you want to find out what k is, then you add 8 to 11.8."
    },
    {
        "video_name": "XD-FDGdWnR8",
        "Q": "why is the audio quality so good from 0:00 all the way to 4:31",
        "A": "it is strange because some are quiet lol"
    },
    {
        "video_name": "QI6x0KNxiCs",
        "Q": "at 5:24 i didn't get it...",
        "A": "Try watching the video again. That s what I do when I get stuck. Good luck! :D"
    },
    {
        "video_name": "QI6x0KNxiCs",
        "Q": "At 7:48 i'snt the answer just switched?",
        "A": "Da of course it s switched because it s kind of the same answer can I get a amen around here please!"
    },
    {
        "video_name": "QI6x0KNxiCs",
        "Q": "at 4:34 sal was circling groups of 2 but how was he if we did not know the number of groups and same thing for 24 divided by 4 and 24 divided by 6?",
        "A": "Because Sal is smart"
    },
    {
        "video_name": "C5Lbjbyr1t4",
        "Q": "03:05 why didn't Sal use rule a^x dx= a^x / ln a +C if a > 0 and a != 0?",
        "A": "it seems to me, he wanted to prove that rule, and he got it right as usual :)"
    },
    {
        "video_name": "C5Lbjbyr1t4",
        "Q": "at 9:47; is ever number connected to 'ln' and raised to the power 'e' is equal to itself?",
        "A": "I think he does this around 3:17 And yes a = e^(ln a) for a > 0"
    },
    {
        "video_name": "C5Lbjbyr1t4",
        "Q": "At around 4:50 onwards Sal shows that the anti derivative of e^au is (1/a)e^au. even though in reverse (taking the derivative) it seems to make sense I can't really understand how he gets the 1/a when taking the anti derivative.",
        "A": "Try making the following substitution: (by int[] i mean the integral of whats inside the square brackets) int[e^ax(dx)] let u = ax, then du = a(dx) then dx = du/a now substitute this back into the original integral: int[e^ax(dx)] now equals int[e^u(du/a)] Therefore: int[e^ax(dx)] = (1/a)int[e^u(du)]"
    },
    {
        "video_name": "C5Lbjbyr1t4",
        "Q": "at 3:19 can't we directly put the formula int{ a^x}=a^x/lna",
        "A": "Of course we could. I think the reason he did not do that, and rather derived it, was to make sure that everyone can follow along. I am convinced that the sudden appearance of the formula \u00e2\u0088\u00ab a\u00e1\u00b5\u00a1 dx = (1 / log a) \u00c2\u00b7 a\u00e1\u00b5\u00a1 + C would confuse somebody."
    },
    {
        "video_name": "C5Lbjbyr1t4",
        "Q": "why is not that 2:46 the integral of 2^u is 2^(u+1)/(u+1) the reverse power rule",
        "A": "The rule is not applicable here since u is a variable. The rule only applies to the situation x^r, where r is a real number."
    },
    {
        "video_name": "glEP04Pzap8",
        "Q": "At 1:27, what does Sal mean when he says that this really is the fraction that are bears? Can anybody explain please?",
        "A": "It means the number of animals that are bears out of the total number of animals. 2 apple 3 orange = 5 Total Fruits. 2/5 really means the fraction that are apples out of the total number of fruits."
    },
    {
        "video_name": "glEP04Pzap8",
        "Q": "1:22 If you just wanted to simplify the fraction of animals that are bears, and you cancelled out the b in the top and bottom, would the numerator become a 1, or would it be a 0, and cease to be a fraction?",
        "A": "When you cancel out factors, you are using division. What is 5/5? 5/5 = 1 Any number divided by itself = 1, not 0. So, b/b = 1 Hope this helps."
    },
    {
        "video_name": "d1O54EQ1DSE",
        "Q": "At 3:44 Sal says that the number type cannot be Real Numbers. But shouldn't 0.5 be a real number? Aren't integers all whole numbers?",
        "A": "The domain is the input values for the function. In this case, the input value (n) is defined as the number of steps. These will be integers because we count steps in whole units. Real numbers wouldn t work because real numbers would mean that we could go up/down fractions of steps, which isn t the case. Hope this helps."
    },
    {
        "video_name": "yb7lVnY_VCY",
        "Q": "What is he saying at 1:02?",
        "A": "at 1:02 he said you also can view this as 1/4 of 1/5. so basically he is saying you can do or look at it in a different way (sorry its. a little late)"
    },
    {
        "video_name": "yb7lVnY_VCY",
        "Q": "At 0:39, Sal says that \"dividing by a number is the same thing as multiplying by its reciprocal\". But from another source on the internet on maths, I found that the definition of reciprocal, is \"1 divided by the number\". Like how the reciprocal of 2 is 1/2, and the reciprocal of 10 is 1/10 (0.1).\nWhich of the two is it?",
        "A": "I don t understand your question. Nothing you have stated is inconsistent with what Sal said."
    },
    {
        "video_name": "yb7lVnY_VCY",
        "Q": "At 0:58, 1/5 divided by 4/1 equals 1/5 x 1/4 ? How is that possible?",
        "A": "When you divide two fractions, it s the same as multiplying by the reciprocal. I guess it s possible because when you re dividing, you re basically trying to figure out what you can multiply something by to get something. Does that make sense? Probably not..sorry..."
    },
    {
        "video_name": "yb7lVnY_VCY",
        "Q": "At 0:41 he mentions recipicles. what are they?",
        "A": "If a \u00c3\u0097 b = 1 then b is the reciprocal of a and a is the reciprocal of b. a \u00c3\u0097 b = 1 So b = 1/a So the reciprocal of a is 1/a A number and its reciprocal multiplied together yields the identity element for multiplication which is 1"
    },
    {
        "video_name": "n8Ic2Oj-zvA",
        "Q": "At 1:04, wouldn' t that be a vector? If you drew it out it has a directon and magnitude of 5 degrees.",
        "A": "Angle and magnitude are completely independent properties. The magnitude will never change regardless of how large or small the angle value."
    },
    {
        "video_name": "S4oOSgTj9C8",
        "Q": "At 0:50, I am sure that Sal never gave us that formula before. I've seen every video in the playlist so far, and I never came across what he calls as \"second fundamental theorum of arithmetic.\"\nDoes anyone else know of any video which explains it? maybe later in the playlist..",
        "A": "From around 1:10 to 1:50 Sal explains where he s coming from. The integral gives you the area under a curve. A rectangle which had the same width as the integral and which had a height of the average value of the curve over that interval would have the same area. Area = (width (dist from a to b)) * (average height) = integral of f(x) from a to b So, if you divide the integral from a to b by the width of the interval you get the average height, or average value of the function on that interval."
    },
    {
        "video_name": "Dy5uDkOoMNc",
        "Q": "Why is there a Please Wait sign at 1:40 or 1:41?",
        "A": "It happens for about a second when you paste something on ArtRage."
    },
    {
        "video_name": "Dy5uDkOoMNc",
        "Q": "At 0:29, how is a ten different from a one?",
        "A": "A ten can be used in other meanings than a one"
    },
    {
        "video_name": "u1SAo2GiX8A",
        "Q": "At 3:13, Is 2y(2y-3)+5(2y-3) also = -2y(-2y+3)+-5(-2y+3)??",
        "A": "Yes because it s the same when double distributed"
    },
    {
        "video_name": "u1SAo2GiX8A",
        "Q": "at 0:47, don't you mean a*c and a+c, and not a*b, and a+b, since the original equation is ax^2+bx+c",
        "A": "Unfortunately, Sal is using variables that are already defined for quadratic equations, he would have better used other variables, two numbers such as e and f that meet the criteria. These a and b are different than the ones in the quadratic."
    },
    {
        "video_name": "u1SAo2GiX8A",
        "Q": "In the video at 2:55, you factored the 4 and 6; you split them both into 2, why did you do this?",
        "A": "That was so simple but I was stuck right there, thank you!"
    },
    {
        "video_name": "u1SAo2GiX8A",
        "Q": "At 0:30, the description should be a*c, not a*b",
        "A": "Actually, Sal is not using the a, b, and c of the equation, he is saying that there are two numbers, a and b, that must multiply to be -60 (as you noted a \u00e2\u0080\u00a2 c)) and add to be 4. It is a bit confusing that he chose a and b as the two variables."
    },
    {
        "video_name": "u1SAo2GiX8A",
        "Q": "At 2:39, why didn't you switch the -6y and 10y?",
        "A": "It will work either way, because you will get the same answer."
    },
    {
        "video_name": "u1SAo2GiX8A",
        "Q": "So, at about 1:30, when Sal multiplies the 4 and -15, which 4 is he multiplying? I just want to know if it's the front term or the middle term so I can apply this model to similar problems.",
        "A": "It s the first one (the front term). However, do note that this model doesn t work for all problems. You ll still have to use the quadratic formula or complete the square for more complicated ones."
    },
    {
        "video_name": "u1SAo2GiX8A",
        "Q": "At 0:22, where would you get the 4 you are multiplying with?",
        "A": "Sal is factoring: 4y^2 + 4y - 15 The 4 used in the 4(-15) comes from the 4y^2. Hope this helps."
    },
    {
        "video_name": "u1SAo2GiX8A",
        "Q": "At 2:18 you put \"-6y+10y\" in this order, but I wrote \"10y-6y\" to get 4y^2+10y + -6y-15\" So I got almost the same result as yourr but inverse: (2y+5)(2y-3) yours is (2y-3)(2y+5). My question is If it does really matters the order in which we write this two middle numbers at the beginning and Could it affect the final result?",
        "A": "The order of the numbers you place in the middle does not matter. The only thing that matters is that the 2 numbers you use must add to the original middle value and multiply to create the -60 (in this problem). As you saw with this problem, you got the exact same answer when placing the middle numbers in the reversed order. The order of the final factors also doesn t matter: (2y+5)(2y-3) vs. (2y-3)(2y+5). The Commutative Property of multiplication tells us they are the same."
    },
    {
        "video_name": "3n0LvI99-KM",
        "Q": "At 7:15, How can you just add x+x+30=180 degrees to solve for the missing angles? How would you which one would be 75 degrees?",
        "A": "They are both 75 because it is an isosceles triangle (two of its sides and two of its interior angles are the same)"
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "At 4:24 is there any other way to get rid of the three? this one seems a bit confusing! Explain more please.",
        "A": "Sal said multiply with -1/3 . It is the same as if you divide with -3. So this is what you start with: -3x = 11 Then you want to get rid of the -3. Divide both sides with -3 (or multiply with -1/3, same thing). -3x/-3 = 11/-3 x = -11/3"
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "At 2:07, isn't -5-3 equal 8 not negative 8?\n\nMy reasoning behind this is because to subtract, you change the operation to addition and change the second integer to its opposite .\nIn this case it would do -5 + -3 which would equal 8.",
        "A": "No. If I lose five pencils, and then I lose 3 more, will I have 8 more pencils? I realize this is a bad metaphor."
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "Thier is a really bad word with subtitles at 1:21 plz fix",
        "A": "Good Find! However, I don t think Sal can do anything about it. He has to contact Youtube in order to change the subtitles. You can also click the Report a Mistake button, and tell Khan academy about it."
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "could i have help with the problem at 2:00?",
        "A": "nope"
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "In 2:11 where did he get 1/10",
        "A": "It is a level 1 step. 10x=-8 To get x, 10x*1/10=-8*1/10 You can multiply the same number on both sides without changing the equation,he multiplies by the reciprocal of the coefficient of x."
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "at 1:13 why is Sal bring the 7x to the other side??? i thought the problem was supposed to be balanced with an x on both sides.",
        "A": "ten commandants of math if you do something on one side of the equation as you do to the other"
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "In 0:56, why did you make the x variable equal to just 1, and then look at the x and 2x together as just 3? Can the x be other numbers and the answer of the equation still be the same?",
        "A": "It s not like like x equals 1, its just 1x, no number in front of x means that we multiply x by one, so we sum x and 2x which results in 3x. All we do is sum x s like apples."
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "What if 5 is squared at 2:57",
        "A": "If the 5 is squared, that term becomes 25x. Then, solve using similar steps to what Sal has done."
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "at 3:18 in the video what if the x had an exponent or if -3 had an exponent\nhow would you cancel it out then?",
        "A": "If the X had an exponent, you would use other techniques to solve the equation. If the -3 had an exponent, you would have to simplify it as your first step."
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "I saw how you solved the problem at 3:33, but how do you work out the problem\n-4=7x+8-8x\nI would appreciate your help. thx",
        "A": "You have to get x on one side and integers on the other. The way to do that is by undoing. -12=7x+8x -12=15x then divide by the integer next to x to free x. -12/15=x Ta Da. Hope this helps"
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "so for the equation at 3:30 i got the same answer but i did it in a different way. what i did was:\n5x-3-7x=x+8\n5x-7x= -2x\n-2x-3=x+8\n+2x +2x\n-3=8+3x\n-8 -8\n-11=3x\n---------\n3 3\nx= - 11/3",
        "A": "What you did is fine. The properties of equality have a lot of flexibility. As you can see from your work vs. the work done in the video, you followed a different sequence of steps. But, you still got the same answer."
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "In 5:12\nhow did -3 became -9/3?",
        "A": "-3/1 * 3/3 = -9/3"
    },
    {
        "video_name": "Zn-GbH2S0Dk",
        "Q": "I dont get how to check the answer at 5:47",
        "A": "Sal goes through it step by step. Where do you get off the track?"
    },
    {
        "video_name": "xdiBjypYFRQ",
        "Q": "At 1:56 he says that the y-intercept is the vertex but this isn't true right?",
        "A": "You are correct, he did make a mistake: The vertex of that parabola did not coincide with the y-intercept. The vertex of that parabola lies at (-\u00c2\u00bd, - \u00c2\u00b2\u00e2\u0081\u00b5\u00e2\u0081\u0084\u00e2\u0082\u0084) ."
    },
    {
        "video_name": "xdiBjypYFRQ",
        "Q": "Yes, i have a problem im following your videos cause im trying to find certain problems and possiable find a solution for my problems on my math evaluation test. Im having trouble with some graphing problems im not sure if this is a quadratic inequalities but the answer is similiar to your example around 2:15 on the video its a (V). The Equation is y=(x+3)(^2)-4 and i have to graph it. and the ^2 means (x+3) is squared if any problems.",
        "A": "probably a bit late but yes this would be the wrong video for graphing that equation. Maybe look at Graphing Functions"
    },
    {
        "video_name": "xdiBjypYFRQ",
        "Q": "At 7:43 where does 12-1 come from?",
        "A": "He meant 12 * 1. (5 + 7)(5 - 4) = 12 * 1 = 12"
    },
    {
        "video_name": "xdiBjypYFRQ",
        "Q": "Why is the y intercept -6? Sal mentions it at 1:52",
        "A": "Because, like in y=mx+b, the constant term is the y-intercept."
    },
    {
        "video_name": "xdiBjypYFRQ",
        "Q": "How do you know that the roots are x=-3 & x=2 at 1:16???",
        "A": "If you have an equation in the form some number times some other number is equal to zero, that means either one number is zero or the other or both. Because to get zero using only multiplication you need at least one zero as one of the factors. So if we have (x+3)(x\u00e2\u0080\u00932)=0, one of the two numbers is x+3 and the other is x\u00e2\u0080\u00932 . The two roots Sal writes down each make one of the numbers equal zero."
    },
    {
        "video_name": "xdiBjypYFRQ",
        "Q": "What does it meant at 9:06?? What is the point on the curve?",
        "A": "At 9:06, Sal highlites the curve under the x-axis, but remember that the entire curve is f(X) (which is also called y). Since you re dealing with a curve rather than a straight line, you use f(x). A straight line is simple, so y works, but curves are more complicated and f(x) is a more advanced tool. Keep it up and you ll gradually understand. It takes time and practice, but it s worth it."
    },
    {
        "video_name": "xdiBjypYFRQ",
        "Q": "How can -3,-4,-5...be greater than 0???!!! :S (at 3:52)",
        "A": "When you plug those numbers into the equation, they give a result that is greater than 0. For example, if x is -4, then -4^2 + -4 + -6 would be 16-4-6, which is 6, which is greater than 0."
    },
    {
        "video_name": "Ll3QzXftlS0",
        "Q": "What is the symbol, :, that you used at 0:53 and why are we using it? Why didn't you use this symbol, ; , instead?",
        "A": "The long hand is a symbol of minutes and the short is for hours. Without them, we would not know"
    },
    {
        "video_name": "JTpXK2mENH4",
        "Q": "Why did he square the terms at 4:31",
        "A": "He s squaring because that s what the question s asking for. For each of the roots of that polynomial, it asks you to square those roots, take the absolute value of the imaginary parts of the resulting numbers, then take the sum of the results of that, and finally take the sum of the components of that number."
    },
    {
        "video_name": "9iaYNaENVH4",
        "Q": "Around 11:47, does it matter whether we say that we calculate a double integral over a \"surface\" (symbol S) over over a \"region\" (symbol R)?",
        "A": "They both mean the same thing in the current context. They both basically mean summing up an infinite number of infinitesimally small area chunks over the entire surface/region."
    },
    {
        "video_name": "9iaYNaENVH4",
        "Q": "At 10:47; by dotting the vector field with the normal wouldn't you get the projection of the vector field perpendicular to the surface? I understand that we need the projection of the vectorfield ON the surface, but I don't see how dotting with the normal vector accomplishes that?",
        "A": "Stokes theorem says that \u00e2\u0088\u00abF\u00c2\u00b7dr = \u00e2\u0088\u00accurl(F)\u00c2\u00b7n ds. We don t dot the field F with the normal vector, we dot the curl(F) with the normal vector. If you think about fluid in 3D space, it could be swirling in any direction, the curl(F) is a vector that points in the direction of the AXIS OF ROTATION of the swirling fluid. curl(F)\u00c2\u00b7n picks out the curl who s axis of rotation is normal/perpendicular to the surface."
    },
    {
        "video_name": "FJIZPvE3O1A",
        "Q": "What is the proof of \"8:14\"",
        "A": "using the same diagram as in the video, imagine the following: draw segment AD and segment BC, angle AED is congruent to angle BEC by vertical angles theorem, angle DAE is congruent to angle BCE because angles that subtend the same arc are congruent (arc DB) so, triangle ADE is similar to triangle CBE by aa similarity postulate. corresponding sides of similar triangles are proportional, so AE/CE = DE/BE, cross multiply and you get the CHORD-CHORD PRODUCT THEOREM, AE*BE = CE*DE."
    },
    {
        "video_name": "FJIZPvE3O1A",
        "Q": "In 1:22 do both numbers have to be squared numbers",
        "A": "I believe you are referring to his work with x^2+x^2... in that case, both numbers have to be squared because he is using Pythagoras s theorem, that is where a^2+b^2=c^2."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 1:05, Does it matter if you put Y1 first before Y2? Same with X1 and X2?",
        "A": "No. However, you need to make sure you subtract them both the same way, which means if you start with the first point minus the second point, on the bottom you start with the first point minus the second point."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "at 0:43 the triangles just mean change in x and change in y, right?",
        "A": "Yes, as slopes with fractions are much easier to graph using the rise over run. The numerator (part of the fraction on top) is how much the line rises or falls over the run, which is the denominator. For example, a slope of 5/4 means that a line rises 5 units over the course of 4 units."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 1:45 Sal mentions \"Delta X \" What does \"Delta X\" Mean?",
        "A": "Delta is a Greek Letter. It is most often seen in math and science as its uppercase symbol, a triangle. It represents change. Slope is delta y/delta x, or change in y over change in x. Delta y over Delta X is a fancy way of saying rise over run."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 2:20, how are you meant to draw that slope (2/3) using this information?\nCan someone help me please?",
        "A": "remember that slope is equal to rise over run or m=y/x so if the slope of a line is 2/3, than you know you must move +2 units on the y axis, and +3 units on the x axis. the only catch is that you have to start from a point that you know is on the line. Did that help?"
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At about 4:20 , they were explaining that it was -6 and -4, how did it become negative? Couldn't they have just gone up?",
        "A": "it is negative because the line had a negative slope instead of a positive slope"
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 0:46, what is the little triangle?",
        "A": "The triangle is the greek letter for delta, which signifies change in in math. Hope this helps!"
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "I understand how Sal got 2/3 at 2:21 but is the slope of the line also called the gradient?",
        "A": "Yes, the slope of a line is sometimes called the gradient of the line. In this case, they mean the same thing."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 1:11, Sal say \" A change in integer coordinates.\" so what does that mean like where the number spots are located? like in the grid pattern? So if I say on the slope of the line x 3 and y 1, would that be integer coordinates?\n\n-Thanks",
        "A": "he just means like whole numbers like 0,1,2,3,4 or any number that s easy to understand rather than fraction numbers and decimal numbers those can be hard to find on a graph some times like is easy to find the number 5 on a grid than it is to find 47/23."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "How did the negatves cancel out? (when you got positive 2/3 out of -4/6) 3:58 to 4:03",
        "A": "No, he had -4 and -6, so (-4)/(-6), can be written as (-1*4)/(-1*6) , -1 cancel out and we get 4/6, then you simplify and get 2/3"
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 2:58 sal names the slope -4 and -6. Why negitive and why not positive",
        "A": "if you are going down on the Y axis, it s going to be a negative number (if going up=positive number. For x if you are going to the left its going to be a negative number (going to right= positive)."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 2:58, why cant you go up and then over instead of going down and over? Would that make a difference to the answer?",
        "A": "You can go up and then over, as long as you still use the mathematical principle of rise/run. Hope this helps! :D"
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 1:45, what does delta mean?",
        "A": "Delta is a Greek letter and looks like a triangle. \u00ce\u0094 In math, it is used to represent change , among other things. For the slope of a line, the definition is the change in Y, over the change of X . In math terms, that looks like \u00ce\u0094y/\u00ce\u0094x."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 3:08 he says it is negative 4, why? is it because he went down 4, but if that was the case then why at 1:59 did he say positive 2?",
        "A": "I think you are asking two questions here. The rate of change is the same in both cases. The difference is he went in reverse so we are looking at a downward slope. thats why its -2/3. Its the same rate of change, what differ is the direction. @3:08 he is using the same rate of change only by a factor of 2. 4/6 or -4/6 is proportional to 2/3 or -2/3. Hope its not confusing."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At about 2:20, he says he could have picked any point. Are there more than one slopes in the line?",
        "A": "there is more than one slope on a line because you can put two points on a graph anywhere to get an x amount of slopes in the world of math."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "Does the order to finding x and y matter because at 1:24 he started with finding x but at 2:52 he started with finding y?",
        "A": "I don t think it does"
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "why can't I see anything that you draw or write? (0:00 to 4:40) I have this problem with almost all the videos",
        "A": "Ask your dad to check it out, the videos should be clear. Might be a problem with YouTube."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 1:19 he drew the arrow to the right. If he drew it to the left would the change in x be negative?",
        "A": "Yes it would but since this graph only has one quadrant, the only to draw it is to draw from left to right."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At about 1:45 Sal says that the triangle thing is a \"delta\" but what is a delta?",
        "A": "Chloe is right. It is the fourth letter of the Greek alphabet. What it means though in mathematics is: the change in . So for example delta x would mean the change in x which is also sometimes known as the run . Also, delta y would mean the change in y which is also known as the rise"
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "I'm a tad confused, at 3:57, He says that the negatives cancel out...why?",
        "A": "Firstly, when you divide a negative number by a negative number, it becomes positive. Same with multiplication. When you have a fraction, you are dividing two numbers. Since the numerator and the denominator are both negatives, they canceled each other out."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "3:25 - 3:38\nHow is the change in x equal to -6?",
        "A": "He went from right to left 6 units. A movement from right to left is a negative change. You could make it positive as long as you switch the sign of the change in y also."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 0:23, can it be :\nchange in y over change of x instead of change in x over change of y?\nWhy do we have to talk about the be y before x?",
        "A": "From what I understood of your question, it IS change in y over change in x. Slope is defined as rise/run. Lets say we wanted to find a slope of 2 lines, one is steeper than the other. Line A Slope is 6/6 as rise over run. Line B Slope is 3/6 as rise over run. If we were to do run over rise then Line A 6/6 Line B 6/3 The less steep line apparently has a higher slope than the steeper line, which does not make sense."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 2:00, they represented the slope in a fraction. How do you plot that again on the chart?",
        "A": "Is it a linear equation? If it is a point-slope formula, like y=mx+b with the fraction as m, then the denominator is how many spaces you would move across the x-axis and the numerator is the number of spaces you would simultaneously move along the y- axis. For example, if the slope has fraction is 1/2x then for every two spaces you move along the x-axis, you move up on along the y-axis. I hope this helped and makes sense."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "at 0:36 what does that mean the delta",
        "A": "Delta (\u00ce\u0094) means change in, or the difference between two values. For example if the temperature yesterday was 27\u00c2\u00b0C and today it was 32\u00c2\u00b0C (t\u00e2\u0082\u0081=27\u00c2\u00b0C and t\u00e2\u0082\u0082=32\u00c2\u00b0C), then the change in temperature from yesterday to today, the \u00ce\u0094t, would be 5\u00c2\u00b0C. \u00ce\u0094t=5\u00c2\u00b0C I hope I answered your question"
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 0:56, So you can choose any point on the graph or is there a guideline or formula?",
        "A": "You can choose any two points on the graph and then apply the formula Sal uses ( [Y2 - Y1] / [X2 - X1] )."
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 1:24 why do you use a delta?",
        "A": "To show change in X and change in Y"
    },
    {
        "video_name": "R948Tsyq4vA",
        "Q": "At 2:17, so if the slope is 2/3 that means the angle formed between the line itself and the x-axis is about \"0,66\"?",
        "A": "That would depend on whether the distance between the same changes ii value on the x and y axes is identical. That is not necessarily the case."
    },
    {
        "video_name": "RKBSX-6pKgY",
        "Q": "At 1:20, why does Sal say \"Two comma one\"? Why the comma?",
        "A": "Points on a cartesian plane can be written as (x, y), and reading that you usually say x comma y . It s just a way to read out load points."
    },
    {
        "video_name": "_E3gx_yQhHQ",
        "Q": "What if the square root didn't come out to a round number? Like at 2:44 what if the number wasn't 64 and it had a bunch of decimals? would you round or is there another step?",
        "A": "You would keep it as a surd, so for example I wanted to find the square root of 23. Well I can t so the answer would be \u00e2\u0088\u009a23. Because \u00e2\u0088\u009a23 x \u00e2\u0088\u009a23 = 23"
    },
    {
        "video_name": "_E3gx_yQhHQ",
        "Q": "At 2:04, Why did sal put 6 as a negative ? It's times 2 right?",
        "A": "Yes, it s times 2 but it s 2*-3. That is why it comes out to -6. There is a - in front of the 3."
    },
    {
        "video_name": "fvC0dm2wzIo",
        "Q": "At 8:42, Sal says \"if x were a google.\" What is a google? Is it a number? How big is it?",
        "A": "It s actually googol, and it is a pretty big number: 10^100. That is one followed by 100 zeros."
    },
    {
        "video_name": "D6yHKOYJiso",
        "Q": "Can the second method at 3:46 work for finding the LCM of 2 numbers??",
        "A": "Yes it can. You can use this method to find LCM of any number of numbers."
    },
    {
        "video_name": "D6yHKOYJiso",
        "Q": "at 5:00 what do you mean",
        "A": "Once you have determined the factors you need to make the Least Common Multiple, you then multiply them together to give the Least Common Multiple. What is the LCM of 4 and 9? 4 has factors 2*2. 6 has factors 2*3. The LCM would then need two 2s from 4, and since we already have 2s, we only need the 3 from 6. The LCM is 2*2*3=12."
    },
    {
        "video_name": "D6yHKOYJiso",
        "Q": "Until 0:50, why does Mr. Khan say that the process is the only solution to getting the least common multiple?",
        "A": "I think you may have misunderstood what he said. He does not say the process is the only solution to getting the least common multiple."
    },
    {
        "video_name": "vr0sTKbV7lI",
        "Q": "At 6:24, what do the square brackets mean?",
        "A": "The square brackets enclose the result of the expression just integrated. The numbers on the right hand side are the limits of integration. We had \u00e2\u0088\u00ab(0,2)dz. The integral of dz is just z, so we write [z](0,2) which reminds us to complete the integration by evaluating the expression inside the [ ] at the given values, thus [z](0,2) becomes 2-0=2."
    },
    {
        "video_name": "vr0sTKbV7lI",
        "Q": "Shouldn't it be the Commutative Property rather than the Associative Property at 3:17?",
        "A": "Yeah, certainly seems so by context. Got the point across anyhow, I s pose."
    },
    {
        "video_name": "MkFbisCyjFc",
        "Q": "At 8:47, since we know 2 sides of the triangle (technically 3) and a side (90 degrees, 45 degrees and hypotenuse=10), even though it might seem unintuitive, could we use AAS (Angle, Angle, Side) to figure out the x- and y-components?",
        "A": "Yes. You can use Pythagorean Theorem, Law of Sines, etc to find the other sides or angles, so long as you have the requirements. Most of the time you can use whatever you wish, or is easiest for you."
    },
    {
        "video_name": "MkFbisCyjFc",
        "Q": "At 5:50, Sal wrote said that x and y component is equal to 2.57 and 3.06 respectively. However, when you divide 3.06 by 2.57, how come you do not end up with 4?",
        "A": "4 is the magnitude of the vector. so if you have seen the previous lesson on magnitude of the vectors, in order to find the magnitude of the vector, you use Pythagoras Theorem (i.e. square root of 2.57^2 + 3.06^2). Then you will get 4 :)) (approx. 4 because the answers are actually rounded to hundredth)"
    },
    {
        "video_name": "10dTx1Zy_4w",
        "Q": "At 0:30, isn't that called diameter or radius?",
        "A": "It is called the diameter. The radius is from the center of the circle out."
    },
    {
        "video_name": "10dTx1Zy_4w",
        "Q": "at 1:14 it explains how triangles always have three sides so that means anything that has three sides is a triangle,\nright?",
        "A": "Yes, as long as all the sides are connected."
    },
    {
        "video_name": "10dTx1Zy_4w",
        "Q": "In 0:28, Sal mentions a line drawn from one point of a circle to another through the center has the same length as another line drawn through the circle (starting and ending at different points). What is this line called?\nAt 0:44, Sal draws a shape and calls it an oval. Isn't it an ellipse? Is there any difference between them?",
        "A": "In 0:28, this line is called the diameter of a circle. And oval can be an ellipse ,but not always."
    },
    {
        "video_name": "K_na_j1Msfg",
        "Q": "At 8:35 (ignoring the lack of negative number) you had B={v_1, v_2}, the way basis work could you swap the set around and have B={v_2, v_1} instead?\nOr does it matter which vector comes first because of the 2D graph shown at 2:35?",
        "A": "You are allowed to change the coordinate system however you like. The matrix D will be slightly different, but it should still be pretty simple."
    },
    {
        "video_name": "PL9UYj2awDc",
        "Q": "1:45 Sal says that because 8 cancelled out each other 3 was alone, but before then Sal multiplied 3 of 3/4, I'm rather confused.",
        "A": "Your question is a little vague. Are you asking why 8 * 3/4 doesn t become 3? Here s the steps: Change 8 into 8/1 Multiply 8/1 * 3/4 = 24/4 Reduce the fraction = 6 Sal is basically doing 8/4 = 2. Then, 2 * 3 = 6. The 4 does not completely cancel out the 8. If this isn t what you were asking, please clarify back in a comment and I ll try to help."
    },
    {
        "video_name": "PL9UYj2awDc",
        "Q": "At 0:29, How do you know whether you add or subtract and what do you add and subtract WITH?",
        "A": "When you have the equation 3/4x + 2 = 3/8x - 4, you can group the x terms and the constants into each side of the equal sign: 3/4x + 2 = 3/8x - 4 3/4x - 3/8x + 2 = 3/8x - 3/8x -4 (subtract 3/8x from both sides to keep it equal) 3/4x - 3/8x + 2 = -4 3/4x - 3/8x +2 - 2 = -4 - 2 (subtract 2 from both sides to keep it equal) 3/4x - 3/8x = -4 - 2"
    },
    {
        "video_name": "PL9UYj2awDc",
        "Q": "At 1:55, how is the 4 in 3(3/8x-4) a *negative four*?",
        "A": "Subtracting is the same as adding a negative. So, the minus in front of the 4 indicates that it is negative."
    },
    {
        "video_name": "PL9UYj2awDc",
        "Q": "At 1:32, Sal says to eliminate the fraction by 3 times 8 over 4. Can't you do it as 8 over 1 times 3 over 4? Or will it be wrong?",
        "A": "They re essentially the same way of doing the problem, since you don t need to change the denominator. You can do the problem whichever way you re most comfortable with, but they might just have not wanted to add another step to the video."
    },
    {
        "video_name": "PL9UYj2awDc",
        "Q": "How do you know when to subtract? at 2:14\nYou can also add but is there a rule of when to add or subtract the variables?",
        "A": "You may be less confused if instead of thinking about subtraction, you think of it as ADDING a negative number. When you re trying to eliminate a term from one side of the equation, you simply ADD the opposite of that term to both sides of the equation (following the rules for the addition of signed numbers, of course). So, if you want to eliminate a term of + 3x, you need to add - 3x to both sides. If you want to eliminate a term of - 3x, then you need to add + 3x to both sides. Hope this has helped!"
    },
    {
        "video_name": "PL9UYj2awDc",
        "Q": "3:16 x minus -16?",
        "A": "please help me explain how to find the product of 13 X9 by using the distributive property."
    },
    {
        "video_name": "PL9UYj2awDc",
        "Q": "At 3:10 He says divide by \u00c2\u00a83\u00c2\u00a8 is it because there is 3x? For example if I had 2x then I have to divide by 2? or I always have to divide by 3?\n\n\n-Edit: Alright never mind I found out that is true for example if I had 6x then I have to divide by 6 or if I had 4x then I have to divide by 4. :P\n\n\nI\u00c2\u00b4m so smart :)\n\n\n\n(sarcasm)",
        "A": "If you found out the answer, then why bothering to ask this question?"
    },
    {
        "video_name": "PL9UYj2awDc",
        "Q": "At 3:07, instead of canceling out the 3x can you cancel out the 6x?",
        "A": "Yes, but then I would cancel out the -32 instead of the 16 so that you have the x and the constant (the non-x side) on opposite sides of the equals. If you cancelled out the 3x you d get this: 16 = -3x - 32 then this: 48 = -3x then this: x = -16 So, you get the same solution, just with different steps."
    },
    {
        "video_name": "PL9UYj2awDc",
        "Q": "i didnt understand how he get the 8 0:48",
        "A": "He multiplied both sides of the equation by 8 to get rid of the fractions."
    },
    {
        "video_name": "PL9UYj2awDc",
        "Q": "At 0:49, Sal multiplied both sides of the equation by 8, then every number in the equation was multiplied by 8. Is it always like this?",
        "A": "Yes, remember, what you do to 1 side of the equation you have to do to the other side of the equation so that the equation stays the same. He picked 8 because it was the least common multiple of 4 and 8 and because it would get rid of the fractions. Yes, you are right, you multiply 8 to both sides of the equation and to every number in the equation. This is always the case."
    },
    {
        "video_name": "PL9UYj2awDc",
        "Q": "At 0:54, can someone elaborate on what he means and the reason why he chose 8?",
        "A": "We want to get rid of the fractions. It just makes the equation easier to work with. We choose 8 because it s the least common multiply of 4 and 8. Multiplying it out, we get 24/4 for the first one, which simplifies down to 6. And we get 24/8 for the second one, which goes down to just 3."
    },
    {
        "video_name": "PL9UYj2awDc",
        "Q": "why is 8 the least common multiple and not 2 or 4.they are both smaller.1:01",
        "A": "2 and 4 both divide into 4&8?"
    },
    {
        "video_name": "Uz0MtFlLD-k",
        "Q": "At 6:50 the video relapsed itself and sal repeated the end again.",
        "A": "Sal repeats himself sometimes so is fine nothing to worry about. (if it s not that then it s your computer or browser)"
    },
    {
        "video_name": "Uz0MtFlLD-k",
        "Q": "at 3:35, Sal says the relation is no longer a function, but what is the proper title for a relation that is not a function?",
        "A": "It is just called a relation or a relationship. There are some specific relations that have specific names, but to be more specific than saying a relation you need to know the details. If the relationship meets the right details, it can be called a multifunction (which should not be confused with an ordinary function)."
    },
    {
        "video_name": "Uz0MtFlLD-k",
        "Q": "at 1:48, how did he give the 1 and the 2 in the domain to the same range? Does that still make it count as a function?",
        "A": "yes it is. You have a snack machine. When you press 1 you get chips when you press 2 you also get chips. This means it is a function (it s always a function if all of the x-values are different). Now say you went to a different snack machine. It has two buttons labeled 1 and one of them gives you pepsi and the other one gives you coca-cola, you wouldn t be happy if you wanted a coke and it gave you pepsi. Since there are two x-values it s not a function. Hope it helps!"
    },
    {
        "video_name": "Uz0MtFlLD-k",
        "Q": "At 6:40, what happens if you have a repeat of an ordered pair? Ex. (2,4), (0,0) (2,4)",
        "A": "You wouldn t ordinarily list the same ordered pair more than once... if you think of a function as a set of ordered pairs, then one ordered pair is either in it or it isn t. For a relation not to be a function, you would need (2,4) and also (2, 3), or 2 as an x-coordinate followed by a different y-coordinate."
    },
    {
        "video_name": "Uz0MtFlLD-k",
        "Q": "At 3:10, 1 is mapped to 2 and 4. This you say is impossible for a function. But why not just: f(x) = x+1 V x+3? (where V is the logical symbol for OR) Seems like a function to me.",
        "A": "No it is not, by definition a function must have a unique (one and only one) output for each input. i.e. there must be no ambiguity when an input goes into a function as to what the outcome is"
    },
    {
        "video_name": "Uz0MtFlLD-k",
        "Q": "exactly at 4:00 and after why one can't be associate with 2 and 4. I 'd say it can be associated with 2 and 4 providing condition. And i'd request you to any of you to explain it.",
        "A": "In this case, Sal is saying that for it to be a function, each part of the domain goes to one, and only one member of the range. Because if you were to graph something where one associates with both two and for, the graph would curve back in on itself."
    },
    {
        "video_name": "Badvask-UDU",
        "Q": "what does he mean by distributing.(1:30\n)",
        "A": "it means to spread it out for"
    },
    {
        "video_name": "Badvask-UDU",
        "Q": "At 1:34 how did you do the steps like that? i did this a different way in school.",
        "A": "5(9-4) 5x9-5x4 45 - 20 = 25 or 5(9-4) 5x9+5x(-4) 45+(-20) = 25"
    },
    {
        "video_name": "hpBBuaiIkrg",
        "Q": "1:40\nCan you have several local Maximum and minimum points in a function",
        "A": "Sure! Just look at f(x)=sin x or g(x)= cos x. They have infinite numbers of local extrema (but no absolute extrema)."
    },
    {
        "video_name": "hpBBuaiIkrg",
        "Q": "1:40\nCan you have several local Maximum and minimum points in a function",
        "A": "yes. like sin or cosine functions"
    },
    {
        "video_name": "hpBBuaiIkrg",
        "Q": "at 1:42, what is a \"local minimum/maximum\" point versus a \"global minimum/maximum\" point?",
        "A": "A local minimum is a point where changing x by a tiny amount in either direction causes f(x) to increase. However, there can be an unlimited number of local minimums, and in between each pair of local minimums there is a local maximum. However there can be only one global minimum, which is the uniquely lowest point the function ever returns."
    },
    {
        "video_name": "hpBBuaiIkrg",
        "Q": "@1:40 Can you have several local Maximum and minimum points in a function?",
        "A": "Yes. There are some videos on the site about local minimum and maximum points. Local minimums and maximums happen when you have wavy-looking functions. The bottoms and tops of the waves would all be local minimums and maximums."
    },
    {
        "video_name": "MeU-KzdCBps",
        "Q": "At 6:02, the video stated that \u00e2\u0088\u0086Y/\u00e2\u0088\u0086X is equivalent to 1. But is the 1 for the blue line or the magenta line?",
        "A": "The magenta line has a slope = 1 The blue line has a slope = 2"
    },
    {
        "video_name": "MeU-KzdCBps",
        "Q": "At 4:48 why is delta a triangle symbol?",
        "A": "The Delta is a letter in the greek alphabet. They use it to show the change in because in Greek it is the first letter in change."
    },
    {
        "video_name": "MeU-KzdCBps",
        "Q": "At 1:51, how did you find the slope?",
        "A": "You can count the rise over run, or use the formula of m= y2 - y1 / x2-x1. Just plug in the numbers for the pairs."
    },
    {
        "video_name": "qIBqvr5_QqQ",
        "Q": "At around 1:30, Sal mentions that z subscript xy = z subscript yx. Is this property called Clairaut's Theorem?",
        "A": "That is Clairaut s Theorem. It states that if the partial second derivatives exist and are continuous, then the partial second derivatives are equal. This extends to higher order differentials as well."
    },
    {
        "video_name": "qIBqvr5_QqQ",
        "Q": "at 2:40 , where did the unit normal vector go? or doesn't it matter?",
        "A": "Sal just forgot the vector sign over the dr. Because obiously it should be a scalar product and you need two vetors in order to compute a scalar product."
    },
    {
        "video_name": "O64YFlX1_aI",
        "Q": "3:38 What video explains \"factor it out by 5\"?",
        "A": "Literally, it means to divide it by 5, since it s clear that it is divisible by 5."
    },
    {
        "video_name": "O64YFlX1_aI",
        "Q": "in 2:45 i was confused how it was A+B=C",
        "A": "What Sal means is that A, B, and C are variables. The are letters that replace an empty area of an equation. Not everyone will use the same numbers to replace the variables, so the answer to the equation will vary. He just means that the answer depends on what numbers are used to replace the variables. Hope this helps!"
    },
    {
        "video_name": "O64YFlX1_aI",
        "Q": "At 1:52 is it ok to put the 9 squared first then put the a squared or does it have to be in order like hes doing?",
        "A": "When you re adding you re allowed to change the order, so in this case it would be okay to put 9 squared first."
    },
    {
        "video_name": "O64YFlX1_aI",
        "Q": "4:15, you said the actual distance is between 10 and 11, isn't that mean that pythagorean theorem isn't efficient if it can't give us the length of every right triangle , that tell us that we can only find the actual distances of number that has perfect square roots?",
        "A": "Between 10 and 11 is an approximation. The Pythagorean Theorem gives you the exact value, which may be an irrational number. Since those numbers have decimal expansions that neither terminate nor repeat, we have a tough time reasoning out exactly what they are in our heads, but we can always get a decent approximation."
    },
    {
        "video_name": "O64YFlX1_aI",
        "Q": "At 3:19 when he draws the two square-root signs over them individually, can't you do it over the whole equation to make it easy, or is that incorrect",
        "A": "You cannot do that. If you are going to do something to one side of an equation/inequality, you MUST do it to the other side to obtain the correct answer. So, a square root can be put on both sides, but not over the whole equation. Putting it over the whole equation is a good way to think about it, though."
    },
    {
        "video_name": "3GL69IA2q4s",
        "Q": "At 2:24, when he's canceling out the y factors, why doesn't this produce y^-1 ?\n\nI thought y^1 / y^2 = y^-1",
        "A": "Yes, y^1 / y^2 = y^-1 which is the same as 1/y. As a result, the y on top was factored out and we re left with one y on the bottom."
    },
    {
        "video_name": "3GL69IA2q4s",
        "Q": "At 8:30, how do you multiply rational expressions with exponents bigger than 2?How do you simplify the expressions by dividing exponents bigger than 2?",
        "A": "Use the distributive law and the laws of exponents."
    },
    {
        "video_name": "3GL69IA2q4s",
        "Q": "At 2:20 why you didn't subtract the exponents of y/y^2 and get y^-1?",
        "A": "To simplify the division of two exponents THAT HAVE THE SAME BASE you can subtract the powers. So y/y^2 = y^(1-2)"
    },
    {
        "video_name": "obZzOq_wSCg",
        "Q": "0:55 why did Sal write J, K, Q shouldn't it be Jack, Queen, King?",
        "A": "He also maybe wrote that because the letters for Jack s, Queen s, and King s came up in his head in that order quickly, so he wrote J, K, Q."
    },
    {
        "video_name": "obZzOq_wSCg",
        "Q": "At 8:10 in the video he said we are counting the jack of hearts twice. Why is that? It doesn't make sense.",
        "A": "We counted all the cards that were jacks, which included the jack of hearts. Also, we counted all the cards that were hearts, which meant we counted the heart with a jack. A heart with jack is the same as a jack of hearts. Then we have counted the jack of hearts twice! Do you see why?"
    },
    {
        "video_name": "obZzOq_wSCg",
        "Q": "At around 8:24, why does Sal subtract 1 and not 2 in the last example?",
        "A": "Because you only have to subtract one card, not two."
    },
    {
        "video_name": "obZzOq_wSCg",
        "Q": "At 9:55 I noticed something interesting: When Sal simplified the fraction, it was 4/13. Notice this: the smaller circle in the Venn diagram was 4, and the numerator of the fraction was 4. Similarly, the bigger circle was 13, and the denominator was 13. So I was wondering if you could use that method to work with these type of problems. Is this possible?",
        "A": "I m pretty sure that was just a coincidence. It might be true for a certain subset of problems, but it is certainly not true in the general sense."
    },
    {
        "video_name": "DkZnevdbf0A",
        "Q": "at 1:49 the ray has 2 points but it still goes through so it is still a ray?",
        "A": "at 1:49 the ray has two points but one of the points is not its end point"
    },
    {
        "video_name": "DkZnevdbf0A",
        "Q": "At 1:54, why is the line only have one end point because there are 2 dots?",
        "A": "Both dots are used to pull the line around"
    },
    {
        "video_name": "MufbvU4tGh8",
        "Q": "at 0:36 sometimes he is hard to understand with his accent can I change this\n\n\n\n\n\n\n\n\n\nhelp me!",
        "A": "Try using subtitles if ur having trouble understanding."
    },
    {
        "video_name": "u00I3MCrspU",
        "Q": "0:25 so do i have to watch some other video first before this one because where before in the chapter om matrices did he mention the + - + rows?",
        "A": "No, this is the first time he introduces us to this concept. Don t worry, just remember the pattern and you ll be fine."
    },
    {
        "video_name": "N8dIOmk_lHs",
        "Q": "At 0:43, the narrator says that you can find the prime factorisation for these numbers. How can you factorize a fraction?",
        "A": "Sal is factorizing the denominators only."
    },
    {
        "video_name": "N8dIOmk_lHs",
        "Q": "At 3:14, I didn't understand how and why did you get 20?",
        "A": "Because you re trying to get the denominator to equal 180 and 9 x 20 is 180. You multiply the top number by that as well because what you do to the bottom, you also have to do to the top."
    },
    {
        "video_name": "VTlvg4wJ1X0",
        "Q": "I'm totally confused when the video gets to 2:34... can anyone help me make any sense of that part?",
        "A": "Well, the expression on the right can be solved by taking square roots, and if you understand the first part of the video, solving it is simple. The left side is simply saying that a function of x, f(x) is equivalent to the expression on the right. Later, it says that the function of x is equivalent to y on a graph. The question asks at what x values does y intersect the x axis. This is asking what x equals when f(x) = 0."
    },
    {
        "video_name": "VTlvg4wJ1X0",
        "Q": "So, would there be an actual way to figure out only one answer. 2:21",
        "A": "No. This is because there are two numbers that satisfy the equation. Leaving one out would mean the solution is not complete."
    },
    {
        "video_name": "VTlvg4wJ1X0",
        "Q": "Wait, at 3:17, he draws the f(x) function and it is curved. So does that mean every function is curved?",
        "A": "No... different functions create different shapes. For example: Quadratic functions like the example in the f(x) in the video create a U-shape called a parabola. A Linear function creates a straight line. An absolute value function will create a V-shape"
    },
    {
        "video_name": "VTlvg4wJ1X0",
        "Q": "At 2:04, he said that x = -1 and -5, but when I did it, I got x = 1 and 5. I looked back, and it looked like I didn't make a mistake. What happened?",
        "A": "3x+3(1-2x)=4-3(x+5)"
    },
    {
        "video_name": "AXasQLqazWU",
        "Q": "At 4:58, how come did he move four and a half?",
        "A": "Sal is trying to pickup another point in the tangent line to draw it properly. The slope is 1/9, so if you move 1 to the right along x axis, you need to move 1/9 up in the y axis. He moved 4.5 to the right along x axis so he needs to move 0.5 (4.5*1/9 = 4.5/9 = 0.5) up along y axis. The dot 4.5 right and 0.5 up was chosen because it s easier to plot than the dot 1 right and 1/9 up."
    },
    {
        "video_name": "oXUJXsoCtDs",
        "Q": "at 7:20 is it not just -1/n rather than what Sal said about it being an alternating series?",
        "A": "Sal is right. (-5)^n = [(-1) * (-5)]^n = (-1)^n * 5^n, by using the power rules."
    },
    {
        "video_name": "aoXUWSwiDzE",
        "Q": "At 7:31,how do you know to know to do the exponents or to simplify first?",
        "A": "You should probably do the exponents first, and then simplify."
    },
    {
        "video_name": "aoXUWSwiDzE",
        "Q": "At 9:38 the cost of 2000 remotes why don't you double the answer of problem a?",
        "A": "Because the Cost of Production = 2000 + 3 x The number of remotes made. 3x means each remote costs 3 dollars to make. The 2,000 is the $2,000 you put into making remotes in the first place. Think of it as if it were the startup cost for your Remote Control factory. So when you just multiply the answer of how much it costs to make 1,000 remotes you are including that 2,000 startup cost twice."
    },
    {
        "video_name": "aoXUWSwiDzE",
        "Q": "At 9:45 Sal says that to make one thousand remotes the company uses 5,000 dollars. But then he said that to make 2,000 remotes the company uses 8,000 dollars. Shouldn't it be 10,000 dollars because it is double the amount of remotes?",
        "A": "No. Because the equation shows a flat weekly fee of $2000 on TOP of the number of remotes made. This fee is unchanging regardless of the number of remotes made. Which will vary your answer depending on the number of remotes made."
    },
    {
        "video_name": "aoXUWSwiDzE",
        "Q": "At 4:52 what does daunting mean",
        "A": "Daunting is simply a word we use to describe something that looks a little intimidating or scary. In this case, Sal used it to describe a problem that looked a little more complicated than the others he had explained."
    },
    {
        "video_name": "aoXUWSwiDzE",
        "Q": "@10:05 the answer to question 21b is 8000, but the answer to 21a is 5000. Since 1000 remote's cost is = 5000, shouldn't the cost of 2000 remotes be = to 10,000? Logically this doesn't appear to make sense. I mean using their formula it does, but in a practical back measure it doesn't.",
        "A": "Interesting, thank you. :)"
    },
    {
        "video_name": "aoXUWSwiDzE",
        "Q": "At 6:30, he added -3-8 and 15-8 before he divided. I thought in order of operations that you do division before addition and subtraction. I'm hoping you can help me out on this.",
        "A": "When you have a problem with a fraction where there s addition in the numerator and/or denominator, you treat it as if there are parenthesis around the numerator and denominator. The fraction is the result of the numerator divided by the result of the denominator."
    },
    {
        "video_name": "aoXUWSwiDzE",
        "Q": "I didn't understand, at 8:00 why did Sal cross out the -1 squared and didn't include it in his final equation?",
        "A": "Any negative number squared is just its positive squared, so -1 ^ 2 = 1. As you know, multiplying 1 to anything does not change it, so Sal crossed it out to simplify the expression."
    },
    {
        "video_name": "aoXUWSwiDzE",
        "Q": "at 6:55 we simplify through subtraction before we deal with PEDMAS... why?",
        "A": "So basically you can always simplify like things first before plugging in a number and the answer will be the same when you re done as it would have been the long way, regardless of what x is?"
    },
    {
        "video_name": "aoXUWSwiDzE",
        "Q": "At 10:51, shouldn't he had subtracted 10 - 2 first, then square the answer first instead of multiplying 4 and 2 because of proper order of operations?",
        "A": "I see your question. It will work either way."
    },
    {
        "video_name": "aoXUWSwiDzE",
        "Q": "During problems at 10:14 he naturally assumes that X is in parenthesis when doing 4x. Is this correct? I understood it to be anything in parenthesis first then exponents then M/D then A/D, is this wrong?",
        "A": "You know, I think it might be wrong. The answer is logical, but why would he do that? I don t see any exceptions."
    },
    {
        "video_name": "aoXUWSwiDzE",
        "Q": "At 7:31, how do you know to know to do the exponents or to simplify first?",
        "A": "Because of the order of operations, PEMDAS parenthesis exponents multiplication division addition subtraction."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "can you turn pi into a scientific notation? 3:00",
        "A": "pi is an irrational number, so the decimals have no end, this would make it impossible to write it in scientific notation."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "What is Avogadro's number 0:57",
        "A": "It is 6.022x10^23. It s basically a constant that is used a lot in chemistry. It is the number of atoms or molecules in 1 mole of a substance."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 00:56, Sal mentions \"Avogadro's Number.\" I saw that someone posted a very long explanation of what it's used for, but does someone just have a quick summary? I'm curious because I love chemistry and he said that's when it's most often used.",
        "A": "Avogadro s number (6.0221413e+23) represents the number of atoms or molecules (constituent parts) in one mole of a substance. For example, it takes 6.0221413e+23 molecules of hydrogen gas to make one mole of it, which is approximately 1.2 grams. And, it takes 6.0221413e+23 molecules of water to make one mole of it, which is around 18.02 grams. A mole is a term used in chemistry that is kind of like a dozen. As a dozen is twelve, a mole is 6.0221413e+23."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 12:05, why does Sal say \"to the minus 1\"? Shouldn't he say \"to the negative 1\" ?\nAren't the exponents in that situation negative?",
        "A": "I would say you are correct, it would have been more formal for him to say to the negative 1 ."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "Why is it that at 18:31 the 40 can stay as a two-digit number? Is it true that in scientific notation the number that you are multiplying to the power of ten has to be one digit?",
        "A": "Looks like Sal didn t quite finish that problem. It is correct as far as Sal took it. However, he should have converted the answer to correct scientific notation. 40 x 10^(-7) = 4 x 10^1 x 10^(-7) = 4 x 10^(-6)"
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 4:23 How did 10 to 100 becam Googol? The big question what is Googol? it may be in the video but I have hearing problems. Would you explain? Thank You.",
        "A": "Googol is a huge, huge number. From what I understand, it is the number 1 followed by one-hundred zeros. If I had a bunch of time to waste, I d type it out for you, but unfortunately I don t. Hope this helped!"
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "How can anyone possibly know how many atoms are in the known universe? ( 4:35 )",
        "A": "It s a figure of speech. Don t read too much into it. (It s a very rough estimate)."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "at 18:32 Sal finishes a problem as 40x10^-7\ncould that be simplified to 4x10^-6 ?",
        "A": "Yes, 40 times 10 to the -7 is the same as 4 times 10 to the -6. Also, as an aside, when using scientific notation on real measurements, it is quite nice to take the exponent to a factor of 3, because they have names . For example, instead of saying 40 time 10 to the minus 7 meters, you would say 4 micrometers."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "What does Sal mean when he says \"whats the largest power of 10 that fits into 6\" at 10:50. Also how come its only 1, what does that mean? Can someone please clarify.",
        "A": "It is supposed to be 0. 10 to the power of 0 is 1. 6 is more than 1. So, 6 can be expressed as 6 x 10^0"
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "Around 20:5 when you were adding 10-23 and 10-22 why did you write it as 10^1 i don't get that bit because before you say add the - so wouldn't you add -23 and -22 to make -45? but you wrote 10^1",
        "A": "You have a sign error. The 1st number is 10^23 (positive 23, not negative 23) So, 10^23 * 10^(-022) = 10^1 or just 10 23 + (-22) = 1"
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 12:56, Sal says that negative powers of 10 are not the number of 0's, like .001 is 10^-3. But if you did it like:0.001, you could do the number of 0's. Am I wrong? Please someone comment.",
        "A": "You are correct, because it is the number of places you move the decimal."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 8:41 Sal says that the Avogadro's number has 20 zeros and Googol has 23 zeros but he writes it 60.22x10^23 so won't the number be 6022 with 23 zeros instead of 20 zeros?",
        "A": "A Googol has 100 zeroes...."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 9:07, is Sal saying that 6.022 x 10^23 will give you 6022 followed by 23 zeros? Isn't it supposed to be followed by 20 zeros because the decimal will shift 3 times for the .022?",
        "A": "I don t think he is saying that, I think he is saying that it will be 6.022 with 23 filled out decimal places. However, you logic is correct, in 6.022 x 10^23 , there will be 20 zeroes (20 decimal places filled by zeroes and 3 filled by 022)"
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "14:03 couldn't that be 70 to the -6",
        "A": "No, because the number has to be more than 1 but less than 10"
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 18:48, he explains how to multiply scientific notation and says to add the exponents. My teacher at school said to minus the exponents! I' m a little confused by this part, do I add or minus the exponents? (and yes both my bases are ten so I can do the exponent trick he mentioned in the video. Thank You!",
        "A": "Your math teacher is right, and so is the man. He s just saying minus instead of negative . He s still telling you to add, but what he s saying is to add the minus 3 and the minus 4, instead of add the negative 3 and the negative 4. The answer is still negative 7 whether one says negative 7 or minus 7."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "@16:36 shouldn't there be a minus sign? I.e 8.192 x 10^-10?",
        "A": "There already is a minus sign there. It already says 8.192x10^-10"
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "how do you do the part in the 13:23",
        "A": "thanks for the help it helped me a lot"
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "at 18:32 Sal finishes a problem as 40x10^-7\ncould that be simplified to 4x10^-6 ?",
        "A": "I believe so. Good eye! I didn t catch that."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 15:30, when Sal writes 5.16x10^-5, could you also write that as 516x10^-7?",
        "A": "You could... but then it would not be in scientific notation. 5.16x10^-5 is in correct scientific notation because 5.16 has only one digit to the left of the decimal point and that digit is not a zero. Your version has 3 digits to the left of the decimal point (it can only have 1 digit to be in scientific notation)."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "3:40 beat a dead horse?",
        "A": "beating a dead horse is an expression that means to rehash the same thing over and over. It is pointless repitition."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 4:41 how did a googol get its name?",
        "A": "the people who made it thought what should we call this number they ended up making it up while voting on letters"
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 8:40, Sal says you immediately know how many \"zeros\" there are. Shouldn't he have said \"you immediately know how many digits before the first there are\"?",
        "A": "Benjamin, Good point. You are correct."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 8:23 in the video , he puts avagodros number into a decimal . When he multiplies it by 10 to the 23 power , your just moving the decimal 23 places so wouldn't you just have a number less than that?",
        "A": "Not really. There are 24 digits in Avogadro s Number. If the 6 is already in the one s place, then when you move the decimal point 23 places, you get 23 digits plus the 6. Ask if it isn t clear, k? ;)"
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "So at 20:15 what is the actual answer when you multiply 6.022 * 7.23? I'm a bit confused on how I should be multiplying the numbers. I should also mention that I'm not using a calculator to figure the answer out either.",
        "A": "6,022*723 then move the decimal once finished"
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 4:08 Sal says aside instead of a side",
        "A": "Yes, he does."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 11:14, how can you express 0 in scientific notation or is it not possible? My idea is the latter. Thx",
        "A": "You re right, it s not possible. 0 doesn t need scientific notation."
    },
    {
        "video_name": "trdbaV4TaAo",
        "Q": "At 1:30, why is the number so long?",
        "A": "There are things that require very large number to quantify. Avogadro s number is used in physics & chemistry to measure / count very small atoms / molecules that make up a unit of measure called a mole. The mass of Earth or the Sun are other very large numbers."
    },
    {
        "video_name": "FXZ2O1Lv-KE",
        "Q": "Do your sample sizes have to be the same size? E.G, at 1:05(ish) there are a bunch of samples with a sample size of four. Would it mess up any calculations if you took a sample of four and then, say, a sample of ten?",
        "A": "Yes, the sample sizes should be the same. The sample size is not considered to be a variable, it s considered to be a constant. The sampling distribution of the sample mean can be thought of as For a sample of size n, the sample mean will behave according to this distribution. Any random draw from that sampling distribution would be interpreted as the mean of a sample of n observations from the original population."
    },
    {
        "video_name": "FXZ2O1Lv-KE",
        "Q": "what does the y-axis represent in the graph of unknown distribution (at 3:30 in video, below the text \"Paint distribution with mouse\")?? Does it represent the frequency of each random variable??",
        "A": "Yes, as per my interpretation, the y-axis does represent the frequencies of the corresponding values for the random variables. Note that there is a single random variable with multiple possible values (the possible values are represented by the x-axis)."
    },
    {
        "video_name": "3UlE8gyKbkU",
        "Q": "What is the formula for determining the sample space (for this question) ?\nLike there were 3 tosses, and possible outcomes (at ~0:50).\nIt must have something to do with factorial, although I don't quite know what.",
        "A": "(possibilities per event) to the power of (number of events) there are 2 possibilities (head or tails) and 3 events (coin tosses) so 2^3 = 8"
    },
    {
        "video_name": "3UlE8gyKbkU",
        "Q": "At 0:22 is there a difference between HHT and HTH? And why didn't he list THH as a possible outcome?",
        "A": "He did, later in the video."
    },
    {
        "video_name": "bWTtHKSEcdI",
        "Q": "at about \"7:30\" what is the definition of similar triangles",
        "A": "the corresponding angles are the same and the ratio of the corresponding sides are the same"
    },
    {
        "video_name": "bWTtHKSEcdI",
        "Q": "Do the arrows on line segments AC and BE at 9:10 represent parallel line segments?",
        "A": "Yes, the arrows on line segments AC and BE do denote parallel line segments."
    },
    {
        "video_name": "bWTtHKSEcdI",
        "Q": "3:32 how does he know it's a square?",
        "A": "The definition of a square is a rectangle with two adjacent equal sides a quadrilateral with four equal sides and four right angles a parallelogram with one right angle and two adjacent equal sides a rhombus with a right angle a rhombus with all angles equal a rhombus with equal diagonals From Wikipedia, the free encyclopedia"
    },
    {
        "video_name": "bWTtHKSEcdI",
        "Q": "Wow, But how does he know at 1:25 that triangle TRP and triangle TAP are congruent?I don't under stand problem 12 at 3:34, Can I have some explanations about problem 12 please ?It would help me out a lot if I could get some explanations for problem 12.Is there also a site that will help me know more about this?",
        "A": "PAT is the same as TAP. Both triangles consist of the same sides, meaning that they are the same. They don t have even one different side or angle."
    },
    {
        "video_name": "Pb9V374iOas",
        "Q": "Is there any significant reason why Sal says \"d'th power\" at 2:12?",
        "A": "I think because he is using a variable called d"
    },
    {
        "video_name": "Pb9V374iOas",
        "Q": "At 3:25, how can a^b^d power become a^bd?",
        "A": "If we also use this with 2, and we have (2^4)^3: We have (2x2x2x2)^3, which is also (2x2x2x2) (2x2x2x2) (2x2x2x2), which equals 16 (for each set) to power 3, which is 4096, or 2^12."
    },
    {
        "video_name": "Pb9V374iOas",
        "Q": "At 2:12 why did u take the equation to the D power ?",
        "A": "Both sides of the equation were taken to the D power. It doesn t change anything because if you raise two equal quantities to the same power, they remain equal. It was done to prove the result."
    },
    {
        "video_name": "uftnBXB98l8",
        "Q": "I'm confused. You are saying, 1:43, \"combine all the people that are on the graph.\" Shouldn't you be saying, \"count all the people whose color is represented on the graph\" or something to that effect? Thanks for any clarification.",
        "A": "we are not counting were adding"
    },
    {
        "video_name": "vRa6XxykfbY",
        "Q": "At 1:08 Sal said that he wont check it because the opposite of E is A and he says that he wont check it because it doesn't say that. Even if it did say that, it's asking \"what can we say about point B\" so it would still be considered incorrect right?",
        "A": "It is actully not."
    },
    {
        "video_name": "vRa6XxykfbY",
        "Q": "What is a hash mark at 0:19 and 0:33?",
        "A": "Good Question! Hash marks are used for many things. They can be used to mark a number on a number line. Like the video, you are watching. They are used a lot when trying to show congruent angles and sides of shapes."
    },
    {
        "video_name": "xBseQOzMB7A",
        "Q": "At 4:19 he multiplied the denominators I thought we don't?",
        "A": "If we are multiplying fractions, we multiply the denominators. I think you are confusing multiplication with addition. When we add fractions, we don t add the denominators."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At around 5:25, why didn't Sal multiply the 2.83 by 4?",
        "A": "Because he cancelled the 4 by the 2 in the denominator which gave him 2 times the square root of 2."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At 2:30. I don't understand how sin30* is equal to 1/2. Please explain? Thanks!",
        "A": "The trig value; sine, cosine, or tangent, of an angle is the shape of the right triangle with that angle in it. Sine = 1/2 means that one side is half as long as the other. Which 2 sides? Just make sure you re thinking of the triangle and the angle formed in the unit circle, a critical tool. Don t just imagine the triangle in space. Specifically, for sine, this means that the vertical leg is half the length of the hypotenuse."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "What does Sal mean by \"unit circles\" around 2:24 ?",
        "A": "A unit circle is a circle defined by radius = 1 (unit = 1). This is a very important circle in understanding trigonometry."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At about 5:02, when the calculator is open, Sal takes 2sqrt(2) to get about 2.83, then proceeds to say that b is approximately equal to 2.83. However, in the equation we were given, b=4(2sqrt(2)) (simplified from 1/4=sqrt(2)/2 / b). Would Sal have to multiply the 2.83 by 4 to get the value of b, or am I missing something?",
        "A": "You made a silly mistake. Let s try solving the equation again. 1/4=sqrt(2)/2/b => 1/4b=sqrt(2)/2 => b=4(sqrt(2)/2) => b=2sqrt(2). You accidentally changed the /2 into a *2."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At 5:17 why didn't he multiply 2\u00e2\u0088\u009a2 by 4? Do you drop that number?",
        "A": "Its 4sqrt(2)/2, which is 2sqrt(2)."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At 5:46, Sal says that, given two sides and an angle, we can solve the triangle. However, isn\u00e2\u0080\u0099t there the infamous Ambiguous Case for the Law of Sines in the SSA situation, when there can be several possibilities like two possible triangles or no triangle at all?",
        "A": "There is a difference between soling for all the parts of a triangle and determining if two triangles are congruent. His statement is true that you can solve for parts in a single triangle, your statement is true in that SSA does not provide proof for congruency of triangles, but these are two different unrelated issues."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At 2:34 Sal calculates the sin and it becomes .5. So why does it become 1/4?",
        "A": "Because there was another 1/2 in the equation"
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "Whoa! At 3:20, Sal say we might know what sine 45 is, and proceeds to write it as \u00e2\u0088\u009a2/2 but here in CBSE system (India) it is mentioned that sin 45 is 1/\u00e2\u0088\u009a2. I want to clarify whether this will cause any exceptional difference in solving advance-level problems or not?",
        "A": "1/sqrt(2) = 1/sqrt(2)\u00e2\u0080\u00a2sqrt(2)/sqrt(2) = sqrt(2)/sqrt(2)^2 = sqrt(2)/2"
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "Is sal just trying to figure out a in that whole part when he circles the 2 parts in white at 2:16?",
        "A": "Yes, you are correct. Sal is solving for a in the white box, although he could have solved for b just as easily by doing sin(30)/2 = sin(45)/b Just know that with the Law of Sines, you need at least 3 components to be able to use it and only two of the components can be the same type. For instance, you can use Law of Sines if you know 2 sides and 1 angle (SSA), or 2 angles and 1 side (AAS). If you know the length of all 3 sides though, and none of the angles (SSS), you need to use the Law of Cosines instead."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At 0:06, Sal says that he can figure everything out given 2 angles and a side length. However, how would you know when to use the Law of Sines and when to use the Law of Cosines?",
        "A": "Excellent question! Law of Sines works when you are dealing with 2 angles and a side length and the law of cosine works out when you are dealing with all the sides and only one angle. So when you are stuck, just check what you already know and what you have to find. Hope this helped."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At 0:53, does he mean \"side\" or \"sine?\"\nThe subtitles say \"sine,\" but I'm guessing that it's \"side.\"",
        "A": "The time you have mentioned 0:53, he doesn t mention any of the words side or sine ! Give a proper timing so I can clear your doubts! :)"
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "What is a Reciprocal? Sal said it at 3:36?",
        "A": "The simplest I can define Reciprocal is this: If you have 3/4, the reciprocal is 4/3. You are flipping your numerator and denominator."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "Why does Sal at 3:53 do the reciprocal of 1/4 = sin 105\u00c2\u00ba/a?",
        "A": "He does it just to show you what the last angle is if you were ever in a position where you had to find all sides and all angles"
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "how did you get 1/4 on 2:55",
        "A": "We know that sin(30\u00c2\u00b0) = 1 \u00e2\u0088\u0095 2 The proof for this is rather simple: Consider an equilateral triangle with side length 1. Drawing the height divides the equilateral triangle into two 30-60-90 triangles where the side opposite to the 30\u00c2\u00b0 angle has length 1 \u00e2\u0088\u0095 2, and since the hypotenuse is 1, we get that sin 30\u00c2\u00b0 = 1 \u00e2\u0088\u0095 2."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At 5:49, Sal says that if you had two sides and an angle of a triangle, you would be able to figure out everything else about the triangle. Is that true of any two sides and any angle of any triangle?",
        "A": "yep, that s right"
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "at 1:27 , for two different triangles, does the law of sines apply and will it still be constant?",
        "A": "The Law of Sines applies to all triangles and the ratio remains constant within the given triangle, though it will be different between the two different triangles."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At 5:16, aren't you supposed to multiply (sqrt2/2) by 4?",
        "A": "He did. He had an intermediate step not written down where he did 4/2 * sqrt(2) and got 2 * sqrt(2), which is equal to about 2.83"
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At 3:34 why did he use the reciprocal of both sides?",
        "A": "The variables he is solving for are in the denominators. Taking the reciprocal of everything moves all the unknown the variables into the numerators."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At 0:55, how does Sal get the 180 degrees? How does this work?",
        "A": "Angles in a triangle add up to 180 degrees. The formula to find the sum of all interior angles of any convex polygon is 180(n-2) where n is the number of sides."
    },
    {
        "video_name": "VjmFKle7xIw",
        "Q": "At 3:51 why did Sal flip sin105*/a to a/sin105*",
        "A": "The law of sines here is that the ratio of the side length and the sine of an angle are the same. As long as the ratio is equivalent, there s no problem if you put the angle above the side length or vice versa (as long as you re consistent throughout the equation)."
    },
    {
        "video_name": "aKAw-DyZa6g",
        "Q": "At 3:34, why would asking the whole class be less efficient? Wouldn't the class be the only relevant demographic in this case?",
        "A": "You are correct. I don t know where you got the idea that it would be less efficient, Sal himself selects it as one of the right answers, saying that asking all 300 students is not unreasonable. He also chooses two of the other options, because selecting 50 random students from said class might also give a pretty good idea of the election results."
    },
    {
        "video_name": "aKAw-DyZa6g",
        "Q": "At 3:21, you say that asking people with a name starting with N or B is bad. I disagree because even though there might be 10 people with N and only 2 people with B, sometimes it might be that a person, who's name starts with N might like Blair much more better than Napoleon.",
        "A": "but that is only 12 people out of 300 people that is 4% of the population"
    },
    {
        "video_name": "aKAw-DyZa6g",
        "Q": "At 4:14, when the instructor agreed to use a computer to generate random numbers, couldn't the computer generate a decimal? Couldn't it generate 5.3647382323 or 34.5748302? If those numbers are generated, how are you supposed to figure out which students were chosen?",
        "A": "When using computers to generate random variables you can choose what kind of number you d like. For example in Microsoft Excel you can use the =RAND() formula to get a probability between 1 and 0, or =RANDBETWEEN( bottom , top ) to get a number between two integers."
    },
    {
        "video_name": "4PPMnI8-Zsc",
        "Q": "For #8 at 04:25 why wouldn't the answer be \"intersecting lines?\" because intersecting lines are connected by one point.",
        "A": "that are straight in the middle"
    },
    {
        "video_name": "4PPMnI8-Zsc",
        "Q": "At 2:03, what are 'compliments' and 'supplements' of triangles?",
        "A": "They are actually compliments and supplements of angles. The compliment of an angle is the angle that makes the first angle add up to 90 degrees. The supplement makes the angles add up to 180 degrees. Example: The compliment of a 40 degree angle is a 50 degree angle. The supplement of a 40 degree angle is a 140 degree angle. Hope this helped!"
    },
    {
        "video_name": "4PPMnI8-Zsc",
        "Q": "at 5:11 what is a counterexample?",
        "A": "A counterexample is some fact or example that demonstrates that a supposed rule is not always true. For example, someone lists 10 thousand prime numbers, showing them to all be odd. They claim that all prime numbers are odd. You need only show them that 2 is prime as a counterexample in order to prove that not all prime numbers are odd."
    },
    {
        "video_name": "ojFuf9RYmzI",
        "Q": "At around 1:45 why did he write (3y^2) SIX times if the polynomial is only to the FIFTH power?",
        "A": "It is written six times even though the binomial is only to the fifth power because for one of the terms, you have to start from 0, not 1. If you count 0, 1, 2, 3, 4, 5, it is six, not five, as opposed to if you started from 1. The exponents on the individual terms should equal the power to which the binomial is multiplied. The first term starts with 5, and 5+0=5. It also has to end completely, hence the sixth term. This is because 0+5=5. Therefore, Sal wrote (3y^2) six times."
    },
    {
        "video_name": "wyTjyQMVvc4",
        "Q": "why at 8:00 is the sqrt of 4 equal to 2, what about -2?",
        "A": "Sal was dealing with the principal square root, not both square roots. The square root was used to find a distance along a curve, and distances are always positive."
    },
    {
        "video_name": "wyTjyQMVvc4",
        "Q": "At about 7:28, you mentioned that the squareroot of 4 is just 2. Isn't it important to remember that it could also be -2?",
        "A": "He was looking for the distance , and distance is always positive."
    },
    {
        "video_name": "0imeUgSxR10",
        "Q": "At 8:30, why isn't it 8(sqrt(m\u00c2\u00b2 + 64)) instead of 8(sqrt(m\u00c2\u00b2 + 1)) ? It looks like it should go from sqrt(8\u00c2\u00b2m^2 + 4\u00e2\u0080\u00a216) to sqrt(m\u00c2\u00b2 + 64) \u00e2\u0080\u00a2 sqrt(8\u00c2\u00b2) to 8(sqrt(m\u00c2\u00b2 + 64)). Thanks!",
        "A": "You need to factor 64 out of both terms. sqrt(8\u00c2\u00b2m^2 + 4\u00e2\u0080\u00a216) = sqrt(64m^2 + 64) = sqrt(64(m^2+1)) = sqrt(64)sqrt(m^2+1). Remember that square roots separate across multiplication, not addition."
    },
    {
        "video_name": "SG4gX-VGzog",
        "Q": "What is 233,548,888 times 60 at 5:54 am",
        "A": "14,012,933,280"
    },
    {
        "video_name": "SG4gX-VGzog",
        "Q": "At 5:31 can the sun dissapear",
        "A": "no, not unless it gets too far from are view"
    },
    {
        "video_name": "SG4gX-VGzog",
        "Q": "At 8:49 what is 233,546,654 times 60 answer",
        "A": "14,012,799,240"
    },
    {
        "video_name": "SG4gX-VGzog",
        "Q": "at 0:00 way is there 4x10 is so easy? 40",
        "A": "Dude you are literally on the video multiplying 1-digit numbers by 10... don t know why you are asking"
    },
    {
        "video_name": "bGNMXfaNR5Q",
        "Q": "@13:50, how do you do this process with an absolute value equation??",
        "A": "absolute value equations are not differentiable."
    },
    {
        "video_name": "bGNMXfaNR5Q",
        "Q": "Around 2:50 he says that the function has to be defined over a closed interval. What about if there are two options, like one function is defined at -1 up until 0 (but not at 0), and then another function is indicated for x=0. Could the MVT still apply over the entire interval [-1,0]?",
        "A": "It depends on how each function is defined. If your first function approaches a different value as x \u00e2\u0086\u0092 0 than when the other function does, then the overall function is not continuous at x = 0, and you can t applied the MVT. If the overall function is continuous at x = 0 and differentiable, you can set a closed interval there and apply the MVT."
    },
    {
        "video_name": "bGNMXfaNR5Q",
        "Q": "At 13:32, shouldn't it be -4/2?",
        "A": "Whoops, never mind. Minus - minus= plus"
    },
    {
        "video_name": "kqU_ymV581c",
        "Q": "5:40 How do you find out that the angle of the triangle is 75?",
        "A": "all he did was make up a random angle. Notice how he said lets assume that this angle is so and so. did that help?"
    },
    {
        "video_name": "kqU_ymV581c",
        "Q": "At 00:51, he gives random angles. If I am playing, how would I know if that angle was possible to find out useing the angels shown?",
        "A": "If the angles were proportional, you would. But since Sal is making reasonable guesses on hat they are, he put numbers.And those 2 angles are enough to find out that it is 59 degrees..."
    },
    {
        "video_name": "kqU_ymV581c",
        "Q": "At 2:24, what is a transversal line?",
        "A": "Transveral line is an interesting line a line that intersects two parallel lines"
    },
    {
        "video_name": "kqU_ymV581c",
        "Q": "at 2:15 , sal mentions that the line is a transversal. What is a transversal?",
        "A": "Actually, a transversal line is a line that crosses across a pair of parallel lines."
    },
    {
        "video_name": "kqU_ymV581c",
        "Q": "who admits that they were scrunching up their eyebrows up to 8:12 and just gets completely mind blown in the next 3 seconds?",
        "A": "I wasn t. I already knew that. But for most people, it might have."
    },
    {
        "video_name": "kqU_ymV581c",
        "Q": "At 2:08, Sal references a transversal. Does transversal mean a line that that passes through two lines on the same plane?",
        "A": "A transversal is a line that passes through two parallel lines"
    },
    {
        "video_name": "kqU_ymV581c",
        "Q": "the problem on 6:15 is 49.\nwhy dont the shapes match up to the angles or is that just a way to trick the tester?",
        "A": "It s a way to make sure the person taking the test knows how to play the angle game, instead of relying on a protractor. (So, yes, you could look at it as a way to trick the test taker.)"
    },
    {
        "video_name": "kqU_ymV581c",
        "Q": "at 1:01 the answer is 59 degrees\nbut why isnt it 124 because the lines are parallel so wouldnt it be equal to the top?",
        "A": "The angle labeled with ? is formed by one of the parallel lines and the horizontal line. The angle labeled 56 degrees is formed by one of the parallel lines and a line that slopes down from left to right. The horizontal line and the line that slopes down from left to right aren t parallel, so the degrees are not equal."
    },
    {
        "video_name": "uXjQ8yc9Pdg",
        "Q": "10:50 the integration by parts is completely unnecessary because sin(t)cos(t) is simply equal to 1/2(sin(2t)) which can be integrated directly! Needless hassle.\n\nGreat otherwise obviously.",
        "A": "I personally find Sal s use of U-Sub to make the integral more straightforward than it might be if he used a trig sub. Also, it shows people who may not have the relationship you mentioned memorized how to do the integral with a more universally applicable method. btw, great video. Thanks!"
    },
    {
        "video_name": "uXjQ8yc9Pdg",
        "Q": "I am totally confused again!! OMG, hold on. at 7:46, should the first part of the integral, underlining x and y, be xy instead of cosx siny ?",
        "A": "Yes, and it is. But we did a change of variables, and now x = cos(t) and y = sin(t) so xy = cos(t)\u00c2\u00b7sin(t). Nowhere in the video does a cos(x)\u00c2\u00b7sin(y) appear."
    },
    {
        "video_name": "uXjQ8yc9Pdg",
        "Q": "At 10:05, did he mean integral when he said square root?",
        "A": "Yes. Sal just made a mistake."
    },
    {
        "video_name": "uXjQ8yc9Pdg",
        "Q": "At 2:53, is the graph z=xy? I googled the image and it looks different. z should be max at max(x) & max(y) and min(x) & min(y). This is so confusing :( Thanks in advance!",
        "A": "Yes, this is the graph z=xy. However, Googling the image will give you different values of min(x), min(y), max(x), and max(y). In Sal s graph, min(x) and min(y) are both zero. In most graphs that you get, you will have a min(x) and a min(y) of something like -5."
    },
    {
        "video_name": "uXjQ8yc9Pdg",
        "Q": "04:34 What if we were to find the volume under the curve ? What would we do? Thanks in Advance.",
        "A": "You would probably want to do a double integral of f(x,y) from y = 0 to y = sqrt(1-x^2) and x = 0 to x = 1."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "At 0:58 he said that the formula for Area of a circle is A= pi * r2 why wouldnt it be A= pi * d??",
        "A": "Thanks for the help."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "At 0:54, when we do the area of a circle why do we square root the radius for the area, but we just multiply the radius by 2 for the circumference?",
        "A": "Reminder that the circumference is the perimeter, which means squaring something for the perimeter would not make sense. Most shapes multiply sides by two If you picture it like a square. Hence multiplying the radius by 2. For area, we do it because if you picture it like a square, the radius is the sides, which means you would square the number. (You square the radius, not square root. A square root would be like if 2 squared is 4, 2 is the square root)."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "At 0:41 the area is PI R squared right?",
        "A": "Yes, it may help to to see it in a slightly less shorthand form. a = (r^2) * Pi You square the radius and then multiply that product by Pi to get the area."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "2:03 Isn't Sal suppose to find the exponent first?",
        "A": "He does, at 1:30. By the time you get to 2:03, the exponent is only on the unit of measurement. He has 64 square millimeters. The value portion has already been squared (8 x 8 = 64) and the unit portion has also been squared (mm x mm = mm^2). There s nothing left to do with the exponent."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "i dont understand what A=pi*r squared means.\n(at 1:00)",
        "A": "That is the formula for the area of a circle. To calculate the area, A, you take pi (which is a mathematical constant, roughly equal to 3.14) and multiply it by r^2, that is (the radius x the radius)."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "At 3:21 does it matter if you put 3.14 or pi ?",
        "A": "The amount of digits you enter, the more precise your calculation will be. So, if you put 3.14, you can only trust two numbers after the point (the 1 and the 4). Try to be as precise as you can! Good luck!"
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "At 1:25, why say pi instead of rounding it to 3.14? Please answer question someone!",
        "A": "Pi is not 3.14. 3.14 is only an approximation to pi. So is 3.14159265359, but it is far more accurate. When we use pi in an expression, we are not sure what degree of accuracy people want when they use it. Leaving it as pi lets them plug in the value that they want to use, according to the amount of accuracy they need."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "At 2:51 what does a squiggly equal sign mean?? I thought it meant the same as s normal on.",
        "A": "A squiggly equal sign means it s the approximate. so pi is approximately 3. It just means it s not the total answer and they rounded some how"
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "isn't the better way to multiply by pi to just stick pi on the end instead of actually multiplying the number by 3.14? 1:32",
        "A": "Yes, you can do it that way. It is also the preferable way as you will get a more precise answer."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "at 1:04, how did he get 64? what did he do? and what does the small 2 mean?",
        "A": "The small two means squared. Squared mean that # multiplied by its self. I hoped that helps."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "At 1:25 how did you get 64 mm?",
        "A": "The radius is 8 mm. That squared is 64 mm."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "At 2:52, what is that little squiggly equal sign he draws? Does it mean \"Approximately equal to?\" I've never seen it before.",
        "A": "Yes it means approximately equal. It s a good symbol to know. The area is approximate because we don t use the full value of pi."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "at 1:30 how did Sal get 64pi i got confused",
        "A": "The diameter is 16, therefore the radius must be half of it, 8. Squared means multiplying that number by itself. 8 * 8 = 64. :D"
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "At 0:54 sal said the formula of finding the area of a circle is pi*r^2. isn't it the same as pi(D/2)^2",
        "A": "yes, the radius is half of the diameter."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "at 1:53 does the area have to be squared",
        "A": "Yes area is always in square units"
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "So basically on 0:40 ,that's how to find the area ?",
        "A": "I think you mean 0:55 but yes, a=3.14*r^2 Remember that 3.14 is equal to pi (not really, I just rounded to the nearest hundredth. pi goes on FOREVER AND EVER!). If you meant DIAMETER (because Sal was discussing about diameter at 0:40),the formula would be d=2r)."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "At 0:26, it would be very hard to be exact with a unit so small like a millimeter; that is why millimeters are used mostly to measure tiny things (like one of those wafers).",
        "A": "Maybe he needs a solid number for a quiz or something."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "I don't get what he did at 1:47, and I've watched this video over and over. Can someone please clarify?",
        "A": "Not sure what you mean. He only rewrote (pi) * 64 as 64 * (pi) at that point in the video."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "at 1:25 why does the 8 mm square become 64 mm square?",
        "A": "okay, thanks:)"
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "I don't understand why r*(pi*r) = pi*r^2 is the area of a rectangle. I need help. Also I don't get 2:00 in the video, and I need an explanation about it.",
        "A": "It doesn t have to be that way. The area of a rectangle is base times height. No pi is involved. Hope this helps, Yamu"
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "1:42, does the answer change if it is millimeters or inches",
        "A": "no it does not change the metric units do not change Jose is wrong"
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "Is the area of a Cricle almost similar to the circumference of a circle and at 0:55 he shows the area of a circle.",
        "A": "No. Area of the circle is the amount of space inside the circle. The formula is A = pi(r)^2. The circumference of a circle is the length around the circle. The formula is C = 2pi(r). Notice one is r^2 and the other one is 2r. If you don t understand the word circumference, just think of circumference of a circle is like the perimeter of a rectangle. It s the length surrounding the shape."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "At 1:15, are you squaring the pi and the 8mm or only the 8mm. confused. Thanks!",
        "A": "thanks! that really helped me"
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "Sal said at 1:22 pi which is roughly about 3.14 times the radius which is 8 mm. He said that it is 64 mm. I thought it was 25.12. What does he mean about 64 mm?",
        "A": "Area is equal to Pi times radius squared. 64 is the square of 8, not the product of 8 times pi."
    },
    {
        "video_name": "ZyOhRgnFmIY",
        "Q": "At around 1:20, when he kept saying meters squared and writing millimeters squared, which one did he actually mean?",
        "A": "it doesn t really matter but i think he means millimeters squared considering the fact that the question stated the units in millimeters"
    },
    {
        "video_name": "DKh16Th8x6o",
        "Q": "At 6:27, How do you get the cube of 512?",
        "A": "It ll come naturally to you as you get more practice. For me, 512 instantly triggers the idea that it is a power of 2 (512 is a common number in computers). So, I know the answer is either a 2, 4, or 8. So the answer must be 8."
    },
    {
        "video_name": "DKh16Th8x6o",
        "Q": "At 2:20, how is it that 512 = 2^9 should give a hint of the cube root of 512?",
        "A": "Recall your work with radicals: 2^9 =(2^3)^3 =8^3"
    },
    {
        "video_name": "DKh16Th8x6o",
        "Q": "Hang on a second... at 1:10, how is the cube root of -1 = -1? Wouldn't it be -i?",
        "A": "No because -1 x -1 x-1 = -1. When three negatives are multiplied together 2 of them cancel out, but on is left so no there is no change to the sign."
    },
    {
        "video_name": "DKh16Th8x6o",
        "Q": "i dont really understand why at 3:00 Sal put them into groups of 3.",
        "A": "Because you are trying to figure out what times what times what equals 512. So you need three things"
    },
    {
        "video_name": "DKh16Th8x6o",
        "Q": "At 1:18, isn't the cubed root of -1 undefined? I know that the square root of -1 is undefined.",
        "A": "Actually, Diego, the cube root of -1 is -1 because -1 \u00e2\u0080\u00a2 -1 \u00e2\u0080\u00a2 -1 = -1. The square root of -1 is undefined because no number multiplied by itself is equal to -1 because getting a negative number in this way is impossible. This is not true for the cubed root,though, because three negatives multiplied make another negative."
    },
    {
        "video_name": "DKh16Th8x6o",
        "Q": "at 0:49 how did sal get negative 1? im confused",
        "A": "Any number times -1 is negative of that number. a * -1 = -a For example 5 * -1 = -5; 348 * -1 = -348 That means any negative number can be represented as positive number with the same value (absolute value) times -1. -a = a * -1 For example -33 = 33 * -1 = -1 * 33; -234 = -1 * 234 = 234 * -1"
    },
    {
        "video_name": "DKh16Th8x6o",
        "Q": "at 0:49 how did he get 512 from -512 3 power and get -1 . do you got any tips?",
        "A": "I m not sure I fully understand your question, but I ll try my best to explain. :) Sal simply breaks the problem down further. He points out that 512 * -1 is equal to -512. if we break the problem down in this way, we can work with a positive number (512) instead of a negative number (-512). this helps to eliminate confusion. After finding the square roots of both -1 and 512, we can multiply the answers together and come up with our ultimate answer. -1 * 8 = -8 -8 is the cube root of -512 Hopefully that helped!"
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "At 4:05, Sal said (x1, x2) but meant (x1, y1) and at 4:40 he meant (x1 -1, y1 +2). Please correct me if i am wrong.",
        "A": "ur right! he wrote it wrong too"
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "At 9:51 Khan shows how you can add vectors visually by connecting the tail of vector b to the head of vector a. He then draws the vector of a+b. My question is, is it allowed to draw the same thing but instead of connecting the tail of vector b to the head of vector a, you do the opposite and connect the tail of vector a to the head of vector b? If I draw the added vector, it still fits in perfectly.",
        "A": "Yes this is fine since by definition vector addition is commutative, that is a + b = b + a for any two vectors a, b."
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "at 4:44, is it x1 + 2 or x2 + 2?",
        "A": "Yes. If you look in the comments, you ll see Nathan has pointed this out."
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "14:30 is -4v co-linear with v. Or does co-linear mean same exact direction.",
        "A": "-4v is co-linear with v. Any vector that s a scalar multiple of another is co-linear with that vector."
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "5:30 at in the video you have x1 as 4 and not -4. Should it be -4?",
        "A": "In the video the first component of the vector is -4, it looks ok?"
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "At the time 4:36 in the video is Sal making a mistake writing \"x1\" again for the y-coordinate of the arbitrary starting point. In other words if the arbitrary starting point is (x1,x2) shouldn't the representation of adding vector A to this arbitrary starting point be written as: (x1-1,x2+2)? Sal wrote (x1-1,x1+2). I completely understand the concept but I want to clarify this for someone else watching so that they are not confused.",
        "A": "I saw the same thing. I think it s a mistake, but I don t have the academic authority to be absolutely sure of it."
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "At 15:35 he says that x - y = x + -1y. Can someone walk me through that?",
        "A": "(-1)y is the same thing as only (-y) and if you add a negative number i become minus.. x + (-y) = x - y"
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "At 21:20, shouldn't the vector (-6,-5) be drawn by going left 6 and down 5?",
        "A": "Yes, that is a mistake."
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "so i got a little lost about 18:30 minutes in. why is it when we do x(2,3) - y(-4,-2) we add -1 to the equation ,could you elaborate on the math there and if we are adding the vectors why does that change? pretty sure i am over thinking it but just to be sure.",
        "A": "subtraction can be defined as adding the negative of something. x-y=x+(-y)"
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "at 4:28, he wrote it wrong!\nits (x1-1, x1+2)\nbut its meant to be (x1-1, x2+2) instead\nGuess everyone makes mystakes! :D",
        "A": "You are correct. They should put a note on the video correcting this."
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "At 23:25, when applying to a vector in 4 dimensions, why does he use 4a - 2b? I thought he was simply finding the difference between the two ends of the vectors.",
        "A": "He was just showing that you can do the same vector operations in R^4 also. By showing the scalar multiplication also, he s reinforcing that it applies in higher dimensions too. He wasn t intending to find the vector difference."
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "At 6:50 he says: \"Sometimes will be an arrow notation.\" Shouldn't it be always, because it means that it is a vector?",
        "A": "Sometimes another notation might be used. A common one is to put a little hat on it to indicate a unit vector, for instance."
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "At 21:40 of the video the vector x-y is drawn from the end of x to the end of y ending up at [-4, -2] yet the calculated value for x-y is [6,5]! Shouldn't the vector x-y be drawn from the end of the x vector to point [6,5]?",
        "A": "No. Firstly, the vector does not point to [-4,-2]: it starts at [-4,-2]. It points to the point [2,3]. Also, the vector only ends up at [2,3] because it starts at the point [-4,-2]. If you were to look at what the vector points to when it starts from [0,0], you would actually get [6,5]."
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "at 24:30,shouldn't it be -2*4,-2*-4 and -2*5?",
        "A": "How do you find an endpoint when a question gives you two points. An endpoint and a midpoint. I haven t been able to find it in my book or been able to retrieve it on this website. Can anyone point me in the right direction to find this equation? Please. Thank you."
    },
    {
        "video_name": "r4bH66vYjss",
        "Q": "At 1:38 how is R 2 bigger than R 1? Aren't they both infinite?",
        "A": "Sal mentioned that the R2 plane is bigger (spatially) because it would cover an infinite number of points on the xy plane whereas the R1 plane covers an infinite amount of points along one of number lines on the plane (x or y). Therefore, although both R1 and R2 are inifinite, R2 covers what R1 covers and even more! Hope that clarified it!"
    },
    {
        "video_name": "OvMBNVi5bLY",
        "Q": "At 6:45 Sal says that g(x) \"is differentiable over the entire interval\". But according to the Fundamental Theorem of Calculus shouldn't g(x) only be differentiable over the intervals (-4,0) and (0,3)?\nBecause the lower bound (zero) of the definite integral (the \"a\"-term) is not an endpoint of the interval--rather, just some point in the middle. See what I mean?",
        "A": "Your first observation is correct, as far as I can tell. At f(0) there is an inderivable point, so it should only be differentiable from (-4,0) and (0, 3). I m insure what you mean by the second portion of your question."
    },
    {
        "video_name": "OvMBNVi5bLY",
        "Q": "06:26 why is the derivative f(x) and not f(t)",
        "A": "The derivative is f(x) because of the second fundamental theorem of calculus (diving board) where the x replaces the t in f(t), so the integral of f(t) from (0,x) would be f(x)* 1, or f(x)"
    },
    {
        "video_name": "xSc4oLA9e8o",
        "Q": "At 9:39, in a word problem when it uses replaced or not replaced does that refer to being a independent or dependent event?",
        "A": "It s true that if you DON T replace the items after choosing them, then subsequent choices will be dependent (since what is left to choose from depends on what was chosen earlier), while if you DO replace items after they are chosen, then each choice is independent, since the conditions are the same each time, regardless of what happened earlier."
    },
    {
        "video_name": "xSc4oLA9e8o",
        "Q": "4:21\nWhen Sal says that \"The fact that I got tails in no way affects the probability of getting a heads on the second flip\"- Does this mean that there is a 1/2 chance of getting tails or heads no matter what? If so, then why are we using this formula to calculate compound probability when there is always a 1/2 chance of getting a heads or tails?",
        "A": "The probability of tails on the second flip, given tails on the first flip, is different from the probability of tails on the first two flips. And does not have the same meaning as given . The probability of tails on the second flip, given tails on the first flip, is 1/2, but the probability of tails on the first two flips is (1/2)(1/2) = 1/4. Have a blessed, wonderful day!"
    },
    {
        "video_name": "xSc4oLA9e8o",
        "Q": "I don't get how they have H H, H T, T H, T T at 2:04",
        "A": "the first letter represents heads or tails (H or T) on the first flip and the second letter represents heads or tails on the second flip. you could get heads for both flips (HH), heads first then tails (HT), tails first then heads (TH), or tails both times (TT). those are the only possible outcomes. hope it helps!"
    },
    {
        "video_name": "GEeKOeGHddY",
        "Q": "at 3:24, when sal raises the 8 to the 1/3, why did he raise it to 1/3 and not 3/1, and do I always have to raise the index over 1?",
        "A": "A root is the same thing as an exponent to the reciprocal of the degree of the root. So, yes, it must be over 1 Thus, \u00e2\u0088\u009aa = a^(\u00c2\u00bd) \u00e2\u0088\u009ba = a^(\u00e2\u0085\u0093) \u00e2\u0088\u009b(a\u00c2\u00b2) = (a\u00c2\u00b2)^(\u00e2\u0085\u0093) = a^(\u00e2\u0085\u0094) \u00e2\u0088\u009ca = a^(\u00c2\u00bc) \u00e2\u0088\u009c(a\u00c2\u00b3) = (a\u00c2\u00b3)^(\u00c2\u00bc) = a^(\u00c2\u00be) and so on."
    },
    {
        "video_name": "GEeKOeGHddY",
        "Q": "what is communitive property? sal mentions it around 0:30",
        "A": "The commutative property states that you can add or multiply in any way or order."
    },
    {
        "video_name": "GEeKOeGHddY",
        "Q": "3:22, 8 to the 1/3 = 2. This is confusing to me.",
        "A": "8^1/3 That s saying what multiplied three times is equal to eight and in this situation it s 2. 2 * 2 * 2 = 8 2^3 = 8^1/3"
    },
    {
        "video_name": "GEeKOeGHddY",
        "Q": "At 4:11 i dont understand how he got that answer.",
        "A": "8^1/3 can be seen as \u00c2\u00b3\u00e2\u0088\u009a 8 which, by simplifying, can be turned into a 2 (x^6)^1/3 can be seen as 6/1 (which is just 6) times 1/3 = 6/3 = 2, which is what he got, and put that two in the place of the ^6 to turn it into x^2"
    },
    {
        "video_name": "GEeKOeGHddY",
        "Q": "Wait im not sure if im missing something but at 2:05 he said that 2 times 4 is 6......",
        "A": "Keep watching. Khan says that he made a mistake. 2 times 4 is 8"
    },
    {
        "video_name": "GEeKOeGHddY",
        "Q": "At 2:50, you have 15 (8x^6)^1/3, why don't you just solve it like this:\n15 (8x^6)^1/3\n15 (8x^6/3)\n15 (8x^2)\nOR you could even solve it like this, I think...\n15 (3sqrt of 2x^2 * 4x^4)\n15 (3sqrt of 8x^3 * 8x^3)\n15 * (8x^2)\nSame answer... Makes me feel like Sal did something wrong although I do get his way of solving it as well...\n\nI get that this gives me another answer, but I don't see why this way of solving it wouldn't work? I'm not breaking any rules, am I?",
        "A": "(pq)^r gives p^r*q^r. Take p=8 and q=x^6 and r=1/3, it gives 8^1/3*(x^6)^1/3=2x^2. You forgot to do 8^1/3 as well as doing (x^6)^1/3."
    },
    {
        "video_name": "GEeKOeGHddY",
        "Q": "at 1:57 he says that 2 times 4 is 6 are you supposed to add or multiply 2 and 6",
        "A": "He made a mistake - he corrects himself a few seconds later. He says, Wait! What was I thinking? Two times four is eight! It s nice to know that even he makes mistakes. :)"
    },
    {
        "video_name": "GEeKOeGHddY",
        "Q": "At 1:56 Sal says that 2x4=6 ?",
        "A": "I agree, it should be 8."
    },
    {
        "video_name": "bamcYQDzVTw",
        "Q": "Can I Do 8(3x^2+30x+5) + x(3x^2+30x+5) + -5 (3x^2+30x+5)? Instead of the way he did it at 2:30 and then add simplify the expression? Would I get the same answer?",
        "A": "I tried just this: 8(3x^2+30x+5) + x(3x^2+30x+5) to see if it equaled 8x(3x^2+30x+5), which it doesn t so I don t think that they are equivalent. However, I m not sure why"
    },
    {
        "video_name": "5DlzmolR8Cg",
        "Q": "0:10 why did he say 1 thousand when the video clearly said 1 million",
        "A": "He probably said the wrong thing on accident, I mean people make mistakes."
    },
    {
        "video_name": "5DlzmolR8Cg",
        "Q": "At 1:00 why did Sal put parentheses around each expression?",
        "A": "Sal put parentheses around each expression to tell you that you have to multiply before you add. These were actually not necessary because we already you have to multiply before you add."
    },
    {
        "video_name": "5DlzmolR8Cg",
        "Q": "At 1:25, how does he get 6/100. Would changing 67/1000 in to 6/100 help you solve the problem. I am very confused. Thank you.",
        "A": "Sal separated 67/1000 into 60/1000 + 7/1000. He did this (and reduced one of the fractions) so that he could easily identify the digit that would go into each decimal place."
    },
    {
        "video_name": "K5rmfHlHy20",
        "Q": "At 1:20 ins't numerals numbers?",
        "A": "Yes they are!"
    },
    {
        "video_name": "Uy_L8tnihDM",
        "Q": "At 07:14, why is a[n] equal to a(n-1)d?",
        "A": "At that point, Sal is saying that a[n] is equal to a + (n - 1)d, not a(n - 1)d. Early in the video, Sal shows that each member of the series consists of the base value a plus a multiple of the increment value d, where the number of d s is one less than the index, so that the number of d s in the nth member of the series is n - 1."
    },
    {
        "video_name": "Uy_L8tnihDM",
        "Q": "At around the 2:00 mark, Sal starts off by reversing the series. My question is, why don't we just realize that a1 + a2 + a3 + a4 is just a1 + (a1 + d) + (a1 + 2d) + (a1 + 3d) = 4a1 + 6d (and the rest is history)?",
        "A": "You still have to be able to add this new series one would generate. By this intuition, For n=50, one would have 50 a + ( (1 d+ 2 d + 3d +.....+ 49d ). How would one do this addition ?"
    },
    {
        "video_name": "Uy_L8tnihDM",
        "Q": "at 3:40 how did\n1d + (n-2)d become (n - 2 + 1)d?",
        "A": "It s an arithmetic factoring, you can see it more clearly if you you first expand the terms in the parenthesis and then take our the common d term: 1d + (n - 2)d 1d + nd - 2d (1 + n - 2) d (n - 2 + 1) d"
    },
    {
        "video_name": "EGNlXtjYABw",
        "Q": "At 15:06, Sal refers to the \"vector A\" -- how come? Isn't this a matrix? Sorry, I'm a bit confused here. Thanks for your help!",
        "A": "That was a mistake on Sal s part and when I watched the video I got a small message that said he meant matrix A"
    },
    {
        "video_name": "EGNlXtjYABw",
        "Q": "The cartesian wich he made at 05:30 is Left-handed orientation, right?\nif yes, why?\nIs this a simple error or some thing that i dont understand",
        "A": "You re correct. It s left handed. Left handed coordinate systems are as valid as right handed ones."
    },
    {
        "video_name": "EGNlXtjYABw",
        "Q": "Why did you change your axes of the 3-d graph at 5:05? Wasn't the original way \"correct\" (i.e. it was right-handed)?",
        "A": "I think that it was changed so accommodate the angle we are looking at the video, so that x-axis is left/right, y-axis is forward/backward, and z-axis as height. To make it right handed, the y-axis could be changed to point forward, or into the screen, but that would make all three axis bars very close to each other, making it difficult to visualize. As Sal said at 5:38, ...you get the idea."
    },
    {
        "video_name": "EGNlXtjYABw",
        "Q": "At 11:11 you said the plane has to intersect (0,0,0) because Subspace always contain the zero vector. But i thinks it a little mistake right? Because you said vector doesn't contain position information? Is the real reason is you draw the vectors at the standard point? I also think any other parallel plane with the one that you found can contain the same set of vector.",
        "A": "I have similar questions. I suspect, but don t expect, the answer will become clear later."
    },
    {
        "video_name": "EGNlXtjYABw",
        "Q": "is the arbitrary vector x a vector that lies on the plane C(A) or is it position vector that starts at the origin and ends at an arbitrary point on the plane C(A)? If it is the latter then wouldn't the difference between vector x and <1,2,3> define a vector that is not in the plane C(A) since <1,2,3> lies on the plane C(A)? and wouldn't the equation at 6:47 be untrue?",
        "A": "x in Ax is an arbitrary vector (x1, x2, ..., xn) where all the xi are unknown, i.e. they can independently be any real number, and so Ax is the column space. The set of equations Ax=b constrains the column space Ax to a single vector, and so restricts the xi to a subset of the possible components in x."
    },
    {
        "video_name": "Qqanbd3gLhw",
        "Q": "At 3:30 is the derivative of y equal to -1 because 2-1=-1, or because the derivative of 2 is 0 and the deriving of t is 1?",
        "A": "The second thought is correct, it is because the derivative of 2 is 0 and the derivative of -t is -1."
    },
    {
        "video_name": "zC_dTaEY2AY",
        "Q": "At 1:44 would it have also been mathematically correct to use the division rule to find the second derivative on the first derivative expressed as a fraction? In other words, would it also have been accepeptable to take the derivative of 4x^3/(x^4 + 27) using the division rule if I wanted to? I know the multiplication rule is easier, but I still wondered.",
        "A": "Yes, both ways are correct, and both will give you exactly the same result (after some simplification)."
    },
    {
        "video_name": "zC_dTaEY2AY",
        "Q": "At 6:50, Sal said that when denominator is 0, it will become undefined while anything divided by 0 is infinite. Am I right?",
        "A": "It isn t infinite. But if you have 1/x and x approaches 0 it will be infinite, since x is infinitelly small and not 0. So 1/0 doesn t have a answer."
    },
    {
        "video_name": "zC_dTaEY2AY",
        "Q": "At 1:35 he uses the product rule for the 3rd expression. Why did he not use the quotient rule ?",
        "A": "Rather than deal with numerators and denominators of the quotient rule, he rewrote the expression in terms of negative exponents, EG (2x+3)/(3x+2) = (2x+3)(3x+2)^-1. Now you can solve via the product rule. You may find this to be less messy in many circumstances."
    },
    {
        "video_name": "zC_dTaEY2AY",
        "Q": "Instead of saying \". . just less/greater than. . .\"\nWouldn't it just be \"just in that sign direction until the critical value?\"\nSo at 13:24, Instead of saying, \"So when x is slightly less than zero. . \" Would it be the same as saying, \"When x is between 0 and -3. . . \"?\n(If there are no more critical points between the value and +/- infinity, then you would just say, \"When x is between 0 and negative infinity. . \"_\nIf not, I would like to know why.\nThanks.",
        "A": "You are correct. I think Sal was just trying to stress the meaning of the inequality. In general there are many ways to say the same thing. Some ways are more formal, some are more standard, some are easier to understand. A good mathematician is always thinking about the best way to say something just like you are."
    },
    {
        "video_name": "TaRPDFqEd-c",
        "Q": "At 1:30 why isn't x=0 a solution? You could determine that x=0 rendering the expression equal to zero.",
        "A": "X can t be equal to 0 because there is no x in the numerator which makes the numerator always equal to 2, if you try to plug in any number to get the second derivative equal to 0, the closest you can get is 0 in the denominator rendering the function undefined. Hope this helps"
    },
    {
        "video_name": "BuqcKpe5ZQs",
        "Q": "At 7:40, he says that A is nxm, isn't it mxn?",
        "A": "I think you re right."
    },
    {
        "video_name": "BuqcKpe5ZQs",
        "Q": "At 13:32, Sal says that a1 belongs to Rn, shouldn't it be Rm",
        "A": "I had the exact same qs.. I think so.."
    },
    {
        "video_name": "BuqcKpe5ZQs",
        "Q": "What does he mean at 5:32?",
        "A": "This is the identity matrix, which Sal has talked about in previous videos. The nxn identity matrix holds the standard basis vectors for the space Rn. To find the matrix form of a linear transformation, you have the transformation act on each of the standard basis vectors, and let those new transformed column vectors be the columns of your matrix."
    },
    {
        "video_name": "BuqcKpe5ZQs",
        "Q": "at 13:27 he means a_i belongs to R^m, right? Jw coz he wrote R_n...\nthx you guys!",
        "A": "a_i belongs to R^m, as m is the number of rows, and n is the number of columns."
    },
    {
        "video_name": "sSE6_fK3mu0",
        "Q": "At 3:40, why does he say that the left hand of the expression (d/dx(e^x)) will remain the same when multiplied by e^x? (The part he underlines in blue)",
        "A": "Thank you good sir!"
    },
    {
        "video_name": "sSE6_fK3mu0",
        "Q": "At 2:10, if ln x=1, then shouldn't ln e be equal to 1 too?",
        "A": "ln(e) IS equal to 1, and that IS what he showed, so if ln(x)=1, then x must be e. But, nowhere did I see ln(x)=1 in the video, so not quite following your logic. Care to expand on your thinking?"
    },
    {
        "video_name": "sSE6_fK3mu0",
        "Q": "Around 2:40, when Sal uses the chain rule, it is necessary to know the derivative of e^x because that is the inside function. But it doesn't make sense to use what you are trying to prove as part of the proof itself. It's like using a word in it's own definition. Can the derivative of e^x be proved any other way?",
        "A": "well, u see, he never actually EVALUATED the derivative of e^x, he just wrote d/dx e^x and he sorta solved for it. he just said that, according to the chain rule, ln(e^x) would be the derivative of the inside function times the derivative of the outside function."
    },
    {
        "video_name": "S7CLLRHe8ik",
        "Q": "On 5:03 and till the rest of the video, he says that the 300th day would be a Saturday. But if you bring Sunday to the first day of the week, won't the answer be Friday? I'm kind of confused.",
        "A": "So Day 1 is a Monday. So then 7 days later, Day 8 is also a Monday, And then Day 15 is. So basically you can model this with 7n+1 is a Monday. You then try the easiest multiple of 7 yo have: 40, because 4 times 7 is 28. 280+1=281, which is also a Monday. Take it two multiples further, Day 295 is a Monday. Day 296 is a Tuesday, Day 297 is a Wednesday, and so on... So Day 299 is a Friday and Day 300 is a Monday. Hope this helps."
    },
    {
        "video_name": "Y2-tz27nKoQ",
        "Q": "near 0:39 he's simply dividing the fractions or is he using another method?",
        "A": "To convert any fraction to a decimal, you use division. In this video, Sal is showing that some fractions create repeating decimals."
    },
    {
        "video_name": "Y2-tz27nKoQ",
        "Q": "at 2:10 sal says 10 10s what does that mean",
        "A": "it basically means 10 times 10"
    },
    {
        "video_name": "Y2-tz27nKoQ",
        "Q": "I am very counfused why were dividind at 0:16",
        "A": "you mean dividend? well if it is, dividend is the result of two dividing numbers"
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "at 0:17 i dont get how he said he had to get the prime factorization of 117. why??",
        "A": "15 radical 13 is a lot simpler than 5 radical 117. In higher math it would be a lot easier to work with a smaller, simpler, radical. And of course he can t simplify 5 radical 117 without knowing what the sqrt. for 117 is. Hope that helps!"
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "At 3:07 why doesn't the number simplify",
        "A": "You can t simplify that anymore. 26 has no factors that will square root. You can write square root of 26 as square root of 2 times square root of 13 but that will just equal to square root of 26 so there is no point of doing that."
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "Around 2:24, Sal explains that 5*3 and the square root of thirteen is 15 times the square root of thirteen. Why would you multiply the numbers 5 and 3?",
        "A": "He is trying to simplify it. 5\u00e2\u0080\u00a23\u00e2\u0080\u00a2\u00e2\u0088\u009a13 is more complex than 15\u00e2\u0080\u00a2\u00e2\u0088\u009a13. The former has 3 steps involved (multiply 5 and 3, find square root of 13, multiply 15 by square root of 13), while the latter only has 2 steps involved (find square root of 13 and multiply by 15)."
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "At 1:27, Why can't i take the prime factors that come up twice and put them outside the radical sign. The prime factors are 3x3x13. So i would get 5x3x3\u00e2\u0088\u009a13 = 45\u00e2\u0088\u009a13",
        "A": "You are half right here, you are taking the factors that appear twice, but since you are moving them from inside the radical, you need to take the square root of anything you move, so you only bring them outside once."
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "At 2:02 in the video, what makes it so that the 3*3 gets \"simplyfied\" to 3? Even though he explained it, I still don't get it.",
        "A": "What the square root function is asking is: What number times itself is equal to me. ie: sqrt(9) What number times itself is equal to 9? When you factor the value inside the squareroot you are effectively answering that. sqrt(3*3) what number times itself is equal to 3*3? Obviously 3. The square root of 9 (3*3) is 3."
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "At 5:07 why did you not write 7 times the square root of 3 times 3 times 3 and instead wrote 7 times the square root of 9 times 3?",
        "A": "Either way can work. Sometimes people don t see that 27 = 9*3 and that 9 is a perfect square. So, if you factor 27 completely (down to prime numbers), you get 3*3*3. Any factor that is multiplied by itself creates a perfect square. So you can group 2 of the 3 s together, and you have found your perfect square. If you can recognize the 27 = 9*3 and 9 is a perfect square, you don t need to factor down to prime numbers."
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "what is that thing you were doing under the problem at 1:01",
        "A": "Prime factorization. This is where you divide a number (117) by other numbers (in this case it was 3) to get an answer (39), and continue on until you get a number that is no longer divisible by anything (meaning you don t come out with a whole number). Example: 105: divide by 5: 21 --> divide by three: 7, which is a prime number. Prime factorization 5, 3, 7."
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "At 1:45 , is there a way to find a GCF using prime factorization? If so, where can I find it?",
        "A": "There sure is! The GCF of any two numbers is the product of all the prime factors they share. For example, the GCF of 117 and 36 is 9 because they both have two factors of 3 and 3*3=9."
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "At 3:04 what does he mean by leave it alone. I don't get it",
        "A": "He meant that you leave it by itself, or do not use it."
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "at 2:03 why did he cross out the 3",
        "A": "Sal didn t cross out 3, he crossed out sqrt3.sqrt3. which is equal to (sqrt3)^2 = 3 [root3 * root3 = 3]"
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "At 0:59 I didn't get why he divided 3 by 117",
        "A": "No, he divides 117 by 3. He is doing this to find 117 s prime factors."
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "At about 2:00, how does this factor method work?",
        "A": "This method works because of the Fundamental Theorem of Arithmetic,in which all numbers can be rewritten in a unique series of prime numbers,for example,45 is equal to 3*3*5."
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "I don't understand what Sal did ant 2:05.",
        "A": "Right before this point, we had this equation: 5 * \u00e2\u0088\u009a(3 * 3) * \u00e2\u0088\u009a(13) After that, Sal just simplified \u00e2\u0088\u009a(3 * 3) into 3. This can be done because 3 * 3 = 9, and the square root of 9 is just 3. So he simplified that to 3, then multiplied it by the 5, and got the answer. It looks like this: 5 * \u00e2\u0088\u009a(3 * 3) * \u00e2\u0088\u009a(13) = 5 * 3 * \u00e2\u0088\u009a(13) = 15 * \u00e2\u0088\u009a(13) Just remember that the square root of 3 * 3 is just 3, and the square root of 5 * 5 is just 5, and the square root of any number times itself is just that number. I hope this helps."
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "At 1:39, he says \"as we know from our exponent properties\" and shows 5 sqrt 3 x 3 sqrt 13. What I never understood was if the 13 can be in its own square root, why does the first two 3's have to be together? I know that 5 sqrt 3 sqrt 3 sqrt 13 does not yield the same answer, but was wondering why the 13 at the end could be on its own, yet the first two 3's had to be together and can't be separate. sorry im not good at maths",
        "A": "First, we must define the property he mentioned, which is Sqrt (ab) = Sqrt(a) Sqrt(b) Now to answer your question, the sqrt (3x3) doesn t have to be together. Sqrt(117) could be written as Sqrt(3x13) Sqrt(3), or Sqrt(3)Sqrt(13)Sqrt(3). The reason he put them together is called grouping so it s easier to simplify. Sqrt(3x3) is the same as sqrt(3)sqrt(3) or Sqrt(3^2), and when we square a squareroot or taking squareroot of a square number, they cancel out. Hope that clarifies it up. :)"
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "At 3:08 Sal says that he can't simplify the problem 3 times the square root of 26. How come?",
        "A": "The square root of 26 can t be simplified because 26 is not a whole-number multiple of a perfect square. Example: 3\u00e2\u0088\u009a48 = 3\u00e2\u0088\u009a(16 * 3) = 3(\u00e2\u0088\u009a16 * \u00e2\u0088\u009a3) = 3(4 * \u00e2\u0088\u009a3) = 12\u00e2\u0088\u009a3"
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "So, in order to simplify a square-root further, there must be an evenly squared number within it? @ 2:46",
        "A": "Yep ...For example the \u00e2\u0088\u009a20 also means \u00e2\u0088\u009a5*4 Since 4 is a perfect square you can square it =).. \u00e2\u0088\u009a4 =2 so you are left with 2\u00e2\u0088\u009a5"
    },
    {
        "video_name": "cw3mp8oNASk",
        "Q": "Wait I still don't understand. Where did Sal get the 15 from at about 2:24? My question is HOW?",
        "A": "The answer is pretty simple. once he had::: 5*\u00e2\u0088\u009a3*3 \u00e2\u0088\u009a13 to simplify we first find what the 3*3is.(which is 9) so we get: 5*\u00e2\u0088\u009a9 \u00e2\u0088\u009a13 now, we know that 9 is a perfect square and teh square root is 3. so: 5*\u00e2\u0088\u009a9 \u00e2\u0088\u009a13 turns into 5*3 \u00e2\u0088\u009a13 now we simplify 3*5 and THAT S how he got 15 at around 2:24 :) Hope This Answers Your Question!!!"
    },
    {
        "video_name": "pWtt0AvU0KA",
        "Q": "At 10:32 you say that c is a function of \u00ce\u0094x.\nWhat do you mean by this? Isn't c is a function of f(c)?",
        "A": "He means that c is determined by the \u00e2\u0088\u0086x because it always within the \u00e2\u0088\u0086x difference. Also, c is a function of f(c) is a common error. Notice that c is in the parenthesis. What really happened is that f is a function of x \u00e2\u0080\u0093 i.e. f(x) \u00e2\u0080\u0093 and c is just a particular value of x."
    },
    {
        "video_name": "pWtt0AvU0KA",
        "Q": "5:45; Is there a video on the mean value theorem for definite integrals?",
        "A": "The Mean Value Theorem for Integrals basically says that under any function, there is at least one rectangle (c) that will have the same height as the average of all the heights."
    },
    {
        "video_name": "pWtt0AvU0KA",
        "Q": "At 5:38 you can skip to the end via intuition without even going into the mean value theorem or the squeeze theorem. The area of a rectangle divided by its length is obviously its height. As \u00ce\u0094X approaches 0, the integral of f(x) better and better approximates a rectangle of height f(x).\n\nNot a question, but I thought more people might see it here than under \"Tips & Thanks\".",
        "A": "This whole thing just clicked for me with your comment. Thank you :)"
    },
    {
        "video_name": "pWtt0AvU0KA",
        "Q": "at 8:14, a 'c' appears to be selected that is an equal distance between x and x + delta(x). does 'c' need to be an equal distance between the two endpoints?\n\nIf I arbitrarily place a 'c' very close to (x + delta(x)) (still satisfying x <= c <= (x + delta(x)) (imagine a sharp downward curve between the two endpoints), then f evaluated at 'c' (height) * delta(x) (rectangle area) does not appear to be equal to the area between the two endpoints, it appears to me to be smaller.",
        "A": "In this video Sal happens to have chosen a value roughly equal distance between a and b to illustrate point c, but it would be coincidental if c (as defined for purposes of this video) actually appeared at that midpoint. In fact, there can be more than one point satisfying the MVT, which merely tells us there is at least one such point."
    },
    {
        "video_name": "Cz_GWNdf_68",
        "Q": "At 11:38, Sal says F(x) is 'an' anti-derivative of f(x). How can a single curve have multiple anti-derivatives?",
        "A": "When you take the derivative, if there is a constant term, it disappears. so you re losing information. When you find the antiderivative, you don t know what the constant term was, which is why a single curve has (infinitely) many antiderivatives."
    },
    {
        "video_name": "Cz_GWNdf_68",
        "Q": "At 8:20 when Sal write n \u00e2\u0080\u0094> infinity , is this +infinity or -infinity or both ?",
        "A": "it is the same way it is with numbers. 5 = +5 10 = +10 infinity = +infinty"
    },
    {
        "video_name": "1uWZNW5PF-s",
        "Q": "Will all due respect, I think your answer might be wrong. I got 8x 5y at 5:00 mins in. Please correct me if I'm wrong.",
        "A": "I made the same mistake during my calculation. At least in my case, it arose because I was trying to evaluate that square by comparing it to the one on the right (the one Sal solves last). You gotta see that the side of the square doesn t quite cover the whole side of the y by y square (the one from the beginning)."
    },
    {
        "video_name": "1uWZNW5PF-s",
        "Q": "Why did you pick 5 for y at 8:40",
        "A": "X and Y are a ratio so they can be any values you want as long as they are constrained by the ratio. Picking whole numbers makes the rest of the problem easier, and at the end you get integer results so you re done."
    },
    {
        "video_name": "oUgDaEwMbiU",
        "Q": "At 2:44, I understand why x cannot equal 2, but why can it not equal -2?",
        "A": "Because he cancelled out an (x + 2) factor earlier. -2 was not allowed in the original equation, so you can t allow it in your simplified equation."
    },
    {
        "video_name": "oUgDaEwMbiU",
        "Q": "Why didn't Sal use the limit way of defining continuity at 3:30?",
        "A": "he did. A func is continuous at c if limit when x goes to c of f(x)= L=f(c) In our case x=-2 and he calculates f(-2) . And then ofc limit when x goes to -2 of f(x) is -3/2"
    },
    {
        "video_name": "XGpEHj43kcc",
        "Q": "At 3:33 Sal says that Theta is between pi and 2pi, so the angle is going to be in the third or fourth quadrant.\n\nI'm confused how Sal came to this conclusion. We're told the cosine of theta is negative, so surely this means the angle is in the second or third quadrant?",
        "A": "Theta between pi and 2pi is given as part of the problem. Without that information we wouldn t be able to tell if we were in the 2nd or 3rd quadrant."
    },
    {
        "video_name": "XGpEHj43kcc",
        "Q": "At 0:10, Sal says \"we can assume [phi] is a positive acute angle...\" What is he basing that statement on?",
        "A": "He realised that the question would not have a unique solution unless this assumption is made. However, somebody mentions below that the word acute may carry with it an implication of positivity, therefore this may just have been a clarification."
    },
    {
        "video_name": "XGpEHj43kcc",
        "Q": "at 0:17 why is fie or whatever that symbol is , can you assume that it is positive?",
        "A": "The formula works for any angles, positive or negative, however you need to know both the cosine and sine of the angles. If phi were in the fourth quadrant, then the sine of the angle would be -24/25 instead of 24/25, and the answer would be different."
    },
    {
        "video_name": "PPvd4X3Wv5I",
        "Q": "Is the first step that Sal does valid in all equations?\nAt 1:10 what is the rule for multiplying the -3 and then adding the x?",
        "A": "At 1:10 he did distribution."
    },
    {
        "video_name": "PPvd4X3Wv5I",
        "Q": "at 0:48 why does Sal put -3x-2 in brackets",
        "A": "hes taking them out"
    },
    {
        "video_name": "iATjsfAX8yc",
        "Q": "At 0:39, he says that the line segment will give the shortest distance between the two points, but isn't the shortest distance between two points zero (by folding space)?",
        "A": "Technicly, it is, but in this question it wouldn t make much sense to fold space. You can t really do that on the Eucliden plane, you must leave it the way it is. Though, you learn about folding and changing properties of spaces in Topology."
    },
    {
        "video_name": "iATjsfAX8yc",
        "Q": "How do i know that the question is about perpendicular? At 0:11",
        "A": "Perpendicular is means any line that is at a right angle with another line. While the two lines in this symbol here are parallel: = the two lines in this one are perpendicular. +"
    },
    {
        "video_name": "SkMNREAMNvc",
        "Q": "At 2:15 sal say \"satisfy\" but, what does satisfy mean",
        "A": "Essentially in this context, satisfy means to agree with the constraint . For example, we can say that a square satisfies the properties of a rectangle as it is a quadrilateral with 4 right angles. In the same way, we can say that \u00f0\u009d\u0091\u00a5 = 2 satisfies the equation \u00f0\u009d\u0091\u00a5 + 3 = 5 as it is a solution to the equation."
    },
    {
        "video_name": "SkMNREAMNvc",
        "Q": "At 2:30, isn't the answer supposed to be yes. It does satisfy all of the equations?",
        "A": "No. It satisfies the first equation, but not the second. In order for an ordered pair to be a solution of a SYSTEM, it has to satisfy both equations."
    },
    {
        "video_name": "kQF9pOqmS0U",
        "Q": "I don't understand what Sal does at 4:03, why does he change the r^2 to r(t)^2?",
        "A": "Sal is just trying to nail home the point that the radius of the circle has some relationship with the time that has passed in our situation. The relationship A=\u00cf\u0080r^2 still stands, but now we know that the radius is dynamic."
    },
    {
        "video_name": "kQF9pOqmS0U",
        "Q": "From 4:29 to 4:38: What does he do there?",
        "A": "Have you ever heard of the chain rule? It says that if you have a function f(g(x)) then you can take the derivative d/dx (f(g(x)) by saying u=g(x) then evaluating df(u)/du * du/dx. Sal is basically doing this, but he is not substituting in the u. So, when he wants to evaluate, d (r(t))^2/dt, that is the same thing as d (r(t))^2/d(r(t)) * d r(t)/dt. If I said u=r(t), I can rewrite it as du^2/du*du/dt. It looks much simpler this way, but Sal just didn t take the time to make the substitution."
    },
    {
        "video_name": "kQF9pOqmS0U",
        "Q": "at 6:20 Sal puts 3 as the value of 'r' (radius)...and that gives us the slope of the tangent line when the radius of the circle is 3cm. And, when the radius increases to a 4cm or 5cm we substitute that as the value of 'r', right?!",
        "A": "Yes. Since A = pi*r^2, that means dA/dt = 2*pi*r*dr/dt, always, for any value of r. If dr/dt remains constant at 1 cm/sec, then as r increases from 3cm to 4cm to 5cm, then dA/dt will also increase, since the r is increasing."
    },
    {
        "video_name": "HKUJkMQsGkM",
        "Q": "At 1:35, I don't get how Sal has to do that to add the rational fractions. Can someone please explain this a bit clearer?",
        "A": "He multiplied each number by the denominator of the other fraction, if that makes sense. He did that to get the denominator of the fractions the same, which makes them easily to add and subtract."
    },
    {
        "video_name": "IjMn7f6bbLA",
        "Q": "Can you right ratios like the following: 3 to 4, 3/4, 3.4, 3:4? I learned those ways in school.",
        "A": "Kylie4706, 3.4 is not a ratio. It s just a decimal. All the other ways correct."
    },
    {
        "video_name": "IjMn7f6bbLA",
        "Q": "at 0:57 how did he automatically get the right answer",
        "A": "The ratio is determined on how many x there are for every y. He simply looked on the table for which numbers the question asked for, and put them in. Since the question asked for the ratio between those in Round-up and Slingshot, he just had to look at how many people where in those two. Since there are 11 people waiting for Round-up and 4 people waiting for slingshot, there are 11 people waiting for round-up for every 4 people waiting for slingshot. This is denoted as 11:4."
    },
    {
        "video_name": "IjMn7f6bbLA",
        "Q": "At 2:05, don't we have to simplify?",
        "A": "No because 4 and 21 don t have any common factors and since the only factor is actually 1 and so we don t need to simplify. Nice question! :)"
    },
    {
        "video_name": "IjMn7f6bbLA",
        "Q": "The ratio of girls to boys in a chess club was 5:4 .these was 32 boys how many girls were there in the club?",
        "A": "Let x be no. of girls no. of girls : no. of boys = x : 32 5 : 4 = x : 32 5/4 = x/32 5/4 \u00c2\u00b7 32 = x Five multiplied by 32 divided by 4 is 40 Therefore, the number of girls in the chess club is 40."
    },
    {
        "video_name": "X6zD3SoN3iY",
        "Q": "@3:07 Why did the denominator get the B?",
        "A": "Because there is a b to the first in there B^1*B^2=B^3 however there is only a B^2 on top which means that you B^-1 which ends up in the denominater"
    },
    {
        "video_name": "X6zD3SoN3iY",
        "Q": "1:17 What happened to the 1? Isn't it supposed to be in the denominator?",
        "A": "Since the result any number that is multiplied or divided by one is the number, (eg 5*1=5 and 4/1=4) the convention is to not write the one. It makes things easier to read."
    },
    {
        "video_name": "X6zD3SoN3iY",
        "Q": "At 3:03 I notice that the final product of the equation is 5a^7/b but could we put a 1 after 5a^7 so that it looks like 5a^7 1/b? I am a bit confused so please forgive me if what I write doesn't add up to what is actually in the seminar. and if this is possible, can I then add a + symbol so that it looks like 5a^7+1 over b??\n\nThanks in advance for the help!",
        "A": "you are correct that 5a^7/b is the same as (5a^7)1/b. But No, you can not use a plus sign. It would be 5 times (a^7) times (1/b). or 5*(a^7)*(1/b). I hope that helps."
    },
    {
        "video_name": "0DBDuNYsciY",
        "Q": "At 2:30, is N equal to the i in the video? N = the base, or start. Right?",
        "A": "Generically: n \u00e2\u0088\u0091 a(i) = a(m) + a(m + 1) + ... + a(n - 1) + a(n) i=m i represents the index of summation a(i) is an indexed variable representing each successive term in the series m is the lower bound of summation n is the upper bound of summation The i = m under the summation symbol means that the index i starts out equal to m. The index, i, is incremented by 1 for each successive term, stopping when i = n"
    },
    {
        "video_name": "WNuIhXo39_k",
        "Q": "At 4:33, when Sal introduces \"length\", is he talking about the magnitude of the vector? He uses the double bar notation ( II A II ), which made me unsure about what he was referring to since the magnitude notation is just the same as the absolute value notation.",
        "A": "||a|| is the same as |a| when dealing with vectors. They both refer to the magnitude of the vector. So yes."
    },
    {
        "video_name": "WNuIhXo39_k",
        "Q": "at 8:20 he says that the length definition (square root of a^2 and so on) is the same as the value of multiplying vector a by itself. How are they the same? The length definition is the square root of a squared number, while multiplying a by itself is just a^2. There's no radical at all. Am I missing something obvious?",
        "A": "He s saying that the inside of the square root is equivalent to a dot a, not that the whole thing is equivalent."
    },
    {
        "video_name": "WNuIhXo39_k",
        "Q": "So at 4:33 we got the number 13. Which is the result of our Dot Product... and the result of our Dot Product was going to be a Scalar. So 13 is a Scalar that is used where? To define the length of our orthogonal line? At Cross Product we learn that (a x b = c. c is orthogonal with a and b) So my question again is: Is 13 going to be used as a *b = 13c?",
        "A": "No, 13 is just a scalar. It is not the length of any line or anything like that. You can think of it as the length of the projection of a onto the span of b, multiplied by the length of b."
    },
    {
        "video_name": "WNuIhXo39_k",
        "Q": "i thought it would be at 3:32 2x7+2x1 5x7+5x1",
        "A": "Not when you accept the definition of a dot product which is simply the sum of the products of vector elements with the same index."
    },
    {
        "video_name": "WNuIhXo39_k",
        "Q": "At 2:50 why the result is AB+...+AnBn? I didn't get it",
        "A": "That is how it is defined. That is not something you derive from something else. It is like defining multiplication of two numbers a and b to be the number a added to itself b times."
    },
    {
        "video_name": "AZroE4fJqtQ",
        "Q": "At 0:46 you talk about seeing if it's Linear by dividing the change in Y by the in change X. I did not understand that?",
        "A": "It means he is dividing the amount between the Y numbers by the amount between the X numbers. Example: X Y 14-11= 3 2-1=1 1 11 2 14 3/1= 3"
    },
    {
        "video_name": "AZroE4fJqtQ",
        "Q": "At 2:04 Sal indicated that the numbers are non-linear because it's going up by huge amounts. Can you explain that to me.",
        "A": "Linear is a multiplicative relationship like, 5 to 10 to 15 and non linear are additive relationships like 3 to 7 to 17"
    },
    {
        "video_name": "SP-YJe7Vldo",
        "Q": "at 1:37 I got curious. can we create a new type of number (like we did with imaginary numbers) to describe an imaginary number multiplied by a real number?",
        "A": "We don t need to. Imaginary * real is imaginary."
    },
    {
        "video_name": "SP-YJe7Vldo",
        "Q": "At 3:00, why do we plot the vertical axis as the imaginary number and the horizontal axis as the real number?",
        "A": "This is the way we usually picture these coordinates, but nothing prevents you from doing otherwise, as long as you remember which coordinate represents which value. When communicating with others, it s probably best to stick with the convention: Otherwise, there could be some serious misunderstandings !-)"
    },
    {
        "video_name": "SP-YJe7Vldo",
        "Q": "At the beginning, Sal explains that I have been learning real numbers all my mathematical life. Then, he gives some examples of them at 0:06: \"zero (0), and one(1), and zero point three repeating(0.3 repeating), and pi (\u00cf\u0080), and e (e)\", and I do not know what \"e\" is. What is e? Does it have an absolute value like 5 or 5.344543 or phi (\u00cf\u0095)? Does it have any connection to i?",
        "A": "e is the limit of (1 + 1/n)^n as n approaches infinity and is approximately 2.71828..., and like pi is irrational and transcendental. You will most likely encounter it when you get into exponents, particularly when calculating compound interest. It has a lot of interesting properties, one of the most useful being that the derivative of e^x is e^x."
    },
    {
        "video_name": "SP-YJe7Vldo",
        "Q": "0:13 what is e?",
        "A": "This is Euler s number, or the natural base. It is the base of the natural logarithm and the natural exponential, and it has a value of approximately 2.718. It is irrational, but that doesn t keep it from making perfect sense! {(-:]"
    },
    {
        "video_name": "SP-YJe7Vldo",
        "Q": "At 2:27, how is '3' the multiple of 'i'? It should be '3i', right?",
        "A": "The imaginary part is the real number that is multiplying the imaginary unit. Both the real and the imaginary parts are real numbers."
    },
    {
        "video_name": "SP-YJe7Vldo",
        "Q": "At 2:50 Sal said that we plot the imaginary part on the Y-axis and the real part on the X-axis. Is it mandatory to plot it like that or can we change the conventions,i.e., can we plot imaginary numbers on the X-axis and real numbers on the Y-axis?",
        "A": "Yeah! Just don t do it on a test, because teachers will be looking for the conventional style."
    },
    {
        "video_name": "SP-YJe7Vldo",
        "Q": "Does it always work that way (At 2:45) when graphing? Is the imaginary axis always the vertical and real always the horizontal?",
        "A": "Yes that is right. It s the same situation as the X-axis always being the horizontal one and the Y-axis always being vertical."
    },
    {
        "video_name": "SP-YJe7Vldo",
        "Q": "at 0:08 Sal says e. I do not know what that really is. Could you please explain?",
        "A": "e is just a constant, like pi. It is about 2.71828, but goes on forever and never repeats."
    },
    {
        "video_name": "SP-YJe7Vldo",
        "Q": "At 1:08, why does the letter Z tend to be the most used variable for complex numbers?",
        "A": "This is just a convention."
    },
    {
        "video_name": "SP-YJe7Vldo",
        "Q": "At 3:02: Would there be a time in which the real number is the vertical axis (y axis) and the imaginary number as the horizontal axis (x axis)?",
        "A": "It s by convention only that we usually draw the real axis on the horizontal, and the imaginary axis on the vertical: we could do it the other way around without changing anything. Likewise, we could use the letters x and y differently. The results would be the same, just as 2+3 would still be 5 even if we used the character & to represent the number 3, and the character % to represent the number 2; in that case we would write &+% =5, and we would be expressing the same mathematical fact."
    },
    {
        "video_name": "Io9i1JkKgN4",
        "Q": "i dont get it at 0:52",
        "A": "You don t get what"
    },
    {
        "video_name": "Io9i1JkKgN4",
        "Q": "Why did Sal, in the 54/81 problem at 1:51 make a dot, erase it, and then write x?",
        "A": "Sal changed the symbol at 1:51 because a dot is used as an x for multiplication in an equation (x might miss-lead you as a number represented as a letter.), but he changed it because he is doing fractions."
    },
    {
        "video_name": "Io9i1JkKgN4",
        "Q": "At 1:51 write x right?",
        "A": "Or you could use the parenthesis or the dot, but yeah."
    },
    {
        "video_name": "Io9i1JkKgN4",
        "Q": "Why did Sal, in the 54/81 problem at 1:51 make a dot, erase it, and then write x?",
        "A": "In algebra, that uses x as a placeholder for a number, multiplication problems are writen with a dot, instead of an x. Sometimes when people are used to doing algebra, they forget, and use the dot instead of an x. That is probably why Sal accidentally wrote a dot. If you are in 3rd grade, or just beginning to study math, use the x instead. It s easier not to get confused."
    },
    {
        "video_name": "Io9i1JkKgN4",
        "Q": "in 0:42 i dont get what he meens about cancling out nummbers?and in 2:05 i dont get it either? and in 2:11 i dont get it how he divides fractions? but i do get it at 2:28.",
        "A": "Cancling out numbers means that you are dividing a number by itself. Ex: 7 divided by 7=1 In 2:05, he s dividing with a common factor for 6 and 9. That gives him the simpler form of 6/9 which=2/3"
    },
    {
        "video_name": "GiGLhXFBtRg",
        "Q": "Whoa!! This video is exceptionally hard, and I am LOST!! I don't even know how the z axis actually works, and then I don't understand why Sal is using the square root formula at \"6:11\"........ Can someone help, please? Thanks.",
        "A": "That formula is the distance formula. It is explain in the video distance formula in the algebra playlist."
    },
    {
        "video_name": "GiGLhXFBtRg",
        "Q": "I don't understand the coordinates at 3:40. Is it 3D?",
        "A": "x,y,z implies 3D space."
    },
    {
        "video_name": "GiGLhXFBtRg",
        "Q": "I can see that Sal is using the distance formula/pythagorean theorem at 6:00. But I thought the pythagorean theorem was only for right triangles. I don't see a right triangle in connection with the orange line. Is there a right triangle?",
        "A": "Well he isn t using the pythagorean theorem, he is using the distance formula. That can be used for any two points in space or in a plane. He isn t doing anything with triangles when he uses this formula. It can be used for any two points."
    },
    {
        "video_name": "GiGLhXFBtRg",
        "Q": "At 3:21, what is Sal talking about? I don't understand what he is talking about?",
        "A": "At that point of the video, Sal is drawing a three-dimensional triangle. He states that he is not making any assumptions of what type of triangle it is because if it were a equilateral or isosceles triangle, calculations might have been different."
    },
    {
        "video_name": "GiGLhXFBtRg",
        "Q": "At 3:02 what were you talking about dimensions?",
        "A": "Can any one prove here that all the median always pass through a single point. Or in other words the every third median will pass through the same point of intersection of first two medians."
    },
    {
        "video_name": "GiGLhXFBtRg",
        "Q": "What does arbitrary mean at 0:11?",
        "A": "Arbitrary means based on random choice."
    },
    {
        "video_name": "GiGLhXFBtRg",
        "Q": "At 6:44 how did he get 4C^2/9?",
        "A": "(C/3) - C = -2C/3 (-2C/3)^2 = (-2)^2 x C^2 / 3^2 so it s 4C^2/9"
    },
    {
        "video_name": "GiGLhXFBtRg",
        "Q": "At 4:20. This proof seems to depend on the coordinates of the centroid being located at (a/3,b/3,c/3). I can't figure this out, and I don't like to take things on faith, especially in math. Can someone tell me why this is so or where I can find the answer?",
        "A": "Each value is divided by 3 because that is the average. He summed all the coordinates for a, which were 3, and divided by that number, which is exactly what you do when you find an average: (a+0+0)/3. He did this for b and c and found the average values for each coordinate. This average value gives the centroid."
    },
    {
        "video_name": "ZiqHJwzv_HI",
        "Q": "At 1:35, why is a line that is tangent to a circle perpendicular to the radius of that circle?",
        "A": "It is because any line that is tangent to a circle only touches one point on the circle and a line from any point on the circle to the center of the circle is a radius. Therefore, any line that is tangent to the circle would also be perpendicular to a radius."
    },
    {
        "video_name": "27vT-NWuw0M",
        "Q": "The pink vector called (x-proj(x)) @4:28, does it matter which way it points? Would it be the same if Sal drew it from x to the line L, instead of the opposite which he actually did in this video?",
        "A": "The dot product, and the value of c that you get from the equation, would be the same."
    },
    {
        "video_name": "27vT-NWuw0M",
        "Q": "At 10:30, the line L is defined as the set of all scalar multiples of the vector [2,1].\nCan we also think of L as the set or subset that is spanned by the vector [2,1]?",
        "A": "I think it s actually a subspace (since it s a span of vectors). The set L is a span of [2, 1], so, yes, except that I wouldn t call it a line. All the vectors in the set are parallel to the line. Also, all their endpoints form a (continuous) line, if all the vectors start at the origin. So I wouldn t say the vectors span a line, but their end points form a line. I m also uncertain about this."
    },
    {
        "video_name": "tZKzaF28sOk",
        "Q": "In 3:26 why can't you have E, out of curiousity??",
        "A": "You could have E but sal was just worried people would confuse it with the irrational number E which is equal to about 2.71..."
    },
    {
        "video_name": "_hrN4rVCOfI",
        "Q": "At 3:17 Sal gives 3 choose 3. This may be more of a question about the combinatorics formula, but 3!/3!(3-3)! should be undefined as it is 3*2*1/3*2*1(0), which I am simplifying as 1/0. Why does this become one for the coefficient? Is it something peculiar to combinatorics where an otherwise undefined fraction equals one?",
        "A": "0! = 1, not 0, so the fraction is defined and equals 1."
    },
    {
        "video_name": "_hrN4rVCOfI",
        "Q": "At 4:14 Sal said that 3 choose 2 = 3. I tried to count and the answer made sense, but the formula doesn't agree. n!/(n-r)! = 3!/(3-2)! = 6/1 = 6",
        "A": "The formula is n!/r!(n-r)! , not n!/(n-r)! so 3!/2!(3-2)! = 6/2(1) = 6/2 = 3"
    },
    {
        "video_name": "UhMM68fq9FA",
        "Q": "can i ask a question dian niovani sleeps at 8:00 pm and wakes up at 5:00 am of the next day. in the afternoon,she naps for an hour.how long does she sleep in minutes plzzz...",
        "A": "Ask yourself how many hours she sleeps. Now multiply by 60 to get the minutes."
    },
    {
        "video_name": "Dq0xFgQB9qo",
        "Q": "in 0:28 how did -2 become -8? .plz answer!",
        "A": "He used the Distributive Property example: 4(x-2) he did 4 * x and 4 * -2 I hoped this helped"
    },
    {
        "video_name": "_QTFeOvPcbY",
        "Q": "In this video, it is said that all squares are rhombus' (at 0:24). In another math book, and in the dictionary (online and Websters), Rhombus is defined as having 4 equal sides and NOT a square (or having right angles). I am wondering why this is different from what Khan is saying. Is a square a rhombus? Is there other texts that also say this? Dictionary.com says a square is a rhombus only in special cases. Any insights on this? Thanks.",
        "A": "one(The one in the math book and dictionary) is the exclusive definition meaning it CANNOT have right angles. Khan uses the inclusive definition of a rhombus and the same thing for a trapezoid. The definition being inclusive means that it includes the extreme case(Like a trapezoid being a parallelogram in some cases or a rhombus being a square if it has right angles."
    },
    {
        "video_name": "_QTFeOvPcbY",
        "Q": "how to do find the area of a rhombus if your given a perimeter 2cm and the angle bisector ratio is 4:3?",
        "A": "The length of a side of the rhombus is 1/4 the perimeter. Angle bisector ratio, in this case, can be used to figure out the length of the diagonal by applying the angle bisector theorem (the bisector of an angle divides the opposite side in the ratio of the sides which constitute the angle). The area of the rhombus, then, will be 1/2(diagonal1*diagonal2) [You can figure out diagonal2 by applying the Pythagorean theorem)."
    },
    {
        "video_name": "_QTFeOvPcbY",
        "Q": "At 3:33, when Sal divided the rhombus into 4 isosceles triangles, are all four of them congruent?",
        "A": "Yes, and it can be shown in several different ways. The diagonals act as transversals to the parallel sides, which makes alternate interior angles congruent. You also have some vertical angles where the diagonals meet. Since all 4 sides are congruent, the 4 triangles are all congruent by either ASA or AAS."
    },
    {
        "video_name": "KibTbfkoPTs",
        "Q": "at 3:04 where does he get the squaring from?",
        "A": "Sal is squaring 3 to get 9 and sqrt(2) to get 2."
    },
    {
        "video_name": "HnDvUaVjQ1I",
        "Q": "At 3:16, it says sin^2 \u00ce\u00b8+cos^2 \u00ce\u00b8=1.\nSo does it mean that sin \u00ce\u00b8=(1-cos^2 \u00ce\u00b8)^1/2 and cos \u00ce\u00b8=(1-sin^2 \u00ce\u00b8)^1/2?",
        "A": "At 3:16 right? yes, it is true that sin \u00ce\u00b8=(1-cos^2 \u00ce\u00b8)^1/2 and cos \u00ce\u00b8=(1-sin^2 \u00ce\u00b8)^1/2, we can achieve that by simple rearrangement of the equation."
    },
    {
        "video_name": "HnDvUaVjQ1I",
        "Q": "At 4:09, Sal mentions that sin^2(\u00ce\u0098) + cos^2(\u00ce\u0098) = 1.\nBut isn't it also true that sin(\u00ce\u0098) + cos(\u00ce\u0098) = 1? (Because when you take the square root of both sides, you get the above statement).",
        "A": "Almost right but if you square root both sides you get sqrt(sin^2(\u00ce\u0098) + cos^2(\u00ce\u0098)) = sqrt(1) which isn t the same as you said. One easy example is 3^2+4^2=25. If we do it your way, we get 3+4=5 which is not right. You have to square root the the whole expression on the left, not just one. Hope it helped!"
    },
    {
        "video_name": "HnDvUaVjQ1I",
        "Q": "At 2:23 why do you have to square sin and cos cant you just keep them as they are?",
        "A": "For one, the Pythagorean theorem calls for squares of the lengths. The other point being that if it were just sin(x) and cos(x), then you would have a/c + b/c, which you can t do anything with."
    },
    {
        "video_name": "feNWZEln6Nc",
        "Q": "At 1:28, it looks like 3 times 7 divided by 21.",
        "A": "That is 1. 3 x 7 is 21. 21/21 is 1"
    },
    {
        "video_name": "feNWZEln6Nc",
        "Q": "what did he say at the time of 1:06",
        "A": "he is saying five but he cuts it off and says seven"
    },
    {
        "video_name": "feNWZEln6Nc",
        "Q": "And at 2:52 why is it 9cm?",
        "A": "You first break it into two parts s either way, then you find the volume for each piece, then you add the number that you find and that is that answer"
    },
    {
        "video_name": "feNWZEln6Nc",
        "Q": "yo sal said a naughty word at 1:06",
        "A": "I think me meant to say five, but you never know..."
    },
    {
        "video_name": "zVsGtU8lIWs",
        "Q": "At about 4:14 Sal says he is talking about the image of our pre-image under S. I thought S was a set. Aren't we taking the image under T, the transformation? I thought when we take a transformation T of S, we call it the transformation of S under T. Am I wrong?",
        "A": "I believe Sal misspoke, and should have said the image under T of the pre-image of S, as you stated."
    },
    {
        "video_name": "9k97m8oWnaY",
        "Q": "At 18:42, how is it that the sum of all the little parallelograms can be compared to finding the area of the region in the s-t plane?",
        "A": "The region in the s-t plane is the surface, so summing up the areas of the little parallelograms on that region would give you the surface area."
    },
    {
        "video_name": "jFd-6EPfnec",
        "Q": "i don,t get 1:49.",
        "A": "How so?"
    },
    {
        "video_name": "jFd-6EPfnec",
        "Q": "At 5:50 Sal comments that the mathematics community should decide on which term to use: GCD or GCF. Why are there 2 terms in the first place? Does any one know if one term is more widely used than the other?",
        "A": "I think it s GCF."
    },
    {
        "video_name": "jFd-6EPfnec",
        "Q": "I still have trouble with 1:31 I don't understand the difference of Greatest common factor(GCF), and Greatest common divisor(GCD).",
        "A": "Thank you, Mr. Khan said you can do either Greatest Common factor(GCF) or Greatest common divisor(GCD) thank you Pennie."
    },
    {
        "video_name": "ukPjc3Oyad4",
        "Q": "I have tried to learn this concept for a long time. Could someone please give me a hint. I feel like I would be a failure in life if I don't get this. I know what the base and height are but this concept just doesn't make sense. Sorry Sal, at 0:05 I don't get inspired, I have before but not anymore.",
        "A": "It s alright, I feel like this too."
    },
    {
        "video_name": "ukPjc3Oyad4",
        "Q": "at 1:53, he breaks up the triangle into two. Is there another place where he could have broken it up?",
        "A": "Yes, there s another place that you could ve cut the triangle. Sal just cut it in that specific place to make the problem more simple, as the only other place you should be able to cut goes into decimals."
    },
    {
        "video_name": "ukPjc3Oyad4",
        "Q": "At 4:31, can you have a rectangle big enough so that that strategy doesn't work?",
        "A": "No you can t. You can always make the grid bigger"
    },
    {
        "video_name": "ukPjc3Oyad4",
        "Q": "At like 1:08 he said it's like you dropped a penny. Do you have to account for wind speed and gravity, and all the other things to find the height?",
        "A": "No, distance does not change based on wind speed and gravity. Velocity changes based on those factors."
    },
    {
        "video_name": "ukPjc3Oyad4",
        "Q": "At 3:06, how do you multiply using 1/2?",
        "A": "just divide by 2 because 10/1 time 1/2 = 10/2 or 5 that is the same as 10 divided by 2."
    },
    {
        "video_name": "SdHwokUU8xI",
        "Q": "At 4:24 why is it 2pi+pi and the other 2pi? They both seem to increase the same, by 2pi(n) right?",
        "A": "Yes they both increase by 2 pi, but the first one will hit all even multiples of pi. Since the other one starts from 1 pi and adds multiples of 2 it will hit all odd multiples of pi. Try plugging in a few numbers for n like 0, 1, 2, 3 ... and see for yourself what happens."
    },
    {
        "video_name": "SdHwokUU8xI",
        "Q": "At 0:58 why does Cos \u00cf\u0080/2 equal to 0?",
        "A": "\u00cf\u0080 is equal to 180 degrees. Therefore 2\u00cf\u0080 equals 360 degrees, which is along the x axis. So looking at the unit circle Cos is equal to X."
    },
    {
        "video_name": "SdHwokUU8xI",
        "Q": "At 1:06, he says the x-coordinate is at 0. However, it looks like x is at 1 and the y-coordinate is at 0. Am I missing something here?",
        "A": "On the unit circle, cos(1/2pi) = 0. Since on the graph y = cos(x) and x = theta, it looks like x isn t equal to 0, but what he s graphing is theta. On the unit circle, though, x = 0."
    },
    {
        "video_name": "SdHwokUU8xI",
        "Q": "at 2:45 how come N cannot be a decimal or fraction because an integer cannot be a fraction or decimal and Mr.Khan describes N as a integer which means N cannot be a decimal or fraction I think Mr.Khan meant a rational or irrational number? What do you think?",
        "A": "He stated it correctly, n MUST be an integer. That has to do with the period of the cosine function. The pattern of the cosine function repeats its self every 2\u00cf\u0080. So the n represents how many periods you go through and only a complete period is enough. So, n must be an integer. In other words: cos (x) = cos [ x + 2\u00cf\u0080 (any integer) ]"
    },
    {
        "video_name": "SdHwokUU8xI",
        "Q": "4:40 Couldn't you also say pi*n, for n of all odd integers?",
        "A": "Yes, but it s a bit clunkier. Sal s solution is more conventional. We tend to like to define n as a member of the integers (a well-known set of numbers) and then write theta=(2n+1)pi, which takes care of the odd numbered multiples of pi we re after."
    },
    {
        "video_name": "SdHwokUU8xI",
        "Q": "At 2:05 about it states that when cos= 1 then theta is 0; I'm confused, I thought cos equalled one at 2\u00cf\u0080.",
        "A": "2pi and 0 are the same thing. 2pi is once all the way around the circle."
    },
    {
        "video_name": "2DBBKArGfus",
        "Q": "He wrote 3/6 at 3:09?",
        "A": "No, he just accidentally made a 5 look like a 6."
    },
    {
        "video_name": "2DBBKArGfus",
        "Q": "At 1:50 in the video, can't you express rates as : bottles/bathroom, if / stands for per?",
        "A": "The symbol / does not necessarily only stands for per. It also stands for dividing, and even the word or ."
    },
    {
        "video_name": "2DBBKArGfus",
        "Q": "At 3:47, why do you need to flip the fraction around, making it become 5/3 instead of 3/5? That changes the entire fraction! 3/5 is less than 1, but 5/3 is almost 2!",
        "A": "As you know, when you flip a fraction you are finding it s reciprocal. An example that might clarify things is 2L divided into 1/4L bottles. You know that dividing by 1/4 is the same as multiplying it by 4 because when you divide 2 into 1/4 pieces, you get 8 pieces and when you multiply 2 by 4/1 (flipped 1/4) , you also get 8. |=|=|=|=| |=|=|=|=| . This is the same for other fractions too, even when they don t really look flipped. I hope this clarifies things a bit for you!"
    },
    {
        "video_name": "BZxZ_eEuJBM",
        "Q": "At 2:38 can i take the conjugate of a+bi as -a-bi because sal says conjugate is the mirror image can the mirror image lie on the fourth quadrant",
        "A": "No, the conjugate is mirrored around the real axis and only around the real axis. The conjugate of a+bi is only a-bi."
    },
    {
        "video_name": "BZxZ_eEuJBM",
        "Q": "at 6:51, shouldn't it be -25 instead of +25? Since \"i\" squared will make it negative.",
        "A": "There are two negatives, so they cancel its 16 - (5i)^2. Inside the parenthesis = -25, so overall its 16+25"
    },
    {
        "video_name": "BZxZ_eEuJBM",
        "Q": "At about 1:19\nWhen Sal says, that b in bi is the scale of i. Does that mean it's like the slope in y=mx+b where m is the slope?\nAlso, is a conjugate just the complex numbers opposite?",
        "A": "When Sal says, that b in bi is the scale of i. Does that mean it s like the slope in y=mx+b where m is the slope? Yes, because like the linear equation y=mx+b, x is multiplied by m and that is why m changes x and scales it. Also, is a conjugate just the complex numbers opposite? According to Google... joined in a reciprocal relation, especially having the same real parts and equal magnitudes but opposite signs of imaginary parts. Sal also explains it at 2:50. Hope this helps!"
    },
    {
        "video_name": "BZxZ_eEuJBM",
        "Q": "At 4:14 Sal says that z+(conjugate) z =2X real part of the complex number, and its also gonna be equal to 2X the real part of the conjugate.\nI kno itz a stupid ques but I am stuck between words, pls help me out...",
        "A": "I got confused when Sal said that a number and it s conjugate are 2 times the real part of the number. It took me a while to realize he was adding them to get this answer, not multiplying. When you multaply them to get a real number you will not get 2 time the real part. Maybe that s where you were confused too?"
    },
    {
        "video_name": "BZxZ_eEuJBM",
        "Q": "@5:33 - Why is -10 the result instead of 10i^2?",
        "A": "i^2 = -1. -1x10=-10. :)"
    },
    {
        "video_name": "BZxZ_eEuJBM",
        "Q": "At time stamp 6:44 why does the speaker state that I squared is negative one for five I sqared. He did not stat that for the numberator.",
        "A": "i^2 is indeed (-1). You ll notice that earlier on he quickly mentions at 5:30 that 2i times 5i equals 10*i^2, or -10. This is because i^2 is (-1), and 10*(-1) is (-10)."
    },
    {
        "video_name": "BZxZ_eEuJBM",
        "Q": "At 2:19, why is the real axis the the x-axis and why is the imaginary axis the y-axis?",
        "A": "By convention."
    },
    {
        "video_name": "5AWob_z74Ks",
        "Q": "At 14:07, when Sal expands ||a-b||^2 to ||a||^2+2(a*b)+||b||^2 why isn't the central term, 2(a*b), squared like the surrounding two terms?",
        "A": "It s part of the FOIL distribution method, and distribution in general. It s supposed to be like that."
    },
    {
        "video_name": "5AWob_z74Ks",
        "Q": "I'm pretty sure the purple diagram @13:00 is not right. C^2 is the hypotenuse. he drew A^2 as the hypotenuse when illustrating the cosine law.",
        "A": "No, it s correct- A, B, and C can be interchanged in the Law of Cosines. C^2 doesn t have to be the hypotenuse, it can be A^2 or B^2 as well."
    },
    {
        "video_name": "5AWob_z74Ks",
        "Q": "At 7:17 why sal changed the direction of vector a-b to b-a ?",
        "A": "Because that s the vector a-b in the opposite direction. He say this at 6:55. To reverse a vector v, you multiply it by negative one, so reversing (a-b) we do: -(a-b) = -a+b = b-a."
    },
    {
        "video_name": "5AWob_z74Ks",
        "Q": "Why it is at 17:19 (A . B) is equal to ||A|| ||B|| cos\u00ce\u00b8 ? not equal to -||A|| ||B|| cos\u00ce\u00b8 ?",
        "A": "Because in the step before, you divided by -2 both sides, giving you (A.B)= ||A|| ||B|| cos\u00ce\u00b8"
    },
    {
        "video_name": "5AWob_z74Ks",
        "Q": "@ 20:17 : Isn't it forbidden for 0 = 0 * cos to bring it straight to cos = 0/0 since you can't divide by zero?",
        "A": "When we divide by a number, we multiply by the multiplicative inverse of that number. By axioms of the real number system (think of them as indisputable rules of a game) there is no multiplicative inverse of 0. So division by zero isn t so much forbidden, but rather it is impossible because it is not structured into the real number system."
    },
    {
        "video_name": "5AWob_z74Ks",
        "Q": "At 23:40 Sal tells us that the zero vector is orthogonal to all other vectors.\nIf the zero vector is orthogonal to all other vectors, doesn't that arise from R0 being orthogonal to R1 in the same way that in constructing the second dimension of R2 one sets the x any y axes orthogonal to each other?",
        "A": "By definition, orthogonal is the name given to the relationship between two vectors described when their dot product is 0. The dot product of the 0 vector with any other vector is 0, so by defintion the 0 vector is orthogonal to every other vector. In other words it just has the right properties for that definition (which someone made up).. there is no more to it than that."
    },
    {
        "video_name": "5AWob_z74Ks",
        "Q": "At 22:33 Sal says that the zero vector is undefined, but it is defined, isn't it?\nI understand that the angle between anything and the zero vector would be undefined since the length of the zero vector is arguably 0, but the zero vector is clearly defined in Rn right?",
        "A": "He means that the line on which the zero vector lies is undefined and the angle between the zero vector and any other vector is undefined. So yes, the zero vector is definitely defined in Rn. Also, the length (or magnitude) of the zero vector is 0, not just arguably."
    },
    {
        "video_name": "5AWob_z74Ks",
        "Q": "If we wanted to solve for the angle between two known vectors, would it not be easier to rearrange the equation Sal explains at 16:54? Instead you could have it as \u00ce\u00b8= arccos((a\u00e3\u0083\u00bbb) /(||a||*||b||)",
        "A": "Yes. that s perfectly valid. I m surprised Sal didn t quickly show that here given the title of the video! For some reason he wanted to focus mainly on the perpendicular case."
    },
    {
        "video_name": "9iulv2QvKwo",
        "Q": "What dose sal mean at 2:07",
        "A": "He means exactly what he said. You have 11 kilometers that you are converting to meters. There are 1,000 meters in 1 kilometer. So he basically just said it two different ways. 11 kilometers, or 11,000 meters. Hope this helps!"
    },
    {
        "video_name": "9iulv2QvKwo",
        "Q": "2:07 I don't know how to kilo to ceni",
        "A": "kilometer = 1,000 meters meter = 100 centimeters You can figure out how to convert using those values."
    },
    {
        "video_name": "jQ15tkoXZoA",
        "Q": "There is one but. In the video (2:43) it said that after the price for topping became 2$, the equation must be 8+2(p+8)/p+8. While I was doing the exercises I got the same problem, but in the hints the answer was that the second part should be 24+2p/p+8. Where did they get 24 and how?",
        "A": "8+2(p+8) = 8+2p+16 = 24+2p"
    },
    {
        "video_name": "jQ15tkoXZoA",
        "Q": "At 7:45 he throws (p+16) (p-8) = 0 I don't understand where these numbers came from. And if they did come from 125 being divided, then where did the 'p' come from?",
        "A": "In the equation p^2 + 8p - 128 equation, he is factoring (breaking down the problem) by using the FOIL method. In short, he has to find out what number will make the middle number (8p) and last number (128) be correct. Google the FOIL method and you will see what I mean. Practice FOIL and you will get it down pat. Good luck with your studies!!"
    },
    {
        "video_name": "jQ15tkoXZoA",
        "Q": "at 4:51 how do I know which set of numbers to add/subtract from each side? Real confused about it :/ and how do you know weather to add/subtract?",
        "A": "It doesn t really matter. Just get everything on one side however you want to do it. Go back and view the Quadratic equations playlist before these, if you need review."
    },
    {
        "video_name": "jQ15tkoXZoA",
        "Q": "At 6:45 why is +64 multiplied by -2 shouldn't 0 be multiplied by -2?",
        "A": "The entire equation must be multiplied by -2 to have an equivalent equation. This includes the 64 and the zero. But, remember, 0*(-2) is still zero. So, the 0 stays 0."
    },
    {
        "video_name": "kT4Mp9EdVqs",
        "Q": "Essentialy does it matter which order you multiply in? Such as at 4:51, instead of 5(-1) + 3x7, can you do 5x7 and 3(-1)??",
        "A": "Yes, you can t change the order because you won t get the same result. If you multiply 5x7 and 3(-1) you will have 32 instead of 16 and you won t have the same matrix."
    },
    {
        "video_name": "rVQ3G9epCjw",
        "Q": "At 7:55, shouldn't proving Associativity be proving that (V.W).X = V.(W.X) (i.e. all three are vectors), rather than that (cV).W = c(V.W)?",
        "A": "Neither of your statements is correct. He s showing the associativity of the dot product over vector addition (with all 3 items vectors): For any 3 vectors a, b, and c, cdot(a+b) = cdota + cdotb ."
    },
    {
        "video_name": "IJWDyPFXGyM",
        "Q": "9:35 what happens when a given value of a piecewise function does not fall in the parameters listed in the piecewise function? I'm really stumped. My first mind is yelling at me that it is undefined or 0 but like I said I am not sure, Any help would be greatly appreciated",
        "A": "If an input is given that does not fall within the parameters of the piece-wise function, then it is simply not a part of the graph."
    },
    {
        "video_name": "nZrxs-U9d8o",
        "Q": "At 2:20, how is the slope of the first graph close to 1?",
        "A": "If you draw the tangent line (as Sal did), it appears to be inclined at about a 45 degree angle. Since the arctangent of 45 degrees is 1, the slope is about 1. The rise and run between any two points are roughly equal, so the slope is about 1. Choosing two points and calculating the slope between them gives about 1. Choose your method for getting the slope, but using any method you like, the slope at the given point is close to 1."
    },
    {
        "video_name": "nZrxs-U9d8o",
        "Q": "at 1:30\nWhy the function needs to have a positive-slope if the Derivative of it F'(x) has a positive slope?",
        "A": "If your function has a positive slope, then the derivative will have a positive value but not necessarily will the derivative a positive slope as well. Conversely: if your derivative has a positive value (regardless of the derivatives own slope), then the function has a positive slope."
    },
    {
        "video_name": "hxZ6uooEJOk",
        "Q": "In the earlier part of the video (around 2:56), he talks about true mean (true difference btw group1 and group2). But later in the video (around 13:06), he seems to assume that the mean difference of a single sample from a single experiment (1.91) is the true mean difference. Why?",
        "A": "This confused me in another video too, but then I understood it. Now it confused me again, but I m confident I understand it, yet again. Thanks for answering your own question."
    },
    {
        "video_name": "hxZ6uooEJOk",
        "Q": "At about 5:20 Sal goes from 95% to 97.5%, why does this work? Doesn't this cause us to include the 2.5% to the left of 95%, so we're taking one tail instead of two tails (and not 95% but 97.5%)? I guess I don't get how we end up with a 95% confidence interval from the 97.5% value in the z-table, I wanted to take the 1.96 value times 2.",
        "A": "We want to find an interval which contains the middle 95% of the distribution, so we want something that is symmetric about the mean. If we took the quantile at 95%, there is 5% left over on the top - but when for the other side, there is 5% left over on the bottom, so we only have 90% in the middle. By taking the quantile at 97.5%, we lave out the top 2.5% and the bottom 2.5%, which means we have left out in total 5% of the area, which is exactly what we wanted."
    },
    {
        "video_name": "hxZ6uooEJOk",
        "Q": "At about 12:16, why doesn't 4.67/10 + 4.04/10 work?",
        "A": "The variance of the sum (or difference) of two random variables is additive. So for random variables X and Y: V( X + Y ) = V(X) + V(Y) V( X - Y ) = V(X) + V(Y) This comes from the mathematical theory. I m not sure where (or if) this is covered on Khan Academy. Then, the 4.67/10 and 4.04/10 are the standard errors (standard deviation) of the two sample means. In order to be able to add them, we first need to convert them into variances by squaring them."
    },
    {
        "video_name": "Wu1KO07WPkk",
        "Q": "what are rational number like 2:5 and 5:4?",
        "A": "Rational numbers are ratios. Ratios aren t numbers, but they show the relationship between then. an example for a question is this : there and 5 more boys for each 2 girls. Make a ratio showing the relationship. The answer would be 2:5 or 5:2"
    },
    {
        "video_name": "Wu1KO07WPkk",
        "Q": "In 2:24 he said rows and are'nt they collums?",
        "A": "No, rows go horizontally and columns go vertically. In the video there are 9 rows and 7 columns (not counting the green blocks). Think of an actual column in real life and how it goes straight up"
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "At 5:40, can someone explain in detail Sal's method of converting -45 degrees into radians?",
        "A": "Watch KA s videos on radians and degrees"
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "4:05 why the angles have to be the same?",
        "A": "We already know that the slope of the ray is -1, so that means that the length of the change to the right is the same as the downward change. Therefore, the triangle has got to be isosceles as well as a right triangle, so it must be a 45-45-90 triangle. (I m quoting Matthew Daly s answer; thanks, Matthew!-)"
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "At 4:42 Sal said (\u00e2\u0088\u009a2/2)^2 + (\u00e2\u0088\u009a2/2)^2= 1^2 but didn't he mean 1?",
        "A": "He s reiterating Pythagorean Theorem, so he said 1^2 to be consistent, (even though 1^2 == 1)"
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "How did you know how to draw a -1 slope at 3:48 ?",
        "A": "First he needs to discover the angle of the triangle, in this case it s 45\u00c2\u00ba, and then he traces the line according to the angle."
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "how did Sal know it was 45,45,90 triangle instead of a 30,60,90 triangle at 4:07?",
        "A": "The gradient of the hypotenuse is 1 or -1 if you draw a right triangle on the x axis with one corner of the triangle at the origin of the circle. So from this fact, the hypotenuse creates an angle of 45 Degrees with the x axis."
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "at 2:28, he says the y axis is sin\u00ce\u00b8 and the x axis is cos\u00ce\u00b8 in the equation:\nsin\u00ce\u00b8/cos\u00ce\u00b8=tan\u00ce\u00b8\nhow come it isn't the other way around?",
        "A": "Look at the standard triangle he draws at 02:34. Where is the y-axis and where is the x-axis, taking the centre of the circle as (0,0) ? This is a circle with unit radius (ie r = 1 ) Now, what is the value of sin\u00ce\u00b8 ?"
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "At 4:29, why isn't the triangle in the 2nd quadrant? Does it make a difference?",
        "A": "That s just a convention - the principal values of arcsin and arctan are in Q1 & Q4 while the principal values of arccos are in Q1 & Q2."
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "During his first example (starting at 3:35), why is the angle considered to be a negative angle instead of a 315 degree angle?",
        "A": "It is Confined..."
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "At around 3:53 Sal says that both angles are equal to 45 degrees, getting this from the fact that the other angle is 90. How can we see that both sides are equal ( I assume this is the reason that both are 45) ?",
        "A": "We already know that the slope of the ray is -1, so that means that the length of the change to the right is the same as the downward change. Therefore, the triangle has got to be isosceles as well as a right triangle, so it must be a 45-45-90 triangle."
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "At 6:50 Sal restricts the range of arctan. Isn't restricting the range to the first and fourth quadrants a bit useless to keep one answer for the function?\nFor example: -45 degrees is in the fourth quadrant. But so is (-45 +360) = 315 which also refers to the same point as 315 + 360.\n\nThere are multiple values for each point, regardless of the quadrant, right?",
        "A": "Sal is not restricting the range of arctan, he is using the defined range of arctan. Arctan has a specific range of (-pi/2, pi/2)."
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "how does Sal know that the triangle at \"3:56\" is an iso. triangle?",
        "A": "Because the line has a slope of -1 so when it is going down, the coordinates for any point on the line will be equal."
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "at around 3:06, what is the triangle thing? (Before the x and y in the circle)",
        "A": "That s a diagram of an angle in the unit circle, with the segments of length sin(\u00ce\u00b8) and cos(\u00ce\u00b8) highlighted."
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "for the problem started at 3:30, why does he use -pi/4 instead of 7pi/4 for the arctan of -1? arent they the same thing?",
        "A": "While the tangent of 7pi/4 is indeed -1, the range of the arctan function (by convention) is the interval [-pi/2, pi/2] in order to make arctan a single-valued (unambiguous) function. Since -pi/4 is in this interval but 7pi/4 is not, the arctan of -1 is defined as -pi/4. This is similar to the situation encountered when finding square roots (radicals); while -3 and 3 are both square roots of 9, the square root of 9 is defined as 3 by convention since 3 is positive. Have a blessed, wonderful day!"
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "At 4:07, Sal says the triangle has to be a 45-45-90 triangle. Why can't it be a 30-60-90 triangle? How exactly do we know the two other angles have to be the same (4:11)?",
        "A": "At this point in the video (starting at around 3:19), Sal is trying to find arctan(-1). Watch this again and try to follow his reasoning. If you follow what he s saying, you will see that the angle he s looking for (Theta) must be 45 degrees. If you get stuck somewhere, ask about that particular problem. (And thanks for the time-stamp: they really help in questions like these!)"
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "At 4:34, how did Sal come up with sqrt(2)/2 on y-axis and x-axis?",
        "A": "Because Sal is working with a unit circle, the length of the hypotenuse of the right triangle is equal to 1. The 45-45-90 Right Triangle Theorem states that the hypotenuse of a 45-45-90 right triangle is equal to the length of one of the legs times rad(2). Applying this in reverse, the legs must be equal to the length of the hypotenuse divided by rad(2). If you rationalize the denominator of 1 / rad(2), you end up with rad(2) / 2 as the final answer."
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "at 6:00 how does sal know to keep pi on top and put 4 on the bottom? b/c we are multiplying it so it would be pretty hard to just know right off the bat what pi/ 180 is.",
        "A": "B/C Sal likes to convert 45 degrees into radiant. So, he multiplied 45 degree with pi/180. Next step, he just did the simplify 45 on top with 180 in the bottom."
    },
    {
        "video_name": "Idxeo49szW0",
        "Q": "I don't understand how to FIND the angle which has a given tangent. Around 3:35 he actually starts to get to work on the problem he made up, but it has a slope of -1 so it's very easy to find the angle. What if you're asked to find the inverse tangent of \u00e2\u0088\u009a3 ?",
        "A": "Sal silently uses the property that y/x=-1, hence y=-x Try that with sqrt(3): y/x=sqrt(3) y=x*sqrt(3) Solve Pythagoras but substituting y: x\u00c2\u00b2 + (x*sqrt(3))\u00c2\u00b2 = 1 x\u00c2\u00b2 + x\u00c2\u00b2 * 3 = 1 4x\u00c2\u00b2 = 1 x\u00c2\u00b2 = 1/4 => x=1/2, y=sqrt(3)/2 You should then RECOGNIZE the square root 3 over 2 value as a side of a 30-60-90 triangle in a unit circle. Since y>x, x is positive and arctan is also positive, it is a 60 degree or pi/3 angle."
    },
    {
        "video_name": "91qVGeyTl44",
        "Q": "From 1:20 to 1:32 - what is the second condition given for?\nIsn't it clear, that if lim(b_n) = 0 when n -> infinity, then {b_n} should be decreasing sequence only? How this sequence could be not decreasing if the limit of b_n (when n -> infinity) is equal zero?",
        "A": "I can t really explain how any of this works, but I can give a counter-example: sin(pi*n/4)/n is not a decreasing series, but still tends to zero as n approaches infinity."
    },
    {
        "video_name": "N1X0vf5PUz4",
        "Q": "1:53-so does that mean any fraction times two is still the same even though the numerator and denominator are different?",
        "A": "Yes. Further explained: if you take a circle and cut it into 4 pieces and shade in 1 of them, you have 1/4, multiply by 2 = 2/8 if you put it on a paper you will notice the same area has been shaded. Hope this helped!"
    },
    {
        "video_name": "N1X0vf5PUz4",
        "Q": "At 2:52, those pieces are equivalent to one right?",
        "A": "All three of the fractions of the pizzas are equivalent to 1. This can be observed from the following: 2/2=1 4/4=1 8/8=1"
    },
    {
        "video_name": "N1X0vf5PUz4",
        "Q": "at 3:35, what does Sal mean?",
        "A": "he means that the more pieces you cut into you double the denominator"
    },
    {
        "video_name": "W7DmsJKLoxc",
        "Q": "In previous videos we learnt that the order does not matter in combinations, then why at 8:07 does Sal use two different orders for 4 shoots out of 5?",
        "A": "If I have got it right , there are 5 different arrangements . The 0.2 beside 5C4 indicates probability of missing"
    },
    {
        "video_name": "W7DmsJKLoxc",
        "Q": "at 9:40 you multiply 0.32768 by 10. I'm not sure why. Shouldn't it be (0.8^5)*5!?",
        "A": "You don t multiply 0.32768. In the video, he converted it into a percentage. You can t multiply by 5! because the free throws aren t different. You can t reorder 5 free throws."
    },
    {
        "video_name": "W7DmsJKLoxc",
        "Q": "At minute 5:58 Sal concludes that P(exactly 3/5) = 20.48%, but at minute 9:46 he concludes that P(exactly 5/5) = 32.77%. How can the probability of getting a perfect 5 out of 5 be greater than getting only 3 out of 5?",
        "A": "If your free throw percentage is 80%, that means you make 8/10 shots, or 4 out of 5 shots. In that case you are more likely to make 5 out of 5 shots than only 3 out of 5 shots."
    },
    {
        "video_name": "W7DmsJKLoxc",
        "Q": "Im a little confused at 4:10 where did you get the 2! from in 5!/3!*2!",
        "A": "it s (5-2)! as in n!/(k!*(n-k)!), the binomial coefficient"
    },
    {
        "video_name": "8JyPzTeA_8w",
        "Q": "what did Sal mean at 2:43, when he said, \"7 and 1/4?",
        "A": "He actually says 7 one fourths , which means 7 pieces that each represent one fourth or 1/4 of a whole."
    },
    {
        "video_name": "gZzsYhyjIXA",
        "Q": "I didn't get that because it says 12pm not 12 am . So surely she was doing it at midnight at midnight and it took her the whole morning to 1:33 the next day to solve it.",
        "A": "you just write the time the a.m and p.m don t matter"
    },
    {
        "video_name": "nFsQA2Zvy1o",
        "Q": "at 3:42 i know that 5 is not equal to 19 so why not just say that 19 is greater that 5?",
        "A": "he does that at 4:37, he only started by saying 5 doesn t equal 19 and then (5 is greater than 19)"
    },
    {
        "video_name": "nFsQA2Zvy1o",
        "Q": "At 0:16, Sal starts a statement that people think that when you see an equal sign, people think that it means give me the answer. Is it true?",
        "A": "When I am in lower grades I do think of it like that but now in higher grades they give you different kinds of questions and it makes me think it as comparing two quantities."
    },
    {
        "video_name": "nFsQA2Zvy1o",
        "Q": "At 1:20 how is 1+1+1=3-2? i'm lost.",
        "A": "The equation is actually 1+1-1 = 3-2. 1+1 = 2, 2-1 = 1 3-2 = 1 So, 1=1 and 1+1-1 = 3-2."
    },
    {
        "video_name": "nFsQA2Zvy1o",
        "Q": "at 5:04, why does the video finish? what is the programming behind it?",
        "A": "it ends at the right time I checked."
    },
    {
        "video_name": "nFsQA2Zvy1o",
        "Q": "At 1:03 to 1:13, how can 1+1-1 = 3-2 ? Couldn't it be something else like 1+1-1= 2-1 ?",
        "A": "Think of this: 1+1-1 = 1, and 3-2=1 too. So they are equivalent. 1 + 1 - 1 = 1 3 - 2 = 1 Same value, same thing. But you re right. Also 2-1 is equal to 1. So the three expressions are equivalent. 1 + 1 - 1 = 1 3 - 2 = 1 2 - 1 = 1"
    },
    {
        "video_name": "nFsQA2Zvy1o",
        "Q": "1+1-1=0 in the video 1:25 it's written that 1+1-1= 1 ?? How is it possible??",
        "A": "1+1=2, 2-1=1 You can do it the other way round too, first 1-1=0, then 0+1 which is 1"
    },
    {
        "video_name": "lT3LRjtSbJc",
        "Q": "At @05:00 The finalists ARE swimming faster because they finished in less average time than the semi-finalists!",
        "A": "Yes. Speed = Distance/ Time. Since here the swimming distance is a constant ( = 100 m ), speed is inversely proportional to time i.e., lesser the time taken to cover the distance implies faster they are swimming. As the data shows that the final round has lower central tendency than semi-final round which means on an average, swimmers in the final round have taken lesser time to swim than in the semi-final. Thus finalist on an average are faster than semi-finalist at swimming. Hope this clarifies :)"
    },
    {
        "video_name": "cvA4VN1dpuY",
        "Q": "When Sal starts explaining how to find the asymptotes at 4:49, it got confusing. Is there a less confusing way to find the asymptotes?",
        "A": "I don t think there is. What helped me, was to first watch the conic section and the hyperbola video. Then I did a few exercises (just going through the motions without really understanding what I was doing). Here I used the hints a lot. Then when I kind of got it, I started with the conic section video again and went through them one at a time. I think now I really got it."
    },
    {
        "video_name": "cvA4VN1dpuY",
        "Q": "When you are converting to standard form, does it usually happen that the two coefficients are swapped and divided by? i.e. at 4:30 what he has before and after he divides by 100:\n4(y-2)^2-25(x+1)^2=100\n=\n(y-2)^2/25-(x+1)^2/4=1\nwith the 25 and 4. Do they usually change positions like that?",
        "A": "No. There is no certainty that it will be that way."
    },
    {
        "video_name": "cvA4VN1dpuY",
        "Q": "At 10:03 shouldn't it be y=3 and y=-7?",
        "A": "No, because he is adding 2 to the right side of the equation, to cancel out the -2 on the left."
    },
    {
        "video_name": "cvA4VN1dpuY",
        "Q": "Why couldn't you set y at zero instead of x? When Sal finds if the hyperbola opens up and down or sideways, he says x could be zero, but couldn't y be zero as well (or y become +2), it would be x = -3, 1? He starts solving for this around 8:34.",
        "A": "If y is zero, then you are left with -((x+1)^2)/4 = 1 Lets multiply both sides by -4, giving: (x+1)^2 = -4 Since anything squared is a positive number, there can be no value of x which works."
    },
    {
        "video_name": "cvA4VN1dpuY",
        "Q": "So, at 1:24, Sal mentions that we can determine what kind of conic the equation will be by the x^2 and y^2 coefficients. WHY is it a hyperbolic equation if one coefficient is negative and the other is positive?",
        "A": "it is just a characteristic of hyperbolic equations. This differentiates hyperbola from the ellipse s equation( having both x^2 and y^2 positive)"
    },
    {
        "video_name": "cvA4VN1dpuY",
        "Q": "At approx 5:50 Sal just removes one of the y(s) from the left side of the equation, but does not show it on the right. Is this because the effect it will have is very small? Then at approx 5:54, he takes the square root of the right side of the equation without a description as to why. Can I assume that it is to simplify the equation, or do you always need to do this?\n\nThanks.",
        "A": "I believe it has something to do with limits"
    },
    {
        "video_name": "cvA4VN1dpuY",
        "Q": "Around 8:42, Sal talks about whether the graph is horizontal or vertical. Can a shortcut be just that when the y^2 value is positive, the graph is vertical and when the x^2 value is positive, the graph is vertical? Or is that not always accurate?",
        "A": "Other than your editing error (when the x^2 value is positive, the graph is horizontal), that is generally true. It comes down to the question Am I trying to take the square root of a negative number? The exceptions I can think of are the sorts of things you won t run into at this level. Trigonometry at the earliest, and maybe not even then."
    },
    {
        "video_name": "j_Taf2Tgggo",
        "Q": "7:57 why is dln(N)/dt equal to dln(N)/dN * dN/dt?\nwhen I enter it in wolframalpha i just get zero, what makes sense to me since the t^0 (that is basically next to the ln(N) turns into t^-1*0.\nEDIT: I see you just multiplied the left side with dN/dN and crossed the parts of the fraction\nanother thing that i dont get is why at 11:30 he replaces 1/N*dN/dt with ln(N) while on the right side it says 1/N*dN/dt ) = ln(N)* d/dt. how can the d/dt just be left out oO?",
        "A": "Why is it frowned upon to algebraically manipulate differentials like any other variable?"
    },
    {
        "video_name": "znmPfDfsir8",
        "Q": "Why did Sal add a 2 in front of 2 * 3 * 3 at 2:49 and what does it mean?",
        "A": "multiply 2 to get 36"
    },
    {
        "video_name": "znmPfDfsir8",
        "Q": "So Sal why will it pop out at me? This is so confusing at you said at 0:14 in your video.",
        "A": "That phrase means it will become very obvious to you."
    },
    {
        "video_name": "znmPfDfsir8",
        "Q": "on time 3:23 Sal said that since we only had one 2(in one space,the number 18),then we have to add another 2...well,on number 18 we have TWO 3's,and on number 12 we have ONE 3. Why do we not have THREE 3's in the factored number addition?",
        "A": "If you re not sure what he s saying, go to settings (the gear icon) and press interactive transcript."
    },
    {
        "video_name": "znmPfDfsir8",
        "Q": "At 0:29, Sal says that 36 (bigger than 12) is A MULTIPLE OF 12! What? That doesn't make sense to me... :/",
        "A": "36 is a multiple of 12... 12. 24 ... 36. 48 Multiple means count by... I think you are being confused by factor."
    },
    {
        "video_name": "znmPfDfsir8",
        "Q": "At 1:14, can anyone please explain Prime Factorization to me? I remember it vaguely from 4th grade, a while back. Can anyone direct me to a video or tell me in the answers how to do this? The video does not make it particularly clear.",
        "A": "So prime factorization is when you try to find all the factors that are prime in a number that, when multiplied together, produce the number that s being factored. For example, the prime factors of 36 are 2, 2, 3, and 3. I know this because 2*2*3*3=36 , and since 2 and 3 are both prime, then we re all good."
    },
    {
        "video_name": "znmPfDfsir8",
        "Q": "I did not understand 1:20 nor 1:30. Can you explain it?",
        "A": "Sal divided the number 18 by 2 and got 9 and then divided 9 by 3 and got 3. He can t divide 3 or 2 so they are prime numbers. So he is just writing the prime factorization of 18. So basically he is saying 2*3*3=18."
    },
    {
        "video_name": "znmPfDfsir8",
        "Q": "The prime factorization way (at 1:01) seems so much easier in the end, but it isn't working for all of my problems. For example, lcm of 3 and 10; I get 36, but the answer is 30. Am I doing it incorrectly, or is this only applicable to certain problems?",
        "A": "The prime factorization method to find the LCM will always work. 10 = 2 x 5 3 = 3 So, LCM(3,10) = 2 x 5 x 3 = 30"
    },
    {
        "video_name": "znmPfDfsir8",
        "Q": "I still don't really get the part around 2:50-2:58. Can anybody help me ? :-(",
        "A": "He is showing how you can use the prime factorization of the two numbers to work out the LCM more easily. 12 s prime factors are 2*2*3. 18 s factors are 2*3*3 As you can see, both numbers have a 2 and 3. So you multiply those. Then, you multiply the numbers they don t have in common (which are the other 2 and 3). Let me know if this still doesn t make sense!"
    },
    {
        "video_name": "rPoqpQcgNv4",
        "Q": "at 2:00 , 2x2 could also be factored right?",
        "A": "Correct! 10 = 2 * 5 after all."
    },
    {
        "video_name": "znE4Nq9NJCQ",
        "Q": "It seems strange that he just guesses that Ae^(2x) is a solution at 7:42. What is the reasoning behind it?",
        "A": "Since we have that the g(x) is 3e^(2x), we now that the particular must be from the family of Ae^(2x) to actually have the y - 3y - 4y equals Ae^(2x)."
    },
    {
        "video_name": "znE4Nq9NJCQ",
        "Q": "At 6:10, I understand how Sal got r = 4 or r = 1, but then I got lost when he went to y (general) = C1e^4x + C2e^-x. Can someone please help me follow how he did that?",
        "A": "the solution for a homogenous linear differential equation (constant coefficients, second order) is just the homogenous solution derived from the characteristic polynomial. a nonhomogenous (general) solution would be a particular solution plus the homogenous solution."
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "At around 1:30, Sal writes the answer as -8(-5 + 4x) + 6. Couldn't you simplify this expression farther as 40 - 32x + 6 and then to 46 - 32x ?",
        "A": "The exercise was to just translate the words into math symbols and thus create the algebraic expression. It didn t ask you to simplify the expression, which is why Sal didn t simplify his result. Can it be simplified? Yes, it can and your version would be correct for the simplified form."
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "At 2:39 he made round parentheses in other round parentheses. Isn't that wrong?",
        "A": "No that is acceptable or you could use brackets. As stuff gets more complicated, the more parentheses you need, for example (7 + 3)/\u00e2\u0088\u009a((2 - 6)/(4 - 20)). you always do the most inside parentheses first and work your way out"
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "at 2:40-2:45, there was a parenthesis around another parenthesis. how does that work?",
        "A": "When there are multiple sets of parentheses in a mathematical equation, the general rule is to work from the inside most set of parentheses outward. That means if you have the equation (5 * ( x * (4 - 2) ) ) where there are three sets of parentheses, you would start with the inside parentheses (4 - 2)... (5 * ( x * (2) ) ) = (5 * (x * 2) ) removing the inside set because we did the operation... ( 5 * (2x) ) = (5 * 2x) = 10x I hope that helps."
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "That first parenthesis at 2:19 was just for didatic porpuses, it wasn't really necessary right? Can't i simply write 4 + 2(7-2x)?",
        "A": "Yes you can! Sal just used the extra parentheses to clear things up."
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "At about 01:40 into the video, he shows this \"7 + -2x = 7 - 2x\" and then he goes on to say, \"it's the same.\"\n\nHow is it the same? Thanks! :)",
        "A": "Subtracting a number(or term) is the same thing as adding a negative number. For example, if you subtract 2 from 5 you get 3. Similarly, if you add -2 to 5 you also get 3. Basically, subtraction is the addition of negative numbers. I hope that makes some sense."
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "Could the expression at 1:35 be written as 4 + (7-2x) squared?",
        "A": "No. You re confusing the word times with squared For examples: 2 x 5 is read as 2 times 5 5^2 is read as 5 squared or 5 to the 2nd power . In other words: 5 x 5 (5 times 5), not 2 x 5."
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "How come at 2:43 he put double parentheses?",
        "A": "The reason Sal put double parentheses is because inside of the parenthesis there was another equation. Let s say you have an equation 4x + 5. Alright so let s just say that you want to add another equation after you solve this one. How do you do that? Well you have to put another set of parenthesis. So it could turn out like this in the end: (5 *(4x + 5) - 2x) Hopefully that helped!"
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "Why did he use two lots of parenthesis at 2:40? Couldn't you just write 4+2(7-2x)?",
        "A": "Because he had to get the answer right"
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "At 1:40 could I write this as 2(7-2x)+4 or do I need both sets of parentheses ?",
        "A": "No you don t need both sets of parentheses, because of the order of operation"
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "At 2:21, could you write the equation as 4+2(7-2x) ? So I am asking if i could have the 2 outside of the parentheses instead of a double parentheses.",
        "A": "Yes, you can write it as: 4+2(7-2x). The extra set of parentheses are not needed."
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "At 2:40-2:45, why did you not remove the second parenthesis in the second problem",
        "A": "Because we are not simplifying the expression, we are just writing it."
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "at 2:23 Salman says 4+(2*(7-2x)). Can't he just say 4+2(7-2x)?",
        "A": "You are correct, these are the same thing"
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "at 0:33 he says that the -5 is quantilty of 4,i got confused can u guys help me plz and thank you alot",
        "A": "We use the word quantity to indicate what we want to put together and what we want to separate - this is used to avoid confusion of what is being asked. If he had just said negative 5 plus four times x, it could mean (-5 + 4) x or - 5 + 4x. He does not say -5 is quantity of 4, he says - 5 plus the quantity of 4 times x (at :17-:29) which gets his expression -5 + 4x."
    },
    {
        "video_name": "xLYVo_k0_us",
        "Q": "At 1:07, do you just put the plus six at the end of the expression?",
        "A": "Yes, did you watch the rest of the video??"
    },
    {
        "video_name": "Y6JsEja15Vk",
        "Q": "at around 0:58, how do you know what number your suppose to multiply to each side? is it the lowest common denominator ? Because some examples my teacher has taught me show completely different numbers. Can you help please?",
        "A": "Claudia, you multiply a number which would make one thing in both equations the same, but with opposite signs! That s why Sal multiplies the upper equation with 2, because that would give 6t in the first equation and -6t in the second equation. When you add the equations, 6t + (-6t) = 0. Hope that helps!"
    },
    {
        "video_name": "Y6JsEja15Vk",
        "Q": "Around 1:41, I still don't get why you can add the two equations together.",
        "A": "You can do whatever you want to an equation, as long as what you do has the same impact on both sides of it. So if I have a=b I can add 5 to one side and 5 to the other. Or I can add 2+3 to one side, and add 4+1 to the other side, because 2+3 = 4+1. Or I can add 5 to one side and add 10/2 to the other side, because 5 = 10/2 So now instead of adding the equation 2+3 = 5, I can add the equation c=d. I can add c to one side and d to the other because they are equal a+c = b+d"
    },
    {
        "video_name": "9ASWQi14FlE",
        "Q": "at 1:31 Sal writes (ra)squared. in (ra) what is a?",
        "A": "a is just the short for circle a, so the radius of circle a"
    },
    {
        "video_name": "9ASWQi14FlE",
        "Q": "At 4:47, Sal uses the term locus. What does it mean? Give an example of it in another situation.",
        "A": "A locus is the set of all points (usually forming a curve or surface) satisfying some condition. For example, the locus of points in the plane equidistant from a given point is a circle, and the set of points in three-space equidistant from a given point is a sphere."
    },
    {
        "video_name": "NLg6hfoKKlE",
        "Q": "At 2:25 he says that transversal intersects 2 lines. Does it have to be 2 lines, or can it be 1, or 3,\nor any other number?",
        "A": "A transversal has to intersect 2 parallel lines because when your talking about same side interior and exterior angles they need to be supplementary they can add up to other random numbers that you would get with 1 or 3 lines."
    },
    {
        "video_name": "NLg6hfoKKlE",
        "Q": "What are Magenta angles at 5:44?",
        "A": "That s just the colour he uses on the angles. They are vertical angles. He coloured them the same to show that they are the same."
    },
    {
        "video_name": "NLg6hfoKKlE",
        "Q": "at 4:11 are the red \"x\"'s and the orange \"x\"'s vertical angles?",
        "A": "Yes. They are vertical angles."
    },
    {
        "video_name": "hZ827mfh1Jo",
        "Q": "at 14:33, every a vector in the matrix A (like a1, a2 etc.) must have m amounts of values right? so a1 is [a1, a2, a3.... am]?",
        "A": "That s right. All those column vectors have m elements."
    },
    {
        "video_name": "hZ827mfh1Jo",
        "Q": "There is something I cannot understand here...\nAt 14:00, I think that it should be {Ax | x is in V} NOT {Ax | x is in R^n}\nThis is because span(V) might not equal R^n\nWhat do you think?",
        "A": "An mxn matrix A transforms vectors x from R^n into R^m vectors. So all the vectors x are from R^n, since @ 14:00, T(x) = Ax, and so any other vector space V is irrelevant."
    },
    {
        "video_name": "hZ827mfh1Jo",
        "Q": "Let's say I have this Im(T), as in 12:30, and I want to equal it to ker(T), what should I do? Just to simplify, consider Rn=Rm=R3.",
        "A": "Yes. I believe so, But just in case, make sure you double check your work, and re-watch that part of the video. I understand that they give you an example then give you something WAY different. But just to experiment, try it to see."
    },
    {
        "video_name": "4Mr7aEHQr8E",
        "Q": "During 6:00 minutes the unit step function stands as 2-2u<pi>(t) + 2u<2pi>(t)\nWhy did you use - sign in 2nd terms but + sign before 3rd term? Please clear this fact.......",
        "A": "The 2nd term was meant to show that the value of f(t) would be 0, past the time of pi. Because of the 2nd term s condition of time being greater or equal to pi = -2, the value of the first term would always cancel out the initial value of 2. With the 3rd term having the range from 2pi; it restored the f(t) s value to 2."
    },
    {
        "video_name": "4Mr7aEHQr8E",
        "Q": "At 1:33 he says something about taking the integral from 0 to infinity\ndoes he have any videos on integrals?",
        "A": "Yes, but look in the Calculus section of Khan Academy Math Tutorials."
    },
    {
        "video_name": "4Mr7aEHQr8E",
        "Q": "Is there a mistake here around 24:20?\n\nL{u_c(t)*f(t-c)} should equal e^{-sc}*L{f(x)}, where x=t-c. Thus L{u_c(t)*f(t-c)}=e^{-sc}*L{f(t-c)} (video shows rhs as e^{-sc}*L{f(t)})",
        "A": "I assume you mean 21:20? If you re-watch the minute or two prior to that you see the point made that using f(x), that is f(t - c),or using f(t) doesn t make a difference as long as both are evaluated at 0 and infinity. In this particular problem the u_c(t) function enables you to licitly evaluate f(t - c) at 0 and Infinity as well as f(t)at zero and infinity (since the function remains on the x-axis between 0 and c)"
    },
    {
        "video_name": "4Mr7aEHQr8E",
        "Q": "why should we put the function multiplied by the unit step function (at 10:38) instead of the function alone?",
        "A": "We want the function f(t) to be shifted by c (i.e we want the function to start at c and below c it should be 0) and that s why a unit step function is used. A unit step function Uc(t) takes the value 0 for t < c and it takes the value 1 for t >= c. Consider the product of the two functions : U_c * f(t-c) . For t<c this function becomes 0*f(t-c) =0 For t>=c this function becomes 1*f(t-c) = f(t-c) thus we get the desired function ! I hope this helps !"
    },
    {
        "video_name": "fyJkXBvcA2Q",
        "Q": "@2:04 How is the anti-derivative of \"Cos at\" is \"1/a Sin at\"??\nShouldn't it be just \"Sin at\" ??",
        "A": "If you take the derivative of Sin(at) the answer is a*Cos(at). What you wanted was to get Cos(at), hence you need to multiply the Sin(at) with 1/a to compensate for the a you get when you take the derivative of Sin(at). I hope that wasn t too messy of an explanation =)"
    },
    {
        "video_name": "gFdh_rE2XgU",
        "Q": "At 6:48, why is Sal able to just take the principal root? Shouldn't there be any mathematical expresion that can let me legitimally rule r= -1/3 out?",
        "A": "Well, for a graph like that, you could just look at when x = 0, it is 3. So a = 3. Then you just pick a point for example, that point (1, 1). You will now have 1 = 3 * r^1 which is the same as 1 = 3r. You can divide both sides by 3 which leaves you with 1/3 = r saving quite a lot of space. But if you just have two points, you ll need to figure out the equation yourself."
    },
    {
        "video_name": "gFdh_rE2XgU",
        "Q": "Didn't sal forget the exponent at 5:26 and r in the a/r be negative",
        "A": "Good question, Sal is using the negative exponent property here. x\u00c2\u00af\u00c2\u00b9 = 1/x\u00c2\u00b9 Any number raised to a negative exponent is equal to the reciprocal of the number raised to the power. In other words, the base number keeps the positive value of the exponent but moves to the denominator, the numerator becomes 1. 2\u00c2\u00af\u00c2\u00b2 = 1/2\u00c2\u00b2 = 1/4 3\u00c2\u00af\u00c2\u00b3 = 1/3\u00c2\u00b3 = 1/27 Sal was working with a \u00e2\u0080\u00a2 r\u00c2\u00af\u00c2\u00b9 which is the same as a \u00e2\u0080\u00a2 1/r\u00c2\u00b9 which equals a/r\u00c2\u00b9, or, a/r. Hopefully this cleared it up a bit."
    },
    {
        "video_name": "gFdh_rE2XgU",
        "Q": "At 3:38, couldn't he just find the y-intercept using the graph?",
        "A": "Yes he can, in this case it would have been very straightforward. for f(x), (0,5) and for g(x), (0,3). Then you can use point (1,1) (or the other point, but (1,1) is simpler) to find m and r respectively. I think the intent was to show another way of doing it, as what you suggested was shown in a previous video"
    },
    {
        "video_name": "gFdh_rE2XgU",
        "Q": "At 3:31 couldn't Sal have just looked at the graph of f(x) to see that the y-intercept (b) was 5? This won't be true in every case, but when the graph passes directly through a number that is marked on the graph you can read the graph at that number. Right?",
        "A": "In this case, you may be allowed to read the graph because the equation is linear. It is a perfectly straight line, and since (0,5) is exactly between (-1,9) and (1,1) it is easy to tell that that is the y-intercept. But yes, you should always check it, if only so your teacher doesn t get mad. :)"
    },
    {
        "video_name": "yZqVJtDO7gc",
        "Q": "3:42 but what if the 25 was bigger than the 100 what would u do?",
        "A": "There would still be a ratio but it would turn out smaller. If you had 100 teachers per 500 students and wanted to find out what the ratio would be for 100 students, then you can see that 100 is 5 times smaller than 500. Since 500 \u00c3\u00b7 100 = 5, so you would do the same to the numerator by dividing it by 5. 100 \u00c3\u00b7 5 = 20, so there would be 20 teachers for every 100 students."
    },
    {
        "video_name": "yZqVJtDO7gc",
        "Q": "from 4:10 to 4:25 he gives it away! why does he give it away!",
        "A": "I think what you call giving away, I would call proving :D In math you have to show your approach is correct aaaallll the time."
    },
    {
        "video_name": "1iSBNSYhvIU",
        "Q": "At 1:31 how did khan 60cm",
        "A": "He multiplied 3, the length, by 20, the height to get 60. To make it simpler, multiply 3 and 2 which makes 6 and add a 0!"
    },
    {
        "video_name": "1iSBNSYhvIU",
        "Q": "On 0:34, why does the end of the sum need a \"squared centimeters\" part?",
        "A": "Since you are finding area, you will use square centimeters because if the length and width are measured in centimeters, and the shape was, for example, a square, it wouldn t make sense to have something just 20 cm , because that would mean 20 cm long or wide, and square centimeters is both length and width. Example: 1 square cm is a 1x1 box. Hope this helps!"
    },
    {
        "video_name": "eLUmpX_h9sw",
        "Q": "@ 3:28 Since \u00ce\u0094E/ \u00ce\u0094T =3/4 does that mean you multiply by 3/4 to go from elevation to time?",
        "A": "You would multiply time by 3/4 and subtract 24 to get the elevation. This is because the function is written as E(t) = 3/4t-24. Also you are not able to directly find the slope by dividing the dependent variable by the independent variable, like in the last example as there is that -24. And so can t simply multiply the independent variable (time) to get the dependent variable, cos you ll miss out that -24."
    },
    {
        "video_name": "eLUmpX_h9sw",
        "Q": "at 5:46,near the end of the video, why is it that the elevation is 0 over the period of time: 32 minutes? (please answer this: is the elevation time minutes or seconds?) If the elevation is 0 shouldn't that mean he didn't make any progress?",
        "A": "It is a point (32,0) which means it took 32 seconds to get to the edge of the volcano. It is not over a period of 32 seconds (not minutes), it is at a point in time. The slope of 24 (change in elevation from 24 feet down to 0 feet) divided by 32 (change in time from 0 to 32 seconds) or slope of 24/32 which reduces to 3/4 meters per second as stated in the problem. Since he started at -24, then 0 is a net change of 24 meters. Hope this helps"
    },
    {
        "video_name": "eLUmpX_h9sw",
        "Q": "At 0:21 why does Sal underline the letter E?",
        "A": "He s just making sure of the definition of the important variables in the problem. Did you notice that he also underlined the t?"
    },
    {
        "video_name": "zRjLZROI7wc",
        "Q": "So at 3:00 he says that whatever you are multiplying with the denominator you have to multiply the numerator. Is that true?",
        "A": "This is totally correct"
    },
    {
        "video_name": "hy_bDS3aHO4",
        "Q": "At 1:34, he starts writing out the subtraction part, but do you have to line up the decimal points when you subtract decimals?",
        "A": "Yes. When adding or subtracting numbers, you always have to line up the place values. You don t add: 653 +9 You line up the 9 underneath the 3 since they are both in the ones place. It works the same way when decimals are involved. You have to line up the place values, which means the decimal points will all line up."
    },
    {
        "video_name": "hy_bDS3aHO4",
        "Q": "At 2:10, why do people use word problems any way?",
        "A": "I see that yo asked this question twice, I hope the other person that answered this question helped you!"
    },
    {
        "video_name": "hy_bDS3aHO4",
        "Q": "0:11 why do people use word problems?",
        "A": "To make you think about what you re doing during the problem."
    },
    {
        "video_name": "hy_bDS3aHO4",
        "Q": "At 1:31 why did you need to add?",
        "A": "Because it said that Leo deposits another $875.50 into his bank account, which means he ADDS that much money to his account. You have to do this to find out how much money Leo has in the end."
    },
    {
        "video_name": "hy_bDS3aHO4",
        "Q": "At 1:17 I do not understand. It ends up confusing me.",
        "A": "Sal subtracts $300 from his total after first adding."
    },
    {
        "video_name": "OwPSuHXmiPw",
        "Q": "At 5:47, why do you divide by (100-1)?",
        "A": "It s because we are working with a sample here. If you want to understand why you have to treat a sample differently, I d suggest you have a look at the videos about variance and standard deviation (descriptive statistics). If any problems with this remain, keep asking ;)"
    },
    {
        "video_name": "OwPSuHXmiPw",
        "Q": "At 8:18 Sal mentions 100 possible values, what do these values represent? And the distribution is sampling distribution of sample mean, why is that, how is this related to 100 discrete values? Thanks.",
        "A": "The values represent the people who answered the survey about which candidate they were going to vote for. If someone indicated that they would vote for person A, then their vote would be assigned a value of 1. Otherwise, they indicated that they would vote for B and thus their vote would be assigned a value of 0. Hope that helps."
    },
    {
        "video_name": "OwPSuHXmiPw",
        "Q": "Around 7:10 I don't understand how if the sample mean was 0.43 and the sample standard deviation was 0.50. Would this not possible result in one standard deviation to the left being a negative value?",
        "A": "I can t watch the video right now, but even if it s impossible for the things being measured to be negative (e.g. rainfall), those sorts of numbers can come out. If there are a lot of data in some cluster around 0.4, but then a few pretty large numbers - outliers - then the standard deviation can get pretty inflated. Some non-normal distributions might exhibit something like this."
    },
    {
        "video_name": "OwPSuHXmiPw",
        "Q": "At 10:20 while estimating the population standard deviation, why is it\n= sample standard deviation/sqrt(100)? Shouldn't it be = sample standard deviation/sqrt(99)?",
        "A": "It should be sqrt(100) because the formula is S.D./sqrt(n)."
    },
    {
        "video_name": "OwPSuHXmiPw",
        "Q": "At 10:50 onwards, I don't understand why we use the S (standard deviation of the sample) as the best estimate for the population's standard deviation? I mean, wouldn't S be equal to Sigma_xbar ??",
        "A": "It s only the best because it s the only one we have. Since we don t have sigma use s/sqrt(n) for sigma/sqrt(n)."
    },
    {
        "video_name": "v3gdX07Q6qE",
        "Q": "There is a cut in the video at 1:19. :)",
        "A": "Ok thanks for letitng us know"
    },
    {
        "video_name": "v3gdX07Q6qE",
        "Q": "At 0:10, why do they use a Chinese thing instead of an American calculator?",
        "A": "I m chinese so i know how to do these and stop being raciest"
    },
    {
        "video_name": "v3gdX07Q6qE",
        "Q": "at 0:23, how does a abacus help with math",
        "A": "It does by helping you count. As you can see, on a abacus there are 10 beads on each row. The first row is the number of ones you have, the second row is the number of tens you have, the third row is how many hundreds you have, the fourth row is how many thousands who have, and so on... The abacus would be useful in counting numerous amounts of things."
    },
    {
        "video_name": "XiAoUDfrar0",
        "Q": "At 4:21 he calls the pentagon a \"not-irregular pentagon\". Isn't it irregular? The definition I believe is that a polygon is \"regular\" only when all angles are equal and all sides are equal, otherwise it is irregular.",
        "A": "it is irregular"
    },
    {
        "video_name": "XiAoUDfrar0",
        "Q": "At 0:54 what does X have to do with this? My teacher never said anything about algebra being part of this?",
        "A": "He is using the point where x = 0 and y = -4 as an example of a point on the shape. X refers to the x-axis."
    },
    {
        "video_name": "XiAoUDfrar0",
        "Q": "In 1:38 Sal said that when we are translation something , you have to move it by the same amount, why is that important in transformation?",
        "A": "Because otherwise you would be changing the original shape. It needs to retain the distance between points, but it also needs to translate to a different point on the graph. For example, you are standing next to the chair. You then move to the other side of the chair. You haven t gotten any taller, or grown another limb. You have simply changed position."
    },
    {
        "video_name": "DRg2VCo8lP0",
        "Q": "At 5:00, why is area considered an important factor in this Khan Academy video? I am an auditory learner, and I've been struggling on this :(. can you explain to me why area is considered important here, and how to find area of rectangles well?",
        "A": "Yes, with what?"
    },
    {
        "video_name": "DRg2VCo8lP0",
        "Q": "at 4:21 when we were multiplying the last figure, when the height and width are the same, is that why people call some numbers \"perfect squares\" ?",
        "A": "Not exactly, their more called perfect squares because taking the square root of the squared number results in a whole number"
    },
    {
        "video_name": "DRg2VCo8lP0",
        "Q": "At 10:45, does an oval have an area?",
        "A": "Ovals do have areas, as Jan said earlier. If you want to know exactly how, it goes like this: Step 1: Find the major radius (the distance from the center of the oval to the farthest edge of the oval) and minor radius (the distance from the center of the oval to the closest edge of the oval). Both radii (radius plural) should form a right angle. Step 2: Multiply the major radius by the minor radius, and then by \u00cf\u0080 (if you don t know, pi is about 3.14). Step 3: You are done (and make sure to say units squared)!"
    },
    {
        "video_name": "Q0tTfe2lKIc",
        "Q": "I have a test on this subject coming up soon, if I am asked a question to draw a diagram similar to the one at 1:39, do the checkpoints and marks have to be precise? There is no way to figure out how much is in between the beginning and the end of the line, so would you just estimate, or is there a certain way to find out where to put the lines? Thank you.",
        "A": "The graphic is just a picture to help understand the word problem. It does not need to be precise. Sal is using it to try and break down the word problem and better understand what the word problem is describing. The actual values are found by setting up the algebraic equations and solving for the respective variables."
    },
    {
        "video_name": "Q0tTfe2lKIc",
        "Q": "In 5:26 how did he get 11?",
        "A": "He subtracted b - 1/12 b so he got a common denominator of 12 12/12 b - 1/12 b put into a common fraction (12-1)/12 b which is the same as 11/12 b he did this process in his head by basically saying if I subtract 1/12th from 1, I get 11/12 (or if a pizza is cut in 12 pieces and you eat one, 11/12 will be left)"
    },
    {
        "video_name": "Q0tTfe2lKIc",
        "Q": "at 5:26, how is B + B/12=11/12B?",
        "A": "In order to subtract -B/12 from B, you need to have both numbers on an equal denominator., this case being 12. Therefore B becomes 12/12B. 12/12B - 1/12B (or -B/12) = 11/12B."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "why is the funtion alwasy like f(x) at 4:30",
        "A": "You plug a number into that (the x slot) and what comes out is the answer."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "at 9:55 how is the domain equal to the set of all integers? and not real nos?",
        "A": "It is equal to the set of all integers because the conditions apply only if the input is an integer. The conditions suggest that the input has to be even or odd, therefore the domain can only be the set of all integers."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "My question is about how to express the domain in (x,y) values. I know that f(x)= x^2 is equal to (-infinity, infinity), but how would you express the domain shown at 4:50 in that way?? (im not sure if the question is clear, i take the class in spanish and it's different)",
        "A": "Since that answer is a disjoint (not all in one piece - draw this out on a numberline to see what I mean) set you need to write it in two pieces like: (-infinity; -3] U [3; infinity). You ll want to use a proper union sign in place of the U there. Same thing what she said"
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "at 4:14 is sal just solving the equation? i think that needs to be explained in successfully giving a domain.",
        "A": "he has formed the equation so that the domain is possible"
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "At 5:35 how is x less than or equal to - 3? Wouldn't that make under the square root negative? Like if x was -6, under the root would be -9?",
        "A": "Kajal, The equation had an absolute value in it \u00e2\u0088\u009a(|x|-3 So if x=-6 then \u00e2\u0088\u009a(|-6|-3) = \u00e2\u0088\u009a(6-3) = \u00e2\u0088\u009a3 When x<-3 and when x>3 then |x|>3 and so the expression is always non-negative when x<-3 or x>3. I hope that helps."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "At 6:39 Can I write the domain as {x=R | -3\u00e2\u0089\u00a5 x \u00e2\u0089\u00a53}?",
        "A": "Sure. That s even better."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "For the first example, at 1:00, he says that the domain can be any real number, but I don't think x can't be any real number. It has to be positive. (because it is x squared)\nsorry, just wanted to point that out, because I was confused.",
        "A": "Thanks, now i understand."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "At 5:25, do you have to keep the absolute value notation? Also at 6:18, shouldn't Sal use the logic notation of or?",
        "A": "1.) Yes, the absolute value is absolutely necessary (pun intended). Without the absolute value, the answer would be x>3, not x>3 or x<-3. 2.) He could, but either way works as long as you get the answer across."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "At 1:34, Sal didn't write out the same thing that he said. He said \"The set of all x's such that x is a real number.\" He wrote this: {x\u00e2\u0088\u0088 \u00e2\u0084\u009d}\n\nThe correct way would be written as domain = {x | x\u00e2\u0088\u0088 \u00e2\u0084\u009d}, because as said, this is said as \"The set of all x's such that ( | means such that) x is an element (\u00e2\u0088\u0088) of all real numbers (\u00e2\u0084\u009d). Am I wrong, or..?",
        "A": "Yep, you re right. Rigorously it should be {x | x \u00e2\u0088\u0088 \u00e2\u0084\u009d} or just plain x \u00e2\u0088\u0088 \u00e2\u0084\u009d sans the braces."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "At 1:19 what is meant by trancendental numbers?",
        "A": "A Transcendental Number is any number that is not an Algebraic Number Examples of transcendental numbers include \u00cf\u0080 and e."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "for 4:39 can you write x is a member of all real numbers, but x cannot be smaller or equal to 3?",
        "A": "I guess you can but that just simplifies to: x>3 (x is greater than 3) Because if x>3, then x is not smaller or equal to 3."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "Why x does not equals 1? 1 can be used in x!!! What does domain means actually? (at 9:53)",
        "A": "@raytib the domain is the set of numbers that don t make a function undefined. we exclude the numbers that do make the function undefined from the domain."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "what does \"absolute no.\" mean? How is it different from a normal variable?(e.g. 5:27)",
        "A": "Both answers are right, but sometimes the following definition is more useful, use whichever one is more helpful to you: |x| = { x if x > 0 { 0 if x = 0 {-x if x < 0"
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "At 0:55 what is the symbol Sal places between the x and the symbol for all real numbers and what does it mean?",
        "A": "That COULD possibly be an arrow pointing to x. It s sorta hard to tell. But if it is an arrow, it means that what the arrow is pointing at is a member of whatever the arrow originates from. Like, if 5 was a solution to an equation, I could write {5 <- answers of my equation}"
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "At 5:03, what is absolute value?",
        "A": "Absolute value is the distance of a value from zero. Put more simply, it makes everything positive, since you cannot be a negative distance from zero."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "At 9:43 the annotation says that x is only defined over INTEGERS (Z) and not as Sal says over REAL Numbers (R). What I know is that the denominator can be any number except 0, so in this example any number (R) except 1 and 2 can be put into denominator. Why is the domain over R not right?",
        "A": "Infinity. If you look back at the original function, the function is defined one way if x is even and another way if x is odd. It does not have any definition when x is neither odd or even. Only integers are even or odd. Non-integers real numbers are neither odd or even. So the function is defined only for integers except the integer 1."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "Regarding the annotation shown at 9:44 Does the annotation basically mean: That functions only apply for integers.\n\nAnd that they do not apply for fractions and decimals?",
        "A": "Basically that is what it means...That function cannot be applied to fractions or decimals.(TRY IT)You usually will get an irrational number (you could though but they will be large)"
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "At 5:50, Sal says that x needs to be smaller or equal to 3. Isn't it the other way round? I'm probably wrong so can someone please explain this to me?",
        "A": "I think if you watch again, you will see that he says it must be less than or equal to -3 OR greater than or equal to 3"
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "At 3:05 , can we also write {x \u00e2\u0082\u00ac R} - {0} ?\nAccording to me this also says that x belongs to all real numbers except 0 .\nPlease correct me if i am wrong anywhere .",
        "A": "I m rusty on the notation to be used, but you need to write the logical statement that x is contained in the union of R and NOT 0, perhaps x\u00e2\u0082\u00ac (R^-0). ( I am not sure what the notation is for not 0. My representation may be wrong, but you should be able to get the idea from what I wrote.)"
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "at 5:08 what is the absolute value of a number?",
        "A": "The absolute value of a number is it s distance from 0. So for example, the absolute value of -5 would be 5. Think of it on a number line."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "How is an 'irrational' number a real number...? (Mentioned at 1:17 in the video)\nJust need a little clarification, thanks!",
        "A": "Irrational numbers are just numbers that can t be expressed as fractions, like sqrt(2) which is 1.414213562... on for infinity. Numbers like this are real numbers. The only numbers being excluded are imaginary numbers, for example sqrt(-1)."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "at 4:17 sal says that 0 (zero) is a valid value for the function. is the square root of zero real?",
        "A": "Yes, the square root of zero is zero. (If you multiply zero by itself, you get zero.)"
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "At 7:35 ,Sai said that x cannot be zero since it forms a zero in the denominator.But you get -3 when you take x as 0 . | 0 | = 0. And 0-3 = -3.",
        "A": "Sal said that x has to be greater than 3, because any values of x less than 3 is unreal or undefined. Therefore, x cannot be 0 either. If x was 0 instead, it would be -3. But currently these numbers are real, and negative square roots do not work with real numbers. And if x was 3, then 3-3 = 0, but you cannot ever have a 0 in the denominator."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "Sal said at around 3:55 you can't take the square root of a negative number, but is it possible to take the cube root of a negative number when finding the domain of a function?",
        "A": "Yes, you can take the cube root of a negative number. Say you have (-3)^3, or negative three cubed. (-3)(-3) = 9. (9)(-3) = -27. Therefore, the cube root of -27 will be -3."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "So the : at 3:01 was supposed to be a vertical line to represent such that?",
        "A": "Yes... that is what the pop-up box is telling you."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "Does one always have to write that x is a real number if it is or can we just state that it's greater than 3? E.g. at 4:43.",
        "A": "It is the convention to write both. A number can be greater than 3 yet not be a real number ( 3+i is complex, for example). 3+i is not part of the domain of f(x) because it is not a real number, even though it is greater than 3. I think I m stating that correctly."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "at 5:11 , why cant the absolute value of x be a negative number , after all he is going to take the absolute value of the number so ... why not ????",
        "A": "x can be a negative number it just has to be less than -3. The absolute value will be positive even if x is a negative number."
    },
    {
        "video_name": "U-k5N1WPk4g",
        "Q": "7:40 Shouldnt be only X > 3? Because if X < -3 wouldnt be possible to have 0 as result?",
        "A": "Think about this line. -3 0 ----|------------|----- Where would x be on that line? Remember, the numbers on the left are smaller than those on the right. -3 0 xxxx|------------|----- If x was larger than -3 (> -3), 0 would be a possible result. -3 0 ----|xxxxxxxxxxxx|xxxxx"
    },
    {
        "video_name": "5xe-6GPR_qQ",
        "Q": "Wow! I never new about \"that trick\" at 0:42 . What are some of the other videos where he explains it?",
        "A": "Can you make that clear."
    },
    {
        "video_name": "5xe-6GPR_qQ",
        "Q": "3:02 what's a multiple? i've herd of it i just don't know what it is.",
        "A": "a multiple is the product of any number and a quantity. for example a multiple of five would be 10, 15, 20,25,30,35..."
    },
    {
        "video_name": "5xe-6GPR_qQ",
        "Q": "At 0:55, Sal says 3 INTO 154.\nBut we are dividing and not multiplying.\nDoesn't INTO mean multiplying ?",
        "A": "Think of it like this: how many times does 3 go into 154. That is division."
    },
    {
        "video_name": "JUgrBkPteTg",
        "Q": "At, 11:25, can it be \"r,1 r2 and r4 form the basis for C(A)\" as N(A) = N(rref(A))",
        "A": "The column space is the span of the column vectors of A in the columns that are pivot columns of rref(A) (yes, in this problem r1, r2, and r4). N(A) is found by translating rref(A) equations into vectors that are a subset of R^n, the domain of x in Ax=0 ."
    },
    {
        "video_name": "JUgrBkPteTg",
        "Q": "At 7:28 you state that r1, r2, and r4 are linearly independent. Are those columns (r1, r2, and r4 ) dependent variables though as they have a leading 1 value ?",
        "A": "Yes, that s right (I didn t notice that). The pivot variables are dependent variables, and the non pivot variables are independent (free) variables. But the vectors in the pivot columns (in both A and rref(A) ) form a linearly independent set, and those in the non pivot columns are in their span, as Sal explains."
    },
    {
        "video_name": "JUgrBkPteTg",
        "Q": "At 10:12, Sal says that to find a basis for the column space of A we should find the pivot columns in the rref(A) and then look at the corresponding columns in A, and those corresponding columns will form a basis. My question is, why take that extra step at the end? Why not just stop with the pivot columns in the rref(A)? Is it because you can't construct column a5 in A with a linear combination of the pivot columns?",
        "A": "The short answer is that you re ostensibly not looking for a basis of C(rref(A)), but rather a basis for C(A). And, as this example shows, the span of the pivot columns in rref(A) is not C(A). Your counter example is correct. You can t generate a3 or a5 or any vector that has a non-zero 4th element with a linear combination of the pivot columns of rref(A)."
    },
    {
        "video_name": "JUgrBkPteTg",
        "Q": "@4:47 you said \"minus 1 plus minus 1 is 0,\" shouldn't it be \"minus 1 plus 1 is 0?\"",
        "A": "Yes, you re right. He said it wrong."
    },
    {
        "video_name": "JUgrBkPteTg",
        "Q": "At time 2:53, it seems there is something wrong with the third row. It is not consistent with the operation that was said will be used.",
        "A": "The third row looks fine. He said he d replace row 3 with row 3 + row 1, and from what I can tell, that s what he did."
    },
    {
        "video_name": "mgNtPOgFje0",
        "Q": "At 3:27 he says I sqaured is negative one. Why is that? How did he come to that conclusion? He says the same thing at 1:17 but I don't understand how he mades that conclusion.",
        "A": "Thank you. Interesting. I see that the other post under this question was removed. In any case thank you."
    },
    {
        "video_name": "mgNtPOgFje0",
        "Q": "At 2:15 he goes to say i^2 is -1, I'm sorry I may be asking a silly question, but how?",
        "A": "i = sqrt(-1); i^2 = sqrt(-1)^2 = -1"
    },
    {
        "video_name": "mgNtPOgFje0",
        "Q": "Can anyone tell me what's wrong here, because I'm confused:\n\ne^(i*pi) = -1 (Euler's Identity, 9:20)\n(e^(i*pi))^2 = (-1)^2 (square both sides)\ne^(2i*pi) = 1\ne^(2i*pi) = e^0 (e^0 =1)\nTherefore, 2i*pi = 0 (false, but seems to follow? e^a = e^b so a=b)\n\nMy guess is that the logarithmic rules go out the window once you introduce an imaginary exponent, and thus the conclusion doesn't follow from the last step, but I can't really explain it. Does anyone know?",
        "A": "Been a while since you posted that, but no that s not an error. (x^m)^n = x^(mn) by standard exponent laws, so it follows that (e^(i*pi))^2 = e^(2i*pi)."
    },
    {
        "video_name": "mgNtPOgFje0",
        "Q": "Am I missing something really obvious here? Starting from 1:34 how are you allowed to just replace terms that contain x with ix?",
        "A": "We have an expression for e^x, and if we wanted to use that expression to calculate e^7, we would replace all the x s with 7 s. In this case, we re going to use this expression for e^x to calculate e^ix (e raised to the power i times x), so we replace all the x s in the expression with ix s."
    },
    {
        "video_name": "mgNtPOgFje0",
        "Q": "At 1:27 in the video: \"so everywhere we see an \"x\" in this polynomial expansion we would write an \"ix\" . Wouldn't it be much better to be consistent with the explanations and say \" the 1st derivative of e^ix = ie^ix, the second derivative is (i)^2e^ix, the third derivative is (i)^3e^ix and so on\" ?",
        "A": "I don t see why that would be better... The entire last video was dedicated to finding a polynomial representation of e^x, or e to the something th power. Now he wants to find a polynomial representation of e^(ix), which is a case of e to the something th power. Doesn t it make a lot of sense then to use the representation he just found and just plug in the ix, rather than to list all the derivatives again, apply to the maclaurin formula again and basically remake the previous video? o_O"
    },
    {
        "video_name": "mgNtPOgFje0",
        "Q": "At about 9:00, Sal says he's assuming that we're working in radians for this identity. Isn't that assumed by the Maclaurin series for sine and cosine?",
        "A": "At this level of study, degrees are almost never used, because the computations are so much more difficult."
    },
    {
        "video_name": "s9t7rNhaBp8",
        "Q": "Anyone else think that Sal is testing our attention to detail by subtracting 36 for 144 and getting 112 @7:19?",
        "A": "Maybe he just goofed- he does that occasionally."
    },
    {
        "video_name": "s9t7rNhaBp8",
        "Q": "How did he get 4 square roots of 7 at 8:16? After you do get your answer of 112(really 108) What do you do next? How do you find the square root of 108?",
        "A": "yeah. it would be equal to 6 root 3. won t it? correct if wrong."
    },
    {
        "video_name": "s9t7rNhaBp8",
        "Q": "At 4:30, the triangle Sal drew doesn't look like it has a hypotenuse. I don't even think it is a right triangle. It looks more like an isosceles. Does anyone know whether or not it is?",
        "A": "It is, in fact, a right triangle - Sal s drawing skills just aren t that exact. It s important to note, though, that a triangle can be right and isosceles."
    },
    {
        "video_name": "s9t7rNhaBp8",
        "Q": "At 7:43, How is 144-36=112 ?? I thought it was 108!",
        "A": "It is 108. Sal probably made a mistake. But honestly, he s human. He s a great man, yes, but he s human and humans make mistakes! It would be impossible to make 4000+ videos perfectly... :)"
    },
    {
        "video_name": "s9t7rNhaBp8",
        "Q": "If the triangle is a right doesn't that mean that the two angles that are not right are 45 degrees?3:57\nHow is it then that there are angles with right angles and the other two angles are not 45 degrees?",
        "A": "The definition of a right triangle is simply that it has a right angle, an angle measuring 90 degrees. As long as the angles add up to 180, and the triangle has a right angle, it is a right triangle. Therefore, you could have a triangle with angles measuring 90, 10, and 80, but it would still be a right triangle."
    },
    {
        "video_name": "u9B-dV6r6rU",
        "Q": "At 3:30, what does Sal mean by translations? What is a polygon?",
        "A": "A translation is basically moving the figure up, down, left, right, or diagonally. A polygon is a closed figure with at least three sides or more. A regular polygon is a closed figure with three or more equal sides."
    },
    {
        "video_name": "u9B-dV6r6rU",
        "Q": "At 0:31, what did he mean?",
        "A": "The y value was 2 in T(5,2) so he moved the polygon up twice after moving it 5 times to the right. From where the polygon is initially, in this problem, you use the T(5,2) and move the polygon 5 places to the right from the starting place, then 2 places up from that place. That is the translation!"
    },
    {
        "video_name": "pDZIvyeqX1o",
        "Q": "At 5:18, is there any reason that one wouldn't be able to calculate that the 6 equal magenta angles are each 60 degrees (360/6) making each remaining angle 30 degrees (180-90-60) allowing you to skip the last step?",
        "A": "you could calculate it but that is not the point of the video. the point is to prove that if the orthocenter and the centroid are at the same point, then the triangle is equilateral. but yes claculating the angles does reinforce the proof."
    },
    {
        "video_name": "3Md5KCCQX-0",
        "Q": "At 4:54, why v2 = a and v3 = b?",
        "A": "Sal is setting the free variables v2 and v3 to be arbitrary real numbers, so he can write v1, v2, and v3 as linear combinations of vectors and the arbitrary real numbers a, b."
    },
    {
        "video_name": "3Md5KCCQX-0",
        "Q": "After 13:25 Sal start to talk about the interpretation of the eigenvectors and how they relate to a vector x being transformed by the matrix A.\n\nHe then goes through showing what happens when x would be in one of the eigenspaces.\n\nMy question is: what is the interpretation of the transformation being applied to a vector outside of any eigenspace, like any regular 3-dimensional x?\n\nHow could I \"visualize\" what happens to such a vector?\n\nIs it related somehow to the characteristic polynomial?",
        "A": "If a vector v in R^n isn t an eigenvector for A (isn t in an eigenspace), then Av isn t equal to a (non zero) scalar times v. I think this means that the vector is either rotated, or transformed to the 0 vector, by A."
    },
    {
        "video_name": "3Md5KCCQX-0",
        "Q": "at 3:58 can you solve for v1, v2, and v3 using an augmented matrix? And if you can, why isn't the solution infinite since the whole bottom row is 0?",
        "A": "Yes you can. In effect that s what he did there. I would say that the solution is already doubly infinite, since it has 2 free variables - and is the span of 2 linearly independent fixed vectors (is that what you mean?)."
    },
    {
        "video_name": "jFSenp9ueaI",
        "Q": "I get a little bit confused at 2:31. How is it 4 yards plus 1/2 yards? Why isn't it just 4 1/2 yards? Or is it, and I just got confused? I don't know. Can somebody help me?",
        "A": "no that mean, that u have 4 1/2 u have 4 and 1/2"
    },
    {
        "video_name": "jFSenp9ueaI",
        "Q": "At 1:59, he says 4 x 36 is 120, +24. Why doesn't he just say 4 x 36 is 144?",
        "A": "He was simply saying aloud what he was thinking, because 4 x 36 is the same as 4 x 30 + 4 x 6 so it is 120+24 so he was doing mental math and to make it easier he split 36 and so yeah 4 x 36=144, but that is harder to do mentally"
    },
    {
        "video_name": "jFSenp9ueaI",
        "Q": "Starting at 00:31, why does Sal have to go through all that trouble when he can just multiply 36 in. times 4 1\u0002 yards?",
        "A": "You could do that, but you re doing several steps inside your head. Sal was just showing the rationale behind making that calculation and breaking it down into its individual steps."
    },
    {
        "video_name": "jFSenp9ueaI",
        "Q": "what did sal mean at 3:22?",
        "A": "At 3:22, he just figured out that the first part of the problem is equal to 4 pints, and he decided to convert it to quarts to be able to simplify the rest of the problem."
    },
    {
        "video_name": "TqAtt3g6Tkc",
        "Q": "0:02 does that mean that this entire video is pre-algebra/algebra for 3rd graders?",
        "A": "Algebra involves solving for unknowns using placeholders for those unknowns (we call those unknowns variables ). In this video, he introduces the idea of these place holders with a blank (__), a ?, a smiley face, a star, an A, and B. This enables 3rd Graders to become familiar with these placeholders long before Algebra introduces them in more complicated calculations."
    },
    {
        "video_name": "_9x2cqO7-Ig",
        "Q": "at ~0:58, I thought that the normal vector always points in the direction the curve is curving. So wouldnt the unit normal vector point in the opposite direction (towards the origin)?",
        "A": "Short version: No. Every plane has two normal vectors, which point in opposite directions from each other."
    },
    {
        "video_name": "jTCZfMMcHBo",
        "Q": "At 2:11, is the 108-54/8-4 like RISE/RUN?",
        "A": "Well yes it is the change in y (which is rise)/change in x (which is run)"
    },
    {
        "video_name": "jTCZfMMcHBo",
        "Q": "How does Sal know at 3:22 that the label is dollars per hour?",
        "A": "The slope is change in y / change in x . All the y values are $ ; all the x values are hours . So the slope is change in $ / change in hours which gives us the label of dollars per hour ."
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "Starting at around 7:25, he starts to draw lines for standard deviations away form the mean. How do you find where those lines should go?",
        "A": "z=+-1: 68% of the area under the normal curve. z=+-2: 95% z=+-3: 99.7%"
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "At 10:42 when Sal is discussing p-values, that's something I don't really understand. Even if we assume 5%, are we always rejecting if the p value is smaller? I've been told it depends on the the null hypothesis. So how can I tell? Thanks!",
        "A": "If the alpha risk is 5%, then reject the null hypothesis if the p-value is <= 0.05 (p is low, null must go) and fail to reject the null hypothesis if the p-value is > 0,.05 (p is high, null flies)."
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "What is meant by 'status quo' at 0:51? I've search for this on wikipedia and it says it means the existing state of affairs, but I don't quite understand it and I can't relate it to 'hypothesis'.",
        "A": "Sal says next that what he means is that the null hypothesis should be that what you are researching has no effect. (He is equating the existing state of affairs to having no effect, that a rat would have the existing response time whether or not you took the drug.)"
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "At around 8:56 Sal say that a Z value of 3 gives you a result of 99.7 % but according to my textbook the correct value is .9987, am I reading the table wrong or did Sal mess up?",
        "A": ".9987 is the cumulative probability: the area under the curve from -infinity to 3. What he was citing was the probability of the area between the two tails: the area between -3 and +3. To get his number find the probability of the right tail: 1 - .9987. Then double it to find the probability of both tails, and subtract from 1. In R: > pnorm(3) [1] 0.9986501 > 1 - 2 * (1 - pnorm(3)) [1] 0.9973002"
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "The formula for z-score is ((x-u)/s.e), so why do you do ((u-x)/s.e)? As in, shouldn't it be 1.05 - 1.2/s.e? This is at 6:25-6:30...",
        "A": "Having a value as a positive or negative would indicate which side of normal distribution it is on wouldn t it? Thus positive = right, negative = left. Z-scores correspond with standard deviations."
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "Q about general definition of pvalue. From the video at 10:25 I get a defintion. - The pvalue is the probability of getting a sample mean this extreme or more given if the Ho is true. What should the general term be for \"this extreme\" - as extreme as the null value or the test statistic value? or something else.",
        "A": "In this problem - as extreme as 3 standard errors (.997) from mu. In general - as stipulated in the problem. What are null value and test statistic value ?"
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "At 8:04, Sal asks, \"What is the probability of getting a result this extreme by chance?\" Why does he ask that question instead of \"What is the probability of getting a result this extreme ON THE LEFT SIDE?\" (I'd use italics but I can't here)",
        "A": "It seems to me that both of these questions are equally valid, and that the answer to the former is .003; the latter .0015."
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "At 0:43, Sal writes the \"null hypothesis\" as mean = 1.2 seconds. Does the null hypothesis always have to be \"equal\" to something, or can it be \"less than or equal to\" or \"greater than or equal to\" as well?",
        "A": "It can. For instance, if I run a training program and want to prove that it is effective, I might offer my students a pre-test and a post-test and compare the scores. In that case, my null hypothesis would be that students scores either went down or stayed the same -- in other words, that \u00ce\u00bc_post - \u00ce\u00bc_pre \u00e2\u0089\u00a4 0. It d be the same process, except that I m only going to measure one tail instead of two."
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "Starting at 4:22, why do you need to estimate the sample standard deviation when you already have it(.5)? He goes on to say that you put a hat on it to show that you estimated the population standard deviation by using the sample but why does the sigma have a hat for population estimate and have an x bar for sample? Is the notation correct on that section?",
        "A": "As you say, the standard deviation of the sample was given. But Sal was estimating the standard deviation of the sampling distrubution of the sample mean, which is smaller by a factor roughly the size of the square root of n. It s easier to differentiate between the two standard deviation types when the other name, standard error of the mean , is used instead."
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "At 6:04, if the computed F-ration was 2.293 with df of (12,8) for p>.05. What may you say about the alternate hypothesis?\n\nWouldn't the alternate be rejected and that there was no differences noted?",
        "A": "We don t like to say Reject the alternative hypothesis, because that s not really how the tests are set up. Hypothesis tests are conducted assuming that the null hypothesis is true. The data then will either present evidence to contradict the assumption (Reject Ho), or will fail to do so (Fail to reject Ho)."
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "at time 09:44 he said less that one in 3 hundred . what does he mean by that? Isn't 0.003 one in 3 thousand.",
        "A": "0.03 is about 1 in 333 or 1 in 334"
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "at 4:40 sal refers to the sample size as being greater than 100.\n\nThat was a mistake right?",
        "A": "Yes, it s a mistake. He meant greater than 30 , which is the threshold for a sample size to be considered good . That is, for the sampling ditribution of the sample mean to be normal."
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "I'm a little confused. Starting around 4:22, Sal estimates the population standard distribution using the sample standard distribution... in order to calculate the standard distribution of the sampling distribution.\nWhy is it necessary to estimate? Can't we directly calculate the standard deviation of the sampling distribution? If I understand correctly, we have enough information to calculate it (square root of [each data point minus the mean squared]), right? Wouldn't that be more accurate than estimating?",
        "A": "The standard deviation of the sampling distribution of the sample mean would require a lot of values of the sample mean. We usually only have one value for this - the mean of our sample. We use the concept of the sampling distribution to understand how the sample mean will behave if we took many samples. But since we just have the one sample, we use the sample standard deviation to estimate the standard error of the sample mean ."
    },
    {
        "video_name": "-FtlH4svqx4",
        "Q": "In the video at around 9:38 it stated something similar to \u00e2\u0080\u009cThe probability of getting a result more extreme than this is called a p-value\"\nI am doing a multiple choice question ware I have not been told an alpha level which asks what the p-value of 0.015 means.\nThe answer states that this means there is a 1.5% chance that the sampling variation is the cause of the observed differences. Could you please tell me how this is true or not true? I am utterly confused.",
        "A": "p=0.015 means it s expected that 1.5% of the possible ways to randomly select n individuals from your population that ll give a sample mean that differs from the population mean by at least as much."
    },
    {
        "video_name": "qsL_5Y8uWPU",
        "Q": "At 5:25, x = -1/-9. Why, then, did it change to x = 1/9? What made it positive?",
        "A": "x= -1 / -9 The negative divided by the negative sign gives you positive sign. or -1 / -9 * -1/-1 =1/9"
    },
    {
        "video_name": "qsL_5Y8uWPU",
        "Q": "At 3:50 ,Sal says,\"Lets do that in green color\",but does it in yellow color.",
        "A": "That really isn t relevant."
    },
    {
        "video_name": "qsL_5Y8uWPU",
        "Q": "Wait.... (Sound of record rewinding) I'm at 0:29, and he says \"We'd get something like x= 5 or 10 or -\u00cf\u0080.\" My question is: Can \u00cf\u0080 really be negative? I'm not sure if the fact that \u00cf\u0080 is infinite changes anything with \u00cf\u0080 being negative, or anything. Thanks in advance!\n- Temp",
        "A": "pi is just a number. It s a value - a very special value, but just a value - so yes, it can be negative. In its capacity as the ratio between the circumference and diameter of a circle, it won t come out negative, since lengths are always positive, but in terms of pure theory and using is just as a number, we can represent it as negative. HTH :)"
    },
    {
        "video_name": "qsL_5Y8uWPU",
        "Q": "4:18 Well, 3 cant equal 2 But 2 1/2 =3 when milt-plied. Right?",
        "A": "no 2 1/2 = 1 beacuse your multiplying 2 and 1/2. so 1/2 plus 1/2 =1"
    },
    {
        "video_name": "qsL_5Y8uWPU",
        "Q": "At 3:04, wouldn't the answer be 1 instead of 0 because x/x = 1. How did he get 0??",
        "A": "When he ends up with the equation: x = x He subtracts x from both sides: x - x = x - x 0 = 0"
    },
    {
        "video_name": "qsL_5Y8uWPU",
        "Q": "At 4:57, after doing 2-3= -1, he wrote that the right side of the problem is 2x-1, but wouldn't it be 2x -(-1) or 2x + 1? (This is in the third problem)",
        "A": "The equation starts as: -7x + 3 = 2x + 2 He combines the constants by subtracting three from both sides: -7x + 3 - 3 = 2x + 2 - 3 which becomes: -7x = 2x - 1 Why? Because 2 minus 3 is -1. Or -3 plus 2 is -1."
    },
    {
        "video_name": "qsL_5Y8uWPU",
        "Q": "At 3:01, how x=x is equal to the 0=0 ? not 1=1?",
        "A": "He subtracted x from both sides."
    },
    {
        "video_name": "qsL_5Y8uWPU",
        "Q": "At 3:30, why did Sal do it differently? I guess my question is this: could he have done it the same way as he did the first one and still get 3=2?",
        "A": "On the 2nd example, he could have subtracted 3 first, then added 7x to both sides. He would get 0 = -1 rather than 3 = 2. The different values still indicate that there is no solution because 0 does not equal -1."
    },
    {
        "video_name": "qsL_5Y8uWPU",
        "Q": "For the third problem at 4:40 do you have to subtract the 3 from the 2 or can you subtract the 2 from the 3 as well?",
        "A": "It is simply a matter of choice. The goal is to get x alone. You could start with adding 7x to both sides, the subtract 2 from both sides. Some orders/choice-paths are easier than others and with practice you will learn to see which way with be easiest before you start."
    },
    {
        "video_name": "Xl724v0CU7E",
        "Q": "At 0:21, it kept repeating. Is there a way to fix that?",
        "A": "Hmmm...looks good on my end; is it still repeating for you?"
    },
    {
        "video_name": "f4MYCepzLyQ",
        "Q": "At 8:49, isn't d(3) = 10? For example, d(3) = 3^2+1 =10. So, shouldn't the slope be (10-5)/(3-2) which equals 5 m/s? Thanks",
        "A": "I had the same question! I m assuming Sal just misspoke?"
    },
    {
        "video_name": "P5y83-ZnQeo",
        "Q": "At 1:24 you placed the fulcrum in the middle of the hypotenuse. If it were to be placed there, the triangle would have fallen on the right. Therefore, the fulcrum should be placed a bit more to the right. But then it will be so close to the median that the distinction will be very hard to make. Our eyes might be playing us. How should we work it out?",
        "A": "Nice catch! you re right the fulcrum should be a bit more to the right, and maybe the mean will be so close to the median but it will be to the left of it, and that s because the curve is a very little bit skewed to the left... If it were more skewed to the left the mean would be further left from the median"
    },
    {
        "video_name": "tVcasOt55Lc",
        "Q": "At 2:35 there is a pop up that says\" Sal said\"obtuse angle\" but meant \"reflex angle.\" What is a reflex angle?",
        "A": "A reflex angle has a measure between 180 and 360 degrees."
    },
    {
        "video_name": "tVcasOt55Lc",
        "Q": "At about 2:20 how is 18pi over 36 equal to pi over 2? Wouldnt be it something smaller because 36 cant go into 18. Help plz.",
        "A": "18/36 is equal to 1/2 You simplify the fraction by dividing the top and bottom by 18. 18pi/36 can also be simplified by dividing the top and bottom by 18 You get 1 pi/2 or pi/2 Hope that s clear."
    },
    {
        "video_name": "_y17uHBwa_8",
        "Q": "at 0:11 why does he want to color the shapes faces when we can clearly see them",
        "A": "Just makes it a little easier to distinguish from the other sides."
    },
    {
        "video_name": "-lUEWEEpmIo",
        "Q": "At 0:55, how do you turn the improper fraction into a mixed number?",
        "A": "you can turn an improper fraction to a mixed number by dividing the top number by the bottom number and that is how you do it :)"
    },
    {
        "video_name": "-lUEWEEpmIo",
        "Q": "At 0:18, is \"of\" a key word for multiplying?",
        "A": "Yes, of can almost always be thought of as multiplication."
    },
    {
        "video_name": "-lUEWEEpmIo",
        "Q": "At 1:17 would you have to show the work on the side?",
        "A": "It depends how strict your teacher is (:"
    },
    {
        "video_name": "-lUEWEEpmIo",
        "Q": "in 0:16 why couldn't you simplify 30/100 into 3/10",
        "A": "You could, and it would still end up with the same answer. If you simplified it first, then you would directly get 18/10 which is still equal to 9/5 and so on."
    },
    {
        "video_name": "R-rhSQzFJL0",
        "Q": "1:15 WHY does a*b= (7)*(-3)\n\nwhy isn't the equation simply speaking a*b = -3\n\nwhy multiply a*c, such that ac= ab. How can it shown to be true, such that ac= ab",
        "A": "Your technique only works if the quadratic starts with a leading coefficient of 1: 1x^2 (or 1k^2 as it relates to this video). The quadratic being factored in this video has a leading coefficient of 7. So, the factoring process needs a different approach. That is what this video is trying to show you."
    },
    {
        "video_name": "R-rhSQzFJL0",
        "Q": "I am confused, because at 2:48 he talks about breaking the 20 into a 21 and a -1 its kind of weird to me that he makes a positive 21 out of a twenty... is there another way to do that?",
        "A": "he is looking for two numbers that when multiplied together equal -21 and when added together equal 20. 21 x -1= -21 and 21 + -1 =20"
    },
    {
        "video_name": "R-rhSQzFJL0",
        "Q": "At 0:30, he says that the common factor is 5, but aside from this problem, if you had to, can you factor out a negative number?",
        "A": "Yes... you can. If the leading coefficient is negative, it is often advantageous to factor out a negative number (even a -1) to convert the leading coefficient into a positive value. It usually makes factoring the rest of the problem easier to do."
    },
    {
        "video_name": "Rsi5mzEaCPw",
        "Q": "What is SAS similarity and what does it stand for? He mentioned it at 3:00?",
        "A": "Actually in similarity the \u00e2\u0088\u0086s are not congruent to each other but their sides are in proportion to each other and angles correspond to each other . Suppose we have \u00e2\u0088\u0086ABC and \u00e2\u0088\u0086PQR. AB/PQ = BC/QR = AC/PR and angle A =angle P,angle B = angle Q and angle C = angle R. Like congruency there are also test to prove that the \u00e2\u0088\u0086s are similar. For example SAS ,SSS, AA. In SAS Similarity the two sides are in equal ratio and one angle is equal to another."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "At 1:40, shouldn't the square root of 52 be sqrt{(1)*(52)}, not sqrt{(-1)*(-52)}?",
        "A": "It is to show that splitting it into negative numbers doesn t work: sqrt(-1) * sqrt(-52) = i*sqrt(-1)*sqrt(52) = i^2*sqrt(52) = -1*sqrt(52), but clearly sqrt(52) is not equal to -sqrt(52) so everything will get messed up."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "You mentioned that you can change the order of i at 3:33. And also clarified that it is easier to read if i is placed after the numbers. My question is, can i be placed anywhere I desire? For example, can i write 2 times i times the square root of 13, rather than writing 2 times the square root of 13 times i?",
        "A": "You can put it anywhere, but it is easier to read if you just put it at the end."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "at 0:48 to 0:50, Sal corrects himself saying he should say principal sqare root not square root, what is the difference?",
        "A": "The principal root is the positive root of a number. For example: \u00e2\u0088\u009a16 = \u00c2\u00b14 Because: 4 * 4 = 16 AND -4 * -4 = 16 The principal square root of 16 would just be 4, disregarding the negative root."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "at 3:32, isn't the sqrt of 4 equal to +/-2(since -2^2 is also 4)?\nHow does that affect our calculations?\nthat is if we do not apply principal square root...",
        "A": "Because in the given question there is no +/-, the answer will be the positive root."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "At 2:29, how does -1 simplify to i? I thought that -1 was i^2, not i.",
        "A": "Think about it. -1 does not equal i. The sqrt(-1) equals i. Or, if you square both sides of the equation, -1 equals i^2. To summarize (-1)^1/2 = i and -1 = i^2"
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "2:05 why doesn't the property work? is there any explanation? and are the normal and principal square root signs the same?",
        "A": "As I understand it, it s because we re working on the complex plane where one number (and one number only) can be complex (i.e. a multiple of root -1). We would have to be working on some weird 3 dimensional volume with two complex planes and, as far as I know, there is no branch of math that has that. Oh and see the answer to the question below... (sort questions by Recent)"
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "2:00 Why is t wrong to say \"sqrt(52) = sqrt(-1) * sqrt(-52)\"? How do we prove that this is wrong?\nif you calculate it you will get the following:\nsqrt(-1) = i\nsqrt(-52) = sqrt(-1*52) = sqrt(-1) sqrt(52) = i * sqrt(52)\nsqrt(-1) * sqrt(-52) = i * i * sqrt(52) =-sqrt(52)\nwhich part of it is wrong that it results in a wrong conclusion that sqrt(52) = - sqrt(52)?",
        "A": "When you are first introduced to simplifying radical expressions, you are shown a rule that says sqrt(a*b) = sqrt(a)*sqrt(b) but only for values of a>=0 and b>=0. If you allow a and b to be negative from an originally positive product, you run into a problem like you pointed out. This restriction is very important but sometimes overlooked."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "at 2:10 can you explain why we cant simplifies the negative numbers ?",
        "A": "Because you ll get answer that doesn t make sense. Also if you continue with that logic you ll never reach an answer. \u00e2\u0088\u009a(52)= \u00e2\u0088\u009a(-1*-52)=\u00e2\u0088\u009a(-1)*\u00e2\u0088\u009a(-52)=\u00e2\u0088\u009a(-1)*\u00e2\u0088\u009a(-1)*\u00e2\u0088\u009a(52), if you continue \u00e2\u0088\u009a(52) will just repeat all over. If you stop with \u00e2\u0088\u009a(52)=\u00e2\u0088\u009a(-1)*\u00e2\u0088\u009a(-1)*\u00e2\u0088\u009a(52 then you get \u00e2\u0088\u009a(52)=i\u00c2\u00b2*\u00e2\u0088\u009a(52)=-\u00e2\u0088\u009a(52). You can not use this because \u00e2\u0088\u009a(52) \u00e2\u0089\u00a0 -\u00e2\u0088\u009a(52)"
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "Is there any difference between just a square root, and the principal square root? At around 1:36 he said square root, then quickly switched to principle square root. Why does he do that?",
        "A": "It is just a matter of not being careful about what you re saying. A lot of people just say square root when they mean principal square root ."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "At 3:24 that is wrong. The 2 2's make the 2 go outside the radical. There would have to be 4 2's for it to be 4!!",
        "A": "At 3:24 Sal hasn t yet taken the square root. All he is doing is saying that the square root of a product (\u00e2\u0088\u009a (4*13)) is the product of the square roots ( \u00e2\u0088\u009a4 * \u00e2\u0088\u009a13). After that he then takes \u00e2\u0088\u009a4.and gets the 2."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "At 2:19, Sal explains that you canNOT separate a square root with 2 negative numbers. He explains that you need one of the numbers to be positive.\nWhen there are 3 numbers inside a square root, how many numbers have to be positive/negative?",
        "A": "Just pick two numbers and consider their product to be a single number. Then you ve reduced the problem back to the two-number case that Sal explained."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "At 3:32, when he changes the order of his answer to 2 times the square root of 13 times i, is that the only way to write that or can you reorder the terms and still keep it correct?",
        "A": "You can write it as 2 times i times square root of 13. He wrote like that because it made it simpler to read ."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "At 0:21 Sal has a negative in the radical does that AUTOMATICALLY make it a imaginary number.",
        "A": "Yes - if the square of the number is less than zero, it s an imaginary number. In this case, the square of sqrt(-52) is -52."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "Principle Square root? 0:30 seconds or so",
        "A": "principle square root is basically a big way of saying perfect squares."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "At 2:52, what is a perfect square?",
        "A": "A perfect square is an integer whose square root is also an integer. For example, 4 is a perfect square because its square root is 2. 9 is a perfect square because its square root is 3. The numbers 16, 25, 36, 49, ... are all perfect squares."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "when you did the factor tree for \u00e2\u0088\u009a52 you got 2*26 and then for 26 you got 2*13.\n\nwhy did you multiply 2 * 2 to get 4 for i * \u00e2\u0088\u009a4*13\n\ncan you multiply 13*2 as well or no?\n\n@3:10",
        "A": "No... you wouldn t also multiply 13*2. If you did, you would be saying that 52 in prime factors = 2*2*2*13. It doesn t. 4*13 = 52. You are trying to use the distributive property. That property distributes multiplication across addition or subtraction. You are trying to use it to distribute multiplication across multiplication. It doesn t work that way."
    },
    {
        "video_name": "s03qez-6JMA",
        "Q": "At 3:15, where does Sal get the 4 from?",
        "A": "4 is a factor of 52: 52 = 4 * 13. So we can factor out a 4 (which is a perfect square) to simplify the answer."
    },
    {
        "video_name": "XChok8XlF90",
        "Q": "Around 8:37, Sal factored with grouping. Wouldn't it be easier to just factor the 3 out of the numerator to make it 3(x^2+x+6), or would that make things more difficult?",
        "A": "yes it would because you would have to distribute the 3 to everything in the equation."
    },
    {
        "video_name": "XChok8XlF90",
        "Q": "How did he get (a+3)(a-3) at 1:34?",
        "A": "You even you may verify it by solving (a+3)(a-3): (a+3)(a-3) = a*a -3*a +3*a - 3*3 = a^2 - 3^2"
    },
    {
        "video_name": "XChok8XlF90",
        "Q": "Why at 2:53 can a not be equal to -9 or 3? Why is the 9 negative and the three isn't, when they were both positive before coming out of the bracket?",
        "A": "If a were either -9 or 3, the denominator would become 0. We cannot divide by 0 in mathematics. So, those are the two numbers that we need to exclude, in order to prevent this from happening. If we plug 3 into the expression, we d get this: (3+9)(3-3) = (12)(0) = 0. Similarly, if we plug -9 into it, we d get this: (-9+9)(-9-3) = (0)(-12) = 0. That is why a cannot be either of those numbers."
    },
    {
        "video_name": "XChok8XlF90",
        "Q": "At 2:54, says that a cannot be equal to -9 or 3; why -9 instead of positive 9?",
        "A": "If a were equal to positive 9, then the bottom would not come out to 0, because that first term is (a+9) which would become (9+9) if a were equal to positive 9. Since the bottom does not come out to 0 at a=9, the function is defined at that point. However, when a equals negative 9, that term becomes (-9+9) , which does equal 0, so then the whole bottom becomes 0 and therefore it is undefined at that point. I hope this helps."
    },
    {
        "video_name": "XChok8XlF90",
        "Q": "on 2:55 of the video why isn't -9 or 3 = to a?",
        "A": "If a equals -9 or 3, the denominator would equal 0, which would be undefined."
    },
    {
        "video_name": "XChok8XlF90",
        "Q": "at 4:11, when you say a can not be equal to 3, you mean -3 or positive 3? I'm confused, thank you",
        "A": "Saying a cannot be equal to 3 always refers to positive 3. If -3 is meant, then we must say a cannot be equal to negative 3 ."
    },
    {
        "video_name": "XChok8XlF90",
        "Q": "At 2:13 where did he simplify 6a into the simplified expression for the denominator?",
        "A": "Sal factored the trinomial in the denominator. If you don t understand how factoring works, check out the videos on factoring. Factoring basically reverses the multiplication done by FOIL. In this case, you find factors of -27 that add to +6. Sal picked -3 and +9 to create the 2 binomials: (x-3)(x+9)"
    },
    {
        "video_name": "9G0w61pZPig",
        "Q": "At minute 09:01 I am sure he ment to divide by 365^3 and not 365^2.",
        "A": "yes indeed :)"
    },
    {
        "video_name": "9G0w61pZPig",
        "Q": "At 11:00, 365 factorial is computed on a Texas Instruments TI-85 simulator? This number is too large for any calculator I have used. Can you explain why this works on this video and if I can duplicate the computation on another device without using mathematical approximations or spelling out the 30 factor multiplication?\nThank you.",
        "A": "365! is a 779 digit number, it would need at least 324 bytes of memory to be stored, so any calculator with less memory than that cannot handle it. You must note, however, that Sal didn t asked for 365! alone, it gave the calculator the whole operation, so it s probable that internally the computation was optimised to avoid using so much memory on a single number."
    },
    {
        "video_name": "X2jVap1YgwI",
        "Q": "at 3:35 why can the answer be 14.25 when its only .75 less than 15 and 95 is 5 less than 100? shouldn't it be 10 (5 less than 14) ?",
        "A": "How many times can 6 2/3 go into 100?. How many times can 6 2/3 go into 95?. Picture it on a number line. Dividing a smaller dividend by the same divisor must yield a smaller quotient. What s 100 * 15/100? What s 95 * 15/100?. What s 75 divided by 5?"
    },
    {
        "video_name": "X2jVap1YgwI",
        "Q": "At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?",
        "A": "At 5:31, how is the moon large enough to block the sun? Isn t the sun way larger?"
    },
    {
        "video_name": "X2jVap1YgwI",
        "Q": "At 2:56 we add then multiply?",
        "A": "What it s basically trying to tell you is to find the percentage of the number by multiplying, then adding the product back to the original number."
    },
    {
        "video_name": "X2jVap1YgwI",
        "Q": "Around 1:24 why does Sal times .15 and 95?? He already added 95 to .15! Please help!!",
        "A": "An amount, 95 dollars, is going to grow by a percent, 15%. You have to find out what 15% of 95 is. To do that, you convert 15% to a decimal. That is .15. Then you have to multiply .15 by 95. The answer is 14.25. That means that 14.25 is 15% of 95. If 95 grows by 15%, then you have to add that percentage to it. (95 + (.15 x 95)), or simplified, that is (95 + 14.25)"
    },
    {
        "video_name": "X2jVap1YgwI",
        "Q": "at 6:16 why is Sal able to get rid of the x and replace it with a 1?",
        "A": "An x by itself is the same as 1(x). 1x+.25x, or expressing it without x distributed, x(1+ .25) = 1.25x."
    },
    {
        "video_name": "X2jVap1YgwI",
        "Q": "7:05 How do we know which number is the numerator and which number is the denominator?",
        "A": "Haha this is going to sound silly, but I remember that the denominator is on the bottom because as I kid, I initially read the word as DEMON inator. I always viewed demons or monsters or whatever as staying down below in a dungeon or something, so the denominator always went below in a fraction. Silly, I know, but I always remembered it!"
    },
    {
        "video_name": "X2jVap1YgwI",
        "Q": "2:30 Sal made a mistake the answer is 14.25",
        "A": "I m pretty sure he INTENTIONALLY made a mistake just so us listeners can remember to check our answers."
    },
    {
        "video_name": "X2jVap1YgwI",
        "Q": "what is the distributive property at 6:16",
        "A": "Given a(b+c) we can use the distributive property to distribute the a term to the b and c terms: a(b+c) = ab + ac. Also, given ab + ac, we can factor out the a to get a(b+c). Factoring is also the reverse of the distributive property, which is what Sal was alluding to."
    },
    {
        "video_name": "X2jVap1YgwI",
        "Q": "About 8:32 concerning the portfolio growing by 25%. Which I interpret as $100 decreasing by 25% to get at its starting point (is that correct thinking?). Using a multiplier 1 (P%as a decimal) which gives me .75. Therefore 100 x .75= $75. Where did I go wrong?",
        "A": "The key word is growing by 25%. Growing means something is increasing by 25%. You actually decreased by 25%. To increase $100 by 25%: 100 + 100*0.25 = 100 + 25 = 125"
    },
    {
        "video_name": "X2jVap1YgwI",
        "Q": "Why is there an extra unknown variable \"X\" added to the equation @ 5:30? I understand\nx +.25=100 but do not understand the second x, the 'of x':\n\nx + .25 \"\"of x\"\"=100.\n\nWhat purpose do each of the unknown variables serve?",
        "A": "x is some number [that you started with] plus 25% of that number ...number = x right? So x + .25x = 100. Remember, you are adding something to the amount you started with. How much is that something? 25% of what you started with."
    },
    {
        "video_name": "X2jVap1YgwI",
        "Q": "at 1:42 why can you multiply a percent and a normal number without changing the value of the number?",
        "A": "A percentage is a fraction. So is the normal number during multiplication. 95 is the same as 95/1 which doesn t change its value. 95*15% = 95/1 * 15/100 = 1425/100 =14.25"
    },
    {
        "video_name": "X2jVap1YgwI",
        "Q": "at 2:32 2 decimals?",
        "A": "Minho: he meant two decimal places. davidann: go back in the videos to a place you understand and work forward."
    },
    {
        "video_name": "4ywTWCaLmXE",
        "Q": "at 4:55 , how is that multiplying 9 times 9 times cancels each other?\n9*9=81....am so confuse",
        "A": "MarinoCabrera92, your right, 9*9=81, but @4:55 that isn t what he does, he multiplies 9*x/9, or = 9*1/9*x. 9*1/9=9/9=1, 1*x=x By multiplying x/9 by the whole number 9 we isolate the variable, or get x by itself. Hope this helps. T. S. P.S. Max Murphy, it doesn t just happens like that :p :0 ;p ;0 ;) :) MarinoCabrera92 is right, 9*9=81, but @4:55 that isn t what he does see above. T.S. (No place for hate)"
    },
    {
        "video_name": "4ywTWCaLmXE",
        "Q": "I understand solving basic proportions, but what about when you have say... 5:6 = 7n+9:9 ?\nThis can still be cross multiplied yes? But not sure what to do with the, 7n+9... 6(7n+9)? 6x9?",
        "A": "5:6 = (7n+9):9 => 45 = 6(7n+9) => 45 = 42n + 54 now its solvable I hope?"
    },
    {
        "video_name": "4ywTWCaLmXE",
        "Q": "is there any reason that around 2:50 he puts the question mark in a box?",
        "A": "He puts a question mark in a box because that is representing an unknown answer."
    },
    {
        "video_name": "4ywTWCaLmXE",
        "Q": "At 0:16 The transcript says \"So 2 cups of flour for every 3 cups of oatmeal\" when Sal says \"So 2 cups of flour, 2 cups of flour, flour, for every 3 cups of oatmeal, for every 3 cups of oatmeal, for every 3 cups of oatmeal, so for every 3 cups of oatmeal, for every 3 cups of oatmeal\", why is that?",
        "A": "That s because the transcript is made to help you understand what Sal is saying, not EXACTLY what Sal is saying. Also, Sal was just repeating what he said to fill the space while he was writing it out."
    },
    {
        "video_name": "gWMTTP58_J0",
        "Q": "at between 2:03 and 2:04 , what does sal mean by either around the triangle or around the circle",
        "A": "I think it means that it is 1/3 of the circumference of the circle."
    },
    {
        "video_name": "gWMTTP58_J0",
        "Q": "At 2:39, how does Sal get the right place to put the compass?",
        "A": "Sal drags the compass so that its center is on the circumference of the original circle and he places it so that it intersects these two points . The two points are the center of the original circle and the point where one side of the equilateral triangle that is already placed touches the circumference of the original circle."
    },
    {
        "video_name": "9xgO-EJ3sr0",
        "Q": "At 7:12, what if L was greater than f(b)?\nAnd if it is not possible so, why not?",
        "A": "The theorem requires \u00f0\u009d\u0090\u00bf to be between \u00f0\u009d\u0091\u0093(\u00f0\u009d\u0091\u008e) and \u00f0\u009d\u0091\u0093(\u00f0\u009d\u0091\u008f). In this case \u00f0\u009d\u0091\u0093(\u00f0\u009d\u0091\u008f) > \u00f0\u009d\u0091\u0093(\u00f0\u009d\u0091\u008e) \u00e2\u0087\u0092 \u00f0\u009d\u0090\u00bf < \u00f0\u009d\u0091\u0093(\u00f0\u009d\u0091\u008f)."
    },
    {
        "video_name": "0z52CLjC2C0",
        "Q": "at 4:40, I don't understand how cosX^2/x^2 multiplied with 2x results in (2cosX^2)/x?\nPlease explain that to me?",
        "A": "You can simplify by canceling the common factor of x. 2x/x=2 (x^2)/x=x"
    },
    {
        "video_name": "u1UKIljUWuc",
        "Q": "I don't understand why he was adding the pink 1. At 7:45 it disappeared from the series side of the inequality but stayed on the function side.",
        "A": "He wasn t actually adding the pink 1 in the first place. Rather he was rewriting the sum of 1/n^2 from n = 1 to infinity as 1 + the sum of 1/n^2 from n = 2 to infinity. That is, the pink 1 is simply the first term of the series (1/n^2 when n = 1). He then uses the rewritten version to link his series to an improper integral that he can evaluate. At 7:45 the pink 1 doesn t disappear; rather, he just returns to writing the sum in yellow, from n =1 to infinity again. Hope that helps!"
    },
    {
        "video_name": "u1UKIljUWuc",
        "Q": "At 0:42 the video mentions f(x) has to be Decreasing.\nWhy can it not be Non-Increasing (i.e. ultimately flat at zero) ?",
        "A": "The only function that can ever be truly flat at zero is y = 0. All other functions, if they are approaching zero, must be decreasing (if approaching from y > 0) or increasing (if approaching from y < 0). If the function is approaching zero, it can never be equal to zero. Eventually, it will get so close so that, for all practical purposes, it is on the x-axis, but it won t actually be equal to zero. I hope this helps!"
    },
    {
        "video_name": "A_ESfuN1Pkg",
        "Q": "At 3:01, can't you factor the 2 out of 4-2i?",
        "A": "Yes, you could factor out 2 and get 2(2 - i). But Sal is pointing us towards the standard form of a complex number. A complex number is made up of two terms, one of them real and one of them imaginary. So to keep to this standard form you won t factor out any common factor between the real part and the coefficient on the imaginary part. This format is going to be important in later videos about complex numbers."
    },
    {
        "video_name": "A_ESfuN1Pkg",
        "Q": "At 3:00 I was wondering how to multiply binomial complex numbers ex. (7+2i)(3+4i) ?",
        "A": "(7 + 2i)\u00e2\u0080\u00a2(3 + 4i) 7\u00e2\u0080\u00a23 + 7\u00e2\u0080\u00a24i + 3\u00e2\u0080\u00a22i + 2i\u00e2\u0080\u00a24i 21 + 28i + 6i + 2\u00e2\u0080\u00a24\u00e2\u0080\u00a2(i\u00e2\u0080\u00a2i) 21 + 28i + 6i + 8\u00e2\u0080\u00a2(i^2) 21 + 28i + 6i + 8\u00e2\u0080\u00a2(sqrt(-1)^2) 21 + 28i + 6i + 8\u00e2\u0080\u00a2(-1) 21 + 28i + 6i - 8 21 + 34i - 8 13 + 34i"
    },
    {
        "video_name": "A_ESfuN1Pkg",
        "Q": "Why in the transcript, there is a 99:59:59?",
        "A": "That s an interesting question. I don t see such a number in the transcript, but perhaps it was once there, and has now been fixed?"
    },
    {
        "video_name": "A_ESfuN1Pkg",
        "Q": "At 3:01 Sal says that we've all been dealing with four throughout our mathematical career. This implies that everyone who's experienced any type of math (ie. counting) knows the number four. My cousin who is three can count to 3 but not 4. Why does Sal say this? My little cousin is crying. Why Sal? Why?",
        "A": "I m sure Sal would talk differently if he knew he was addressing a three-year-old. :-) I m sure he would not want to discourage your cousin--or anyone, for that matter--much less upset them. Perhaps you could help your cousin understand that Sal means no offense: He s simply addressing a different audience, and he s assuming that everyone in his audience has had some experience dealing with real numbers like 4."
    },
    {
        "video_name": "MC0tq6fNRwU",
        "Q": "At 4:47, how can x be equal to zero ?",
        "A": "At this point, Sal is merely eliminating as impossible the case where x is negative, so he states that x \u00e2\u0089\u00a5 0. A minute or so later in the video he points out that x = 0 is also impossible, and eliminates that case."
    },
    {
        "video_name": "MC0tq6fNRwU",
        "Q": "at 5:06 sal said 20-2x> or equal to 0 , with that same logic 30-2x is greater or equal to zero right ?",
        "A": "No, by that same logic, 30-2x > 0. No equality is possible anymore."
    },
    {
        "video_name": "MC0tq6fNRwU",
        "Q": "At 4:27: 30 - 2x is the width, isn't it?",
        "A": "The width is 20 - 2x, the depth is 30 - 2x. But it doesn t really matter in this case, because if you turned the box 90 degrees, the depth would become 20 - 2x and the width would become 30 - 2x."
    },
    {
        "video_name": "MC0tq6fNRwU",
        "Q": "At 4:52 I don't understand when V(10)=0",
        "A": "So, we have the equation: V(x)=x*(20-2x)*(30-2x) What he s saying is that when x=10, V(10)=10*(20-2[10])*(30-2*[10]) V(10)=10*(20-20)*(30-20) V(10)=10*0*10 V(10)=0 He put 10 in for x in the volume equation. Hope this helps you out!"
    },
    {
        "video_name": "M3FuL9qKTBs",
        "Q": "I belive that the phrase (19:13) is wrong. It's possible to have N(A)\u00e2\u0089\u00a0{0} and non particular solution for the equation Ax=b. So a transformation can be one-to-one with N(A)\u00e2\u0089\u00a0{0}. Am I right?",
        "A": "Then show us how - show us an example."
    },
    {
        "video_name": "M3FuL9qKTBs",
        "Q": "isn't the definition of null space, N(A)= 0 where 0 is a 0 vector? what does he mean by @16:23 that the function is injective if and only if N(A) = 0 vector?",
        "A": "the def of null space is N(A) is the solution set of vectors x that satisfies the condition Ax=0 so for the transformation to be one-to-one the solution set N(A) can only have one member {zero vector}."
    },
    {
        "video_name": "M3FuL9qKTBs",
        "Q": "The proof at 7:00 min is the most confusing proof I've seen out of all of Sal's videos thus far. First Sal says that let xp + xn be a solution to Ax = b and then he asks is that a solution? He then assumes that x = xp + xn, which is the only way Axp = b, because Axn is 0 by definition. But this proves nothing because x= xp + xn was the original assumption (circular proof). In addition there appears to be some abuse of notation going on with vector arrows dropping and appearing.",
        "A": "No, Sal did not assume that xp + xn is a solution. He instead plugged in xp + xn into Ax, and found that he got b, proving that xp + xn is a solution."
    },
    {
        "video_name": "M3FuL9qKTBs",
        "Q": "So, At 3:33, after performing row operations, Sal has rref(A)x=b', but how is b' a valid x?!\nIs there any intuition behind it? Or I should take one's word for it?",
        "A": "That is a mistake (unless your rref(A) happens to be the identity matrix). More accurate would have been to say that from b and rref(A), you can derive another vector p which is a particular solution to your system."
    },
    {
        "video_name": "M3FuL9qKTBs",
        "Q": "At 13:35 you say x is a particular solution plus any member of the null space. xh would be the general solution to the null space, right? So, doesn't x have to equal a particular solution plus the general solution to the null space? Or am I thinking the general solution x is equal to the particular solution plus the general solution of the null space? I'm confused.",
        "A": "If A*(xp)=b and A*(xh)=0 then A*(xp+xh)=A*(xp)+A*(xh)=b+0=b. So if x = xp+xh then Ax=b."
    },
    {
        "video_name": "iqpDFKIREBo",
        "Q": "At 1:11 what is a \" liter?\"",
        "A": "a metric unit of capacity, formerly defined as the volume of 1 kilogram of water under standard conditions, now equal to 1,000 cubic centimeters (about 1.75 pints)."
    },
    {
        "video_name": "qyYSQDcSNlY",
        "Q": "In 5:32, it is said that the question \"Find the difference in rainfall in Seattle and Singapore in 2013\" is not a statistical question. However, to find the amount of rainfall, you have to measure the rainfall on each day. It is data with variability, so shouldn't it be a statistical question?",
        "A": "Yes I believe so"
    },
    {
        "video_name": "qyYSQDcSNlY",
        "Q": "At 7:21 Sal adds 2013 to the last question. Would the question be a statistical question if it DIDN'T have 2013?",
        "A": "No. Adding a year only made it more specific. If it didn t have 2013 then it would refer to the current year."
    },
    {
        "video_name": "qyYSQDcSNlY",
        "Q": "at 1:26, sal says you have to find central tendency. what does central tendency mean??",
        "A": "Measures of central tendency include mean, median, mode. How centered is your data set."
    },
    {
        "video_name": "qyYSQDcSNlY",
        "Q": "At 3:10, aren't wolves and dogs both kind of the same thing?",
        "A": "Not all dogs are wolves, but all wolves are dogs. So, wolves are a type of dog. The question should have used two things that aren t so similar, but it was just an example, so lets give Sal or whoever wrote the question a pass."
    },
    {
        "video_name": "qyYSQDcSNlY",
        "Q": "At 4:00 and 7:00 the question is asked about two specific animals and two specific persons........... but to answer the question I need two data points (not one, remember the previous video) and there maybe variation. So why don't these two questions (5, 10) qualify to be a statistical question",
        "A": "You do need two data points, but there is only one answer with no variability."
    },
    {
        "video_name": "qyYSQDcSNlY",
        "Q": "Regarding the \"difference in rainfall\" question (starts at about 5:10), wouldn't this in the real world (and even meteorology textbook versions of the real world) be a statistical question? Both Seattle and Singapore are large enough that there will be multiple locations of collection with lots of interesting outlying points--it's not as though all the rain at all possible locations will simply be added up with no averaging or rounding.",
        "A": "Sal is presumably talking about total rainfall (or total precipitation). And yes, over a large region, there will be slightly varying amounts, generally there is some final measure reported. Like City A experienced B inches of rainfall this year or There has been Z inches of snow this year in such-and-such city. That final value is likely what Sal is talking about, not any individual measurements that go into calculating the final value."
    },
    {
        "video_name": "C7ducZoLKgw",
        "Q": "At 6:55 Sal says \"Indefinite integral with upper boundary x\" but does he mean Definite Integral",
        "A": "Yes, he did mean to say definite integral . There should be a text box in the lower right corner of the video correcting Sal."
    },
    {
        "video_name": "C7ducZoLKgw",
        "Q": "at 4:10 sal says that every continous function has an anti-derivative.but there are continous functons like sin(x)/x which is undefined only at x=0 .but still it is not possible to find the anti-derivative of this function over any interval which excludes 0....why is that?",
        "A": "According to the Squeeze Theorem, I believe that the limit as x approaches to zero of sin(x)/x is actually 1. There is a proof out there for this I believe Sal may have a video about Squeeze Theorem. Dont know if that helps at all."
    },
    {
        "video_name": "C7ducZoLKgw",
        "Q": "At 3:38, Why can't you say that it's also equal to f(t)?",
        "A": "Here x=t (if you look at the graph). So f(x) and f(t) are the same thing."
    },
    {
        "video_name": "C7ducZoLKgw",
        "Q": "@1:59, why there is a dt there?\nI think i missed a entire topic, what topic it is about dt?",
        "A": "Oh think of t as being another variable like x and y, since f(t) is a function of t (or a function in terms of t, for example f(t)=2t, again like x and y) you would have to integrate it in terms of t and hence the dt. If it was f(x), then you would use dx or f(y) with dy, f(u) with du, so on and so forth, catch my drift."
    },
    {
        "video_name": "7Mo4S2wyMg4",
        "Q": "At 14:51, why does Sal refer to the \"inverse\" transpose?",
        "A": "I think inverse transpose means to undo the transpose, so that: The inverse transpose of the transpose of a is a ."
    },
    {
        "video_name": "7Mo4S2wyMg4",
        "Q": "At 1:05 Sal means a_m1 not a_mn.",
        "A": "Just keep watching. He corrects it later in the video."
    },
    {
        "video_name": "7Mo4S2wyMg4",
        "Q": "Do the number of variables (columns) in the matrix denote the space it's in? For example, the matrix at at 16:07 has four columns (variables.) Does this denote R^4?",
        "A": "The matrix itself does not live in R^4 or R^3. One way to think about it is a transformation from R^4 into R^3. It s input will be vectors in R^4 and its output is vectors in R^3."
    },
    {
        "video_name": "7Mo4S2wyMg4",
        "Q": "At 16:25, Sal rewrites the matrix A as a row vector whose components are column vectors. Earlier on, he also implies that a matrix can be seen as a column vector of row vectors. Generally speaking, can you always treat a matrix this way? Can you simply \"take out\" or \"add\" brackets in these situations?",
        "A": "Yes, generally you can treat a matrix as a bunch of vectors x1, x2, x3, etc Usually you treat a matrix as a row vector made up of column vectors [x1 x2 x3] for computational purposes: If you are adding matrices [x1 x2 x3] and [y1 y2 y3], you can rewrite it as [(x1 +y1) (x2+y2) (x3+y3)]."
    },
    {
        "video_name": "7Mo4S2wyMg4",
        "Q": "At 1:04, did Sal mean am1?",
        "A": "Yes, he made a typo"
    },
    {
        "video_name": "7Mo4S2wyMg4",
        "Q": "At 1:05 shouldn't the last value in the first column be a,m1 rather than a,mn since the value isn't the same as the final value on the leading diagonal, which is already a,mn.",
        "A": "Apologies, Sal, you have corrected yourself later in the piece at 6:16. I hadn t watched it all the way through before I posted my question."
    },
    {
        "video_name": "7Mo4S2wyMg4",
        "Q": "0:03 a matrix is a 2D array of numbers-- what does that mean?\nPlease reply ASAP.\nThank you!",
        "A": "It means the numbers are arranged (arrayed) in two dimensions - in the plane. array: to put in order, from Vulgar Latin *ar-redare, from Latin ad to (see ad-) + Frankish *r\u00c3\u00a6d- ready"
    },
    {
        "video_name": "7Mo4S2wyMg4",
        "Q": "At 14:19 shouldn't the equation read as follows: = [ a1T dot x]\n[a2T dot x] ? Otherwise you will be multiplying a 4X1 matrix times a 4X1 matrix.",
        "A": "The way Sal wrote it is correct. The dot product is defined for vectors, not matrices, so dotting an R\u00e2\u0081\u00b4 vector with another R\u00e2\u0081\u00b4 vector is perfectly valid. a\u00e2\u0082\u0081 is a 4x1 matrix or an R\u00e2\u0081\u00b4 vector, a\u00e2\u0082\u0081T is a 1x4 matrix (or a row vector)."
    },
    {
        "video_name": "rcRg_gO7-7E",
        "Q": "At 1:06 Sal says that a (sub n) = 1(1/2)^(n-1). Why is the exponent (n-1)? Why can it not be (n)?",
        "A": "The first index number of a sequence is n=1. If we define a_n as 1(1/2)^(n), then the first term of the sequence in the video would be 1(1/2)^(1)= 1/2. But the first term of the sequence in the video is given as 1. If we define the sequence as Sal did, then we get 1(1/2)^(n-1) = 1(1/2)^(1-1) = 1(1/2)^0 = 1, as required."
    },
    {
        "video_name": "rcRg_gO7-7E",
        "Q": "at 0:32 in the video what did Sal mean when he said some stuff about a sub n equals infinity?! I don't get it one single bit! X_X please help!",
        "A": "Be careful. Sal didn t say that a sub n equals infinity. He said that this is an infinite geometric sequence . That means that the sequence will have an infinite number of terms, not that any of those terms equals infinity."
    },
    {
        "video_name": "DK5Z709J2eo",
        "Q": "Is there such a word as infiniter? 3:35",
        "A": "I believe the actual term is more infinite because that is how many other english words are expressed if the -er ending is incorrect."
    },
    {
        "video_name": "DK5Z709J2eo",
        "Q": "3:47 What does she mean with an infinite series that aproaches one? How can something infinte approach a normal integer?",
        "A": "An infinite series is not necessarily infinite. An infinite series is really a sum of an infinite number of terms. Some terms may be negative, causing cancelling. Some terms may be small enough to fit within the area remaining."
    },
    {
        "video_name": "DK5Z709J2eo",
        "Q": "At 1:16, what is a Apollonian Gasket?",
        "A": "It is a lot of shapes that fill up some two-dimensional object that somebody drew, for example the circle that is filled with other circles."
    },
    {
        "video_name": "DK5Z709J2eo",
        "Q": "At 0:45... who is Mr. Tusks?",
        "A": "Mr. Tusks is an elephant in Dinosaur Comics. He is afflicted by island dwarfism and is the Vice-Mayor of Tiny Towne."
    },
    {
        "video_name": "DK5Z709J2eo",
        "Q": "What is an \"alu\" at 3:42?",
        "A": "An arbitrary length unit."
    },
    {
        "video_name": "DK5Z709J2eo",
        "Q": "What did she say at 4:30? Was it, \"I can say the next word fast as that and the next one even faster!\"?",
        "A": "I believe she said but I can still fit it in the next 5 seconds if I say the next phrase twice as fast, and then the next phrase twice as fast as that and then it gets too fast and high-pitched to understand."
    },
    {
        "video_name": "UJQkqV2zGv0",
        "Q": "9:30 Usage of '(' and ')' after and before the infinity denotes its included. However, how can we handle infinity which is just an idea. So is it better to use '[' and ']' brackets before and after the infinity?",
        "A": "No... you have the symbols reversed. The square brackets indicate the numbers are in the set. For example: [5, infinity) is the same as x >= 5. Hope this helps."
    },
    {
        "video_name": "UJQkqV2zGv0",
        "Q": "Referring to 6:24 and 8:00,how do you call these kinds of intervals?",
        "A": "they are called Half-closed or Half -open Intervals .You can call them either way thumbs up!"
    },
    {
        "video_name": "UJQkqV2zGv0",
        "Q": "At 2:22, what is the sign he draws that looks like an e and what does it mean/do?",
        "A": "The e is read as is an element of . It is a symbol used with sets. So, the first example would be read as x is an element of Real Numbers where x is greater than or equal to -3 and less than or equal to 2 . Hope this helps."
    },
    {
        "video_name": "UJQkqV2zGv0",
        "Q": "At 8:49, why isn't there a bracket next to negative infinity in the interval notation? Doesn't the set of numbers include negative and positive infinity?",
        "A": "Watch a little beyond that point. He explains it."
    },
    {
        "video_name": "UJQkqV2zGv0",
        "Q": "-3 is > or < which sign? 2:02",
        "A": "-3 is less than(<) 2, though could you clarify your question this answer might not answer your question. Sincerely, CaptainMellow"
    },
    {
        "video_name": "UJQkqV2zGv0",
        "Q": "so 1:08 he mentions that closed intervals are that you include the end points? what? So the open end points are the one without the end points not included?",
        "A": "Open intervals exclude the end points. These occur with the inequalities of > or < . Closed intervals include the end points. These occur from the inequalities of >= or <=> Hope this helps."
    },
    {
        "video_name": "UJQkqV2zGv0",
        "Q": "At 9:25 can you write {xER/1>x<1/}",
        "A": "Good ole Kim she answers every siingle question"
    },
    {
        "video_name": "UJQkqV2zGv0",
        "Q": "At 8:43, why is 'or' used instead of 'and'? I can't remember why you would use one over the other.",
        "A": "When or is used, then 1 of the 2 (or both) statements has to be true for the whole entire statement for it to be true. With and, both statements have to be true for the whole entire thing to be true, or else it s false. Remember, or needs at least 1, and needs both."
    },
    {
        "video_name": "UJQkqV2zGv0",
        "Q": "At 8:23,Sal writes the interval that mentions the infinity,minus infinity and so on.He was going through that part too quickly,and I couldn't understand it.Can anyone give a detail explanation of that interval?",
        "A": "What he is doing is creating a union of the answer. We see that every real number is a solution EXCEPT 1. So we can write the answer as (-infinity, 1) U (1, infinity) this is another way to write that interval correctly. What it is saying is that every number from -infinity to 1 (but not including 1) is a solution AND every number from 1 (not including 1) to positive infinity is a solution."
    },
    {
        "video_name": "UJQkqV2zGv0",
        "Q": "At 2:22 when they say \"x is a member of the real numbers\", are they referring to all the numbers in the interval?",
        "A": "Yes... if x is a member of the real numbers and an interval is specified, then they are referring to every possible number within that interval on the number line."
    },
    {
        "video_name": "UJQkqV2zGv0",
        "Q": "At 8:58, Sal said that x is a member of a set of real numbers such that x is a member of the series of numbers between -ve infinity and 1, 1 and +ve infinity. But why didn't he use the [ ] brackets instead of ( ) parenthesis when denoting that ? Can't x be a part/member or itself infinity if we use the [ ] brackets ?",
        "A": "Using brackets indicates that the endpoint is included,. infinity has no endpoint. That s why parentheses are used. (Note that Sal does mention this a little after 8:58 in the video.)"
    },
    {
        "video_name": "0671cRNjeKI",
        "Q": "At 2:30 how can 3 be the negative of -3? Or how can 3 be the negative inverse of -1/3? And what's the difference between reciprocal and inverse?",
        "A": "The multiplicative inverse, or reciprocal of x is 1/x. The additive inverse of x is -x."
    },
    {
        "video_name": "0671cRNjeKI",
        "Q": "where'd the x go in 1:25",
        "A": "The x was irrelevant at that point. He was just showing that for two lines to be perpendicular, the product of the coefficients by the x must be equal to -1. This can happen only if the coefficients are negative inverses of each other (in other words, m and -1/m)."
    },
    {
        "video_name": "0671cRNjeKI",
        "Q": "At 3:18, Sal says that Line \" C \" slope, which is 3, is the inverse of Line \" B \" slope, which is -1/3 - but not it's negative inverse. It got me confused - isn't a positive is the negative of a negative? by this logic - lines B and C are perpendicular.\nor did i got it wrong?",
        "A": "Ben, You are correct that Sal probably should have said, negative inverse or maybe even negative reciprocal rather than just inverse. As far as your logic goes, though, it s perfect. If two lines have slopes that are the negative reciprocal of each other (like 3 and -1/3), then they are perpendicular to each other."
    },
    {
        "video_name": "0671cRNjeKI",
        "Q": "At 2:35 in the video the negative inverse of 3 was -1/3 so is the negative inverse for 4 is -1/4?",
        "A": "yes what you do is you take your number ex. 4 which= 4/1 and flip the numerator and denominator and then change the negative or positive sign to its opposite in this case positive 1/4 to negative 1/4"
    },
    {
        "video_name": "0671cRNjeKI",
        "Q": "At 0:30 Sal tells me that perpendicular lines must intersect with a right angle.\nParallel lines never meet because they have the same slope...\nSo what exactly is it when to lines cross, but not with 90 degree angle?",
        "A": "You would call them intersecting lines."
    },
    {
        "video_name": "G2WybA4Hf7Y",
        "Q": "At 4:10, how did 2^n+1/ 2^n= 2",
        "A": "When we divide with exponents, we subtract them. For example: 2^5 / 2^4 = 2^(5-4) = 2^1 = 1. Sal did essentially the same thing: 2^(n+1) / 2^n = 2^(n+1-n) = 2^1 = 2 Hope this helps."
    },
    {
        "video_name": "G2WybA4Hf7Y",
        "Q": "At 4:10, how did 2^n+1/ 2^n= 2",
        "A": "correctly written, add a parentheses 2^(n+1)/2^n When I divide with the same base, I subtract the exponents, so 2^(n+1-n) = 2^1 = 2 because n-n is 0"
    },
    {
        "video_name": "4JqH55rLGKY",
        "Q": "1:50 and 2:12 are typos and many many many many more. If you find anymore typos for the captions, please comment them below!!",
        "A": "Thanks so much! Yeah I ve been using YouTube s"
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "At 2:32, why does the right side absolute value cancel out with the -6? Why does the left side stay?",
        "A": "Tle left side stays because 6 is the exact opposite of -6, therefore getting 0 when added and cancelling out. 8 is not the opposite of 6, and will combine to make 14 |x+7|."
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "at about 3:00, Sal divided both sides by 14. Why didn't he divide the | x+7 | as well?",
        "A": "He did... On the left side of the equation is: 14 * |x+7| If you divide that by 14, the 14 outside the absolute value cancels out and you are left with the |x+7| Another way to think of it is: I m going to swap out |x+7| and replace it with a . 14a / 14 = a. So, if a=|x+7|, we re left with just |x+7|. Hope this helps."
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "At 1:40 you say that the easiest way to move the 4 to the opposite side of the equation is to subtract 4. Is there another way?",
        "A": "No, there is no other way. That is the fastest way to get 4 to the other side."
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "At 1:07, by what property (ex. distributive) or law does Sal do this?",
        "A": "It looks like combining like terms to me. Not sure if there s a proper name for that. The absolute value is an expression which can be viewed as a coefficient in the case where it is multiplied by a number."
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "4:13 why did he rewrite 7 as 49/7",
        "A": "So that it would have a common denominator with the -1/7"
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "At 1:20 Khan adds 6|x+7| to one side to cancel out. Couldn't he divide by -6|x+7| to solve this equation? Doing this yeilds |x+7| = 8/3 which is different from the result on 3:20. Is there a video to explain why you subtract instead of divide? Thanks!",
        "A": "8|x+7|+4=-6|x+7|+6 If you divide both sides by -6|x+7| then you get (8|x+7|+4)/(-6|x+7|) = (-6|x+7|+6)/(-6|x+7|) That reduces to 2|x+7|-24|x+7| = 1+6/(-6|x+7|) If you divide by -6|x+7|, you have to divide each term by that amount."
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "At 2:15, why is it 14 Ix+7I and not 14 I2x+7I?",
        "A": "When we add, we only look at the coefficients. Take this equation for example 2(x+1)+3(x+1) If we just add the two things together, we see that 2 and 3 are the coefficients and x+1 is the variable. So that equals 5(x+1). We do not add everything in the variable in this case. Same thing with absolute value equations."
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "At 4:15, how did you get 49/7 ? How come 7 is equal to 49/7 ?",
        "A": "Because 49/7=7 and because 7 squared is 49."
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "At 1:46, Why couldn't he have subtracted -8|x+7|-4 from both sides, since that also cancels the first equation?",
        "A": "If Sal had subtracted -8|x+7|-4 from both sides, the equation would become: 0 = -14|x+7| + 2 While this can be done, it actually creates more work. When solving equations with one variable, you always want the variable to be on the opposite side from the constant. Your method places them on the same side. You would then have to add 14|x+7| to both sides to be able to solve the equation: 14|x+7| = 2 Then, you have the same equation as Sal. Hope this helps."
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "At 2:04, why did he only add the 6 and 8.\nHow come he didn't add the x +7 and the other x+7 to get 2x+14??",
        "A": "Act as if you had a variable, |x+7|=y. You add 8y and 6y to get 14y. Do not add what is in the absolute value, only what it is multiplied by."
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "in the video at 4:00 he says that you subtract the 7 from the 7. he writes it out as x = 1/7 - 49/7. what I don't understand is how did he get the 49?",
        "A": "One whole would be 7/7. But there are 7 wholes, so you have to multiply the top number by 7, thus 49/7."
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "at 2:09 shouldn't it be 2x not just x",
        "A": "The absolute value signs act as parentheses or brackets, and so the number in front multiplies the whole absolute value term, not just part of it. Since it is in brackets, the |x+7| term can be treated as a single unit, so when you add 6 of them to 8 of them, you get 14 of them."
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "At 4:09, how is 7 able to be written as 49?",
        "A": "7 can t be written as 49. Sal changed 7 into 49/7 He is just creating an equivalent fraction by multiplying by 7/7 7/1 * 7/7 = 49/7 Hope this helps."
    },
    {
        "video_name": "UvtWf4TVawE",
        "Q": "At 4:11 Why did he not write 7 but instead wrote 1/7",
        "A": "If you mean when he subtracts 7 from both sides, he rewrites it as 49/7. 49/7 is the same thing as 7 (it s just a different expression with the same value), and he s writing it that way to share a common denominator with 1/7, so it will be easier to subtract. If you watch at 4:28 , you will see he does the same thing again.."
    },
    {
        "video_name": "g6nGcnVB8BM",
        "Q": "At 0:34 why do we not do (2x) \u00c2\u00b7 (2x) = x\u00c2\u00b2 + 4x + 4 ?",
        "A": "Because: (2x)\u00c2\u00b7(2x) = 2\u00c2\u00b7x\u00c2\u00b72\u00c2\u00b7x = 4x\u00c2\u00b2 which is \u00e2\u0089\u00a0 x\u00c2\u00b2 + 4x + 4"
    },
    {
        "video_name": "g6nGcnVB8BM",
        "Q": "Why does he multiply by negative 1 at around 1:30?",
        "A": "When isolating a variable, you want to undo the operations that are done to the variable, in the reverse of the order of operations. The two operations being done to x there are squaring and multiplying by -1. To undo the multiplication by -1 you could either multiply or divide by -1 (since -1 is its own reciprocal). Then proceed to undo the squaring."
    },
    {
        "video_name": "g6nGcnVB8BM",
        "Q": "at 0:27, isn't five x squared, ten x to the fourth and for negative eight, negative 16????????",
        "A": "huh"
    },
    {
        "video_name": "g6nGcnVB8BM",
        "Q": "At 3:52 why is 2 x (2 sqrt2) = 4 sqrt 2?",
        "A": "Because from the associative law 2*(2*sqrt(2)) = (2*2)*sqrt(2) = 4 sqrt(2)"
    },
    {
        "video_name": "TljVBB7gxbE",
        "Q": "Can someone please tell me where all of the \"e\"s came from at 7:55?",
        "A": "It s the exponential form of a complex number. if z=a+bi then z=r*e^(i*phi), where r is the modulus (or size ) of the complex number z, ie r=sqrt(a^2+b^2) and phi is the argument (or angle between the real axis), ie tan(phi)=b/a"
    },
    {
        "video_name": "TljVBB7gxbE",
        "Q": "Just wondering, at 7:35, why did Sal not just factor x^24-1=0 using difference of squares?",
        "A": "If you use difference of squares and sum or difference of cubes, you will end up with: (x^4+1)(x^8-x^4+1)(x^2+1)(x^4-x^2+1)(x+1)(x^2-x+1)(x-1)(x^2+x+1) All of these factors have imaginary roots , factors like (x^8-x^4+1) are probably solved easier by Sal s method anyway"
    },
    {
        "video_name": "ZYn8MhrwHr4",
        "Q": "At 1:01, when Sal mentions to multiply both sides by -1 does it mean -1*a+b and -1*0 making b = -a? He didn't show that part.",
        "A": "No, actually he meant multiply both sides of a = -b by -1, which results in b = -a. That was a little confusing for me, too. He starts waving his mouse around a = -b at 0:54, so that cleared up the misunderstanding for me."
    },
    {
        "video_name": "eLyLbaXfJXo",
        "Q": "At 4:06 is it really safe to assume any number of salary you want? Or do you have to have a certain number(s)? \"_\"",
        "A": "Why should you, you have four clear ranks, 1.Unemployed, the left extreme 2.Superstar, the right extreme. 3.Average in the middle (for the civil engineering). 4.HIGH paying jobs, which is midway between the average and the superstar, hope this helps"
    },
    {
        "video_name": "4OEeVLo5V1o",
        "Q": "At 3:02, cos^2 + 1 + sin^2 equation is I think incorrect since cos^2 + sin^2 = 1 , and if we transfer the other 1 from the left side to the right side, we get the opposite sign which is negative which would make the equation cos^2 + sin^2 = 1 - 1 or 0. Right?",
        "A": "This expression is not an equation. In other words, it does not have two expressions separated by an equals sign. So, there is no left side or right side. That s why the 1 from the (cos^2 + sin^2) stays with the other 1 to make 2."
    },
    {
        "video_name": "4OEeVLo5V1o",
        "Q": "at 3:00 sal adds cos^2 with sin^2 to get 1 i don't understand that",
        "A": "That is a very important identity that comes directly from applying the Pythagorean theorem on the unit circle. In the video, he used the Pythagorean theorem to say x\u00c2\u00b2+y\u00c2\u00b2 = 1, but in the graph, x = cos \u00e2\u008a\u009d and y = sin \u00e2\u008a\u009d. Thus (cos \u00e2\u008a\u009d)\u00c2\u00b2+(sin \u00e2\u008a\u009d)\u00c2\u00b2 = 1 and this is often written as cos\u00c2\u00b2 \u00e2\u008a\u009d+ sin\u00c2\u00b2 \u00e2\u008a\u009d = 1."
    },
    {
        "video_name": "4OEeVLo5V1o",
        "Q": "At 3:00 Sal adds cos^2 with sin^2 to get 1, with 1 that equals 2. Ok I get that..........however, I would not have seen that......\n\nI would have looked at +1+sin^2 and thought to take the negative of the 1+sin^2. which would be cos^2-1-sin^2. Wouldnt that be cos^2-cos^2=0?",
        "A": "Scottyboy1, Like wmj259, I am not certain what you did or what you mean by \u00e2\u0080\u009ctaking the negative\u00e2\u0080\u009d. In Sal\u00e2\u0080\u0099s problem, cos^2\u00ce\u0098 +1 + sin^2\u00ce\u0098, you could replace the sin^2\u00ce\u0098 with 1-cos^2\u00ce\u0098. That would leave you with cos^2\u00ce\u0098 +1 + 1-cos^2\u00ce\u0098. The cos^2\u00ce\u0098 and the -cos^2\u00ce\u0098 would cancel out leaving 1+1 which equals 2."
    },
    {
        "video_name": "gRntusF_tVI",
        "Q": "At 3:29 when you plug \"y = 1\"into the first equation (x+y = 15) it is x+1 = 15, this simplified is 14 there fore are apples not $14? Thought that you could use either equation to find your x value. Just thinking out loud...",
        "A": "But the two equations are 5x + 4y = 10 and 5x + 5y = 11, putting in 1 for y in either equation, you get 5x + 4 = 10 and 5x + 5 = 11, 5x = 6 or 5x = 6. Either equation is fine, but I do not know where you are getting this other equation."
    },
    {
        "video_name": "gRntusF_tVI",
        "Q": "At 2:50, why did he multiply the equation by a negative?",
        "A": "He multiplied it by a negative to make one of the 5x s to be a negative, to cancel out the 5x."
    },
    {
        "video_name": "gRntusF_tVI",
        "Q": "3:30 Isn't it kind of obvious that price of oranges equals $1 ? When the price and number of apples doesn't change, and the price and number of oranges goes up by one, then oranges must be $1 an orange.",
        "A": "You need to think again. It s kind of obvious that oranges are $1. But, apples can t be $1. If they were, then 5 apples and 4 oranges would = $9, not $10."
    },
    {
        "video_name": "aIjzkiijGnA",
        "Q": "Where did you get the 0.47 in your explanation at time, 1:33",
        "A": "47% of 50, for the percent you move the decimal over. correct me if I m wrong i did not watch the whole video."
    },
    {
        "video_name": "aIjzkiijGnA",
        "Q": "How did Sal simplify \"50*(1.47)*(0.47)+50*(1.47)\" to \"50*(1.47)^2\" at 2:32?",
        "A": "Think of it as starting with the base, 50, then multiplying by 1.47 each time. 50 + 50*(0.47) , is the same as multiplying 50*(1.47). Just skipping the step of adding. I think he did it to try and break down the question. Hope that helps"
    },
    {
        "video_name": "uhxtUt_-GyM",
        "Q": "At 10:28, Sal says to take the arithmetic mean of the 2 middle numbers, why do you do that?",
        "A": "Because there are an even amount of numbers, if you picked one number there would be (for example) 6 numbers on one side and 7 on the other. Instead of having 2 numbers as the mean we average them. I hope this answered your question!"
    },
    {
        "video_name": "uhxtUt_-GyM",
        "Q": "Is it true that a set is a group of unique numbers, and therefore, by definition, does not have a mode? I think the group of numbers at 3:45 is a list, but not a set, of numbers.",
        "A": "That s a fair point. I would argue that you d do better referring to that as a multiset of numbers, since we aren t concerned about the order the way we would for a list. But statistics doesn t really pay much attention to the vocabulary of set theory, so you may as well get used to statisticians talking about a data set that has duplications."
    },
    {
        "video_name": "uhxtUt_-GyM",
        "Q": "at 5:23 he says we need to put them in order. Is this to make the problem easier to solve or just look better?",
        "A": "When you re looking for median, placing the numbers in order tends to make it easier to solve...but it is not necessary."
    },
    {
        "video_name": "Zn2K8UIT8r4",
        "Q": "@8:00 he says vector B\u00e2\u0082\u0082'... does he mean \"vector space\" ?\nIs he writing vectors with capital letters? Or are they vector spaces? Aren't vectors written in lower case bold?",
        "A": "He has B2 = {a1, a2, b2, ..., bm} . {...} is set notation, and the elements are vectors, so it s a set of the named vectors."
    },
    {
        "video_name": "Zn2K8UIT8r4",
        "Q": "Starting at 17:08, I don't understand how he resolves the contradiction. I see that there is one, but I don't know how he can draw the conclusion he does. If he can devise a subset that spans V and is smaller than the originally conceived span, why wouldn't he simply have been wrong that A is linearly independent?",
        "A": "The information we re originally given is as follows: A is a subspace created as the span of n basis vectors B is a subspace created as the span of m basis vectors m>n B spans A Since there s a contradiction once given these things, we can say that all of the above statements can t be true at the same time. Since the first two statements are just definitions, it s the bottom 2 statements that are incompatible."
    },
    {
        "video_name": "Zn2K8UIT8r4",
        "Q": "2:50\n\nI don't quite get that, that every linear combination of the set A is in V because every member of V can be represented as a linear combination of the set A.",
        "A": "Every linear combination of the set A is in V because A is a basis for V. Thus the inverse is also true that every member of V can be represented as a linear combination of the set A. In other words span(A) = V and V = span(A). Does that help at all?"
    },
    {
        "video_name": "f7cX-Ar2cEM",
        "Q": "0:04 Why does Sal assume that 3 variables/ 3 equations, means 3 planes intersecting, & not 3 lines, in 3 dimensions, intersecting?",
        "A": "Each equation has 3 variables in it therefore it represents a 2 dimensional manifold residing in 3 dimensional space aka a plane. If there were two variables in an equation, it would represent a line."
    },
    {
        "video_name": "f7cX-Ar2cEM",
        "Q": "At 3:04, he says to eliminate the y variable by multiplying by 7. The equations are:\n16x+7y=-2\n8x-y=-10\nWouldn't it be easier to eliminate the x variable by multiplying the second equation by -2?",
        "A": "Absolutely! You would be on your way to get the correct value for y if you chose your method. There are many approaches to eliminating the variable terms when solving using the elimination system."
    },
    {
        "video_name": "f7cX-Ar2cEM",
        "Q": "At 5:07, where did he get the -1 from??",
        "A": "He added -3 and +2"
    },
    {
        "video_name": "f7cX-Ar2cEM",
        "Q": "At 4:15, why are you taking the \" 8x-y= -10 \" equation to use to find the rest of the variables?",
        "A": "I think Sal is doing that because it s an equation that has only 1 unknowns. If you plug the x value into any of the original three equations you will still have 2 unknowns and it ll be hard to work out the other unknowns. However, since he knows the value of x already he can plug that into 8x-y=-10 and then algebraically manipulate it to find the value of y."
    },
    {
        "video_name": "f7cX-Ar2cEM",
        "Q": "at 5:16, Sal says to multiply both sides by negative 1. I think he meant divide by neg 1.",
        "A": "Thomas is right"
    },
    {
        "video_name": "4tdyIGIEtNU",
        "Q": "at 2:43, I m lost why instead of saying [1 1], it is changed to [ sqrt(2)/2 sqrt(2)/2)]?",
        "A": "He wants a vector with unit length, that is length 1. The vector <sqrt(2)/2, sqrt(2)/2> has length one. You can can imagine this vector on a 2d plane - it is sqrt(2)/2 long in the x-axis, and sqrt(2)/2 long in the y axis. We can then use pythagoras to find the length of the vector = sqrt((sqrt(2)/2)^2) + (sqrt(2)/2)^2) = sqrt(2/4 + 2/4) = 1."
    },
    {
        "video_name": "_60sKaoRmhU",
        "Q": "In my Engineering Math course instead of Sal's bringing in of the dt at 16:01, I immediately see:\nds = [ ( g ' ( t ) ) ^ 2 ) + ( h ' ( t ) ) ^ 2 ] dt which is equal to the formula at 16:01. Does anyone else recollect this?",
        "A": "It makes sense. Sal is deriving the formula before using it. Especially in an engineering course you would be less focused on how to derive these equations than on how to use them. Just different emphasis not better or worse. It s just like you can either remember the quadratic formula or learn how to derive it by completing the square. The first is more practical while the second can help you understand and remember it better."
    },
    {
        "video_name": "_60sKaoRmhU",
        "Q": "So obviously at 4:17, since the curve goes from a to b, then the point at which the curve ends would be given by (g(b),h(b))?",
        "A": "Exactly right!"
    },
    {
        "video_name": "_60sKaoRmhU",
        "Q": "At 13:20, Sal states that f(x,y) = f(x(t),y(t)) even though x is parameterized as g(t) and y as h(t) along the interval a<=t<=b. This is incorrect, it should be f(x,y) = f(g(t),h(t)).",
        "A": "g(t) is x, and h(t) is y. So, x = g(t) = some function of t, and y = h(t) = some function of t. He just rewrites it to make more sense. It s the same thing as using z instead of f(x, y), since z = f(x, y) = some function of f and y."
    },
    {
        "video_name": "_60sKaoRmhU",
        "Q": "At 12:17, shouldnt it be d^2x instead of dx^2?",
        "A": "It is most certainly (dx)^2. (d^2)x would imply a second derivative, and there are no second derivatives involved."
    },
    {
        "video_name": "_60sKaoRmhU",
        "Q": "Can You please tell me what is the difference between \"delta x\" and \"dx\"? (mentioned between 0:35-0:45)",
        "A": "Delta x is the change in x, with no preference as to the size of that change. So you could pick any two x-values, say x_1=3 and x_2=50. Delta x is then the difference between the two, so 47. dx however is the distance between two x-values when they get infinitely close to eachother, so if x_1 = 3 and x_2 = 3+h, then dx = h, if the limit of h is 0. I guess basically dx is the special case of delta x when delta x is infinitely small."
    },
    {
        "video_name": "_60sKaoRmhU",
        "Q": "At 2:45, couldn't Sal just have written x and y as x(t), y(t) instead of g(t), h(t)?",
        "A": "Yes. He probably should have, it would have avoided some confusion later on."
    },
    {
        "video_name": "_60sKaoRmhU",
        "Q": "What does this symbol mean \u00e2\u0088\u00ae ? Also at 3:13, how would it work to use parametric equations for the positions?",
        "A": "If one considers a line integral along a closed path, the symbol \u00e2\u0088\u00ae is sometimes used to emphasise that the path is closed. At the time you reference, he does use a parametric equation for the position."
    },
    {
        "video_name": "11dNghWC4HI",
        "Q": "at 11:20 we can also solve using synthetic division what is the procedure using synthetic division?",
        "A": "I think it s the other way around - Sal uses long division here, and he explains synthetic (just numbers) division in the videos SteveSargentJr gave the links to."
    },
    {
        "video_name": "11dNghWC4HI",
        "Q": "At 9:30, wouldn't it have been way easier to factor the quadratic polynomial by grouping? We could have factored out a [lambda]^2 from the first two terms and a -9 from the last two, giving us ([lambda]^2 - 9) * ([lambda] - 3) = 0 which easily leads us to find that [lambda] can be 3 or - 3.",
        "A": "Yes, As Sal mentions there is (long and tedious) formula to factoring but in practice it is more of an art. The method you describe is elegant and when you can you should do it. The way Sal describes might be a little more algorithmic so if you can t see a clever way to do it is method is nice to know. But by all means use your way and continue down the path to becoming an algebra ninja."
    },
    {
        "video_name": "11dNghWC4HI",
        "Q": "At 5:30, shouldn't that be -4 -4?",
        "A": "No it s the product of all the elements on the diagonal (you can go back and review the rule of Sarrus)."
    },
    {
        "video_name": "9_VCk9tWT0Y",
        "Q": "At 0:58 wouldn't 2 and 3 be separate terms?",
        "A": "yes it will be but he is using a number instead of variable so when there is a number and a variable for example 2y and the value of y is 3 so in that case you will multiply number by variable = 2x3 so that s what sal is meaning i guess"
    },
    {
        "video_name": "9_VCk9tWT0Y",
        "Q": "at 1:04 Sal said that the second term is 4 . Why is the second term just 4?",
        "A": "I didn t watch the video but the second term is probably just four because a term is separated by addition or subtraction."
    },
    {
        "video_name": "9_VCk9tWT0Y",
        "Q": "at 1:00 why is 4 a term?",
        "A": "A term is anything that is added or subtracted from a number or variable. 4 is a term because in that case, it was being added to 2*3. Hope I helped!!"
    },
    {
        "video_name": "9_VCk9tWT0Y",
        "Q": "at 1:53, the meaning of co-efficient is any non-variable that multiplies the rest of the term. At school, we learnt it as any factor or a group of factors [in a monomial] of a term is called the co-efficient of the remaining part of the monomial[a expression that contains only 2 terms]. I'm getting confused with this concept, can someone help?",
        "A": "The Khan Academy definition is correct. And a monomial has one term not two hence the prefix mono which means one. A binomial has two terms. Hope this helped :) Happy holidays!!"
    },
    {
        "video_name": "9_VCk9tWT0Y",
        "Q": "At 5:43, when Sal adds 1 to the expression, would it still be a term if it were a Zero rather than a 1?",
        "A": "as in this example Sal adds a one to x because there is no coefficient number to multiply with x. X in this term is by itself which means that it is I times itself or I X itself. Hopefully this helps with the confusion"
    },
    {
        "video_name": "9_VCk9tWT0Y",
        "Q": "At 0:58 why would 2*3 be a term? Wouldn't 2 and 3 be separate terms? Please help!",
        "A": "The definition of a term is section in a expression that is separated by subtraction or addition."
    },
    {
        "video_name": "9_VCk9tWT0Y",
        "Q": "At 2:17 when he starts talking about coefficients I had trouble understanding what he was talking about. Could you help please?",
        "A": "Co effecients are simply normal numbers when there is a factor that has both a number and a variable EXAMPLE 7x 7 would be the coefficient x would be the variable"
    },
    {
        "video_name": "9_VCk9tWT0Y",
        "Q": "(4:20 in video)\nWhy wouldn't \"x+1\" and \"y\" be seperate factors",
        "A": "A factor is a part of a product. In the expression (x+1)(y), there are two parts in the product. The parentheses SEPARATE the two variable expressions, x+1 and y, which is why the x+1 and y are SEPARATE factors. Without the parentheses, the factors would be x and y, since x is considered only one part of a product and so is y."
    },
    {
        "video_name": "9_VCk9tWT0Y",
        "Q": "At 2:07 at the bottom of the page it says he meant -7 but 7 is what it said in the problem. Is the (-) being covered up by the parenthesis?",
        "A": "By -7 I think they mean 7 is being subtracted, not the 7 actually being in the negatives. I hope this helped."
    },
    {
        "video_name": "9_VCk9tWT0Y",
        "Q": "at about 0:39, how is (2x3) a term",
        "A": "Individual terms are separated by addition and subtraction. Since 2 and 3 are separated by multiplication, they are considered a single term. Being simplifiable doesn t negate being a term, it just means you should simplify it."
    },
    {
        "video_name": "qZlBjnC3iro",
        "Q": "At 2:15 how come when we take the derivative of the vector valued function on the left side we get a vector of the respective derivatives of the variables, but when we take the derivative of the parametric equation on the right side we get a dot product of the gradient with the vector of the derivatives of the variables? I though the vector valued function and the parametric equation were just the same thing in different notation? I'm missing something somewhere!",
        "A": "Hi Jordan, as you are saying vector valued function v(t) = x(t) i + y(t) j, and parametric equations x(t), y(t) which define xy plane, both are same because they both defining position of a point on xy plane. But at right hand side, there is a new introduced function f, inputs are the xy plane points, so in the video grant first take parametric equation of x and y coordinates and write the chain rule equation, then transform it as input of into vector valued functions."
    },
    {
        "video_name": "EvvxBdNIUeQ",
        "Q": "0:40 how do you get 2rp",
        "A": "He multiplies p by 2r to both sides of the equation. That is the beauty of Algebra, where you can do thinks to that."
    },
    {
        "video_name": "EvvxBdNIUeQ",
        "Q": "At 3:20, the equation 4 times 3.5 equals 14 is written to demonstrate how 4-pi is less than 14. Where does the 3.5 come from? Thanks.",
        "A": "3.5 is more than 3.14159265357898..... {4pi=(approx)12.56637061431592}"
    },
    {
        "video_name": "EvvxBdNIUeQ",
        "Q": "Ok, I'm sure this is going to sound dumb and maybe I wasn't listening close enough, But where did Pi come from at around 1:00? I don't really understand Pi):",
        "A": "Pi is just a number, but since it s both an important number and hard to write down as a decimal, we give it the symbol \u00cf\u0080. In geometry you learn that the area of a circle is given by \u00cf\u0080r\u00c2\u00b2, where r is the radius of the circle. If you want to see where this formula comes from, check out a geometry course."
    },
    {
        "video_name": "McINBOFCGH8",
        "Q": "At 3:56, how did Sal get square root 2 times x instead of 2x?",
        "A": "Because (x) was squared and could be shown as (x) instead of (square root of x). The (2) couldn t be taken out of the square root unless he wanted a never-ending decimal."
    },
    {
        "video_name": "McINBOFCGH8",
        "Q": "I thought that c would be x in 4:15",
        "A": "Maybe you are used to seeing hypotenuse as the right side. Hypotenuse is the one that does not have a right angle on it. C in Pythagorean equation is always the hypotenuse. In case of the ratios for 45 45 90 triangle the x s will be the ones that are 90 angle and the x\u00e2\u0088\u009a2 will be hypotenuse. Sorry my answer is mangled."
    },
    {
        "video_name": "McINBOFCGH8",
        "Q": "at about 4:04, why does c equal x times the square root of 2? if it was 2x^2 before, why isn't it 2x now?",
        "A": "You take the principal square root of 2x^2 which means square root of x^2 which is x as well as square .root of 2 which is root2"
    },
    {
        "video_name": "McINBOFCGH8",
        "Q": "4:02\nwhy is 2x^2 = x times the square root of two",
        "A": "You have misread the equation: x^2 + x^2 = c^2 2\u00e2\u0080\u00a2x^2 = c^2 sqrt(2\u00e2\u0080\u00a2x^2) = sqrt(c^2) sqrt(2)\u00e2\u0080\u00a2sqrt(x^2) = sqrt(c^2) [sqrt(2)\u00e2\u0080\u00a2x = c]"
    },
    {
        "video_name": "McINBOFCGH8",
        "Q": "@ 4:10 why do we take the principle root of both sides of the equation?",
        "A": "So that the equation is equal to C."
    },
    {
        "video_name": "McINBOFCGH8",
        "Q": "at 5:03 why is it 3 times the square root of two, i understand 3 comes from the other 2 leg lengths but why square root of two, also why not simply solve for the hypotenuse ( 4.24) instead of labeling it 3 times square root of two",
        "A": "Because 4.24 is an approximate value. That may be fine in certain applied problems, but in an abstract mathematical setting, we prefer exact results."
    },
    {
        "video_name": "McINBOFCGH8",
        "Q": "In 4:16 mins why is the square root of 2x squared x and 2 in the square root....... The exponent of x should cancel the whole square root right?",
        "A": "no, because the x is the only number that is squared. The 2 is multiplying the squared x. So the x is canceled out and 2 is left in the radical."
    },
    {
        "video_name": "McINBOFCGH8",
        "Q": "At 5:00, why is it 3 square root 2? Not 27?",
        "A": "Because c = the square root of ( a^2 + b^2) since a and b are 3 c = sqrt(3^2+3^2) = sqrt(9+9) = sqrt(18) 18 is not a square number, but it is equal to 9x2 and 9 is a square number (with root 3), so sqrt(18) is the same as 3*sqrt(2)"
    },
    {
        "video_name": "McINBOFCGH8",
        "Q": "In the start of the video at 1:05 why did Sal write the shortest side as root 3, but not root 3 by 2, as 1 * root 3/2 = root 3/2.\nConfused, pls help",
        "A": "Sal was simply writing out the ratios here, not the actual numbers. He wrote this: 1 : \u00e2\u0088\u009a3 : 2 What this means is that the second side (the one across from the 60 degree angle) is \u00e2\u0088\u009a3 times the length of the first side. And the third side (the hypotenuse) is 2 times the length of the first side. So here, Sal was not actually writing out the side lengths; he was just writing out the ratios to remind you what they were. I hope this helps."
    },
    {
        "video_name": "McINBOFCGH8",
        "Q": "Can you find the ratio of the 45:45:90 triangle another way?",
        "A": "No. There are only two ratios: leg/hypotenuse and leg/leg."
    },
    {
        "video_name": "McINBOFCGH8",
        "Q": "0:25 I don't really get this, why did Sal write square root of 3 instead of square root of 3 divided by two?",
        "A": "Look at the details written out at 0:55 in the video 1 = the 30 degree angle (so Sal is using 1 = x/2) sqrt(3) = the 60 degree angle. Sal basically got to sqrt(3) by substituting in the value of 1 for x/2 in the original value x sqrt(3)/2 2 = the 90 degree angle (so Sal is using 2 = x) Hope this helps."
    },
    {
        "video_name": "McINBOFCGH8",
        "Q": "At about 3:56 he gets sqrt(2x^2) is equal to xsqrt(2), but how did he get that? I don't understand.",
        "A": "sqrt(2\u00e2\u0080\u00a2x^2) = sqrt(2)\u00e2\u0080\u00a2sqrt(x^2) = sqrt(2)\u00e2\u0080\u00a2x"
    },
    {
        "video_name": "9wOalujeZf4",
        "Q": "At 10:53, the equation is y=0x+3.75. Sal says that as the x changes a (horizontal) line will be created. Regardless to what number x changes to, both x and y will be (0,0) so how can a line be created? As long as the y-intercept is 3.75, y=0x+3.75 will always be a point above (0,0) won't it?",
        "A": "No, it won t be (0, 0) always because we are changing the value of x. The value of y will always be the same (3.75), therefore creating a horizontal line."
    },
    {
        "video_name": "9wOalujeZf4",
        "Q": "in 1:09, why do they replace the x by zero?",
        "A": "It s a first guess point"
    },
    {
        "video_name": "9wOalujeZf4",
        "Q": "Between 4:15-4:30, How did Sal sir find out that Y-int is going to be 4/3 ? I don't get it.\n\nPlease Help!",
        "A": "The line intercepted the Y axis at 1 and 1/3, converting this from a mixed number to a fraction gives you 4/3."
    },
    {
        "video_name": "9wOalujeZf4",
        "Q": "Why -1/5 and 1/-5 are equivalent numbers? 8:48",
        "A": "If we have two rational numbers p\u00e2\u0082\u0081/q\u00e2\u0082\u0081 and p\u00e2\u0082\u0082/q\u00e2\u0082\u0082 then they are equivalent if p\u00e2\u0082\u0081\u00c2\u00b7q\u00e2\u0082\u0082 = p\u00e2\u0082\u0082\u00c2\u00b7q\u00e2\u0082\u0081 In this case we have p\u00e2\u0082\u0081 = -1, q\u00e2\u0082\u0081 = 5, p\u00e2\u0082\u0082 = 1, and q\u00e2\u0082\u0082 = -5. So p\u00e2\u0082\u0081\u00c2\u00b7q\u00e2\u0082\u0082 = (-1)\u00c2\u00b7(-5) = 5 and p\u00e2\u0082\u0082\u00c2\u00b7q\u00e2\u0082\u0081 = 1\u00c2\u00b75 = 5 Hence they are equivalent."
    },
    {
        "video_name": "9wOalujeZf4",
        "Q": "Ok,so at 0:15 Sal was putting up y=mx+b. What if the equation is something like 4(3+3/2y)-6y=12? How would you graph that?",
        "A": "If you were to solve that, you would get 12=12. All values of y work in this case, and no graph is necessary."
    },
    {
        "video_name": "9wOalujeZf4",
        "Q": "At 3:26 he said the slope was 3/-2. Where did he get the -2? Wouldn't it be 3/2?",
        "A": "Since your going down the graph You go negative so its -2 Please like this"
    },
    {
        "video_name": "9wOalujeZf4",
        "Q": "why didn't Sal go over by two and up by one 6:07",
        "A": "He could have... but it doesn t matter. The slope is constant. Sal has slope = 2/4 = 1/2 It s an equivalent fraction to 1/2 Hope that helps."
    },
    {
        "video_name": "9wOalujeZf4",
        "Q": "Just after 10:15, he says that, since there's no X, we can say that the slope is 0x. That makes sense. But then he says that that means the slope is 0/1, so we get a perfectly flat line. Couldn't 1/0 also be 0? In other words, couldn't the slope be 1/0, in which case we would get a line that goes straight up and down?\n\nHope that makes sense.",
        "A": "No. Those are two different things. Since you can t divide anything by zero, 0/1 = 0 and 1/0 = undefined."
    },
    {
        "video_name": "9wOalujeZf4",
        "Q": "I'm certainly over looking something here but in the equation (y = -x), why does Sal say at 9:27 that when X is equal to 0, y is also equal to 0?",
        "A": "If you replace x with 0 you will end up with y=0 y = - x / put 0 for x y = 0"
    },
    {
        "video_name": "qs0hRHfnzTQ",
        "Q": "When Sal introduces the density function at 1:15, he separates it from the scalar functions M and N. In the general theory of divergence, could density be considered implicit in M and N? That is to say, did he just separate density for clarification of the concept?",
        "A": "What you said is correct. The product of density times velocity in this video is just one example of how the divergence theorem can be applied in physics."
    },
    {
        "video_name": "qs0hRHfnzTQ",
        "Q": "At 7:32 and onwards, isn't Sal wrong in differentiating the i component with respect to x and not with respect to y?",
        "A": "Nope, refer to the video before this where he explains the 2D version of the divergence theorem. :)"
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "At 1:16, where does Infinity come from? I understand the n, but not the infinity.",
        "A": "Infinite means without end . If a sequence is infinite, it never ends. Like {1, 2, 3,....}"
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "At 0:37 , what is meant by each successive term is a fixed amount larger than the previous one?",
        "A": "It means that to get from any one term to the very next term, the same amount is added (or subtracted) each time."
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "When Sal rights {A sub N} \\/n=1^(infinity) what is being the significance of writing n=1 such as here: 3:45\nWill n always equal 1?",
        "A": "We use n, but it can be any letter. It s convention I guess, in the same way we use x s for unknowns. Here n represents the term where you are starting. Its significant is that we almost always start to count from the first term (in the context of sequence). Hence n = 1. It can t be a negative number, because you can t have negative terms (like a -2nd term). It can only be positive integers."
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "At 4:56 if it was multiplied and the number you were multiplying was the same for each term would that be arithmetic or not?",
        "A": "That would be a geometric sequence. They re discussed later on."
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "Is there a way to define the sequence at 5:31 explicitly?",
        "A": "Yes it can: an = (n(n+1))/2 Note that this is the same formula for the sum of positive integers from 1 to n."
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "At 1:57, Sal establishes that: a sub n = -5 + 2(n - 1). I paused the video and tried the problem before Sal did, and got: a sub n = 2n - 7. Would mine also be right?",
        "A": "Yes, yours is just a simplified version of Sal s"
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "I couldn't understand (in 2:30 ) why n\u00e2\u0089\u00a52 ? Why couldn't be n\u00e2\u0089\u00a5 1??",
        "A": "Because Sal already defined the term when n is equal to 1. We see that a1 is -5. So the recursive formula only works for when n is greater than or equal to 2."
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "Why is Sal saying \"for n has to be greater than or equal to 2\" at 5:20",
        "A": "Because in order to create a arithmetic sequence, you generally need to know the first term and the common difference. So n=1 should be defined, then you can calculate a n values for n\u00e2\u0089\u00a52."
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "1:56 where does (n-1) come from?",
        "A": "n-1 is the number of times 2 is added, to get the nth term."
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "At 4:47 when Sal is writing the (n-1) part of the equation shouldn't him have written (n-K) since he was writing the equation with the constant K?",
        "A": "k is the first term, = a(1) . d is the common difference. So, a(n) is the first term plus (n-1) times the common difference. a(n) = a(1) + (n-1)d."
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "At 5:19, why are the words \u00c2\u00a8for n is equal to or greater than 2\u00c2\u00a8 continuously written after the recursive formula?",
        "A": "In order to do a recursive formula, you have to start somewhere, and in this case you are given the first term and have to find all the nth consecutive terms after that (2nd, 3rd, ...). So n has to be greater than or equal to 2 because 1 is already defined."
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "At 2:30, why does n have to be greater or equal to 2?",
        "A": "Oh... It is because if n=1 then n-1 would be zero and n sub zero does not exist. Right?"
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "At 2:30 he mentions that for the first sequence when written recursively the it would be for N greater than or equal to 2 why is this ? Confused by how he got to that conclusion.",
        "A": "The first term can be anything. The common difference or factor only applies to the terms after term #1."
    },
    {
        "video_name": "_cooC3yG_p0",
        "Q": "At 2:01, when defining the sequence explicitly, instead of a(n) = -5+2(n-1), why can't we just write a(n) = -7+2n? Wouldn't it simplify the expression?",
        "A": "Yeah. No problem. -5+2(n-1)=-7+2n. But just to mention the first term of the sequence and to show that the later terms can be obtained from the first term, we define a(n) as -5+2(n-1)."
    },
    {
        "video_name": "r3SEkdtpobo",
        "Q": "At around 1:25, Sal divided \"a\" by everything... but i thought it was illegal to divide by \"a\" because it might be 0!! (And you can't divide by zero.) So why can Sal do this?",
        "A": "If a=0 then it s not a quadratic equation any more, a has to be more than or equal to 1 in order for it to be a quadratic equation, because the x^2 would dissapear if a=0, so in that way, you can always divide by a and there won t be any problems."
    },
    {
        "video_name": "r3SEkdtpobo",
        "Q": "1:15 why do we wanna divide the equation by a?",
        "A": "Dividing by a gives a coefficient of 1 for the x^2 term. This makes it much easier to complete the square ."
    },
    {
        "video_name": "r3SEkdtpobo",
        "Q": "At 4:38, why does (b/2a)^2 lead to b^2/4a^2. Shouldn't the exponent be gone since it was used to produce the result of b^2/4a^2?",
        "A": "What is being squared is b/2a. In other words: b/2a x b/2a which is b^2 / 4a^2"
    },
    {
        "video_name": "r3SEkdtpobo",
        "Q": "how did sal get x^2 + (b/a)x + (b/2a)^2 = x + (b/2a)^2 at around 2:50",
        "A": "never mind I got it, it is the square root of the last term"
    },
    {
        "video_name": "r3SEkdtpobo",
        "Q": "1:16 ive always done my maths by equating\n(ax^2 + bx + c = 0)/a => x^2 + bx + c = 0/a\nnow my whole foundation of maths is shaky. what have i missed??",
        "A": "When you divide an algebraic expression by something, you need to divide each term of the expression, not just the one with the coefficient you want to get rid of. So (ax^2 + bx + c)/a is the same as (ax^2)/a + (bx)/a + c/a."
    },
    {
        "video_name": "r3SEkdtpobo",
        "Q": "At 6:35 sal did the square root of the both sides, but why did he not do the numerator b2-4ac?",
        "A": "sqrt(b^2 - 4ac) doesn t simplify without substituting in the numbers."
    },
    {
        "video_name": "r3SEkdtpobo",
        "Q": "At 4:30 How did the (b/2a)^2 Become b^2/4a^2? I understand the numerator. but wouldn't (2a)^2 become either 2a^2 or 4a and not 4a^2? Seeing as 2^2 = 4\n\nOh! I think I just answered my own question!\n2^2 = 4\na^2 = a^2\nand if 2a^2 = 2^2 * a^2 it would equal 4a^2 Right? or am I just confused?",
        "A": "You got it right: (2a)^2 = (2a)(2a) = 2^2 a^2 = 4a^2"
    },
    {
        "video_name": "r3SEkdtpobo",
        "Q": "How is it that sal can multiply the -c/a by 4a without having to multiply everything else by 4a to keep the equation equal? this is at roughly 5:10 or so",
        "A": "-c/a is exactly the same as 4ca/4a^2 just like 1/2 is exactly the same as 4/8. If you multiply the numerator and the denominator by the same number, you will get the exact same value. I can multiply the numerator and the denominator in 1/2 by anything and get the same value."
    },
    {
        "video_name": "r3SEkdtpobo",
        "Q": "At 4:44, when Sal multiplies -c/a times 4a, doesn't he have to multiply 4a times the right side of the equation as well, in order to preserve the equation?\nThanks a lot!",
        "A": "He multiplies -c/a by 4a/4a, which is equal to 1. He s using the identity property of multiplication, not the multiplication property of equality. Hope I have helped you."
    },
    {
        "video_name": "r3SEkdtpobo",
        "Q": "at 3:59, how does b/2ax + b/2ax = 2b/2ax?",
        "A": "Consider b/2ax to be one thing, like one apple or one Chuck Norris. If you have one b/2ax and another b/2ax and you add them together, you get two of them, or 2b/2ax. Another way to look at this is that you have two fractions with the same denominator so when you add them you get (b+b)/2ax, which is 2b/2ax."
    },
    {
        "video_name": "r3SEkdtpobo",
        "Q": "At 6:08, does it matter whether the +/- is on the top or bottom of the fraction?\n\nThanks,",
        "A": "Yes: if it s in the bottom, you can t add the fractions."
    },
    {
        "video_name": "r3SEkdtpobo",
        "Q": "at 6:30 where did he get the +/- from?",
        "A": "In a formula, whenever you square root something you put a plus or minus sign next to it. Example: x= +- sqrt(16) x= 4 or -4. It can be either."
    },
    {
        "video_name": "S5_5KyCVjrU",
        "Q": "What's the spreadsheet formula to find the exact z score of a given percentage?\nIn the video, at 3:07 we that the percentage=0.7019 so the z score=0.5 (but we actually want the z score of a 0.7)",
        "A": "The exact z score for a given cumulative percentage, in Excel in Office 365, is either =NORMSINV(percentage) or =NORM.S.INV(percentage) So the exact z score for a cumulative percentage of 0.7 is either =NORMSINV(0.7) or =NORM.S.INV(0.7) Have a blessed, wonderful day!"
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "5:11 am i right in assuming that cos^2 a+sin^2 a= 1 because the radius of the unit circle is one?",
        "A": "yes! thats the Pythagorean identity"
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "0:00 what is a easy way to remember sin, cos, and tan?",
        "A": "Think SOHCAHTOA. Sin=opposite/hypotenuse, cos=adjacent/hypotenuse, and tan=opposite/adjacent. I hope that is what you are asking about."
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "Why is cos2(a) = both cos and sin squared but sin(2a) is = 2sin(a)cos(a)? 4:00 vs 10:15",
        "A": "You should not expect cos(2a) and sin(2a) is somewhat similar. This is trigonometry, not addition or subtraction in Arithmetic."
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "At 6:04 got lost a bit how is 2cos^2-1 = cos2a, can someone give me a refresher?",
        "A": "In this section, Sal starts with the formula cos(2a) = cos^2(a) - sin^2(a). (He derives this formula around 4:40.) Do you see where this comes from? Did you follow the steps from there? If not, which step was not clear to you?"
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "on 5:02, Sal starts writing formulas, but what are they exactly?",
        "A": "Sal just tried to prove cos(2a) finally equals 2cos^2 (a) - 1. That s it. And then he moved on to manipulate the identity to prove cos^2(a) = 1/2 (1+cos (2a). If it seems to be too confused to you, then I suggest you watch the video over and over, pause it when necessary, play back part by part, until you well understand the identity of cos(2a)."
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "at 7:43 Sal converted cos^2 a = 1 - sin^2 a to cos (2a)= (1 - sin^2 a) - sin^2a\nHow did you do this?\nWhere did Sal get it from?\nThank you for help",
        "A": "Earlier we have an identity written in yellow with a box around it: cos(2a) = cos^2(a) - sin^2(a) At this point he is using that identity, but replacing cos^2(a) with 1 - sin^2(a)."
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "At 4:35, why is it cos^2 a and not (cos a )^2?",
        "A": "cos^2a and (cosa)^2 mean the same thing, which is that cosa is mutlplied by cosa. However, the former notation is standard because it is easier to write down."
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "At 6:59, Sal says that cos^2a=1/2(1+cos2a) is a power reduction identity when it should be a half angle formula. A power reduction identity, to my knowledge, is cos^2a= (cos(2a)+1)/2. Is Sal wrong or am I?",
        "A": "They are the same formula written differently. Multiplying by 1/2(Sal s) is the same as dividing by 2(Your s)."
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "I didn't really know what happened at 8:15. Can someone explain this to me?",
        "A": "He just replaced cos^2 x with 1-sin^2 x, and subtracted and he got that equation"
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "Why wouldn't you just add the angles in sin(A + A) and then solve? 4:37.",
        "A": "Because if you want to solve using the Unit Circle, you d do it that way. If you want to/can use a calculator, you can do it either way, but usually your teacher won t let you use a calculator on addition and subtraction identities."
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "At 0:40 it says that Sin (a+b) = Sin a +Cos b.\n\nBut Look At This:\n\nSin (5+10) = 0.26\nSin 5 + Cos 10 = 0.09+0.98 = 1.07\n\nWhat does this mean then?",
        "A": "Sal gave the following formula for sin(a+b): sin(a+b)=sin a cos b + sin b cos a This is the correct formula. To apply it, you have to multiply the sine of the first angle times the cosine of the second, and also multiply the sine of the second angle times the cosine of the first, and then add those two products together. In your example, you re adding where you should be multiplying, and also leaving out the second part of the formula."
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "At 6:09, why is 2cos^2a - 1 equal to cos2a?",
        "A": "He began that with the first yellow line cos(2a) =... then working through the trig identities he was able to simplify to 2cos^2(a)-1 which equals the original expression: cos(2a)"
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "1:56 in what video does he talk about the cosine being even and sin being odd?",
        "A": "I don t remember him mentioning it in a video. My recommendation is to go to the unit circle exercise and just play with it for a while and watch the values change as you go around in the positive or negative direction. (Don t even bother to answer the question until you get a feel for how it works). Its like skating or throwing a ball. You just have to spend a little time doing it but eventual it just becomes instinct."
    },
    {
        "video_name": "a70-dYvDJZY",
        "Q": "At 6:05, how is 2cos^a - 1 = cos2a? Where did the -1 go? I understand everything up until this point.",
        "A": "I m not sure I understand your question, but maybe this will help. I\u00e2\u0080\u0099m just rewriting what Sal did. Starting from this identity: cos(2a) = cos^2(a) - sin^2(a) Substituting for sin^2(a): cos(2a) = cos^2(a) - ( 1 - cos^2(a) ) Then just combining terms: cos(2a) = cos^2(a) - 1 + cos^2(a) cos(2a) = 2 * cos^2(a) - 1 This last equation is the identity for cos(2a). I hope this helps."
    },
    {
        "video_name": "mIx2Oj5y9Q8",
        "Q": "Can someone explain to me how he got y^2-2y(mx+b)+(mx+b)^2 at 1:13?",
        "A": "Actually, (A-B)^2=(A^2-2AB+B^2), not -B^2"
    },
    {
        "video_name": "o6KlpIWhbcw",
        "Q": "What is a khamentashen? 1:55-2:02 ish",
        "A": "Hamentashen are pastries filled with fruit stuff. Where she colors in around 1:59 is where the fruit would be peeking out of the dough. It is kind of like a fruit pie but a different shape."
    },
    {
        "video_name": "o6KlpIWhbcw",
        "Q": "At 2:40, Vi says that the triangle doesn't exist, but won't it exist if you go the other way? And if that's true, is there no such thing as an actual Bermuda Triangle?\n\nThanks to anyone who answers.\nJack.",
        "A": "Thank you, Jack, for bringing this to my attention. I don t know the answer to your question, but I am sure the web has your answer."
    },
    {
        "video_name": "o6KlpIWhbcw",
        "Q": "In 4:16 are the kotch snowflakes real?",
        "A": "Since she can t draw a real, infinitely recursive Koch Snowflake because she is only a human, it is not a real Koch Snowflake, but it works for demonstrating the shape."
    },
    {
        "video_name": "o6KlpIWhbcw",
        "Q": "At 2:09 those trigonometry references sines, cosines, tangent.",
        "A": "The entire video is a trigonometry reference. The teacher is talking all about trigons."
    },
    {
        "video_name": "o6KlpIWhbcw",
        "Q": "3:02 why is she drawing on an apple?",
        "A": "2:56 To demonstrate that angles on a sphere always add up to more than 180\u00c2\u00b0"
    },
    {
        "video_name": "o6KlpIWhbcw",
        "Q": "at 2:02 what is a cake triangle",
        "A": "At 0:45 she says cheese slice triangles which is the same as a cake slice triangle. They re symmetric like a slice of cake but they re still pointy (acute)."
    },
    {
        "video_name": "o6KlpIWhbcw",
        "Q": "At 2:10, what was the second sign?",
        "A": "It is a Biohazard sign, which basically means an organic or synthetic material that can cause harm to those around it. Biohazard signs are typically seen in disease research centers. Diseases, parasites, basically any dangerous micro-organisms are biohazards. Edit: I think it s either a caution high voltage sign or a slippery when wet sign.. either one."
    },
    {
        "video_name": "o6KlpIWhbcw",
        "Q": "At 2:13, why did Vi say, \"But enough with this tangent\"?",
        "A": "Tangent is one of the trigonometric identities, like sine and cosine."
    },
    {
        "video_name": "o6KlpIWhbcw",
        "Q": "Is she going to eat that marker stained apple in 3:06? If she is EWWW!",
        "A": "1: It s washable and/or erasable. 2: She probably wouldn t eat the skin. That kind of marker doesn t leak (much less through apples, one would think) so she would just have to peel it off."
    },
    {
        "video_name": "o6KlpIWhbcw",
        "Q": "at 2:00, what does she say? They look more like......",
        "A": "She said: These look more like Komentashen . Whatever that is."
    },
    {
        "video_name": "o6KlpIWhbcw",
        "Q": "At 3:40 in why was it so bright?",
        "A": "Maybe the flash had tuned on"
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "1:30. A neg x a positive but lets say -1x0 its neither positive or negative its zero",
        "A": "If you multiply a negative and a positive, you DO get a negative. Your example stated -1 x 0, but then you said zero is neither positive nor negative. So you re not multiplying negative by positive anymore; you re multiplying negative by neither."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "@ 1:41 what does Sal mean by the same \"property?\"",
        "A": "They use the same mathematical property, the commutative property. The commutative property states that in addition and multiplication problems, the answer will be the same no matter which way the numbers are facing. 3 times 2 is the same as 2 times 3, even though the numbers are mixed up."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "at 1:17 i don't understand how a negative times a positive is a negative, can someone help me?",
        "A": "Let s say you have -4x6. You re basically saying you want to have 6 sets of -4, right? This means you ll have (-4)+(-4)+(-4)+(-4)+(-4)+(-4). When you add these, you end up getting a negative. Remember: Multiplication is just addition made fast."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "the answer for negative and negative 2:32 is going to be a negative",
        "A": "Nope sorry... a negative * a negative = a positive. You may be thinking of addition. If the problem was -2 + (-3) = -5, we do get a negative."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "-48 @ 4:21, correct?",
        "A": "12 * (-4) = (-) * 48 = -48 The answer to that expression is -48."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "At 0:44 Sal says that -2 * 3 = -6, or -2+-2+-2 couldn't we write this as -2+(-2)+(-2)? Thanks!",
        "A": "Yes. That is correct."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "At 0:32 he starts telling us about positives and negatives. Does my saying work; If a Bad thing happens to a bad person (-x - -x) its a good thing (answer would be a positive)?",
        "A": "Mathematically, we would get a neutral or positive. We wold probably represent it more as an addition problem in math, but the abstract concept of a negative negative will always give a positive still stands in your saying. :) It s kind of saying like : I will not not do something... Interesting Question. :) Sylvia."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "At 0:46 , he told to add negative 2 three times . How can we explain that on a number line in words of stepping forward or backward? I can't get this.Well, I come up with an idea instead of adding negative ,why can't we subtract a positive?! That is start at -2 subtract 2 from it twice we end up at -6.",
        "A": "Yes, you do it your way , it is absolutely right. Instead of adding -2 three times you can subtract 2 three times. Sal is just explained it in that way. Its your choice."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "@ 4:07 how does 0 times -1 = o and not -o?",
        "A": "0 is a neutral number. If you put zero on a number line, it would be the middle of all the numbers."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "at 1:99 he siad 3+2=6 but it is 5",
        "A": "Actually, the problem being shown is 3 TIMES -2. And 3 x -2 is -6."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "Why is that if you mulitiply a negative number by 0, it doesnt equal -0. At 4:10 it explains it but it still doesnt make any sense!",
        "A": "Zero is neither positive, nor negative. But, anything times zero is zero because if you have nothing, then more of nothing is still nothing."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "so at 2:05 he said that they were the same, so -90*6 = 6*-90?",
        "A": "yes, that s right :)"
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "on 1:29, sal said that it doesn't matter\nwhy?",
        "A": "The commutative property of multiplication states that no matter what order two numbers are multiplied together, you will get the same answer. For example: 3 \u00e2\u0080\u00a2 2 = 6 and 2 \u00e2\u0080\u00a2 3 = 6 Furthermore, if you have multiple numbers, all being multiplied by each other, you can arrange them in any way you want to. 1 \u00e2\u0080\u00a2 2 \u00e2\u0080\u00a2 3 \u00e2\u0080\u00a2 4 will give you the same answer as 4 \u00e2\u0080\u00a2 2 \u00e2\u0080\u00a2 1 \u00e2\u0080\u00a2 3. Both of these will give you 24. Sal was saying that it didn t matter what order you multiplied 3 \u00e2\u0080\u00a2 -2 because you will still get -6 as an answer."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "At 0:25 why does a positive plus a positive always equal positive?",
        "A": "A positive plus a positive always equals a positive because there is no way that it can decrease if positives are added."
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "4:28 what would happen if you had a number with the ( ex. ) sign? for an example how would you multiply -9 x (-3)?",
        "A": "the parentheses are added to avoid confusion"
    },
    {
        "video_name": "47wjId9k2Hs",
        "Q": "At 3:03 how does a negative and a negative cross out each other and become a positive number.",
        "A": "it is just a rule that states when numbers are same like negative and negative or positive and positive, it will always be positive, and when they are different like negative and positive, it will always be negative."
    },
    {
        "video_name": "YahJQvY396o",
        "Q": "at 1:12, can't we just write 7( x^2 - y^2 )?",
        "A": "No. Note that the 7s are inside the brackets, and the contents of the brackets are squared. This includes the 7s. We can write either 7^2(x^2 - y^2) or 49(x^2 - y^2)."
    },
    {
        "video_name": "YahJQvY396o",
        "Q": "At 1:11, the purple and blue color, shouldn't it be (7x^2)-(7y^2) instead of (7x)^2-(7y)^2?",
        "A": "No, it is correctly (7x)^2-(7y)^2 because it was originally 49x^2-49y^2. 49 is 7^2, so you want to square both 7 and the variable."
    },
    {
        "video_name": "YahJQvY396o",
        "Q": "at 2:17, when sal said (7x+7y)(7x-7y) is equal to 49(x-y)(x+y), how is Factoring 7 out of (7x+7y)(7x-7y) not\n7(x+y)+7(x-y)\n(7+7)(x+y)(x-y)?",
        "A": "Because the notation (a+b)(a+b) means (a+b) times (a+b) (and NOT (a+b) plus (a+b) So if we wrote the multiplication sign the steps would be (7x+7y)*(7x-7y) = 7*(x+y)7(x-y) . You can put multiplication factors in any order that you want (2*3*4 is the same as 3*4*2). So it we put in a convenient order it says 7*7*(x+y)*(x-y) which makes 49*(x+y)*(x-y) Written in common notation (where we don t add the multiplication sign when not necessary) that makes 49(x+y)(x-y)"
    },
    {
        "video_name": "xKNX8BUWR0g",
        "Q": "Thank you Sal for the very instructional video. At around 9:00 you talk about when the function isn't injective nor surjective. Is a such function called anything and what does that mean to the im(f)?\nMarc",
        "A": "a non injective/surjective function doesnt have a special name and if a function is injective doesnt say anything about im(f). a function thats not surjective means that im(f)!=co-domain"
    },
    {
        "video_name": "xKNX8BUWR0g",
        "Q": "@ 1:17 Sal writes that my function is surjective if f(x) = y. Does this mean that all my elements in my domain have to map to atleast one element of my codomain. Or is it possible that my domain has unmapped elements?",
        "A": "For every y in Y, y = f(x) for some x in X (every element of Y is mapped to by f)."
    },
    {
        "video_name": "gv9ogppphso",
        "Q": "what is \"This thing\" at 1:19 he says \" as X gets very very large this thing is going to approximate\" is \"this thing\" just a squiggly 'equals' sign?",
        "A": "The \u00e2\u0089\u0088 sign means approximately equal to . I assume you know that \u00cf\u0080 is an irrational number with infinitely many decimals, and that it is approximately equal to 3.14159265\u00e2\u0080\u00a6 etc. Then we can write \u00cf\u0080 \u00e2\u0089\u0088 3.14159265. It is not equal to that value, but the approximation works for most practical situations."
    },
    {
        "video_name": "gv9ogppphso",
        "Q": "At 2:42 , just wondering even though it's a dumb question (probably for newbies like me) but....what's an acentode?xD (And how do you even spell it? :D ) First time hearing that term lol",
        "A": "I think you mean asymptote. It means the distance between the line and the curve approach 0 as the function goes to infinity. Basically the line that the curve gets closer and closer to."
    },
    {
        "video_name": "gv9ogppphso",
        "Q": "3:35 in video. Why does the limit also approach 2/3 as x moves towards -infinity based on the idea that those exponents dominate the function? In this case they both exponents (on top and bottom) are 5, but what would happen if one was 5 and on 4? Would the result be a -2/3?\nPlease help! Thank you in advance.",
        "A": "It depends on which is which - if the numerator was 5 and the denominator was 4, then that would mean that the top would get larger much more quickly than the bottom - so the expression would increase without bound (i.e. approach infinity). - if the numerator was 4 and the denominator was 5, then that would mean that the bottom would get larger much more quickly than the top - so the expression would get closer and closer to zero."
    },
    {
        "video_name": "_-a-yNubpk8",
        "Q": "At 2:35 Sal says to increment y by -2k. Why do we just add -2k instead of going by the slope..?",
        "A": "a more specific answer"
    },
    {
        "video_name": "egNq4tSfi1I",
        "Q": "At 4:35 wouldn't 4lxl*\u00e2\u0088\u009a2 + 10\u00e2\u0088\u009a = 4lxl+11\u00e2\u0088\u009a2 (meaning wouldn't the 10\u00e2\u0088\u009a2 + \u00e2\u0088\u009a2 =11\u00e2\u0088\u009a2",
        "A": "Not quite, because 4|x| is multiplied by \u00e2\u0088\u009a2, not added. If it was 4|x| + \u00e2\u0088\u009a2 + 10\u00e2\u0088\u009a2, you would be correct. If you are trying to factor out the \u00e2\u0088\u009a2, you have to add together the ENTIRE numbers being multiplied by \u00e2\u0088\u009a2. So you would add 10 and 4|x|. Since these are not like terms, you get: (4|x| + 10) * \u00e2\u0088\u009a2"
    },
    {
        "video_name": "egNq4tSfi1I",
        "Q": "I don't understand how at 3:51 he got from 5 to 10. Why did he multiply the 5 by the 2?",
        "A": "because the square root of 4 is 2 and the original square root was being multiplied by 5 so now its the 2 times the 5 so its 10."
    },
    {
        "video_name": "egNq4tSfi1I",
        "Q": "At 2:54, how do we know that x squared needs a positive square root? Why can't it be a negative square root too?",
        "A": "For any square root, the default is to use the principle root (the positive root). If a problem expects the negative root, you will see a minus sign in front of the radical. For example: sqrt(9) = 3, which is the principle root. - sqrt(9) = -3, which is the negative root. The minus in front tells us to use the negative root."
    },
    {
        "video_name": "egNq4tSfi1I",
        "Q": "3:30 If square root of x^2 is |x|, why isn't the square root of 4 |2| ?",
        "A": "Because |x| is not 2 or -2 or |2| or |-2|"
    },
    {
        "video_name": "egNq4tSfi1I",
        "Q": "In 1:30 , why don't we just add the x squared so we can get the problem correct?",
        "A": "bc each problem is not like this therefore if we do this the problem would not be correct it would be very much the opposite"
    },
    {
        "video_name": "egNq4tSfi1I",
        "Q": "Where does he get the one from at 0:36? I'm slightly confused about that.",
        "A": "Because you know when you have lets say x, 2x, 3x, 4x ect. We just say x but we really mean 1x. Same with the square root, he has square root, but he really means he has 1 square root."
    },
    {
        "video_name": "egNq4tSfi1I",
        "Q": "At 4:33, why couldn't Sal simplify it further?\nCouldn't he have done: 14|x|\u00e2\u0088\u009a2? adding the 4 and 10 and keeping the absolute value bars around the x?",
        "A": "No. Sal can t simplify further because these are unlike terms. One term has the X and other doesn t. We can never add unlike terms. Hope this helps."
    },
    {
        "video_name": "epgwuzzDHsQ",
        "Q": "At 4:49 Sal says that the function, \"has the same first derivative\"- I guess he means the 1st derivatives of p and f, and shortly before that (3:46) he says the first derivative is zero. That indicates p and f have slopes of zero, but that does not appear to be the case when inspecting the curve of f on the graph, and why should it be the case?",
        "A": "Shortly before 3:46, Sal didn t say that the first derivative is zero, he only said that the first derivative term is zero. Because the term is f (0)x, what he meant there is that when x=0, that term f (0)x would become zero. And yes, at 4:49 by has the same first derivative , Sal meant the first derivatives of p(x) and f(x) are equal."
    },
    {
        "video_name": "epgwuzzDHsQ",
        "Q": "At 10:04 why is it (1/2) multiplied by (1/3)? According to Sal's logic from before, shouldn't it just have been (1/3)? I know there was an answer to this question later on, but I didn't understand at all from that... ><",
        "A": "We want p\u00c2\u00b4\u00c2\u00b4\u00c2\u00b4(c) to be equal to f\u00c2\u00b4\u00c2\u00b4\u00c2\u00b4(c). Let s look at that one term. p(c) = terms + f\u00c2\u00b4\u00c2\u00b4\u00c2\u00b4(c) * 1/(2*3) * x^3 Let s look at its first derivative. p\u00c2\u00b4(c) = terms + f\u00c2\u00b4\u00c2\u00b4\u00c2\u00b4(c) * 1/(2*3) * 3 * x^2 p\u00c2\u00b4(c) = terms + f\u00c2\u00b4\u00c2\u00b4\u00c2\u00b4(c) * 1/2 * x^2 Let s take the second derivative. p\u00c2\u00b4\u00c2\u00b4(c) = term + f\u00c2\u00b4\u00c2\u00b4\u00c2\u00b4(c) * 1/2 * 2 * x^1 p\u00c2\u00b4\u00c2\u00b4(c) = term + f\u00c2\u00b4\u00c2\u00b4\u00c2\u00b4(c) * x Then let s take the third derivative. p\u00c2\u00b4\u00c2\u00b4\u00c2\u00b4(c) = f\u00c2\u00b4\u00c2\u00b4\u00c2\u00b4(c) * 1 * x^0 p\u00c2\u00b4\u00c2\u00b4\u00c2\u00b4(c) = f\u00c2\u00b4\u00c2\u00b4\u00c2\u00b4(c) You have to have both 1/2 and 1/3 so that the coefficients eventually cancel each other out."
    },
    {
        "video_name": "epgwuzzDHsQ",
        "Q": "10:47 - the n must be bracketed to indicate a derivative. otherwise it will imply an exponent.",
        "A": "I dont think this is true...we are doing this in Calc 2 right now and my professor writes it without the bracket just like he does. Also, if you are raising a function to a power dont you write it as (f(x))^n outside a second set of parenthesis"
    },
    {
        "video_name": "epgwuzzDHsQ",
        "Q": "At 3:57, p'(x) should be f\"(0)x + f'(0) instead of just f'(0).\nThen p'(0) = f'(0).",
        "A": "No, p(x) = f(0) + f (0)x so, p (x) = f (0) Therefore, whatever x you chose you get f (0), including if you choose x = 0: p (0) = f (0) remember f(x) is a function, but f(0) is just a number, the value of f when x = 0."
    },
    {
        "video_name": "epgwuzzDHsQ",
        "Q": "On 0:40, if f(0),f '(0), what is the difference by f and f '?",
        "A": "f(0) is the y value with respect to x, and f (0) is the rate at which the function changes at that point."
    },
    {
        "video_name": "epgwuzzDHsQ",
        "Q": "At 10:05, why do we put 1/6?",
        "A": "because the first derivative will take out the 3 and the second derivative will take out the 2 (so 2*3=6:P )(example): p(x) = f (0) 1/(2*3) x^3 p (x) = f (0) 1/(2*3) * 3 * x^2 p (x) = f (0) 1/(2*3) * 3 * 2 * x I hope it is the answer you are looking for"
    },
    {
        "video_name": "epgwuzzDHsQ",
        "Q": "from 6:40 onwards I don't understand where the factorial out the front of each derivative comes from? what is is purpose?\nsomething to do with removing the coefficient of each derivative or something... but how dose it relate back to approximation of the polynomial?",
        "A": "think of it like this. for each nth derivative, you re antideriving it n times to make the approximation"
    },
    {
        "video_name": "epgwuzzDHsQ",
        "Q": "a little question why is that 8:15 the constant dissapears? :D I almost remember something about a dx/dy and something was cancelled",
        "A": "the derivative of a constant is 0"
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At 6:50 how does one come to know that the graph is supposed to be a parabola ?",
        "A": "A graph of a quadratic polynomial which is a polynomial of degree 2 like in this case x^2 is always a Parabola."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At 9:20, Sal says we are approaching 4. I'm confused by this answer. Why isn't he approaching 2 since that is where we are heading in terms of the function.",
        "A": "you are looking at the y axis instead of the x axis"
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At around 4:51, Sal says that as you're getting really close to 1, you were really at 1 the entire time. How is that if F(x) is undefined at 1?",
        "A": "Because it s not actually 1. It s just close enough that we can consider it to be 1. This allows us to use 1 and not have to worry about F(x) being undefined."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "9:12 -THEORETICALLY THE FUNCTION GRAPHED IS A SET OF FUNCTIONS?\nG(X)=(X-2\\X-2)(x^2)\ng(X)=heres the tricky bit asuuming that all is undefined g(X)={0}*X\u0000- besides when g(1)=2\nthen you can say that\nG(X)=g(X)\nis this mathamatically sound? (is there a way of writing a function that graphs a point by undefining evrything else but the point?)",
        "A": "Well, yes. Define g(x) = 2, and define the domain of g to be {1}. Then g(1) = 2, and undefined elsewhere."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "Can you ask what a limit is for every number in a function? For example, from function g(x) at 5:24, would lim g(3) be equal to 9? Or would the concept of a limit not apply?",
        "A": "The function g(x) evaluated at x = 3 equals 9. This lesson is in learning how to evaluate the function as the limit of x=>2."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At 1:13, Sal says that anything divided by zero is \"undefined\". What exactly does \"undefined\" mean? And wouldn't any number divided by zero equal infinity, since a number divided by nothing is infinitely large?",
        "A": "Anything divided by 0 is undefined. It is called this because anything you put in there will have some other problems with other parts of math. For example: log2 of 16 = 4 would be 4^2 = 16 But what if 16 is negative? Then it will not work, and therefore it is undefined. Undefined means that it doesn t really have a meaning, and will mess with other rules. Hope that helps!"
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "Maybe this question gets answered in a later video, but at about 10:50 he says the function has to approach the same number from below and above. What is the answer when that is not the case? For example, say f(x) is defined as = 2 when x < 2, and f(x) = 1 when x > 2. What is the limit as x approaches 2 for this function? Maybe that is 'undefined'?",
        "A": "Not quite. Insofar as I understand it, if the limit as you approach from the left doesn t match the limit as you approach from the right, then f(x) is said to have no limit. :) Do you see? By your problem saying when x < 2 , what it s really saying is as X approaches 2 from the left. Conversely, when x > 2 refers to all the numbers LARGER than 2 and thus actualy means as X approaches 2 from the right. Kinda confusing, but there you go."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "Why is the variable f in f(x) at 0:26 used? When or how did the variable f start to be used?",
        "A": "f is not a variable, it is the funciton operator, much like how the multiplication or addition signs are operators. typically, y=f(x), so you can have an equation like y=x+2 be expressed as f(x)=x+2. You can just as easily write g(x), h(x), etc."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "i recently started this topic and don't know much about it so i was a little confused where u said that the function is approaching 1 but at 1 so how s it approaching 1 and at 1 ? this was at 4:30",
        "A": "Visually you can see that as x approaches 1, f(x) is already equal to 1. Taking the limit just defines it mathematically while the graph defines it visually, together you can say that without a doubt, the limit as x approaches 1 is 1."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At 7:20, is it possible to graph a function that g(x) = {x^2 , x is not equal to 2 and 4 , x=2}? what would be the limit?",
        "A": "Yes, it is possible. Then, the limit would be 4, as x^2 at x=2 approaches 4 and 4 at x=2 is 4."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At 7:53, why is it just at (2,1)? Shouldn't there be a line y=1 that goes through the point (2,1), since part of the function is g(x)=1?",
        "A": "because g(x) = 1 is only for one point, when x = 1 and a line itself is actually a set of points, so of course you don t create a line there."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "at 3:26 i got confused what does sal mean by f1 is not defined",
        "A": "f(1) is not defined simply means that the value of the function is not defined at x=1, since 0/0 is undetermined."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At 7:06 should Sal say g of x instead of f of x?",
        "A": "Yes, seems like it should be g(x). We just say f(x) a lot, so it s a common mistake."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "In the video, at the point ish 8:07, Would that be closed filled circles on both sides? I mean for showing that it is exactly that (greater than or equal to/less than or equal to) or near that. I have just never seen my teacher draw a circle to break functions.",
        "A": "An empty circle can be used to indicate that a function is undefined at a particular point, or (as in this case) to indicate that although all nearby points are on this curve, this particular point is not (because this function specifies that g(x) = 1 for x = 2)."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At 0:50, Sal said that the function he was going to take a limit on was f(x)=(x-1)/(x-1).\nCould have Sal done the function like this: f(x)=(x-1)^0?",
        "A": "f(x)=(x-1)^0 equals one. Taking the limit of a constant is equal to that constant."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "about 8:00 minutes you have drawn a parabola, which represents what? and then a dot showing g(x) at 1 when g = 2. What about the parabola?",
        "A": "The parabola is the sketch of the function of equation y=x^2 or f(x)=x^2. There is an asymptote (a hole) in the graph when we evaluate g(x) but the limit as x approaches 2 is 4. Hope that helped."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "Starting at 04:05, you say that \"limit as x approaches 1 of f(x). This is equal to 1 (as we get closer, we can get infinitely closer, as long as we aren't at 1).\" I'm confused about that considering that it almost contradicts itself. So basically saying that x ->1 is the same thing as x=1? I thought that we weren't allowed to touch 1 since its an open circle? Maybe I'm just going in nonsensical loops.",
        "A": "He did somewhat contradict himself with that statement. I believe what he means by this is that as long as we are close enough to x=1 that we can simply state that f(x)=1."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "09:28\nSo Is it that the limit is what it WOULD HAVE been if we didn't have the 1,x=2",
        "A": "Yes that s what the limit is"
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At 1:54 what does he mean by y=f(x) axis? Wouldn't that just be the y axis??",
        "A": "It would be counted as the y-axis but we want to graph in terms of the equation and the 2 variables in the equation are x and f(x)."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "1:49 So it's basically just F(x) = 0?",
        "A": "No, F(1) = 0/0 is not the same as F(1) = 0. The zero in the denominator makes the whole term undefined, rather than zero."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At 5:00 , when sal defined g(x), why didn't he put a closing brace } on the other side of g(x)?",
        "A": "You don t put a closing brace there. The brace is for grouping the different pieces of the function."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At 6:08, why is the graph a parabola? How do we know which graph to draw?",
        "A": "you should know from pre-calc that an equation of form ax^2 + bx + c graphs to a parabola"
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "1) How do you graph functions? Is there any method to do it?\n2) What are continuous and discontinuous functions?\n3) At 09:31, why is the limit of the function g(x) equal to 4 and not 1?",
        "A": "Though I am not allowed to use a calculator for graphing functions in my school, I hope I will be able to learn to graph them at a later stage. As for now, thanks for the explanation on continuous functions :-)"
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "If f(2) is equal to 1, at 6:59\nthem how come at the end of the video the value of f(x) is approaching 4 as x is approaching 2?",
        "A": "the limit of f(x) as x approaches 2 is not affected by f(2)"
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "I'm a little confused. At 3:54 and following, why is function said to be \"approaching\" 1 from both sides around x=1? I mean, to the right and left of the f(1) the function is already 1. If we use the method that Sal uses to explain limits with the parabola in the second half, if you plugged in the numbers into the formula, it would equal 1. I'm sure this question has been asked already, so sorry in advance for replicating it (I couldn't find a similar question). Thanks in advance! :D",
        "A": "The function is f(x)=(x-1)/(x-1), if you try to evaluate f(1)=0/0, and that is not defined, that s why in the drawing Sal represents that point with an open circle. The other function that Sal uses: f(x)=1 ; x!=1, you cannot evaluate on 1, since the second part of the function says that the function is not defined when x=1, and so it would be illegal to try to evaluate on that point."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "Sal shows at 7:25 that there is a break in the graph and the point is one. Can this happen but there is no value? Just a hole.",
        "A": "Yes it can happen. When there is just the hole a at given point, say x=a, the value of the function f at a, that is f(a), is undefined, but that does not mean that the limit of f(x) as x approaches a does not exist. This is the subtle but important point you will be learning about limits."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At 4:41 it talks about that the limit is 1. I still don't get this concept!!! Help me please!!!",
        "A": "At 3:25, Sal says What if I were to ask you what is the function approaching as x equals 1? This is the limit! The limit is what x equals as it approaches a number. In this case, x equals 1 is undefined. But as x gets closer and closer to 1, the y axis is approaching 1, so the limit is 1. You can watch the next few videos in the Precalculus section to get a better understanding of limits. Then you should do the skill Limits 1. After that, you should have a really solid understanding of limits."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "at 2:32, I didn't understand how any value of x other than 1 could possibly give the value of y as 1.",
        "A": "Yeah! that totally makes sense. thanks a lot"
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At 2:51 how is it that at 1 f(x) is undefined (0)?",
        "A": "Because when x=1 the expression (x-1)/(x-1) becomes 0/0 and division by zero is undefined. Division by zero is an operation for which you cannot find an answer, so it is disallowed. You can understand why if you think about how division and multiplication are related. 12 divided by 6 is 2 because 6 times 2 is 12 Now consider: 12 divided by 0 is x would mean that 0 times x = 12 But no value would work for x because 0 times any number is 0. So division by zero doesn t work."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "If f(x) simplifies to 1/1, then why should we worry about the denominator being 0, which Mr Khan does at 0:42?",
        "A": "That is a good question P.C. we should worry about it being zero because if (x = 1) then the formula (f(x) = (x - 1 / x - 1) ) would be (1 - 1 / 1 - 1) which simplifies into (0 / 0) which is undefined. And that would not solve the equation. I hope I helped you!"
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "Wait, at 6:15 if x=2, g(x) should equal 4, why did he write 1?",
        "A": "Take a look again at the definition of g(x). According to the video g(x) = x\u00c2\u00b2 for all x, so as long as x\u00e2\u0089\u00a02 When x=2, then the alternate definition of g(x) is applied, where if x=2 then g(x)=1."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "So at 4:54 Sal mentions that we are trying to find the limit of f(x) as x -> 1. I am confused as to whether we are watching the x variable or y variable.\nThis is my understanding. We are looking at x -> 1 for the x value and then we are replacing it in f(x) so that it becomes f(1). After this we check to see what f is when x is one.\nIs this correct?\nWhat are the errors or [no pun intended] holes in my current understanding?\nThanks1",
        "A": "The limit of a function is what the function approaches as the input approaches a given number, not the function of a given number. So the limit as x ->1 of (x-1)/(x-1) is what the function approaches as x approaches 1. Since the function is 1 everywhere besides x=1, the function approaches 1. You can also solve this algebraically. For example, the limit as x ->1 of (x-1)/(x-1) can be simplified by dividing the numerator and denominator by x-1. Now you have the limit as x ->1 of 1/1 =1/1=1."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "at 5:20 is it what we call peace wise function.?",
        "A": "Could please explain briefly what it is?"
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "At about 8:49, Sal mentioned something about approaching 4 when the x value gets infinitely close to 2, but at two, the value drops straight down to 1. That would be just one point, so how would you graph that?",
        "A": "the way to graph a split function like this is to realize that you are graphing two equations...but one of them is only defined at a single point. The first equation is a line with a gap in it because it s not defined at x=2. You indicate the gap (undefined point at x =2) with an open circle, but connect the line up to that point on either side. For the second equation, when x = 2, you plot that as a separate point, and it s not connected to the first line at all"
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "at 9:29, if the graph at 2 falls down at 1, how can the limit be 4?",
        "A": "You re not concerned by the value of the function at that point, but infinitely close to it."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "Points don't have length, right? So if we graph a function like the one at 6:51, the gap in the line doesn't actually have a length. How does that work? If you take a point out of a line, does it divide the line into two rays or is the line still continuous?",
        "A": "It does have a length of 1 point. It depends on us, how we scale this one 1 point. It will have a gap like demonstrated in the video. It won t necessarily become 2 rays but it would split the graph into 2. A ray is linear line with an end point. As demonstrated in the video, the graph was a parabola, not linear. If we zoom it out so much we can t see a gap there but we know it s there. And because there is a gap, no matter how large or tiny it is, the graph isn t continuous at that point."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "I think i understand the limit idea. however in the 2nd example, which starts at ~05:20, our dear sal says that at x=2 g(x)=1. my problem with this is that x^2 at x=2 is 4 and not 1, right?? thanks for your help",
        "A": "That is how function g is defined, on two branches, the first one says that if x is not equal to 2 then g(x) = x^2, but when x is exactly 2, then the function is not x^2 anymore, it is g(x)=1 therefore g(2)=1."
    },
    {
        "video_name": "riXcZT2ICjA",
        "Q": "What does Mr Khan mean at 3:57, when he says 'as long as you're not at 1, f(x)=1'?",
        "A": "Sal means that as x approaches one, the y value will be 1. however, x cannot equal 1 exactly, for that would make an undefined y value."
    },
    {
        "video_name": "k9MEOgcc5KY",
        "Q": "Sal, at 5:54, can't I simplify 33/220 and right it as 3/20? Why would be wrong in doing so?",
        "A": "As you go higher in Calc., it is actually not recommended to simplify unless it lessens the amount of work needed in the future. AP graders will not ding you for not simplifying. However, if you do attempt to simplify and do it wrong, you ll get points off. It s best to stay in the safe side at times."
    },
    {
        "video_name": "ArcrdMkEmKo",
        "Q": "At 0:20, what is meaning of det(c)?",
        "A": "det is short for determinant"
    },
    {
        "video_name": "ArcrdMkEmKo",
        "Q": "At 2:33 I thought in the first method of finding a determinant for a 3x3, Sal took the total sum of the first set of vectors (circled in purple) and then subtracted the TOTAL sum of the green circled vectors, not individually subtracted each green circled result from the total of the first purple set, What is the correct order of operations here ?",
        "A": "They would both produce the same result. Subtraction is simply adding a negative, and addition is both commutative and associative so order and grouping don t change the answer."
    },
    {
        "video_name": "7BFx8pt2aTQ",
        "Q": "minute 7:12 , hello there, big fan of you... how come you say \"it can span V\"? isn't it have to span V form the first place? - isn't the definition of orthonormal set is a standard indipendent set of vectors in a slicly difrrent angle from the normal x\\y\\z\\... axis?\nhopefully i explained myself ok with my louzy engilsh, thanks in advance keep up the incredible work, you make me love math, simply love it.",
        "A": "he said it spans a subspace not entire space"
    },
    {
        "video_name": "7BFx8pt2aTQ",
        "Q": "Is it called \"Orthonormal bases\" or \"Orthonormal basis\"?\nIt was \"bases\" in the title, but he said and wrote (as at 11:12) \"basis\"",
        "A": "When bases is the plural of base, it is pronounced bay-sez. One base, several bay-sez. When bases is the plural of basis, it is pronounced bay-sees. One basis, several bay-sees. (ie you never have several basises)"
    },
    {
        "video_name": "jzn4rTPajT0",
        "Q": "At 6:55, how come g(x) is defined when x=3+ amd x=3- are not equal?",
        "A": "A function does not need to be continuous to be defined. When x = 3 there is a coordinate point that is defined that satisfies the situation, even though the limit does not exist."
    },
    {
        "video_name": "j3-XYLnxJDY",
        "Q": "At 1:10 Sam divides the trapezoid into a smaller rectangle. Is that what you are supposed to do? The video was just a bit unclear.",
        "A": "I guess so, cause I saw at 1:17 it was divided into a small rectangle. So to answer your question, yes."
    },
    {
        "video_name": "j3-XYLnxJDY",
        "Q": "I'm confused at 2:11, what does he mean by \"half the difference between the smaller and the larger on the right hand side\" ?",
        "A": "It comes from the formula for trapezoids: 1/2 (b1+b2) which is what he is trying to get to in the end."
    },
    {
        "video_name": "j3-XYLnxJDY",
        "Q": "at 7:47 Simone Said,\" I didn't know a trapezoid is the same as a parallelogram!\" it blows my mind",
        "A": "where does it say that"
    },
    {
        "video_name": "j3-XYLnxJDY",
        "Q": "At 0:26, how does Sal figure out the height of the trapezoid?",
        "A": "It was given in the start of the problem. Height refers to perpendicular height. This may mean that the height of a shape is not in the actual shape itself, as shown in the trapezoid, and all obtuse triangles. The height is from a point on the base to the line segment(or extension of the line segment) so that it is perpendicular."
    },
    {
        "video_name": "j3-XYLnxJDY",
        "Q": "4:13 Umm..How? I am Confused! Somebody Help Me!",
        "A": "you see when you find the base 1 and base 2 then you add it... after that you multiply it by the height after that you divide the product by 2. Then you get the answer :)"
    },
    {
        "video_name": "j3-XYLnxJDY",
        "Q": "I still didn't understand why he took the average of the total rectangle that is 6 into 3 and the smaller rectangle present in the trapezoid that is 2 into 3 at 2:36.",
        "A": "Description: Area of a trapezoid is found with the formula, A=(a+b)/2 x h. So the area is the average of the 2 bases times the height. Another way to look at it is by subtracting the empty space from the rectangle 6x3, which is 3(6-2)/2 (triangle). So 18-6=12"
    },
    {
        "video_name": "j3-XYLnxJDY",
        "Q": "I have a question: in 3:47 how did Sal Khan get 4 as the average of 6+2 over 2?",
        "A": "Yes 6+2 over 2"
    },
    {
        "video_name": "j3-XYLnxJDY",
        "Q": "What if you don't have the height of the trapezoid At 0:35",
        "A": "If you didn t have the height then you wouldn t have enough information. It s similar to a rectangle. If you know the length but not the width, you wouldn t know what to multiply the length by. It s the same with triangles and parallelograms. I hope that answered your question."
    },
    {
        "video_name": "j3-XYLnxJDY",
        "Q": "At 3:08, why do the two 3s factor into one 3? I've never learned that before.",
        "A": "It is because of the distributive property, if you work it out, 6*3+2*3=24 which is the same as 3(6+2)=24. Because you are multiplying both 6 and 2 by 3 you can put them in parentheses together like 3(6+2)."
    },
    {
        "video_name": "j3-XYLnxJDY",
        "Q": "At 3:33 , I didn't get how Sal got the area of the trapezoid. I need some help.",
        "A": "The formula of the trapezoid is base1 + base2 multiply the height then divide by 2 ."
    },
    {
        "video_name": "j3-XYLnxJDY",
        "Q": "At 4:03 he wrote it three ways. Are those all the formula?",
        "A": "NO none of those are the formulas the only 4mula is A=2(b1+b2)h"
    },
    {
        "video_name": "TdLD2Zh-nUQ",
        "Q": "At 0:23, how do we know that z=xy^2 is a surface? Other things I imagine in R3 space are three dimensional shapes of various sorts -- what would equations of those look like?",
        "A": "It can t be a three-dimensional shape because there is only one z for each (x,y) pair. As a result, all equations of the form z=f(x,y) will be surfaces. If you want a three-dimensional object in R3, you would need an inequality. A sphere of radius 1 would be something like x^2+y^2+z^2 <= 1."
    },
    {
        "video_name": "TdLD2Zh-nUQ",
        "Q": "at 6:45... does it matter if the integral is dxdy, or can it be dydx?",
        "A": "In this case it does not matter except that if you do switch form dxdy to dydx you need to remember to switch the order of the integrals as well. In other cases your bounds of integration depend on one of the variables and in those cases you need to be very careful about switching the order because it will require you rethinking your bounds of integration."
    },
    {
        "video_name": "TdLD2Zh-nUQ",
        "Q": "at 6:45 , he said we consider y as constant, isn't y changing with x so that we must get the relation between them and substitute for y ? like in 2D?.....thanks in advance :)",
        "A": "for each of those frame environments in the x plane that the integral uses, y is a constant"
    },
    {
        "video_name": "Yz2OosyMTmY",
        "Q": "There is something still missing in this proof for me. At ~4:00, Sal says \"rref(A) = an nXn Matrix where every column is a linearly independent pivot column.\" -- this is not the definition of the Identity matrix, because an rref(A) matrix does not necessarily have 1 along diagonals.\n\nWhat about a rref(A) like this:\n0 1 0 0\n1 0 0 0\n0 0 1 0\n0 0 0 1\n\nThis is not the same as the Identity matrix! What am I missing?",
        "A": "That matrix isn t reduced row echelon form. It s reduced, but it s not echelon. In rref, if you start from the left and move right, the first 1 will be in the first column, the second 1 will be in the second column, and so on. Thus, if you have an nXn matrix and every column is a pivot column, you get an identity matrix."
    },
    {
        "video_name": "ljzn4SADuZc",
        "Q": "after 3:25 | So, basically when someone gives me a certain range around the limit, do I have to find the range of the function? or the domain of the function?",
        "A": "You are given the domain (the range around the limit), so all you need to find is the range of the function, basically plug in the upper bound and the lower bound and you will have your range"
    },
    {
        "video_name": "ljzn4SADuZc",
        "Q": "at 4:41 there is no explanation as to why .25 is chosen. I understand that it is within .5 so does that mean that .4 would have worked as well, or .1? How do you decide what to chose as c- x to be?",
        "A": "The delta can be any number where L-epsilon<=f(a)<=L+epsilon and where a is every value between c-delta and c+delta, where L is the answer to the limit, epsilon is the range that you are given which in this case is .5, and c is the x-value at which the limit is taken. In this case the .25 came from an arbitrary function with arbitrary values. It could have been .2 or .001."
    },
    {
        "video_name": "ljzn4SADuZc",
        "Q": "At 4:34 was it necessary that if one end of the range was c-0.25 then the other end must be c+0.25. couldn't it be like c+0.15 or anything like that",
        "A": "Ya, but Sal chose it to be c centred, because it is algebraically easier"
    },
    {
        "video_name": "ljzn4SADuZc",
        "Q": "In 5:18 Sal says that we need to be able to do \"this\" for any range from C. What if requested range maps to the part of the function that also is not defined (For example if there is no \"x\" that would equal L + 12)?",
        "A": "Sal says that for any distance from f(c), we need to be able to find an x-value that maps to something within the given distance of f(c). We are not being given a radius around c in the domain."
    },
    {
        "video_name": "lGQw-W1PxBE",
        "Q": "What does \"as X approaches positive or negative infinity...\" means? (at 3:43)",
        "A": "as X approaches positive or negative infinity... = what happens when X becomes larger and larger or smaller and smaller, try going to the Limits video."
    },
    {
        "video_name": "lGQw-W1PxBE",
        "Q": "At 8:48, isn't x=-3?",
        "A": "LOL yea, just keep watching the video..."
    },
    {
        "video_name": "Iz6IVf8frjw",
        "Q": "At 2:52 Sal said to assume b is not zero. But, what if b is Zero ?",
        "A": "In that case the operation (0^0) is undefined."
    },
    {
        "video_name": "Iz6IVf8frjw",
        "Q": "at 5:58, why did you end the video",
        "A": "Maybe he hit a speed bump at 5:58."
    },
    {
        "video_name": "uYXhga17q1g",
        "Q": "At 2:40 you can see that the orange dot and black dot do not overlap each yet they should. I was just wondering if this was just a simple computer problem or actually ha something to do with math?",
        "A": "This problem is about the angles rather than the points. Points C and E are there only to establish a line that is parallel to line AB. The question never states that segment AB is the same length as segment CD. So when the translation is made carrying point D to point B, there is no reason to believe that the position of point C should translate onto the position of point A. Hope this is of help to you!"
    },
    {
        "video_name": "uYXhga17q1g",
        "Q": "At 2:59, Sal talks about a bisector. What is a bisector?",
        "A": "A bisector cuts something evenly in half."
    },
    {
        "video_name": "uYXhga17q1g",
        "Q": "what did Sal say between 5:01-5:03? I couldn't understand.",
        "A": "That theyta should be equal to phi, as in the angle, Hope this helped!"
    },
    {
        "video_name": "NJRyxmommqQ",
        "Q": "At 4:01, Could you also take out 3 in the 6a+3b=15 equation.\n6a+3b=15\n(6a+3b=15)/3\n2a+b=3",
        "A": "Yes you can do that! HOWEVER you made an error in your factoring 6a+3b=15 when you factor out the 3 becomes 2a + b = 5"
    },
    {
        "video_name": "4ThUMRAvUkY",
        "Q": "Can you use point-slope form for the equation at 0:35?",
        "A": "Yes you can and it be much easier than y=mx+b"
    },
    {
        "video_name": "Xy8NKUoyy98",
        "Q": "At 1:00. Why say you are going to multiply it one way and then multiply it the totally opposite way? TOTAL OPPOSITE from the way you drew the arrows.",
        "A": "It s not the opposite way of multiplying it. He is actually doing exactly what he says. I can see why you think that, but what he is doing is multiply the number (x-4) by x and then multiplying (x-4) by 7 and adding the sum, which is exactly what he stated."
    },
    {
        "video_name": "zltgXTlUVLw",
        "Q": "@ 3:03 Sal says a way to visually determine if a function is odd includes checking if it goes through the origin, but in \"Recognizing odd and even functions\" Sal creates an example of an odd function that does NOT intersect the origin @ 9:28 (only a function for 0 being undefined I guess). Can some one clarify if an odd function HAS to intersect the origin?",
        "A": "f(x) = 1/x is defined for all x except x = 0. Yet for x=a (a!=0), f(-a) = 1/-a = -1/a = -f(a). So, over its domain, it satisfies the definition of an odd function. And visually you can see that it s the same when rotated 180 degrees. However, I think checking to see whether the graph goes through the origin is a good way to see if a function is odd - it just doesn t always work."
    },
    {
        "video_name": "d1oNF88SAgg",
        "Q": "I don't understand what he did in 3:15",
        "A": "it s kk"
    },
    {
        "video_name": "d1oNF88SAgg",
        "Q": "I'm confused at the part when Sal writes at 3:00, 0.7x = 12.60\nHow does that happen and why?",
        "A": "Sal used 1.0x to represent the original price of the guavas. He then subtracted the discount price from the original price (1x - 0.3x) = 0.7x. Once you simplify it to 0.7x = 12.60 then you can solve the problem for the original price of the guavas."
    },
    {
        "video_name": "d1oNF88SAgg",
        "Q": "at 2:20. i don't get how he got .7x = 12.60??",
        "A": "wats up with letters in math dont like it but still things happen for a reason"
    },
    {
        "video_name": "d1oNF88SAgg",
        "Q": "At 2:24 what does Sal mean by \"taking 30% off of the full price, off of the full price.\"?",
        "A": "he meant to say of the full price"
    },
    {
        "video_name": "d1oNF88SAgg",
        "Q": "problem: 340 students took an english test, 238 passed. what percentage passed the test?\nIf i have this sort of problem. im not sure how to find the percentage.12:23pm",
        "A": "238 out of 340 students passed the test, so this is a fraction: 238/340. Just convert it to a decimal with a calculator, and turn that into a percent."
    },
    {
        "video_name": "d1oNF88SAgg",
        "Q": "2:52 how do yo get .7 from subtracting 1x to 0.30?",
        "A": "1.00 - 0.30 = 0.70 which is the same as .7. If you had 1 dollar and spent 30 cents, you would have 70 cents left. Same math."
    },
    {
        "video_name": "d1oNF88SAgg",
        "Q": "at 2:58 how is the answer 0.7 nd not 0.6?",
        "A": "The original price = 100% The new price is 30% lower, or 100% - 30% = 70% Change 70% into a decimal and you get 0.7 It would not be 0.6 Hope this helps."
    },
    {
        "video_name": "d1oNF88SAgg",
        "Q": "From 0:00 to 6:17 he talks about guavas on sale. What are guavas???",
        "A": "Guavas are a tropical fruit. If you re not familiar with them, you can think of apples or oranges. The math is the same."
    },
    {
        "video_name": "d1oNF88SAgg",
        "Q": "at 6:17 how does 1x-3x =7",
        "A": "Well that s not exactly what the equation says, they are saying a whole x is equal to 100% and 0.30 is the same as 30% so if you take 100%-30% you will get 70%. 70% is equal to 0.70 and 0.7 is also the same thing. I hope that helps you understand it!! :)"
    },
    {
        "video_name": "d1oNF88SAgg",
        "Q": "ALGEBRA? At 1:34, he actually says, \"So let's do a little bit of algebra here.\". This is SIXTH GRADE MATH! NOT PRE-ALGEBRA! And at 1:37 through 1:40, he says, \"Pick a suitable color the algebra. Maybe this grey color.\" WHAT! Sorry, it's just- is this normal?",
        "A": "Well, everyone knows that he probably picked a random color/colour."
    },
    {
        "video_name": "d1oNF88SAgg",
        "Q": "At 2:55 , I am not really understand why should I minus 1 x - 0.30 x. Can someone explain for me pleases ?",
        "A": "x = price of 6 guavas at full price and there is a 30% discount which means 30% cheaper than x. So 100% less than 30% = 70% since it is 30% cheaper because of the discount. That s why 1x - 0.3x."
    },
    {
        "video_name": "d1oNF88SAgg",
        "Q": "At 3:20 in the video Sal goes on to explain how we find the full price of 6 guavas by dividing $12.60 by .7, how (or rather why) exactly does that get us to our answer of 18?\nnote: i'm asking why does it work.",
        "A": "The reason it works is because he found the number that when multiplied by the original, non-sale price gives us 12.6. Division is opposite of multiplication so 0.7 \u00e2\u009c\u0096\u00ef\u00b8\u008f 18 = 12.6 12.6 \u00e2\u009e\u0097 0.7 = 18 In the video problem we did not know the original price or the number multiplied by the original price. So Sal set up a one step equation to solve for X using what he did know.: 1X \u00e2\u009e\u0096 0.3X = 12.6 Also you can look at it this way: 18X = 12.6 What times 18 is equal to 12.6? 12.6 \u00e2\u009e\u0097 18 = 0.7"
    },
    {
        "video_name": "ROpbdO-gRUo",
        "Q": "At 6:11, where Sal states \"these are just theories\"; surely he means \"these are just hypotheses\"?",
        "A": "Yes and/or no. The difference between a theory and a hypothesis is the latter is untested and then is supported or not by concluding results. A theory is a collection of hypotheses whose project has been repeated multiple times and data and results have supported and ended up being similar to the previous work."
    },
    {
        "video_name": "KyHvVJWjW6Y",
        "Q": "at 2:44 why does he cross out the x",
        "A": "Ah, because you only want x on ONE side of the equation...eliminating the x on the right side of the equation leaves you with x on the left, which is what you want!"
    },
    {
        "video_name": "KyHvVJWjW6Y",
        "Q": "in 0:27 Why is x standard why not y?",
        "A": "Ren\u00c3\u00a9 Descartes began using x to represent an unknown in La G\u00c3\u00a9om\u00c3\u00a9trie, published in 1637."
    },
    {
        "video_name": "2EwPpga_XPw",
        "Q": "the value for m = -4w+ 11 @1:49 essentially the equation for the line which you could then use to find the answer to any combination/ variable to the problem",
        "A": "yep that s correct"
    },
    {
        "video_name": "2EwPpga_XPw",
        "Q": "In 1:05, how did he get 100m/100=-400w/100+1100/100?",
        "A": "The equation he s dealing with is: 100m + 400w = 1100 Move the 400w to the other side: 100m = -400w + 1100 Divide by 100: 100m/100 = -400w/100 + 1100/100"
    },
    {
        "video_name": "2EwPpga_XPw",
        "Q": "At 4:05, Sal puts a cross on positive 2200 and -2200 to show that both values cancel each other out, but according to my teacher, work shown like tells the teacher not to grade the work since an x mark over the work means that there was a mistake made. So, my question is, is it appropriate to draw a cross over work to show the canceling of two values or just simply draw a slash over it?",
        "A": "It is simply a matter of convention. When in doubt, use the convention your teacher has established."
    },
    {
        "video_name": "p0jCw2sqZgs",
        "Q": "0:47 is he doing partial products?",
        "A": "Yes he is doing partial products."
    },
    {
        "video_name": "p0jCw2sqZgs",
        "Q": "at 1:50 and 4:22 did sal meant to say \"fourty two thousand\".?",
        "A": "Yes, he should have said forty-two and I\u00e2\u0080\u0099m guessing that\u00e2\u0080\u0099s what he meant."
    },
    {
        "video_name": "rCGHUXSd15s",
        "Q": "4:07 Why are you allowed to factor out the negative signs and make b into +b? That part is really confusing me.",
        "A": "He brings the -b inside the parentheses which has neg sign in front. So neg. b has to be pos. b to be correct."
    },
    {
        "video_name": "rCGHUXSd15s",
        "Q": "At 3:47 ,\nThere is a number line with x and -x\nBut what I don't get is that x is to the left of the number line\nBut usually is the left negative and right positive ?",
        "A": "well it depends on what kind of # line it is. Another thing to think about is that he is talking about absolute value. In absolute value, it doesn t matter which side it is on. I t will always be that whole #"
    },
    {
        "video_name": "rCGHUXSd15s",
        "Q": "im a bit confused what in 5:03 meant. that long line of letters and lines are confusing.",
        "A": "Rewatch the video and make sure you understand everything leading up to 5:03."
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "At 4:03 he talks about perpendicular bisectors. Do the two line segments have to be equal in length?",
        "A": "The line that is a perpendicular bisector will divide the other line segment into two equal parts indeed."
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "@ 2:30 when Sal indicates the new figure, he mentions it as a parallelogram? Understanding that all Rhombus are Parallelograms. Why didn't he indicate it more specific as a Rhombus? Or am I missing something and it is not a Rhombus?",
        "A": "It isn t a rhombus because all the sides have to be the same length. That shape doesn t have all the sides the same length."
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "At 2:51 isn't the 2 orange sides of the kite right angles too?",
        "A": "It is possible for that angle to be 90 degrees. And it appears to be close in this example. However, that is not a requirement for kites, and the video shows other examples of kites that do not have any 90 degree angles where the sides meet."
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "At 4:59 isn't it rhombuses, not rhombi?",
        "A": "Dictionary says: Rhombus (plural rhombuses or rhombi)"
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "At 5:14 Sal said that a square is also a rhombus so is a kite a square?",
        "A": "All squares are rhombuses. All squares are kites. All rhombuses are kites. Only some rhombuses are squares. Only some kites are rhombuses. Only some kites are squares."
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "At 3:01, isn't the second shape not a parallelogram, but a rectangle because it has right angles? I know that rectangles are a specific version of a parallelogram, but you did every other shape specifically but this one.",
        "A": "The shape is a parallelogram and a rectangle. A parallelogram is a shape with parallel sides right? A square has that, and so does a rectangle. it also must have 90 degree angles, or right angles, as well. This is because non-adjacent opposite sides are congruent to each other, and therefore must be parallel and form at right angles."
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "At 3:48 what is 'perpendicular bisector'",
        "A": "It s a line that is perpendicular to a line segment, and also cuts that line segment into two equal parts."
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "At 1:10, Sal said 'congruent'. What does that mean?",
        "A": "congruent means similar. For example, two triangles are congruent if all the sides and angles are equal. This applies to all shapes"
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "At 1:39, Sal Khan says adjacent, what does that word mean?",
        "A": "next door or next to each other also parrallel"
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "At 0:50 how can we say those two sides are congruent?",
        "A": "That s what most kites are. Two pairs of congruent sides."
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "at 2:20 I am awsome",
        "A": "Sal, he isn t doing related things."
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "At 3:13, Sal Khan says bisector, what does that word mean?",
        "A": "It means to cut in half. Bi is a prefix meaning two , and sect is a root that means to cut . So a bi-sector is a Two-pieces-cutter ."
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "At 3:05 what does bisecting means?",
        "A": "Bisects means cuts in half. Happy studying! \u00e2\u0080\u0094CT-2/002-24"
    },
    {
        "video_name": "inlMrf2d-k4",
        "Q": "At 1:40 why is it that the congruent sides are adjacent, isn't it suppose to be hypotenuse? I'm confused, someone please explain it to me.",
        "A": "adjacent means next to hypotenuse means the longest side of a triangle"
    },
    {
        "video_name": "J08-L2buigM",
        "Q": "I don't get exactly what happens at 2:42. It seems kind of chain rule, but since it's considering cos(x) as constant, wouldn't its derivative be 0 ?",
        "A": "In this case, cos(x) is a constant which is coefficient to your variable y^2. If you were to derive 3x with respect to x, you would get 3. Similarly, deriving (y^2)cos(x) with respect to y would keep your constant (cos(x)) as is, while taking the derivative of your variable (y^2 becomes 2y)."
    },
    {
        "video_name": "J08-L2buigM",
        "Q": "at 3:25, why the second derivative written like this (in the Numerator : the superscript of 2 is written after the partial derivative symbol, while in the denominator it is written after the x ).. what is the intuition of this writing ?",
        "A": "when you take a second derivative and are using Leibniz notation, think of it as the d s in the numerator getting squared and the dx s in the denominator being squared. So d/dx(dy/dx)= d*dy / dx*dx = d^2y/dx^2"
    },
    {
        "video_name": "NqbkvEK46xM",
        "Q": "On the first problem discussed at 0:18, why are there any other ways to measure heat in other units?",
        "A": "You can solve the first problem discussed at 0:18 using different units that measure heat. There are lots of these. Some of these units are Kelvin, Delisle, Gas Mark, Leiden, Newton, Planck, Rankine, Reaumur, Romer, and Wedgeood."
    },
    {
        "video_name": "NqbkvEK46xM",
        "Q": "at 0:24 sal says that he has made another video on how to derive the formula...\nmay i know what the title of the video is?",
        "A": "I believe he has said he might make a video on it to explain the concept...i don t think he made one. You can always look at other videos similar to deriving the formula."
    },
    {
        "video_name": "NqbkvEK46xM",
        "Q": "@ 2:30 wouldn't it be remainder 3",
        "A": "No. Its a remainder of 1 because 3 go into 10 3 times, thats 9 which you subtract from 10 to get a remainder of 1. Written as a fraction that 1 correspond to 1/3."
    },
    {
        "video_name": "MiyFReTAdCA",
        "Q": "at 6:07 x was added to 7/4x and it came out as 11/4x. but if x was added to 7/4x wouldn't it be 7/4 * 2x so it wouldn't it become 7/2?",
        "A": "7/4 x is 7/4 TIMES x. x is 4/4 Times x 4/4 + 7/4=11/4 Add the x in, 11/4 x. View Sal s adding fractions video for more info, as usual!"
    },
    {
        "video_name": "ldLdWj6DLTw",
        "Q": "at 4:02 i don't understand why he can just subtract?",
        "A": "That s a good question Vivian! See, he is not actually subtracting; he is putting it in terms that are easier to understand. Cosine is a periodic function with a period of 2\u00cf\u0080. This means that every 2\u00cf\u0080, the function repeats. This makes f(0) = f(2\u00cf\u0080) = f(4\u00cf\u0080) and etc. It is a ton easier to know the cosine of a more familiar value, namely those that are 0<x<2\u00cf\u0080. Sal simply rewrote 11\u00cf\u0080/2 as 3\u00cf\u0080/2 + 4\u00cf\u0080. Just like f(0) = f(2\u00cf\u0080), we can also say that f(3\u00cf\u0080/2) = f(7\u00cf\u0080/2) = f(11/2) and etc. I hope that helped!"
    },
    {
        "video_name": "UTs4uZhu5t8",
        "Q": "2:50 39.999.. repeating is the same as 40, not below it. 0.333.. repeating = 1/3, therefore 0.999.. repeating = 3/3 = 1 right? idk if i understand this correctly soo yeah plz help.",
        "A": "so u do not round up at all?"
    },
    {
        "video_name": "UTs4uZhu5t8",
        "Q": "At 7:36, he calls the problem and \"original inequality\" why can't he just say inequality?",
        "A": "He s just differentiating from one inequality to another"
    },
    {
        "video_name": "UTs4uZhu5t8",
        "Q": "Did Sal screw up at about 2:50 ?\n\nWouldn't 39.9 repeating be the same as 39 9/9 which equals 40?",
        "A": "Yes, Sal was wrong about 39.9 repeating as satisfying the [strict] inequality of x less than 40. As Vi Hart cleverly shows in one of her videos, 39.9 repeating equals 40!"
    },
    {
        "video_name": "UTs4uZhu5t8",
        "Q": "At 3:37, I thought you where supposed to flip the greater than around so the answer would have been x<=-75 if you are bringing over a negative #. It is clearly wrong, but I thought in the previous videos that was how it was taught? Is it only if you dividing both sides with a negative number? Thanks",
        "A": "not when adding or subtracting... Sal specified this at the start of the video. You only flip the sign if you are <b> multiplying or dividing by a negative number </b> substitute values for x and try solving the equation if you still dont get it."
    },
    {
        "video_name": "UTs4uZhu5t8",
        "Q": "what did sal mean that it could be a google at 4:55 ?",
        "A": "By google, Sal was most likely referring to the Googol. The Googol is a very, very large number. 10 to the 100th power to be exact. That is the number one followed by 100 zeros. That is... LARGE."
    },
    {
        "video_name": "UTs4uZhu5t8",
        "Q": "At 2:47 we are told that 39.9999999 repeating is as close as you can get to 40 because 39.999999999 repeating is less than 40\nBut isn't 39.9999999 repating equal to 40 just as 0.99999999 repeating is equal to 1",
        "A": "Yes it is! But you have to remember that 39.9999999 to 40 is rounding."
    },
    {
        "video_name": "UTs4uZhu5t8",
        "Q": "At 2:47 sal says that the closest you can get to 40 is 39.99999999 repeating\nBut Vi hart says that 3.999999999 repeating is the same as 40 please explain",
        "A": "If you round 39.9999999999 etc., you get 40; 39.99999999 is an infinite series of 9s that make it more infinitely close to forty, so you could infer that 39.9999999999999999=40"
    },
    {
        "video_name": "UTs4uZhu5t8",
        "Q": "3:37 so you change all the sign why?",
        "A": "When you deal with negative numbers, you have to turn the sign around."
    },
    {
        "video_name": "UTs4uZhu5t8",
        "Q": "is there a notable difference between an equation and inequality? i saw there was a correction at 1:21",
        "A": "Whenever you multiply or divide both sides of an inequality by a negative number than you have to switch the inequality sign for the inequality to remain equal, which you don t have to do for an equation."
    },
    {
        "video_name": "UTs4uZhu5t8",
        "Q": "I don't get where we get the positive 5 from at about 1:25.",
        "A": "By adding 5 and subtracting five on the other side it contains the same value."
    },
    {
        "video_name": "UTs4uZhu5t8",
        "Q": "In other videos, when you use a negative sign, you flip the > or < sign. In this video, in the second problem, (around 4:12) you don't. Why is there no flipping??",
        "A": "Yea i noticed that and though why he didn t change it! but then i remembered that the flipping of > or < only if its multiplication example -4x<20 x>20/-4 <-------only in similar cases you flip the inequality"
    },
    {
        "video_name": "UTs4uZhu5t8",
        "Q": "At 2:52, it says that 39.99999.........(repeating) is equal to 40, isn't it just approaching 40, extremely close to 40 but not equal?",
        "A": "No, Sal never said that. He said what you did, actually, that 39.999 (repeating forever) is as close as you can get to forty without actually having forty, since the inequality is <40."
    },
    {
        "video_name": "s9ppnjgmiyk",
        "Q": "at 4:17, i don't get how the square root of 2 times the square root of 2 is just 2. isnt it 4?",
        "A": "square root of 2 * square root of 2 = square root of 4 Now simplify the square root. You get 2, not 4. Hope this helps."
    },
    {
        "video_name": "s9ppnjgmiyk",
        "Q": "in 2:47, sal says five times two, why did he say that",
        "A": "Hi friend, because, Sal is just doing prime factorization of 200, and product of 2 * 2 * 5 * 5 * 2 is 200, therefore Sal says last two factors 5 * 2."
    },
    {
        "video_name": "s9ppnjgmiyk",
        "Q": "At 0:56 Sal says that \u00e2\u0088\u009a2 \u00e2\u0080\u00a2 100 is the same thing as \u00e2\u0088\u009a2 \u00e2\u0080\u00a2 \u00e2\u0088\u009a100 and that the simplified version is 10\u00e2\u0088\u009a2. But what if you had \u00e2\u0088\u009a2 \u00e2\u0080\u00a2 4 \u00e2\u0080\u00a2 2 should you have \u00e2\u0088\u009a2 \u00e2\u0080\u00a2 \u00e2\u0088\u009a4 \u00e2\u0080\u00a2 \u00e2\u0088\u009a2? And what should it equal?",
        "A": "Note: \u00e2\u0088\u009a2 is not a perfect square. But, you have two of them. So, multiplied together they create a perfect square \u00e2\u0088\u009a(2 \u00e2\u0080\u00a2 4 \u00e2\u0080\u00a2 2) = \u00e2\u0088\u009a(2^2 \u00e2\u0080\u00a2 4) = \u00e2\u0088\u009a(4 \u00e2\u0080\u00a2 4) = \u00e2\u0088\u009a(16) = 4 OR, you could do: \u00e2\u0088\u009a(4 \u00e2\u0080\u00a2 4) = \u00e2\u0088\u009a(4) \u00e2\u0080\u00a2 \u00e2\u0088\u009a(4) = 2\u00e2\u0080\u00a22 = 4 FYI.... you need parentheses to show what is inside your radicals. In your version: \u00e2\u0088\u009a2 \u00e2\u0080\u00a2 100, the 100 is not inside your radical. Hope this helps."
    },
    {
        "video_name": "s9ppnjgmiyk",
        "Q": "At 2:25 Why doesn't Sal Just Write it as 2^2 * 5^2 * 2 Where it would be easy to see that it would just be simplified.",
        "A": "You are completely correct! I believe he may have just been wary that by adding a square within a square root sign may have started to confuse some people and so also wanted to show each of the prime factors and how it breaks down into the final answer."
    },
    {
        "video_name": "s9ppnjgmiyk",
        "Q": "Wait, at 3:42 he said \"This is the square root of 200\" and points with the mouse at the answer to a problem that is the square root of 1/200, not 200. Can someone say why?",
        "A": "Sal is solving a 2-part problem. When he says this is the square root of 200 he is referring to the solution of just the denominator, though both parts of the problem are in the actual solution he s pointing to."
    },
    {
        "video_name": "j_kI7OcaLKI",
        "Q": "From 4:18 onwards, the area that is described is at x = 2. Isn't it supposed to be at x = -2?",
        "A": "You are right. It should have been x=-2 for the relative maximum in the curve."
    },
    {
        "video_name": "Z6vANdo1L68",
        "Q": "At 4:54 you find out the value of probability for values within twice the standard deviation. So, it includes Probability of the fuel ranging from values (mean-2sd) to (mean+2sd). Subtracting probability from the table from 1 will give us the value of P( fuel consumed<15L)+ P(fuel consumed>25L). What we want is just P(fuel consumed>25L). So, shouldn't you divide the final answer by two?",
        "A": "Actually, Sal is right. We are used to the 68 - 95 - 99.7 rule, which tells us the percentage occupied by +- sd, +- 2sd ... So that leaves area on both sides. But, What the Z table gives us, is not that!! it gives us the percentage, that is below whatever sd you are at. So when you subtract that value(0.97..) you are actually ONLY left with the tiny area only on the right hand side."
    },
    {
        "video_name": "-Dg9da1BGsM",
        "Q": "At 1:43, I don't understand how you can create the fraction 9 over 5.\n5 can fit 1.8 times in 9, were does the 9 over 5 come from?",
        "A": "I agree with you"
    },
    {
        "video_name": "PKh5B9xyzSc",
        "Q": "How come when you have prime numbers, the smallest common multiple is them times each other? (Example: 5 and 2.) At 0:49 - 1:00, Sal said that when both numbers were prime, you just needed to multiply them times each other. Why does that work?",
        "A": "That works because the 2 numbers he s using have only 2 factors each(1&5) and (1&2) and since these 2 numbers have only these factors or they are considered prime,you will find that the smallest common multiple is just multiplying those 2 numbers together."
    },
    {
        "video_name": "PKh5B9xyzSc",
        "Q": "From 0:46 to 0:58 Sal says that the LCD (Lowest Common Denominator) of 5 & 2 is the product because they are prime numbers. Does this mean that the LCD of the two denominators will always be the product of the two as long as they are prime numbers?",
        "A": "Yes... when both denominators are prime numbers, the LCD would be the product of the 2 numbers."
    },
    {
        "video_name": "F5Nb6cIRZLU",
        "Q": "I thought standard for was when the x and y are on the same side of the equation, but he said that 2y+4=6 is standard form, so now i'm a little confused. (2:10)",
        "A": "pretend there is 0x on the left."
    },
    {
        "video_name": "F5Nb6cIRZLU",
        "Q": "At 4:55 I don't understand why having the same slope means that they will never intercect. I understand how to solve systems of equations using methods shown above, but how does the slope help?",
        "A": "Thank-you. I think that I ll expeirament some with slope. I think I get it."
    },
    {
        "video_name": "xuAiQOzIkWY",
        "Q": "At 5:52, why is the expression approaching negative infinity?",
        "A": "Think of slopes as rise over run. In this case it is negative infinity because you start of with a change in x which is negative (c being bigger than x) and a change in y which is positive (being f (x) larger than f (c))..... therefore dy/dx is negative, and as x approaches c the limit of the function from the left is equal to negative infinity....."
    },
    {
        "video_name": "xuAiQOzIkWY",
        "Q": "At 3:36, why is the slope zero? isn't the left-sided limit approaching the point c?",
        "A": "Remember the slope is rise/run \u00e2\u0080\u0093 i.e. the steepness of a line (or curve). The left sided limit of the function is f(c), but the line is horizontal and therefore has a derivative (slope) of zero."
    },
    {
        "video_name": "NCF4m8BDs7w",
        "Q": "At 1:58, how can the area of the rectangle exactly match the area under the curve from a to b? Is it an approximation of the actual area?",
        "A": "How do you choose this height? Is it the average height? Meaning that the average height times the interval will always equal the integral of a to b?"
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "4:52 you have 20x-20=540 how do you get x-1=540?\nare you taking 20 from the 20x ?",
        "A": "Actually we have to isolate the x and for that reason we take 20 from 20x."
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "At 2:44, wouldn't the formula be 180 (n-2) where n is the number of sides the figure has?",
        "A": "that formula, 180(n-2) is to find the total measure of the angles in that shape."
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "4:48,isn't (20x-20)/20=x-20?",
        "A": "No. You must divide both of the terms in (20x - 20) by 20."
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "8:46 how are there exterior angles of a hexagon aren't those called vertecies",
        "A": "The points in which two lines, that make up the hexagon, meet are called vertices. The lines that make up the shape aren t specified, so one can assume that the lines go on forever. Which means that every interior angle of the shae has an exterior/vertical angle (which are reflecteded on the hexagon s vertices), even if such angles aren t clearly shown on a figure"
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "at 4:53 how did you get x-1?",
        "A": "20x - 20 = 540 Divide both sides of the equation by 20: (20x - 20)/20 = 540/20 (20x/20) - 20/20 = 540/20 x - 1 = 27"
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "At 4:41,i didn't understand why 20x minus 20 is equal to 540?",
        "A": "If you are confuse on the 540 he explain it on @2:50 . Basically it is the addition of all the angles in a pentagon (triangle - 180, square - 360, pentagon - 540, hex -720, etc) If you are confuse on the 20x-20 that is the addition of all the angles (which is given in the problem statement as 6x + 2, 2x -16, 4x-6, etc...) So if we add all the angles they should equal to 540 degrees for a pentagon."
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "At 9:00, since all shapes add up to 360 degrees, could you find how many sides are in the shape and then divide it by 360 degrees? Like in this situation, a hexagon has six sides so you could take the 360 degrees then divide it by six so it would equal 60 degrees still?",
        "A": "Actually, only quadrilaterals interior angles add up to 360\u00cb\u009a. Hexagons are 720\u00cb\u009a and each interior angle is 120\u00cb\u009a. if N = sides, It s (N-2)180\u00cb\u009a for the total degrees. Hope this helps, eslack487! =)"
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "can someone explain why and what sal is doing in 4:56- 5:04 ? Like how did sal got from 20x-20+ 540 into x-1=27 and get 28?",
        "A": "x-1=27 +1 +1 So -1+1 equals zero so they cancel each other out then all you re left with is x. After that you add 27 and 1, then x=28."
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "At 7:00, how do you know that angle 1 + 54 + 92, equals 180?",
        "A": "The sum of all the inner angles of a triangle are always 180 degrees, no matter what the triangle looks like."
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "at 4:56, why does Sal divide the number 20 with 20x. Isn't he supposed to add 20 to 540 then divide by 20x??",
        "A": "at 4:56 he used one method of solving for x. He divided all the terms on both sides of the equation by x to isolate x. Then he carried over the -1 to the right hand side and got 27 + 1, which would be 28. Another method would have been adding 20 to the right which would have given you: 20x = 560, then divide both sides by 20 to isolate the x: x=28."
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "On question 50 couldn't it be 3 triangles . Time 9:32",
        "A": "I have been trying to figure out if this is possible, but I don t think it is. Can you sketch it and show"
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "In 4:58, why doesn't Sal just add 20 to both sides, then divide by 20, getting 28? Because that is A.",
        "A": "because it is easier and less work to divide by twenty. besides, if we did it your way, it will be longer and we might get the wrong answer."
    },
    {
        "video_name": "_HJljJuVHLw",
        "Q": "Hi . At 7:12 problem 48 why we didn't say that angle 1 is 88 ? shouldn't it be equal as we said in the parallel video ?? Thank you .",
        "A": "Not quite. I believe you are recalling the opposite angles theorem, where any two opposite angles formed by the same two perpendicular lines are equal. Angle 1 in this case, is not opposite to the 88 degree angle."
    },
    {
        "video_name": "CecgFWTg9pQ",
        "Q": "At 0:02, what is the difference between a geometric progression and a geometric sequence?",
        "A": "There is no difference; they are the same thing."
    },
    {
        "video_name": "CecgFWTg9pQ",
        "Q": "at 4:12, shouldn't it be a(r)^k, since only the r is being raised to the k and not a? Or doesn't that matter?",
        "A": "Remember your order of operations - exponentiation before multiplication, thus ar^k means implicitly a(r^k). If you saw this in a polynomial 3x^2 you won t get 9x^2, right? Same deal here. Keep on asking! Keep on Studying!"
    },
    {
        "video_name": "CecgFWTg9pQ",
        "Q": "what he is saying at (1:00)",
        "A": "From about 0:55 to 1:05, he says, What I now want to focus on in this video is the sum of a geometric progression or a geometric sequence, and we would call that a geometric series. More concisely expressed, a geometric series is the sum of all the terms in a geometric sequence."
    },
    {
        "video_name": "ypxHVqE26gI",
        "Q": "At 1:30 in the video Sal says you can do this : 3(56+7) instead of 3*(56+7). can you also use this expression? (56+7)3?",
        "A": "Yes, as long as you do the parenthesis first, you can still do it either way because you are multiplying. It only works that way if you multiply or add. Not subtract or divide."
    },
    {
        "video_name": "ypxHVqE26gI",
        "Q": "at 5:03 what does quotient mean?",
        "A": "The quotient means the answer to a division problem"
    },
    {
        "video_name": "ypxHVqE26gI",
        "Q": "At 3:41 , doesn't the order of operations go like this: Parentheses, Exponents, Division, Multiplication, Addition, Subtraction?",
        "A": "Yes! In this case, Sal just removed the Parentheses to do the problem, because in that problem, you would have to do the multiplication first anyway."
    },
    {
        "video_name": "ypxHVqE26gI",
        "Q": "At 5:24, is there only one kind of division sign? Thanks. :)",
        "A": "no, there are multiple division signs. the line with two dots is one and the slash, / is another division sign also, a number over a number, or 9 -- 5 can also indicate division, unless it s a fraction"
    },
    {
        "video_name": "ypxHVqE26gI",
        "Q": "At 0:38, Sal says parentheses are brackets. So why do people call them parentheses when brackets are easier to remember? Complexity?",
        "A": "Parentheses are actually officially called round brackets ( ), and most people just call the square brackets [ ] brackets. It s really just another way to separate the two. In higher levels of math, people use parentheses ( ), brackets [ ], and braces { }."
    },
    {
        "video_name": "ypxHVqE26gI",
        "Q": "At 1:31, Sal said you can just put the 3 without the multiplication sign.Can you do this with other operations,like subtraction,division or addition, as well?",
        "A": "as long as you know what operation you are doing, you can do this with any thing math-related"
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "For the first graph around 5:30, Sal is saying that the limit as x approaches 2- and x approaches 2+ are not equal. Thus, making the point that the general x approaches 2 neutral does not exist. But since there is a point where x=2, wouldnt f(x) simply just equal 9 because that's the point where x=2?",
        "A": "It s true that if you input x=2 into the function, you d get the value 9, because that s what the graph shows us. But make sure to see that Sal is writing lim f(x) each time, not just f(x). Since he s finding a limit and not a solution to the basic function, you re going to get those two limits (approaching 2- and approaching 2+). Limits are different from solutions, as indicated by the filled-in and empty circles!"
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "At 3:35, Sal talks about how the limit of f(x) as x -> 2- and x -> 2+ are not equal, so the limit of f(x) as x -> 2 does not exist. Do the individual limits still exist then? Can the limits of f(x) as x -> 2- and x -> 2+ exist as two separate, one-sided limits?",
        "A": "Yes. One-sided limits can still exist, even if the combined limit does not exist. However, it is still important that one-sided limits are not true limits. They are of lesser status than a regular limit where we approach x from both directions."
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "Why, at 5:30, does the limit as f(x) approaches 4 not equal to -5, because that is what you get when you approach 4 from both the negative and the positive direction?",
        "A": "You re right it should be -5 I think that Sal just forgot the minus sign. Hope this helps."
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "I was thinking that @5:28, Is lim->4 f(x) = 5 or -5 ?",
        "A": "It s -5. Just a little earlier Sal slipped, called it 5, corrected himself to say -5, then slipped again and said (and wrote) 5. I guess he was just in a positive frame of mind."
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "At 5:30, why is f(x) as x reaches 0 = 5 and not -5?",
        "A": "He made a mistake"
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "In the second example at around 6:20 why did he choose 3 as the limit instead of 1 and the same for when he chose 1 for approaching from the positive direction and not 3?",
        "A": "Because in order to check whether the limit of f(x) as x approaches 8 exists, the function must approach the same value (y value if you like) coming from both left and right. Whe approaching x = 8 from the left, f(x) approaches 3 and when doing so from the right, f(x) approaches 1. Hence the (two-sided) limit at that point does not exist."
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "At 5:58, Sal means -5, right?",
        "A": "Yes! That has been noticed a lot"
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "At 5:29, Sal says the limit of f(x) as x approaches 4 is 5 but shouldn't the limit be -5?",
        "A": "Yes it is supposed to be -5, not 5. His brain tripped again :p. If you paid attention from 4 :30, a couple times he said 5 (but meant -5), he noticed once and corrected it. at 5:29 he didn t notice it he said 5. But yes, he meant -5."
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "In the problem presented at 7:19 , let's go to another part of the graph and where lim x->7 is, the graph is divided in its left side and right side but shouldn't lim x-> 7 be equal to -4 ? Its limit is defined, even though its not continuous.",
        "A": "The value of f(x) is defined at x = 7, and is equal to -4 as we can see from the graph. Lim f(x) as x approaches 7 from the negative side also appears to equal -4. However, because lim f(x) as x approaches 7 from the positive side is actually equal to -2, the general limit of f(x) as x approaches 7 is not defined. Like Sal said at 3:10 , if the two one-sided limits are not equal, the general limit does not exist. I hope this helps."
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "at 6:40 why there is a red line on the graph?",
        "A": "A vertical dashed line indicates an asymptote. It doesn t have to be red, that s simply a choice they made here. The asymptote here indicates the value goes up to infinity as the function approaches 3 from above and down to negative infinity as the function approaches 3 from below, with no value assigned to the specific input 3."
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "At 5:30 in the video, should the limit as f(x) approaches 4 be negative 5? I'm really bad at math, so I'm confused.\nThanks for any help!",
        "A": "I think you are right, Sal just messed up there and the answer should be -5."
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "At 5:28, Sal says 5 instead of -5.",
        "A": "That mistake is corrected now"
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "Not actually a question. but at 8:12, a correction pops up on the lower right-hand corner of the screen saying \" Sal said 2 but meant -2\". But Sal actually said \"..as x approaches 2 from the negative direction...\" Therefore, the correction shouldn't be since this expression is the same as -2. Isn't it? :)",
        "A": "No it is not the same. Approaching 2 from the negative direction means you are getting closer and closer to 2, moving to the value 2 from the left, that is, 1.5, 1.7, 1.9, 1.99, 1.9999 etc. Approaching -2 from the negative direction means you are getting closer and closer to -2, moving to the value -2 from the left, that is, -2.5, -2.3, -2.1, -2.01, -2.001, -2.00001 etc. Do you see the difference? Hope that helped!"
    },
    {
        "video_name": "nOnd3SiYZqM",
        "Q": "At 0:01 Sal shows a graph what's the function of that graph",
        "A": "I believe most of it is just a rough hand sketch of a piece-wise function, but this is what you can tell from the sketch: f(x){ 9, as x=2 -3x+7, as 2<x<=4 As for x>4 it seems to be around 2((x-4)^.5)-5 but this is not an exact match"
    },
    {
        "video_name": "qvyboGryeA8",
        "Q": "Shouldn't the equation at 7:35 be x2+2x3+3x2 = 0 be x2+2x3+3x4 = 0?",
        "A": "nvm he corrected it at 9:30"
    },
    {
        "video_name": "qvyboGryeA8",
        "Q": "When writing the solution set (~9:20) Sal defines the variables in terms of X3 and X4, which makes sense, but then jumps to the claim that this solution set (vectors scaled by X3 and X4) equals the span. How does this equate to a general span when it was previously defining only a relation to X3 and X4?",
        "A": "The rref gives x1 and x2 (which are vectors in the directions of the x1 and x2 axes) both in terms of x3 and x4 (which are not vectors). As x3 and x4 vary over all the possible pairs (x3, x4), x1 + x2 assumes all the possible points in the x1x2 plane, and the span of x1 and x2 is the x1x2 plane."
    },
    {
        "video_name": "qvyboGryeA8",
        "Q": "Is it necessary to subtract R1 - R2 at 6:29? Will it still be considered a Reduced Row Echelon Form matrix, if you skip this action?\n\nThanks!",
        "A": "By definition of reduced row echelon form, the matrix would not satisfy the requirement that every leading coefficient is = 1, AND is the only nonzero entry in its column."
    },
    {
        "video_name": "qvyboGryeA8",
        "Q": "Why is that (at 6:02) we want to get rid of the 2nd entry in the first row when we are reducing the matrix, but not the second and third?",
        "A": "We want the second entry to be a pivot entry. To be a pivot entry, it needs to have a 1 in one row and 0s in all the other rows."
    },
    {
        "video_name": "qvyboGryeA8",
        "Q": "At 7:29, why is the coefficient of X4 -1, not -2?",
        "A": "Oh, never mind...he fixes it."
    },
    {
        "video_name": "qvyboGryeA8",
        "Q": "At 7:51, I'm noticing that the second and third rows of the original matrix contain the same numbers, just ordered differently. Does that explain why the 3rd row of the rref of the matrix is 0 0 0 0? Does it always come out that way -- with a final row containing all zeroes in rref-- regardless of the particular numbers used?",
        "A": "In general that statement doesn t hold true. You can get matrices in rref which do not have a final row of all zeros. Unless you know beforehand that one of the rows can be expressed as a linear combination of the others, you cannot automatically say that it will become all zeros."
    },
    {
        "video_name": "cTyNpXB92bQ",
        "Q": "at 9:20 he says a couple of times the project of v onto x.\nShouldn't it be the projection of x onto V ? or are they one and the same thing?",
        "A": "yup, you are right, he confused it that one time."
    },
    {
        "video_name": "cTyNpXB92bQ",
        "Q": "At 10:36, why can't you just say that x is equal to Ay? This would imply that x is a member of V so it's projection onto V would just be equal to itself. If x and Ay are not equal that would mean that multiplying by A^T is not a linear transformation; it would be equivalent to a relation outputting the same value for multiple inputs which precludes it as a function. Why can this be ignored? I have to be missing something here.",
        "A": "If ab = ac, then we can say that b = c, but only if a \u00e2\u0089\u00a0 0. A similar rule applies to matrix-vector multiplication. If A x\u00e2\u0083\u0091 = A y\u00e2\u0083\u0091, then x\u00e2\u0083\u0091 = y\u00e2\u0083\u0091 only if A is invertible. Since we re dealing with an nxk matrix, we don t know that it s invertible in the context of the video (and it probably isn t since n would have to equal k)."
    },
    {
        "video_name": "mols6pMKrto",
        "Q": "At 5:47, why can't we say that the limit is infinity?",
        "A": "In order for a limit to exist, the function has to approach a particular value. In the case shown above, the arrows on the function indicate that the the function becomes infinitely large. Since the function doesn t approach a particular value, the limit does not exist."
    },
    {
        "video_name": "33TYoybjqPg",
        "Q": "2:07 you began to show the anti derivative at 246 you are complete in writing it out however in the |e^((a-s)t) DT anti derivative the DT just vanishes. I will go back and study my anti derivatives however I don't see how it is just solved out.",
        "A": "That is always true of integration. If integration sums all of the tiny DT s then after integration its no longer a tiny DT it s the entire interval."
    },
    {
        "video_name": "33TYoybjqPg",
        "Q": "at 2:30, Sal takes the integral of e^((a-s)t) dt, but isn't e^(a-s) a constant? Can't we put it outside of the integral so that it becomes e^(a-s)*integral of e^t dt?",
        "A": "(I will denote e\u00e1\u00b5\u00a1 by exp(x).) Observe that exp(a - s) \u00c2\u00b7 exp(t) = exp[(a - s) + t], whereas exp[(a - s)t] is equal to [exp( a - s )]\u00e1\u00b5\u0097. Because of this, one can not simply factor out exp(a - s) and integrate exp(t)."
    },
    {
        "video_name": "33TYoybjqPg",
        "Q": "At 3:49, since the limit diverges, does this mean the limit goes to infinity?",
        "A": "@Hollerdog: yes, to infinity and beyond! ;-) e^((-constant)*(a very large number)) --> 0 Don t misinterpret alwick1234 s comment. He is not talking about exponential function going to -infinity but some other function. The exponential function will not go to -infinity, all you can diverge to are 0 or infinity (or 1 in the case of e^0)."
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "At 2:46, Sal maps the x value of -1/2 on the unit circle. He then begins drawing a line going up from (-1/2,0). Why does he draw it going upwards into the second quadrant instead of going downward in the third quadrant?",
        "A": "That is because arccos is only defined for first and second quadrants, the reason for that is because the cosine value of an angle lies on the x axis of a unit circle."
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "At 2:44 to 2:47, Sal says \"when we intersect the unit circle, the x-value is minus 1/2\".\nHow is the x-value of minus 1/2 determined when Sal makes a point on that specific part of the circle? Why has Sal chosen that specific area on the circle?",
        "A": "Sal is just giving an example, and he chose x = -1/2. He could have chosen another value for x. He says (at around 1:28): Let s say I m told to evaluate the arccos of -1/2. Once this value is chosen, then the point on the unit circle is determined: the point with the x coordinate equal to -1/2."
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "At 2:52, why does Sal take theta as an angle outside the triangle and not as 60 itself?",
        "A": "When talking about angles on the co-ordinate axes. by convention the initial ray for all angles is the positive X-axis. The terminal ray may end up in any of the quadrants and by doing so, will affect the sign of the trig function."
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "I don't quite understand how to get the range of the function. In the segment from 6:55 to 7:05, Sal describes the range of theta to be greater than or equal to 0 and less than or equal to 102 pi. Would someone mind explaining how those values were deduced?",
        "A": "Sal misspoke when he said 102 pi then corrected himself to say between 0 & pi. Remember cos is the x value which is a horizontal line. So it s essentially like a protractor with x being the flat part. This is what he is restricting the range to, and remember 180* = pi radians. Make sense?"
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "At 2:18, Sal said that what he drew looks more like an ellipse than a circle. Isn't a circle an ellipse?",
        "A": "A circle is one special type of ellipse - namely, the ellipse whose width equals its height. The converse is not always true - not all ellipses are circles, as not all ellipses have a width that equals its height."
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "how does he know that the triangle in \"3:35\" is a 30 60 90 triangle?\nare there any rules for that?",
        "A": "i guess you can call it a rule because in a 30 60 90 triangle the side opposite of 30 degree angle is always 1/2. And side opposite of 60 degree angle is always square of 3 divided by 2. Sal had 1/2 so he knew it was a 30, 60, 90 triangle."
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "I am lost in this video from 4:43 to the end. Is someone able to explain how to write the restrictions for the answer, and also let me know what the last half of this video has to do with the first? Thanks:)",
        "A": "Did you watch this video Restricting domains of functions to make them invertible , and if you still have a problem after watching, please let me know, I ll help you"
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "At 2:50 you explain that theta is the angle that when intersected with the unit circle gives an x value of -1/2. However wouldn't that mean that theta equals 60 instead of 120? Please explain why we chose theta as 120. I'm having difficulty figuring out how to identify which angle is theta.",
        "A": "Hello Seahawks, The angles are always measure counter clockwise with respect to the X axis. For this problem think of a clock. Start at 3 o clock and move the hour hand backwards to 11 o clock. This is where the 120 degrees came from. You could work the problem starting at 9 o clock as you mentioned. You will need to do more mental work to keep things straight... Regards, APD"
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "At 6:40, Sal says he will restrict the range to the upper half-circle (0 <= theta <= pi). Can we restrict the range to the lower half-circle, so (pi <= theta <= 2pi)?\n1. If yes, is the upper half-circle chosen as a matter of convention only?\n2. If no, why not?",
        "A": "By convention, the arccos is restricted to the upper half circle, that is, 0 <= theta <= pi. Theoretically, there are an infinite amount of angles that will satisfy the arccos function, since the cosine function is periodic. However, as a matter of convention, mathematicians define the arccos of a function as being in the range 0 to pi."
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "How does Sal know it is a 30, 60, 90 triangle at 3:36? It looks like he just eyeballs it and calls it.",
        "A": "The hypotenuse is twice the length of one of the legs. That only happens in 30-60-90 triangles."
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "At 3:35 Sal says the hypotenuse is 1, how did he get that?",
        "A": "Notice what he says at 1:46: so let s draw our unit circle . He s using a circle with radius 1, so the distance from the origin to any point on the circumference is going to be exactly 1. That s why the hypotenuse of that triangle is 1."
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "At 2:52, how does Sal know that theta is the larger angle on the right and not the smaller angle on the left?",
        "A": "All angles are measured beginning at the positive side of the x-axis."
    },
    {
        "video_name": "eTDaJ4ebK28",
        "Q": "At 4:18, Sal establishes the angle opposite the \u00e2\u0088\u009a3/2 side as a 60\u00c2\u00b0 angle, then converts it into radians. Is there a way to calculate this angle without first using degrees?",
        "A": "IF any sci calculator you can go to mode and switch to rad, then input 60\u00c2\u00b0. The cache is it gives only the decimal form, not as pi."
    },
    {
        "video_name": "6Kw2nIwWYL0",
        "Q": "At 2:46, he says the side is 3(sqrt3)/2. Why? Where did the sqrt of 3 come from?",
        "A": "The sqrt(3) comes from the fact that this is a 30/60/90 triangle. You can search google for 30 60 90 triangles and see more clearly (with pictures) why this is such."
    },
    {
        "video_name": "6Kw2nIwWYL0",
        "Q": "at 7:54 in the video sal puts a title over the 'j' instead of a 'hat'.is it a mistake?",
        "A": "Yes, this is a mistake (something like a typo in a book). It would be better (to my way of thinking) if Sal always said j-hat when talking about this unit vector: writing simply i instead of \u00c3\u00ae (i-hat) is a bit like saying i instead of i-hat."
    },
    {
        "video_name": "6Kw2nIwWYL0",
        "Q": "At 2:43 sal said perpendicular of triangle is 3/2\nDid he use sin",
        "A": "Yes, because sin(x)= opposite/hypotenuse, where the hypotenuse is 3 and x=30. Let s solve for the opposite: sin(30)=opposite/3 3*sin(30)=opposite, and sin(30)=0.5, so the opposite side, or the perpendicular side is indeed 3/2. To get the side adjacent to the angle s, where x represents the angle, length: hypotenuse*cos(x) To get the side adjacent to the angle given the opposite side: opposite=adjacent*tan(x) Divide both sides by the opposed and adjacent side lengths to get: adjacent= tan(x)/opposite"
    },
    {
        "video_name": "6Kw2nIwWYL0",
        "Q": "At 5:52 how would we know that sine or cosine of 45 degrees is equal square-root of 2 over 2 (if Sal did not tell us) if we get decimal numbers on our calculators? How do we figure this out on our own for similar problems?",
        "A": "Sines and cosines of certain degrees should be memorized. The cosine of 45 degrees in particular can be easily found to be \u00e2\u0088\u009a(2)/2 with just elementary geometry if you know the side ratios of a 45-45-90 triangle (which are 1 : 1 : \u00e2\u0088\u009a2)."
    },
    {
        "video_name": "6Kw2nIwWYL0",
        "Q": "pls can someone explain to me why the vector B became the square root of -2 at 7:27, was it because the direction is moving backwards or what?",
        "A": "That is correct, the magnitude of the i component of the original vector is sqrt(2), and because it is in the negative direction (right to left) it is - sqrt(2)."
    },
    {
        "video_name": "kbqO0YTUyAY",
        "Q": "at 4:05 what are the red lines?",
        "A": "Khan is simply splitting his equation into groups there. So if i said divide rrr (one group of THREE r) by three, you might split it up to look like this r l r l r . There I have three separate groups of ONE r, rather than one group of THREE r. What he does is just a visual representation. Those aren t fancy math lines or anything! :)"
    },
    {
        "video_name": "kbqO0YTUyAY",
        "Q": "At 3:38, Sal says to divide by 3. But it wouldn't really work if x #1 weights 2 kg; x #2 weights 3 kg and x #3 weights 4 kg.",
        "A": "Yes, that s true. But in this video all the 1 weights weigh the same, and all the x weights weigh the same. And in math, all ones are the same and the x s in a problem are all the same."
    },
    {
        "video_name": "kbqO0YTUyAY",
        "Q": "at 3:15, is it also because the xs have to be on 1 side of the equation?",
        "A": "Yes. Otherwise you don t have any actual value for x."
    },
    {
        "video_name": "kbqO0YTUyAY",
        "Q": "at 1:28 why is khan writing 9 ones instead of writing something shorter.... I dont know if you can simplify but if you can, isn't it easier?",
        "A": "Yes, you can simplify, but Sal is just doing it in ones to explain it easier, so everyone understands! Hope this helps!"
    },
    {
        "video_name": "kbqO0YTUyAY",
        "Q": "at 4:05 what are the red lines",
        "A": "The red lines he is drawing at that point in the video are just dividing his group of X s and 1 s into groups of three or 1/3. He is visually showing you how he got the answer x=3 by taking one third of each group."
    },
    {
        "video_name": "kbqO0YTUyAY",
        "Q": "to do this easier at 1:56 couldn't we just have figured out that 3 on the one side and nine on the other side would've just been 3 on the side with the X's becuase 3x3 is nine so that would've been easier\nthanks.-Layla small",
        "A": "Yes. But in future problems that are Much more complex, you will not be able to do that so I would recommend practicing this with the simpler problems."
    },
    {
        "video_name": "kbqO0YTUyAY",
        "Q": "at 4:00 in the video why dosen't he just do this?\n3x=9\n------\n3",
        "A": "He drew out the three x s and the nine 1 s earlier, and now wants to multiply by one third, so he can take one third of the x s and one third of the 1 s and have his answer. It makes it easier to see and visualize."
    },
    {
        "video_name": "kbqO0YTUyAY",
        "Q": "in 3:44, Why wouldn't you subtract 3 from both sides?",
        "A": "Because on side of the scale has xs and the other side has kilograms..they are different sizes so you can t just subtract the number of one from the other."
    },
    {
        "video_name": "paQ10POrZh8",
        "Q": "Hi, at 0:03 when you mentioned that it was Arthur Stone that first invented/discovered the hexaflexagon, how come it is Martin Gardner's birthday that is celebrated and so praised? Shouldn't it be Arthur stone and Richard Feynmann or even Tuckerman?",
        "A": "Well, since Arthur Stone, Richard Feynman, and Bryant Tuckerman all forgot about it because of the war, and Martin Gardner made it super famous, he s celebrated instead of the people who actually invented the concept of the hexaflexagon."
    },
    {
        "video_name": "paQ10POrZh8",
        "Q": "There's a purple side on her six-sided hexaflexagon which she did not color in the previous video! She colored them:\n\nYellow\nOrange\nGreen\nBlue\nPink\nNeon Yellow\nYet there is a purple side in the one she has now at 0:22. Just thought I'd point that out. Great videos! :D :D",
        "A": "This is a good point. I think it belongs as Report a mistake in the video rather than Questions, though. (That way the KA staff will see it, and it will not be in the queue of questions for other users to answer.)"
    },
    {
        "video_name": "paQ10POrZh8",
        "Q": "2:56 Is this the same Feynman that the Feynman point is named after!?!",
        "A": "Yes, they are both named for Richard Feynman, a physicist who won the Nobel Prize in 1965."
    },
    {
        "video_name": "paQ10POrZh8",
        "Q": "Did anyone else pause at 3:43 and bust out laughing?",
        "A": "And I thought I was the only one."
    },
    {
        "video_name": "paQ10POrZh8",
        "Q": "Did anyone notice at 0:22 the purple only has 3 instead of 6? How does it do that?",
        "A": "It also happens with the green one at 0:27........."
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "Is there any real reason that he uses C at 1:00 ?",
        "A": "The Pythagorean Theorem usually uses the equation a^2 + b^2 = c^2. You can use different variables, but it is easier to just use what everyone else uses so no one gets confused."
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "At 0:46 and on all the exercise problems, why do they claim the shorter leg to be longer than the oter one when it is actually smaller? That really confused me.",
        "A": "See, it does not matter which leg is smaller or bigger. Only the hypotenuse matters - Example - A triangle with sides of measurement 3,4 & 5 than you can know that the hypotenuse is the longest leg. But if we had to find hypotenuse than we can add in any way but in subtraction we have to subtract the smaller length from bigger!"
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "At 2:47, is the length of the unknown side sqrt91?",
        "A": "Yes, that s right. Good job on figuring that out!"
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "A 90 degree angle always has two 45 degree angles and a 90 degree angle. You said that if 2 angles are equal in a triangles then the sides they are not sharing with each other are equal as well. But in my math textbook it says that sides of right angled triangle are always in ratio 3:4:5.",
        "A": "The sides of a right triangle are not always in the ratio 3 : 4 : 5. That is just one example of a primitive Pythagorean triple. There are an infinite number of possible ratios for right triangles and 3 : 4 : 5 is just one possibility."
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "How do you figure out the number to the right of the radical sign? At 8:52 you said that 50 squared equaled 5*2 squared.",
        "A": "okay so basically 50 squarerooted equals (25x2) squarerooted right? and 25 squarerooted equals five. so 50squared would equal five times 2squarerooted"
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "6:45. Sal it's called an isoceles triangle :) (I dont know about spelling)",
        "A": "Really? It looked pretty equilateral to me. But it s just about the same rules for both of them."
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "In the last example. at 9:20, why is C = 5 * sqrt of 2? The sqrt of 2 1.3 or something but 5 * 1.3 = like 6 or 7 ( just guessing).5 * 10 = 50 so its like hes saying the square root of 2 = 10! Please help! I am confused!",
        "A": "think of roots as sqrt(root) x*y. sqrt 50 is sqrt 25*2, and sqrt 25 is 5. so sqrt 50 is 5 sqrt 2."
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "How do we know when to go ahead and do the square root of the answer instead of just leaving it? My father recently had me do a problem like this and he said to further reduce the answer by finding its square root. Whereas you at 2:17 did not do the square root of the answer. Help!",
        "A": "Sal says that to estimate the square root of 85 (which is an irrational number) You have to find which two numbers the awnser is between.)"
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "Now i'm further confused. At 8:55 Sal further reduces the answer (50) into 5 square root of 2. Now I don't know when to reduce, when not to reduce, and how to reduce.",
        "A": "\u00e2\u0088\u009a50 = \u00e2\u0088\u009a(25 x 2) = 5 x \u00e2\u0088\u009a2"
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "At 4:09, it says that we must not use the Pythagorean Theorem for other types of triangles. Is it applicable to right-angled triangles only? If so, why?",
        "A": "because the sum of the squares of two sides is equal to the hypotenuse squared, only right triangles have hypotenuses. If you compare the longest side squared to the sum of the two smaller squares, you can determine if the triangle is right triangle (c^2 = a*2 + b*2), an obtuse triangle (c^2 > a^2 + b^2) or an acute triangle (c^2 < a^2 + b^2)."
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "At 0:29, is the length of the unknown side sqrt85?",
        "A": "That s also right! You re good at this!"
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "At 2:15, Sal's answer was 85 route, but can't we simplify it into 5 route 3?",
        "A": "\u00e2\u0088\u009a85 = \u00e2\u0088\u009a5 * \u00e2\u0088\u009a17, which can t be simplified. You need two of the square roots of the same number, like: \u00e2\u0088\u009a325 - \u00e2\u0088\u009a5 * \u00e2\u0088\u009a5 * \u00e2\u0088\u009a13 - 5\u00e2\u0088\u009a13"
    },
    {
        "video_name": "nMhJLn5ives",
        "Q": "at 6:54 what Sal is referring to is called the Equilateral Theorem.",
        "A": "Gracias."
    },
    {
        "video_name": "4uQtJDf7drU",
        "Q": "At 5:00, you calculated there to be 56 cups in a gallon. Before I tried it on my own, but I got 16. This is what I did:\n1 gallon * 4 = 4 quarts\n4 quarts * 2 = 8 pints\n8 pints * 2 = 16 cups\nI know I got the numbers right, but I don't know if I did it right. I also don't understand the way sal khan did it.",
        "A": "You are suppose to multiply 7/2 gallons by 16 cups/gallons and divide the product you get by 2 because you are multiplying 7 over 2 by 16. I hope that helps you a lot, talk to you latter."
    },
    {
        "video_name": "4uQtJDf7drU",
        "Q": "at 3:30- \"but we haven't even answered our question yet..\" LOL",
        "A": "Yes, but we get there eventually!"
    },
    {
        "video_name": "4uQtJDf7drU",
        "Q": "at 5:01 it looks like wps...",
        "A": "t is confusing"
    },
    {
        "video_name": "4uQtJDf7drU",
        "Q": "At 2:20 why do you cross out the quarts and the gallon?",
        "A": "You actually don t cross out the quarts and the gallon, just the quarts. You cross out quarts twice, once from the first conversion and one from the second. The reason you cross them out is because one is in the numerator and one is in the denominator. It would be similar to crossing out both 4 s if you were multiplying the fractions 4/5 * 3/4. Hope that makes some sense."
    },
    {
        "video_name": "4uQtJDf7drU",
        "Q": "at 3:01 what does he mean by \"get his real estate back\"?",
        "A": "He means that he would like to have the space back. Real estate is basically property, property is a space where you build your house. In his case he is writing on his space. He would like to get that space back."
    },
    {
        "video_name": "lA6hE7NFIK0",
        "Q": "At 3:55, she talks about countable numbers making lists. Would irrational numbers be on this list?",
        "A": "No, you cannot make an infinite list of all irrational numbers. You can see this by the fact that she showed that the rational numbers can be listed and the real numbers cannot be. Since the irrational numbers are simply all of the real numbers that aren t rational, the irrational numbers also cannot be listed."
    },
    {
        "video_name": "lA6hE7NFIK0",
        "Q": "at 2:12 in the video,she says that 5+6 is two numbers, and 12 is only 1.how do we know 2 numbers is more than 1?",
        "A": "Even IF we were taught it, how do we know that this video is telling us the truth about infinity? Think about it. 5+6 is two numbers and 12 is one. It only counts when the number that is the product has two different numbers. 1+1 does equal two, but not bigger."
    },
    {
        "video_name": "lA6hE7NFIK0",
        "Q": "Ok, a little help here. \"Metaphorically Resonante\" @ 1:11? So what is that exactly? Thanks, T.S.",
        "A": "She is referencing The Fault in Our Stars. That is a quote from the book."
    },
    {
        "video_name": "lA6hE7NFIK0",
        "Q": "At 3:08, I thought there was an infinite counting numbers. Why is it Aleph Null? It can surely be much larger than that, and of course there are more than Aleph Null.",
        "A": "Aleph null is infinite, namely the smallest infinity. The set of all counting numbers, therefore, has size aleph null. Of course, there are numbers larger than aleph null, which are also infinite, but their sets cannot be put into a one to one correspondence with the counting numbers."
    },
    {
        "video_name": "lA6hE7NFIK0",
        "Q": "So at 0:32 she says that the real numbers extend across aleph null decimal places. So, since there are ten arabic numerals, there are ten possibilities for each decimal place, and so aleph one (or whatever it is), is 10^aleph null? We are discussing infinity here, so it probably doesn't work that way.",
        "A": "You can say that. In fact, the way beth 1 is most often defined is as 2 ^ aleph 0. Of course, that is exactly the same value as 10 ^ aleph 0. We, however, have not proven that beth 1 is actually equal to aleph 1. In fact, it has been proven that we can t prove that either way."
    },
    {
        "video_name": "lA6hE7NFIK0",
        "Q": "@ 4:36 What does she mean by a number that can't be on the list even though all numbers are on the list?",
        "A": "She started by assuming that all numbers are on the list, and then proved that a number was missing, and so the initial assumption must be wrong."
    },
    {
        "video_name": "lA6hE7NFIK0",
        "Q": "5:10 If it's an infinite list, you can't say that number x is not on the list because it's an infinite list. By definition, infinity includes ALL of the numbers.",
        "A": "No. There is no rule that infinite sets contain all numbers. The only rule, as Vi pointed out in this video, is that an infinite set must contain at least as many elements as there are integers. This list clearly does follow that rule, because we made sure that each element is paired with an integer."
    },
    {
        "video_name": "ftndEjAg6qs",
        "Q": "How do you know if it is 15 min past or to at 0:00",
        "A": "You check the moving line on the bottom of the screen and check for how long the video s been running."
    },
    {
        "video_name": "ftndEjAg6qs",
        "Q": "For the first exercise, Sal says we're in the fourth hour, doesn't he mean fifth hour? (and then again in the second example he says we're in the ninth hour, doesn't he mean tenth hour, after having completed nine hours since 12:00)",
        "A": "I think that s one of those technicalities that just confuse things (like when you re 15 years old, you re actually in your 16th year, but everyone just says they re 15). It s much easier to say in the fourth hour when it s 4:xx just to make things easier, even if it s not technically correct."
    },
    {
        "video_name": "ftndEjAg6qs",
        "Q": "it gets funny right after 1:22 to me",
        "A": "OK that s fine."
    },
    {
        "video_name": "ftndEjAg6qs",
        "Q": "how can the time be something like 11:26",
        "A": "If the minute is less than 60 and the hour is less than 13 it is a time. Like this: 7:50"
    },
    {
        "video_name": "ftndEjAg6qs",
        "Q": "what is the circle thing at 0:02",
        "A": "Definitely a clock."
    },
    {
        "video_name": "ftndEjAg6qs",
        "Q": "whats 9:40 add 7:80 what time is it",
        "A": "First of all 7:80 is more than an hour so it s 8:20. So 9:40 + 7:80 or (8:20) equals 6:00. It depends if your doing am or pm."
    },
    {
        "video_name": "h9ZgZimXn2Q",
        "Q": "At 2:34 on the second scales you placed x+y=5 kilograms. How did you calculate this when we do not know what x and y is. Thanks.",
        "A": "On the right hand side of the scale there is 5 one kilogram blocks, which equals five kilograms"
    },
    {
        "video_name": "h9ZgZimXn2Q",
        "Q": "At 2:52, how did you determine x+y on one side of scale is equal 5 kg on other side of scale?",
        "A": "I didn t see the hole video, but for me is just an example, he could put that x + y is equal to 6, for example..."
    },
    {
        "video_name": "h9ZgZimXn2Q",
        "Q": "At 5:22 he was talking about taking away X and Y but how is it the same thing as changing the mass, how is that possible?",
        "A": "yes because at 7:14 he takes X away so the mass of X changes to 3 which combines with the mass of Y which is 7."
    },
    {
        "video_name": "h9ZgZimXn2Q",
        "Q": "At 2:20 Khan adds x and y to the other scale and 5 blocks until they balence. However in algebra in school you don't get a scale and blocks during a test. How would you figure a problem like that out without a scale?",
        "A": "You are right. You can t solve that problem without real scales and real weights. In a test on that problem through, you will be told that x + y = 5. When you are doing algebra at school, you can always draw some scales, just like he did. Good teachers like to be able to see how you work things out in a test. Just make sure you always do the same thing to both sides and your scales will always balance."
    },
    {
        "video_name": "h9ZgZimXn2Q",
        "Q": "At 2:55, how did Sai know to add 5 to the right hand side?",
        "A": "It was just part of the equation presented by the problem. If you were solving this as an exercise yourself then you wouldn t have to come up with the second equation. You would be presented with the first equation and the second equation and then asked to solve for x and y based on both of them."
    },
    {
        "video_name": "h9ZgZimXn2Q",
        "Q": "in 2:20 he adds x and y with the scale,but i do not understand how he gets 5.It's not like they are going to gave us a scale on a test.",
        "A": "You may not be given a scale, but you must be given enough information to solve the problem. 2x + y = 8 is not solvable, on its own. You need more information, and in this example, the second given is that x + y = 5. Once you have the two different equations with two unknowns, you have enough information to solve for each of them. 2x = 6 could have worked for the second equation, and would lead to the same solution."
    },
    {
        "video_name": "h9ZgZimXn2Q",
        "Q": "At 7:54 what do you do after you get x=2 y=2",
        "A": "Ana, at that point you have found what you were wanting to know: what numbers can I replace x and y with that will make BOTH equations true?"
    },
    {
        "video_name": "h9ZgZimXn2Q",
        "Q": "At 3:45 how did you know x + y = 5?",
        "A": "JP, You had an x block and a y block on one sides of the balance scale and five 1 blocks on the other side of the balance scale. Because the scale was balanced, you know that x + y = 1+1+1+1+1 which is x+y = 5 I hope that helps."
    },
    {
        "video_name": "h9ZgZimXn2Q",
        "Q": "In 3:11, how is 1x+1y = 5 yellow blocks equal to 2x+y=8?",
        "A": "It isn t, x+y is equal to 5 and 2x plus y is equal to 8. Solving x and y using simultaneous equations: x+y = 5 2x+y = 8 By subtracting 2x+y = 8 by x+y =5.. 2x + y = 8 - x + y = 5 \u00e2\u0080\u0094\u00e2\u0080\u0094\u00e2\u0080\u0094\u00e2\u0080\u0094\u00e2\u0080\u0094\u00e2\u0080\u0094\u00e2\u0080\u0094 x = 3 After solving for x, we can plug x in for one of the equations to solve for y. I m going to use the equation x+y = 5 3+y = 5 y = 2 ( x , y ) = ( 3 , 2 ) I hope that made sense. xD"
    },
    {
        "video_name": "h9ZgZimXn2Q",
        "Q": "How do you know it balanced out at 5 blocks that are equal to y and x?\n(3:01)",
        "A": "First, you put on X and Y on one side. To figure out how much this equals, you put on one block at a time until the scale is balanced. In this video, we re assuming that these are real scales. We re also assuming that Sal balanced the scale off video to find out how much it equals."
    },
    {
        "video_name": "h9ZgZimXn2Q",
        "Q": "At 2:31, how did you know that x + y = 5 (on the second scale). In the problem 2x + y, couldn't this occur: If x = 1, then y=6; if x=2, then y=4; if if x=4, then y=1. I don't get why you knew that x + y = 5.",
        "A": "It sounds like Sal is using a real scale to weigh two items of unkown weight and when five 1 kg weights are added the scale balanced. Or, Sal just pulled that out of thin air."
    },
    {
        "video_name": "h9ZgZimXn2Q",
        "Q": "At 2:26 to 3:00 couldn't sal have just as easily said x=2 and y=4 therefore x+x+y=8?",
        "A": "Hey, how do you get a timestamp?"
    },
    {
        "video_name": "gMcs6RF_LrQ",
        "Q": "he deleted one t he thought is + in 9:40",
        "A": "u = e^(-st), and u = -(1/s)e^(-st) [= \u00e2\u0088\u00abu ]"
    },
    {
        "video_name": "gMcs6RF_LrQ",
        "Q": "Where does he get a^2/s^2 from at 9:53? I thought it would just be a/s, I don't understand where the squared part comes from.",
        "A": "He pulls 1/s and a out of the integral because those are constant. He also has a/s outside the parenthesis. So he distributes inside and gets (a^2)/(s^2). Hope that helps!"
    },
    {
        "video_name": "gMcs6RF_LrQ",
        "Q": "@5:29 shouldn't [(-1/s) e^(-st) sin(at)] also be evaluated from zero to infinity ?",
        "A": "that s what I m also wondering :/"
    },
    {
        "video_name": "rU222pVq520",
        "Q": "At 1:45, couldn't you use synthetic division? It's a lot easier to me.",
        "A": "Sure, but not everyone has learned it, and it s just a short-cut version of the polynomial long-division above."
    },
    {
        "video_name": "rU222pVq520",
        "Q": "At around 3:47 why can't you factor out the numerator x^2+x+1 to be (x+1)(x+1)? And then have the entire equation be [(x+1)(x+1)]/(x+1) and be left with just x+1 then input 1 into the x value and have the final answer be 2?",
        "A": "The problem with that approach is that we can t factor x^2 + x + 1 the way you indicate. The expression (x + 1)(x + 1) multiplies out to x^2 + 2x + 1 (notice 2x in the middle)."
    },
    {
        "video_name": "rU222pVq520",
        "Q": "At 1:40, why didn't he just use difference of cubes (since he used difference of squares in the denominator). It's way faster, easier (Algebra I) and it should give the same result? Or am I wrong?",
        "A": "That s a better and faster way to do it. Much faster than long division."
    },
    {
        "video_name": "rU222pVq520",
        "Q": "at 1:31 for x cube minus 1,can't I just simplify to x square(x-1)?",
        "A": "that would be x cube - x square"
    },
    {
        "video_name": "rU222pVq520",
        "Q": "Why does Sal swap signs in the long division of x^3-1 and x-1 in 2:30 in the video? Don't get it",
        "A": "To help computing easier. The next step is to subtract (x\u00c2\u00b2-x) from x\u00c2\u00b2. It very easy to make mistake when you subtract a negative. Since subtracting (x\u00c2\u00b2-x) is the same as adding -x\u00c2\u00b2+x, he swapped the signs then he carried out addition instead of subtraction. Pretty neat trick. I don t swap the sign like he does, and sometimes made the mistake. Like subtracting -x will give you x, the mistake would be -x for the answer. Hope that explains it."
    },
    {
        "video_name": "rU222pVq520",
        "Q": "At 3:13 , Sal writes when x =/= 1. But after simplification, even x=-1 will give an undefined answer. Why do we not take that into consideration? How do we know what to put conditions for and what not to put conditions for.\nI understand that since the question deals with limit of x going to 1, -1 is not that important, but my question is still how do we generally know what to put conditions for.",
        "A": "He only stated the restriction for the factor that cancelled. Since it cancelled it is no longer obvious so you must state it explicitly. Since the x+1 factor is still there, the x=/= -1 restriction is obvious by looking at the problem."
    },
    {
        "video_name": "rU222pVq520",
        "Q": "At 3:16, what is the difference between evaluating at x equals 1 and evaluating as x approaches 1?",
        "A": "evaluating x=1 means ,you are just putting the value x=1 in the given function whereas x approaches to 1 means you are putting the values in the function very close to x (approaching) and checking to which value y is getting closer to ."
    },
    {
        "video_name": "rU222pVq520",
        "Q": "At 3:16 in the video, Sal canceled (x-1) in both the numerator & the denominator & said (x^3 -1)/(x^2-1) is equal to (x^2+x+1)/(x+1) where x=/= 1 but for the x+1 in the denominator, you substitute x for 1 & get 1+1=2, not 0. you can divide by 2\n\n\n\n\n\n\n\nP.S.\nhere is a Chuck Norris joke : Chuck Norris CAN divide by zero.",
        "A": "That s because we re not dividing by 0 - we re getting so close to 0 that in practice, it counts as 0. But because it s still a number (albeit a small one), we can divide by an expression equal to (in a sense) 0."
    },
    {
        "video_name": "XMJ72mtMn4Y",
        "Q": "in 5:36 why the answer of 18 ??why help me plz",
        "A": "So he is trying to subtract 6- (5/3). The way to subtract or add fractions is to have the same common denominator. To convert 6 to have a common denominator you can multiply by (3/3) which equals to one. 6* (3/3) = (18/3) Now, it s much easier to do. (18/3) - (5/3) = [(18-5)/3] = (13/3)"
    },
    {
        "video_name": "XMJ72mtMn4Y",
        "Q": "At 1:14, in the y=mx+b, what does x stand for? Thanks!",
        "A": "y=mx+b is the equation of a line. Every point on the line can be written as a coordinate pair (x, y) So given an x value, evaluate mx+b, and that tells you the y value, and now you have the point on the line that corresponds to the given x value. Alternatively, you might be asked, is the point (x, y) on the line c=mx+b, where c is a given value for y. In this case, plug x into mx+b. If the result equals c, then it is on the line, otherwise the point (x, y) is not on the line."
    },
    {
        "video_name": "XMJ72mtMn4Y",
        "Q": "at 3:46 there are 2 numbers that are not zero for my question the cordinates are (3,6) and (-2,6) when you plug that in you get a zero so it would be m=0/-5 is that hwo i leave it or what do i do",
        "A": "y=mx+b..... I can t stress this equation enough for these particular problems. (3,6) and (-2,6) lets let our first set of co-ordinates be the starting point y2-y1/x2-x1 6-6/-2-3 0/-5 (any number divided by zero is zero) we just solved for the slope only if we want to solve the whole equation we must do more y = (-0/5)x + b we can plug in one set of co-ordinates we had earlier to solve for the y-intercept 6 = 0(3) + b 6 = b hope this helps"
    },
    {
        "video_name": "XMJ72mtMn4Y",
        "Q": "at 5:23 how does he get 18? does he multiply the 6 by the three?",
        "A": "6 = 6/1 = 6/1 * 3/3 = 18/3 (equivalent fractions). he does that so that he can subtract 5/3 from 6."
    },
    {
        "video_name": "XMJ72mtMn4Y",
        "Q": "At 3:44, why didn't you put parentheses between the minus and the negative 1? I was always told to put parentheses around two operations that are next to eachother",
        "A": "You can absolutely do that: 5-(-1). This makes it helpful and less sloppy to do that, so it doesn t end up like someone wrote 5-1. Typically in school, they teach it that way to keep us organized. As you progress, teachers won t care anymore, and kids like to take the shortcut and write 5- -1 which equals to 5+1 (that s what I write)"
    },
    {
        "video_name": "XMJ72mtMn4Y",
        "Q": "At, 4:01 Sal simplified the gradient. Will you get a different answer if you forget to simplify the gradient?",
        "A": "No because if you simplify a fraction, it has the same value as it had before you simplified. For example, 1/2 = 2/4"
    },
    {
        "video_name": "XMJ72mtMn4Y",
        "Q": "At 5:25 in the video, how did Sal go from 6 to 18/3? Thanks!",
        "A": "This is probably a lot easier than you think it is! In order for you to subtract 5/3 from 6, they have to have common denominators. 6 is actually 6/1, so you need to get the denominator to a 3. Since you multiply the denominator by 3 (1 x 3 = 3), you must also multiply the numerator by 3. 6 x 3 = 18 and your fraction is now 18/3, which if you think about it, simplifies to 6. Now that you have common denominators, you can subtract 5/3 from 18/3! Hope this helps! Tell me if you have any other questions!"
    },
    {
        "video_name": "XMJ72mtMn4Y",
        "Q": "At 5:25, why does 6 turn into 18/3? I don't really get that.",
        "A": "18 divide by 3 equals 6"
    },
    {
        "video_name": "XMJ72mtMn4Y",
        "Q": "At 1:39, why did Sal use a triangle before the y and the x?",
        "A": "That triangle is the Greek symbol Delta, which means difference. So, the triangles mean the change in y divided by the change in x."
    },
    {
        "video_name": "XMJ72mtMn4Y",
        "Q": "At 4:20 Sal uses 6 as his y-value. Does it matter which one we use?",
        "A": "No, it does not matter. You may use which ever y value you think will make the problem easier for you to solve."
    },
    {
        "video_name": "XMJ72mtMn4Y",
        "Q": "Does it matter which one I pick to subtract around 3:00? How did he choose the start and end points? Like, he chose to subtract 6 from -4, but why did he decide to do that instead of 6 - (-4)? Is there a way to decide which to subtract from which or did he just randomly decide that and it doesn't matter which? I'm having a lot of trouble with this.",
        "A": "The order doesn t matter. Just pick one point and label it as (x1, y1), then label the other as (x2, y2). Put the values in the right places according to the formula to calculate the slope. You will get the same answer for the slope regardless of which you use for (x1, y1) vs (x2, y2). Just so you can see, I ll redo Sal s work but use the points in the reverse positions. Sal calculated a slope = -5/3. Here s my calculation: m = [6 - (-4)] / [-1 - 5] = [6+4] / [-6] = 10/[-6] = -5/3. See... same slope! Hope this helps."
    },
    {
        "video_name": "7UISwx2Mr4c",
        "Q": "At 8:30 he says that on and isosceles triangle if you have two angles that are the same then the line that connects them is a congruent line. Is this true with other kinds of triangles?",
        "A": "No, because the whole proof starts with the fact that there are two equal sides, ie it s an isosceles triangle"
    },
    {
        "video_name": "7UISwx2Mr4c",
        "Q": "At 5:30, Sal says that we have a lot of triangle congruency theorums to use. But in earlier videos, Sal calls them postulates because they can't be proven. Which one is right???",
        "A": "They are all theorems. To be precise, SAS is Proposition 4, SSS is Proposition 8, and ASA and AAS are combined into Proposition 26. Sal may have been thinking that Euclid never formally defined what he meant by congruence."
    },
    {
        "video_name": "7UISwx2Mr4c",
        "Q": "At 5:16 how come Sal called AC and AB segments aren't they supposed to be together as angles?",
        "A": "He s referring to the congruency of the segments as opposed to the angle they make. AB and AC are being compared for their length."
    },
    {
        "video_name": "7UISwx2Mr4c",
        "Q": "When Sal says 6:34 that you can always construct and altitude, does this mean if i get a triangle and construct and altitude the 2 sides will be 90 degrees? I dont understand that part? Please help!! Like when can assume that one side is 90?",
        "A": "All altitudes form 2 right angles with the base of the triangle. If you draw an altitude straight down to the triangle s base from one corner, you will form two 90 degree angles at the base on either side of the altitude."
    },
    {
        "video_name": "7UISwx2Mr4c",
        "Q": "At about 7:43 he says the bottom triangle is AAS but what about the very top acute angles, both of them are the same\n.",
        "A": "You re correct, the very top angles are also the same. Good job! However, we only need two pairs of congruent angles, so we can ignore the top angles for now."
    },
    {
        "video_name": "7UISwx2Mr4c",
        "Q": "can I do the second proof (at around 5:30 ) by construction? that is, if I take a compass and can make a circle with center A and radii AB and AC, would this be a way of proving the statement AC = AB? I assume so, since all radii in a circle are equal in length to each other, right?",
        "A": "Yes, you can. Technically, you can construct any triangle, and it really helps me figure out difficult geometric equations. Also, constructing a triangle helps show your work so you can refer to it later."
    },
    {
        "video_name": "7UISwx2Mr4c",
        "Q": "@6:00 are the midpoint and the point intersected by the altitude not always the same thing? I think @8:54 he says this is only true of isosceles triangles?",
        "A": "No they aren t always the same thing. Only in isosceles triangles like Sal said."
    },
    {
        "video_name": "7UISwx2Mr4c",
        "Q": "At 2:29, when Sal separates triangle ABC into 2, when you separate a triangle, are they always congruent triangles?",
        "A": "Triangles are not always congruent when you separate them, it depends on whether you can prove if they are congruent (SSS, ASA, SAA, AAS, or CPCTC). However, in this case, it has been split congruently because of SSS, or three congruent sides."
    },
    {
        "video_name": "6vUjGgI8Dso",
        "Q": "At 3:10 Sal says #2 in the video cannot be written as a product of x and function of y, therefore its not separable. If the criteria for an equation to be separable is to isolate x and y as stated in previous videos, why does it have to be a product? In other words, why couldnt we manipulate #2 to be dy + y = dx - x + (1/2) and that be separable?",
        "A": "I didn t understand this at first until my professor asked us to try it. In the case of the second problem, he simplifies it to dy/dx=-x-y+1/2 and you are correct that we can add the y term to both sides getting us y(dx)+dy=(-x+1/2)dx (I just reordered the terms in your equation and multiplied the dx last). Now with this form, y(dx)+dy=(-x+1/2)dx, try to integrate both sides and you will see that you can t. In fact, you would have to come up with a new mathematical proof for a way to solve this problem."
    },
    {
        "video_name": "xpzt0wqMT6Y",
        "Q": "3:30 i dont know concept of \"canceling out\" where does he talk about this ?",
        "A": "You want to treat them as algebraic values. You re trying to get the dollars alone. If you did 16 g * 1g / 3d, you would get 16g^2 / 3d. //where g is gallons and d is dollars."
    },
    {
        "video_name": "xpzt0wqMT6Y",
        "Q": "I'm still confused about how, at 1:54, Sal got 400/25. Can somebody please tell me how this works?",
        "A": "to see how much gallons he needs to buy for the 400 mile trip i hope this helped!!"
    },
    {
        "video_name": "xpzt0wqMT6Y",
        "Q": "At 1:10 and 3:02, how is it clear whether to use gallons per mile or miles per gallon and dollars per mile or miles per dollar? How do you know which unit to put as the numerator and the denominator?",
        "A": "OneEye, the word per means divided by. So if we say gallons per mile we mean gallons divided by miles. In a fraction, the numerator is divided by the denominator. Therefore gallons would be on the numerator and miles on the denominator. If we say miles per gallon , miles would be on the numerator and gallons on the denominator. Tell me if you need more explanation!"
    },
    {
        "video_name": "xpzt0wqMT6Y",
        "Q": "At 2:23 isn't it way easier to just do 400/25 and then multiply the quotient by 3? Its much simpler and takes less time. So why do it the long way?",
        "A": "It s perfectly fine to do that, probably even faster- you still get the same answer. It may be easier for you(and it seems so to me as well :) ), but for people just learning this stuff it s much more helpful to see it done the detailed way. After a while, most of these elaborate steps are skipped by people. These videos are really elaborate to help people who are just learning."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 6:42 Sal says that angle w can be calculated using the 180 degrees theory, 90 and 65 was solved for the two angles and W would have to be 25 degrees. Is there another way to solve ANGLE W without using this \"180 degrees method\"?",
        "A": "I don t know why you wouldn t want to solve for the 3rd angle using the fact that the angles of a triangle add up to 180, but you could probably use a trig function to solve for that angle."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": ".As mentioned in the video @5:08 sec Sal says that both the sides are divided by cos 65 degrees.\nMy question is that instead of doing all this can we find the value of cos65 degrees and then equate its value with 5/b. and get the answer (value of b) ?",
        "A": "Yeah, as he said at the start of the video there are many ways to go about finding the answer. It s all about preferences."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 2:48, the equation 5*tan65 is entered. I entered this into my TI-83, 5*tan(65) and it keeps giving me -7.350191288. Is my calculator on the wrong setting?",
        "A": "Yes, your calculator is set in radians. To change it to degrees, go to MODE and scroll down to RADIANS | DEGREES and click DEGREES"
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 4:49, Sal multiplies both sides of the equation by b, but couldn't he just multiply by 1/5? I also do not understand why I get 0.0845236523 when I do this and not 11.83100791. Thank you so much!",
        "A": "Interesting strategy, but if you multiply both sides of the equation by 1/5, then you wind up with (cos 65)/5 = 1/b, whereas you seem to have assumed that you would get (cos 65)/5 = b. And, indeed, you can see that your answer is the reciprocal of the correct answer that Sal got."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 0:04 the transcript says links, does it mean lengths?",
        "A": "Yes, it was a mistake. It was supposed to say lengths."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "I did the same thing in the calculator at 2:45 but instead got -7.350",
        "A": "Change your mode to degree mode.....it is probably on radian mode right now."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 2:51, when Sal enters the expression 5*tan(65), he gets (Let's say we're rounding to the nearest tenth) 10.7.\nHowever, when I enter 5*tan(65), I get -7.4. Is it because of what calculator I have? I have a TI-84 Plus Silver Edition. I wonder if my calculator is acting weird or if I'm inputting the wrong expression.",
        "A": "Do you have it set to degrees?"
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "For the problem starting around 4:30, cos65degrees = 5/b, why could you not divide by 5 or multiply by 1/5 for both sides/equations? Why do we have to simplify by getting rid of the denominator, b?",
        "A": "whenever they hit the down button and kellogg is coming for the right of necromancy and shiblouni of the rock nations with the spicy pizza back at it again with them white vans."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 5:36, the answer Sal got is 11.831..... By using the Pythagorean theorem i got 11.810.... I did this using Windows Calculator app.\nCould i have done it wrong or will the answer be marginally different by using the Pythagorean Theorem??",
        "A": "It doesn t work because the example problem is using rounded side lengths and/or angle measurements. He rounded and cut off a few decimal places to simplify the problem."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 2:55, for 5* tan of 65, my calculator said -7.35. Why did it do that and how do I fix it?",
        "A": "change from radians to degrees on you calculator"
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 2:47 Sal inputs the calculation into the TI-85, when I try the same inputs with the TI-83 it gives me a completely different result (-7.3501xxxxx). I'm inputting 5*tan(65), any help would be great.",
        "A": "Are you in degree or radian mode?"
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 2:50 Mr. Khan types 5 times tan 65. I used to google calculator to follow along and I (the calculator) got -7.35. I typed the same thing. Why did I get a different answer? I will try to figure this out on my own until I get an answer. At 5:36 again I type in the same things on google calculator and I got -8.89.",
        "A": "The Google calculator you are using is in radians mode. There is probably a way to switch it to degrees. If you are a math student, you should get a real calculator and become familiar with how to use it. They are not expensive, and using a calculator is an important skill to develop."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "at 4:15, when Sal chooses to do cosine instead of sine, is it just that triangle that can switch around sohcahtoa, or can all right triangles do it?",
        "A": "Sal decides chooses cosine over sine at 4:15 in the video it wasn t because of that triangle if you know the values. like in the video the value of sides a and c and angle c were known so we could switch around SOHCAHTOA but you can switch between sine and cosine if you find the correct values. It is a part of the trig ratios. tangent = sine / cosine."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "When the hypotenuse of the triangle in worked out in the video the answer given is 11.8 (5:35 minutes) but when I worked out the hypotenuse myself using the Pythagorum theorum, I get an answer of 139.49. Later in the video (5:56 minutes), it tells you that the answer should be the same.",
        "A": "Your 139.49 came from the sum of the squares of both sides. But you still need to take the square root of 139.49 to get the answer!"
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "how did he get the 5 at 2:10 ?",
        "A": "5 is the length of the adjacent side."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 4:50, Salman starts to work out what the value of B was. Why didn't he times both sides by 5 like he did when he was trying to work out what the value of A? Instead hi divides both sides by cos 65(degrees). I multiplied both sides by 5 and worked it out the same way as he did A, and i got a completely different answer. Why is this? Thank you : )",
        "A": "When he was doing the tangent, for A, 5 was in the denominator, so to get rid of that he multiplied by five. When he was doing cosine, for B, 5 was in the numerator so he couldn t divide by five to get rid of anything."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 2:05 you multiply the number 5 (the adjecent) with tan of 65 degrees, but when you figure out the cos of 65 degrees at 5:15 you divide it. Why is that? Is there a rule of thumb for this?",
        "A": "It s just algebra. Because tan65=a/5 he had to multiply by 5 to isolate and solve a. Same for cos65=5/b he multiplied by b to get b*cos65=5 and then divided by cos65 to get b=5/cos65 so he could solve for b."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "2:30, Sal takes the calculator out to determine length of A by using the tangent of 65. But how does he really do that (without the calculator) if the tangent is A over 5?",
        "A": "Recall SOH-CAH-TOA. tangent = opposite/adjacent."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "The measures of the angles of a triangle are in the ratio of 5:6:9. What is the measure of each angle?",
        "A": "Ok, so you know your angles are 5x, 6x and 9x. But we also know that the angles of a triangle add up to 180 degrees. So we must have 5x + 6x + 9x = 180. Now you just need to simplify and solve this equation!"
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "at 2:52 (5 x tan65=7.77) on my calculator?",
        "A": "You re using radians. Switch your mode to degrees:)"
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "at around 2:10, how does he know to put a/5 and not 5/a when doing the tan of 65 degrees?",
        "A": "He had labeled a as the opposite side and 5 as the adjacent side. Since tan is opposite divided by adjacent, he correctly divided a/5."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "but the value of tan is not defined how can we evaluate at 2:33",
        "A": "The value tan is not defined, however it the ratio remains constant for any certain angle. For example, due to angle-angle similarity, the ratio of the opposite over the adjacent side for any 65 degree angle in a right angle triangle will be the same. The calculator is simply accessing the value that ratio has for that certain angle and converting it into decimal form"
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 2:58 he solves a for about 10.7 using his calculator. But when I typed the same problem 5*tan(65) into my calculator I got -7.35. The only thing I can figure out that's different is my calculator is newer. It's a TI-84. Please help!",
        "A": "Did you make sure that your calculator is on degrees? Because if your calculator is on radians, you will most definitely get a different (and wrong!) answer."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "At 5:41, Sal solves for the hypotenuse with the Trig method. I for the hypotenuse using both the Trig and Pythagorean Theorem method and I got slightly different answers when using the different methods.\nPythagorean: 11.81058847\nTrig: 11.83100792\nIs this difference because I rounded?\nThanks!\nThe #1 Tau Proponent",
        "A": "The exact value for a is 5tan(65), but Sal wrote down 10.7 instead of 10.7225346... because the problem told him to round to the nearest tenth. If you enter in sqrt(5^2+(5tan(65))^2) instead of sqrt(5^2+10.7^2), the result will equal the one from the trig method."
    },
    {
        "video_name": "l5VbdqRjTXc",
        "Q": "How did the calculator transform tan (65 degrees) into a number at 2:47? It seems like the sin, cos, tan of any random degree is already known. Can you actually calculate them (by hand)?",
        "A": "Your calculator uses something called a polynomial expansion to calculate the sine and cosine (and some other) functions. (It does not need to do that for tan because tan = sin/cos) It s not realistically possible for you to do this by hand on a routine basis, but you can try it out to see how it works. Check out Sal s vids on the Taylor Series if interested. Before calculators, people used tables to look up sin and cos. Those tables were calculated using the Taylor series for those functions."
    },
    {
        "video_name": "hoRISaqp1Po",
        "Q": "Sal says @ 1:54 that the slope is -2... but if you're approaching 3 from the positive direction wouldn't the slope just be 2?",
        "A": "By writing a limit without + or - we mean that both left-sided and right-sided limits are equal. Approaching 3 from the positive direction doesn t change the value of the limit because both numerator and the denominator change signs."
    },
    {
        "video_name": "hoRISaqp1Po",
        "Q": "I don't understand how Sal found the value of h around 2:27.. Since we are looking at value as h approaches 0, how come our h is around 7 on the x-axis?",
        "A": "Sal did not explain that very clearly, and did misspeak. It was not the value of h, but the value of x+h (before applying the limit h\u00e2\u0086\u00920). Sal was attempting to explain how a secant line becomes a tangent line when the two points the secant line connects are infinitesimally close to each other."
    },
    {
        "video_name": "hoRISaqp1Po",
        "Q": "At 2:14, why are we approaching 0 from the left? Shouldn't it be 8 from the left? Or are we looking at Y values instead of X?",
        "A": "We re approaching x = 8 from the left, but instead of saying x approaches 8 we re setting up a variable h to act as the difference between our x value and 8 and looking at what happens when that variable approaches zero. Because h acts as the difference between x and 8, making h go to zero is the same as making x go to h."
    },
    {
        "video_name": "hoRISaqp1Po",
        "Q": "At 0:43 , why is the arbitrary X ahead, not behind?",
        "A": "It s only for sake of convenience when using the formula that we choose a point that is ahead. You certainly can choose a point that is behind if you want."
    },
    {
        "video_name": "hoRISaqp1Po",
        "Q": "I see how Sal found the slope equal to 1 at 3:25; but when I watch at 1:48 I am having trouble understanding how Sal got the slope equal to -2?",
        "A": "Did you see Sid s answer to Xen s question above?"
    },
    {
        "video_name": "4b8YnmUUIfI",
        "Q": "@3:26 Does this mean that the slope of the tangent line at of x=2 is also 4?",
        "A": "Yes, very much so."
    },
    {
        "video_name": "TsEhZRT16LU",
        "Q": "At 1:52 how is it that when you multiply -1/2 bt 6 you get 3?",
        "A": "you have the equation -1/2 multiplied by 6. you want to think of 6 as 6/1 multiply both the denominator and numerator together. you then get -6/2. then divide the numerator of -6 by that 2 because you simplify it you get your answer of negative 3"
    },
    {
        "video_name": "UJxgcVaNTqY",
        "Q": "Would it make a difference if you did xc-x at 11:45?",
        "A": "If you used x0-x you d end up with the same equation, and it s just as simple to calculate. But x-x0 shows more clearly and simply what s being done"
    },
    {
        "video_name": "UJxgcVaNTqY",
        "Q": "@2:23, does a normal vector mean a perpendicular vector to a certain plane from a certain point?",
        "A": "Yes, a normal vector is a vector that is perpendicular to the plane in every point of the plane. You can put the normal vector in any point of the plane, and it will continue to be perpendicular."
    },
    {
        "video_name": "UJxgcVaNTqY",
        "Q": "@6:38 a bit confused that the green line (x-x0) actually (quote:) \"lies along the plane\" a vector originates from zero, no??, x - x0 gives a vector in the same direction as the plane (but this point [i.e. vector] does not lie on the plane??)\n\nanyway redoing it in R2 was easier to visualize (for me) for Ax +By = C; --> n1(x-x0) + n2(y-y0) = 0 (it's the x0 and y0 values that raise it back to the actual line)",
        "A": "x-x_0 does lie in the plane. But, a vector does not have to originate from 0. It can originate from anywhere. When you draw it from zero, that s drawing it in standard form. A vector without any other information simply means a direction and magnitude, not position. Even in R2, the (x-x0), (y-y0) vector would lie along the line, not necessarily from the origin."
    },
    {
        "video_name": "UJxgcVaNTqY",
        "Q": "What is the difference between the two different symbols above a vector? I'm familiar with the one above x0 and x, but what does the small triangle above the perpendicular vector n mean? He draws one at 7:25.",
        "A": "That triangle is called a caret or hat. What it is is the unit vector of a vector. If there was a vector v_, the unit vector of _v is called v hat . It is defined like so. r = <x, y, z> -- vector r |r| = sqrt(x^2 + y^2 + z^2)-- magnitude of r rhat = r/|r| -- unit vector of r You can treat unit vectors the direction of the vector. The magnitude of a unit vector is always equal to 1. You can regain a vector by taking the product of its magnitude and unit vector."
    },
    {
        "video_name": "UJxgcVaNTqY",
        "Q": "At 13:35, why is this method of defining a plane in R^3 preferred by computer game programmers?",
        "A": "The reason this is a useful solution as it uses less memory. For example, you could define a plane using 3 points contained on the plane. This would use 9 double values at 4 bytes each. Using a point and a vector (or just two points one of which is off the plane) takes up 6 doubles. Its also useful to have the perpendicular vector for the plane handy. For example, if you want to find the minimum distance of a point to a plane, you will need the plane vector."
    },
    {
        "video_name": "UJxgcVaNTqY",
        "Q": "At 1:43 Sal starts talking about \"naught\" or \"not\" in relation to X0. He never defines what not/naught means. Does it just mean that a variable is perpendicular to it's non-not/naught counterpart? (Meaning X is perpendicular to X0)",
        "A": "It s x naught . x\u00e2\u0082\u0080 usually refers to the initial variable of something. It could be the initial value in an iteration or the starting position of an object, etc.. In this case, it s just a variable."
    },
    {
        "video_name": "UJxgcVaNTqY",
        "Q": "I'm confused. The synopsis of this video was in essence defining a plane with a point (i.e. not a vector) and a normal vector. Why then @10:10 does Sal give his example with with the use of vectors without specifying a point on his example plane? I was under the impression that this video would use a point in its example to illustrate how to define a plane with a POINT and a normal vector.",
        "A": "You are right. There s a fixed point P0 = (x0, y0, z0), a variable point P = (x, y, z), and a vector N = (n1, n2, n3). The equation is (P-P0) dot N = 0; which is OK, because the directed line segment P-P0 is also a vector, since it has direction and magnitude, and also a fixed tail (P0) and variable head (P) . Personally I feel that calling P and P0 (Sal s x and x0) vectors is confusing, since their location as points is important, not just their direction and length."
    },
    {
        "video_name": "UJxgcVaNTqY",
        "Q": "@10:38 why is it that the normal vector and the point don't necessarily have to intersect ?",
        "A": "Because vectors don t necessarily have a location at all, so they don t necessarily have locations relative to each other. But they do have a direction and direction relative to each other. Directed line segments (from one point P1 to another P2 ( P2-P1 ) have a location, and are also vectors (because they have magnitude and direction)."
    },
    {
        "video_name": "anlOhNHlqwg",
        "Q": "At 1:10,why did he have to first find the solution to how many he made in one game?",
        "A": "He probably had to find the solution to how many he made in one game because in the word problem he is modeling, it asks us to find the number of field goals Blair makes in each game, and we re under the assumption that Blair scored the same number of field goals each game. So he s just doing what the problem tells him to do."
    },
    {
        "video_name": "anlOhNHlqwg",
        "Q": "At 1:12 what does Sal mean?",
        "A": "He means that Blair scored 144 points total in the period of all 16 games. And this being field goals."
    },
    {
        "video_name": "yDB-191sa2M",
        "Q": "At 3:57, I think Sal means they're not smooth at the corners, not edges.",
        "A": "I guess so since it is where two lines meet."
    },
    {
        "video_name": "ar9SeJpbPI8",
        "Q": "At 8:10, since v1+dv=v2, shouldn't the tip of the second pink vector touch the tip of the differential blue vector?",
        "A": "If they were in the vector space, yes. If the initial points of v1 and v2 were the same and dv started at the tip of v1, the tips of dv and v2 would be touching. In the xy plane, however, the initial points of the vectors are shifted to the xy coordinates where they are evaluated - v1 and v2 start at different points."
    },
    {
        "video_name": "qv7DM5Ph0vU",
        "Q": "I'm confused as to how the integral of 1/(x^2) became -(1/x) at 1:52 . Can someone please clarify?",
        "A": "It may be easier to see if you think of it as how the integral of x^(-2) became -x^(-1) What is the derivative of -x^(-1) ?"
    },
    {
        "video_name": "itQEwESWDKg",
        "Q": "At 2:31, did he say the answer is e)?",
        "A": "He said the answer was NOT e) because you can still have a z-score even if the distribution isn t normal."
    },
    {
        "video_name": "e4MSN6IImpI",
        "Q": "At 2:50 how did she draw that? I don't get it.",
        "A": "She drew a binary tree by splitting the lines and when 2 lines met she stopped drawing lines from them. Then she filled in the resulting triangles with the red marker."
    },
    {
        "video_name": "e4MSN6IImpI",
        "Q": "At 1:03 she says that you might run out of space if you keep going on for ever. Is that true?",
        "A": "eventually yes. you would."
    },
    {
        "video_name": "Zr72qWNMk7E",
        "Q": "At 1:48, can you guys explain to me where the f'(x) came from? Why did it suddenly get pulled out as a derivative?",
        "A": "That is just the chain rule: the derivative of the composite function g(f(t)) = g (f(t)) * f (t)"
    },
    {
        "video_name": "Zr72qWNMk7E",
        "Q": "At 3:55 the change in x is 1.5 not 3.\nand change in y is 1 .Right ?",
        "A": "Yes, that s true; however it means the same thing. 1.5/1 * 2/2 = 1.5*2 / 1*2 = 3/2."
    },
    {
        "video_name": "LEFE1km5ROY",
        "Q": "@2:15 Sal said \"9 and 9\" but meant \"7 and 9.\" Right?",
        "A": "Yes, he did mean 7 and 9."
    },
    {
        "video_name": "LEFE1km5ROY",
        "Q": "at 4:51 khan said 22+7=27 but it is 29",
        "A": "okay then that happens"
    },
    {
        "video_name": "LEFE1km5ROY",
        "Q": "What are some applications of this? Never seen it used in real life? @0:00 do baseball statisticians use this?",
        "A": "I ve seen it used on a Ferry timetable with the stem having the hours and the leaf having the minutes... Actually gave you a decent sense of the peak running times when there were extra services... at a glance. Then you could look closer and get the exact times."
    },
    {
        "video_name": "LEFE1km5ROY",
        "Q": "Wasn't Sal supposed to say seven not nine at 2:14 ?",
        "A": "haha you re right"
    },
    {
        "video_name": "LEFE1km5ROY",
        "Q": "At 0:00, what other sports would stem-and-leaf plots be useful for!",
        "A": "Any sport what might require to track scores, maybe the ones, where points can be scored quickly."
    },
    {
        "video_name": "LEFE1km5ROY",
        "Q": "Sal never worked as a waiter:):)! He would not have the mistake adding at 5:13 !:) Am I wrong!?",
        "A": "Everyone makes mistakes, Jean, even Sal. :P I make addition mistakes all the time. My excuse is that I m so used to math that required deep-thinking like trig and pre-calc that it no longer understand easy problems like 2 + 2. :P Sylvia."
    },
    {
        "video_name": "LEFE1km5ROY",
        "Q": "At 2:13 doesn't he say 9 instead of 7?",
        "A": "After making so many videos, he will probably make some mistakes."
    },
    {
        "video_name": "LtpXvUCrgrM",
        "Q": "At 6:30, is slope-intercept the same as solving for y? Why is it so important to know both slope-intercept and slope-point form (like written around 4:30)",
        "A": "If you re trying to solve a problem in real life, you probably won t get presented with a neat graph containing all the information you need. You ll probably have only partial information, and you will have to convert that information into a form you can use to solve the problem and interpret the results. The ability to look at information, work out what form it s in, and then convert it to a form you can use, is an essential part of real-life problem solving."
    },
    {
        "video_name": "LtpXvUCrgrM",
        "Q": "at 5:20, sal did y-9=-4(x-4), because he subsitituted with the ordered pair (4,9). But can I substitute with (6,1)?",
        "A": "Yes you can. You can use any point that you know is on the line. And, if you convert both versions of the equation to slope-intercept form, you will see that you get the exact same equation."
    },
    {
        "video_name": "LtpXvUCrgrM",
        "Q": "I understand that you can find point-slope form from any point on the same line. In this video, the point-slope form for the point (4,9) was found. I found the point-slope form for the point (6,1). However, the point-slope form was converted into slope-intercept form starting at around 5:20. So I decided to also find the slope-intercept form from my equation, and I got something entirely different. So how do you know which point to use??",
        "A": "If your math is correct, the slope-intercept form will be identical regardless of the point that you use. Point-slope form: y - y1 = m (x - x1) where (x1, y1) is your point on the line. Let s use slope m = -4 and the point (6, 1) from the video y - 1 = -4 (x -6) Change to slope intercept form: Distribute -4, then add 1 y - 1 = -4x +24 y - 1 + 1 = -4x +24 + 1 y = -4x +25 (same version as Sal got) Hope this helps."
    },
    {
        "video_name": "LtpXvUCrgrM",
        "Q": "@4:20 without needing to plug in y - y1 / x - x1 = m, can't you just shortcut and plug in y1 * y + x1 * x = m? My teacher taught me that that works.\n\nNVM, this is for standard form. Ignore this post",
        "A": "You will get the same answer whether you choose x2 = -7.5 y2=18 or if you choose x2=-4.5 y2=9.5"
    },
    {
        "video_name": "LtpXvUCrgrM",
        "Q": "At 1:54, how do you know if you go up or down first?",
        "A": "If you are asking: how does he know to do 1-9 rather than 9-1 , the answer is it doesn t matter You can calculate the exact same slope if you do: (9-1) / (4-6) = 8/(-2) = -4 Hope this helps."
    },
    {
        "video_name": "R-Qu3QWOEiA",
        "Q": "At3:31, why does the cylinder become 2piy instead of 2piy^3?",
        "A": "Because That s is just the surface area, depends only on y coordinate (not y^3)"
    },
    {
        "video_name": "dOr0NKyD31Q",
        "Q": "At 2:20, when he's talking about the zoo, isn't the mass of an animal in the zoo discrete? I get what he's saying that all possible masses is a continuous variable. However, there are only a finite number of animals in the zoo. Each animal has an exact mass (even if we can't measure the mass exactly, it does have an exact mass). So if we have a finite list of masses, doesn't that make the variable discrete?",
        "A": "Let s weigh the gorilla. Suppose he s between 500 kg and 501 kg. Is there any number in between those, to any number of decimal places, that CANNOT be the gorilla s mass? If not, then mass is a continuous variable."
    },
    {
        "video_name": "dOr0NKyD31Q",
        "Q": "at 2:10 he says that X is a discrete variable why is it not a continous variable?",
        "A": "discrete variables are whole #s only (ex: 1, 2, 3, etc) continuous variables can be any number (ex: 1, 1.1, 1.2, etc) I hope that helps!"
    },
    {
        "video_name": "dOr0NKyD31Q",
        "Q": "It is said that continuous random variables cannot be listed.But in a question based on time (like the one at (08:42) ) , if we have a very advanced device that can measure time very accurately like an atomic clock , will we able to list it?",
        "A": "Not if you consider time to be a continuous variable. Between every 2 femtoseconds there is half a femtosecond."
    },
    {
        "video_name": "dOr0NKyD31Q",
        "Q": "At about 10:20 Sal explains that variable X is continuos because you can't count the exact times that it could be, but I thought time wasn't infinitely divisible. Am I wrong or what's going on here?",
        "A": "I ve been studying math now for over a month with the assistance of Khan academy. I begun from basic arithmetic and now I m here. No problem so far and math has never before been this easy for me. Thank you so much for the work you do, the lessons are really educative."
    },
    {
        "video_name": "xiIQQNufFuU",
        "Q": "On 0:56 how did you get 2(0)",
        "A": "The x-intercept of a line will always have a y-value of zero. Sal then used y=0 and substituted it into the equation to solve for x. Hope this helps."
    },
    {
        "video_name": "xiIQQNufFuU",
        "Q": "Why did Sal at 0:47 of the video multiply 2 by 0. Wouldn't the y be 1?",
        "A": "He tried to find the x-intercept. Therefore, the y value must be 0, so he could isolate the x value on the x-intercept. The x-intercept means what is the value of x when y equals 0. That s why Sal substituted the y value with 0."
    },
    {
        "video_name": "xiIQQNufFuU",
        "Q": "At 0:47, I was like, its 2 1/3",
        "A": "intersecting point"
    },
    {
        "video_name": "xiIQQNufFuU",
        "Q": "Around 1:11 through 1:20 why do you break apart 7/3 to get the answer instead of just turning 7/3 into a decimal?",
        "A": "It doesn t matter. To graph 7/3, you can change it into the mixed number 2 1/3 like Sal did. Or, you can change it into its decimal value of 2.333... with the 3 repeating. They are the same number. It won t change the graph."
    },
    {
        "video_name": "Zm0KaIw-35k",
        "Q": "At 0:42, why does Sal write 189 over 3? That is an improper fraction, and can't you work with 3/189? He just flipped it! Is it easier or something? Please tell me! I need all the help I can get",
        "A": "3 by 189 would be completely wrong also because you are trying to find out how many papers she delivers in a hour, and there are 3 hours, so you would need to divide 189 by 3 instead of 3 by 189."
    },
    {
        "video_name": "Zm0KaIw-35k",
        "Q": "At 1:10, flipping the dominator and numerator is not equal, right?",
        "A": "I don t think so I think he just simplified it."
    },
    {
        "video_name": "Zm0KaIw-35k",
        "Q": "What did he do when he added 1+8+9 (1:09)? I've never heard of that.",
        "A": "He uses the rule of divisibility any thing dividable by 9 must add up to nine if you do the math 1+8=9 +9=18 and 9 goes into 18 2x"
    },
    {
        "video_name": "Zm0KaIw-35k",
        "Q": "Is 2:5 the same as 5:2",
        "A": "it is not always the same. if it states 2 oranges and 5 apples as a ratio, then it will be 5:2, not 2:5"
    },
    {
        "video_name": "Zm0KaIw-35k",
        "Q": "I understand why he flipped the values at 0:42 , but how is it possible? Because 1/2 isn't the same as 2/1.",
        "A": "It s not literal, he s just making an example to show you how to get the rater per HOUR, not rate per NEWSPAPER. Get it?"
    },
    {
        "video_name": "Zm0KaIw-35k",
        "Q": "Woah, wait a minute! How'd you do that trick at 1:07 to figure out if it's divisible by 3?",
        "A": "By adding all the numbers in that one number up so for example: 539 all the numbers added up make it 17 so you know that, that numbers is not divisible by 3. But if you did 465 all the numbers added up equals: 15 and 15 IS divisible by 3"
    },
    {
        "video_name": "Zm0KaIw-35k",
        "Q": "At 1:10 he does the divisibility trick. Does that trick always work?",
        "A": "Yes, that divisibility test for 3 always works."
    },
    {
        "video_name": "Zm0KaIw-35k",
        "Q": "If after you have solved a problem, and you get the ratio of 1:3, will 3:1 be the same?",
        "A": "If I have a bag of M&Ms, and find that the ratio of the number of yellow M&Ms to the number of red M&Ms is 1:3, this would mean that for every one yellow M&M I have, there would be three red M&Ms. This would -not- be the same as saying that for every three yellow candies I have, I have one red candy. However, I could say that for every 3 red candies I have, I have one yellow, and that would be a correct statement. Hope that helps your understanding."
    },
    {
        "video_name": "9uwLgf84p5w",
        "Q": "what is a gnu? at 1:50",
        "A": "Its a animal it kind of looks like a deer and it is a wild animal and can not be a pet it lives in plain biome"
    },
    {
        "video_name": "9uwLgf84p5w",
        "Q": "11:49\nhow do you determine exactly how long each side is?",
        "A": "i get a ruler and measure from then on"
    },
    {
        "video_name": "5mMLaK1ByZc",
        "Q": "At 9:50 - formal definition of directional derivative part.... we should take unit vector along v' (v' cap) and not v' ... I guess there should be a correction",
        "A": "No you don t. If you took the unit vector, the magnitude of the vector plays no role in the final derivative. The directional derivative is not exactly the same thing as the slope of a function. What we need here is the directional derivative of the function as we are on the way to find the derivative of f with respect to t. The said vector v is just an intermediate step."
    },
    {
        "video_name": "erMopRd-MXg",
        "Q": "At 4:24. Shouldn't it be 3x^2 - 3y^2?",
        "A": "Yes. A small box pops up in the lower right corner of the screen with the correction (at least it does when I watch the video)."
    },
    {
        "video_name": "4tWFdI7DkJ4",
        "Q": "At 2:52, Sal says that evaluating definite integrals with a calculator can be useful when you \"can't actually evaluate them analytically\". Are there integrals that are impossible to evaluate analytically? If so, can I see some examples?",
        "A": "\u00e2\u0088\u00abe^(x^2) dx cannot be expressed as an elementary function."
    },
    {
        "video_name": "GDcVdBAnBdU",
        "Q": "At 3:13, it states that all the angles add up to 180 degrees in a triangle. If a quadrilateral is 360 degrees, and a triangle is always 180 degrees, then how can you split quadrilateral into 4 triangles if the quadrilateral is still 360 degrees?",
        "A": "Its the interior angles that add up to 180 degrees. When you put in the 2 diagonals you are creating 4 more interior angles which are 90 degrees each."
    },
    {
        "video_name": "GDcVdBAnBdU",
        "Q": "At 0:50, Sal states that \"we know that the diagonals of a parallelogram bisect each other.\" Yet that was something he proved in an earlier video. In writing proofs, how do we know when we can state \"we know that . . .\" versus having to prove it before using it?",
        "A": "Don t over think it. You only have to prove what you have been asked to prove. Here, Sal was asked to prove that AC was perpendicular to BD. He was not asked to prove that the diagonals of a parallelogram bisect each other, therefore he can assume it to be true without proving it."
    },
    {
        "video_name": "GDcVdBAnBdU",
        "Q": "At 3:05 how does Sal know that those 2 angles are supplementary? It seems like he is jumping to a conclusion.",
        "A": "The reason those angles are supplementary is because together, the line segments that make them up can make up another line segment. If you look closely, AC is made of AE, part of the first angle, and EC, part of the second angle. Because a line is considered to be 180 degrees, two angles that ca make a straight line collectively are 180 degrees. This means that the angles are supplementary. I hope this helps!"
    },
    {
        "video_name": "MyvRxKM0xns",
        "Q": "what is that symbol next to the tangent at 7:50?",
        "A": "If you mean the sign in the upper right hand corner next to tangent, that would be theta. Theta is a Greek letter used in trigonometry to represent a missing value."
    },
    {
        "video_name": "Ix8Nne-a-KQ",
        "Q": "At 2:51, couldn't you just turn the equations x+2y=13 and 3x-y=-11 into slope-intercept form? would that still work?",
        "A": "sure sure u can in either way"
    },
    {
        "video_name": "D2sMsmL0ScQ",
        "Q": "At 2:00, Sals mentions that we will Reject the Null Hypothesis if the probability of a sample mean 17.17 is less than 1%. I don't understand why we would reject the Null Hypothesis if we confirm the probability of getting the sample mean is less than 1%. Would that not cause us to accept the Null hypothesis?",
        "A": "Usually the null hypothesis is rejected if the p value is less 5%."
    },
    {
        "video_name": "7JPIX3odZrY",
        "Q": "At 5:16, don't you have to add a zero to the quotient when you add a zero to the dividend?",
        "A": "No, not at all. I used to do that sometimes, and I got the wrong answer. In this problem in the video, you would get the answer of 25.02, which is not right. The reason you can add it to 630, is because, as Sal says, 630.0 is the same thing as 630. You can add as many zeroes as you want! At the zero store, we never run out!"
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "Here's a question, at 4:47, is it truly necessary for you to factor out the five before continuing the problem? Or can you simply take the anti derivative of the expression with the 5 in the expression?",
        "A": "Since the 5 is a constant, you can factor it out or leave it inside the integral. It won t change the answer in the end. However, it is common practice to pull constants out of the integral because it (usually) makes taking the integral easier to read and you can find any mistakes easily if you made them. tl;dr version, it s completely optionally"
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "At 3:10 you begin to solve for the antiderivative of x^5dx and eventually solve for x^6/6 + c. My question is why is x^6 over 6? Using the product rule, a derivative of x^5 could also result in the antiderivative of just x^6 + c. So why the fraction with x in the numerator and n in the denominator?",
        "A": "There is a coefficient of 1 in front of the x. By power rule x^6 becomes 6x^5. but that s not what we want. we need some number to multiply with the six out front to make 1. 1/6 times 6 = 1. That s what we need. so (1/6)x^6 + C is the antiderivative of x^5. This can be rewritten as x^6/6 + C. Does this help?"
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "at 2:32 how do we know that the anti-derivative was over n+1",
        "A": "When we take a derivative we use the form of d/dx x^n becomes nx^(n-1) So, when we take the anti derivative we add 1 to our power instead of subtract 1. After we ve changed from nx^(n-1) to nx^n we have to divide the coefficient nx by our new power n. So now we have the term (nx^n)/n which will simplify to x^n. As you can see from here our original answer now looks just like our anti derivative of the derivative we took at the beginning. Hope that helps."
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "At 5:30, why can't Sal just write 5c?",
        "A": "He can, but that is just another constant, there is no x in it, so he can simply write c2 to make it less confusing."
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "4:20 What is a scalar?",
        "A": "The scalar is another word for the coefficient of the variable in this instance."
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "At 1:03 Why don't we take the derivative of the denominator? Does it always just stay the same?",
        "A": "Its because he is taking the derivative with respect to x (look at the d/dx ), not n. This means that n+1 can be treated as a constant. When you have (x^n+1) / (n+1) you can write it as (1/n+1) * (x^n+1). Remember, when you have a constant in front you can take it out of the derivative: d/dx[ (1/n+1) * (x^n+1) ] == (1/n+1) * d/dx[(x^n+1)]"
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "at 4:36, Sal says that it's the power of -2+1 on the denominator. shouldn't he leave out the \"power\" part?",
        "A": "Yes. You are correct."
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "At 1:03 why dont we take the derivative of the denominator? does it always stay the same?",
        "A": "The denominator does not contain any x terms, so there is no derivative. It is effectively the coefficient of the x term. If you are taking the derivative of (1/4 * x^2), that is the same as 1/4 * derivative of (x^2). You don t take the derivative of the 4. It is a little more confusing when using the general case and Sal uses n+1 instead of 4."
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "1:15 would this not simplify to (n+1)/(n+1) * (X^n)/(n+1)?",
        "A": "No, that would be true for addition but not multiplication: ( (n + 1) + x^n ) / (n+1) = n+1 / n+1 + x^n / n + 1"
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "At 0:25, why Sal didn't use the quotient rule?? is n treated as a constant??",
        "A": "Yes, n is just a constant. Quotient rule would have worked but would have been much more work."
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "At 1:03: why don't you differentiate n+1?",
        "A": "Because n isn t changing with respect to x."
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "2:33 I see what you did there :)",
        "A": "Classic integral. :)"
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "At 2:01 he says the indef integral is the symbol for the antiderivative. Are they the same thing?",
        "A": "Thankyou for the clarification That helps"
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "At 3:30 could the solution also be written as 1/6*x^6",
        "A": "Yep ! it is correct"
    },
    {
        "video_name": "QxbJsg-Vdms",
        "Q": "AT 5:40, can I further simplify it to -5/x + C as in negative 5 over x ?",
        "A": "Yep! Those are identical."
    },
    {
        "video_name": "7aGEvpHaNJ8",
        "Q": "At 7:00, Sal said that y-x/y is the same as 1-x/y. Can someone explain to me why this is possible?",
        "A": "y-x/y = y/y - x/ y .... Got this ? = 1 - x/y ..... So we got the ans."
    },
    {
        "video_name": "7aGEvpHaNJ8",
        "Q": "At 8:12 x over y is circled in purple pointing to ce over de = x over y how come",
        "A": "We had already seen that x/y equals CE/DE, so Sal was able to replace x/y in his other equation with CE/12 (since DE equaled 12)."
    },
    {
        "video_name": "7aGEvpHaNJ8",
        "Q": "At \"7:30\" how does he get the x/y+x/y or 1-x/, wouldn't i just simplify to -x?",
        "A": "He s just working with the side opposite the green CF/9 , the blue & orange ( y - x ) / y These are the steps: ( y - x ) / y => y / y - x / y => 1 - x / y You can test that these 3 equations are equivalent by putting in values for x and y and checking that they give the same result. You can test your own reasoning of rearranging ( y - x ) / y by putting in values for x and y and checking that they give the same result."
    },
    {
        "video_name": "7aGEvpHaNJ8",
        "Q": "at 5:45 how do we know that CF/9=x/y",
        "A": "Are you able to recognize that triangle ABE is similar to triangle CFE?"
    },
    {
        "video_name": "EJjnEau6aeI",
        "Q": "at 0:14, why is the fraction so simple?",
        "A": "So that it would be easier for viewers to understand. If the fractions were humongous, the audience might complain that their minds are bursting. Remember, some people don t even know how to add fractions or what fractions really mean. Hope this helps!"
    },
    {
        "video_name": "EJjnEau6aeI",
        "Q": "what does it mean at 0:17.",
        "A": "When adding fractions with the same denominator, one only adds the numerators and leaves the denominators the same."
    },
    {
        "video_name": "EJjnEau6aeI",
        "Q": "At 2:28 he changes colors from red to blue WWHHYY",
        "A": "because he is representing different fractions and he is trying his best to avoid confusion"
    },
    {
        "video_name": "9ZYoIliBXYE",
        "Q": "At 1:27, how can you assume that the lines which the vertical transversal passes through are parallel? I know that you claim that they are in the beginning, but what if that information is not given in a problem?",
        "A": "If the lines are parallel, certainly the exercise will tell you."
    },
    {
        "video_name": "MwG6QD352yc",
        "Q": "At 6:35, why didn't Sal need to find out what x equaled? How could he just ignore x^2 and x?",
        "A": "Sal evaluated the function 3x^2 - 4x + 7 for 1. 1^2 is one, and 4(1) is also one. Therefore, Sal simply evaluated 3 x 1 - 4 x 1 + 7 which equates to 3 - 4 + 7 or 6."
    },
    {
        "video_name": "MwG6QD352yc",
        "Q": "At right before 2:16, why is a -1. Is it always -1 or is there a way to know what a is?",
        "A": "There is a way to know what it is! The rational root theorem allows you to generate a list of possible zeroes for a polynomial. Even with tricks to narrow down your options, it can get to be a really long list. An efficient and painless method like this lets you get through them really fast."
    },
    {
        "video_name": "MwG6QD352yc",
        "Q": "At 2:21 they us long division to divide, but in this case wouldn't it be easier to use synthetic division?",
        "A": "Yes, it would be. :P"
    },
    {
        "video_name": "MwG6QD352yc",
        "Q": "at 6:25, he says that if you only cared about the remainder, that would be the easy way to find it. In what case would you only care about the remainder?",
        "A": "If you re doing a maths question that explicitly asks for the remainder and nothing else is a quick example off the top of my head."
    },
    {
        "video_name": "MwG6QD352yc",
        "Q": "3:35-3:41. Did we really get taught long polynomial division in 3rd or 4th grade? That seems earlier than I remember.",
        "A": "You would have been taught long division of numbers in 3rd or 4th grade. The process used to long division of polynomials is the same process used for long division of numbers. You would not have learned long division of polynomials until you learned Algebra."
    },
    {
        "video_name": "yYGf7xn7TyM",
        "Q": "How do I know if (n-1) should be an exponent or not? As you can see in 5:11, (n-1) is not an exponent and it is in fact multiplied with 3.",
        "A": "You need to find the pattern in the sequence. If you are adding/subtracting the same value as the sequence progresses, you have an arithmetic sequence and (n-1) becomes a multiplier like you see at 5:11 in the video. Multiplication is the same as repetitive addition/subtraction. If the pattern you see is that the sequence progresses by multiplying the same value, then it is a geometric sequence and the (n-1) would be an exponent. Exponents are used to represent repetitive multiplication. Hope this helps."
    },
    {
        "video_name": "DLno3b-Xsmo",
        "Q": "At 1:06 he has the answer 9 hours at the top, and the remainder is 9 minutes at the bottom. Why didn't he just stop right there and say the answer is \"9 hours 9 minutes\"?",
        "A": "Because the question states: How many hours are in 549 minutes. The answer is only correct if it is expressed as a decimal or fraction. So the answer could either be 9 9/60 => 9 3/20 or 9.15."
    },
    {
        "video_name": "DLno3b-Xsmo",
        "Q": "At 1:06 he has the answer 9 hours at the top, and the remainder is 9 minutes at the bottom. Why didn't he just stop right there and say the answer is \"9 hours 9 minutes\"?",
        "A": "Because he wants us to understand the concept"
    },
    {
        "video_name": "w9jEq6dmqPg",
        "Q": "What if you just put any letter with an arrow over it to designate the direction? Would you even need the second point like he says you do in 1:55 and 3:51? Or could you just create a point for the purpose of having a direction?",
        "A": "You need two points - one to define where you start, and one to indicate what direction you are going. If you just put any letter with an arrow over it, you have an infinite number of possible rays."
    },
    {
        "video_name": "w9jEq6dmqPg",
        "Q": "At 4:18, how come Sal writes FE/FC? Can't I just state FE or FC?",
        "A": "You can, but Sal wants to point out all the possibilities."
    },
    {
        "video_name": "w9jEq6dmqPg",
        "Q": "Well, when Sal was talking about ray FE in 4:00, I think ray FE is just a line segment",
        "A": "A ray has a starting point, a 2nd point, and an arrow indicating that it continues. FE meets this definition. F is the starting point. E is the 2nd point and the line continues beyond point E. This makes it a ray. AE would be a line segment. It starts at point E, ends at point A and no line extends beyond point E. Hope this helps."
    },
    {
        "video_name": "w9jEq6dmqPg",
        "Q": "At 4:35 couldn't he have said line FC also ?",
        "A": "Yes but the problem asked to identify rays only, not lines :)"
    },
    {
        "video_name": "w9jEq6dmqPg",
        "Q": "At 2:34 you said CF is a ray. But since in both directions it is expanded shouldn't it be a line.",
        "A": "this really help me"
    },
    {
        "video_name": "w9jEq6dmqPg",
        "Q": "at 3:28, why isn't CE a line?",
        "A": "I believe it is because it doesn t count as a line if there are multiple points (more than two that is) on the line? That was my impression anyway."
    },
    {
        "video_name": "w9jEq6dmqPg",
        "Q": "At 3:32, can we conclude that ray EF is the same length as ray CF?",
        "A": "Rays do not have finite length as by definition they keep going straight on their indicated direction on to infinity. It is only a line segment that does so. Therefore, both EF and CF have infinite length"
    },
    {
        "video_name": "w9jEq6dmqPg",
        "Q": "At 2:33, Ray CF has the E in it, would it still count as a ray even though there are 3 points in the \"ray\"?",
        "A": "A ray has infinite points. So that there are three specific points shouldn t be a problem."
    },
    {
        "video_name": "w9jEq6dmqPg",
        "Q": "At 0:20; why does a ray need a point on it for it to be a ray? Doesn't it just need an endpoint and then go on forever in one direction?",
        "A": "I think that if you just had one point then you couldn t know what direction the ray was going. like the sun is the end point but it puts out rays everywhere, so when you have two points you can know in what direction a single ray is going. This is just how i think of it. I don t know if it ll make sense to anyone else :-/"
    },
    {
        "video_name": "w9jEq6dmqPg",
        "Q": "At 3:03, could you also write ray EC instead of CE? Because it is going from left to right, and the ray starts with E on the left.",
        "A": "Yes, Sal did add ray EC but it is different from ray CE because the first letter of the ray is always the endpoint and the second letter always tells in which direction the ray goes on infinitely."
    },
    {
        "video_name": "w9jEq6dmqPg",
        "Q": "At 0:34, Sal says XY is a ray, but wouldn't it be a line segment?",
        "A": "A ray is a line with one segment. One side of the ray goes on forever while the other stay in place. The arrow you saw next to the x represents the fact that x is the part that goes on forever while the dot you saw next to y represents that that is the endpoint of that side. Therefore, it is a ray."
    },
    {
        "video_name": "LrTn4cvsewk",
        "Q": "At 0:22, The line cutting through the shape doesn't make it symmetrical. Isn't it supposed to be reflective?",
        "A": "if (x < 0) { return; }"
    },
    {
        "video_name": "LrTn4cvsewk",
        "Q": "I don't understand what reflectional symmetry is. I know it is the same as line symmetry. Sal talks about reflections at 1:05, but is that a type of symmetry?",
        "A": "They re synonymous."
    },
    {
        "video_name": "LrTn4cvsewk",
        "Q": "At 0:16 isn't the polygon actually a heptagon because it has seven sides?",
        "A": "Yes, but it is also still a polygon"
    },
    {
        "video_name": "LrTn4cvsewk",
        "Q": "At 2:09 Sal just eyeballed the line of symmetry. So, can I just estimate the line of symmetry? Or is there a better way to determine the line of symmetry?",
        "A": "That depends on the context you re working in. Oftentimes, for symmetry, it s enough to just know that the line of symmetry exists. We know that it s halfway between the endpoints of a shape. If we re working in a coordinate system, we can apply the distance formula to find the equation for a line of symmetry."
    },
    {
        "video_name": "LrTn4cvsewk",
        "Q": "so, to check for a line of symmetry, should the line of symmetry must be perpendicular to the blue line? For example, at 1:00 of the heptagon",
        "A": "The blue line is the proposed line of symmetry. Sal is using the greenish dotted line to check that points on the top fold over exactly to match points below. And yes, I suppose the green dotted line would be perpendicular."
    },
    {
        "video_name": "oO3nWnJppqg",
        "Q": "At 4:15, Sal says that x^2=-4 does not have a solution. However, couldn't you say that x=(plus or minus)(sqrt(-4)), which can be simplified to x= (plus or minus) 2i, where i is the square root of -1?",
        "A": "That s true, but the x-axis only has real numbers on it, so we are only concerned about real values of x and y that satisfy the equation."
    },
    {
        "video_name": "jb8mFpA1YI8",
        "Q": "At 0:33 isn't it harder to say 3x6x10, than to say 3x60?",
        "A": "He s explaining a way to think about it to simplify larger and larger multiplication problems. For example 9 x 800 = 9 x 8 x 10 x 10 = 72 x 10 x 10 = 720 x 10 = 7200. Yes it s a bit more complicated but it explains why all you need to do is just take off the zeros, do the multiplication, and then re-add the zeros. Once you get what s going on, you can follow why that s the easy way to do it."
    },
    {
        "video_name": "jb8mFpA1YI8",
        "Q": "1:49 why didn't you just multiply the thing then just working it out the long long long way!?",
        "A": "He was just showing how the long way does technically work, and then just after he shows the easier way. He is just making sure that we are aware of the different possible methods."
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "At 8:25, the person describes multiplying 10^3 then 10^-3.\nIn a simpler method, can I just cancel them out?",
        "A": "He is trying to show how scientific notation would work in that problem. Also ,are you Kevin from my Math class"
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "At 11:14 Sal said 0.000029 2.5 x 10^-5. Did he really mean 2.5 or did he mean 2.9?",
        "A": "I think he meant 2.9-unless he meant to say something else and didn t. Someone else had the same question more recently and I answered it in greater detail."
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "at 11:19 did sal mean 2.5 or 2.9",
        "A": "Yeah, he meant 2.9"
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "at 2:10 I don't do not understand how I know how much I have to switch the decimal point. basically don't understand where to put the decimal point at? ..... please help?",
        "A": "Examples: 7.4 x 10^4 = 74,000 = You would move the decimal to the right 4 places 7.4 x 10^9 = 7,400,000,000 = You wold move the decimal to the right 9 places 7.4 x 10^-4 = .00074 = You would move the decimal to the left 4 places 7.4 x 10^-9 = .0000000074 = You would move the decimal to the left 9 places"
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "at 1:56 sal has a 10 with a power of two. why?",
        "A": "because that is the number there in scientific notation, there is always 10 to a certain power in scientific notation."
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "At 11:19 shouldn't it be 2.9 times 10 (to the negative fifth) not 2.5?",
        "A": "yes you are correct, but Sal is not a robot and therefore he messes up in his videos. there is usually something that tells you when Sal made a mistake."
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "in 5:07 i do not get why he put the decimal there in the 1.2",
        "A": "He didn t just randomly choose that spot. With scientific notatian you are trying to have a number expressed as a number from 1-9.99 x 10e. So he put the decimal there because that is where it would go to make the number between 1 and 9.99."
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "wait, sal said 2.5 but isn't It supposed to be 2.9 at 11:15",
        "A": "Yes, it is suppose to be 2.9, not 2.5. Thankfully, someone has already pointed that out, and there is a correction box in the video around that time in the bottom right corner. Mind you, if you put the video in full screen, you will not be able to see the correction box pop up at that time. Any further questions?"
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "At 4:15, why can't he just say \".00175\" instead of \"0.00175\" ?",
        "A": "They mean the same thing, so either is correct. This is mainly an issue of personal preference."
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "At 8:30, how would that work? They are multiplying * 10^3 then * 10^-3, also 10:57",
        "A": "x^\u00c2\u00b1a*x^\u00c2\u00b1b = x^(\u00c2\u00b1a \u00c2\u00b1 b) 10^3*10^(-3) = 10^(3 - 3) = 10^0 = 1"
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "At 7:19ish why would the leading digit be the 2 and not the 1??",
        "A": "The leading digit would be 2 because basically: x.zz \u00e2\u0080\u00a2 10^z x (leading digit) should be: 1 \u00e2\u0089\u00a5 x < 10."
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "I didn't understand that why did you multiplied 0.000029 with 10^5 and then with 10^-5 at 10:47",
        "A": "He was attempting to explain that to write 0.000029 in scientific notation, shown here 2.9 * 10^-5 . You have to multiply 0.000029 by 10^5 to get 2.9, like this 0.000029 * 10^5 = 2.9 . Then to make it the same number only in scientific notation you multiply by 10^-5, like this 2.9 * 10^-5 . Now Sal here explained poorly and made a simple mistake by putting 2.5*10^-5 at the very end, lol. Hope this helps. ^_^"
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "At about 7:00, we deal with a problem that leads to a result of 2.81*10^-3. However, for some reason, a 10^3 winds up in Sal's explanation, even though that moves the decimal in the other direction. He then says that to cancel that, we multiply it by 10^-3, but wouldn't that cancel out the 10^-3 as well? My point is that the detour only confused me, and I'm still trying to understand the logic behind it. Can someone please explain what was going on there?",
        "A": "Yes, if you are getting the correct answer just by counting how many places the decimal moves you could ignore that extra step. The point of it is that *10^3*10^-3 doesn t change the value of the expression. Multiplying by 10^3 moves the decimal to the right and then multiplying by 10^-3 keeps the value of the expression the same, but remains part of the scientific notation."
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "At 5:10, when he is doing the example with 120,000, can't it be written like this?\n\n12 * 10^4\n\nit is much easier to write for me and is also correct. So why did Sal write this?\n\n1.2 * 10^5\n\nI do not understand, and hope someone will clarify.\nSincerely,\nHer Enderness, the great Endercheez:)",
        "A": "Yes, both of these are correct, however 1.2*10^5 is in the most simplified form for traditional convention purposes."
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "At 11:14 doesn't he mean 2.9 x 10 to the negative 5?",
        "A": "Yes he does. It is just a mistake that he probably made. You were right, it is 2.9 x 10 to the negative 5. Just ignore that small typo."
    },
    {
        "video_name": "i6lfVUp5RW8",
        "Q": "Is this truly Algebra 1?! Sal said at 0:12",
        "A": "No, this is pre-Algebra. He said he is using an Algebra book for examples, but most Algebra books will also include some pre-Algebra as refresher material."
    },
    {
        "video_name": "q20v5ZBsTk8",
        "Q": "At about the 4:00 mark, Sal writes that the equation is approximately equal to 1.35 times 1.051^t. but shouldn't it be equal to 1.35^5 times 1.05^t? It doesn't make a difference to the answer for the question, but it would make a difference to actual calculations.",
        "A": "Yup. Sal made an error."
    },
    {
        "video_name": "q20v5ZBsTk8",
        "Q": "At 3:51, I repeated the last part many times yet I still don't get it, I just don't understand how 5% is the answer",
        "A": "The common ratio is 1.051, which means that every time t is increased by 1, M(t) is multiplied by 1.051. Since 1.051 is larger than 0, M(t) will be increasing. 1.051 = 105.1%, and since anything multiplied by 100% is itself, M(t) will increase by 5.1% (105.1-100) every day. Hope this helps!"
    },
    {
        "video_name": "q20v5ZBsTk8",
        "Q": "At 3:52, Did Sal forget to write 1.35^5? Or was that on purpose?",
        "A": "It was a accident, and a clarification has been made to notify students. ;)"
    },
    {
        "video_name": "jNUz0P5MG9M",
        "Q": "I am confused how will the the answer be 5:3 when i did it on my paper i got 3:5",
        "A": "I have to say, I think the exercise in the video is unclear as it doesn t specify the ration you need to test. Sal did the ratio of tomatoes to servings. You did the ratio of servings to tomatoes. You got the right numbers which is great. Given the vague instructions, I would say you have a good ratio. If the exercise explicitly tells you the ratio, follow the sequence of the words to place your numbers in the right position. Hope this helps."
    },
    {
        "video_name": "jNUz0P5MG9M",
        "Q": "I don't what she said at 0:40",
        "A": "it s if you divide by a same number, example: 10 divided by 2 and 6, diveded by 2 they get 5 to 3. example: 15 divided by 3, 9 divided by 3 they also get 5-3 example 25 divided by 5, 15 divided by 3, they ALSO get 5-3 so, everything you divide is the same so the table IS porportional"
    },
    {
        "video_name": "CA4S7S-3Lg4",
        "Q": "At 3:53, why does he do a dotted line? Is a dotted line for greater or less than, and a solid line for greater than/less than or equal to?",
        "A": "That is correct, a dotted line is for greater than, and less than. A solid line is for greater then than or equal to, and less than or equal to."
    },
    {
        "video_name": "CA4S7S-3Lg4",
        "Q": "At 0:42. what does the boundary line mean?",
        "A": "The boundary line can be thought of as the fence surrounding a garden. One can plant inside the fence, but cannot plant on the fence. The boundary line is just like that, as you can shade on the side that is supposed to be shaded, but the line is not included in the shading. That is what Sal means at 0:42 in the video"
    },
    {
        "video_name": "CA4S7S-3Lg4",
        "Q": "At 1:10 why is the slope of one(1)?\nPlease answer.\nThank you.",
        "A": "The slope-intercept form of an equation is y=mx+b, correct? In this equation, there is no coefficient in front of the x, so m (the slope) is 1."
    },
    {
        "video_name": "CA4S7S-3Lg4",
        "Q": "At about 1:20-Sal says that if y is = to zero, x is = to 8? Is that because you would add a positive 8 to both sides and you are left with 8 = x? Thanks.",
        "A": "Yes, 0 = x - 8 Add 8 8 = x or 0 = x - 8 Subtract x -x = -8 Multiply by -1 x = 8 Hope that helps!"
    },
    {
        "video_name": "ysdY1iX_XCs",
        "Q": "3:22 Couldn't the answer be he can water 8 plants and then he has a little bit of extra water left to partially water a 9th plant?",
        "A": "Yes, in the real world that s exactly what we d do, but the question was asking for how many plants can be watered. That is, how many whole plants can be watered. If the question was adjusted we could give the answer as 8 4/7."
    },
    {
        "video_name": "ysdY1iX_XCs",
        "Q": "at around 3:41 Sal converts 0.7 into 7/10 then while dividing 6/0.7 flips 7/10 to 10/7 & multiplies is there a name for this process?",
        "A": "He is just using the standard steps for dividing fractions. There is no special name for it."
    },
    {
        "video_name": "ysdY1iX_XCs",
        "Q": "2:38 Why is 6 over 7/10 the same as 6 times 10/7 ?",
        "A": "6 over 7/10 is the same as 6 divided by 7/10 Change 6 into a fraction: 6/1 divided by 7/10 Flip the 2nd fraction: 6/1 * 10?7 = 60/7 If you don t know or have forgotten how to divide fractions, I suggest you review the lessons on that topic. Go to Subject: Arithmetic; Pick fractions, then division. Hope this helps."
    },
    {
        "video_name": "BLQNL_UGONg",
        "Q": "2:00 i dont get it...",
        "A": "Hi Kevin, Are you asking about angle B? Recall the sum of the angles in a triangle is 180 degrees. Also observe that (for now) these trig problems involve a right triangle. If A = theta it follows that B = 90 - theta. If I missed your point please leave a comment below. One more thing. Please search cofunction identity and observe how the 90 - theta fits in the equations. Regards, APD"
    },
    {
        "video_name": "bQ-KR3clFgs",
        "Q": "at 4:40, what are these other conceptual ways of undefined?",
        "A": "\u00e2\u0088\u009e/\u00e2\u0088\u009e is undefined because it does not say which infinities are involved. Any place the rules of math do not always get the same answer is called undefined because it does not fall within the definitions of the rules."
    },
    {
        "video_name": "bQ-KR3clFgs",
        "Q": "At 3:33,\npos/neg=neg\nneg/pos=neg\nneg/neg=pos\npos/pos=pos\nI really don't get why is there is the same sign, it will always be positive, while when there are different signs, it will be a negative, is that ALWAYS the way?",
        "A": "Think of it this way, if a negative number is the opposite of a positive number so it changes a positive to its opposite, a negative. So a negative can do the same to another negative and turn it into a positive."
    },
    {
        "video_name": "bQ-KR3clFgs",
        "Q": "at 4:34, it says that if anything (that's not 0) is divided by 0, it's undefined. if anything that's not 0 is MULTIPLIED by 0, then it's 0. why is that?",
        "A": "You have the division part wrong. He says anything divided by 0 is undefined. What you said, 0 divided by anything, is 0. Just like anything times 0 is 0."
    },
    {
        "video_name": "bQ-KR3clFgs",
        "Q": "at 1:43 how can a positive divide a positive",
        "A": "You can always divide a positive by a positive. like 8 / 2, you should have learned this in 2nd, 3rd, or 4th grade, is 4. 8 and 2 are both poritive."
    },
    {
        "video_name": "bQ-KR3clFgs",
        "Q": "How come at 0:50, Sal wrote -16 over 4 as a division problem? is that another way to write a division problem?",
        "A": "Yes. -16 over 4 is the same as -16 / 4 = -16 \u00c3\u00b7 4 = -4"
    },
    {
        "video_name": "bQ-KR3clFgs",
        "Q": "at 4:29 why is zero undefined",
        "A": "In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a (assuming a\u00e2\u0089\u00a00), and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 also has no defined value; when it is the form of a limit, it is an indeterminate form"
    },
    {
        "video_name": "bQ-KR3clFgs",
        "Q": "at 3:41 how did you get _7",
        "A": "It s just part of the initial problem. Sal wrote out a new problem of: -2 * -7 * -1"
    },
    {
        "video_name": "bQ-KR3clFgs",
        "Q": "At 4:50, why can't 0/0 just be 0, since everything divided by 0 is 0?",
        "A": "0/0 is undefined if 0 is in the denominator the answer is undefined! The same thing applies with 0/0 I hope this helps!"
    },
    {
        "video_name": "bQ-KR3clFgs",
        "Q": "At 2:23, Sal says that in future algebra, we will use dots to represent a multiplication sign so it wouldn't be confused as a variable of \"x\". In algebra, \"x\" isn't the only variable, there is \"a\", \"b\", \"c\", or \"t\". Because in algebra there will also be addition signs, wouldn't the variable \"t\" be confused as the addition sign?",
        "A": "Since x xylophone and 5 x 3 look identically the same and t eam and 5 + 3 don t. I believe that the amount of confusion is the matter here. x => Very confusing t => t and + look different, so less confusing. What s more, the variable x is used far more often than the variable t for writing expressions in mathematics."
    },
    {
        "video_name": "bQ-KR3clFgs",
        "Q": "At 4:43 Sal says that there are still arguements for what you get when you divide by zero.\nMy question is How can there even be arguements on \"how to divide by zero\". It's just all confusing !",
        "A": "Anything divided by zero is UNDEFINED."
    },
    {
        "video_name": "bQ-KR3clFgs",
        "Q": "At 1:21 I don't understand how negatives cancel each other out. Can someone help me understand how? I know it does I just don't understand how.",
        "A": "let say you tell your dog to not not eat. That is an example of two negatives cancelling each other out (each negative is referring to not ). So you are trying to tell your dog to eat, which is positive."
    },
    {
        "video_name": "bQ-KR3clFgs",
        "Q": "Around 4:40 it says zero can make undefined numbers when divided, but how come someone didn't invent that years ago? Seriously! \u00f0\u009f\u0099\u0080",
        "A": "If you divide a number by zero, there will be no answer. The problem is impossible to do. If you were dividing 15/3, you would find that number to multiply with 3 that will = 15. The answer would be 5 because 5*3 = 15. Now, do the same problem dividing by zero. 15/ 0: You need to find a number to multiply with zero that will = 15. The number does not exist. Every number you multiply with zero will = 0. You can never create the 15. This is why division by zero is undefined."
    },
    {
        "video_name": "n-iEqLhGfd4",
        "Q": "at 3:00 how do we identify when we should do integration pay parts twice?",
        "A": "You know that when you ve integrated by parts once and it hasn t fully simplified the problem."
    },
    {
        "video_name": "n-iEqLhGfd4",
        "Q": "4:04 is a mistake I guess! he converted f'(x) to f(x)!\nshouldn't f'(x)=x? the anti-derivative of it is equal to x^2/2 or you can manipulate the functions when you wish?",
        "A": "No, I don t see a mistake. But it s true that with integration by parts you always can choose which part you want to be f(x) and which part is g (x). And so you could make: g (x) = x g(x) = x^2/x exactly like you said. The only problem with this approach is that it makes the integral harder and doesn t lead to something you can easily integrate. If you do it Sal s way you end up with the integral of e^x which is just e^x."
    },
    {
        "video_name": "n-iEqLhGfd4",
        "Q": "@2:57, you don't necessarily have to move the '2' outside the integral, do you? even if you leave it in, you'll get the same answer; just wondering if this is a coincidence with this example, where it worked out that way..",
        "A": "As with differentiation, you can pull out constants without changing the value."
    },
    {
        "video_name": "n-iEqLhGfd4",
        "Q": "At 0:36 Sal said that taking the anti derivative of x squared gives 2x which might be a slip of tongue",
        "A": "Yep, and four whole seconds later a box pops up to that effect. Sal said antiderivative of x^2 but meant derivative of x^2"
    },
    {
        "video_name": "3Xcae0OGavk",
        "Q": "At 0:10 he said \"and\". Isn't \"and\" only said when there is a decimal in a number?",
        "A": "That s what I was taught. The correct (I guess) way to say it is without the and unless you re specifying a decimal, but you will hear people pronounce it incorrectly all the time, so you will have to be the judge."
    },
    {
        "video_name": "3z-M6sbGIZ0",
        "Q": "Is there a specific video for e, or for the limit approching infinity, that Sal mentions at 9:10?",
        "A": "Check out the Precalculus section of exponential and logarithmic functions. Go to the videos on compound interest and study those. He explains how to derive e. it helps to use a calculator from your end to follow sals logic"
    },
    {
        "video_name": "3z-M6sbGIZ0",
        "Q": "at 07:26, is the \"e\" natural logarithm?",
        "A": "Natural logarithms use e as their base"
    },
    {
        "video_name": "3z-M6sbGIZ0",
        "Q": "Around 3:30: I'm understanding the equation to be stating, let the expected number of cars to pass in a given hour be equal to 60 min/hr (n) * (lambda/60)(p). Isn't the assumption here that lambda cannot exceed 60, that is, you can't have more than 60 cars pass by in an hour?",
        "A": "Yup! That s why he then says, Well, we can have more than one car a minute, so let s try one car per second. No, wait, let s have one car in an infinitesimally small division of an hour! (That s when we have the limit as x goes to infinity.)"
    },
    {
        "video_name": "gYv8sttBIqs",
        "Q": "At 3:30, Sal mentioned that in the last couple of videos, we learnt that det(Sij)= -det(A). May I know where specifically I can find the proof of this result?",
        "A": "I ve submitted a request for a video on Determinant after swapping two rows in the request video/features link on the right hand side of the page."
    },
    {
        "video_name": "VhtFSSq3lXo",
        "Q": "At 6:15 when Grant starts to look at the partial of Q with respect to y, how is the magnitude of y not changing? It sure looks like it is. Is this just a consequence of Grant evaluating to far away from the original input (x,y) coordinate of (2,0)?",
        "A": "Yes. The derivative is only instantaneously zero."
    },
    {
        "video_name": "95logvV8nXY",
        "Q": "Would it be correct to say that the sum of the interior angles of a polygon (assuming that you can divide it into triangles as he does at 3:40) is equal to 180 degrees (the sum of the interior angles of a triangle) times the number of triangles in the polygon?",
        "A": "You guys are both right. It just so happens that you can always make 2 less triangles than the number of sides, that s where 180(n-2) comes from."
    },
    {
        "video_name": "95logvV8nXY",
        "Q": "The video is stoping at 6:34. Please fix it.",
        "A": "i believe that that is suppose to be the end, mine ends at 8:36, because after 6:34, it just does non needed letters, and it cancels it out at the end"
    },
    {
        "video_name": "95logvV8nXY",
        "Q": "at 2:30 sal said \"i am factoring out negative signs \" what did he mean by factoring out",
        "A": "If you have something like: (-x - y) We can see that both variables have a coefficient of -1. We can factor out that -1 and put it outside the parenthesis: (-x - y) = -(x + y) Notice, if we distribute the negative on the right equation, we get back to where we started on the left."
    },
    {
        "video_name": "95logvV8nXY",
        "Q": "At 1:46, Sal said that \"We can write 180-h because angles b and h are supplementary. I see that, but what about the angles that aren't supplementary?",
        "A": "Ah, we re looking for some known number to use in our equation...like, if the angles were complementary, we d know that they sum to 90 degrees. Does that help/make sense?"
    },
    {
        "video_name": "95logvV8nXY",
        "Q": "I pause at 3:48 and do this\n3 Triangles X 180 deg = 540\n(and yes, I did check that each angle belonges to only one triangle)\n900 - 540 = 360\n\nOr we can use the formula\n180 (n-2) where n is number of angles\n180 X (5-2) = 540\n900 - 540 = 360",
        "A": "either way works"
    },
    {
        "video_name": "iX7ivCww2ws",
        "Q": "7:45 , hi can any one plz tell me wat does decimal powers mean? for example 2^3 =2.2.2 similarly wat does 2^(1.25) mean/??....how can yo expand it??",
        "A": "2^(1.25) 2^(1 + 0.25) 2^(1)\u00e2\u0080\u00a22^(0.25) 2^(1)\u00e2\u0080\u00a22^(1/4) 2\u00e2\u0080\u00a24thrt(2)"
    },
    {
        "video_name": "iX7ivCww2ws",
        "Q": "At 2:26, Sal says that the root sign \u00e2\u0088\u009a indicates the principal root. Is there a sign that indicates the negative root of a number?",
        "A": "it would be -\u00e2\u0088\u009a, but other than that, there really isnt any sign that indicates the negative root. I hope that helps!"
    },
    {
        "video_name": "iX7ivCww2ws",
        "Q": "At 1:18 Sal Says That 3*3*3 Equals 29 When It Clearly Equals 27 Times 3!",
        "A": "Your calculation are right, but I believe that he is saying 3 * 3 * 3 is 27; 9 * 3, 27"
    },
    {
        "video_name": "iX7ivCww2ws",
        "Q": "I'm confused about 4:30 ... how does 2 times 3 to the 1 fifth power translate into 2 times the fifth root of 3? I'm just very confused about this whole thing.",
        "A": "2 times 3 to the 1 fifth power is the same as 2 times the fifth root of 3. Its just written in a different form. Don t worry about it too much! \u00f0\u009f\u0098\u0080"
    },
    {
        "video_name": "iX7ivCww2ws",
        "Q": "Are there any steps for canceling because i got a little lost at 6:08 seconds in the video since there is no canceling",
        "A": "How???"
    },
    {
        "video_name": "iX7ivCww2ws",
        "Q": "at 5:39, how did you get the 2 to the 6th power onto the other side of the radical",
        "A": "Oh... I see whats up He didn t pull 2^6 (2 to the 6th power) out of the radical and place it in front. He s taking the 6th root, of 2^6 The Sixth root, and the sixth power cancel out and you re left with 2, which is still a number, but now that it s an easy number pull it out of the radical and keep pluggin and a chuggin"
    },
    {
        "video_name": "iX7ivCww2ws",
        "Q": "At 8:23 how did Sal get X to the FOURTH power when he only added X to the FIRST power and X to the ONE-THIRD power?",
        "A": "Good question! The answer is: common denominators. In order to add 1 and 1/3 you need to find the common denominator between them. We can rewrite 1 in rational form as 1/1. Now we have 1/1 + 1/3. The common denominator is 3 so we multiply 1/1 by 3/3 to get 3/3 + 1/3. Now it should be clear, 1 = 1/1 = 3/3 and 3/3 + 1/3 = 4/3, therefore 1 + 1/3 = 4/3."
    },
    {
        "video_name": "iX7ivCww2ws",
        "Q": "At 2:20 ,He told it could also be ' -2 ' .But if we are to take x^1/57, here 57 is odd, so if 'x ' is negative ,our answer is negative.My doubt is what sign we should consider for all the factors ? Should we consider all negative ones or the combination of both - and+? Is there any sort of convention?",
        "A": "Radicals like square root, 4th root, 6th root, etc. have 2 roots: the principal root and the negative root. Unless you are solving a quadratic equation, the sign in front to the radical tells you which root to use. \u00e2\u0088\u009c16 has no sign (so assumed positive). This means you want the principal root = +2 -\u00e2\u0088\u009c16 has a minus sign in front. The minus tells you to return the negative root = -2. Hope this helps."
    },
    {
        "video_name": "iX7ivCww2ws",
        "Q": "At around 5:27, how is 2^6 just 2?",
        "A": "I was not just 2^6, he pulled the 2 out of the radical. So really what he did was (2^6)^(1/6), taking a power to another power is the same as multiplying the powers so (6)*(1/6)=(6/6)=1. Thus the new power is 1 and the number becomes 2^1 or just2."
    },
    {
        "video_name": "tnkPY4UqJ44",
        "Q": "About 3:03, what if instead of 1/2, it was 3/7?",
        "A": "4/(3/7) = 4*(7/3) = 28/3 = 9 1/3"
    },
    {
        "video_name": "tnkPY4UqJ44",
        "Q": "At 0:30, Sal said that you take 1/2, and change it to it's reciprocal. How come he didn't change 3/5 to it's reciprocal? How do you know which fraction to change?",
        "A": "The second one. Always flip the second fraction."
    },
    {
        "video_name": "tnkPY4UqJ44",
        "Q": "5 divided by 6 that's what it said at 1:05 I thought 6 divided by 5 now I am just mixed out.",
        "A": "how do you divide fractions david i really need help"
    },
    {
        "video_name": "tnkPY4UqJ44",
        "Q": "Exactly how are they in two groups?3:10",
        "A": "because each group is split into two grops.There is four in the group so to make to grops you split it in to two."
    },
    {
        "video_name": "tnkPY4UqJ44",
        "Q": "at 0:22 can you just cross multiply?",
        "A": "Yes, with any fractions, and proportions. With this example: 3/5*1/2 you put 3*2 on top and 5*1 on bottom. Doing this is basically dividing fractions."
    },
    {
        "video_name": "tnkPY4UqJ44",
        "Q": "at 1:35 what was the mixed number",
        "A": "A mixed number is a number where a whole number and a fraction in it, for example... 1 2\u0003 356 50/4532 5 3/9 72 11/12"
    },
    {
        "video_name": "tnkPY4UqJ44",
        "Q": "3:02 helps a lot. Thanks for the diagrams, Sal!",
        "A": "no problem thanks"
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "at 0:34 i thought you were suppose to divide not multiply",
        "A": "Imani, Dividing by -5 and multiplying by -1/5 is the same thing. Multiplying by the reciprocal always works whether the coefficient with the x is an integer or a fraction. If you remember to multiply by the reciprocal, it always works and you don t have to think about when to divide and when to multiply. I hope that helps."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 1:05, aren't you supposed to Divide both sides by -5? Why is he multiplying both sides by 1/5?",
        "A": "Dividing by 5 is the same thing as multiplying by 1/5."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 01:31 how did that person got a -3 ?",
        "A": "He was just simplifying the equation. From -1/5*-5c >= 15*-1/5, it simplifies down to c is greater than or equal to -3. The -1/5*-5 cancels out for c and 15/1 * -1/5 equals -3. (-15/5 = -3.)"
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 0:53, Sal says that if you are multiplying an inequality by a negative number, you need to flip the sign around. but when i was doing this problem\n10x <_ -17 (do you get it? i couldn't find a way to make a sign)\ni flipped the sign around and i got it wrong. so now i am sort of confused about when to flip. Could you please tell me when the correct time to flip the inequality is and give an example?",
        "A": "When you multiply or divide by a negative number, you need to flip the sign. In that problem, you aren t dividing by negative, you re dividing by 10, to get rid of the coefficient on the x. For instance, if the problem was: -10x <_ 17 then you would divide by negative 10 and flip the sign."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 0:30, isn't it supposed to be reciprocal, not inverse? Isn't 5 the inverse of -5? If it were -5x, the inverse would be -x/5. But Sal is not talking about functions here, so in this case wouldn't inverse mean the opposite?",
        "A": "I think you are right it is a reciprocal i think it was an error. like you said the inverse of -5 is 5 so you are approximately right"
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "can you explain, 0:17 through 0:38",
        "A": "He just says that -5 divided by -5 is 1 in a bit of a complicated way."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "at 0:34 i was sure you divide not multiply",
        "A": "To solve this type of problem, you can divide by -5 or you can multiply by the reciprocal of -5 = -1/5. They both achieve the same result. So, go ahead and use division."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 0:52, Sal tells you to multiply both sides by -1/5. Can't you just divide both sides by -5 instead?",
        "A": "Yes, jgerald, you could do that. Sal just chose do multiply by -1/5, the inverse of -5. Dividing by -5 would be just as valid and right."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "in 0:32 where does the 1/5 comes from",
        "A": "It comes out of nowhere. But as long as you do the same thing to both sides of an equation/inequality (i.e. in this example, multiplying -1/5 to both sides) then it would still be balanced and the equation still valid."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "Why is it at 1:05 Sal says that dividing it by 1/5 is equal to dividing it by -5?",
        "A": "Yes, if you have learn exponents you should known these kind of sums"
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 0:37, aren't you supposed to divide instead of multiplication.",
        "A": "dividing by -5 and multiplying by - 1/5 are the same thing. I always just divide. Make sure you flip the inequality when you multiply or divide by a negative number."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 1:48, why is the circle bubbled in?",
        "A": "The circle is bubbled in because it is greater then or equal to. It would also be bubbled in if it were vice-versa; less than or equal to. So basically to help you remember, bubble it in when there is a line under the value sign. :P"
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 2:24 th inequality was switched around,why? Is it because c=1 which is greater than -3?",
        "A": "It s because if the first number you divide is a negative, you have to switch the sign. It s the rule. If you don t do it, your answer would be incorrect."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 1:44 can't you just divide instead of multiplying by the inverse?",
        "A": "The two operations are equivalent."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 00:36 why not just divide by -5 instead of going through the additional work of making a fraction?",
        "A": "Well Manav, some people interpret and learn things differently. So while some people may see that you can divide by -5 right away, but others may need to simplify the equation in order to understand the concept."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 0:35 , why we didn't dvide it over 5 or we didn't multiply it with -5 ?",
        "A": "Dividing by five is the same as multiplying by 1/5. Think about this; if I divide something by five, I am finding one fifth (1/5) of that something."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "at 0:39, is putting a negative sign on both the top and the bottom of a fraction different from putting it on just one number, and if so, why?",
        "A": "evanhebert03, Yes, it is different. For example, -1/3 is actually -1 divided by 3. -1/-3 is actually -1 divided by -3. And by rules of a negative divided by a negative, it would equal a positive. So -1/3 is a negative fraction while -1/-3 is a positive fraction."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 0:28 it mentions that -1/5 is the inverse of -5. What is the inverse and how do you figure it out?",
        "A": "He is finding the recipricol (multiplicative inverse ) of -5. To find the recipricol, flip the numerator (top of fraction) with the denominator (bottom of fraction). Since -5 is equal to -5/1, the multiplicative inverse is 1/(-5), or -1/5."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "When you talked about solving and inequality with multiplication at 0:20 what about division? Is that the same, or do you have to divide the inverse by the inequality?",
        "A": "The procedure is exactly the same as isolating an equation. If you have c/8, you multiply through by 8. The only difference is if you multiply or divide by a negative number, you have to flip the inequality around."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 0:35, to isolate the C, why can't you just divide both sides by 5, instead of multiplying both sides by the inverse?",
        "A": "Either way works fine. Multiplying by the 1/5 and dividing by 5 have the same effect. You ve got it!"
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 0:40,im confused why you are suppose to multiply because my teacher told me to divide it by the first number which in this case is -5 so please helpppppp",
        "A": "You can do either one. Multiplying by -1/5 creates the same result as dividing by -5. So, pick the method that you are most comfortable using."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "at 2:30, when 20 is compared to the inequality do we use the original or the flipped inequality. just a bit confusing because if you substitute -4 to make: -5(-4) \u00e2\u0089\u00a4 15 .... 20 \u00e2\u0089\u00a4 15 .... 20 is greater than 15 which is 20 \u00e2\u0089\u00a5 15...just not sure how to use the solution",
        "A": "never mind. i missed the statement that -4 doesn t work."
    },
    {
        "video_name": "D1cKk48kz-E",
        "Q": "At 0:43, how did you get -1/5?",
        "A": "An inverse is taking a fraction (in this case -5/1) and flipping it, thus -1/5."
    },
    {
        "video_name": "TX_mx3qULpw",
        "Q": "at 6:42 why sal is pertaining to x-intercepts when he is talking about zeros of the function?",
        "A": "A zero of a function is the value of x that makes the function value zero. All points with a function value of zero lie on the X-axis. In other words, the function intercepts the X-axis at those points."
    },
    {
        "video_name": "TX_mx3qULpw",
        "Q": "What exactly is a removable discontinuity? I see it mentioned at various times such as at 2:55.",
        "A": "To be accurate, a removable discontinuity is one that can be removed by redefining a finite number of points. Most of the time the only removable discontinuity you need to worry about is a hole in the graph."
    },
    {
        "video_name": "TX_mx3qULpw",
        "Q": "How does Sal pick g(x)=(x+3)(x+1) at 5:25?",
        "A": "g(x) can be any function. He s choosing factors for g(x) that cancel out factors in the numerator."
    },
    {
        "video_name": "EX0Ha42WQ24",
        "Q": "At 0:25, what is the point of writing the vector as (-2i - 4j)? Because writing it as (-2, -4) is the same thing, except without the useless letters...?",
        "A": "Why write a number such as 345 as 3.45 x 10\u00c2\u00b2? The latter is scientific notation - it has its place. Why write a vector, such as (2, 4) as 2i + 4j? The latter is engineering notation - it has its place. Some conventions may seem pointless to you now, but if you ever get into the areas they are used, they will make total sense."
    },
    {
        "video_name": "sCVyvfOLI6U",
        "Q": "At 1:31 you didn't add the remainder of the decimals into the decimal answer, so you wouldn't have to add the zeros to the decimal all the time or just in this equation?",
        "A": "Well it depends on this problem\u00f0\u009f\u0098\u008d\u00f0\u009f\u0098\u008d\u00f0\u009f\u0098\u008d\u00f0\u009f\u0098\u008d"
    },
    {
        "video_name": "sCVyvfOLI6U",
        "Q": "00:17 Isn't he dividing 11 INTO 25? This doesn't sound right to me",
        "A": "The reason, Joe, that he did that is because he needs a decimal instead of a whole number. Hope that helps! :) Raeshelle"
    },
    {
        "video_name": "4PlkCiEXBQI",
        "Q": "What does \"compute\" mean at 2:08??",
        "A": "It means to calculate something"
    },
    {
        "video_name": "4PlkCiEXBQI",
        "Q": "Can someone explain why Sal is cutting the boxes into halves at 1:33?",
        "A": "He was showing the 5 boxes and how the 5 was being multiplied by 1/2. He then showed how many halves there were by numbering each one from 1-5."
    },
    {
        "video_name": "4PlkCiEXBQI",
        "Q": "0:48 where did you come up with the three",
        "A": "The problem in the video only uses 1/2 and 5. At 0:48 he s simply counting each portion up to 5."
    },
    {
        "video_name": "4PlkCiEXBQI",
        "Q": "At 2:30, why don't you find the lowest common denominator similar to adding fractions?",
        "A": "To multiply and divide with fractions, you do not need a common denominator. The common denominator is only needed when you add & subtract fractions."
    },
    {
        "video_name": "4PlkCiEXBQI",
        "Q": "At 2:59, the denominator is still 2. How can make 5/2 a proper fraction or a mixed number?",
        "A": "you divide 5 by 2 2 goes into five twice with 1 left over therefore you answer is 2 1/5"
    },
    {
        "video_name": "5r2FYszC_sU",
        "Q": "at 0:29, what does i@#x27 mean? I don't think that Sal actually said those symbols.",
        "A": "He didn t! That s unicode. Your computer should have put ONE symbol there."
    },
    {
        "video_name": "fYQ3GRSu4JU",
        "Q": "At 3:41 shouldn't the terminal side rest on the on the line since 1 radian is a 180 degrees (Straight Line) and 2 radians (Full rotation). And if it has 3 radians shouldnt it be a full rotation and a straight line?",
        "A": "Your conversions are incorrect. 1 radian is about 57.3\u00c2\u00b0. 2 radians is about 114.6\u00c2\u00b0 3 radians is about 171.9\u00c2\u00b0 It is \u00cf\u0080 radians that equals 180\u00c2\u00b0 2\u00cf\u0080 radians = 360\u00c2\u00b0"
    },
    {
        "video_name": "fYQ3GRSu4JU",
        "Q": "At around 2:35 he says \"pi/2 here would be 3.5 pi over 7\". I have no clue what he means by that.",
        "A": "He s trying to figure out if 2pi/7 is less than or greater than pi/2. By multiplying the numerator and denominator of pi/2 by 3.5, he gets 3.5pi/7. 3.5 is greater than 2, so 3.5pi/7 (or pi/2) is greater than 2pi/7, which means 2pi/7 would not pass pi/2 which puts it in the first quadrant."
    },
    {
        "video_name": "fYQ3GRSu4JU",
        "Q": "2:36 Kindly explain about why pi/2 is would be 3.5pi/7",
        "A": "What s 1/2 - 3.5/7? (Come up with an answer by yourself). 3.5/7 = (2*3.5)/(2*7) = 7/14 = 1/2."
    },
    {
        "video_name": "fYQ3GRSu4JU",
        "Q": "at 1:15 how does sal know that it is pi/2?",
        "A": "A full circle is 360 degrees, or 2pi radians. Half a circle, or 180 degrees, is 1pi radians, or just pi radians. A quarter circle is 90 degrees, or pi/2 radians."
    },
    {
        "video_name": "fYQ3GRSu4JU",
        "Q": "at 1:00 Can someone explain to me how big 1 radian is? This is very confusing so can someone suggest a lesson i should watch to understand this more? what is pi/2 supposed to mean? if the positive X axis is pi/2 then what is the negative Y and negative X?",
        "A": "1 radian represents the central angle needed to produce an arc along a circle with length equal to its radius. It is equal to 180/Pi or about 57.3 degrees. Pi/2 radians is equal to 90 degrees. Pi = 180, 3Pi/2 = 270, and 2Pi = 360"
    },
    {
        "video_name": "D2xYjiL8yyE",
        "Q": "what does all the stuff that she writes at 2:18 mean?",
        "A": "she is showing the pythagoras theorem... and yes, thats what shes saying"
    },
    {
        "video_name": "D2xYjiL8yyE",
        "Q": "Hey, Vi! Are you actually IN Math Class recording these, or do you just use \"oops, teacher's walking 'round, better draw some axes...(Axes, yeah.)\" as a transition between the two, uh, parts, of the video near 4:11...?\n\nP.S. If you are that is really cool, but I somehow doubt it...",
        "A": "Well, she isn t in math class, because if she was there would be a lot of background noise. As you say she is probably saying that the teacher is walking around as a transition between two parts, or to end the video as she does in Doodling in Math Class: triangle party."
    },
    {
        "video_name": "D2xYjiL8yyE",
        "Q": "At 2:28, is tau the same as pi?",
        "A": "No, tau is pi(2)"
    },
    {
        "video_name": "D2xYjiL8yyE",
        "Q": "At 1:47, Vi casually throws out that circles have the most possible area given their perimeter (which is tau times the radius!). Can someone explain...?",
        "A": "Let s say that you have a string and you want to create an enclosed area that has the most area possible without changing the length of the string. To do that, you need to create a circle. Given any constant length of perimeter, the circle with that perimeter has the most area out of all of the two-dimensional figures with that perimeter. I hope this clarifies what Vi Hart said!"
    },
    {
        "video_name": "D2xYjiL8yyE",
        "Q": "At 1:08, how would she ever get a circle if she kept on going? Wouldn't she be able to finish it? How long would it take? And what's up with the hypotonues?",
        "A": "She would approach a circle, but for any finite amount of zigs, she would never get a circle. She would need to do it an infinite amount of times."
    },
    {
        "video_name": "D2xYjiL8yyE",
        "Q": "How can you ziz zag until infinity if the perimeter of the polygon is finite? 3:19 - 3:20",
        "A": "Because you are not zig-zagging a set amount each time, but rather less and less with each zig or zag. In other words: The zig-zags are infinite because the sequence 1/2 + 1/4 + 1/8 + 1/16 + 1/32... never quite reaches 1"
    },
    {
        "video_name": "D2xYjiL8yyE",
        "Q": "0:32 Could this work with any circumference?",
        "A": "Yeah, I guess so."
    },
    {
        "video_name": "D2xYjiL8yyE",
        "Q": "At 0:15 what is that thing she drew??",
        "A": "It s the letter y ."
    },
    {
        "video_name": "D2xYjiL8yyE",
        "Q": "4:21, so it's kind of like a binary tree with a different approach than the doodling in math class doodle game",
        "A": "It s similar. Try doodling both and comparing them. The only major difference is that the binary tree has little squares inside the bigger squares. Great discovery!"
    },
    {
        "video_name": "D2xYjiL8yyE",
        "Q": "at 1:22, what is zigfinity?",
        "A": "/\\/\\/\\/\\/\\/\\/\\/\\/\\/\\/\\ (etc). I made zigfinity! :3"
    },
    {
        "video_name": "v9Evg2tBdRk",
        "Q": "Sal says at 5:48 that there is only one way to get a^2, which makes sense to me, but I count to ways of getting to it on Pascal's triangle; one straight down and one looping around the two. Can someone clarify this?",
        "A": "That looping up thing was just a way to remember the Pascal s triangle"
    },
    {
        "video_name": "v9Evg2tBdRk",
        "Q": "1:32 pm i don't really understand pascals triangle.can i get help.",
        "A": "For the first 2 rows, write 1 on the top and two 1 s on the next level. On the side of the triangle, keep on writing 1. In the middle, you put the sum of the two numbers above it. So it is: 1 11 121 1331 14641... The side is 1, and 1+1=2, 1+2=3, 1+3=4, and 3+3=6."
    },
    {
        "video_name": "v9Evg2tBdRk",
        "Q": "Starting at 3:49, what would happen if the binomial was something like (3a+4b)^4? How would he write the terms in?",
        "A": "Using Pascal s triangle, you would do: 1 (3a)^4 (4b)^0 + 4 (3a)^3 (4b) + 6 (3a)^2 (4b)^2 + 4 (3a) (4b)^3 + 1 (3a)^0 (4b)^4 I ll let you clean up the exponents and simplify. Hope this helps."
    },
    {
        "video_name": "LOf8bfjiLow",
        "Q": "At 3:23, in the question you gave us, you multiplied the matrix (containing elements 1,2,3,4,5,6) with the zero matrix (with dimensions 3x2). Now my question is, cant the zero matrix have its dimensions as nx2? Because the matrix multiplication would still be valid, right? If so, then the matrix product would have dimensions as nx3. Therefore, unlike inverse matrix, which is unique for every matrix, there can be 'n' number of zero matrices for a given matrix. Am I correct?",
        "A": "Yes. For a nxm matrix A, and zero matrix Z, both AZ and ZA are defined for an infinite number of zero matrices."
    },
    {
        "video_name": "T971zHhZ3S4",
        "Q": "at around 1:19 i got lost, could someone explain it a little better please?",
        "A": "Once Sal knows that the triangle is a right triangle, he applies the Pythagorean Therorm (a^2 + b^2 = c^2, where c is the length of the hypotenuse and a and b are the lengths of the other 2 sides) So he squares each of the shorter sides (8 and 15), adds the results together (getting 289), and then takes the square root to find the length of the hypotenuse of the triangle (which is the diameter of the circle) as 17."
    },
    {
        "video_name": "gD7A1LA4jO8",
        "Q": "at 4:10 why did he flip the left side?",
        "A": "What you do to one side you do to the other. He flipped the right to get the answer in terms of T. Because he did that, he had to flip the left as well."
    },
    {
        "video_name": "gD7A1LA4jO8",
        "Q": "at 4:06, why did he flip the left side?",
        "A": "The equation was 15/8=1/t. So he decided to take the inverse 1/t and put it into the right side. But whatever you do to the right side, you have to do it to the left side."
    },
    {
        "video_name": "gD7A1LA4jO8",
        "Q": "at 4:59, why does Sal multiply 60 by 7/8? What happened to the 1 7/8?",
        "A": "The time needed if they both worked together (t) was 1 7/8 HOURS. So Sal kept the 1 hour separate and then converted 7/8 of an hour to minutes by multiplying by 60 min/hr. After he got the answer of 52.5 minutes, he then combined that with the 1 hour to get the final answer of 1 hour 52.5 min."
    },
    {
        "video_name": "uvNkvT6XgBo",
        "Q": "around 7:40 how come the time interval where the amount of water is at a minimum happens to be where the two lines intersect? ,is it because at that time the area between the curves (representing the amount of water inside) is at a minimum?",
        "A": "The points of minimum and maximum are the points where the derivative of the function is equal to 0 (0 raise over any run, or just a flat horizontal line). For more detailed description, I would advise to review Optimization in Differential calculus section on Khan Academy."
    },
    {
        "video_name": "uvNkvT6XgBo",
        "Q": "When you are finding where there is a local minimum (about 8:00), how do you know that the value t=3.272 is a local minimum and not a local maximum? When the derivative of a function is 0 the function can have either a max or min. Wouldn't you have to check to make sure it is indeed a min and not a max?",
        "A": "I feel like that is a point of interest, because I found that a little bit strange as well. One thing to point out is that one item to look at would be in part b, where we found that at t = 3, the graph was decreasing. While this isn t to say that this is always 100 % accurate, you can look at the trend and see that when t = 3.272, the derivative = 0 and the graph was decreasing moments before it became 0, you can reason that the point at t will most likely be equal to the minimum point on the graph."
    },
    {
        "video_name": "uvNkvT6XgBo",
        "Q": "At 10:04, and during the entire rest of the problem, wouldn't it be easier simply to find the intercept between the two functions R(t), D(t)? Is it only a coincidence that it worked for this problem.",
        "A": "I think your question needs a bit more elaboration. Wouldn t it be easier to find what exactly? There are 3 parts to this question and all are slightly different in what they re asking."
    },
    {
        "video_name": "uvNkvT6XgBo",
        "Q": "Around 1:20, Sal uses R(x) - D(x) in the integral rather than R(t)-D(t). Can someone please explain?",
        "A": "As he explains, he as used the variable t for the upper bondary of the integral (W is expressed as a function of time so t is a common choice here), so another variable name has to be chosen for the integration. He s using x, for it s the most conventional choice, but any other letter could do the job."
    },
    {
        "video_name": "96uHMcHWD2E",
        "Q": "At 5:07, if we're just given a function g(x) = x without knowing that it came from x^2/x, the domain is different right? I don't understand why the domain of x2/x is the same as the domain of x when clearly the latter one is just x/1 so it should be all real numbers.",
        "A": "The difference between x and x^2/x is that no matter what, you cannot divide by 0, so the difference in domains is that the first is all real numbers and the second is all real numbers except for 0. This is true despite the fact that x = x^2/x."
    },
    {
        "video_name": "96uHMcHWD2E",
        "Q": "at 2:22, why does it turn out to be squared? I was just thinking x.",
        "A": "It was just an example - he could have chosen any expression with x. He chose x^2 to show that the domain and range values need not be the same. In this case the domain is (-\u00e2\u0088\u009e,\u00e2\u0088\u009e) and the range is [0,\u00e2\u0088\u009e)."
    },
    {
        "video_name": "96uHMcHWD2E",
        "Q": "At 3:48, since the range of f(x) is every non-negative integer, couldn't you say that {f(x) \u00e2\u0088\u0088 W} f(x) is a member of whole numbers?",
        "A": "Sal has {f(x) \u00e2\u0088\u0088 R | f(x) >=0}. This is not the same as non-negative integers. Sal s range is much larger. It contains all of the positive fractions and decimals as well as the positive integers. Your version excludes the positive fractions & decimals. Hope this helps."
    },
    {
        "video_name": "96uHMcHWD2E",
        "Q": "At 2:22 Sal says that the definition is F(x) is going to be equal to x^2. Does that mean that if I have a function notation such as f(x)=x+4 and a given x is 2, do I have to square the 2?",
        "A": "Yes - that is how it works, if you have f(x)=x\u00c2\u00b2 and are asked what is f(2), then you replace every instance of x in the function definition with 2 so given f(x) = x\u00c2\u00b2, that means f(2) = 2\u00c2\u00b2 = 4. Here is another example: If f(x) = x\u00c2\u00b2 + 5x then f(2) = 2\u00c2\u00b2 + (5)(2) = 4 + 10 = 14"
    },
    {
        "video_name": "96uHMcHWD2E",
        "Q": "At 5:23 he says that x cannot be equal to 0. Why is this?",
        "A": "Well the function g(x) is x^2 /x. So if x were 0 then you d have to divide by zero, which would be undefined. Whenever we have a function including a fraction, we always have to make sure that the bottom can t equal zero as the maths doesn t work otherwise."
    },
    {
        "video_name": "96uHMcHWD2E",
        "Q": "Umm, around 5:21 Sal says, X can not be equal to 0. But if X was 0, wouldn't that be 0^2/0 = 0/0 = 1?\n\nOr is there a rule that 0/0 doesn't equal 1? Is it undefined?",
        "A": "0/0 is indeed undefined."
    },
    {
        "video_name": "96uHMcHWD2E",
        "Q": "Why did Sal say that f(x) is not going to be negative at 4:23?",
        "A": "because there is no x that results in x^2 being a negative number. Any positive number times itself is a positive number. Any negative number times itself is a positive number"
    },
    {
        "video_name": "96uHMcHWD2E",
        "Q": "at 3:20 why did he create parabola for x square??",
        "A": "a graph is simply a representation of y for every x.... therefore f(x)=x^2 then for x=2 then y=4, x=10 y=100....in such manner small values of x will give very high values of y....hence the graph cannot be a straight line....it is a a group of infinite points that have high diff in values hence it is a curve. or parabola also it cannot be -ve...hope u got it"
    },
    {
        "video_name": "96uHMcHWD2E",
        "Q": "AT 4:13, It is said that range cannot be less than zero,but in my book there are some range and domains those are negative.",
        "A": "Sal s comment at 4:13 related to the specific function he was talking about, namely, y=x^2. The range / domain of other functions may be all positive, all negative, or combinations of both, depending on the given function."
    },
    {
        "video_name": "96uHMcHWD2E",
        "Q": "At 4:06, can we say that the range is all squares",
        "A": "The function f(x) is defined as f(x)=x^2, which means you can input any value for x not just whole numbers(even irrationals like pi or e!),these numbers,if you square them will give you all sorts of exotic results and not just square numbers,and since any real number squared has to be positive, the range has to be any real number greater than or equal to zero."
    },
    {
        "video_name": "qk69pR91R00",
        "Q": "at 1:11 why is -4 x",
        "A": "So when you have an ordered pair, which represents a point on a graph, it is shown as (x, y), which means that the first number found in the parenthesis will be the number that represents x on the graph and the second number in the parenthesis will represent y on a graph. so for this problem, we have the numbers (-4, 4) in our parenthesis, which means that our x value is the number -4, and our y value is 4. I hope this helps, and if you have any other questions I d be happy to help if I can!"
    },
    {
        "video_name": "ualmyZiPs9w",
        "Q": "I'm wondering why @ 12:09 he's using \"approximately equal\" for cov(x,y) and (xybar-xbar*ybar). Why is there any possibility that the 2 are not the same thing (especially based on our work here so far)?",
        "A": "He is thinking of covariance as a population parameter (this seems to be the standard), while the xy-bar etc. are sample statistics. Sample statistics can be thought of as estimators of population parameters. Consequently, the covariance would be equal to: E[XY] - E[X] * E[Y], which is estimated by: xy-bar \u00e2\u0080\u0093 x-bar * y-bar. He talks about this at 10:34."
    },
    {
        "video_name": "ualmyZiPs9w",
        "Q": "@ 14:00 Khan says that the denominator is the \"variance\" but is it not just the \"variability\" in this case? Should not the variance be divided by the population number? Is it in this case, just the total variance and not the mean variance that I am implying to?",
        "A": "The denominator is the same expression as the simplified formula for population variance, which = cov(x,x). (What is variability ? population number ? total variance ? mean variance - are you comparing a lot of different populations - ? )"
    },
    {
        "video_name": "ualmyZiPs9w",
        "Q": "2:41\nWouldn't this just be 0? Isn't the mean of the product of two variables the same thing as the product of the means of those two variables? Or am I wrong about this.",
        "A": "The mean of the product is not the same as the product of the means. For example if x = [1,2,3] and y = [4,5,6] then the mean of the product of [x,y] would be (1 * 4 + 2 * 5 + 3 *6)/3 or (4 + 10 + 18)/3 = 32/3 = 10.666... Alternatively, the product of the means would be ((1+2+3)/3) * ((4+5+6)/3) = 2*5 = 10 So they are not equal. Hope this helps!"
    },
    {
        "video_name": "ualmyZiPs9w",
        "Q": "When Sal at 11:39 speaks about approximations, doesnt this require atleast some minimum number of observations (n) ?",
        "A": "There are ways to calculate how accurate such an estimate is, but you dont really need a minimum of obervations, you just get a very bad estimate (approximation) if you dont have many."
    },
    {
        "video_name": "ualmyZiPs9w",
        "Q": "At 6:25 to about 6:50, Sal says that E[E[x]] is just E[X]. I just can't intuitively see why this is true. I don't understand.",
        "A": "For any constant c, E[c] = c. Then E[X] = \u00c2\u00b5, where \u00c2\u00b5 is just a constant, so E[\u00c2\u00b5] = \u00c2\u00b5. Hence, E[ E[X] ] = E[ \u00c2\u00b5 ] = \u00c2\u00b5 = E[X]."
    },
    {
        "video_name": "AtbZZiSLemQ",
        "Q": "At 6:40, Sal says that 1/2^n converges because it is a geometric series with a common ratio of less than 1. But there are other series which also have these properties and are not convergest, like 1/n.",
        "A": "Observe that the harmonic series \u00e2\u0088\u0091 1/n is not geometric."
    },
    {
        "video_name": "AtbZZiSLemQ",
        "Q": "At 4:00, is zero consider a positive integer ? If [an/bn = 0] can we conclude on the convergence of the series ?",
        "A": "No, zero is not considered a positive integer. In fact, some mathematicians consider zero to not even be finite. As for the answer to your question, the test would be inconclusive."
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "I don't get what's going on at 2:22. Can someone explain to me where 13/2 comes from?",
        "A": "He made it up as an example of how you have done this before when converting from improper fractions to mixed numbers."
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "I dont get..around 5:50, where did the X+3 go? Was it substituted by the A and B?",
        "A": "If you are talking about when he writes the yellow text, then the x+3 isn t supposed to be in there. What he was doing was taking the (A/x+5)+(B/x-8) and getting a common denominator. At around 7:00, when he gets the common denominator, he sets it equal to the (x+3/(x+5)(x-8))."
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "at 1:00 why didn't Sall divided P/Q ? I saw he divided Q/P . Isn 't more logical the first one ?",
        "A": "I m not sure what you re asking. There s only one way to do the division here. We re given that the numerator (I ll call it N) is (x^2-2x-37) and the denominator (I ll call it D) is (x^2-3x-40). We re given N/D, so that s the way we have to divide, and that s what Sal does in this video: D divided into N gives us N/D."
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "Why you use the \"short method\" to make the \"B\" term disappear, i.e. you assume x = -5. Does this not makes the previous expression an invalid statement? Looks at 8:05 the term in magenta. Given the value x = -5 this evaluates to division by 0. Why this is legal?",
        "A": "Yes the division by 0 does make the previous expression undefined. However while we are trying to find the value of A and B, we re not actually dividing and then re-multiplying by 0 to eliminate the denominators, but we are comparing the numerators directly, so no division by 0 occurs."
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "How do we know what order to put the factors of the denominator polynomial in? For example, at 4:02, he determines the order as plus 5, minus 8. Are the factors supposed to be ordered by least to greatest by the absolute value of their constants, or from greatest to least on the number line (not absolute value)?",
        "A": "It does not matter in which order they are, 1/(x+5)(x-8) is the same thing as 1/(x-8)(x+5). This is acording to the rule a * b = b * a"
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "At 1:52 doesn't he have to divide x^2-3x-40 into -2x-37 to finish the division?",
        "A": "The goal of the decomposition was to get a numerator with a smaller degree than the denominator. The denominator s degree is 2, so once we ve gotten a remainder with degree 1, we re done! In fact, if you tried to continue on and divide again, you would get the quotient to be 1 + 1/x, which is not a polynomial and a dead end for this kind of math."
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "At 10:13 what does he mean when he says that you could bring the 13 down? How would the fraction be rewritten like that?",
        "A": "(a/b)/c = a/(bc) a/(b/c) = (ac)/b so Sal s final answer could be written as 1+{2/[13(x+5)]}+{11/[13(x-8)]}"
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "How do you tell what the denominator for A and B are? At 4:45, he put A over x+5, but how do you tell if you are supposed to put it over x+5 or x-8? Does it matter which one goes over which?",
        "A": "Of course it doesn t matter whether A is above x+5 or x-8."
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "7:56 why is he substituting Zero ? Any explanation ? it looks like magic",
        "A": "He isn t substituting 0, he is substituting x= -5, so we will have B(-5+5) which is 0, so B will disappear and we can find value of A."
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "The main problem where I see a real leap to something new is 5:10 in the video. (x+5)(x-8) becomes \"magically\" (x+5) +(x+8). This is very confusing and there is no previous video that I can find where this is ever done.",
        "A": "I take it you meant that (x+5)(x-8) becomes (x+5) + (x-8) ? That s not what is going on. Those are the denominators he has decided to use, that is what this video is about, and he has to go through the steps outlined in the rest of the video to make that happen for his chosen denominators."
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "The numerator is smaller than the denominator!!! How do you know the degree is lesser,larger,same?? at-0:31",
        "A": "A degree of a polynomial is equal to the exponent of the variable in the leading term. So for example: x^2 + x + 1 has a degree of 2"
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "3:02\n\n6(1/2) =/= 6+(1/2)\n\n3 =/= 6.5\n\n\nSix and a half does not equal six plus one half.\n\nor am I wrong?",
        "A": "Six AND half IS equal to Six PLUS half which is 6.5. Your confusion probably arises because you re mistaking this for six TIMES half (which is 3)"
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "At 8:05, how can Sal assume that x equals anything? You don't know if A really IS 2/13.",
        "A": "A and B will be the same for all x. Therefore, if we find A for x is one value, it is the A for x is any value. I hope this helps!"
    },
    {
        "video_name": "S-XKGBesRzk",
        "Q": "At 5:12, he says that he will have \"decomposed the fraction\" and then said that he didn't know if that was the correct term. Does anyone know an official term for that?",
        "A": "correct the official term is break it down"
    },
    {
        "video_name": "wRBMmiNHQaE",
        "Q": "Sal said that vertical angles are always equal to each other at 0:44 and they're on opposite sides so would they also be considered an opposite angle?",
        "A": "The complete name for these angles is vertically opposite angles."
    },
    {
        "video_name": "wRBMmiNHQaE",
        "Q": "At 1:43 Sal says supplementary what does that mean",
        "A": "It means that two angles add up to 180 degrees."
    },
    {
        "video_name": "Z1zdkcwosD4",
        "Q": "at 5:55 can't you just do 3! (3 factorial)",
        "A": "Yes, but Sal Was trying to make a point and explain how the problem works."
    },
    {
        "video_name": "dh__n9FVKA0",
        "Q": "@1:22 the way you switched from taking the derivative of that function to take its integral confuses me! exactly where you didn't write the integral function sign (lower sigma) in front of the function, why?!",
        "A": "What he did at 1:22 was that the derivative of f(x)*g(x) is equal to f (x)*g(x)+f(x)*g (x). That is the product rule. If you do the anti-derivative of both sides, then you would get f(x)*g(x)=Sf (x)*g(x)+Sf(x)*g (x). Basically what he did was set up the problem so that you could solve for Sf(x)*g (x). Does that help out?"
    },
    {
        "video_name": "dh__n9FVKA0",
        "Q": "at 2:15 why does sal choose to solve for the anti derivative of f(x)g'(x) dx?",
        "A": "Sal could have solved for the anti derivative of f (x)g(x)dx but the answer would be the same. You can call those functions whatever you want, they could be d(x)r (x) or book(x)tree (x). Those names are just symbolic."
    },
    {
        "video_name": "w7NhLkQynS8",
        "Q": "At 3:22, I don't understand if the one affects anything. I also don't understand why instead of a-b it is b-a.",
        "A": "The b-a comes in from doing: 1/a - 1/b. So, let s do thru that part. 1) Find the common denominator = ab 2) Convert each fraction to the common denominator. 1/a (b/b) - 1/b (a/a) = b/ab - a/ab = (b-a)/ab Hope that helps."
    },
    {
        "video_name": "w7NhLkQynS8",
        "Q": "I don't get what he means at 2:01?",
        "A": "he has to multiply that side by b so that the denominators are equal."
    },
    {
        "video_name": "_sQ3dqhLsIw",
        "Q": "at 3:00 she meant meters .right?",
        "A": "Yes Lindsay meant Meters"
    },
    {
        "video_name": "xUDlKV8lJbM",
        "Q": "at 1:41 what if you kept getting a remainder for the number? would it go on forever?",
        "A": "Yes, but like 2/3 as a decimal it is .666666666666666666666 but you can just put a line over .66 which represents the repeating symbol."
    },
    {
        "video_name": "xUDlKV8lJbM",
        "Q": "at 1:18, Sal says that he could just as a many decimal as he wants, but isn't that improper?",
        "A": "No but as you see there comes a point where there is no need for more as it makes no difference. Just like it might not be big of a deal if someone says they run a max speed of 10 miles per hour or if they say they run 10.00 miles per hour..."
    },
    {
        "video_name": "xUDlKV8lJbM",
        "Q": "At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?",
        "A": "The sun is much bigger, however, the moon is much closer.... To put it in perspective, hold up your thumb to the sun (if the sun is out), your thumb will cover the entire sun even though your thumb is about 1/1,000,000,000,000,000th the size of the sun. Just because it s SO much closer it appears that way."
    },
    {
        "video_name": "xUDlKV8lJbM",
        "Q": "At 0:06, Salman says he will divide 60 by 35, but this tutorial is for dividing decimals, not whole numbers",
        "A": "I think he means the whole number of the decimals."
    },
    {
        "video_name": "lHdlHTsXbZg",
        "Q": "From 6:40: Here's an idea: why not define the mathematical expression 0/0 = k as a multiple answer equation, where the set of k values is (-\u00e2\u0088\u009e, +\u00e2\u0088\u009e)",
        "A": "That seems like just a very math-y way to say undefined"
    },
    {
        "video_name": "lHdlHTsXbZg",
        "Q": "0:52 What about x/y*y ? That would be x/(y^2), correct?",
        "A": "Technically: x/y*y = (x/y) * y/1 = x Why? Because of PEMDAS rules. We multiply & divide from left to right. To create your version x/(y^2), the original expression needs parentheses around the y s to show they are both in the denominator of the fraction: x/(y*y) = x/y^2 The parentheses also shift the order operations to do the multiplication then division. Hope this helps."
    },
    {
        "video_name": "lHdlHTsXbZg",
        "Q": "what does he mean around 1:30 x times 0 equals 0 OR any x? I get the x times 0 = 0, but how does he get the any x part?",
        "A": "okay, I get it now. Thanks =)"
    },
    {
        "video_name": "lHdlHTsXbZg",
        "Q": "1:19 If x*0=0 for any x then that would make x*0=1 impossible. And you can not make and equation with an unknown variable impossible to answer. Because, as long as there is an unknown variable, the problem has a solution. So how can x*0=0 for any x? And if x*0=0 for any x then that would make 0*infinity=0, and 0*infinity has been labeled 'undefined'. So again, how can x*0=0 for any x?",
        "A": "Zero times infinity =0."
    },
    {
        "video_name": "lHdlHTsXbZg",
        "Q": "At 0:38 Sal introduces an equation: x/y * y = x so long as x and y are defined, but I found an easy way to break this: x/0.7 * 0.7! Go ahead and try it. With every number I've tried, you end up just short i.e 4/0.7 * 0.7 =3.99... Any thoughts?",
        "A": "3.9... repeating infinitely is equivalent to 4. Therefore, you have not broken the rule ."
    },
    {
        "video_name": "lHdlHTsXbZg",
        "Q": "At 1:05, how can 10 divided by 5 times 5 equal 10?",
        "A": "Go in steps. 10/5 is 2, right? Now multiply 2 times 5. What d ya get? 10! Easy math. Take your problems slow."
    },
    {
        "video_name": "lHdlHTsXbZg",
        "Q": "2:23 you could also use a/b on the right side if you want to say x/0 is rational is x is not 0.",
        "A": "x/0 can t be a rational number because rational numbers never have a denominator = 0. This restriction is in the definition of rational numbers."
    },
    {
        "video_name": "lHdlHTsXbZg",
        "Q": "At 6:40, Sal says K can be anything. But, since we are multiplying K by 0, wouldn't it always come out to be 0 no matter what the value of K is?",
        "A": "Exactly Sal s point- since K can be anything and still result in 0, we have no proof for what 0/0 is."
    },
    {
        "video_name": "lHdlHTsXbZg",
        "Q": "X / Y * Y = X right. So why dosent X = 5 and Y = 2 so 2 * 2 = 4. And 5 divided by 4 is not 5 so what does sal mean?(1:00)",
        "A": "Multiplication and partition should be done just left-to-right, because one has no priority above the other. So we read x / y * y as ( x / y ) * y. So 5 / 2 * 2 = 2.5 * 2 = 5"
    },
    {
        "video_name": "lHdlHTsXbZg",
        "Q": "at 1:32, how is x*0 equal to 0 or any x i don't really understand it",
        "A": "multiplication is just quick addition. 3*0=0+0+0=0 It really does not matter what you multiply by zero, because no matter how many times you add 0 s the sum will stay 0"
    },
    {
        "video_name": "lHdlHTsXbZg",
        "Q": "So at 1:01 u said that x divided by y multiplied by y would equal x. Is it always like that or can it change??",
        "A": "It would always be like this....(x/y)*y equals (xy/y) which equals (y/y)*x which equals x."
    },
    {
        "video_name": "lHdlHTsXbZg",
        "Q": "At 1:00 Sal sir defines a formula (x/y)* y =x , here it should be noted that y never equals zero ,then at 2:50 how can we take the zero ? I was confused . Sorry if it's not a proper doubt.",
        "A": "The point of the video is to explore what happens if we allow division by zero and examine the 0/0 case. To do that we make some assumptions that we normally would not make to show that if we do try to allow these undefined operations, the results make no sense."
    },
    {
        "video_name": "5B8XluiqdHM",
        "Q": "At 16:23 the projection is calculated as [-27/13, -18/13]. But the projection vector has a positive horizontal component (it's pointing to the right). Am I missing something? I'm assuming that vector is w.r.t to the original space (vs. the null+row space) since the projection is calculated using vectors from that space.",
        "A": "The projection is [27/13, -18/13], not [-27/13, -18/13]. The correct projection does point to the right along the rowspace of A."
    },
    {
        "video_name": "ibeyn2QGjCM",
        "Q": "At @1:03 which is just when we isolate \u00e2\u0088\u009a(3x-7) on the left by subtracting \u00e2\u0088\u009a(2x-1) from both sides of the equation, to have\n\u00e2\u0088\u009a(3x-7 =  - \u00e2\u0088\u009a(2x-1)\nWouldn't we know right then that there is no solution?\nUnder what conditions could we have the square root of one expression equal the negative of the square root of the other.",
        "A": "It would still be possible for there to be a solution if both 3x-7 and 2x-1 were equal to 0. Obviously that is NOT the case, but the negative sign on the one radical doesn t eliminate the possibility of a solution by itself."
    },
    {
        "video_name": "ShpI3gPgLBA",
        "Q": "At 3:45, how did the (-1) only interact with the du on the right side of the equation, and not also make (e^-u) into (-e^-u)?",
        "A": "Sal merely multiplied by -1 twice, which es equivalent to multiplying by 1. Since the coefficient, the integrand, and the differential are all being multiplied together, you could make several choices about which two to multiply by -1 (or more generally, by any number and its reciprocal), but Sal s choice makes sense in this context."
    },
    {
        "video_name": "ShpI3gPgLBA",
        "Q": "at 2:40 why cant we take the anti derivative then?",
        "A": "Hm? You can, and one method to take that antiderivative is through substitution."
    },
    {
        "video_name": "ShpI3gPgLBA",
        "Q": "At 0:21, wouldn't it just have been a lot easier to set u = -sin(5x) ?",
        "A": "Yeah but he is approching the problem from both sides cos5x/exp(sin5x) and changing the expression to cos5x*exp(-sin5x). I guess is bc he wants to empathized in two substitution in the video and u cant get that from the 2nd expression."
    },
    {
        "video_name": "ShpI3gPgLBA",
        "Q": "at 2:50 why should it pop out to me that I should do another u substitution why couldn't I have just taken the antiderivative of 1/e^-u which I don't really know but couldn't I have just done that?",
        "A": "We don t know the anti-derivative of 1/e^u (which is the same as e^-u), but we do know the anti-derivative of e^x , where x is any variable (a, b, c, ... u, w, etc.), so we let the -u equal one of those variables and carry on with the known formula for e^(any variable) and do the extra stuff (i.e. multiplying by -1)."
    },
    {
        "video_name": "ShpI3gPgLBA",
        "Q": "2:22 - why didn't he just take the anti derivative of 1/[e^u] as ln|e^u|??\ni tried to get the answer that way but it didn't work. so why doesn't it work?",
        "A": "It doesn t work because derivative of ln(e^u) isn t 1/e^u, as it should be is ln(e^u) was the integral function. d/du ln(e^u) = d/du u = 1 =/= 1/e^u"
    },
    {
        "video_name": "q7eF5Ci944U",
        "Q": "at 3:07 Sal says triangles are congruent because of RSH congruence but we have studied till now that it is RHS congruence not RSH",
        "A": "For this the order of letters really would not matter."
    },
    {
        "video_name": "q7eF5Ci944U",
        "Q": "At 6:10 and 6:14, is the principle root of a number simply the positive square root?",
        "A": "Yes, the the positive square root and the principal square root are the same thing. Although, strictly speaking, we should say that the principal square root is the non-negative square root (because 0 is non-negative and non-positive and is its own square root)."
    },
    {
        "video_name": "q7eF5Ci944U",
        "Q": "what does rsh stand for in 2:42",
        "A": "From Wiki.Answers.com, It is a postulate concerning congruent triangles. Two triangles are congruent if they are both right angled (R), their hypotenuses are the same length (H) and one of the sides of one triangle is congruent to a side of the other (S)."
    },
    {
        "video_name": "q7eF5Ci944U",
        "Q": "I apologize if this is a silly question, but when you had the two right triangles (and he does this at 2:20 in the video) AMO and CMO, and knew their hypotenuses- AO and CO- were equal... can't you prove the triangles are congruent without the RSH postulate thing, just by knowing the right angles and hypotenuses are equal? I was just wondering. Thanks!",
        "A": "No, there exist multiple pairs of legs that produce the same hypotenuse, given that the length of the hypotenuse c is given as the sqrt(a^2 + b^2). You need at least one leg and the hypotenuse."
    },
    {
        "video_name": "q7eF5Ci944U",
        "Q": "on 2:55, i kno where you got the RSH from, but on the circle, weren't you supposed to prove that side AM is congruent to MC. I mean i can easily say they aren't congruent. Can you explain to me how side AM and MC are congruent. By the way, I already know how you got the hypotenuse and right angle, so no help on that.",
        "A": "AM and MC are corresponding parts of congruent triangles. Congruent triangles have congruent sides and congruent angles, too. Because AM and MC are the same parts of congruent triangles, they re congruent. I hope this helps!"
    },
    {
        "video_name": "q7eF5Ci944U",
        "Q": "At 6:00, what is the principle root of both sides? How can you take the square root of everything? If you can do that, cant Pythagorean theorem be A + B = C? It doesn't seem algebraically correct.",
        "A": "The principal root of both sides is the non-negative square root of both sides. Taking the square root of A^2+B^2 means adding A^2 and B^2 together then taking the square root of the sum. You only get A+B=C if you take the square root of A^2 and B^2 separately then adding it together. If you take A+B=C and square both sides you would get A^2+2AB+B^2=C^2 instead of A^2+B^2=C^2."
    },
    {
        "video_name": "q7eF5Ci944U",
        "Q": "At 2:40 could you also prove it as angle angle side. Because the angles opposite the congruent sides (radii) are congruent. So the smaller triangle share one side, the 90 degree angle, and angle A and Angle C are congruent?",
        "A": "According to you info you are sort of correct. there is a different name for if you use the right angle. You have to use like hypotenuse leg or leg leg or their direct corresponding angles"
    },
    {
        "video_name": "aSokFEpoJFM",
        "Q": "At 9:52, I don't understand how the cube ends up the way it does. Doesn't the cube end up different? Throughout the way he is cutting the cube?",
        "A": "at 9:52? There is only 7 minutes in this video"
    },
    {
        "video_name": "aSokFEpoJFM",
        "Q": "Notice how at 5:03 the pentagon almost looks like a regular pentagon (equal sides). Do you think a regular pentagon is possible, or does it always have to be an irregular pentagon?",
        "A": "At 5:03 the pentagon is not a regular pentagon. It is impossible to get five perfectly distanced points on a cube, which is necessary for a regular pentagon."
    },
    {
        "video_name": "qQAhhithHa8",
        "Q": "At 2:00, he could have written (b+a*cos(t)) outside of determinant. Result would be the same, but there would be less writing, as you can divide one row (or a column) by some number, and than multiply determinant by the same number (in our case (b+a*cos(t)) )",
        "A": "Is it just me or can no one see the cursor in this and the previous video cause without it, it gets kind of hard to understand which term Sal is referring to when he says this guy and that guy"
    },
    {
        "video_name": "NC7iWEQ9Kug",
        "Q": "At 2:46 - If the range for arcsin is supposed be within the 1st and 4th quadrant, then why would Sal consider the value 2pi - 0.34 as a suitable answer for X in the equation sinX=1/3 ?",
        "A": "That s for calculators, a true arcsin function would give you infinite answers."
    },
    {
        "video_name": "NC7iWEQ9Kug",
        "Q": "I understand why we subtract the angle from pi at 3:38 in this example, but does this always work to get the answer, like for cosine and tangent equations?",
        "A": "For tangent equations, typically yes since the y=tan(x) function has a period of pi. However, for cosine, it is often 2pi, as the period for y=cos(x) is 2pi."
    },
    {
        "video_name": "NC7iWEQ9Kug",
        "Q": "I don't understand where Sal got the 2pi from at 4:19",
        "A": "Sal get s the 2*pi here from the understanding of the unit circle. We know that a unit circle is 2pi radians. So if we want to be somewhere on the unit circle, we identify the point and then we can add 2*pi*n to it to be at that same point where n is any integer."
    },
    {
        "video_name": "NC7iWEQ9Kug",
        "Q": "at 4:06, he says that he adds any multiple of 2\u00cf\u0080n. Why does he add the 2\u00cf\u0080n? In the other answers, only a \u00cf\u0080n is added? He does not specify why he adds it? Is it some kind of trig identity that he mentioned before or something?",
        "A": "2\u00cf\u0080 radians is 360 degrees, so adding 2\u00cf\u0080n just means adding a full rotation, which will put you back where you started. In the answers with \u00cf\u0080n, that is just adding a half-rotation, or 180 degrees."
    },
    {
        "video_name": "NC7iWEQ9Kug",
        "Q": "At around 3:30, where did he get 2.80? I don't understand why we are subtracting .34 from pi.",
        "A": "There are two angles that = .34, which is arcsin 1/3. The first is just 0 +.34 = .34. The easiest way to find the second angle is to subtract .34 from 180*, which is half of the circle. Remember 180* = pi radians, since we are dealing in radians we get pi - .34 = 2.80 radians for the second angle. So our 2 angles are .34 & 2.80 radians. Hope that helps!"
    },
    {
        "video_name": "NC7iWEQ9Kug",
        "Q": "I'm confused at 5:36\n-0.34+pi by my understanding would be a correct answer since even though it start at the -1/3 then it switches to the second quadrant because you rotated it by only pi\n\nSal marked it wrong though?",
        "A": "You re answer only works for -0.34+\u00cf\u0080 but the answer in the video was -0.34+\u00cf\u0080n. Whenever n is odd, sin(-0.34+\u00cf\u0080n) = 1/3 but whenever it is even, sin(-0.34+\u00cf\u0080n)= -1/3."
    },
    {
        "video_name": "NC7iWEQ9Kug",
        "Q": "hey we just set a limit to values of sin from pi/2 to 3pi/2 then why are we considering the angle in 2 quadrant at 3:19",
        "A": "The value of sin is set to 1/3, which is within the domain of sin values. However, the angle measures that make up a sin value can be in any quadrant, because sin is only a y value."
    },
    {
        "video_name": "wUNWjd4bMmw",
        "Q": "at 3:26 he says that H of 2 is equal to one, how is that Possible?",
        "A": "Find x=2, then go up to the graph and see how high it is there."
    },
    {
        "video_name": "wUNWjd4bMmw",
        "Q": "At 2:24 where did you get (-3)^2-1 from?",
        "A": "Sal is plugging in (-3) for x in f(x). f(x) = x^2 - 1, so plugging in x makes f(x) = (-3)^2 - 1. He also gets the -3 from g(2) because g(2) is the input into the function f(x)."
    },
    {
        "video_name": "pkGrXzakRFs",
        "Q": "At 4:02, why can't one cancel the log base 3 from both sides of the equation and get 27x= 27+x? Wouldn't one be able to solve for x?",
        "A": "No you cant, you have to abide wit the rules, theres no exception dude"
    },
    {
        "video_name": "pkGrXzakRFs",
        "Q": "After 1:46, couldn't you just substitute the x,y, & z instead of taking those extra steps?",
        "A": "Yes, but not 100% of the listeners would be able to understand it properly."
    },
    {
        "video_name": "pkGrXzakRFs",
        "Q": "At 1:35 Sal starts converting the expressions to exponent form to explain the problem. So why do I have to fuss around with logarithms? Can't I just convert any particular problem into exponent form and then solve from there?",
        "A": "some problems you can. like log(base 10) 10 = x but then you ve got more complex problems too, which are easier to solve in logarithmic form moreover, logarithms have interesting properties like change of base, sum of log (see video) and many more."
    },
    {
        "video_name": "xkg7370cpjs",
        "Q": "what shoud be 40 minutes later than 4:25 pm.?",
        "A": "Let me do this in 2 ways-- 1) We know every hour has 60 minutes, and 40 minutes is 20 short of an hour. So let me think of this as 4:25 PM + 60 min - 20 min = 5:25 PM - 20 min = 5:05 PM 2) How about we add 40 min directly to the current time of 4:25 PM. This would make it 4:65 -- but hey, 60 minutes make an hour, so we can take away the 60 from the 65 and increase 1 hour. What are we left with? 5 more minutes. Again that means the time is (4+1):(65-60) i.e. 5:05 PM"
    },
    {
        "video_name": "xkg7370cpjs",
        "Q": "in 6:15 why did you put a dot instead of X",
        "A": "A dot dot also represents multiplication. For example, if you had 3dot3 is the same as saying 3 times 3."
    },
    {
        "video_name": "xkg7370cpjs",
        "Q": "At 2:35 Sal said that there were sort of the same thing, but aren't they exactly the same?",
        "A": "sal is the teacher but sometimes he says the wrong words in a way. so yes they are tecnically the same."
    },
    {
        "video_name": "xkg7370cpjs",
        "Q": "what does the dot at 6:04 mean?",
        "A": "No, the dot is the multiplication sign."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "What does \"subtends\" mean at 3:33 ? I tried looking it up but the definition confused me. Is it just the distance between two rays or is it more specific than that?",
        "A": "It is just a way of saying how the two things are related. For a particular central angle less than 180 degrees, there is only one minor arc that it cuts out on the circle. For any minor arc of the circle, there is only one central angle that yields that arc. Subtend is basically just a word describing that relationship."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "How do you find out how many radians are in a angle?11:43 AM",
        "A": "Divide the degrees by 180 and multiply by \u00cf\u0080 to get radians."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "At 0:59, Sal says that using degrees is not the only way to measure angles. Besides degrees and radians, are there any more units of angle measure?",
        "A": "Yes, there are even more ways people measure angles. I ve heard that some engineers and architects use gradians, which are 1/400th of a turn. Some people measure using tau units, where one full turn equals 1 tau."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "At 0:15, that angle would be 45 degrees right? Cause that's about half of 90 and half of 90 is 45 correct?",
        "A": "The angle is not drawn to scale, so it can measure whatever he says it measures."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "4:00 Is this definition applicable to circle with any radius? If I fix the angle and increase the radius, the length of the arc that subtends the angle also will increase (to the same length of radius)? How we are sure about that?",
        "A": "By definition the arc intercepted by a radian has a length of the radius. This applies regardless of the radius of the circle."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "Don't you mean \u00cf\u0084 at 7:33 because 2\u00cf\u0080 is equal to \u00cf\u0084?",
        "A": "You could say \u00cf\u0084 but 2\u00cf\u0080 is the same thing so it doesn t matter."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "At 7:46, Sal mentioned the symbol for degrees but is their a symbol for radians? Or do we have to write out radians each time we work a problem involving it.",
        "A": "If there would be ambiguity, it is sometimes customary to use the abbreviation rad. to denote radians. But usually it is assumed that you are working in radian measure, and since radian measure is dimensionless, you would not even have to write a symbol."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "What does he mean by radiuseseses? 7:06",
        "A": "I believe the appropriate term Sal should use is radii , but I might be wrong."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "9:18 - 9:46 I am confused. my mathbook says to convert radians = degrees * pi/180 and degrees = radians * 180/pi?",
        "A": "This is correct. When converting to radians, you want the pi to be in the numerator (pi/180); this way you will have pi in your final answer. When converting to degrees, the pi should be in the denominator (180/pi), so that they will cancel each other out."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "At 5:00, if we change the radius of the circle, will a radian stay the same (in degrees)?",
        "A": "There are always 2pi radians in a circle, just like there are always 360 degrees. If you change the radius of a circle, it affects the circumference; only for a unit circle is the arc length equal to the angle in radians."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "at 2:18 can someone explain how something can be more 'mathematically pure ?'",
        "A": "I know I m pretty late, but mathematically pure means that it is analytic a priori, that is, true by virtue of the properties of angles and circles, as opposed to the impurity of the degree system, which relies on empirical observation (astronomy theory), or culturally specific numbering systems (Babylonian theory). Maths is meant to be by true regardless of physical laws, as demonstrated by such noumena as the Banach-Tarsky paradox"
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "So at 10:42, he explains that as long as we remember 2 pi radians = 360 degrees or pi radians = 180 degrees, we should be fine. However, does this mean this is the universal conversion?",
        "A": "Yes, the degree is 1/360th of a circle, which is 2\u00cf\u0080 radians."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "Somewhere around 6:21 in the video Sal starts talking about pi radians/pi degrees or some such. Where does the pi come from, and exactly how many degrees are in a radian?",
        "A": "Pi comes from the ratio of the circumference to the diameter."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "why at 3:12 does Khan call that angle theta",
        "A": "Theta is just referring to any angle. For example, let s say you have angle x . You can call x theta."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "At 9:15, when converting radians into degrees, is it common to have the degrees answer be in weird forms like 180/pi?",
        "A": "You might, but usually your radian measure will have pi in it, so the pi will cancel out when you convert to degrees. For example, it is unusual (but not impossible) to talk about 1 radian or 2 radians. Usually we would be talking about pi/3 radians or pi/4 or pi/2 or pi/6."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "1:30\nWhat and is there a relationship of 360 degrees and 365 days ?\nWas this just a miscalculation on the astronomy's side like an\ninaccuracy ?",
        "A": "As far as I know, the Egyptians were the first to know that there were 365 days in a year, and As far as I know, the whole Babylonian base 60 thing where they got degrees from was before that, so At the time degrees were invented, they thought that there where 360 days in a year (I think). Even the Egyptians, who knew that there were 365 days in a year, had a legend where there were originally 360, but a god or something added 5 days onto the end for some reason or another."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "What is theta?\n(3:16)",
        "A": "Theta (\u00ce\u00b8) is a greek letter which is commonly used in math to represent the value of some angle. So, theta is a variable just like x, except it s almost always used to represent an angle."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "At 4:49, I don't know why here the angel delta is 1 radian. How does he get that? Does that mean radian always equal to radius?",
        "A": "So angle theta is the angle that makes the subtended arc equal to the radius. right? What I think he is saying is that this the amount of radii that fit in here would be one. So what that means is that the angle is the same as the radius so theta equals one radius. he later says that 360 degrees is equal to 2 pi radians. So a quick recap. Radians is the angle in which the radius equals the arc length. here theta equals 57.29577951 degrees. That many degrees is equal to one radian."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "3:22. This question might be a little random, but why do we use theta to represent an angle's measure?",
        "A": "It s just convention that we use lower case greek letters to denote angles. Phi, psi, eta, alpha, beta, and gamma are also relatively common variables choices for angles. (\u00cf\u0086, \u00cf\u0088, \u00ce\u00b7, \u00ce\u00b1, \u00ce\u00b2, \u00ce\u00b3 respectively). I have heard that this is a holdover from ancient Greece, whose intellectual culture was revered in the West for a long time, but I can t confirm it."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "At 6:35, Mr. Khan says that the circle's circumference is 2 pi r. Is this just a fact in math, or can it be proven? More simply, WHY is it 2 pi r?",
        "A": "All facts in math are proven. It can t be a fact if it cannot be proven. This formula is easy to derive. By definition, \u00cf\u0080 is the ratio between any circle s circumference and diameter. Hence: \u00cf\u0080 = \u00f0\u009d\u0090\u00b6/\u00f0\u009d\u0091\u0091 Rearranging to solve for the circumference: \u00f0\u009d\u0090\u00b6 = \u00cf\u0080\u00f0\u009d\u0091\u0091 And since the radius is half a diameter: \u00f0\u009d\u0090\u00b6 = 2\u00cf\u0080\u00f0\u009d\u0091\u009f All formulas have proofs. How can we call it a fact if we aren t 100% it is true? Comment if you have questions."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "At 8:08, Sal writes the full word 'radians'. Is there any short form for it?",
        "A": "There is no commonly used symbol for radians because it is a ratio, not a unit. Thus, if an angle is expressed without a unit symbol, then that means the angle has been expressed with radians."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "at 8:19, what is pi radian is equal to?",
        "A": "180 deegree it is a standard rule applied in interconvrsions"
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "In 2:15 to 2:33 as Sal said that Radian measure is much like pure mathematics and is not based on cultural artifacts can you give me some other examples where more pure mathematics is used other than cultural artifacts,like why do we have only 1-10 why not 1-100 or even why only numbers are divided upto 10(aren't they based on cultural artifacts) why not in something more pure form just like Radians . Why not Time,Day,Year,Our whole calender system, Measurement system or are they really a sort of pure Mathematics?",
        "A": "e, i, and pi are all pretty much independent of measurement system"
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "( 4:39 - 4:41 ) Is Sal talking about the arc itself, or the line segment between the ends of both radii?",
        "A": "He s talking about the arc itself. A length equal to one radius is superimposed on the circumference of the circle, and that arc subtends an angle of one radian."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "In 7:33, why did you say \"2 pi radians\" instead of \"2 radians times pi\"",
        "A": "because it is actually 2pi, or tau radians. If you say 2 radians times pi, that is confusing."
    },
    {
        "video_name": "EnwWxMZVBeg",
        "Q": "4:10 subtends or sublends",
        "A": "I think he drew the horizontal line of the _t_ too low..."
    },
    {
        "video_name": "yUaI0JriZtY",
        "Q": "At 2:20, can the probability change if the bag is maybe smaller?",
        "A": "probably not because it depends on what you take out not whats in the bag so much"
    },
    {
        "video_name": "yUaI0JriZtY",
        "Q": "at 0:32, what is a outcome",
        "A": "Outcome: one of the possible events that could happen from the list of everything that could happen (the more specific a scenario, the more specific the definition of outcome can be)"
    },
    {
        "video_name": "yUaI0JriZtY",
        "Q": "At 1:38, is a \"sample space\" another name for a population, or is it just similar?",
        "A": "Technically, no. A population encompasses an entire set (for example, all women ). A sample space is a subset of a population. To get some idea of a population trait, you would observe a smaller subset (i.e., a sample ) of that population."
    },
    {
        "video_name": "av947KCWf2U",
        "Q": "I didn't get how Sal just wrote down n+1 term dividing by nth term at 3:15: ((n+10)^10/(n+1)!)/(n^10/n!). How did he get those?",
        "A": "In this video, Sal was performing the ratio test on the sum of n^10/n!. In doing this, he took the (n+1)th item and divided it by the nth item. The nth item is : n^10/n!. For the (n+1)th item, you substitute (n+1) for n giving you (n+1)^10/(n+1)!. Therefore, dividing these two terms gives the ratio test resulting in: ( (n+1)^10/(n+1)! ) / ( n^10/n!). I hope this helps. Please note that your original question must have mis-typed the answer."
    },
    {
        "video_name": "v7ZZ1ctT1k0",
        "Q": "In 1:49 how do you know at wich point to start at",
        "A": "You can tell because it is the point... not the arrow. In vectors, direction matters. The point represents the starting point."
    },
    {
        "video_name": "v7ZZ1ctT1k0",
        "Q": "In 1:49 how do you know at wich point to start at",
        "A": "Look at the point with the v-shape. That tells you the end point of the vector or the terminating point. Follow the components in reverse to find the initial point. That s how you know at which point to start."
    },
    {
        "video_name": "v7ZZ1ctT1k0",
        "Q": "At the time 1:31, Sal said \"down by three (3).\" Did he mean \" to the left three? Just wanted to make it clear because i got confused",
        "A": "Yes, he meant to the left. Good Catch. Thanks."
    },
    {
        "video_name": "fFdOr8U4mnI",
        "Q": "where did the 526.90 come from at 2:02? did he add the numbers did not hear him say",
        "A": "Yes, he added two deposits, one of $426.90 and another one of $100.00"
    },
    {
        "video_name": "fFdOr8U4mnI",
        "Q": "at 1:08 why does it get so complicated?",
        "A": "Try thinking of it this way: He starts with $526.90 and removing $15.08 from that. That s another way you could look at it. If you re adding to a negative, try subtracting from the added value if you don t get it. Another way to look at what I just said: -$15.08 + $526.90 --> $526.90 - $15.08 Hope that helps."
    },
    {
        "video_name": "fFdOr8U4mnI",
        "Q": "Is it possible (at 0:07) to have a negative amount of money in your account?",
        "A": "Yes, it would mean you spent more money than you had, thus putting you into the negatives which means you would owe the bank money."
    },
    {
        "video_name": "fFdOr8U4mnI",
        "Q": "how dose he ow then money at 0:07",
        "A": "He could be in debt or he could have taken out a loan and not payed it off yet."
    },
    {
        "video_name": "fFdOr8U4mnI",
        "Q": "At 1:32 is there any other ways on how to solve this problem without using the number line?",
        "A": "better in a number line than in your head"
    },
    {
        "video_name": "fFdOr8U4mnI",
        "Q": "super video just I dont really know how he got that at 1:40?",
        "A": "Just imagine that your friend lent you $15 dollar and 8 cents to buy something. Then you worked very hard and got 2 pay cheques of 426.90 and 100.00 dollars. This money adds up to 526.90 dollars. Now you have to pay back your friend. So you take it out of you savings: 526.90-15.08=511.82 dollars you have left."
    },
    {
        "video_name": "fFdOr8U4mnI",
        "Q": "2:30 Can't I just subtract 15.08 from 100.00, add 426.90, and get the same answer?",
        "A": "Yes, and this is just because you have a list of positive and negative numbers that you are adding, you can always change the order when adding signed numbers."
    },
    {
        "video_name": "fFdOr8U4mnI",
        "Q": "At 0:07 Sal mentions that Stewart has a negative amount of money in his account; how is that possible?",
        "A": "This can also happen if you deposited a check from someone and that check then bounces , which means the person who wrote it did not have enough money to cover it. You think you have the money though, so you write checks or use your debit card and because the money is not there, you go into a minus balance (also called in the red or in the hole."
    },
    {
        "video_name": "fFdOr8U4mnI",
        "Q": "At 1:22, he uses a Kerala Marlboro line. How does that work?",
        "A": "it means he owes money to the bank"
    },
    {
        "video_name": "fFdOr8U4mnI",
        "Q": "At 0:09 when he deposited the first check did he pay the bank the money he owed it?",
        "A": "Yes, he paid the deficit of -$15.08 using the $426.90 check. By doing so, he has $411.82 in his bank account. -15.08+426.90=411.82 Just to help prevent confusion, remember that in the video he combined the two checks into $526.90 for simplification of solving the problem."
    },
    {
        "video_name": "0lOpqmTdtzk",
        "Q": "At 3:35, why did Sal have to multiply by ten on one side, but divide by ten on the other?",
        "A": "Inverse operations to get the same answer."
    },
    {
        "video_name": "0lOpqmTdtzk",
        "Q": "At 0:40, Sal wrote \"5 (liters) x 40%\". Can someone show me how to work this out? ( You don't have to include the liters)",
        "A": "Sal is using a Part=Percentage \u00e2\u0080\u00a2 Whole equation. The Part represents the part of the body s blood that is red blood cells. 40% is the percentage of any liquid amount of blood that is made up of red blood cells. 5 (liters) represents the Total amount of blood in the human body. So, when he wrote (5 liters) (40%) he is just going to multiply Percentage(Whole). Note: 40% is 40/100 = 2/5 so, (5 liters) (40%)=5\u00e2\u0080\u00a22/5=2 (liters)"
    },
    {
        "video_name": "0lOpqmTdtzk",
        "Q": "What does he mean at 1:57",
        "A": "In order for something to be properly written in scientific notation, it has to be no less than one and no greater than ten. So since 90 is way bigger than ten, it s not properly written. You need to move the decimal place over so that the number becomes 9.0 does that make sense? I hope that answers your question lol."
    },
    {
        "video_name": "0lOpqmTdtzk",
        "Q": "At 1:48 in the video, you start to mention that 90 isn't in scientific notation because it isn't less than ten, which I agree with, but under that understanding, wouldn't 10^-15 technically not be in sci.n. because it isn't less than ten?",
        "A": "90 isn t in scientific notation because the coefficient isn t less than 10. Since 90 is just a number, 90 is the coefficient which isn t less than 10. With 10^-15, the coefficient is 1 which is less than 10. However, it should be noted that to be in scientific notation, the coefficient must also be greater than or equal to 1. I hope this clarifies what Sal meant!"
    },
    {
        "video_name": "0lOpqmTdtzk",
        "Q": "At 1:01 sal said that the volume of red blood cells is 90 times 10 to the negative 15 liters. If it is that small, then how come they fill up the human body?",
        "A": "What else is in the blood Kaycee2418? Just asking I have a test tomorrow."
    },
    {
        "video_name": "RLyXTj2j_c4",
        "Q": "At 0:26, Sal moves the reflecting tool line to the line y=x+2. How does he know to move it at that point? How does he know where y=x+2 is?",
        "A": "He knew that by using two pieces of information: 1- the slope of the line/reflecting tool which is the coefficient of variable x. 2- the y-intercept which is the intersection of the line/the tool with y-axis. The value of y-intercept is aways the constant in the right-hand side of the equation. Hope this helps!"
    },
    {
        "video_name": "RLyXTj2j_c4",
        "Q": "in 1:02 what does he mean by \"when x is equal to zero y is equal to 2\"does he mean that every time you go one to the right , you go 2 to the y direction?",
        "A": "No. He s talking about the y-intercept (the point where the line intercepts the y axis). Points on the y axis have an x value of 0. So if y is 2, x will be 0."
    },
    {
        "video_name": "lCsjJbZHhHU",
        "Q": "At 5:25 why is the vector no longer considered a real valued 3-tuple and if its not what is it known as?",
        "A": "What Sal did is he made the vector (i,1,0) where i is the square root of negative 1. The square root of negative 1 is not a real number , it is an imaginary number . So that vector is an imagainary 3-tuple ."
    },
    {
        "video_name": "lCsjJbZHhHU",
        "Q": "at 5:44, does the e-looking symbol mean 'a member of'? what is the symbol called?",
        "A": "yes, the symbol is used that the left side (element) is part of the right side (set). There is no real name to it other than membership sign , but that could refer to subsets as well."
    },
    {
        "video_name": "lCsjJbZHhHU",
        "Q": "At appx. 5:40 says that any n-tuple containing an imaginary number could not be a member of R^n, but are there more spaces like I^n that contain imaginary numbers? Like the imaginary plane for complex numbers.",
        "A": "Yes, there is such a thing as \u00e2\u0084\u0082\u00e2\u0081\u00bf, which take complex numbers as components instead of only real numbers. However, \u00e2\u0084\u0082\u00e2\u0081\u00bf will tend to follow some slightly different rules."
    },
    {
        "video_name": "lCsjJbZHhHU",
        "Q": "At 4:47, I could vaguely understand the meaning of \"\u00e2\u0088\u0088\". Which math subject should I take to learn more about a symbol like this ?",
        "A": "It means is a member of . It s from set theory."
    },
    {
        "video_name": "lCsjJbZHhHU",
        "Q": "at around 2:31 when you put the arrow on the fourth space, you had it pointing out. Could you point it in? Or would there be something mathematically wrong with that? Please explain.",
        "A": "I m not totally sure what you mean by pointing in, but if that means the arrow would end up pointing in a different direction , then yes, that would be incorrect. Or, more precisely, it would be a graph of a different vector than (3,4), which must always go over three and up four no matter where it starts from. Hope that helped."
    },
    {
        "video_name": "lCsjJbZHhHU",
        "Q": "At 4:24 Sal assigns x as a vector for 3 tuple [0,0,0] but in the previous video states that a vector has to have both magnitude and direction. Can x be a vector even though it has no magnitude or direction? Wouldn't that just be a point?",
        "A": "the zero vector (0, 0, 0,...0) has a magnitude of 0 and points in every direction, AKA it is orthogonal (perpendicular) to every other vector"
    },
    {
        "video_name": "lCsjJbZHhHU",
        "Q": "At 0:36 he says that real coordinate space is the 2D space in our regular coordinate plane. Does that include all four quadrants?",
        "A": "Yes. It includes all real numbers in the 2D coordinate plane, which means all the real numbers in all the quadrants."
    },
    {
        "video_name": "lCsjJbZHhHU",
        "Q": "at 4:06, what is the purpose of doing all of the various boldface R^3 etc. What situations would require the use of R^a large number?",
        "A": "If we lived in a many dimensional world then R^(large number) could be used to describe positions."
    },
    {
        "video_name": "lCsjJbZHhHU",
        "Q": "At 3:14 , is it possible for the R^2 a.k.a 2-tuple to be shown in the matrix form of [3 4] instead of what it is shown? Besides, R^2 can also be (-3,4)? So it doesn't have a fixed reference point?",
        "A": "A vector in R2 is any vector [a, b] where a and b are real numbers. If you re going to write a vector in matrix form, it must be vertically oriented, which makes it look exactly like vector form anyways. For the sake of typing, you can write [[a][b]] for the matrix form or simply [a, b] for the vector form."
    },
    {
        "video_name": "lCsjJbZHhHU",
        "Q": "At 4:53, Sal writes an equation like x (with an arrow over it) E IR^3\nWhat does the E mean?",
        "A": "It means that the vector x is a member of R3. So if R3 is every possible combination of 3 real numbers, x is one of those combinations."
    },
    {
        "video_name": "lCsjJbZHhHU",
        "Q": "At 2:27, sal said that we only care about a vector's direction and magnitude, so if i were to draw a vector not from the origin point, what should i do?",
        "A": "What do you mean what should i do? . Where you draw a vector doesn t change what the vector means, because the coordinates in linear algebra do not mean the same as they do in normal coordinate systems. That is, normally a coordinate defines a position in a space, but a vector has no knowledge of its own position, it only knows magnitude and direction."
    },
    {
        "video_name": "lCsjJbZHhHU",
        "Q": "At 5:55, why is R with a superscript of n still a real coordinate space when n is an imaginary dimension?",
        "A": "What do you mean by imaginary dimension? n is only allowed to be a positive integer, it s not allowed to be a complex number like i. Do you mean since humans only live 3D, things like R^5 can only be imagined in our heads? If you mean that question just know that math and the physical world are two separate things. We are allowed to talk about whatever is logical in math, not just whatever is physical. R^5 for example is not an imaginary land. It is a very real and important thing in mathematics."
    },
    {
        "video_name": "Dru0RHgfp2g",
        "Q": "on tha 6:38 minute of this video. Why does 0=-2(x+6x+8) becomes to 0=-2(x+2) if 2*2=4?\ni think its have to be 2*3=6 => 0=-2(x+3), but then is not going to be a parabola.\nplease answer..",
        "A": "It does not become 2(x+2) it actually become 2(x+2)(x+4). He actually factored the expression. Look again at the video!"
    },
    {
        "video_name": "Dru0RHgfp2g",
        "Q": "At 6:38, Sal factors out a -2. However, when he rewrites his equation, the x^2 paired with the -2x^2 becomes x. Why is that?",
        "A": "He just forgot to write the exponent. Notice that he went on to factor it as though it said x\u00c2\u00b2. There s a popup box addressing this."
    },
    {
        "video_name": "Dru0RHgfp2g",
        "Q": "At 6:37, when he is factoring, shouldn't it be (x+3) since 6/2=3? Maybe I am just confused.",
        "A": "you multiply the secondary terms to the eight, not the six. the equation (x+2)(x+3) is actually x^2+5x+6"
    },
    {
        "video_name": "fNzM0lQHZsk",
        "Q": "At 2:40, you determine that the number of ones is 36. What would have happened if we had added 15 ones to the 9/10 so that it looks like this : ...=50+21+150/10+9/10+3/100? In other words, can the numerator in the tenths be more than 10? I would tend to say no but I'll let you explain the reasoning behind it.\nthank you.",
        "A": "Sal explains it very well"
    },
    {
        "video_name": "UFWAu8Ptth0",
        "Q": "At 8:57 Sal concluded that if g(x) and h(x) are both solutions, adding them together also is a solution. Please let me know if I'm understanding this correctly. So when he says one of those functions \"is a solution\", in this case, since he's speaking of homogenous equations, he's basically saying \"is equal to 0\"? So, then, when he's saying \"g(x) + h(x) is a solution\" he is pretty much saying \"0+0=0\"? Am I correct in my understanding? Thanks!",
        "A": "Yes, I think you have it exactly right. You can see that when you plug solutions in the terms can be easily separated which really simplifies things because you can just sum all your solutions. You ll use this same idea later with non-Homogeneous equations."
    },
    {
        "video_name": "UFWAu8Ptth0",
        "Q": "At 0:09 wat dose he mean 2nd ordeal diferntial equations",
        "A": "The order of a differential equation is the highest-order derivative that it involves. Thus, a second order differential equation is one in which there is a second derivative but not a third or higher derivative. Incidentally, unless it has been a long time since you updated your profile, you might be in over your head on this one. I might recommend taking a while to learn differential and integral calculus before you try to tackle differential equations."
    },
    {
        "video_name": "UFWAu8Ptth0",
        "Q": "Im confused at 2:52. Can you please explain again why is it called a homogeneous differential equation?",
        "A": "This is called a homogeneous differential equation because the the term on the right-hand side of the differential equation is zero. In other words, d(x)=0."
    },
    {
        "video_name": "mI84WDfhuYA",
        "Q": "at 00:11 what is decemters i am not good at uhhhhhhhhhhhhh at convetion plz help",
        "A": "If you are saying decametre it is 10 times of a metre or if you are saying decimetre it is 1/10 of a metre."
    },
    {
        "video_name": "sOluh1zX5tE",
        "Q": "g'(x) is the slope of the function g,right? so when we get g'(x)=0,it would mean that the slope is parallel to x axis.but if the function has to go on after that point where sloe is 0,in this casewhen x=0,there should be a turning of the graph.\nfor eg,if the fn was decreasing frm (-infinity) till x=0,and at x=0,the slope is 0,so from x=0,shouldnt the graph increase??\nhere at 4:57 the graph at both intervals are decreasing.how is that possible?",
        "A": "When g (x)=0, x is a critical number (c.n.) A c.n. is where the graph could have min/max (turning point where the graph change from decrease to increase or vice versa) but it does not guarantee that it will have a min/max. That s why we have to do what we call the first derivative test like Sal does in the video. An example of this would be f(x)=x\u00c2\u00b3 then f (x)=x\u00c2\u00b2 f (x) = 0 at x = 0, but f(x)=x\u00c2\u00b3 is increasing for all x because at x=0 the slope is 0 but it s neither a min or a max."
    },
    {
        "video_name": "sOluh1zX5tE",
        "Q": "at 3:10, is there a difference between (negative infinity, 0) and [-negative infinity, 0]\nis there a close interval contains negative infinity?",
        "A": "Parentheses ( indicate non inclusive. Brackets [ indicate inclusive. So (-inf,0) means it is neither negative infinity or zero, but everything in between. [-inf,0] means that it could be zero or negative infinity along with everything in between. Since something can never equal infinity, we use parentheses by them. therefore, [-inf,0] is technically impossible. (-inf,0) also does not equal (-inf,0] because it could be zero in the second one but not the first one."
    },
    {
        "video_name": "sOluh1zX5tE",
        "Q": "At 3:50, he says that the interval would be decreasing in the interval from 0 to negative infinity. However, wouldn't the function be increasing if we took the interval from negative infinity to 0? Because if we take it that way, g prime of x becomes less and less negative as we approach zero from negative infinity",
        "A": "We always consider intervals going from left to right (i.e. smaller to larger values). It doesn t matter that g (x) is becoming less negative because it is still negative and therefore the slope of g(x) is still downward \u00e2\u0080\u0093 i.e. g(x) is still decreasing."
    },
    {
        "video_name": "Za7G_eWKiF4",
        "Q": "At 1:56, can we say the events are not independent because we remove a possible outcome without replacing it?",
        "A": "Yes that is exactly what Sal said at 1:58"
    },
    {
        "video_name": "Tal_fgREll0",
        "Q": "1:21. triangle ABD does not share that point though? i'm confused",
        "A": "You are looking for two similar angle. Angle abc and angle dbc. Do nothing with abd. Each of the angle have a right angle and one comman angle c. It will not work on angle adb, because it didn t share angle c."
    },
    {
        "video_name": "Tal_fgREll0",
        "Q": "At 4:10, couldn't you just cancel out the BC's and then find out what AC over DC? Help!",
        "A": "i don t think so. Here: AC BC AC+BC --------- = ----------- is a proportion. If it was this-> ----------------------- = something BC DC BC+DC Then you cross cancel. But you can t cross cancel over an equal sign (or any other sign like +,-,/.*, unless the denomenator is the same)."
    },
    {
        "video_name": "Tal_fgREll0",
        "Q": "At 8:40, is principal root same as the square root of any number?",
        "A": "The principal square root is the nonnegative square root -- that means the principal square root is the square root that is either 0 or positive."
    },
    {
        "video_name": "pxX07gUbIQQ",
        "Q": "At 0:31 why did Sal add the whole numbers 1st?",
        "A": "Because he doesn t need to convert them into a mixed fraction, it s easier that way."
    },
    {
        "video_name": "rmYlCuiC5uY",
        "Q": "At 3:36, why can't you simplify 12x^2 and 14x^2 by 2 to be 6x^2 - 2 / 4x^2 - 9?",
        "A": "No, we can t simplify any further. When we reduce fractions we cancel out common factors (items being multiplied. For example: 10/15 = 2/3 because 10 = 2*5 and 15 = 3*5 and they share a common factor of 5. In the video, 12x^2 and 14x^2 are terms (they are being added/subtracted with other values). We can t cancel terms. Hope this helps."
    },
    {
        "video_name": "rmYlCuiC5uY",
        "Q": "On video at 1:18 why is it -2-3x and not -3x-2 instead? Many thanks in advance.",
        "A": "The 2 expressions are equivalent. -3x-2 would be preferred as it is in standard form. But, it isn t required."
    },
    {
        "video_name": "W_VhVHRal6o",
        "Q": "At about 1:46 why would he put a zero in front of 99061? Wouldn't it be the same thing with or without it?",
        "A": "A zero is not put in front of 99061, but .99061. The zero is placed for you not to forget that the number is a decimal. A lot of test are now in computer format so you need to place that decimal in front for your answer to be correct. It s a nice reminder."
    },
    {
        "video_name": "W_VhVHRal6o",
        "Q": "at 1:05, is their anyway to remember that trick?",
        "A": "how many zero s there are in like th 100 u move it like he showed move it twice"
    },
    {
        "video_name": "_npwsLh0vws",
        "Q": "AT 11:39, is x+y (to the second power)=3 a function?",
        "A": "x+y (to the second power)=3 represents a function of x in terms of y because x can be easily solved for to get x=3-y^2. This means that every real value y is associates(or mapped) to the real value 3-y^2. So by definition, x=g(y)=3-y^2 is a function of y. However, the equation x+y (to the second power)=3 cannot be said to represent a function of x because for any x less than 3, y=(3-x)^0.5 and y=-(3-x)^0.5 both satisfy the given equation. So every x less than 3 is mapped to two values, so x is not well defined."
    },
    {
        "video_name": "JXVGPEOQCb8",
        "Q": "Sal you said at 5:37 that ln(2) (.693147...) is going to be some fraction. And that fraction, if greater than 1/2 will result in 8*ln(2) being more than 4. While ln(2) * 8 is greater than 4 (5.54518), I wanted to follow your reasoning and got lost at this point.\n\nBeing more than have will eventually give you a negative number when added with -4.",
        "A": "If p > 1/2, then 8*p > 8*1/2 = 4. Multiplying both sides by the same positive number leaves the direction of the inequality unchanged. (Or is this not what you were asking?)"
    },
    {
        "video_name": "JXVGPEOQCb8",
        "Q": "When Sal was testing out the endpoints and the critical points around 4:11 in the video, is there another way to find out the maximum value ? Like can I use the line test for that?",
        "A": "Yes You can try to do it, but it is easier to do it in Sal s way and you would need to do it later on. Really good point though! Im gonna try it rn!"
    },
    {
        "video_name": "JXVGPEOQCb8",
        "Q": "At 4:00 Sal looks for where f'(x) is undefined but surely he should be looking where f(x) is undefined - no?",
        "A": "No; you re trying to find where the slope of the function is undefined (a point on the curve where you can draw multiple tangent lines, making the slope undefined) to see if there is a critical number there that would be a candidate for the absolute max. The function itself is continuous and is defined on every point in the given interval. You do that to see if the pointy curve he drew last was indeed what the function looked like at the given interval."
    },
    {
        "video_name": "BsBH8nAv5l4",
        "Q": "at 6:00 can we write a - b = c ?",
        "A": "You were almost correct. It s c = - a - b"
    },
    {
        "video_name": "BsBH8nAv5l4",
        "Q": "I thought the vectors connect from tail to tip only (1:30)",
        "A": "they do, but the resultant vector is showing you a final direction, so that resultant vector is not connecting , it s the final vector with a new magnitude (length) and direction. so if add vectors with an equation that define that relationship as Sal said, you could say that: a+b+c=0 a+b=-c"
    },
    {
        "video_name": "BsBH8nAv5l4",
        "Q": "Is there any way we can use vector subtraction to write an equation for the figure at 4:51 (the last example)?",
        "A": "Yes, but it adds unnecessary negative symbols. For example, we can rearrange a+b=-c to be b=-a-c. This uses subtraction, but adds more negative symbols than we need. It makes more sense to multiply both sides by -1 to get -b=a+c. Using addition just makes everything tidier."
    },
    {
        "video_name": "vaOXkt7uuac",
        "Q": "On problem 39, at about 7:30, would it be easier if you used 30-60-90 triangles to find the height?",
        "A": "I m sorry can explain that a little easier"
    },
    {
        "video_name": "vaOXkt7uuac",
        "Q": "I know this is simple, but I just wanted to reassure myself. At around 8:27 when Sal multiplies the triangles, the formula is b times hieght divided by 2. Now does that only count for that one half of the triangle? Is that why he had to multiply it once more? Also, hypothetically, if I were to encounter a triangle such as this again, would I do the method shown here to find its area?",
        "A": "The easier way to do it was to split the trapezoid in half so that they would be two triangles. Then he found the height using the pythagorean theorum. That way, he could use the formula for the two triangles using formula 1/2*b*h and add the sums together. You could also directly find the area of a trapezoid using the formula average width times altitude. Hope tis helped."
    },
    {
        "video_name": "vaOXkt7uuac",
        "Q": "At 2:25 can you combine all the parallelograms to make a big parallelogram?",
        "A": "Sure, you could paste them together but it really wouldn t make the problem easier to solve."
    },
    {
        "video_name": "vaOXkt7uuac",
        "Q": "where did u get 144 on 03:54?",
        "A": "The area of the entire figure is 12 x 12 = 144 square units, then subtract the area of the parallelograms to find the shaded area."
    },
    {
        "video_name": "eTwnt4G5xE4",
        "Q": "At 4:09, Sal says 129 instead of 121.",
        "A": "Yes if you notice the text box at that point, this mistake has been found and corrected"
    },
    {
        "video_name": "eTwnt4G5xE4",
        "Q": "Go to 3:57 for the funniest moment in this video. :)",
        "A": "lol it is funny at 3:57"
    },
    {
        "video_name": "eTwnt4G5xE4",
        "Q": "At 2:05, Sal says the question mark angle equals to 92 degrees, but isn't the 2 lines that made up the angle perpendiculer to each other ? So isn't that angle suppose to be 90 degrees ?",
        "A": "No. Interior angles in a triangle must sum to 180. The angles derived at that point are 29, and 59. 180 - 29 - 59 = 92 The math must dictate the actual nature of picture; an imprecise picture cannot dictate the math."
    },
    {
        "video_name": "eTwnt4G5xE4",
        "Q": "I don't understand the video at 2:05",
        "A": "We know that the sum of interior angles in a triangle equal 180\u00c2\u00b0. Let s say the third angle is a . So 59\u00c2\u00b0 + 29\u00c2\u00b0 + a = 180\u00c2\u00b0 a + 29\u00c2\u00b0 = 180\u00c2\u00b0 - 59\u00c2\u00b0 a = 180\u00c2\u00b0 - 59\u00c2\u00b0 - 29\u00c2\u00b0 a = 180\u00c2\u00b0 - 88\u00c2\u00b0 a = 92\u00c2\u00b0 We also know that vertical angles are equal so the angle on the opposite side is 92\u00c2\u00b0. We can use this logic to also find the third angle there as well. We get 59\u00c2\u00b0 Knowing that vertical angles are equal we know that x must be equal to 59\u00c2\u00b0. Hope this helps."
    },
    {
        "video_name": "eTwnt4G5xE4",
        "Q": "AT 1:49 i got confused. why do you have to subtract all 3? cant you add 2 of them and then subtract by 180?",
        "A": "Yes you can! It is the same thing after all!"
    },
    {
        "video_name": "eTwnt4G5xE4",
        "Q": "at 2:30 he says sides but he actually meant to say angles ? Is there a difference between the two ?",
        "A": "Yes he should ve said angles. He corrected himself a bit later. An angle is created by two lines, while a side (of a geometrical figure, here: a triangle) is (a part of) a line."
    },
    {
        "video_name": "eTwnt4G5xE4",
        "Q": "At 1:10, Kahn gets that the supplementary angle of the 121\u00c2\u00b0 angle is 59\u00c2\u00b0. So from there couldn't you just reason that since the two intersecting lines of the only horizontal line are both parallel, that the angles would be the same for where the other parallel line intersects? He gets there eventually, but I just thought this way was much faster and easier for me to understand. Is it wrong to do what I did?",
        "A": "You are correct...there is often more than one way to reason through these problems."
    },
    {
        "video_name": "eTwnt4G5xE4",
        "Q": "129 said instead of 121 in 4:30",
        "A": "Good point, but please only post questions in this tab. Comments like the one you posted here should go under Tips & Thanks. Thanks"
    },
    {
        "video_name": "0oGJTQCy4cQ",
        "Q": "So at 4:02, when Sal says algebra of matrices, does he mean that instead of a number you can have a variable in a matrix?",
        "A": "Yes. Sometimes, problems will ask you to solve for some variable that is in a matrix."
    },
    {
        "video_name": "0oGJTQCy4cQ",
        "Q": "what exactly is dimension you mentioned in 2:55?",
        "A": "If a matrix has 3 rows and 2 columns, its dimensions are 3x2."
    },
    {
        "video_name": "0oGJTQCy4cQ",
        "Q": "At 2:24, Sal draws a 3 * 2 matrix. is there a 4 * 4 matrix?",
        "A": "Yes, there is a 4*4 matrix. There is, in fact, an infinite amount of matrix dimension possibilities. However, the dimensions MUST be whole numbers. You can t have a negative amount of dimensions or a fraction of a dimension."
    },
    {
        "video_name": "0oGJTQCy4cQ",
        "Q": "At 2:48 in the video, It says that the matrix is a 3 by 2 matrix,( or 3*2 ). So because of the multiplicative inverse of multiplacation and addition, Could it also be written as a 2 by 3 matrix?",
        "A": "No: in matrices, the first number is the height of the matrix and the second number is the width of the matrix. The multiplicative inverse of multiplication and addition is not used here because you are not multiplying 3 by 2."
    },
    {
        "video_name": "0oGJTQCy4cQ",
        "Q": "Didn't Sal spell the word \" columns\" wrong at 1:21?",
        "A": "Nikki, Yes, at 1:21 he misspelled columns carelessly omitting the n . Sal makes mistakes just like you or I do. None of us are perfect, even Sal. But at Khan Academy, we can keep trying to get closer to perfect. My careless math mistakes dropped substantially when I was forced to get 10 questions in a row correct to be proficient in a skill at Khan Academy."
    },
    {
        "video_name": "d5UCZ9hO8X4",
        "Q": "At 1:00 it says it is AAS, but if you do it the other way, wouldn't it be SAA?",
        "A": "They are the same"
    },
    {
        "video_name": "l9ft9jpriNA",
        "Q": "At 0:13, I think the problem assumes that those three people are in the club?",
        "A": "Yes, exactly."
    },
    {
        "video_name": "l9ft9jpriNA",
        "Q": "I don't understand the notation being used in this video at 1:49.\n\nIn earlier videos, as well as at the beginning of this one, the numerator is said to be the number of possibilities that meet the given constraints and the denominator is said to indicate the total number of possibilities.\n\nBut here, Sal wrote 9/P * 8/VP * 7/Sec. How are President, Vice President and Secretary the total number of possibilities?\n\nShouldn't it be the other way around? P/9 * VP/8 * Sec/7?",
        "A": "9/P, 8/VP, 7/Sec here its nt written in fraction terms, he just wants to say tht thr r 9 contenders for prez, 8 for vp so on.the desired outcome is only 1 and the total no of possible outcomes are 9.8.7=504. so P=1/504. hope it helps!!"
    },
    {
        "video_name": "l9ft9jpriNA",
        "Q": "ex: Why does Sal NOT use shorthand notation at 5:43?!",
        "A": "Nice."
    },
    {
        "video_name": "Dpo_-GrMpNE",
        "Q": "At 0:17 Sal said \"e\" was approx. 2.71..., but wouldn't it be 2.72 because the next digit is 8?",
        "A": "Yes, i think so... e is defined as 2.718281828."
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "At 6:53, why not worry about the right angle? Why aren't we solving for it instead?",
        "A": "Since the right angle is given, it is unnecessary to solve for it instead."
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "At around 3:43, Is the circumference the inner part of the circle and the outer part or outline of the circle the perimeter? Also, do you divide 360 degrees by the denominator of the fraction?",
        "A": "the circumference is the out line of the circle( like the bark of a tree) and yes to your second question like he said 1/6 of 360 degrees is 60 degrees or 1/4 of 360 degrees is 90 degrees for example"
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "at 5:08\nwhy does he do so many examples",
        "A": "To help you understand better"
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "What's radians at the time 1:11?",
        "A": "Not exactly, radians are not like degrees. One degree is pi/180 radians. 2 radians is 360 degrees."
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "at 8:40 why is a 90 degree angle called a right angle?",
        "A": "The right in right angle is based on a Latin word and means upright . It s not that the angle is correct or turns to the right . It means that if you put a stick in the ground at 90 degrees, it will be upright or perpendicular to the ground. Many of the words and terms we use everyday come from Latin and Greek."
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "At 0:46, if Sal didn't put those white curves there could we have chosen which way the angle opens?",
        "A": "Yes, I think so. But, he always put the white curve to define where is the angle he is talking about."
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "What is the Angle called inside of the circle but outside of the vortex, at 7:06",
        "A": "It is called a reflex angle: greater than 180* but less than 360* Google it and you ll find the names for other types of angles."
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "At 1:14 what are the lines around the circle?",
        "A": "It s representing the rotation of the ray in 360 degrees to create an angle that s circular."
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "At 3:55 how did he know that it was 1/6 of the circle?",
        "A": "He doesn t really know that. He just said what if so that he could use it for an example to show that one sixth if a circle is 60 degrees."
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "at 7:29 why he chose 270-degree instead of 90 . this point is confusing me a bit",
        "A": "becuase it is talking about the exterior, not the interior part. Hope this helped"
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "At 4:16 Sal said, \"if the other ray were here it would be 0 degrees.\" Why couldn't it also be 360 degrees?",
        "A": "yes, you are right for all angles, but if you have a reflexive angle which goes the other way around which is 360 degrees."
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "1:19 what are radians",
        "A": "the radius being laid along the circumference Take the Radius of the circle and slap it on the curved edge."
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "Why does Sal when writing \"circumference\" make a dot at the top of the word, if there is no \"i\"?. At 6:16 you can see that he has written all the circumferences with a dot in top, in the first one it is over the\"u\", in the second one it is in top of the \"e\" and on the last one he wrote it above the \"m\".",
        "A": "Perhaps this is just to prove his point, as an exclamation."
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "Why did Sal write 360=365 when 360 isn't equal to 365 ?(at 2:32)",
        "A": "also, he wrote it with squiggly equal sign, while he said 360 days is almost, it s just about 365 days... and did not refer to 360 being equal to 365. You can write the squiggly equal in math and it means approximately, it helps to have the context."
    },
    {
        "video_name": "D-EIh7NJvtQ",
        "Q": "For the angle drawn at 7:30, if you were to label one ray as AB-> and the other as ray AC->, how would you write that so then you do not ask for a 90 degree angle, but instead for the 270 degree angle?",
        "A": "I am not relay sure but i think it would be 3/4 of the circumferences of the circle"
    },
    {
        "video_name": "EV57jv7JKCs",
        "Q": "At 4:53 why did he choose to make X equivalent to 2?",
        "A": "Sal is finding the line of symmetry or the x value for the vertex. It will always be half way between the 2 x-intercepts. Sal does the calculations at about 3:55. Hope this helps."
    },
    {
        "video_name": "EV57jv7JKCs",
        "Q": "At 1:49 why would i want to put the equation to equal 0?",
        "A": "If you set y=0, you can find the x-intercepts for the equation by solving for X. These are also sometimes called the roots or the zeroes of the equation. -- We can do this in many cases by factoring the quadratic. This requires that we use the zero product rule, which only works if one side of the equation is zero. -- If the quadratic is not factorable, then we would likely use the quadratic formula. This also requires that one side of the equation be zero. Hope this helps."
    },
    {
        "video_name": "ilWDSYnTEFs",
        "Q": "At 5:33, there's something that's called an \"open circle\" and at 5:43 there's something called a \"closed circle\" on the numberline, what's the difference between the two?",
        "A": "When there is a closed circle on the number, that means that the value could be equal to the number it is on. When there a an open circle on a number, that means that the value is not equal to the number it it on. For example, if there is a closed circle on 6 and the arrow is pointing to the right, then the value could be greater than or equal to 6. If there is an open circle on 6 and the arrow is pointing to the right, then the value is definitely greater than 6."
    },
    {
        "video_name": "ilWDSYnTEFs",
        "Q": "Why didn't he count from 0 to 1500 at 2:31 ?",
        "A": "Cause that IS a lot of WORK"
    },
    {
        "video_name": "ilWDSYnTEFs",
        "Q": "at 7:07, who is Sal? is Sal the person who is talking during the video? If so, why did is he called Sal?",
        "A": "Why are you called Talia? It s your name. Sal is his name. Sal is the founder of this great learning site, and many of the videos were made by him."
    },
    {
        "video_name": "ilWDSYnTEFs",
        "Q": "at 3:05 and 3:56, what is the difference between the number lines?",
        "A": "Hello, gittensnina! The difference is the little circle. One is filled, and the other is open. The filled circle means either greater than or equal to or smaller than or equal to. The open circle means < or >. I hope this helped!"
    },
    {
        "video_name": "ilWDSYnTEFs",
        "Q": "4:31 He missed up what he said. I heard at least and then he ended up showing us, greater than 64 instead of equal to or greater than 64, like he read. Just throwing that out there and i know its not a big deal.",
        "A": "But right after he said he heard we need at least 64 ounces a day, at 4:39 on the transcript, he says there s one way I could think about, where I always want to drink more than 64 ounces, so that would be W is greater than 64. So he hears one thing, and says a different thing which matches what he writes."
    },
    {
        "video_name": "ilWDSYnTEFs",
        "Q": "At 2:13 to 2:15 he says \"15, 1500\" I am confused. Can you fix this?",
        "A": "he wasn t finished writing what he was saying, so he did that"
    },
    {
        "video_name": "ilWDSYnTEFs",
        "Q": "at 2:03 how is it less than or equal to?",
        "A": "Because you put the line underneath it, he wants to eat 1500 or less calories a day, it is including 1500. If it is just <1500 it means ONLY less than 1500. I hope this answered your question."
    },
    {
        "video_name": "ilWDSYnTEFs",
        "Q": "What does the line under the greater or less than sign that Sal puts under the number at 1:48 mean?",
        "A": "The symbol \u00e2\u0089\u00a5 means greater than or equal to. For example: 7 \u00e2\u0089\u00a5 2 is true because 7 is greater than 2. 7 \u00e2\u0089\u00a5 7 is also true because 7 is equal to 7. 7 \u00e2\u0089\u00a5 11 would be false because 7 is not greater than, nor does it equal 11. The symbol \u00e2\u0089\u00a4 means less than or equal to For example: 7 \u00e2\u0089\u00a4 9 is true because 7 is less than 9. 7 \u00e2\u0089\u00a4 7 is also true because 7 is equal to 7. 7 \u00e2\u0089\u00a4 3 would be false because 7 is not less than, nor does it equal 3. Hope this helps."
    },
    {
        "video_name": "Pb04ntcDJcQ",
        "Q": "As sign error at 8:55? f(y) should be -1 not 1?",
        "A": "A ok I was too quick with my question"
    },
    {
        "video_name": "Pb04ntcDJcQ",
        "Q": "In 5:10 , Why do we plus f(y) , not plus C ? Why do we have this ideal ?",
        "A": "We are treating y as a constant. So, that must mean that f(y) is a constant too, with respect to x. The reason we use f(y) instead of C is because we can t be sure that the constant doesn t actually depend on y, so we try to be more general."
    },
    {
        "video_name": "Pb04ntcDJcQ",
        "Q": "at 3:30 how is the anti-derivative of f'(y)=1 is f(y)=y+c instead of f(y)=c?",
        "A": "The antiderivative of a constant is a polynomial of the first order, remember that the power rule when integrating adds one power to the variable, and a constant is like having a variable to the 0 power, so after integrating you would get the variable to the 1 power. That is why the 1 becomes y."
    },
    {
        "video_name": "Pb04ntcDJcQ",
        "Q": "At 8:56, the right side should be -1 not 1, right? Sorry if it has been addressed in the previous comments already.",
        "A": "Yes, and he corrected himself a little later."
    },
    {
        "video_name": "Pb04ntcDJcQ",
        "Q": "At 9:10 , why isn't f(y) = -y plus some function of x?",
        "A": "It can t have x in it by definition. Because f(y) was defined as a sole function of y @6:15 , as it was referring to constant value of the partial integral of psi with respect to x."
    },
    {
        "video_name": "Pb04ntcDJcQ",
        "Q": "What happened to y prime after taking derivative with respect to x (Nx) 12:10",
        "A": "If y has no coefficient function of x attached to it, it can be considered to have a coefficient of 1 (from algebra). The derivative of 1 with respect to x is 0, so the derivative of (1)y with respect to x is (0)y = 0. So basically it disappears."
    },
    {
        "video_name": "Pb04ntcDJcQ",
        "Q": "is the second part of the eqn sinx + x^2e^(y-1) or is the -1 not in the exponent? And at 8:55 you wrote f'(y) = 1. But isn't it -1?",
        "A": "You are right and later on the video he fixed it :)"
    },
    {
        "video_name": "Pb04ntcDJcQ",
        "Q": "how come nobody pointed out that M(subscript y) = cosx + 1x e^y should be M(subscript y) = cosx + 2x e^y at 2:42 that is green that threw me off.. i was how are they equal... i was lost just by one mistake. i was huh? but i started watching other videos and understood it was just human error but everything else is great! thank you, you guys are awesome",
        "A": "He said it was 2, and wrote a funny looking 2 which totally looks like 1, but is actually a 2. So that s why nobody pointed it out."
    },
    {
        "video_name": "Pb04ntcDJcQ",
        "Q": "At 5:31, how does the antiderivative of psi x equals to psi ?",
        "A": "The x in Psi x is a subscript. Psi x represents the partial derivative of Psi with respect to x. Then obviously the antiderivative of the derivative is the original. Since it is a partial derivative, that is why he adds the H(y). In order to account for and figure out the functions of y lost by only having the partial derivative with respect to x."
    },
    {
        "video_name": "Pb04ntcDJcQ",
        "Q": "At 3:30, he says we can rewrite the original equation as d/dx Psi = 0. Aside from the 0, I see no relationship to the original equation.",
        "A": "Look for the similarities between the multi-variable chain rule derivative of psi with respect to x (d/dx Psi). You will notice that the form of d/dx Psi is the same as the original equation."
    },
    {
        "video_name": "8V4_4M90RfA",
        "Q": "4:34 I can't understand in this Video what for we are computing second deriative for T.",
        "A": "The first derivative regards to the tangent of the curve, and this is not what we are looking for; we are looking for the change of the direction. In other words: the change of the slope (first derivative). So, derivating the tangent (first derivative), you get an idea about the change in the direction of the slope. Try to see the second derivative like a tangent of the tangents, this is what we are looking for. Sorry about my english."
    },
    {
        "video_name": "H0q9Fqb8YT4",
        "Q": "At 1:09 I thought that multiplying negative nombre by a even positive nombre is a positive nombre",
        "A": "A negative x a negative = a positive. (-)(-) = (+) A negative x a positive = a negative. (-)(+) = (-) A positive x a negative = a negative. (+)(-) = (-)"
    },
    {
        "video_name": "H0q9Fqb8YT4",
        "Q": "Around 3:10 you said you could rewrite -1/2 as -2/1 or 2/-1, correct? If we wrote it the latter way, we'd end up with 8/-1. Is that equal to 8 or negative 8? Thank you.",
        "A": "You cannot rewrite -1/2 as -2/1 or 2/-1. But -2/1 and 2/-1 are equivalent. If the number of negative signs in a multiplication or division problem is odd, the result will be a negative. If the # is even, the result will be positive"
    },
    {
        "video_name": "H0q9Fqb8YT4",
        "Q": "He lost me at 2:54 all the way to the end",
        "A": "re-watch the video if you don t get it the first time."
    },
    {
        "video_name": "H0q9Fqb8YT4",
        "Q": "At 1:16 is it supposed to be 3/4, and then be changed to 4/3?",
        "A": "yes because when your dividing for example 3/2 / 3/4 you would switch the second fraction to 4/3"
    },
    {
        "video_name": "H0q9Fqb8YT4",
        "Q": "At 1:35 why didn't Sal just cross reduce, i mean like 6 and 4 can both be divided by 2 so it would save him a ton of extra work that you don't have to do.",
        "A": "The concept might still be new for some people and Sal didn t want to confuse you by introducing so much to you at once"
    },
    {
        "video_name": "H0q9Fqb8YT4",
        "Q": "why at 1:40 dose he divide both of the numbers by 2",
        "A": "what you do at the bottom you do at the top"
    },
    {
        "video_name": "H0q9Fqb8YT4",
        "Q": "hi i really don't get why he did that at 3:00",
        "A": "he was dividing fractions when you divide you take the second fraction and flip it around and then multiply so if it was -3/7 divided by 4/8 it would be -3/7 times 8/4"
    },
    {
        "video_name": "H0q9Fqb8YT4",
        "Q": "at 1:02 can we put the- at 6",
        "A": "If you mean is -5/6 equal to 5/-6 ? Sure it is equal. A fraction is definitely negative if either its numerator or its denominator is negative . Take the negative sign outside , solve the fraction taking both numerator and denominator as positive and than put the negative sign on the answer."
    },
    {
        "video_name": "H0q9Fqb8YT4",
        "Q": "What does he mean by Negative? 2:33",
        "A": "Have you not learnt about negative numbers yet? If you haven t I can tell you some videos to watch to learn about negatives (:"
    },
    {
        "video_name": "33J1JHSTv2k",
        "Q": "For the third example, at 7:46, I understand that the graph is not continuous as x approaches c; however, would the graph be continuous if you specifically said as x approaches c from the left? That statement satisfies the definition relayed at the beginning of the video",
        "A": "The graph would still not be continuous as a whole. BUT, the CLOSED INTERVAL from negative infinity to c would be continuous, which is what the definition of continuity implies."
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "At 3:45 sal said a area with 0 is a ray is that true?",
        "A": "I m not sure I get what you re asking, but an angle that measures 0 degrees is basically a ray."
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "At 6:43-7:44 the video says obtuse and acute angles. How do remember that acute angle is the small one and obtuse angles are the big one.",
        "A": "Like Sal said, an acute angle is cute and therefore small."
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "At 6:10, If a 180 degree angle is the same thing as a line, in any other circumstance could we label a line as an a 180 degree angle instead?",
        "A": "Well, if you re in a situation involving angles, label it as an angle. On the other hand, if it involves lines, call it a straight line. But whichever way is easier for you works."
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "At 00:13,can there be a 200 degree angle?",
        "A": "An angle between 180 degrees and 360 degrees is called a reflex angle."
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "at 5:54 he says that Z, X, and Y are COLLINEAR. What does that mean??",
        "A": "Colinear jut means that those points are on the same line."
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "At 3:58 it says that a 0 degree angle is just a ray at that point, but could it also be just a line segment?",
        "A": "Yes, but usually an angle has rays, not line segments. But it can."
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "At around 1:00, Sal refers to the lines of an angle as rays. But wouldn't these lines be line segments?",
        "A": "These would be considered rays because they continue on past where Sal has drawn them. You can see this because Sal has drawn arrows on the end of the lines."
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "what does m stand for at2:18",
        "A": "m means measure ."
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "Are there any times when anybody would use the little box at 5:10 other than in a right angle? Also, why do we need to learn about angles? Is it important in the modern day?",
        "A": "No, that box always signifies right angles. Angles are incredibly important in architecture, physics, chemistry and a whole bunch of other subjects."
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "What is ''triometry'' and ''radiuns'' at 2:50 ?",
        "A": "Do you mean trigonometry"
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "At 2:49, Sal mentions radians. Could someone please explain this?",
        "A": "Radians are just another unit for measuring angles in. 2pi radians is the equivalent of 360 degrees and to convert from radians to degrees you multiply your radian measurement by 180/pi, and to go from degrees to radians you multiply your degree measurement by pi/180. Radians are used in a lot of upper level math courses for measuring angles."
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "At 9:49 how is the protractor able to measure 245 degree angles?",
        "A": "you cuold use it twice to get to 245. Or you could use a 360 degree protractor."
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "At 5:47 an horizontal line can be called an 180 degree angle?",
        "A": "yes of course !"
    },
    {
        "video_name": "92aLiyeQj0w",
        "Q": "at 8:22, when he said when you get a 180 degrees, shouldnt he have said straight angle, because i believe the actual name for an 180 degrees angle is called a straight angle!",
        "A": "In the same way as saying both 90 degree angle or right angle are equivalent, 180 and straight are both correct and can be used interchangeably, although calling something a straight angle is not common in more sophisticated mathematics."
    },
    {
        "video_name": "0_VaUYoNV7Y",
        "Q": "At 3:26 Sal multiplies both sides of the inequality by b and because he knows b < 0, he says you have to switch the direction of the inequality. What if he didn't know whether b was negative or positive? Does this mean you cant multiply both sides of an inequality by a variable unless you knows it's sign?",
        "A": "No, you can not multiply or divide both sides of an inequality by a variable if you are not sure if it is positive or negative."
    },
    {
        "video_name": "0_VaUYoNV7Y",
        "Q": "At 3:01, Khan said 'absoulute value'.\nWhat is absolute value?",
        "A": "The absolute value of a number is its distance from zero, IRREGARDLESS OF ITS DIRECTION. So, the absolute value of 5 is 5 and the absolute value of -5 is also 5."
    },
    {
        "video_name": "0_VaUYoNV7Y",
        "Q": "at 4:04 why is b squared",
        "A": "Jonathan, You had a/b < ab You multiplied each side by b to get a*b/b < ab*b and the b/b is 1 and disappears so you get a < ab * b which is a < a *b * b and b*b is b\u00c2\u00b2 so a < ab\u00c2\u00b2 That is where the b\u00c2\u00b2 came from. Then when you divide both sides by a you get 1 < b\u00c2\u00b2"
    },
    {
        "video_name": "eGo8C2Jshzs",
        "Q": "At around 3:00, Sal probably should have factored out a 16.",
        "A": "You should have posted this in tips"
    },
    {
        "video_name": "xZRRPOneU2c",
        "Q": "At 2:41, why cant you use hours per meter as a rate ?isn't that the same thing as meters per second?",
        "A": "The accepted convention for measuring speed is to do distance per time rather than time per distance. Could you use meters per hour rather than meters per second? Yes. If you already have meters per second, you can convert it to meters per hour. Though, you may also want to increase the distance measurement to kilometers per hour."
    },
    {
        "video_name": "GAmzwIkGFgE",
        "Q": "At 3:30 and 4:04, why do you divide the \"Y's\" and the \"X's\" by 3? Where did the \"three\" come from?",
        "A": "he was just calculating mean by adding up all the numbers then DIVIDING BY THE TOTAL NUMBER OF DATA POINTS, which is 3"
    },
    {
        "video_name": "GAmzwIkGFgE",
        "Q": "1:40 so that entire hairy equation gives us b = the mean of y - m * the mean of x?",
        "A": "Yup! Absolutely craazy!!"
    },
    {
        "video_name": "8Eb5MWwcMMY",
        "Q": "At 0:12, why is the \"missing\" sign a + and not a x like it would in something like \"7y\"?",
        "A": "For example; 1 1/2 we say as one and a half or, 1 + 1/2 1 multiplied by a half would just give a half Much like decimals, 3.5 would be 3 + 0.5, mixed fractions work in that way, 3 1/2 = 3 + 1/2 Does that clarify it for you?"
    },
    {
        "video_name": "8Eb5MWwcMMY",
        "Q": "At 1:18 where does he get the 6 to multiple 2 and 3 with??",
        "A": "The common denominator needs to be 18. The fraction with 3 in the denominator needs to be converted to 18. What times 3 will create 18? The number is 6. That is where the 6 is coming from. Hope this helps."
    },
    {
        "video_name": "8Eb5MWwcMMY",
        "Q": "At 1:54, I don't get how to simplify the fraction. What does he mean by divisible by 3? Is it like a common denominator?",
        "A": "lets take 9/12. we can divide 9 by 3. It equals 3. then we can take 12 and divide it by 3. Its 4. put them together to get 3/4."
    },
    {
        "video_name": "gcnk8TnzsLc",
        "Q": "Hey Sal when you get the answer at 3:59, it is acceptable to write the answer as 4p+12/5 or do we have to leave in the factor state?",
        "A": "I think you can reduce it to 4p+12/5 like you said. He just probably said to himself that it was simplified enough to end the video there."
    },
    {
        "video_name": "gcnk8TnzsLc",
        "Q": "Why doesn't Sal just factor out the (p+5) before he multiplied? What he seems to do at 2:50 is multiply the 2 fractions then reduce. Why doesnt he cross-cancel before he even multiplies?",
        "A": "because then you wont have to simplify at the end."
    },
    {
        "video_name": "gcnk8TnzsLc",
        "Q": "I just noticed this at 3:40 but im sure it happened earlier. When Sal factored 4p+20, it looks like he made it 4(p+6). Is that really 4(p+5)?",
        "A": "Yes, that s supposed to be a 5."
    },
    {
        "video_name": "gcnk8TnzsLc",
        "Q": "at around 3:56 in the video, Sal said and wrote the final answer is 4(p + 3) over 5. Could one actually distribute the 4 to get 4p +12 over 5 and still be accepted as an answer?",
        "A": "Do you care? What s the difference?"
    },
    {
        "video_name": "gcnk8TnzsLc",
        "Q": "At 2:57 when Sal is writing the denominator, why can't he just distribute the 10 to the P+5?",
        "A": "He wants to keep the (p+5) as a separate factor so he can cancel it out with the (p+5) in the numerator."
    },
    {
        "video_name": "2RnS3fSHVV8",
        "Q": "Why did Sal say at 1:47 that x>-4 is positive? It could be negative (i.e. -3).",
        "A": "He didn t. He said x + 4 > 0 is positive, if x = -3 then x + 4 > 0 -3 + 4 >0 1 > 0 The expression is still positive."
    },
    {
        "video_name": "2RnS3fSHVV8",
        "Q": "I'm not quite sure why the inequality was swapped at 04:12? I understand that when you divide by a negative this is done but I don't see the negative-- is there a video that goes over this?\nThanks in advanced",
        "A": "If x < -4, When we multiply the inequality by (x + 4), we are actually multiplying the inequality by a negative number. How? ... When we assume x < -4 then the greatest value that can be attributed to x is -5. Substituting we have: (x + 4) ---> (-5 + 4) = -1 We are multiplying the inequality by a negative number. Hope this helps!"
    },
    {
        "video_name": "2RnS3fSHVV8",
        "Q": "At 6:55 Sal says the second method only works when a/b > 0 and then rearranges the equation as so. Why does this only work when one side of the inequality is a zero? Surely in the original problem a/b > 2 the same logic applies, both a and b must be positive or negative to reach a number greater than 2?",
        "A": "It is because inequalities have to have zero to balance the equation."
    },
    {
        "video_name": "2RnS3fSHVV8",
        "Q": "does sal say \"gecko\" at 9:15? i don't understand english very well. so forgive me.\nthanks.",
        "A": "we could ve written that from the get go As in we could ve written that in the first instance. Or straight away."
    },
    {
        "video_name": "2RnS3fSHVV8",
        "Q": "At 7:25 doesn't he mean when you multiply or divide by a negative number you switch the signs, not just multiply?",
        "A": "yes. But technically i guess, you can think of dividing as multiplying a fraction. Either way, yes. When you multiply or divide the inequality by a negative number, you switch the sign."
    },
    {
        "video_name": "ZCnsO65rWX0",
        "Q": "At 3:40, you said that the correct answer is x=-3 but isn't this the graph of g'(x)?? The question is asking for the left most point of inflection in the graph of g(x) not g'(x)",
        "A": "That is true, however don t forget that they are all still the same function. Which means that g (x) represents g(x), so if you understand how the graph of g (x) works you can know what the graph of g(x) will generally look. Also when you are trying to find inflection points the way you find it is by understanding at which points the slope changes (As Sal has already mentioned in the video) and since g (x) shows the slope of g(x) we can find at which points the slope of g(x) changes. I hope you understood it :)."
    },
    {
        "video_name": "hJOqGkWElNs",
        "Q": "At 3:02, Sal says you could have solved this with substitution. How could you do that?",
        "A": "EpicOne, The system of equations was -3y+4x=11 y+2x=13 If you subtracted 2x from both sides of the second equation to get y=13-2x Now you can substitute (13-2x) for the y in the first equation. -3(13-2x)+4x=11 Then you can solve for x. After finding x, Then substitute that answer for x in either equation and solve for y. The substitute both answers into the other equation to double check. If the equation is in balance, you have the correct answers. I hope that helps."
    },
    {
        "video_name": "hJOqGkWElNs",
        "Q": "why do you mutiply by three at 1:06",
        "A": "I hope I can make my answer understandable: It doesn t necessarily need to be a three, it can be any number, positive or negative, as long as both x or y values match."
    },
    {
        "video_name": "S34NM0Po0eA",
        "Q": "At 4:48, why is 3/2*3/2=9/2? Please help?",
        "A": "It is not 9/2 it is 9/4. This is because 3*3 is 9 and 2*2 is 4."
    },
    {
        "video_name": "S34NM0Po0eA",
        "Q": "At 5:42 why does (27^1/3)^2 = 3^2? and the same for (8^1/3)^2?",
        "A": "Think about it: What is (1/3)*2? When you raise exponents to exponents in this situation you need to multiply them. So 27^(1/3)^2 = 27^(3/2) and the same for your other example."
    },
    {
        "video_name": "S34NM0Po0eA",
        "Q": "At 4:48, why is 3/2*3/2=9/4? Please help?",
        "A": "First, lets multiply the numerators, which are the numbers at the top. 3*3 = 9 This is your new numerator. Next, lets multiply the denominators. 2*2 = 4 This is you new denominator. This is why 3/2*3/2=9/4."
    },
    {
        "video_name": "S34NM0Po0eA",
        "Q": "at 4:53 did Sal mean numerator",
        "A": "Yeah... I think he meant numerator."
    },
    {
        "video_name": "S34NM0Po0eA",
        "Q": "Where does that ''4'' comes from 1:57 ?",
        "A": "The cube root of 64, written as 64^(1/3) is 4, since 4\u00c2\u00b3 or 4x4x4=64."
    },
    {
        "video_name": "S34NM0Po0eA",
        "Q": "Can we simplify 9/4 more ,if we divide by 2. 5:47",
        "A": "Yes, 9/4 could be simplified to 2 1/4"
    },
    {
        "video_name": "vRw4ovZK1CQ",
        "Q": "At 2:23 Sal makes a typo, writing \"b\" instead of \"m\", intending slope.",
        "A": "Yup well spotted!However in the previous video he explained local linearity without making any mistakes...Therefore, he has been forgiven :)"
    },
    {
        "video_name": "vRw4ovZK1CQ",
        "Q": "At 4:16, the denominator should be 2, not -2",
        "A": "Sal is only saying that (-1 - 1) = -2. He then squares the -2 which results in a positive 4 in the denominator."
    },
    {
        "video_name": "3_caioiRu5I",
        "Q": "At 0:43, what if the decimal is something like 1.28? Can I write 128%, or am I missing something?",
        "A": "I didn t watch the video, but yes, if it is 1.28 then the equivalent percent is 128%."
    },
    {
        "video_name": "3_caioiRu5I",
        "Q": "at 1:23 Sal says \"As 150.1 percent, which is the same thing as 150.1%.",
        "A": "Yea, he s just writing all the ways you can display it."
    },
    {
        "video_name": "3_caioiRu5I",
        "Q": "At 1:12 i don't get how he put 150.1/100??\n\nPlease help me!",
        "A": "Sal created an equivalent fraction by multiplying both the numerator and denominator by the same value: 1.501 /1 (100/100) = 150.1/100"
    },
    {
        "video_name": "3_caioiRu5I",
        "Q": "at 0:25 does it always have to be times 100?",
        "A": "Most of the time. If you are finding a weird percent that s not per CENT and you have to multiply it by, like, 73, no. These are very rare,though, so you probably won t see them for a while."
    },
    {
        "video_name": "640-86yn2wM",
        "Q": "At 1:10 and at 1:32, for those questions I write the answers as :7(r) and 10(u).\nIs that write because that's how I learned it in school.",
        "A": "7(r) and 10(u) do mean 7 times r and 10 times u , so they do answer the question correctly. However, in mathematics, we usually want to write things as simply as possible, especially when parentheses are involved. That s why 7r and 10u are preferable answers."
    },
    {
        "video_name": "640-86yn2wM",
        "Q": "At 1:08, Sal says \" Lets do a couple more of these. This is a lot of fun! \" How is math a lot of fun?",
        "A": "Why do you think there are those of us who sit around and try to answer these questions? We are not being paid to do it, we do it because Math is fun."
    },
    {
        "video_name": "uk7gS3cZVp4",
        "Q": "0:18 In the UK it's y = mx + c instead. Why is it different?",
        "A": "The letters used for the variables and constants are arbitrary. Some places use different letters than others. Some even use Greek letters."
    },
    {
        "video_name": "uk7gS3cZVp4",
        "Q": "On 2:01, is change in y over the change in x mean the same thing as rise over run?",
        "A": "Dear Anshu, It means the same thing"
    },
    {
        "video_name": "uk7gS3cZVp4",
        "Q": "At 0:43 and 1:23, is b still the y-intercept if x is not zero?",
        "A": "Emma, For the equation y= mx + b, the point (0,b) is the y-intercept. Though we often say that b is the y intercept, what we really mean is that the y intercept is a the point (0,b). It would be more correct to say the point (0,b) is the y=intercept. In math, we often talk in shorthand calling b the y intercept. That can cause some confusion if someone does not understand what we say and what we really are slightly different. I hope that helps."
    },
    {
        "video_name": "uk7gS3cZVp4",
        "Q": "At 2:50 you go up on the 4th co-ordinate to plot a point but on the 3rd co-ordinate you go down. Why can't we go up in the 3rd co-ordinate too?",
        "A": "The slope of this particular line is 1/3. In order to maintain a positive slope, we either need to go forward and up or back and down (effectively, -3 back and then -1 down to have slope of (-1)/(-3)= 1/3). If you went back and up, you would have a negative slope: 1/(-3)= - 1/3."
    },
    {
        "video_name": "uk7gS3cZVp4",
        "Q": "at 1:33, what does sal mean by \"how much we change in y for a change in x\"?",
        "A": "It basically means, how many units do we go up for every unit we go right."
    },
    {
        "video_name": "uk7gS3cZVp4",
        "Q": "At 0:40 what does he mean when x is equal to zero? How do you know its the y intercept?",
        "A": "When the x coordinate is 0, you haven t moved sideways at all from the y axis, so the coordinate with x = 0 in it will be on the y axis somewhere."
    },
    {
        "video_name": "uk7gS3cZVp4",
        "Q": "At 2:10, why do you go 3 to the right?",
        "A": "with a slope of 1/3, change in y is 1 and change in x is 3, so while I would prefer for students to learn to go up 1 and right 3, he does the x first and goes right 3 (positive direction) and up 1 (positive direction), and later he went left 3 (negative direction) and down 1 (negative direction) because a negative number divided by a negative number is also positive."
    },
    {
        "video_name": "uk7gS3cZVp4",
        "Q": "At 1:17, are m and b the same as x and y in this context?",
        "A": "No, they aren t. In order to make it fairly easy to graph a straight line, its equation is often given in the slope-intercept form of y = mx + b. The m represents the slope of the line and is the coefficient of x. The b represents the y-intercept (where the line crosses the Y-axis) and is the constant term."
    },
    {
        "video_name": "uk7gS3cZVp4",
        "Q": "What is x used for? he said the formula in 1:1",
        "A": "Thank you. Its actually does help me"
    },
    {
        "video_name": "uk7gS3cZVp4",
        "Q": "12:18 when the slope is Y = 1/3 X, shouldn't that mean that 1 movement of Y equals 1/3 movement of X (or 3Y for 1X). How come it is the opposite ?",
        "A": "Slope is defined as the Change in Y over the Change in X. In other words, it is Y/X. So if the slope is 1/3, it means to 1 move for Y (one unit up) and 3 moves for X (3 units to right)."
    },
    {
        "video_name": "uk7gS3cZVp4",
        "Q": "At 2:15, he graphs the run and then the rise. Shouldn't you be graphing rise first and run second?",
        "A": "That is not necessary. Both will give same results and it is completely up to you what to do first."
    },
    {
        "video_name": "uk7gS3cZVp4",
        "Q": "At 3:41, how could you use the slope and y-intercept to graph the function f(x)=3?",
        "A": "What is the slope of y = f(x) = 3? Slope is rise over run , or \u00ce\u0094y/\u00ce\u0094x, correct? There is no rise, so \u00ce\u0094y=0, giving 0/\u00ce\u0094x. But 0 divided by anything is 0, so the slope is 0. Now, the y intercept happens when x=0. When x=0 the value of y=3 (actually for ANY value of x y=3) so the intercept is 3. So in slope intercept form we have: y = mx + b with m=0 and b=3, which means that mx + b = (0)x + 3 = 3. Thus the graph is a horizontal line parallel to the x axis passing through y=3"
    },
    {
        "video_name": "yUpDRpkUhf4",
        "Q": "Why can i put ln in front of the limit ? at 8:00\nThanks",
        "A": "ln(1+u) is continuous"
    },
    {
        "video_name": "yUpDRpkUhf4",
        "Q": "at 8:45 Sal equates (1 + 1/n) to the nth power to (1 + u) to the 1/u power but if u approaches zero you have an exponent with the denominator approaching zero. How does this effect the (1 + u) to the 1/u power expression.",
        "A": "Kane, what it really comes down to is limits.. if you are looking at your 3x(x-1)/(x-1) and you plug in x=1, it is undefined at that exact point. But if you take the limit from the left you approach 3, and taking the limit from the right yields the same (e.g. 3). x/0 is necessarily undefined for that reason. You must be approaching 0 in the denominator for the expression to yield some value. So lim a -> 0 (x/a) is always going to be infinity."
    },
    {
        "video_name": "yUpDRpkUhf4",
        "Q": "What happened to the 1/x exponent at 7:00, how did it disappear?",
        "A": "it comes to the front of the natural log. try n concentrate. its just there.. anmol."
    },
    {
        "video_name": "yUpDRpkUhf4",
        "Q": "At 8:30, can I just do the limit?\n\nlim u->0 (1+u)^(1/u)\nlim u->0 (1+0)^(1/0)\n\n(1+0)=1\n(1/0)=infinity\n\n1^infinity is equal to 1...",
        "A": "Oh, now I m seeing that the lim u->0 1/u is different when I approach from both sides, so the limit does not exist, I didn t see it yesterday, I m sorry to disturb you"
    },
    {
        "video_name": "yUpDRpkUhf4",
        "Q": "At 8:06, how can log be taken outside limit? I don't understand from 8:06 to 8:24. How is the whole equation in red equal to e?",
        "A": "You can factor out anything from a limit that does not contain the limit s variable. The expression in red was in fact the definition of e e is defined as either of these two equivalent expressions: lim h\u00e2\u0086\u00920 (1+h)^(1/h) or lim h\u00e2\u0086\u0092+\u00e2\u0088\u009e [1 + (1/h) ]^h"
    },
    {
        "video_name": "yUpDRpkUhf4",
        "Q": "At 5:23 on the video, should delta x not be replaced by xu, not just u?",
        "A": "If we just substituted directly, yes. But we re not. Sal noticed that letting \u00e2\u0088\u0086x go to 0 is the same as letting u go to 0, since u is just \u00e2\u0088\u0086x times some factor."
    },
    {
        "video_name": "yUpDRpkUhf4",
        "Q": "When Sal makes the substitution\nlim as u --> 0 u = lim as n --> \u00e2\u0088\u009e 1/n\nat about 8:35 I understand that the lim as n --> \u00e2\u0088\u009e of 1/n approaches 0. But the lim as n --> -\u00e2\u0088\u009e approaches 0, too, doesn't it? Would the lim as n --> -\u00e2\u0088\u009e of (1+(1/n)^n still be e?\nWhy do we assume that 1/u approaches \u00e2\u0088\u009e, and not -\u00e2\u0088\u009e?",
        "A": "The substitution should be u = 1/n, and lim_{u->0+} u = lim_{n->infinity}1/n Key is the approaching 0 from the right, otherwise we wouldn t know whether it s + or - infinity."
    },
    {
        "video_name": "yUpDRpkUhf4",
        "Q": "Is it correct to say that e=1^infinity?\nIf yes, this cud've ended at 4:33, with the substitution that delta x appches zero, but then we'd get the ans as 1. This is wrong. Can anybody pls explain?",
        "A": "e != 1^infinity. 1 raised to any power = 1. e = (1 + 1/n)^n"
    },
    {
        "video_name": "yUpDRpkUhf4",
        "Q": "So I think reality just exploded before my eyes. At 6:56, Sal does ln((1+u)^(1/(xu))) = ln((1+u)^(1/u))^(1/x), which just blows my mind completely. Isn't that impossible? Furthermore, can't it be disproved by simply subbing in one example of say u = 2, x = 3:\nln((1+2)^(1/6)) to ln((1+2)^(1/2))^(1/3), which, according to the video, would mean that 0.1831 = 0.819?",
        "A": "But if you go to 6:56, you will see that he did in fact shift the (1/x) power outside of the natural log, which is incorrect. He must have missed a third pair of brackets inside the natural log, which would have made it correct."
    },
    {
        "video_name": "yUpDRpkUhf4",
        "Q": "What do the semicolons mean at about 5:07?",
        "A": "Interesting question! I do not know the answer for certain , but my math teacher uses a semicolon in an equation when there are multiple variables; So perhaps here as well, since he is showing us that they are equivalent to each other, just a different way of writing it he decided to use a semicolon."
    },
    {
        "video_name": "yUpDRpkUhf4",
        "Q": "at 8:50 Sal explained how the whole expression (lim u->0 (1+u)^(1/u) ) equals to e.\ncan somebody help me understand that?",
        "A": "Assume for a moment that u = 1/n. What then does (1+u)^(1/u) equal? (1+(1/n))^n And that equals e. Next, Ln e = 1. (e^1 = e). Thus, d/dx ln e = 1/x."
    },
    {
        "video_name": "vdpyWeiHXmU",
        "Q": "Is rotating something around the line, like at 1:41 to 2:17 , the same as rotating something about the line, or is it different?",
        "A": "Rotate [thing] about the line and rotate [thing] around the line mean the same exact thing. You can think of it sort of like the word roundabout , if that helps."
    },
    {
        "video_name": "btGaOTXxXs8",
        "Q": "Isn't it dV = A*dx (6:45)",
        "A": "It depends on what name you give to the volume of that infinitesimally thin disk. If you re willing to call the volume of the entire shape as V , then dV is appropriate. Otherwise, it s better to call it V"
    },
    {
        "video_name": "btGaOTXxXs8",
        "Q": "Where's the +C? Is it unnecessary for solids? I would expect it at 8:56.",
        "A": "+c is unnecessary for definite integrals because when you evaluate it at it s limits you will have one +c and one -c which will cancel out."
    },
    {
        "video_name": "st6D5OdFV9M",
        "Q": "At 4:38, if you wanted to check if Col(A) is a valid subspace wouldn\u00c2\u00b4t we have to check if it has the zero vector as well?",
        "A": "He says (and explains why) it contains the zero vector at 1:35"
    },
    {
        "video_name": "st6D5OdFV9M",
        "Q": "At about 4:00, when you were proving it was closed under addition, why are you adding by c and not a?",
        "A": "In the green text, a = c1 v1 + c2 v2 + ..."
    },
    {
        "video_name": "dvSa_tx04hw",
        "Q": "At 6:53, I have noticed that if you calculate a z-value without making the P-values the same (leaving the variances at .230 and .242 respectively), you wind up with the same answer. Why is this the case?",
        "A": "I think there actually is a difference of .0003 which simply got lost with all of the rounding up at each step."
    },
    {
        "video_name": "dvSa_tx04hw",
        "Q": "The explanation on the critical Z value is confusing at about 14:15 of the video. So if you Z value is greater than the Critical Z value, what does that mean? Thanks in Advance!",
        "A": "If the Z value is greater than the Critical Z value (which it was), then it means the result would only happen 5% or less of the time. Since 5% was chosen as the significance level, then the results tell us that the data disagrees with the null hypothesis, and we should reject it."
    },
    {
        "video_name": "dvSa_tx04hw",
        "Q": "8:00: How is it fair to estimate the SD of the difference in distribution using the samples, since we ultimately use this to validate/reject the null hypothesis. Suppose I get a particularly skewed sample of the population, wont this affect the Z score?",
        "A": "It would be unlikely to get a particularly skewed population but this is why we can say 90% or 95% confident and not 100% confident of the results."
    },
    {
        "video_name": "K759mIqpvOU",
        "Q": "At 7:00 I know what he meant but what was the reason for 1y instead of y?",
        "A": "for when the absolute value comes how do u know u know how to do it"
    },
    {
        "video_name": "K759mIqpvOU",
        "Q": "At 10:29, couldn't he have used a delta symbol to define a change in a value?",
        "A": "Sure, but he probably just wanted to keep things simple"
    },
    {
        "video_name": "K759mIqpvOU",
        "Q": "At 3:06 it says that the absolute value of negative 4 is equal to 4 does that mean that all negative numbers are equal to the opposite number? For example the absolute value of 10 would be equal to 10?",
        "A": "you seem confused. absolute value is the total value from zero.negative 100 s absolute value would be 100 because it is 100 digits from 0. similarly, 50 s absolute value would be a 50 because it is 50 digits away from 0. you said the absolute value of 10 would be equal to 10 well, the answer is yes but it can be the absolute value of negative 10 too. i hope that worked out"
    },
    {
        "video_name": "K759mIqpvOU",
        "Q": "At 0:35 to 0:38 he says the absolute value of a-b is the same thing as b-a. I don't understand how that is possible.",
        "A": "It is possible! A simpler way is to think of actual numbers! For eg: a=2, b=5. Let s tackle the first one. l a-b l= l 2-5 l = l-3l= 3 ( I l means absolute value) You agree that this much is correct? Then lb-al, * l b - a l= l 5-2 l = l3l = 3! * And I m sure you agree that 3 = 3!! If you still can t understand, try using a number line. The absolute value is just the distance between 0 and the number on the number line. It HAS to be positive! Hope this helps and you understand! :)"
    },
    {
        "video_name": "i8r7ZTUqB9w",
        "Q": "and what does he mean at 2:14?",
        "A": "Because later you will have to regroup."
    },
    {
        "video_name": "i8r7ZTUqB9w",
        "Q": "9:38\n357 - 251 = 106",
        "A": "that answer is correct"
    },
    {
        "video_name": "smtrrefmC40",
        "Q": "At 4:46, Sal says M-coordinate. Is this \"x-coordinate\"?",
        "A": "Yes, I believe he means the x-coordinate of point M."
    },
    {
        "video_name": "smtrrefmC40",
        "Q": "At 10:50, its cos(x)=sin(90-(x)) cos(theta+90), so why does it become sin(90-(theta-90))? Shouldnt it be sin(90-(theta PLUS 90)) instead?",
        "A": "If you look at it this way, it should help: cos(theta+90)=sin(90-(theta+90)). Which is the same as sin(90-theta-90)."
    },
    {
        "video_name": "smtrrefmC40",
        "Q": "i dont understand how he got 10:37 he got cosx-sin90-x",
        "A": "This is a trigonometric identity. The cos(x) = sin(90\u00c2\u00b0-x) and sin(x) = sin(90\u00c2\u00b0-x). An example using 30-60-90 triangles: cos(60\u00c2\u00ad\u00c2\u00b0) = sin(30\u00c2\u00b0) = 1/2."
    },
    {
        "video_name": "smtrrefmC40",
        "Q": "At 5:21\u00ef\u00bc\u008cI really can't get why line DM is perpendicular to line AB",
        "A": "AD = BD and AM = BM, so both D and M are equal distances from A and B, which makes DM perpendicular to AB."
    },
    {
        "video_name": "FaDtge_vkbg",
        "Q": "at 1:10 when he says percent = per 100, what if the decimal was in the thousandths?",
        "A": "It would be per mille"
    },
    {
        "video_name": "FaDtge_vkbg",
        "Q": "Like in 1:02, why cant it be 16/4?",
        "A": "Because it would be an improper fraction"
    },
    {
        "video_name": "FaDtge_vkbg",
        "Q": "at 1:27 where did u get the 25 from to multply",
        "A": "25 is the percentage of 1/4 100 divided by 4 is 25 Hence, 25%"
    },
    {
        "video_name": "FaDtge_vkbg",
        "Q": "At 2:11 can't you just simplify the fraction?",
        "A": "This is just an example to show you how to do more complicated problems."
    },
    {
        "video_name": "FaDtge_vkbg",
        "Q": "I am sure it is obvious, but I don't see it. At 0:41, how is 4/16 the same as 1/4?\n\nAlso, at 1:24, sure, 4 fits evenly into 100 with no mess, but what about when you have a number that doesn't fit in so nicely? Like here (I had this problem on a test and was unable to do it, that's why I came here): \"What percent of 52 is 12? Round your answer to the nearest tenth.\"",
        "A": "4/16 is the same as 1/4, because if 1/4 is just a simplified version of 4/16. You see, if you divide the numerator and the demoninator by the same number, it won t change the value of the number. Think about it this way. If you split a pizza into 8 pieces, and eat 2, it would be the same thing if you split a pizza into 4 pieces, but ate 1. This is because you are using double of half of an item. Hopefully I clarified this for you."
    },
    {
        "video_name": "FaDtge_vkbg",
        "Q": "at 0:59, how is it one over four equal to sixteen over four?",
        "A": "Sal did not say that 1 /4 = 16 /4. He has 1/4 = 4/16. These are called equivalent fractions. The fraction 4/16 is not simplified. You can reduce it by dividing both the numerator and denominator by the common factor of 4. 4/16 = (4/4) / (16/4) = 1/4 The fraction 1/4 is the fully reduced version of 4/16. Hope this helps."
    },
    {
        "video_name": "FaDtge_vkbg",
        "Q": "At 2:23 can you just leave that as 0 R4? its just a question.",
        "A": "Yes you can leave it as 0 R4 but in later mathematics you don t use remainders, you would keep going by adding a decimal point. when problems tell you to do this they also sometimes tell you to round to the nearest tenth or hundredth. but anyways the reason Sal does not leave it as 0 R4 is because this does NOT answer our question. we want a percentage. Hope this helps!"
    },
    {
        "video_name": "FaDtge_vkbg",
        "Q": "At 1:50 Sal kinda skips over the fact that he is simplifing a fraction of 25/100. HE just says it is 25%. I thought about it like this:\n25/100=1/4=.25 times 100= 25%",
        "A": "Well, since the denominator of the fraction is 100, it sorta makes sense to do so, since you do not need to simply it to make it easier for writing it in decimal or percentage form."
    },
    {
        "video_name": "FaDtge_vkbg",
        "Q": "How do you convert time into a 3% increase. Example 1:55 with a 3% increase in time",
        "A": "Are you referring to a time of 1;55 pm, as in a time on the wall-clock? If you want to multiply time, you need to establish a start time. You could use midnight, but that s rather arbitrary. If instead you meant 1 minute 55 seconds , well then that s just a number, and a 3% increase would mean multiplying it by 1.03. So the answer is (1*60 + 55) * 1.03."
    },
    {
        "video_name": "FaDtge_vkbg",
        "Q": "At 2:00,can't you just convert the 4/16 to one fourths then convert into a decimal then find the percent?",
        "A": "That is what he does @00:40. Just another way to do it."
    },
    {
        "video_name": "FaDtge_vkbg",
        "Q": "At 1:03; What do you do if you can not convert the fraction to a fraction over 100?",
        "A": "Everything can a fraction of something if it s a lesser or greater number. Lets say you have 150 of something, and there are a 1000 of them then that would be 15%. (150/1000)*100 If the fraction is 125/100, the calculation is the same (125/100)*100 = 125% More complex example is 321/980 = 0.32755, multiply that by 100 and the answer is 32.755% If that s not what you mean get back to me"
    },
    {
        "video_name": "FaDtge_vkbg",
        "Q": "at 5:31 how is the moon over the sun if the sun is bigger",
        "A": "Have you ever held up your hand to shade your eyes because the sun is too bright? How can your hand cover the sun when the sun is so much bigger? You hand is MUCH closer than the sun so it appears much bigger. The same is true with the moon."
    },
    {
        "video_name": "MABWdzmZFIQ",
        "Q": "At 2:37, how do you know the slope of the tangent is one?",
        "A": "It s not the slope of the tangent. It s the slope of the line that forms an angle with the positive x axis, which is equal to one. Tangent is equal to opposite/adjacent. In this case, the opposite is y and the adjacent is x. Thus, tangent = y/*x* = slope. If the tangent is equal to 1, then the slope is equal to 1."
    },
    {
        "video_name": "XuwldEyWjH0",
        "Q": "near the end at 3:40, can -1(2t + 3)^2 be written as (-2t -3)^2? why or why not?",
        "A": "Yes, it can be rewritten by simply distributing the -1. This is a technique used throughout most of algebra 1. I would recommend watching the earlier videos for more information on it."
    },
    {
        "video_name": "XuwldEyWjH0",
        "Q": "didn't Sal make a mistake at \"1:38\"?\nif we say a^= 4t^, a equals 4t, not 2t.\ni think he was mistaken that a^ was 4t not 4t^.",
        "A": "thank you"
    },
    {
        "video_name": "XuwldEyWjH0",
        "Q": "0:50 why does sal mention factoring by grouping in this video if grouping is the next section?",
        "A": "He was just mentioning it for those of us that already knew how to group, or for those that wanted to know what an alternate method would be."
    },
    {
        "video_name": "3-xfmbdzkqc",
        "Q": "What happens when you have:\n10\n01\n\nfor your matrix instead of your matrix of:\n1 1\n00\n\nat time 9:52",
        "A": "Re-check your work. If you re doing calculations properly, you shouldn t get that as your matrix."
    },
    {
        "video_name": "o-ZbdYVGehI",
        "Q": "at \"0:44\" isn't Sal supposed to write (((+-5)^)^) just like he did for 9?",
        "A": "Good observation! (At first I instinctively replied No, he wasn t. What he wrote is correct. , but that is because I was too focused on your un-mathematical notation. ( ^ does not equal \u00c2\u00b2 , you need to write it as ^2 if you don t have access to the \u00c2\u00b2 symbol. The symbol ^ stands for to-the-power-of, not for to-the-power-of-two.) Also, (\u00c2\u00b15\u00c2\u00b2)\u00c2\u00b2 would equal 625, which is not the 25 we need. But he could (and should) have wrote \u00c2\u00b15x\u00c2\u00b2. So you are right."
    },
    {
        "video_name": "o-ZbdYVGehI",
        "Q": "At 2:13, why didn't Sal write +3? In what type of problem should I write +3?",
        "A": "Because if he put +3 in it and factored it out he would have ended up with; 25x^4 + 30x\u00c2\u00b2 +9 instead of 25x^4 - 30x\u00c2\u00b2 + 9. You can make it +3 if the quadratic equation was 25x^4 + 30x\u00c2\u00b2 +9."
    },
    {
        "video_name": "o-ZbdYVGehI",
        "Q": "At 1:25, where does the 2 in \"2*5x\u00c2\u00b2*-3 = -30x\u00c2\u00b2\" come from?",
        "A": "Consider a binomial a+b, squaring it gives: (a+b)\u00c2\u00b2 = (a+b)(a+b) = a\u00c2\u00b2 + ab + ab + b\u00c2\u00b2 = a\u00c2\u00b2 + 2ab + b\u00c2\u00b2 Now maybe you see where the 2 comes from. When you square a binomial you get the first term squared, the second term squared, and two lots of the product of the two terms. If your binomial is Ax+B, squaring it gives (and we can use our previous result): (Ax+B)\u00c2\u00b2 = (Ax)\u00c2\u00b2 + 2(Ax)B + B\u00c2\u00b2 = A\u00c2\u00b2x\u00c2\u00b2 + 2ABx + B\u00c2\u00b2 So we can see that the coefficient of the x term will be 2AB."
    },
    {
        "video_name": "o-ZbdYVGehI",
        "Q": "At 1:07 he says that it has to be two times the two parts of the problem. Where does that two come from?",
        "A": "He is following the formula from the last video."
    },
    {
        "video_name": "o-ZbdYVGehI",
        "Q": "At 1:44, what if you have a problem like: x\"^4\" \u00e2\u0080\u0093 x\"^2\" + 16? The middle term does not work out if you do it your way.",
        "A": "Technically, x^2 is a perfect square. x^2 can be written as 1x^2, and 1 is the square of 1."
    },
    {
        "video_name": "o-ZbdYVGehI",
        "Q": "At 0:56 how come it can be + or - 3? I understand how that is useful later on to get the result, I just don't understand the justificiation for the possible -3.",
        "A": "It is possible that it can be \u00c2\u00b13\u00c2\u00b2 because 3 \u00c3\u0097 3 = 9 AND -3 \u00c3\u0097 -3 = 9. In order to find out which one you would use, you have to look at the coefficient on the x-term (b). If that term is negative then the constant number would have to be negative. If the b term is positive, then C would also be positive. (Note, this is for factoring perfect squares specifically)"
    },
    {
        "video_name": "o-ZbdYVGehI",
        "Q": "Don't exponents come before multiplication?\nat around 1:40 he says:\n2 * 5x\u00c2\u00b2 = 10x\u00c2\u00b2 * -3 = -30x\u00c2\u00b2 = 900x\n\nBut if the exponents come first, wouldn't it be\n2 * (25x) = 50x * -3 = -150x",
        "A": "A good observation. This is a common mistake. The exponents here apply only to the variable x, not to the coefficients. So, a number such as 5x^2 would be read as Five multiplied by x, which is raised to the second power . Now, if the number were something like (5x)^2, then you would get 25x^2. Parenthesis make the difference. Gargamel"
    },
    {
        "video_name": "o-ZbdYVGehI",
        "Q": "1:42 didn't sal forgot to apply the order of operations?",
        "A": "He s multiplying by negative 3, not subtracting negative 3. He should have put parenthesis around the -3 to make it more clear."
    },
    {
        "video_name": "o-ZbdYVGehI",
        "Q": "At 1:08, sal says that -30x^2 needs to be 2x the product of 5 and 3. Why two times? I thought it was just a product of two numbers that added up to the constant?",
        "A": "Sal says 2(5*3) =2(15) =30"
    },
    {
        "video_name": "dbxJ6LD0344",
        "Q": "@ 3:01 how is i^2 equal to -1?",
        "A": "\u00f0\u009d\u0091\u0096 is defined as: \u00f0\u009d\u0091\u0096 = \u00e2\u0088\u009a-1 Squaring both sides: \u00f0\u009d\u0091\u0096\u00c2\u00b2 = -1"
    },
    {
        "video_name": "VPz4bCQDGgk",
        "Q": "At 0:30, why is ds equal to Sr x S theta dr d theta? Thank you!",
        "A": "The vector ds is the product of the unit normal vector to the surface n and the scalar area element ds (as Sal has written at @0:30). It can be shown that n = (Sr x Stheta)/mod(Sr x Stheta) and ds = mod(Sr x Stheta) dr dtheta so the magnitudes cancel and you re left with the vector ds = (Sr x Stheta)dr dtheta. I think this is covered in the Introduction to the surface integral video, you should check that out if you re unsure on where these come from."
    },
    {
        "video_name": "pj1VvscsV0s",
        "Q": "How will you know the value of f'(x)? example @ 0:38 seconds, how did you know the derivative is a constant -2?",
        "A": "Pick any pair of y and x points in the local range and put them into the slope formula m = (y_2 - y_1)/(x_2 - x_1) and you will see the slope always computes to -2."
    },
    {
        "video_name": "pj1VvscsV0s",
        "Q": "0:38 how u take the derivative a straight line parallel to x-axis?",
        "A": "A straight line would be y=k, where k is a constant. The rate of change of a straight line is zero, This is reflected in the derivative of a constant, d/dx[y=k] yields dy/dx = 0."
    },
    {
        "video_name": "roHvNNFXr4k",
        "Q": "at 4:46 , why is 27x-10,000 and not 27-10,00=x?",
        "A": "x represents the number of computers sold in a month, each computer sells for 27. So the value of the sales is 27 * x, or 27x. To ensure that sales exceeds costs, the solution to 27x - 10000 = 0, tells how many computers must be sold to meet the costs."
    },
    {
        "video_name": "roHvNNFXr4k",
        "Q": "Wait how would you do that at 4:44 it says 370.3 but wouldn't you round down to 370? Why did he round up >)> Help me please.",
        "A": "She can t sell just part of a computer. So, the answer needs to become 370 or 371. In normal situations, we would round, and you are right in that case, we would round down to 370. The key point in this problem is that she has to make $10,000 in profit. If you round down, she won t make that amount of profit. Her profit will be a little low. This is why the value was rounded up to 371 to make sure the goal of $10,000 in profit was met. Hope this helps."
    },
    {
        "video_name": "roHvNNFXr4k",
        "Q": "Charlie has p stamps.. He puts the stamps into two albums in the ratio 2:5. There are 15 more stamps in the second album than the first one.\nWrite an equation in terms of p and solve it.",
        "A": "So you have three pieces of information, let x be first album stamps and y be second album stamps. We know that total stamps = p, so p = x + y. Next, x/y = 2/5, and cross multiplying you have 5x = 2y Also, y = x + 15, we can solve the second two equations by substitution, so 5x = 2(x+15) Distribute 5x = 2x + 30, Subtract 2x, so 3x = 30, divide to get x = 10, y = 10+ 15= 25 Substitute into first equation p = 10+25 = 35 I cannot think how to do it all with just one equation."
    },
    {
        "video_name": "roHvNNFXr4k",
        "Q": "at 1:58 why would you add 10000",
        "A": "If you change one side of an equation, you MUST change the other. So in this equation, you want to get rid of (+)10,000 so to do that you take away that 10,000 (which equals 0) but you have to take away 10,000 of the other side as well. Does that help?"
    },
    {
        "video_name": "vBlR2xNAGmo",
        "Q": "At 5:21 why is it 20/5 not 16/5",
        "A": "If you make 20/5 a whole number it equals 4, and that enables 12/5 and 4 to be combined if they re made to have the same denominator."
    },
    {
        "video_name": "vBlR2xNAGmo",
        "Q": "At 1:34, shouldn't that third green number be (4/9) to the third and not (4/3) to the third?",
        "A": "you can see that he wrote it accidentally because he said 4/9 even though he wrote 4/3"
    },
    {
        "video_name": "vBlR2xNAGmo",
        "Q": "at 1:57: just wonder why did you time S with 4/9???",
        "A": "It s not intuitive, but he multiplies the series by 4/9 in order create a finite difference between S and 4/9S, so that he can solve for S. That is, he can t solve by adding up all the numbers in an infinite series, because it would take an infinite number of steps, but he can assert that the difference between the infinite series S and the infinite series 4/9S is 4/9, which allows him to solve for S. It totally blows my mind, too."
    },
    {
        "video_name": "vBlR2xNAGmo",
        "Q": "i dont know if anybody cares but the equation is false. Whe nKhan writes out (1:50)S= the third number is (4/3)^3 which is incorrect. It might be just my bad eyes but i am pretty sure its there.",
        "A": "You are correct (unless my eyes fail me as well). Sal wrote (4/3)^3 instead of (4/9)^3, but the later evaluation uses it as (4/9)^3, so the final result is correct. --Phi \u00cf\u0086"
    },
    {
        "video_name": "VX7K8iqoiRc",
        "Q": "6:59 Is there an equivalent of sigma sum notation for multiplication, i.e. everything in the series is multiplied with each other?",
        "A": "Yes, it s capital pi \u00ce\u00a0."
    },
    {
        "video_name": "jRrRqMJbHKc",
        "Q": "why is it that in 3:48 that the diagonals always split each other?",
        "A": "Carrie, all parallelograms are just that-opposite sides are parallel by definition. Therefore, a segment drawn from opp. corners automatically bisects(and is perpendicular to) the opposite corners segment."
    },
    {
        "video_name": "jRrRqMJbHKc",
        "Q": "What was he doing at around 2:41? I don't understand.",
        "A": "He was writing an equation to solve the problem."
    },
    {
        "video_name": "jRrRqMJbHKc",
        "Q": "1. At 4:23, what does 'Lateral area of a cone' MEAN..??",
        "A": "I am not sure, but my best guess would be it s the surface area of a cone."
    },
    {
        "video_name": "jRrRqMJbHKc",
        "Q": "I don't understand what lateral area is...question 28 at about 6:00. Can anyone explain it to me (besides what the formula is b/c that's given)?",
        "A": "notice the shape, height, base, l is the hypotenuse of a pythagorean triangle. So a squared (base) + b squared (length) = c squared (L). So 5 squared is 25 and b squared is 64 add them it is 89."
    },
    {
        "video_name": "Pt-Tn6L60-o",
        "Q": "At 1:03, what happens if the change in Y is not completely consistent but there is still some consistency in the numbers such as the change in X also follows a consistency, is it linear or non-linear?\n\nEX: 1, 2, 4\n3, 6, 12",
        "A": "It s still non - linear. The change in the x and y values must be completely constant."
    },
    {
        "video_name": "aTjNDKlz8G4",
        "Q": "At 0:43 and 7:11, Sal draws an arrow to the right of the equal sign. Can anyone tell me what it means?",
        "A": "It s the symbol for implies. He wants to find out that a speed of 28 feet/minute implies a speed of how many inches per second. Or, given the first statement, how do we complete the second statement so that it is true."
    },
    {
        "video_name": "aTjNDKlz8G4",
        "Q": "Why does Sal use shorthand at 5:48?",
        "A": "It saves space."
    },
    {
        "video_name": "aTjNDKlz8G4",
        "Q": "how does Sal know which numbers to put in the numerator spot? why didnt he put 60sec in the numerator instead of the denominator? how can I know where to put them? at 4:25 he said he put minutes in the numerator to cancel it out, but for me it is more intuitive to say 60 sec = 1 min instead of 1 min = 60 sec so i would have put the 60 on the top. am i just missing it?",
        "A": "yes?"
    },
    {
        "video_name": "YFqeMzfcyg4",
        "Q": "If JK+KL=114 (1:28) does that mean KL+JK also equals 114?",
        "A": "Yes, it is the commutative property of addition. X+Y=Y+X or 1+2=3. 2+1=3"
    },
    {
        "video_name": "YFqeMzfcyg4",
        "Q": "At 3:04 , why do you divide both sides by 16? Why is that so necessary?",
        "A": "Since first u need to solve for X your are going need get to be alone so it equals such case X=......"
    },
    {
        "video_name": "oTKVXJ7TcbA",
        "Q": "at 0:16 Sal says that the associative law is associating the numbers but how do you do that?",
        "A": "Sal means by associating numbers that you can group numbers in any way in addition. Like 2+3+4 = (2+3)+4 that is 2 and 3 have been grouped or associated in it and will be calculated first . It can also be written as 2+(3+4) in which 3 and 4 a grouped and will be calculated first. Basically the Associative Law of Addition means that the order that the numbers are added does not matter."
    },
    {
        "video_name": "oTKVXJ7TcbA",
        "Q": "At 1:45 in the video, he practically says it doesn't matter how you associate the numbers... is that always true?",
        "A": "Yes it does not mater how you group the numbers it will always come out with the same number. 6+5+2=13 // 5+6+2=13 // 2+6+5=13 // 2+5+6=13 It will always work out the same no matter how you group them."
    },
    {
        "video_name": "oTKVXJ7TcbA",
        "Q": "0:35\nwhat's makes the associative different from commutative? Is it just the grouping versus the order?",
        "A": "The Commutative Property says you can do any number in a different order, but that they have to stay the same. example: 4 + 5 = 5 + 4 The Associative Property just means you do the equation in the parentheses first. example: (4 + 6) + 7 You would do 4 + 6 first."
    },
    {
        "video_name": "HLNSouzygw0",
        "Q": "0:5 till 0:11 maybe. Does a difference of squares have to be subtraction or can it be addition, because judging of the name \"difference\". So I'm thinking subtraction only, but I just want it clarify whether it can also be addition",
        "A": "Difference means subtraction, so it can only be a subtraction. If it is an addition such as x^2 + 1, think what that would do to the discriminant, b^2 - 4ac = 0^2 - 4(1)(1) = -4 which would mean you would have to take the root of a negative number, not allowed in the real domain. With c being a negative perfect square, the determinant is 0^2 - 4(a)(-c) so two negatives cancel, and since each of 4 and a and c are perfect squares, the root would be a whole number."
    },
    {
        "video_name": "wTkH288r84s",
        "Q": "At 1:21 why did he multiply by 1/2 i don't understand please explain.",
        "A": "You do this because 2 triangles make a rectangle, and 1 triangle is half of a rectangle. Plus, B times H is the formula for a rectangle so take half of that, a BOOM! You got the formula for a triangle."
    },
    {
        "video_name": "wTkH288r84s",
        "Q": "at 1:21 why did he multiply 1/2 i don't really understand what he is doing in that part of the video",
        "A": "He did not multiply he divided."
    },
    {
        "video_name": "YJdCw2fK-Og",
        "Q": "At 3:13 (or 3:12) Sal says it's called an exponent. What the heck is that?",
        "A": "The exponent is the power that a base number is getting raised to. For example, in 3^4 (or Three to the power of four ) the base is the three and the exponent is the 4."
    },
    {
        "video_name": "YJdCw2fK-Og",
        "Q": "At 1:05 Mr.Khan asks what is the pattern here I figured the s=answer would fall within the lines of adding a zero each time you multiply ny another ten for example:\n2x10=20\nif I multiply 2 again by another 10 watch\n2x10x10= 20 and then one more zero",
        "A": "the answer statement that you have made was very good"
    },
    {
        "video_name": "YJdCw2fK-Og",
        "Q": "At 1:00, if you multiplied 10x10x10 enough times would you get past googolplex?",
        "A": "Yes. But it will be really hard if you try it"
    },
    {
        "video_name": "lohMwoq3WFA",
        "Q": "At 0:45, what does a unique triangle mean?",
        "A": "A special triangle"
    },
    {
        "video_name": "KxOp3s9ottg",
        "Q": "At 1:38, isn't it more accurate to say \"and\" instead of \"or\"?",
        "A": "Actually, no. And would mean that x would have to satisfy both inequalities while or means that it just needs to satisfy either one. In this case, x just needs to be either one, and it does not have to satisfy both."
    },
    {
        "video_name": "KxOp3s9ottg",
        "Q": "From about 4:15, why aren't we including d and e when we talk about the increasing and decreasing intervals? Since d and e are single points, I would think we would include them as 'increasing' and 'decreasing' are about a difference between two points.",
        "A": "Sorry, that is simply not true. If the derivative at a point is zero, this does not imply the function is neither increasing or decreasing at that point. Take f(x) = x^3 for instance. It is increasing for all x. But, f (x)=0 at x=0."
    },
    {
        "video_name": "fD7MbnXbTls",
        "Q": "Hi at (2:00 to about 2:48) Sal starts by expressing that he wants to use \"1+tan^2(theta) = sec^2(theta)\" My question is: What do I look for in the starting integral, in order for me to find the best trig substitution.\n\nIn this example Sal chose to use \"1+tan^2(theta)\" but why couldn't he use \"sin^2(theta)+cos^2(theta) = 1\"",
        "A": "My textbook has a table for this and it goes like this: a^2 - x^2 substitute: x = (a)sin(theta) Because: 1-sin^2=cos^2 a^2 + x^2 substitute: x = (a)tan(theta) Because: 1+tan^2=sec^2 x^2 - a^2 substitute: x = (a)sec(theta) Because: sec^2-1=tan^2"
    },
    {
        "video_name": "fD7MbnXbTls",
        "Q": "At 0:25 he says we cant use U-substitution\n\nWhat I dont get is couldnt we make:\n1/2x\u00e2\u0088\u00ab1/(36+x^2)*2xdx\nthen we could use U-substitution....\nu=36+x^2\ndu=2x*dx\n\n1/2x\u00e2\u0088\u00ab(1/u)du\n1/2xlog(u)\n1/2xlog(36+x^2)\nWhy is this incorrect?",
        "A": "No. You can t. You can only pull constants, not variables, out of the integration. The x on the outside is not integrated, while the x on the inside is integrated. Thus, this is not a valid substitution."
    },
    {
        "video_name": "y5yNi08cr6I",
        "Q": "isn't 0x equal to 0 at 1:04 ?",
        "A": "Yes... 0x does = 0. Sal is forcing an X term into the equation to relate it to slope-intercept form. Since he started with y = 6, and he added something = 0, he hasn t changed the underlying value of the equation. Hope this helps."
    },
    {
        "video_name": "y5yNi08cr6I",
        "Q": "Shouldn't the equation at 1:52 be representing point 2, -2?????",
        "A": "I don t think he did, the point slope form is y - y1 = m(x - x1) so since the x1 is positive it means it must be a negative number, and since the y1 is negative it must be a positive number, (-2, 2)"
    },
    {
        "video_name": "VYbqG2NuOo8",
        "Q": "At 0:20, There is nothing wrong with saying \"alphabets\". Why was this even corrected when in reality it should be appropriate to say?",
        "A": "Alphabets is plural, so talking about alphabets means that there are more than one alphabet being talked about, what Sal is clearly alluding to are the 26 letters in the English alphabet which is what is shown in the correction, not about the English and say the Greek alphabet which are often used for variables."
    },
    {
        "video_name": "5GOhM3jq9Fc",
        "Q": "at 1:53, i didn't understand it",
        "A": "you don t have to put a question mark."
    },
    {
        "video_name": "5GOhM3jq9Fc",
        "Q": "at 1:10 how do we know that dividing the 63 and 7 will tell us how tall the stories are?",
        "A": "We know the building is 63 feet tall and that the building is made up of 7 floors. We are assuming that each floor is the same height. 7 floors = 63 feet. What does one floor equal? To put it another way, 7 floors x (height of one floor) = 63 feet. How many feet does each floor have to be if there are seven of them?"
    },
    {
        "video_name": "DtWovvMnPrk",
        "Q": "2:16- 2:21\n\n4/8 divided by 4 does equal 1/4, but 4/8 can be divided by 2 and equal to 2/4.\nWhy didn't Sal use the two instead of the four?",
        "A": "if you want to add the the trail mix, you should simplify the least fraction as possible"
    },
    {
        "video_name": "qH4IQfDD1Nw",
        "Q": "at 5:06, wouldn't you subtract both values of the absolute value of x from eaxh other?",
        "A": "You can! 6*(x)^1/4 |x| - x^1/2 |x| = [ 6*(x)^1/4 - (x)^1/2 ] * |x| But this is unnecessary, making the answer much more complex. :)"
    },
    {
        "video_name": "qH4IQfDD1Nw",
        "Q": "\"From 1:53-1:54 , in the video, Sal mentions that there has to be a '4th exponent sign beside the 3' after completing the factor tree. My question is, Why does the exponent, which is four, have to be beside the number 3?\"",
        "A": "3^4 is equal to 81, so he has to include that to keep the radical the same. For ex., 8 is also equal to 2^3, not just 2. All he is doing is keeping the expression equivalent."
    },
    {
        "video_name": "heKK95DAKms",
        "Q": "At 1:29, for the Ouroborromean rings, where does the concept come from?",
        "A": "The Ourpborromean rings is a portmantaue of ourobori and the borommean rings. Vi made a video on borommean rings on youtube, called boromean onion rings."
    },
    {
        "video_name": "heKK95DAKms",
        "Q": "At 3:54 did she mean to say e-edge?",
        "A": "Probably not. I think it was a problem with the recording system she used."
    },
    {
        "video_name": "heKK95DAKms",
        "Q": "At 0:44 to 0:52, vi explains that the weaving game will always work, and i even assumed it was because i had an odd number of crossings but i made one with six that did not work so please tell me what is happening!",
        "A": "Vi was right. The weaving game always works out perfectly. It is possible to have any number of crossings, either odd or even, but it makes no difference. It would be best if you looked over your curve. Did you actually alternate going over and under throughout the entire curve, or did you miss one? Did you make all the crossings distinct?"
    },
    {
        "video_name": "heKK95DAKms",
        "Q": "what happened at 3:55 ?",
        "A": "at 3:55 there is a problem with the video..."
    },
    {
        "video_name": "heKK95DAKms",
        "Q": "What did she say at 3:16?",
        "A": "Pick an outside section and color it in. Now you want to alternate colorings so that no two faces of the same color touch. Curiously enough, much like the weaving game, this game also Mathemagically works out."
    },
    {
        "video_name": "heKK95DAKms",
        "Q": "at 1:00 what is a plane?",
        "A": "In Geometry, a plane is basically a flat surface; usually used with lines, rays, and line segments to indicate that essentially, they are on the 2nd dimension, and are on the same face."
    },
    {
        "video_name": "heKK95DAKms",
        "Q": "AT 2:05, who else saw the dark mark??",
        "A": "Um, if you re eyes were watching the video, EVERYONE saw the dark mark!"
    },
    {
        "video_name": "heKK95DAKms",
        "Q": "At 1:23 how can the rings come together but still stay linked?",
        "A": "The rings can t actually be put together, unless you can work in the fourth dimension. Although you could also use broken rings in which case they might not stay linked."
    },
    {
        "video_name": "heKK95DAKms",
        "Q": "at 1:29 with Ouroborromean Rings, how are no 2 snakes not linked?",
        "A": "look up borromean rings and you will understand it more"
    },
    {
        "video_name": "heKK95DAKms",
        "Q": "What are Duborromean rings at 1:30?",
        "A": "Borromean rings are three rings that are linked together, but if one ring is removed, you are left with two unlinked rings. This means that no two rings are linked with each other, but all three are linked."
    },
    {
        "video_name": "Iq7a2vEsT-o",
        "Q": "At 0:25, why does x have to be a positive integer?",
        "A": "This is just the definition of the function. While you can still figure out the result of g(x) for non-positive integers, the function is not defined for these values. Imagine the function g(x) is used to determine how many grams of soap you would need to clean x plates. You could still determine the result of the function for fractional or negative or complex values of x, but these values wouldn t be within the domain of the problem, since you can t have -3.4 plates ."
    },
    {
        "video_name": "Iq7a2vEsT-o",
        "Q": "At 0:28, couldn't he have written it like x > 0 or x > 0, x = I(nteger)?",
        "A": "Yes! Your notation is more explicit, and works better in more formal contexts. You could also use the symbol N, for the natural numbers."
    },
    {
        "video_name": "Iq7a2vEsT-o",
        "Q": "At 0:26, positive doesn't have a dot above one of the i",
        "A": "OMG - O! MY! GOD! How could he ?"
    },
    {
        "video_name": "Iq7a2vEsT-o",
        "Q": "@ 0:25 can x be negative?",
        "A": "It would be possible, but the function is defined only for positive x values."
    },
    {
        "video_name": "Iq7a2vEsT-o",
        "Q": "1:43 you said that 9*8 raise to x-1 is equal to 9 but how",
        "A": "At about 1:13 Sal says that, when x = 1, that means we have 9 * 8^(1-1) which equals 9 * 8^0 which equals 9* 1 = 9."
    },
    {
        "video_name": "Iq7a2vEsT-o",
        "Q": "at 1:22 when x is one how is g(x) 9 because x-1 where x -1 will be = 0 hence 9^0 = 1 ? Am i wrong ?? Please tell me i am confused ...I even went back to watch a video on exponents",
        "A": "\u00f0\u009d\u0091\u0094(\u00f0\u009d\u0091\u00a5) = 9 \u00e2\u0088\u0099 8^(\u00f0\u009d\u0091\u00a5 \u00e2\u0088\u0092 1) \u00e2\u0087\u0092 \u00f0\u009d\u0091\u0094(1) = 9 \u00e2\u0088\u0099 8^(1 \u00e2\u0088\u0092 1) = 9 \u00e2\u0088\u0099 8^0 = 9 \u00e2\u0088\u0099 1 = 9"
    },
    {
        "video_name": "XUvKjC21fYU",
        "Q": "At 3:17 why is a=1?",
        "A": "Because if there is not a number before X it is 1"
    },
    {
        "video_name": "XUvKjC21fYU",
        "Q": "at 1:50, is it possible for the quadratic to have one real and one imaginary solution? From what I understand, it must have two solutions, because there is a square in the equation, as a cubic curve would have three solutions. I am confused because if this is true, then what is the other solution?",
        "A": "The question: Is it possible to have one Real root and one imaginary root? Answer: no. The root of a number is a number that when multiplied by itself always yields a positive number if the root is Real and always yields a negative number if the root is imaginary. x-1 and x-I are two different numbers."
    },
    {
        "video_name": "DL-ozRGDlkY",
        "Q": "Is a further explanation of 3:54 + possible, please? (I can't quite understand what's happening there with the -2 before the x and the 1/2 etc ..and the subtitles ain't helping) Thanks in advance",
        "A": "You are integrating: \u00e2\u008c\u00a0 -x\u00c2\u00b2 \u00e2\u008e\u00ae -x e dx \u00e2\u008c\u00a1 To integrate it you use the substitution u = -x\u00c2\u00b2, and it s differential du = -2x dx, which reduces the integral to 1/2\u00e2\u0088\u00abe^u du = 1/2 e^u Replacing the u substitution you get 1/2 e^(-x\u00c2\u00b2)"
    },
    {
        "video_name": "DL-ozRGDlkY",
        "Q": "At 6:45, why couldn't we multiply through by two to simplify the equation, and having the constant absorb the 2, making C=1/2 ?",
        "A": "0.5 = 0.5 + c ---- X 2 1 = 1 + 2*c 2*c = 0 c = 0"
    },
    {
        "video_name": "DL-ozRGDlkY",
        "Q": "around 8:09, Sal sets y equal to the principle root of the left side, because the initial condition only gives a positive y value. But why is this necessary? Wouldn't it be just as correct to say plus or minus?",
        "A": "I agree with Raviv; I m fairly certain that the y^2=e^-x2 would also work. The question didn t specify that we had to find a function - since we can differentiate a conic section and end up with an expression for dy/dx, then if working backwards yields a curve that is not a function, that curve should still satisfy the conditions set forth in the problem."
    },
    {
        "video_name": "DL-ozRGDlkY",
        "Q": "At 8:22 why does it become e^[(-x^2)/2]? Shouldn't it be e^[-x] because i cancel the 2? Properties of powers, isn't it?",
        "A": "This is a common mistake that I have seen. The correct answer is indeed e^[-x^2/2]. sqrt (e^[-x^2] ) = (e^[-x^2] )^(1/2) <-- in this instance, you multiply the exponents e^[-x^2 * 1/2] = e^[-x^2/2] From EEweb.com a^[n/m] = (a^[1/m] ) ^n = (a^n)^[1/m] --------------------------------------------------------------------------------------------------------------------------------------------"
    },
    {
        "video_name": "DL-ozRGDlkY",
        "Q": "Can someone explain the integration part right after he separates the equation at 3:30? where did he get y^2 /2? what did he substitute into?",
        "A": "y^2 /2 is the antiderivative of f(y) = y w/ respect to y. This is essentially the same idea as taking the integral of f(x) = x, only with the variable of integration interchanged."
    },
    {
        "video_name": "DL-ozRGDlkY",
        "Q": "on 3:33 you integrate on both sides, left side along y, right side along x. Can you explain why this allowed?",
        "A": "Both sides of the separated differential equation are equal to each other, so when you add up pieces from each side (integrals are really just sums), the two sides continue to be equal (within a constant of integration, at least)."
    },
    {
        "video_name": "MbpmP1esh-Q",
        "Q": "Hi, everybody. I'm a Chinese-speaking student. There's a verb I can't hear clearly, which sits between two sentences between 00:54 and 01:00. These two sentences are \"We\u00e2\u0080\u0099ll subtract 32 hundreds from 37 hundreds, we\u00e2\u0080\u0099re getting 5 hundreds. But we (?) back just this traditional long division mechanics. But that\u00e2\u0080\u0099s what is really going on here.\nThank you who'll help me to overcome this conundrum. Thank you very much.",
        "A": "You can look at the subtitles to see that the word is revert. It means to return to a previous state, condition, practice, etc. according to the definition."
    },
    {
        "video_name": "e7Nel0UoXR8",
        "Q": "At 8:30 when we confirm the output vector of the partial derivative [2, 1, -1], is that output vector a position vector with the origin of (0,0) pointing to (2,1,-1), or is the output vector pointing from (0, 1, 0) (the red dot)?",
        "A": "The (2; 1; -1) vector is pointing from the red dot, because this shows how the red dot will change along the red line."
    },
    {
        "video_name": "8GEGnSEJA2s",
        "Q": "So basically what Sal is saying at 4:20 is if f(g(x)) is equal to g(f(x)), they're inverse functions, right?",
        "A": "if they equal to x"
    },
    {
        "video_name": "8GEGnSEJA2s",
        "Q": "FYI: At 5:44 Sal draws the mapping under f from left to right but it should be the other way around. For g and f to be inverses, the range of g has to be the domain of f and the range of f has to be the domain of g. At 5:10 he defines the \"right set\" to be the range of g and thereby making it the domain of f.",
        "A": "Ohmygosh, I did not see that coming! Well, here s a new idea you can tell your teachers: When two functions are inverses, then the domains and ranges are the same! Then the teacher will be like: No, that s not right. Who taught you that? And you would say: Sal Khan!"
    },
    {
        "video_name": "-xYkTJFbuM0",
        "Q": "at 10:11 you said:\n\"if he has not blue, then these guys are going to leave after tree turnings on of the light, but if they dont leave, then he must be blue.\"\n\nLets say, there is 3 blue and 1 red head. Light will be turned off and on 3 times, 4 times or endless?\n\nAfter 3 times light goes on and off, nobody leaves the room. Red guy think, because no one leaves the room, i must have blue head, and he can leave the room.",
        "A": "Every person would see a number of blue foreheads. if I HAVE a blue forehead I see one less than the total number of blueforeheads. If I have other than blue forhead I see the total number of blue foreheads. Everone would leave on the number of blue foreheads seen plus one. So all blue foreheads would leave on the total number of blue foreheads time the light goes out, while the other foreheads are wating for the total number of blue foreheads plus one."
    },
    {
        "video_name": "-xYkTJFbuM0",
        "Q": "At 5:58, what does vice versa mean?",
        "A": "It means if this happened the other way around."
    },
    {
        "video_name": "-xYkTJFbuM0",
        "Q": "At 2:00 why do they leave?",
        "A": "every person would see a number of blue foreheads. if I HAVE a blue forehead I see one less than the total number of blueforeheads. If I have other than blue forhead I see the total number of blue foreheads. everone would leave on the number of blue foreheads seen plus one. So all blue foreheads would leave on the total number of blue foreheads time the light goes out, while the other foreheads are wating for the total number of blue foreheads plus one."
    },
    {
        "video_name": "-xYkTJFbuM0",
        "Q": "At 9:00, after the two blue foreheads walk out, why wouldn't the last person in the room, even though he saw that he didn't have blue earlier, now assume that he DOES have blue because he is to only dude left in the room? Just wondering.",
        "A": "Since he s the only person left in the room, there would be nobody else to compare with and that leaves a tiny chance of his forehead being painted blue."
    },
    {
        "video_name": "-xYkTJFbuM0",
        "Q": "Sal said that 0:58 that the lights would go on/off 100 times, then all 100 would leave the room.\n\nDoes that mean, if you scale it up to 999,999,999 people in the room (all with blue foreheads), would they leave the room after the lights go on/off 999,999,999?",
        "A": "Yes, although they may end up forgetting how many times the lights have gone off at that point."
    },
    {
        "video_name": "-xYkTJFbuM0",
        "Q": "at 9:36 what if all the three people have a blue forehead?",
        "A": "then the question will be wrong"
    },
    {
        "video_name": "-xYkTJFbuM0",
        "Q": "I'm at 4:26 and i found the answer, if the other person does not leave the room, then i must have a blue forehead",
        "A": "The other person meant forehead."
    },
    {
        "video_name": "-xYkTJFbuM0",
        "Q": "At around 1:15,why do they have to turn the lights on and of 100 times?Couldn't the logicians who realised their forehead was blue all leave at once?",
        "A": "To determine for certain that their foreheads were blue, they would need to see each other 100 times."
    },
    {
        "video_name": "-xYkTJFbuM0",
        "Q": "At 10:00, can somebody please help me understand how 3 people with blue foreheads will leave after 3 light turning ons?",
        "A": "From the point of view of person 1, if he were not blue, person 2 and person 3 should leave the 2nd time the light turns off. The reason is that if person 1 were not blue, than person 2 would see only one blue forehead, and person 3 would see only one blue forehead. As a result, they should leave the 2nd time the lights go out. Since person 2 and person 3 don t leave, that cues to person 1 that he should leave the next time the lights go out."
    },
    {
        "video_name": "-xYkTJFbuM0",
        "Q": "I do not understand the lights turning on and off 100 times (like 00:32.) Does it mean that the lights stay on for a few seconds and turn off again?",
        "A": "They probibly stay on for a minute or so that way everyone can look around."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "First of all, on 4:02 , infinity divided by zero is undefined! So, if you have 4 friends and zero apples, in how many ways can you distribute the apples? IT IS UNDEFINED!\n\nAnd also, so if you say you are ninety nine point nine nine nine nine nine nine nine nine nine nine nine percent sure, you are one hundred percent sure? I thought it meant you are just a very little tad bit unsure?\n\nSo, 0.9999999999999999...=1 is VERY controversial...",
        "A": "If you say you are 99.99% sure, you say it with a finite number of nines, usually just one or two. I ve never heard someone say it and mean an unending string of nines. And 0.99... with any finite number of nines is strictly less than 1. But that s not what we re considering. The claim is that the nonterminating decimal 0.9... (that is, after every 9, there is another 9) is precisely equal to 1. It s only controversial among people who think that infinity is a mystical, impossible-to-define notion. It isn t."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "A 4:53, she said that 1 - 9 repeating is one. i agree with that but what if it was .9 repeating + x? What would that equal?",
        "A": "Well, 2, becuase if x = 0.999... = 1, x + x = ?, 1 + 1 = 2"
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 5:00, what about 1.0000000....1? pronounced \"One point zero continuing one\"?",
        "A": "you would just pronounce it one because the zeros that come after the decimal are absolutely pointless. You cannot have INFINITE zeros and still have a number after those infinite zeros, unless the zeros are not infinite."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At around 02:34: Usually, you divide by the number instead of the x, don't you?",
        "A": "Only if you are trying to solve for x, but Vi already knew that x=0"
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "Vi, you said \"0.9 not equal to TEN is un9.9...able\" at 8:45, but I think you meant 'ONE' instead, right? Otherwise, 0.9... not equal to ten is quite 9.9...able!",
        "A": "She said the view that 9.9-repeating does not equal ten is simply un-9.9-repeating-able. 9.9999... does in fact equal 10, not 1."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At about 4:57, Vi says that the difference between 1 and 0.9999... is 0. I'm confused on this whole topic. How can 1 minus a number slightly less than 1 equal nothing? I consulted my calculator, and it also said that the answer is 0. Can anybody explain?",
        "A": "I know its quite hard to believe, but 0.999.. till infinite does equal 1. This is just for theory, in reality, you will never encounter .99999..... It is hard to imagine that .99999.... equals 1, but u can view it as numbers with the same value can have different look. for example, there are many different representations of the number e. look it up on wolfram, you ll see."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "7:47 she sed tht their is no number hiher thn infinity. but what if humans never discovered 1000? we would think there was no higher number than 999",
        "A": "There are several levels of infinity, so one could say that one infinity is higher than another. In fact, if someone discovered a number higher than infinity, it would just be added to the many other levels of infinity."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "Starting at about 1:23, I think Vi says that .999... is a real number, but I think she then continues on and says that 1 is not .999..., and doesn't that contradict the whole point of the video? Maybe she was just talking too fast for me to understand. Can somebody please help me understand Vi's Reason Three?",
        "A": "No, she says that .999 repeating equals one, but there are numbers that are infinitely close to 1 and not one, and that .999 repeating is a real number, but that number infinitely close to 1 and not one is not."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 2:45, Vi Hart tries to show that 42 is equal to one. I don't get why she writes that when she didn't give any mathematical proof showing a possibility in which it is true (which it isn't).",
        "A": "She proved it, using perfectly legal methods, other than dividing by 0. She was trying to show that allowing dividing by 0 breaks algebra."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 2:10 I don\u00c2\u00b4t get the equation. How can 9.999=10x --> (subtract 1 or 0.9999...) give you --> 9=9x? Where did the 0.000...01 go?",
        "A": "She did 9.999...-0.9999... on the left side and 10x-x on the right side. That leaves 9 = 9x. Note that the 9 s repeat infinitely, and so there is no 0.000...01 anywhere, because the zeroes repeat infinitely."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "In 1:40 to 2:12 when you're giving your 3rd reason that .999...= 1, you're using circular reasoning. You're assuming in the first place that .999...= 1",
        "A": "First, 1:40 to 2:12 is not the 3rd reason, but the 3.999....th reason. There, she is not using circular reasoning at all. She is defining x to be .9999..., then multiplying by 10, subtracting x = .9999..., and dividing by 9. None of those operations assume that .9999... equals 1."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "I am confused. At 3:00 she says to subtract .999 from both sides but clearly she subtracts .999 on one side and 1 on the other, Why? I thought what you do to one side you must to the other?",
        "A": "she subtracts 0.999...... from the 9.999...... side and 0.999.....=x so she subtracts x from the 10x side making 9x so she gets 9 = 9x hope i helped!"
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "Based upon the logic discussed at 4:41 would the closest number to zero be one less than an infinite amount of zeroes, then a one at the end? But isn't that mixing hyperreals with integers? Where is the breaking point where you can start adding on zeroes instead of subtracting from infinity?",
        "A": "There s no closest number to 0. As Vi said, if you can t introduce a number between two different numbers, they re not different. If you find a number that has no numbers between 0 and itself, its value is 0."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At around 2:14 or so, she says that if you subtract 0.999...from 9.999 you get 9. Then, that means if you subtract 0.999...from 10, it equalls 9.001. 9.001/9.001=1. 9/9.001=a yucky number like 0.9998...Could you please explain? Why did she round it to 9x instead of 9.001x?",
        "A": "Because she set 0.999 . . . equal to one"
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 2:13 in the video, she claims this: x=.99999 repeating, and also x=1, but, at any point, you should be able to substitute x for .999999 repeating. When she has 9.99999=10x, it works, but (magically) subtracting x and .99999 repeating, 9x cannot be substituted by 9. Why is this?",
        "A": "You are missing the point. What she is saying is that 9x can be substituted by 9. She is saying that they are the same value."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 2:00 she subtracts .9999... from one side but not the other.I'm only doing Algebra 1,so I am probably missing something,I just thought what you did to one side you did to the other.Could someone please explain this to me?",
        "A": "9.999999999999...=x. 10x -1x = 9x"
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 3:50, VI HART says that infinity is not a number but if you were to find the biggest number than how would u find it?",
        "A": "Let s play a game to see who can find the biggest number. You go first. I win. According to the rules of algebra and common sense, all I have to do to win is to add one to your number. But I could add a billion or a trillion. Numbers have no end."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 2:13, if x=.999..., how could 9=9x because 9 would equal 9 *.999...?",
        "A": "9 does equal 9 * .9999......... - if you do the multiplication by place value you get 8.1+0.81+0.081+0.0081 etc, which equals 8.999999....., or 9."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 0:17: What is a \"Number - Venn - Graph - O - Chart\"?",
        "A": "It is Vi s name for the Venn Diagram of the different types of numbers."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 1:23 Vi said 0 = -0 so there is a -0?",
        "A": "In computer science, where we have to convert all the numbers to binary, including the negative sign, it is possible to have a +0 and a -0, depending on how you encode the numbers. They represent the same quantity of things (there are both 0 elephants and -0 elephants standing on my toes right now), but to a computer that doesn t know any math and is just blindly running electricity through transistors, they evaluate as different."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 2:03 in the video she said subtract 9.9999... From 9.9999... And from 10. 10-.99999.... We know equals one,but if this is proof to prove that .9 repeating equals one then we can't use it as one yet.why does she use it as one in that instance?",
        "A": "What she is doing is subtracting x from 10x, not 0.999... from 10 and at the same time subtracting 0.999... from 9.999... Therefore you receive 10x - x = 9.999... - 0.999... Note that we have already defined x to be 0.999..."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "So based on 6:46 how are you able to cross a room? This is blowing my mind!",
        "A": "As Vi said, infinity has our backs. We may not be able to cross a room given a finite number of those steps, but with an infinite number of those steps, it can be done. Note here that the time taken to carry out each of those steps reduces at the same rate."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 2:00, how does subtracting .9999.... from 10 give you 9?\nWouldn't it become 9.111111...?",
        "A": "If you subtract .9 from 10, you get 9.1. Now subtract .09 from that and you come up with 9.01. Then subtract .009 from that and you get 9.001, etc etc etc. You are subtracting .8888...."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 9:53, Vi had a giant graphic about the groups of numbers (naturals, reals, etc...) and there was a group made in yellow. What was it? I can't read. Can someone tell me?",
        "A": "The yellow was Algebraic Numbers."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "At 8:55\nPlease, correct if I wrong:\n\n0.999... = 1\nso,\n4.999... is equal to 5?\n\nright?",
        "A": "Yes, it is. Take into account that 4.999.... is equal to 4+0.9999..., which is 4+1, which is 5."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "So does that mean that .666...=2/3 and also 1/2? I kinda lost track after 1:40",
        "A": "0.666... does equal 2/3, but not 1/2. 0.4999... = 1/2"
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "at 1:56 she multiplies by ten, so 9.999........0 should have a zero at the end of the nines just as at 4:48 a 1 is at the end 0+1 is less than 10, there for 9.999........ is less then 10",
        "A": "You are talking about an end to an infinite set of nines. You are not grasping the definition of infinity, which is unending. You can t put a 0 beyond all of the nines, because there is no end to the nines. Such an end does not exist."
    },
    {
        "video_name": "TINfzxSnnIE",
        "Q": "what is vi talking about at 1:00 and are you kidding me I still don't get it",
        "A": "She was explaining why 0.99999999... equals 1. She later says that 0.99999999 repeating does not equal 1. It makes sense that you don t get this because it is completely false."
    },
    {
        "video_name": "TcIDXqmt74A",
        "Q": "At 7:15 \"We saw this in the last video\".\nwhere is the last video?\nI can't find the lecture content in last videos.\nSo I don't understand lecture content at 7:20.\nPlese help me. . .",
        "A": "I think it s this video: Variance of differences of random variables under the Random Variables section of Statistics and Probability"
    },
    {
        "video_name": "DopnmxeMt-s",
        "Q": "wait 1:23 -15-3= 18? no 12? -15-3 3 have invisible positive.",
        "A": "-15-3 is -18 a negative subtracted by a negative is a negative"
    },
    {
        "video_name": "DopnmxeMt-s",
        "Q": "how does Sal check his work at 5:14",
        "A": "You should always get in the habit of checking your work. If you can find and fix errors, you get better grades."
    },
    {
        "video_name": "DopnmxeMt-s",
        "Q": "at 5:28 1+9/12 where did the 9 come from?\nI would do 1/12+3/4=4/15... what am I doing wrong here?\nIs it right to ask what I could multiply 3/4 with in order to get a common number (?/12) ?\nto make it 1/12+3/4(3) or 1/12+3x3/4x3 = 1/12+9/12 = 1+9/12 =10/12>5/6\ndoes that make sense or am I complicating things?",
        "A": "Joleen, Yes, you did it correctly. 1/12+(3/4)*(3/3) or 1/12+3x3/4x3 = 1/12+9/12 = 1+9/12 =10/12=5/6"
    },
    {
        "video_name": "DopnmxeMt-s",
        "Q": "At 1:31, why do you leave out the x in the reciprocal? If the reciprocal is the opposite, shouldn't the x be included?",
        "A": "The goal is that you re wanting to turn the 2 into a 1, so that the only thing remaining on the left side is an X. If you had used 1/2x, the equation would have turned into 1 = -18/2x. While the equation would still be correct, you wouldn t have figured out X!"
    },
    {
        "video_name": "DopnmxeMt-s",
        "Q": "im confused because at 1:01 you said what you do to one side you have to do to the other my algebra teacher told me thats only in some cases is that true or false",
        "A": "yes, you always have to do it to both sides otherwise you change the equation"
    },
    {
        "video_name": "DopnmxeMt-s",
        "Q": "at 4:24, why do you multiply both of them by the reciprical? I thought you just divided the fractions. Well i know you switch one fraction when you divide them but im not sure which one, the first or second? You get what im saying?",
        "A": "say you have 4x = 4 which is same as 4x/1 = 4/1 now we know that 4/1 times 1/4 = 1 so we multiply both sides by 1/4 . 4x times 1/4 = 4 times 1/4 = x = 1 if you 2(pi)xyz = 2(pi)yz and were asked to solve for x it is super easy to multiply both sides 1/2(pi)yz which leave x = 1"
    },
    {
        "video_name": "KouDAzYl_bc",
        "Q": "At 4:34, Can we use the identity (a^2 - b^2) = (a-b)(a+b) for the terms (x-1)^2 and 9?\nCan't we just factor these terms as (x - 4)(x + 2)?",
        "A": "Very clever! Yes, you can! In fact, as you ve shown the factoring, you get the same answer."
    },
    {
        "video_name": "yOqhKOicqhU",
        "Q": "at 1:42 81/36 squared so it can also be 9/6 instead of 9/4?",
        "A": "Not exactly. Be careful about what those fractions equal. 9 / 4 = r ^2 ( which means that r = 3 / 2 ) 81 / 36 = r ^2 ( which means that 9 / 6 = r ) But 9/ 6 can be simplified down to 3 / 2 which is the same answer! Hope this is of help!"
    },
    {
        "video_name": "yOqhKOicqhU",
        "Q": "1:00 Why does it have to be H(4)/H(2)? Why not H(2)/H(4)?",
        "A": "Because he divided 324 by 144, which is h(4)/h(2). If he wrote 144/324, it would have been h(2)/h(4)."
    },
    {
        "video_name": "UmSe4rh__iU",
        "Q": "so at 0:42 you mentioned the area. do you always have to put in the cm2 so then it also can be feet2 or m2 or inch2?",
        "A": "Yes. If the length of the square is 1 foot and the width of the square is 1 foot, then the area of the square will be 1 foot squared."
    },
    {
        "video_name": "CLQRZ2UbQ4Q",
        "Q": "at 2:53 you say it is not enough info. to solve. What if you aren't allowed to write that on your paper?",
        "A": "Then it is most likely solvable. Most teachers do no give unsolvable questions on tests."
    },
    {
        "video_name": "CLQRZ2UbQ4Q",
        "Q": "At 1:10, why did sal cross out the expression divided by 3?",
        "A": "He s dividing both sides of the equation by 3. 3 divided by 3 = 1. And, if you multiply anything by 1, it doesn t change the other value (5*1 = 5; x*1 = x; etc). You can also look at it as he is reducing the fraction. This process of cancelling common factors is done all the time when reducing fractions."
    },
    {
        "video_name": "CLQRZ2UbQ4Q",
        "Q": "At 2:15, instead of factoring, why can't you divide the 3a and the 2 by 3 to isolate the \"a\"? For example a+5b= 2/3. Then divide the 5b and the 2/3 by 5 to get a+b=2/15. Then just multiply 2/15 by 15 to get 30/15 which equals 2.",
        "A": "Good Question. Remeber that when you have (a+b)/3 that is the same as a/3 +b/3. When you had 3a+5b=2 and you divide both sides by 3 you should get (3a/3)+(5b/3) = 2/3. whcih is a+(5/3)b=2/3 and you cannot find a solution of a+b becasue you do not have enough information. When you did it you did not divide the 5b by 3 but left it as 5b. That is why you came up with an answer."
    },
    {
        "video_name": "vO1Ur38PGCY",
        "Q": "At around 3:30, couldnt he just have used -1/2 as the scale?",
        "A": "-1/2 would be the opposite of 1/2, which would make the shape twice as large. Dilating by 1/2 makes a shape smaller."
    },
    {
        "video_name": "vO1Ur38PGCY",
        "Q": "what do you even do for dilate.i don't get it.oh and reflect i cant seem to figure out what do the things at the top mean I've tried to use this as a guide for defining congruence through rigid transformations .The first video didn't work either.In the video 1:50-3:43 and3:57-4:33 that part seems pretty confusing I've also watched this video multiple times and it isn't giving me energy points.Why don't they give me energy points?",
        "A": "I d suggest rewatching some of the earlier videos about dilation/scaling around a point. That usually helps me. Regarding your energy points, sometimes that s happened to me too. It usually doesn t last for more than a day or so (for me). Good luck!"
    },
    {
        "video_name": "vO1Ur38PGCY",
        "Q": "Hi Im just confused about something. At 4:19, i don't get how y=1 goes to (0,1) (1,0).",
        "A": "(0,1) and (1,0) are just point on the y=1 line. They weren t purposefully chosen, these are just arbitrary points on the Mirror line."
    },
    {
        "video_name": "vO1Ur38PGCY",
        "Q": "at 0:05 he meant W instead of U!",
        "A": "That s right! There is now a correction. I guess even geniuses like Sal can make mistakes. ;)"
    },
    {
        "video_name": "vO1Ur38PGCY",
        "Q": "At 0:38, how does Sal figure out how much he's going to move it?",
        "A": "He took point E and the equivalent point on the second figure, and determined the distance between them in both the X and Y directions."
    },
    {
        "video_name": "kpEJwpemL2Q",
        "Q": "So 1 + 4 is equal to 4 + 1 in 0:45. Is this the commutative law?",
        "A": "Yes, very good observation. It is also called the Commutative Property of Addition."
    },
    {
        "video_name": "UAcRCt9pEDI",
        "Q": "At 2:41, Sal multiplies both sides by y^-2 instead of y^2, why is that?",
        "A": "He is multiplying both sides of the equation by y^(-2) so that the y^2 cancels out on the right hand side. This is to get the equations in separable form (the y terms on one side and the x terms on the other)."
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "So at about 1:54 he says multiply by 1/5 how do you do that because i'm really confused and i'm guessing that i'm really going to know how to do that.",
        "A": "When you multiply something X by a fraction A/B, you treat X as the fraction X/1 since X/1 = X. Then you are simply multiplying two fractions in which you multiply the numerators and place the product over the product of multiplying the denominators. so X * A/B = (X*A) / (B*1)."
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "2:59 what's a \"reciprocal\"? the dictionary descriptions aren't good enough.",
        "A": "To reciprocate is simply to inverse."
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "At 6:15 when you flip it over shouldn't it be a positive fraction?",
        "A": "No, a reciprocal stays with the original sign."
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "what is a variable? 3:33",
        "A": "A variable is a letter that represents a unknown number."
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "I do NOT uderstand how you simplify @ 7:02 in this vidio.\nDo you cross divide?",
        "A": "You are suppose to divide both side by two, then that cancels out the two, and the four becomes a two. So x=2"
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "yes why didnt you just multiply straight forward instead of simplifying...that just threw me off there? 6:37",
        "A": "It is usually factor to simplify first then multiply. The reason is that before multiplying, the numbers are smaller and it is easier to spot common factors that should be cancelled out. If you prefer, you can multiply 1st, then simplify the fraction."
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "At 4:36, why do you express the multiplication sign with a dot? I kind of got confused",
        "A": "It s just conventional, because in subjects like algebra, you use the variable x a lot, which can be confused with the multiplication sign. In order to replace the x in these cases, we can use a dot or an asterisk."
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "0:00 through 1:39 was not very helpful but later it gets a lot more complicated... why do you multiply 1/5?? dont you just divide and you just get you answer instead of making it more complicated?! ugh! help me understand! PLEASE!",
        "A": "ok... thank you! Im SLOWLY getting it"
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "uhhhhhhhhhh . . . wait, isn't the reciprocal of -3/4 +4/3, not -4/3? (See 3:09)",
        "A": "No reciprocal keeps its sign."
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "In the first fraction example when he is checking the answer couldn't you also cross simplify the -3 and the 39 since 3 goes into 39(around 4:50 in the video)? The negative in the numerator and the denominator would cancel each other out and make the fraction positive.",
        "A": "-26/25"
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "How come at around 6:37 you divided the 6 and 8 by 2 when the 7 and 5 couldn't be divided by 2? Also how do you know when to do that and when not to because they make different answers, right?",
        "A": "Fractions always have to be reduced. Sal could have multiplied 1st: 7/8 * (-6/5) = -42/40 then reduced by a common factor of 2. Or, since the 2 fractions are being multiplied, you can actually reduce prior to performing the actual multiplication. This is just the basic process for multiplying fractions. It you don t understand the process, I suggest you review the section of videos on multiplying fractions."
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "Around 5:20 how come you didn't simplify the 3s in the fraction?",
        "A": "he does it in the next step after he folds together the 3 x 10 so then 30/39 all divided by 3 at once"
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "at 4:58 sal said -3/4(-40/39)= 10/13. how?",
        "A": "Negatives cancel out => 3/4(40/39) 4 s cancel out => 3(10/39) 3 s cancel out => 10/13."
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "At 3:43 , how come the reciprocal of -3/4 isn't 4/-3. Or is it negative 3/4?\n(Sorry if this question is confusing)",
        "A": "Either way is fine. The negative sign can be moved freely from the numerator or denominator."
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "at 6:48, why were the 8 and the -6 divided by 2 instead of being multiplied straight across?",
        "A": "You have to simplify it first to make it easier"
    },
    {
        "video_name": "bAerID24QJ0",
        "Q": "at 4:20 I don't understand how he inverted the fraction 3over 4, what happened there?",
        "A": "The reciprocal of -3/4 is -4/3. -3/4 times -4/3 is 1, so he is trying to make the coefficient of x as 1. We can of course multiple both sides of an equation by the same number."
    },
    {
        "video_name": "h5BmWo5_sc8",
        "Q": "At 3:40, What does Grant mean by putting vectors anywhere in space? Wont the rotation always take place around the axis of rotation (which is why the right hand rule works) which is essentially the vector arrow will be pointing?",
        "A": "He s talking about the location of the arrow that represents the vector, as a whole... not about the tip of the arrow. You can place the arrow at the origin, or at the poles of the sphere, or anywhere, as long as the arrow has the same direction and magnitude, it is the same vector."
    },
    {
        "video_name": "sSyPAAyL8nQ",
        "Q": "At 4:32 you take the derivative of P to y first, but Greens theorem says that you have to start with Q.",
        "A": "Because in this example we are going clockwise (rather than counter-clockwise, which gives the form usually seen in Green s Theorem) around a curve we have to multiply the double integral by -1, or we can essentially just switch the order and start with the derivative of P with respect to y."
    },
    {
        "video_name": "sSyPAAyL8nQ",
        "Q": "At 6:15, Sal says we can do the hard way by setting the bounds of integration for y as y=+/-sqrt(x^2-1). I'm not sure why don't we just take the bounds for y as +/-1 as we do for x.",
        "A": "If you took the bounds for y as +/-1, and x as +/-1, you would get the area of a square with four corners at (1,1), (1,-1), (-1,-1), (-1,1), not a circle with radius 1."
    },
    {
        "video_name": "JtsyP0tnVRY",
        "Q": "on 3:41 im confused there is no 0th term and when he does the 4th term why does he put a 3",
        "A": "It s always the term minus one. So for example, when finding the first term he would do (n-1), and if we were finding the first term it would be (1-1) which equals 0. Same goes for the fourth term when you do (n-1), (4-1) equals 3."
    },
    {
        "video_name": "JtsyP0tnVRY",
        "Q": "1:40 - 2:00 I think Sal should not say \"subtracting negative 6\" but either \"add a negative 6\" or \"subtract 6\"",
        "A": "I agree"
    },
    {
        "video_name": "JtsyP0tnVRY",
        "Q": "Why did Sal say subtract negative six at 1:40?",
        "A": "If you see the correction text box, it notes Sal says subtract negative 6, but means subtract 6 ."
    },
    {
        "video_name": "JtsyP0tnVRY",
        "Q": "at 2:18 isnt it supposed to be: 15-(0)(-6) the (-6) looks positive in the video",
        "A": "I think it suppoes to be poistive"
    },
    {
        "video_name": "JtsyP0tnVRY",
        "Q": "I don't understand what Sal did at 2:10",
        "A": "He s defining the first term as the 0th term. The pattern appears to be: 15-(n)6 (Starting term 15, each subsequent term [aka value of N] subtracts six) Where N is the term. By defining the first term as N=0, the equation becomes: 15-(0)6 Or, in other words, just 15. If you wanted to define the pattern in a bit more of an intuitive way, you could reformat the formula for the Nth term as 21-(n)6 Then, the first term is N=1 and you can work with it like that."
    },
    {
        "video_name": "JtsyP0tnVRY",
        "Q": "At 2:16 wasn't he saying it wrong? instead of \"minus negative . . .\" shouldn't he have said just \" minus\"?",
        "A": "Minus a negative is the same as adding a positive."
    },
    {
        "video_name": "JtsyP0tnVRY",
        "Q": "at 4:19 we see the equation 15-99 times 6, can you just put it in the calculator like that and end up with the correct answer?",
        "A": "Yes.You will get the correct answer."
    },
    {
        "video_name": "JtsyP0tnVRY",
        "Q": "At 3:40 why does Sal use (n-1) ....... how did he get that to work out the rest.....",
        "A": "because 15 is the first so you exclude it that is why it is (n-1)"
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "At 2:37, why does pi=4pi, pi/2=2pi, and pi/4=pi? He doesn't explain it and i can't figure it out.",
        "A": "He found a common denominator for the three terms and added them into one term."
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "Could sal have used the side ratios for 45:45:90 triangles to find the side lengths?",
        "A": "what do you mean by 45:45:90"
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "At 5:03, Sal say (x^2)+(x^2)=1^2, but then says 2x^2=1. Why is the 1 no longer squared?",
        "A": "Because 1\u00c2\u00b2=1. He just simplified it."
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "At time 5:35, wouldn't the x^2 = 1/2 become x= (the square root of) 1/2, instead of 1 over the square root of 2? doesn't quite make sense to me what he did there...",
        "A": "\u00e2\u0088\u009a(1/2) = \u00e2\u0088\u009a1/\u00e2\u0088\u009a2 = 1/\u00e2\u0088\u009a2"
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "at 3:16, Sal says that the length of hypotenuse is 1. Was he measuring the line segments in terms of radians ? If not, what did he mean by 1? 1 radian? 1 radius? 1 unit?",
        "A": "Careful; I believe radians are only used to measure angles. In case I am wrong though, someone please correct me. In this case, when Sal says the length is 1, then he means 1 unit, in respect to the coordinate plane he is using. You could also say the hypotenuse is the radius of the unit circle. Hope this helps!"
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "in min 2:30, why did he refer to pi as pi times 4?",
        "A": "we refer to pi as 4pi/4, pi/2 as 2*pi/4 pi - pi/2 - pi/4 = 4*pi/4 - 2*pi/4 - pi/4 = (4pi - 2pi -pi)/4 = pi/4 It is the same rule as with fractions: 1 - 1/2 - 1/4 = (4 - 2 -1)/4 = 1/4"
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "At 6:30, Why the point B is taken (cos\u00cf\u0080/4,sin\u00cf\u0080/4).",
        "A": "cosine \u00cf\u0080/4 (45\u00c2\u00b0) is the Y coordinate and sine \u00cf\u0080/4 is the X coordinate where the terminal ray crosses circle."
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "When he said he dropped a perpendicular around 0:10, did he mean an altitude?",
        "A": "Correct. the dropped perpendicular is the y coordinate aka altitude. On this unit circle the y coordinate is calculated as SIN(pi/4). Regards, APD"
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "At 7:36, how is the tangent equal to the slope of the hypotenuse? Is it always the case?",
        "A": "If you look at the soh cah toa definition of the tan function, we see that the tan is the opposite divided by the adjacent. This is the exact definition of a slope: the amount of rise (opposite) for distance traveled (adjacent)."
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "At 5:00 Sal writes th sq. rt. of x^2+x^2=2x^2; really?",
        "A": "Yes, really. x\u00c2\u00b2+x\u00c2\u00b2 can be rewritten as: 1*x\u00c2\u00b2+1*x\u00c2\u00b2 then you can use the distributive property to get: (1+1)*x\u00c2\u00b2 which of course is 2*x\u00c2\u00b2 or 2x\u00c2\u00b2"
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "at 2:31 he gets the common denominator the same, the first term is 4/4 x pi = 4 pi/4 (makes sense). How does he evaluate the second and last term?",
        "A": "If I understand your question correctly, he doesn t evaluate the three terms individually. He instead simplifies the numerator to a single term, which is just pi (4x - 2x - x = x) and puts that over 4."
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "at around 5:43 how did sal know that the 2 sides were both\nthe same thus labeling both as x?",
        "A": "He justified that claim by noting that the angles opposite to the two equal sides are also equal. Whenever that is the case, you have an isosceles triangle, in which the sides opposite the equal angles are always equal to each other."
    },
    {
        "video_name": "KoYZErFpZ5Q",
        "Q": "On 5:30 . he multiplied sqrt(2)/sqrt(2) to 1/sqrt(2). But isn't he supposed to multiply sqrt(2)/sqrt(2) to both side of x = 1/sqrt(2)?",
        "A": "No. sqrt(2)/sqrt(2) equals 1, so multiplying it to one side doesn t actually alter the equation."
    },
    {
        "video_name": "2CsQ_l1S2_Y",
        "Q": "@ 2:42 Sal uses the chain rule. Do you always have to use the chain rule in implicit differentiation?",
        "A": "Yes, implicit differentiation is a special application of the chain rule. It s how we take the derivative of an expression involving y with respect to x, which otherwise doesn t sound possible (we normally need a function of x in order to differentiate with respect to x)."
    },
    {
        "video_name": "2CsQ_l1S2_Y",
        "Q": "Sorry for programmer format, but surely - x / 2*y^(-1/2) at 5:20 is wrong?\n\nSurely x * (1/2) * y^(-1/2) = x*y^-(1/2) / 2 ?\n\nNot x/2*y^(-1/2)!\n\nIf it is right, and I am wrong though then please explain why, I'll be very grateful..",
        "A": "Now I get it. Both mean (x/2) * y^-1/2. I originally thought Sal meant x/2 * 1/sqrt(y) Thanks, the wolfram page looks helpful.."
    },
    {
        "video_name": "2CsQ_l1S2_Y",
        "Q": "at 0:59, is dy/dx the same thing as d/dx [x^-2] because y = x^-2?",
        "A": "yes that s how you write the notation. You can also write dy/dx as (y) or f (x). There are many ways in which you can indicate a derivative of a function."
    },
    {
        "video_name": "2CsQ_l1S2_Y",
        "Q": "At 5:12 in the video, does the sign of the exponent matter when Sal rewrites\n1/2(y)^-1/2 as x/2sqt(y)?",
        "A": "Recall that, as a property of negative exponents, a^(-b) = 1/(a^b). In other words, you can move a factor of a rational expression from the numerator to the denominator (or vice versa) by changing its exponent s sign. Sal uses this to change y^(-1/2) to 1/(y^(1/2)) = 1/sqrt(y)."
    },
    {
        "video_name": "2CsQ_l1S2_Y",
        "Q": "At ca. 3:51, why do you multiply with dy/dx",
        "A": "We want to find d sqrt(y)/dx so we use the chain rule and compute d sqrt(y)/dy and then multiply by dy/dx."
    },
    {
        "video_name": "2CsQ_l1S2_Y",
        "Q": "At 7:34 from where does Sal substitute that value.",
        "A": "Sal s already figured out that y = 1 / x^2 when he did implicit differentiation, so he simply just substituted for 1 / x^2 for y"
    },
    {
        "video_name": "2CsQ_l1S2_Y",
        "Q": "At ca. 2:07 Sal says \"...with respect to x...\" What does \"with respect to\" mean in this context? Thanks for any insight.",
        "A": "With respect to x in this context refers to the rate of change in the graph of the function for a certain value of x. Since 1 is a constant and does not change no matter how big/small x gets, the rate of change with respect to x is zero."
    },
    {
        "video_name": "6kGi1dvGZNY",
        "Q": "At 4:48, why take the square root of everything now instead of at the end?",
        "A": "It would save a step (or more) by eliminating the process of having to calculate the whole value of what s inside and then taking the square root of that (which with a calculator would require a second step)."
    },
    {
        "video_name": "x5EJG_rAtkY",
        "Q": "I have a question about the fraction @3:45. How does (5/2)(4)=10?",
        "A": "5/2 is equal to 2.5, and if you multiply that by 4, you get 10. OR 5/2 times 4/1 equals (5)(4) over (2)(1) equals 20/2 = 10."
    },
    {
        "video_name": "x5EJG_rAtkY",
        "Q": "I'm confused on why Sal brings up x in 0:14, when the variable mentioned in the beginning is r. Is x just an example, or is it something else?",
        "A": "Correct. Sal is just using X to make a generalization."
    },
    {
        "video_name": "x5EJG_rAtkY",
        "Q": "After 3:10\n2r-13/4<5/2\n2r-3.25<2.5 I converted the fractions to decimals instead of multiplying as Sal.\n2r<5.75\n2r< 575/100\n2r<23/4\nr<23/4(2/1)\nr<46/4\nr<23/2 Sal got r<23/8\nWhy?",
        "A": "Your error is in the line where you divide both sides by 2. 2r<23/4. You divide both sides by 2, so it will be 23/8 NOT 46/4."
    },
    {
        "video_name": "x5EJG_rAtkY",
        "Q": "2:38 shouldnt it be 4 times -3 plus 1 which is -11?",
        "A": "It is 3*4+1 which is 13, not -3*4+1."
    },
    {
        "video_name": "x5EJG_rAtkY",
        "Q": "Where did he get the 4? at 3:21 ~ 3:25",
        "A": "No.. he multiplied 4 to the entire inequality to get rid of the fraction. Whatever you do to one side you do to the other side. By multiplying 4 to the inequality, you can do it without fractions that may confuse you or complicate things."
    },
    {
        "video_name": "NzIoHQfStoM",
        "Q": "At 0:00 why did they pick 2.75",
        "A": "Its just a example. He could of chose 2.93 if he wanted to."
    },
    {
        "video_name": "u1HhUB3NP8g",
        "Q": "at around 9:15, when dividing the first equation by x-bar, why doesn't Sal cancel out an x-bar from every component of the equation like he did with b(x-bar)?\n\nshouldn't m(x-bar)^2 + b(x-bar) = (xy)-bar become m(x-bar) + b = y-bar?",
        "A": "No, it s not x-bar*y-bar, but xy-bar (the average of their product)."
    },
    {
        "video_name": "u1HhUB3NP8g",
        "Q": "at 4:15 how did he get 2bn , shouldn't it be just bn by taking out one b",
        "A": "Correct, when you take the derivative of a value (let s say x( you reduce the power by 1, making the original power a factor of that same value. So x to the power of 2 (i.e. x^2 or x-squared) becomes 2x, 2x^3 (or 2x to the power of 3) becomes 6x^2 (6x squared)."
    },
    {
        "video_name": "u1HhUB3NP8g",
        "Q": "Likewise, at 4:07, how did Sal take the partial derivative of nb^2 and get 2nb (or 2bn)? Also, I thought the object here was to factor out the b?",
        "A": "It s just a basic rule for derivatives. Deriv. of x^2 is 2*x. He s finding the derivative w.r.t. b when everything else in the term is held constant."
    },
    {
        "video_name": "u1HhUB3NP8g",
        "Q": "Thanks for the extremely helpful video series. At 9:08 Sal is dividing the equation by mean of x. What happens when mean of x's is zero. Is the derivation/formula valid is such cases?",
        "A": "If x-bar was zero, then both sides of the equation would be 0 so there s no need to even do it."
    },
    {
        "video_name": "u1HhUB3NP8g",
        "Q": "at about 6:03 the bit about -@y+(m@x+b)=0 where @y is y-bar and @x is x-bar well m@x+b is y(@x) which is probably @y and -@y+@y is ALWAYS 0!",
        "A": "So what? And how do you know y(@x) is @y, and what difference does that make?"
    },
    {
        "video_name": "u1HhUB3NP8g",
        "Q": "I don't understand the part at 7:14. xy-bar does not equal x-bar * y-bar. How is he dividing out an x-bar to obtain this y-bar?",
        "A": "oh I got it. He didn t derive it like that. He used the other partial derivative starting point."
    },
    {
        "video_name": "McOMtxI_Jzs",
        "Q": "At 2:11 i be like um.... what? This is gibberish to me",
        "A": "Hey Danielle, You should be like this... He is multiplying both sides by 3 to remove the denominator of 3 on the left hand side of the equation. Hope that helps! - JK"
    },
    {
        "video_name": "tV8cMNYi4q4",
        "Q": "What if you had an exponent on the outside of the parenthesis? How would you solve for it? Just like in 0:35 he showed a problem without the exponents on the outside.",
        "A": "the parenthesis were around the negative 2 so it didnt look confusing like this 3+-2 but he put 3+(-2)"
    },
    {
        "video_name": "N5kkwVoAtkc",
        "Q": "5:40. Why does Khan multiply 1/y by dy/dx? I don't see where the dy/dx comes in.",
        "A": "This is a common question. well let me ask you: what is the derivative of lnx ? simply it is 1/x, what is the derivative of y simply y or dy/dx. combine both: what is the derivative of lny ? it is not only 1/y, why? because y is given as a function in x, and the derivative of y is not 1 (instead it is dy/dx). Thus derivative of lny is (1/y).(dy/dx). I hope this is helpful"
    },
    {
        "video_name": "N5kkwVoAtkc",
        "Q": "In the original problem x^x is substituted for y. Wouldn't you substitute x^(x^x) in for y in the more complex problem instead of only x^x? Should the solution at 7:10 be x^(x^x)(lnx+1)lnx+1/x*x^x?",
        "A": "At the end, he substitutes x^(x^x) for y, but other than that, he doesn t make any substitutions for y. The right hand side of the equation is x^x*(ln(x)+1)*ln(x)+(1/x)*x^x because the derivative of x^x is x^x*(ln(x)+1). (the bold is a mistake)"
    },
    {
        "video_name": "N5kkwVoAtkc",
        "Q": "at 0:45 Sal call taking the 'x' from the power as a product of ln(x) like: x*ln(x). that is straight out of the logarithm properties, isn't it?!",
        "A": "You are correct sir. Basic logarithm properties. log (a^b) = b * log (a)"
    },
    {
        "video_name": "N5kkwVoAtkc",
        "Q": "At 1:41 why is the chain rule used? Isn't lny just one function?",
        "A": "Y is x^x which is a function. So ln y isn t one function."
    },
    {
        "video_name": "N5kkwVoAtkc",
        "Q": "At 0:35, Why Sal use natural log, instead of common log? And why y = x^x is defined only for x>0",
        "A": "Common log is fine. But natural log is easier to take a derivative of"
    },
    {
        "video_name": "N5kkwVoAtkc",
        "Q": "At 4:00 onwards, while solving x^x^x why can't we make it x^x^2 because when have something like x^3^2 we make it x^6 (3 and 2 multiply right)?\nSo if x ^ x ^ x so if x and x multiply we get x^x^2.\n\nWhats the fault in my logic?",
        "A": "With 3\u00c2\u00b2 you do not multiply 3 and 2. The 2 tells you how often to multiply 3 with itself. So in fact 3\u00c2\u00b2 = 3*3 = 9. The same reasoning applies to x^x = x*x*x ... *x (multiplied x times rather than 2 times). So x^x^x not equal to x^x^2 (except for x=2)"
    },
    {
        "video_name": "N5kkwVoAtkc",
        "Q": "At 7:09, I just factored out an x^x on the right side of the equation. Doing that, I ended up with the answer of (x^x)(x^x^x)* [Lnx(Lnx+1) + 1/x]. Would this also be correct?",
        "A": "Yes, that would work. You can check by letting x = 2."
    },
    {
        "video_name": "N5kkwVoAtkc",
        "Q": "At 6:35 shouldn't the derivative of x^x be lnx+1 ? why is Y added to it?",
        "A": "The derivative of x^x is: d/dx(x^x) = x^x [ln(x)+1] But, remember that this video is about the derivative of x^(x^x) which is more complicated."
    },
    {
        "video_name": "R486L0M5cWk",
        "Q": "at 6:47 how do you get 108",
        "A": "all you have to do is subtract 268-160=108 try it"
    },
    {
        "video_name": "R486L0M5cWk",
        "Q": "at 0:03 i dont get it",
        "A": "It is a division problem. What is it that you don t get?"
    },
    {
        "video_name": "R486L0M5cWk",
        "Q": "at 5:18 sal said 13 minus 5 but meant 13 minus 8",
        "A": "That s already been fixed...."
    },
    {
        "video_name": "R486L0M5cWk",
        "Q": "at 3:37 you subtracted 108 minus 96 shoould you be subracting 108 minus 86",
        "A": "The video is correct. You should subtract 96, not 86. It is the result of multiplying the 6 (from the dividend 8 6) times 16 (the divisor). So you get 6 times 16 = 96."
    },
    {
        "video_name": "lh2mp0aqSh8",
        "Q": "At 1:47, it states that 1/4 is equal to 25/100. so how can you change other fractions to a different fraction?",
        "A": "if you have 1/2, multiply the top and bottom by two and you get 2/4. Multiply by two again and you get 4/8."
    },
    {
        "video_name": "lh2mp0aqSh8",
        "Q": "at 4:33 how can u recognize it easily because i dont understand the way u said it?",
        "A": "0.02 = 2/100 0.2 = 2/10"
    },
    {
        "video_name": "lh2mp0aqSh8",
        "Q": "at 4:10 does 0.75 have to be 3/4\u00ef\u00bc\u009f",
        "A": "To represent 0.75 as a fraction you would write 3/4, yes. These two numbers are equal. Of course the fraction 6/8 also represents the value 0.75, but you can just simplify this fraction to 3/4."
    },
    {
        "video_name": "lh2mp0aqSh8",
        "Q": "At 4:27 can the answer also be 2 and 1/2?",
        "A": "Yes, 2 and 1/2 is also the same as the other answers Sal talked about at the 4:27 mark in the video."
    },
    {
        "video_name": "Uc2Tm4Lr7uI",
        "Q": "At 0:59 he wrote \"slore\". Didn't he mean \" store\".",
        "A": "Didn t even notice that. Thanks!"
    },
    {
        "video_name": "Uc2Tm4Lr7uI",
        "Q": "at 0:59 you put \"gift slore\" instead of \"gift store\"",
        "A": "Thanks, for the information! I reported it."
    },
    {
        "video_name": "Uc2Tm4Lr7uI",
        "Q": "Between 5:27 & 5:36 you added the fractions to get a whole number, but what if the fractions doesn't make whole",
        "A": "Seriously, couldn t you have put a better answer rsdefenbaugh?? \u00f0\u009f\u0098\u0092"
    },
    {
        "video_name": "suwJmCrSDI8",
        "Q": "Did Sal make an error at 2:25, where there was a negative sign before the -sqrt5/sqrt35, but at 2:36 he says the simplified version is simply a positive sqrt1/sqrt7?",
        "A": "Yes he did! Good eye. He forgot to bring the negative over when we was simplifying! Oops! There should be a little bubble that pops up in the bottom corner that shows the correction though. The correct answer should be -\u00e2\u0088\u009a(1/7)"
    },
    {
        "video_name": "suwJmCrSDI8",
        "Q": "Wait at 2:24, the answer is actually -sqrt1/-sqrt7?",
        "A": "You are correct that the answer needs a minus sign. But, it should only have ONE minus, not 2. The correct answer would be -\u00e2\u0088\u009a(1/7) or -\u00e2\u0088\u009a(1) / \u00e2\u0088\u009a(7)"
    },
    {
        "video_name": "suwJmCrSDI8",
        "Q": "At 2:37 in the video the answer is sqrt(1/7), but shouldn't it be -sqrt(1/7). Seems like the negative got dropped when it was reduced.",
        "A": "There is a correction box that pops up... yes, it should be negative."
    },
    {
        "video_name": "suwJmCrSDI8",
        "Q": "isn't te awnser to the first question negative -sqrt(1/7) not sqrt(1/7)? at 2:14-2:39",
        "A": "Yes, you are correct. Good catch. Looks like Sal dropped the minus sign."
    },
    {
        "video_name": "7MKA2QlKvHc",
        "Q": "At 5:40, are we allowed to just factor whatever we want out without worrying about finding a GCF for the entire expression? How can we just casually decide, \"oh let's just pull out a1 from the entire thing even though it isn't a gcf for the entire thing?\" Are we allowed to do that?",
        "A": "Yes, absolutely. We are allowed to factor anything out of anything. For example, how would we factor 2 out of 2x+1? Beginning students would say it is not possible because 1 does not divide 2. But we are working with the real number system here, a number divided by any (non zero) number is still a number. Factoring 2 out of 2x+1 is simply 2(x+1/2)."
    },
    {
        "video_name": "7MKA2QlKvHc",
        "Q": "At 17:35, why don't we have to do \u00c2\u00b1 when square-rooting?",
        "A": "Because the length of a vector is always positive. For example, suppose we have the vector Z, in R\u00c2\u00b2, with x = -3 and y = 4, his length, ||Z|| is = squareRoot(-3\u00c2\u00b2 + 4\u00c2\u00b2). So, for each negative element of the vector, when it is squared, we know, it became positive, for that the length of a vector is ALWAYS positive."
    },
    {
        "video_name": "kdEQGfeC0SE",
        "Q": "About the interval at 1:15, the first interval with the jump continuity, doesn't this mean that if you had just the right side, i.e. your interval stopped at c as its left endpoint, would the endpoint then be continuous in that interval alone, and if so, can this be useful/a nuisance?",
        "A": "If you were looking at some interval on that graph from c to c+z, then yes it appears that this segment would be continuous. On the other hand if you tried to do an interval to the left, from c-z to c then it would be discontinuous over the interval as the point c would not be on the left curve. (where z is any positive, non-zero value) -John"
    },
    {
        "video_name": "kdEQGfeC0SE",
        "Q": "But by the logic around 5:05, the graph y=1/x is continuous because at 0, the limit is undefined, as is the y value. Can anyone explain?",
        "A": "If a function is undefined, you can t put it into any equation, even if the other side is also undefined, so when a function is undefined, it is discontinuous."
    },
    {
        "video_name": "kdEQGfeC0SE",
        "Q": "At 10:33 , Sal shows the limit approaches the function c from the negative side, although it being on the first quadrant where x and y both are positive and even the diagram represents that the function is starting from positive. How is that possible?",
        "A": "Should that even apply to the first quadrant, even there being no negative sign whatsoever?"
    },
    {
        "video_name": "qS4stB9LfJA",
        "Q": "On 2:46 there is a mistake. 72-10 isn't equal to 62-2. It's incorrect. I know what you meant but anyway, it's definitely incorrect.",
        "A": "I completely agree, ShevrillHD! That part of the work shown needs to be refilmed. We should never write run-on sentences in math, since false equations result. 72 - 10 \u00e2\u0089\u00a0 62 - 2 = 60"
    },
    {
        "video_name": "qS4stB9LfJA",
        "Q": "So at 0:14, she is talking about the green shape when she says shaded figure?",
        "A": "Yes. She is trying to identify the area of the green shape."
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "At 5:16, what's the point of multiplying by 1? Isn't the answer the same?",
        "A": "Sal is making the answer less confusing (dividing C by the square root of 2 = MIND BLOWN). C/sqrt2 is equal to C*sqrt2/2 but is less confusing."
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "How come at 9:03 the answer is not 4root2, wouldnt that be in lowest radical form?",
        "A": "it s correct because he didn t mean that c is divided by 2, he meant that c has to be subcutaneously divided by 2 because that way, the hypotenuse is equal to the leg when subtracted by radical 2. In other words, in 45-45-90 triangle, the leg is equal to the hypotenuse when added by radical 2."
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "Ok..at 3:29, he say to substitute B for A...what does that mean? And after the whole time he worked out the problem I don't get what he is doing..from\n3:29 t0 4:03..please explain it to me more. Thanks!(Please be specific :)",
        "A": "A=B because 45=45"
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "At 6:17 when Sal finds the hypotenuse he says the answer is (c*sqrt2)/2 but i got different answer. I did 2b^2=c^2 and i took the square root of each side. So i got B*sqrt2=c. Wont this be better for finding the hypotenuse?",
        "A": "You got a different result because you solved for c . Sal solved the equation for b . b represents the legs, not the hypotenuse. If Sal had solved for c , then his equation would match yours."
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "I don't get it! On 6:28, the formula A=B=the square root of 2 times C????\nHow did he get that formula what is this formula supposed to do?",
        "A": "The formula is actually A (or B) =C*the square root of 2/2. It is used to solve for an unknown side in a 45-45-90 triangle. For instance: because A=B, one could find the value of C when either A or B was known. All they would have to do is substitute the known value for A and B in the formula and solve."
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "At 5:11 how does he know that square root of two times square root of two equals 2",
        "A": "the square root of two is approximately 1.4. so if it was the square root of two then when you squared it it would be two. so the square root of two times the square root of two or 1.4 times 1.4, well that equals 2 (approximately). if that doesn t make sense, try this. instead of using the square root of two use the square root of 4. well that s 2, isn t it? and 2 times 2 is 4. it is the same thing with the square root of 2 and it works with all square roots. Did I make sense?"
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "At 5:31. Sal says that the hypotenuse of a 45-45-90 triangle will always equal c*((/sqrt2)/2). Couldn't he have simplyified it to c/sqrt2?",
        "A": "Yes, but the former is in rationalized form."
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "At 8:45, why does he not just divide 16 by the square root of 2? I get that 2 cannot square cleanly, but when I plug 16/sr2 into my calculator, it gives the answer 11.3, and that seems acceptable.\nThanks in advance!",
        "A": "The question is always whether you want an exact answer or an approximate answer so the exact answer is 8 SQRT(2). If you round, then you have to know how many places to round to - is the tenth place okay, or do I need more accuracy? Often in pure Math, we want an exact answer. In real life situations, we will probably round to get an approximate answer, but the accuracy we want will affect how many places we round to."
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "At 0:36 he mentions a theorem from the last video,I watched that one but it does not give the theorem's name.What is the name of that theorem and who made it?",
        "A": "I don t think there isn t a name for this theorem , as it is just a special relationship for a special right triangle, namely the isosceles (45-45-90) that happens to be derived from the Pythagorean Theorem. I guess you can call it a corollary to the Pythagorean theorem, by Pythagoras. However, if you really need to search up this theorem look up something along the lines of 45-45-90 special right triangles"
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "At 1:43: If the hypotenuse is opposite the right angle, how do you find the hypotenuse on a triangle without a right angle?",
        "A": "Without a right angle, there is no hypotenuse. By definition, the side opposite the 90\u00c2\u00b0 angle is called the hypotenuse."
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "At 8:52 Sal says that, 16 square roots of 2 over 2, is the same thing as ,8 square roots of 2. I don't understand how this is when, 16 square roots of 2 over 2, has the square roots of 2 in the numerator. I know that 16 divided by 2 is 8 but wouldn't you divide the square roots of 2 by 2 as well since it is also in the numerator?",
        "A": "OR... you can just remember that: in a 90-45-45 triangle, the size of the hypotenuse is always the size of the other sides (remember, both sides are equal) times the square root of two. So if the side is 8, the hypotenuse is 8 times the square root of two. If it is 23.023450723945, than the hypotenuse is 23.023450723945 times the square root of two. If the size is 1, it is just square root of two."
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "Why is it 45:45:90? I mean can't just be 45:45?",
        "A": "it has to have the 90 to make it add up to 180 because the angles in a triangle is equal to 180"
    },
    {
        "video_name": "tSHitjFIjd8",
        "Q": "at 8:15 why does he multiply by the inverse of \u00e2\u0088\u009a2/2? Should he not have multiplied 8 by \u00e2\u0088\u009a2/2, or would he have gotten a wrong answer?",
        "A": "Sal multiplied by 2/\u00e2\u0088\u009a2 so he can cancel out the \u00e2\u0088\u009a2/2 and isolate the variable, C. Also, it s reciprocal, not inverse."
    },
    {
        "video_name": "QTcSBB3uVP0",
        "Q": "So the final transformation matrix at 12:07 is symmetric. Is there anything significant about this? Will it always be symmetric?",
        "A": "One way to quickly prove this is to just take the transpose of the projection matrix, P = A inv(A A) A . P = ( A inv(A A) A ) = A (inv(A A)) A = A inv( (A A) ) A = A inv( A A ) A = P, where several properties of inverses and transposes have been used that were demontrated several videos ago. It s a good exercise to see that each step of the above is justified. In the end we get P = P, in other words, P is symmetric. Hope that helps!"
    },
    {
        "video_name": "AIb5x7fmtvE",
        "Q": "At 1:20, why did Sal say 28 feet instead of 78 feet?",
        "A": "Yeah, it s meant to be 78. There used to be an option to report mistakes like these but I can t find it now."
    },
    {
        "video_name": "AIb5x7fmtvE",
        "Q": "at 1:20 Sal said 28 and meant 78",
        "A": "That means 28 times 78"
    },
    {
        "video_name": "PiQxA9O7Rd8",
        "Q": "What do those two extra tiny lines mean on the triangle at 1:05?",
        "A": "I think it means the side is equal to one of the other sides on the triangle."
    },
    {
        "video_name": "PiQxA9O7Rd8",
        "Q": "at 1:30,the obtuse triangle has 2 acute angles so doesn't that mean it's a acute triangle?",
        "A": "really? Wow, I didn t know that! :P"
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "At 4:35, Sal says y2=-b2+b2/a2*x2. What happened to the 1 from the previous step?",
        "A": "In order to isolate the y\u00c2\u00b2 term on the left, he multiplied both sides by -b\u00c2\u00b2. 1 times -b\u00c2\u00b2 = -b\u00c2\u00b2."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "At 10:35, shouldn't it be +- a instead of +- sqrt(a)?",
        "A": "Yes, it mentions that in an annotation (red, off to lower right area) at 10:36."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "At about 10:30 Sal states (in purple) that for x\u00c2\u00b2=a\u00c2\u00b2 x = +/- sqrt(a). I don't get this? If you square root both sides you get x=a, right?",
        "A": "This is a mistake in the video. I m not sure why it has not been corrected. (Thanks for the time-stamp: they really help with a question like this :-). Notice that Sal doesn t use this mistaken result: when he locates his solutions on the x-axis, immediately after making his mistake, he labels them, correctly, as a and -a"
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "at about 7:20, won't the second line's slope be -a/b instead of -b/a? i learned that perpendicular lines were negative reciprocals, not just negative opposites.",
        "A": "the asymptotes are not perpendicular to each other."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "At 4:19 how does it become b squared over a squared x squared?",
        "A": "Because he multiplied both sides of the equation by -b^2, and he distributed it on all the terms on the right side of the equation, the negatives cancelled out on that term and he put b^2 in the numerator and took x^2 out beside it so that it was multiplied by the fraction b^2/a^2, I think he wrote it that way so that at the end \u00c2\u00b1b/a would be the slope of the asymptotes."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "at 7:05 why cant you take the square root of the green equation but you can take the square root of the orange equation",
        "A": "The green square root contains 2 terms. That binomial can t be factored into 2 of the same factors. The orange square root contains only 1 term and can be factored into 2 of the same factors ( bx/a ) * ( bx/a )."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "At 07:39 Where does raidcal -b^2 gone?",
        "A": "Oh.. Thank you ^^"
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "At 7:07 what happens to the minus b ^2 when you find the approximation of y?",
        "A": "it is assumed to be very small so it is assumed to be 0"
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "at 4:21, shouldn't the result of -b^2 timesing the- x^2/ a^2 be x^2 b^2/ a^2, not the fraction b^2/ a^2 timesd by x^2 ? Wouldn't the x^2 still be by the b^2?",
        "A": "I think so but that s a good question."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "I just learned about hyperbolas today. At 2:31, he says that y^2/b^2, but I thought that a and b always stay in the same place. Am I mistaken, or.........? Please and thank you!",
        "A": "Watch on, at 2:50 he explained it. I agreed with him."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "at 10:33, how does x^2 = a^2 turn into x = +- a^1/2?",
        "A": "This is an error in sal s calculation. It is meant to be +/- sqrt(a^2)"
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "I noticed at 10:24 he said, it you get x squared is equal to a squared and then you get-\nat 10:29, x is equal to + or - the square root of a. Shouldn't it be x = a ?",
        "A": "Yes, you are right. Sal writes x = + or - square root of a but when he draws the intersections with the x-axis at 10:33 he draws them correctly: two points, (a, 0) and (-a, 0)."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "He says a few times in the video (like at 13:00) that \"we're not dealing with imaginary numbers.\" How does a person know when it is appropriate to use imaginary numbers?",
        "A": "Great question: when graphing sections or and anything for that matter, imaginary numbers are not used. However, if solving for solutions, like what x equals, they are used."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "At 10:35 you have placed that x equals plus or minus the sqrt of a, then a box pops up saying the x=-a not sqrt(a), I am confused. I was taught that when you take the sqrt of a square they cancel out thus making this statement x=+/-a being that a squared number can be a positive or a negative.",
        "A": "I think the pop-up box is supposed to read: x = +-a (plus or minus). Your logic is correct."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "math question - 4:20\nwhen Sal multiplies each side by -b^2 can you please walk me through the math of how we end up with x^2 on the side like a mixed number?\n\nI understand it is b^2/1 * (x^2/a^2) and the implicit 1 before the x but why does b/1 * x/a move the X to the side of the equation\n\nThank you",
        "A": "It is not a mixed number. In this problem, having nothing there means multiplication. So, b^2x^2/a^2=b^2/a^2 x^2."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "At around 6:42, why does -b^2 get taken out of the equation when x gets closer to infinity?",
        "A": "At that time, x is HUGE! So -b^2 is very small compared to x^2. Think of it like a billion minus one-billionth. You might as well say that that s equal to one billion."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "At 7:40 where did the -b^2 go in the \" y is approximately equal to\" equation.",
        "A": "When talking about asymptotes they describe end behavior, so the x values get really large ( +/-) which means that compared to x^2, the b^2 is really no longer significant. That is why it dropped out of the formula."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "At 2:46 he said parabola when referring to two equations he was presenting. He meant to say Hyperbola right?",
        "A": "yep"
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "When Khan said at 3:30 that whenever he has a hyperbola, he likes to solve for y, does that mean you can't solve for x at all?",
        "A": "of course you can solve for x if you wanted to but he finds it convenient to do so. cheers."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "At 4:30 why (+)or(-) before the square?? I have been explained but i just dont get it @_@",
        "A": "6*6 = 36 and -6 * -6 = 36 So the \u00e2\u0088\u009a36 = +6 and \u00e2\u0088\u009a36 = -6 We use the \u00c2\u00b1\u00e2\u0088\u009a36 to remind us that we have two answers. \u00c2\u00b16 which means +6 and -6 I hope that helps make it click for you."
    },
    {
        "video_name": "pzSyOTkAsY4",
        "Q": "At about 3:43 he says \"what [he] like[s] to do is solve for y.\" But is it just what he likes to do? Or is that the way you always do it, and if so why? Could you solve for x? I guess I just want to hear a little more why, or an added step in there. Thank you!",
        "A": "i think tht he wants y on one side so he can solve the other side of the equation"
    },
    {
        "video_name": "rOftmuhGLjY",
        "Q": "At about 2:50 Sal says that\nd/dx 3f(x) =\n3d/dx f(x)\nCan anyone explain to me why this works? Or point me to a video that explains this? Or give me the term for this property so I can google it? Thanks!",
        "A": "It s just one of the basic properties of derivatives Let c be a constant, (i) (cf) = cf (x) (the derivative of a constant multiplied by a function is the constant multiplied by the derivative) (ii) (f+g) = f (x) + g (x) (or minus) (iii) c = 0 (derivative of a constant is 0)"
    },
    {
        "video_name": "rOftmuhGLjY",
        "Q": "What is d/dx? I know dy/dx is a derivative of a point and the d is a infinitesimally small change in x and y but what does d mean on its own like at 0:59?",
        "A": "You can read d/dx as the derivative with respect to x. So, for example, you re familiar with seeing: y = x\u00c2\u00b2 dy/dx = 2x We can say essentially the same thing without introducing the variable y: d/dx x\u00c2\u00b2 = 2x In the first case above, we re saying, given that y = x\u00c2\u00b2, the derivative of y with respect to x is 2x. In the second, we re simply saying, the derivative of x\u00c2\u00b2 with respect to x is 2x."
    },
    {
        "video_name": "rOftmuhGLjY",
        "Q": "At 8:26 It is said that g prime of x is equal to the slope of the graph when x is equal to 9, Can anyone explain how?",
        "A": "g prime [g (x)] at any x is the slope of the line tangent to the function g(x) at that x value. At x=9 the value for g (x) is the slope of the tangent line at x=9. Thus g (9)=1."
    },
    {
        "video_name": "rOftmuhGLjY",
        "Q": "At 2:42 what does he mean by the term 'scalar'? I have done vectors before, but this doesn't make any sense to me.",
        "A": "In this context, Sal is using the term scalar to mean a multiplying constant. It is a factor by which the function scales up or down."
    },
    {
        "video_name": "OM3e-J3euXo",
        "Q": "At 7:00, I understand why Sal crosses out the third integral for his situation (the cylinder). But isn't that taking out some generalization? What if the sides of the cylinder bulged out? Sort of like a barrel. The normal vectors on the third surface would not be orthogonal to the k vectors, right?",
        "A": "yep, besides if the cylinder was a type of bulged barrel it would not be a type 1 region, since the side of the barrel cannot be expressed as a function of x and y"
    },
    {
        "video_name": "VzWxvDe8TUQ",
        "Q": "At around 6:00, he talks about how the variables are totally arbitrary. Could someone explain to me how this is true. Like y=g(x) right? So, x numbers are the domain and y numbers are the range. So the inverse of g(x) cannot be the inverse of g(y)! Am I wrong? Like, how does he just swap the variables out with each other? They already have meanings.",
        "A": "Yeah, that variable swapping confuses me a lot at the beginning. It sounds like the input of f is the input of f^-1 at the same time, but really it is 2 different variables. I think it would be more intuitive to rename the variable x of f to something like i, right from the start so we never have to deal with two different variable with the same name, and then we could say that x = f(i) since we are asked to find f^-1 in term of x."
    },
    {
        "video_name": "VzWxvDe8TUQ",
        "Q": "At 4:00 in the video, it should be -1? Correct?",
        "A": "If you look without full screen you ll see that it says that he made a mistake. Whenever you think he made a mistake, disable full screen and see if he did."
    },
    {
        "video_name": "-fIsaqN-aaQ",
        "Q": "Thank you very much for the amazing video.\nThe big question is: why people create models describing processes?\nAt 2:45, you (assumed) some number, but maybe it's not a fact and then the whole model is wrong. I would assume different numbers.\nThe only benefit I see is that you expect (not even sure) the number of population at any time. Could you please tell me what are the real benefits of modeling in general?",
        "A": "Sal assumed the numbers just as an example. In a real scenario, you would probably count the population at given points in time. As for modeling in general, it can be used to show how things are going to grow or change in number. Even if it s not an exact number, it can still provide a pretty close estimate."
    },
    {
        "video_name": "1ct7LUx23io",
        "Q": "At 3:35, I really dont understand how to get du. Someone please help help. Thanks!",
        "A": "du is the differential of u with respect with x. First, you simply take the derivative of u -- you will get du/dx = u Then multiply both sides by dx to obtain du - u dx"
    },
    {
        "video_name": "1ct7LUx23io",
        "Q": "At 4:00, why isn't the chain rule applied when taking the derivative of x^3ln(2)?",
        "A": "The chain rule applies when one function applies to the output of another. Here we have only one function, x^3, multiplied by a constant, ln(2). I realize ln(2) looks like a function, but it s a constant like 7 or \u00cf\u0080. And if it were a function, we still wouldn t apply the chain rule, we d apply the product rule, because then we d have two functions multiplied together instead of one function applying to the output of another."
    },
    {
        "video_name": "1ct7LUx23io",
        "Q": "At 6:21, could the exponential function be simplified back into the one given in the question, i.e (e^(ln2(x^3) = (2^x^3)?",
        "A": "Yes that works. (e^(ln2(x^3)) = 2^(x^3)"
    },
    {
        "video_name": "1ct7LUx23io",
        "Q": "at 7:52 how is e^ln2 =2?",
        "A": "ln x and e^x are inverse functions, i.e. they undo each other. To see this, you can rewrite this exponential in log form. e^(ln2) = y log (base e) y = ln 2 ln y = ln 2 so, y = 2"
    },
    {
        "video_name": "yUYDhmQsiXY",
        "Q": "At 1:26, why does Sal rewrite 5/3 & 2/5 to 5/5 & 2/3?",
        "A": "It s a division equation; so it s not 5/3 & 2/5 but (5 * 2) / (3 * 5). Since multiplication is commutative, we can reorder the numbers to make them easier to deal with without changing the equation [5/5 * 2/3 = 1 * 2/3 = 2/3] = [(5 * 2) / (3 * 5) = (10) / (15) = 2/3]"
    },
    {
        "video_name": "yUYDhmQsiXY",
        "Q": "at 4:19 i dont get the blocks part",
        "A": "he s just giving a visual example of what those fractions would look like if they were an object"
    },
    {
        "video_name": "yUYDhmQsiXY",
        "Q": "what did he say at 5:00",
        "A": "So this expression is the largest, 8/7 times 2/3."
    },
    {
        "video_name": "yUYDhmQsiXY",
        "Q": "what did he say at 2:00",
        "A": "let s say the height of this bar right over here is 2/3."
    },
    {
        "video_name": "OMA2Mwo0aZg",
        "Q": "Regardless of the number of rows/columns in a matrice, would it always result in a 2 x 2 answer as represented in 5:10 ? Or does the final answer vary from equation to equation?",
        "A": "No, by multiplying two matrices A of dimension mxn and B of dimension nxt you will end up with a mxt matrix."
    },
    {
        "video_name": "OMA2Mwo0aZg",
        "Q": "At 1:50, why can't ED always equal DE? When would the order not matter in multiplying matrices, because Sal said that ED wouldn't always equal DE?",
        "A": "The problem with turning around ED is that the multiplications won t be the same anymore: You always multiply the rows from the first matrix with the columns of the second, if you were to switch the two you wouldn t be doing the same multiplications anymore. An easy example of matrices where the order wouldn t matter are a matrix with it s inverse matrix (later on in matrices) because multiplying the two always leads to an identity matrix. Hope this made a bit of sense."
    },
    {
        "video_name": "OMA2Mwo0aZg",
        "Q": "At 0:57 Sal says that the first thing that we need to do is see if \"it is even a valid operation.\" What does that mean?",
        "A": "You need to check the dimensions of the matrices being multiplied. You can only multiply matrices if the number of columns of the first matrix is the same as the number of rows as the second matrix. For example, say you want to multiply A x B. If A is a 3x1 matrix, B has to be a 1xY matrix (Y can be any number), because A only has 1 column. This also leads to an interesting result, by the way. A x B may be defined, whereas B x A may not be defined."
    },
    {
        "video_name": "MDav5OMpCto",
        "Q": "At 4:50, why did Sal shift the function from the lower one when right before, he shifted the higher one.",
        "A": "The question said to write a formula for function g in terms of function f. That s why Sal had to look at f(x) to see how to shift it to get g(x). In the first problem, f(x) was above g(x). In the second problem, f(x) was below g(x)."
    },
    {
        "video_name": "MDav5OMpCto",
        "Q": "In 4:41 and a little before that, why didn't Sal just use the general form (sort of like a formula for most functions): y= a | x-h |+k? (x-h being in absolute value) Wouldn't this be easier to remember instead of using the words \"Horizontal shift\" and \"Vertical shift\", where he could define h and k as shifting commands?",
        "A": "I was looking at that, too. I think he s trying to help the newbies out, since they may or may not have taken an Algebra class in the past. I agree that it may be easier to use h and k because most teachers (or my past teachers) use them."
    },
    {
        "video_name": "MDav5OMpCto",
        "Q": "at 2:23, why did Sal do x minus the horizontal shift and x plus the vertical shift? is there some kind of rule that explains why the horizontal shift must be subtracted and the vertical shift must be added?",
        "A": "By shifting this function, you are not trying to get the exact point to shift, you are trying to get y to be equal. y=g(x)=(f(x+4)-7)=y"
    },
    {
        "video_name": "pInFesXIfg8",
        "Q": "How do you decide whether the function is increasing or decreasing toward or after the critical point? 4:18",
        "A": "thumbs down on confusing video."
    },
    {
        "video_name": "pInFesXIfg8",
        "Q": "How did you get x= -2 at 1:28?",
        "A": "We have x^2=4, and there are two solutions to that equation, 2 and -2. Both of these numbers, when squared, equal 4."
    },
    {
        "video_name": "pInFesXIfg8",
        "Q": "Around 8:04 it is said to regard maximum points as having first a slope which is increasing and positive and then a negative and decreasing slope, however, shouldn't that entire part of the function be considered decreasing since the slope originally starts as extremely positive and then approaches zero, only to become extremely negative? Or are the terms \"increasing\" and \"decreasing\" referring to something else?",
        "A": "As the function approaches its maximum point, it increases, which means that its derivative is positive. You are right, the derivative is constantly decreasing in the critical points vicinity. It is the functions value that increases when approaching and decreases after reaching maximum point."
    },
    {
        "video_name": "pInFesXIfg8",
        "Q": "At 2:08 , what does sal meant by saying that the function takes the minimum values or the maximum values ?",
        "A": "A maximum value is like a hump; the curve goes up, reaches a maximum and then comes down again. A minimum value is like an upside-down hump. There can be a number of these humps depending on the curve."
    },
    {
        "video_name": "57BiI_iD3-U",
        "Q": "At around 2:08 what does ((sin^2)(y)) mean?\nDo you square the y and square the sin of y^2,\nOr do you just find the sin of y and square it? I hope my question makes sense?",
        "A": "Yeah sin^2(y) is just like finding sin(y) then squaring the result ((sin(y))^2"
    },
    {
        "video_name": "cBi4a1iSaPk",
        "Q": "at 2:00, he called the constant multiplied by the 'x' term, a scaling quantity. what does this scalar mean? does this have to do with vectors?\nthank you!",
        "A": "No, nothing fancy here. This is just a term sometimes used for a coefficient -- in other words, for a constant used to multiply a variable. In this case, 7 can be called a multiple or a coefficient or a scaling quantity, all the same thing."
    },
    {
        "video_name": "cBi4a1iSaPk",
        "Q": "At 5:57 can I further simplify it to 7x^4/4 - 10 sqrt(x^3)/3 + 12/sqrt(x^3) - 39/x^39 ?",
        "A": "Yup, that would be perfectly acceptable, but make sure you remember to add +C at the end!"
    },
    {
        "video_name": "cBi4a1iSaPk",
        "Q": "At 2:18, we take the anti-derivative of 5*sqrt(x). Does this mean that the domain of x is now restricted to all non-negative x values? If not, please explain. Thanks.",
        "A": "Before obtaining the antiderivative, if you wanted real results, you would have to restrict x to positive values. After obtaining the antiderivative, nothing has changed, if you still want real results, you will have to continue to restrict x to positive values. The antiderivative operation changes nothing in the range of the function."
    },
    {
        "video_name": "cBi4a1iSaPk",
        "Q": "Correct me if I'm wrong, but when you apply the sum rule to break the antiderivative into smaller pieces, you would technically have an added constant for each piece. At 4:40 when you add the constant, that constant would technically represent the sum of all smaller constants found when taking the general antiderivative of each of the smaller pieces, correct? It would be almost impossible to find a particular solution given an initial condition, since it we would be solving for multiple constant C's.",
        "A": "Technically, you are right, but you re overthinking it a bit, I think. Say you ve broken the antiderivative into 2 parts. Each piece would have a constant. We can call them C1 and C2. Now, if we combine the solutions together (as they should be), our constant becomes (C1+C2). That s just one big constant, and we normally call it just C. I hope that makes sense- I think it might be easier to understand when it s written out."
    },
    {
        "video_name": "cBi4a1iSaPk",
        "Q": "you said at around 4:05 that -5 halves plus 1 is -3. shouldn't It be -4?",
        "A": "It would be, but we re dealing with halves and whole numbers all mixed together. So -5 halves plus 1 (which you can write as 2 halves ) is equal to -3 halves. Another way to write it is -2.5 plus 1, which makes it a lot easier to see that the answer is is -1.5"
    },
    {
        "video_name": "cBi4a1iSaPk",
        "Q": "At 2:41 Sal said that we could increment the power of 1/2 by 1 to get 3/2. What does this mean?",
        "A": "1/2 + 1 = 1/2 + 2/2 = 3/2 f(x) = sqrt(x) F(x) = \u00e2\u0088\u00ab(sqrt(x))dx F(x) = \u00e2\u0088\u00ab(x^(1/2))dx \u00e2\u0088\u00ab(x^n)dx \u00e2\u0086\u0092 1/(n + 1)\u00e2\u0080\u00a2x^(n + 1): F(x) = 1/(1/2 + 1)\u00e2\u0080\u00a2x^(1/2 + 1) F(x) = 1/(1/2 + 2/2)\u00e2\u0080\u00a2x^(1/2 + 2/2) F(x) = 1/(3/2)\u00e2\u0080\u00a2x^(3/2) F(x) = [2/3\u00e2\u0080\u00a2x^(3/2)]"
    },
    {
        "video_name": "cBi4a1iSaPk",
        "Q": "At 5:48 why is this negative? Isn't it supposed to be plus 1/39x^-39?",
        "A": "The way Sal wrote it the first time was + x^-39/-39, presumably writing it this way because that s the direct result of applying the power rule for antiderivatives. Cleaning up later he moves the minus sign out of the denominator to write - (1/39)x^-39, same thing."
    },
    {
        "video_name": "cBi4a1iSaPk",
        "Q": "At 1:55, when suddenly the x^3 becomes (x^4)/4, does that have to do with the x^(n-1)/(n-1) from the previous video? How is it that that works, again?",
        "A": "Since the derivative of x^n is n*x^(n-1), the antiderivative of x^n is x^(n+1)/(n+1)."
    },
    {
        "video_name": "cBi4a1iSaPk",
        "Q": "At 4:23, at the top rights we get 18x^-5/2, but when he writes it at the bottom, he gets 18x^-3/2. How is this so?",
        "A": "In right-up of the screen, he adds the exponents, 1/2 and -3. So 1/2 + -3 is -(2+1/2), which is equal to -(4/2 +1/2) = -(5/2). Then he integrates \u00e2\u0088\u00ab x^-(5/2), by reverse power rule definition, x^(5/2)+1 / -(5/2)+1. -5/2 + 1 IS THE SAME AS -5/2 + 2/2, which is equal to -3/2."
    },
    {
        "video_name": "cBi4a1iSaPk",
        "Q": "at 3:20 how is 18x^1/2.x^-3= 18x^-2.1/2",
        "A": "It s the multiplication of exponent with the same base. When multiplying same base, you add the exponent power. And subtracting if you divide. a^m * a^n = a^(m+n) a^m / a^n = a ^(m-n)"
    },
    {
        "video_name": "cBi4a1iSaPk",
        "Q": "At 2:45, what does he mean by \"incremented exponent\"?",
        "A": "Incremented as in the exponent rule of integration, where the exponent has 1/1 added to the exponent. Incremented by +1/1."
    },
    {
        "video_name": "HJgjAZ7JkHQ",
        "Q": "at around 2:00 how does he know it will be x to the fourth and x squared?",
        "A": "When you multiply (x^2 + a) with (x^2 + b) , you get x^2(x^2 + b) + a(x^2 + b) using the distributive property. x^2(x^2 +b) = x^(2*2) + bx^2 a(x^2 +b) = ax^2 + ab Adding these two will give us the value of (x^2 +a)(x^2 +b). x^(2*2) +bx^2 + ax^2 + ab = (x^2 +a)(x^2 +b) x^4 + x^2(a+b) + ab = ( x^2 +a)(x^2 +b)"
    },
    {
        "video_name": "BSrJi9uEJp0",
        "Q": "At 1:02, is there any particular reason he's using a natural log and not log base 10?",
        "A": "Differentiating natural log is a lot easier than differentiating log base 10 for L Hopital s rule thing. Also math people just like natural log because 1. its about everyone s favourite number e , 2. its the original one, and 3. you don t have to wright the o and the 10 and can write an n instead of a g"
    },
    {
        "video_name": "m1FhUjMMv30",
        "Q": "Around 8:50, Grant was talking about the mixed partial derivative, and he said you could take the derivative of fx with respect to y (so fxy) OR you could take the derivative of fy with respect to y (so fyx) because either calculation would give the same answer. Is this true of any problem regarding maxs/mins/saddle points of a multi variable function? Or does it only apply here because fx and fy are similar?",
        "A": "This is due to Clairut s Theorem: If fxy and fyx are continuous at (a, b), then fxy(a, b) = fyx(a, b)."
    },
    {
        "video_name": "eVme7kuGyuo",
        "Q": "Throughout the video you discuss discontinuity. At about 3:33, you say that there is no way that the slope can be defined because you can't define a slope with discontinuity. So, my question is, how can you take the slope of an undefined slope (whether because of discontinuity or not) and thus taking the derivative of a function where that function is undefined? Thanks.",
        "A": "You cannot take the derivative where the function is undefined."
    },
    {
        "video_name": "eVme7kuGyuo",
        "Q": "In 4:47 the orange line ( the last section of the derivative graph) have an open or closed hole?",
        "A": "This piece wise function is discontinuous at the boundaries of each piece, hence the derivative does not exist at the boundaries, hence, the orange graph of the derivative of f is not defined at the boundary, hence the interval is open on the left hand side, hence the hole is open (as are all the intervals (holes) of each f for each piece of f)."
    },
    {
        "video_name": "eVme7kuGyuo",
        "Q": "How is the slope, the way it is drawn, positive when Sal is talking exactly at 1:23 to 1:28? I know I am wrong, but I want to know why? The slope is clearly in a decreasing fashion, reaching towards zero, then how could it be positive? Can anyone please help me out here? Again it is during the time 1:24.",
        "A": "I think there is a little confusion going on. For me, Sal is talking about the curve when he describes the slope, because the slope of the derivative is as you said clearly negative. Since the derivative is a direct representation of attributes of the curve, the decreasing positive values represent the decreasing positive slope of the curve. So he explains, that as long as the slope of the curve is still positive, the representing values of the derivative will still be positive."
    },
    {
        "video_name": "eVme7kuGyuo",
        "Q": "At 4:25, why does Sal not shade in the second blue circle? Why can't the slope be defined at this point?",
        "A": "Because there is a corner meaning that the derivative from the left hand side does not equal to the derivative from the right hand side (Slope of tangent line are not equal from the left and right). Since derivatives come from limits, in order for a derivative to exist the left hand derivative must equal to the right hand derivative. At the corner this is not true and so therefore, the derivative is undefined at that point"
    },
    {
        "video_name": "eVme7kuGyuo",
        "Q": "At about 3:51 Sal starts evaluating the derivative of the third segment. If you took that (red) segment as an interval alone and evaluated it, wouldn't the f'(x) of its left endpoint still be defined because the limit would only include evaluation from the positive side?",
        "A": "The problem is, that to calculate the slope you need to draw a tangent line. But in that left most extreme point a number of different tangent lines can be drawn. For each line a different slope value will be present. That is why f (x) cannot be calculated."
    },
    {
        "video_name": "eVme7kuGyuo",
        "Q": "Why the last jump of the derivate (at 4:46) is a defined point and not undefined?",
        "A": "it s an undefined point, as both circles at that x-value are open - also, if you re trying to find the slope of the point between the blue & orange lines (the x-value at the open circle), it would be unknown because there are too many possible slopes at that x-value, which is why it s undefined"
    },
    {
        "video_name": "8EA5wv-kQPw",
        "Q": "At 9:49 pm,Why y can't be equal to 3 or 2 or 1 I mean less then 4?",
        "A": "Because your initial condition started at 6. The range is based off of the initial condition."
    },
    {
        "video_name": "8EA5wv-kQPw",
        "Q": "Why at 0:48, the line y=4 behaves like an asymptote? You can see that at other values of y, the slope remains at one number. So why there aren't asymptotes at other values of y?",
        "A": "There is another asymptote at y=0."
    },
    {
        "video_name": "ODNCH3gSKdA",
        "Q": "At 0:12, it says the trophy is covered in gold foil. What would the shape of the gold foil have to be to exactly cover all of the trophy?",
        "A": "Try to follow through with the video, but what they re looking for is the surface area, assuming the gold foil is infinitely thin (i.e. no area for the sides) and the minimum it would to take perfectly cover every inch of the pyramid. In other words, it s a fancy word problem saying to find the surface area of the pyramid, which is easy by finding the net first. Follow through with the video to find the actual calculation"
    },
    {
        "video_name": "Q3wfb0CPhIY",
        "Q": "At 0:55, I kinda already figured out that it was either 9 * (8+12) or 9 * 20 or 9 * 12 + 9 * 8.\u00f0\u009f\u0098\u009d\u00f0\u009f\u0098\u0089\u00f0\u009f\u0099\u0082\u00f0\u009f\u0098\u008a\u00f0\u009f\u0090\u00bb",
        "A": "What do you mean by that."
    },
    {
        "video_name": "UyCwAFQt4v0",
        "Q": "at \"3:54\" did sal say 3 or t",
        "A": "Sal said the number three"
    },
    {
        "video_name": "UyCwAFQt4v0",
        "Q": "At 5:57, I get C1 = -5 and C2 = 7 which is a also a solution though different from Sal's. Does anyone else get different constants than Sal's which are solutions?",
        "A": "that wouldn t work for 3 = -2C1 - 3C2"
    },
    {
        "video_name": "2oGsLdAWxlk",
        "Q": "But I don't get it, at 5:11 why is the right side less? WOuldn't the opposite side of where he shaded be \u00e2\u0089\u00a5y?",
        "A": "Sal had the inequality y <= 5x/3 - 25/3. In words, that says y is LESS than .... and so everything underneath the line needs to be shaded. You are correct in saying that the opposite side of where he shaded is >= y, but in words, what that statement says is that that is the area that is GREATER than y."
    },
    {
        "video_name": "2oGsLdAWxlk",
        "Q": "At 5:12, when you're graphing the line, how do you know which side of the line has the inequalities and which has the solutions?",
        "A": "The shaded side is the side that satisfies the inequality, the solutions will be in the shaded area as well."
    },
    {
        "video_name": "2oGsLdAWxlk",
        "Q": "at 0:24 what did you mean by try for 1 and negitive 7?",
        "A": "What he says is that and then we can try it for 1 and negative 7, What Sal means is that you substitute 1 in for x and -7 in for y and see if the equation is true. 5x - 3y < 25, so 5(1) - 3(-7) < 25, 5 + 21 < 25, 26< 25, which is false, so point is not within the inequality."
    },
    {
        "video_name": "2oGsLdAWxlk",
        "Q": "around 5:18 when the problem is divided by three cant you also simplify -25/3 to get the starting point?",
        "A": "-25/3 can t be simplified, but it can be converted to a mixed number. -25/3 = -(8 1/3)"
    },
    {
        "video_name": "VidnbCEOGdg",
        "Q": "ok... why did Sal say at 0:57 \"we subtract 45\" why on earth did he say that?",
        "A": "He actually said we subtract 54 . He wrote -5 and should have said subtract 5 . I am glad you were paying attention and saw the issue. Do you understand how he solved the problem? I am sure that, like all of us, Sal wishes he never made a mistake while speaking. This will never be true."
    },
    {
        "video_name": "VidnbCEOGdg",
        "Q": "Did Sal mean to say \"yellow\" instead of \"green\" at 1:38?",
        "A": "It s reasonable to say that colors look different on different screens, on computer/mobile software, and in general, colors look different to different pairs of eyes. Depends on who is seeing it. But I wouldn t consider it a huge, significant mistake..."
    },
    {
        "video_name": "VidnbCEOGdg",
        "Q": "at 1:40, how come you have to substitute 49 for a ?",
        "A": "You do it as a check to make sure you are correct."
    },
    {
        "video_name": "VidnbCEOGdg",
        "Q": "At around 0:40 Sal says that in the expression a + 5, the plus 5 is the same as a positive 5, and therefore, a positive five and the negative five cancel out. However, aren't the addition operator and the positive sign two different things in the equation? Wouldn't the way to go about the equation be to say that x + 5 -5 = x + (5 -5) = x? How does he group the addition sign with the 5 to make it a positive 5?",
        "A": "A number is positive unless it has a negative sign. The plus sign in this case is to show addition, the 5 is a positive even without the plus sign. I hope this helps you! :)"
    },
    {
        "video_name": "VidnbCEOGdg",
        "Q": "why is there such bad audio in 0:00 to 1:53",
        "A": "Lol. Good answer. A vote for you!"
    },
    {
        "video_name": "VidnbCEOGdg",
        "Q": "Hey! why is this video so short! I mean 1:54 really!",
        "A": "Well, you really don t need that much time to explain how to solve the most basic equations, when all you need to do is perform one operation. 1:54 is long enough."
    },
    {
        "video_name": "VidnbCEOGdg",
        "Q": "At 1:50 what did he mean",
        "A": "It means that both sides equal 54 when 49 is put back into the equation. So, therefore 49 must be the correct answer. It s just to check whether you got the right answer."
    },
    {
        "video_name": "VidnbCEOGdg",
        "Q": "at 0:45 sal says to get a on one side by itself. Isint it easier just to subtract 5 from 54 to get the answer instead of all the high definition terms and steps?",
        "A": "If you just subtracted 5 from 54 the answer would be a+5 = 54 -5. You are not just subtracting 5 from 54. You are subtracting the five from both sides, which is getting a by itself . Hope this helps."
    },
    {
        "video_name": "VidnbCEOGdg",
        "Q": "I think this is a mistake. Sal said subtract 54 from the right at 0:57",
        "A": "Very observant of you. Well done. What he said was different from what he wrote and did. I make the same mistake often when teaching my own classes. Thanks for helping them clean up the video. They usually put a correction into the video later when people notice discrepancies like that."
    },
    {
        "video_name": "VidnbCEOGdg",
        "Q": "At 0:59 Sal said, \"So we also have to subtract 54 from the right.\" But we have to subtract 5, right?",
        "A": "Yes, and a box pops up at 1:01 in the video and tells you that Sal meant to say subtract 5"
    },
    {
        "video_name": "9eWyvWq9CCs",
        "Q": "At 00:47, why did Sal divide each side by A? Can anyone explain?",
        "A": "to get the leading coefficient to be 1. it doesn t change the polynomial at all, but puts it in a nicer format for us."
    },
    {
        "video_name": "9eWyvWq9CCs",
        "Q": "At 3:38 Sal said we have a hunch that the part that matters is the coefficient which is one less than the highest degree. Couldn't we also have hunched that the coefficient that matters is the coefficient in front of the x term?",
        "A": "Only in the case of quadratics. In cubics, it s the coefficient of the x^2 term. In quartics, the x^3 term. In quintics, the x^4 term. And so on."
    },
    {
        "video_name": "GVZUpOm3XUg",
        "Q": "in 4:40 thats actually not membership right? it is called belongs to isnt it?",
        "A": "If we write a \u00e2\u0088\u0088 A, we could say either of the following: Element a is a member of set A. Element a belongs to set A. Those two statements are equivalent."
    },
    {
        "video_name": "GVZUpOm3XUg",
        "Q": "Would you actually say, \"-5 is a memebership of C\"? at 4:45",
        "A": "No, it is not a member of C, but would be C or C compliment. -5 and 5 are two totally different numbers."
    },
    {
        "video_name": "GVZUpOm3XUg",
        "Q": "At 4:38, In the NCERT isn't it written that the symbol is the greek letter epsilon?",
        "A": "No..it s not. Sal said it right. It s a derivation of lower case of epsilon, but it s not Epsilon. I hope you understood. If x is a member of a set A, then it is also said that x belongs to A, or that x is in A. This is denoted by x \u00e2\u0088\u0088 A. The symbol \u00e2\u0088\u0088 is a derivation from the lowercase Greek letter epsilon, \u00ce\u00b5 , introduced by Giuseppe Peano in 1889 and shall be the first letter of the word \u00e1\u00bc\u0090\u00cf\u0083\u00cf\u0084\u00ce\u00af (means is ). The symbol \u00e2\u0088\u0089 is often used to write x \u00e2\u0088\u0089 A, meaning x is not in A ."
    },
    {
        "video_name": "GVZUpOm3XUg",
        "Q": "in 1:41 why do we subtract U from A U - A and whats that slash symbol there",
        "A": "That is an alternative to the subtract symbol. Khal explains it in the previous video, Relative complement or difference between sets."
    },
    {
        "video_name": "GVZUpOm3XUg",
        "Q": "at 4:16 why does Q mean rational?",
        "A": "I would assume that Thomas reasoning that R was already taken for the set of real numbers is correct. In terms of where Q came from: rational numbers are numbers that can be expressed as the ratio of two integers. When we divide two numbers, we call the result a quotient. Hence, the Q to denote the set of rational numbers (though the mathematician who introduced the notion was presumably considering the Italian word, which thankfully also begins with a Q)."
    },
    {
        "video_name": "xVWCfMe97ws",
        "Q": "At 1:28, Sal doesn't divide dy/dx by y... Why not?",
        "A": "y and dy are not the same value. We put dx on the other side so it is easier to integrate both sides and solve for y."
    },
    {
        "video_name": "xVWCfMe97ws",
        "Q": "at 1:20, when Sal divides both sides by y, I understand that he distributes 1/y to (1+2y^2), but how come he does not also distribute to dy/dx?",
        "A": "[(cos x) ln(2y-8)+1/x] dx + sinx/y-4 dy"
    },
    {
        "video_name": "xVWCfMe97ws",
        "Q": "At 1:15 Sal makes y^2/y into just y by canceling out the bottom y. However, isn't this only correct if y cannot equal zero? and the initial condition shows that y in fact does exist at zero, so how can you justify canceling out the bottom y? I know this is kind of geeky, but I am interested to know the answer.",
        "A": "@sethsm,yes,y^2/y=y,only in case y is not equal to zero,and here it is true,caz in the solution we get natural log of absolute value o f y(ln/y/!!!)which is valid for y not equal to zero.and in the initial condition there is nothing said about y=0,it says y(0)=1.i.e for x=0,we get y=1,for y=0,no comments!!is there any need to bring l Hospitals rule??"
    },
    {
        "video_name": "xVWCfMe97ws",
        "Q": "how can i solve this question??\nQ: dy/dx = (y + 9x)^2 (y+9x = v)\n\nat 3:58 pm sunday 21st november 2010\nplease reply soon",
        "A": "Let y+9x = v =>dy/dx =dv/dx - 9 =>from the original equation dv/dx - 9 = v^2 =>dv/dx = (v^2 +9). Carry on from here"
    },
    {
        "video_name": "xVWCfMe97ws",
        "Q": "At 2:51 ins't the derivative of cos x= -sin x , not sin x ?",
        "A": "you re right that the derivative of cos x=-sin x, but here he is taking the integral of cos x which is sin x"
    },
    {
        "video_name": "xVWCfMe97ws",
        "Q": "At 3:22, Why did Sal simplify ln l 1 l to 0?",
        "A": "Definition of natural log of 1 is zero. It is the same as saying, To what power (or value of x) does e^x = 1? Knowing that any number raised to the zeroth power is 1, so must e^0 be equal to 1. So then ln(1) = 0."
    },
    {
        "video_name": "xVWCfMe97ws",
        "Q": "At 1:26, when he is dividing both sides by 'y', why does nothing happen to the 'dy/dx' part?",
        "A": "He never divides both side by y. He multiplies by dx."
    },
    {
        "video_name": "PiuhTj0zCf4",
        "Q": "11:53 He draws a box around [x]B and another around [Ax]B, then calls them the same? How can that be, if one has matrix A in it while the other doesn't?\n\n12:45 How does he come up with C-inverse*Ax?",
        "A": "On 11:53 you have also D on the left. So it s D[x]B = [Ax]B, not [x]B=[Ax]B. On 12.45 he just used the fact that [Ax]B = (C^(-1))*Ax (without []B, so in standard basis). C^(-1) is just the inverse of change of basis matrix. So it changes basis from B to the standard one. I think you may want to watch previous videos again, because those are concepts explained in them."
    },
    {
        "video_name": "PiuhTj0zCf4",
        "Q": "at 13:17 the description becomes so muddled I can't tell what is going on. First, it should have been made clear whether C is the came as the basis vectors B. Then the whole account of how Ax sub-b is equivalent to to Cinverse*A*C is unclear. I think the idea is that C^-1*A*C will map a vector in standard basis into nonstandard basis, but I'm not sure. The other videos have been great, but this one is unintelligible after many viewings.",
        "A": "C^-1*A*C is a matrix isomorphic to A, AKA it represents the same thing but (maybe) has a different basis."
    },
    {
        "video_name": "c4fUaD4g4mc",
        "Q": "At 0:16 Sal said that Nadia was 21.5 inches long. Why didn't they include Venessa length ? Could be a distraction ?",
        "A": "It is just a distraction so you you focus."
    },
    {
        "video_name": "tfjQVtOyoaQ",
        "Q": "at 1:11 he has a really long problem to multiply the fraction 1/5 * 3 + 1/5 * 3/1 couldn't he just turn the mixed number to an improper fraction then multiply and reduce ?",
        "A": "Yes. you can solve the problem that way. Sometimes there are multiple ways to do the math and still get to the same answer. This is one of those situations. Your way works. Sal elected to use the distributive property, which also works."
    },
    {
        "video_name": "VrB3LaSD_uo",
        "Q": "At 12:52 Sal says x_j did he mean x_ij? shouldn't the i still be part of the summation even though he described i as a row separately? Also should the |A_ij| be |(X,Y,Z)_ij| respectively?",
        "A": "It s okay not to write x_ij because x terms only appear in the i-th row anyway the way he wrote his matrix. x_j = x_ij. It s slightly unclear because he is overloading the letter x; X is the entire matrix, but the variables of the form x_j represent only the individual values of that special row (the i-th row). I agree that it should be |(X,Y,Z)_ij|, but his ~interesting~ A_ij notation does have the benefit of showing that |X_ij| = |Y_ij| = |Z_ij| and can therefore make like terms for adding."
    },
    {
        "video_name": "VrB3LaSD_uo",
        "Q": "At 15:20 when he says you can immediately see that det(Z) = det(X) + det(Y), is there math you can perform on the sigma notation to express that? Can x_j and y_j factor out somehow?",
        "A": "Since each x_j and y_j is different, they can t be factored outside the summation symbol. but for each j you can check that: (x_j+y_j)|A_ij| = x_j|A_ij + y_j|A_ij|, You then have (leaving out some detail) sum(x_j|A_ij|) + sum(y_j|A_ij| and sum(x_j|A_ij + y_j|A_ij|) , which you can verify are equal for each j (for every term in the summation), which makes the summations equal."
    },
    {
        "video_name": "XIsPC-f2e2c",
        "Q": "At 2:23, did Sal mean domain, or am I being too literal?",
        "A": "At 2:23 he said range or the y-values. Domain is for the x-values."
    },
    {
        "video_name": "3-aryZYsoxU",
        "Q": "at 3:02pm why is the number line not standard: negative numbers to the left and positive numbers to the right? The video accompanies the exercise of opposite number placement on number lines. The exercise questions reverse negative numbers to the right and positive to the left. I don't understand",
        "A": "I suppose negative on the left is just a convention, but it shouldn t do that. Try Report A Mistake on the question that does that."
    },
    {
        "video_name": "3-aryZYsoxU",
        "Q": "what does it mean when he says at 4:21, the negative of the negative of negative 2?",
        "A": "He is just trying to explain that negative is like the opposite. So the negative(opposite) of negative 2 is just 2 because you change -2 to a positive number, which is the opposite of the -2. By adding another negative to that statement means that it would flip back again(be the opposite of) the number you just got. Since the negative of negative 2 was just (positive) 2, then a negative of that turns it back into -2(since that is the opposite of 2). I hope that isn t more confusing."
    },
    {
        "video_name": "3-aryZYsoxU",
        "Q": "At 3:20, How come the negative sign means opposite? Wouldn't The negative sign still keep the number negative, because you're only putting a negative sign in front of a negative sign?",
        "A": "The signs cancel out."
    },
    {
        "video_name": "f_DTiXZpb8M",
        "Q": "I have two questions:\n\n1. At 11:50 he says that A will be l x n. That makes sense except how do we know which subset of R^n (vector x) or R^l (vector z) will be the column, and which will be the row?\n\n2. This seems awfully familiar to the g(f(x)) and F(g(x)) stuff that I did in college algebra/Algebra 2. Is this related at all? Was the stuff they showed us in algebra kind of a precursor to this stuff?",
        "A": "For an mxn matrix, the matrix is m tall and n wide, so m rows and n columns. An lxn matrix would be n wide and l tall, giving the transformation A x\u00e2\u0083\u0091 = z\u00e2\u0083\u0091. And yes, it s very similar, just with more variables."
    },
    {
        "video_name": "f_DTiXZpb8M",
        "Q": "What does he mean at 5:32?",
        "A": "What does (ToS)(x) mean? The meaning is a little vague - could be more precise. We re putting T and S together (T composed with S) somehow. T(S(x)), however, can only mean one thing, which is: 1) Find y = S(x) 2) Then, find z = T(y) = T(S(x)). This is what he s explaining around 5:30."
    },
    {
        "video_name": "2IFItASxDVo",
        "Q": "At, 3:52- 4:17, why aren't we taking the y-term exponents into consideration?",
        "A": "If you take the y term and try to make a general formula, ... it will not match up with the common formula for the coefficient but if you do as sal sir is doing and try to make a general formula ... it will be equal to the common formula...."
    },
    {
        "video_name": "2IFItASxDVo",
        "Q": "At 2:40, what does Sal refer to when he says \"divided by which term that actually was, so divided by that term?\" Is he referring to the previous term which is 1?",
        "A": "Sal is referring to the term before the one that you are trying to find the coefficient of (In this case, the first term. He uses the phrase which term that actually was because he has been referring to the coefficient & exponent of the previous term."
    },
    {
        "video_name": "dw41PMWek6U",
        "Q": "At 4:26 what if the protractor doesn't show up like it does in the video?",
        "A": "I will always if you have a good protractor"
    },
    {
        "video_name": "dw41PMWek6U",
        "Q": "At 1:13 in the video, I don't now which side is supposed to be at the bottom of the protractor.",
        "A": "The flat side is always on the bottom, with the rest of the angle inside the clear part of the protractor."
    },
    {
        "video_name": "dw41PMWek6U",
        "Q": "At 1:18,why do all of the angles have to be at 0 degrees?",
        "A": "They don t. You can start at any angle but it would be harder because you would have to subtract the small number from the larger one; so that s why most people start at the 0 degree mark: you don t have to any extra math."
    },
    {
        "video_name": "dw41PMWek6U",
        "Q": "At 2:31, the vertex isn't overlapping the centre of the protractor.",
        "A": "That would be why the protractor was measuring slightly more than 80 degrees even though 80 degrees was the correct angle. This is actually a good example of why you should make sure to line your protractor up correctly, or your measurement will be incorrect."
    },
    {
        "video_name": "dw41PMWek6U",
        "Q": "At,3:15 it's 71 degrees not 70.",
        "A": "It shouldn t really matter. The only important thing is that you know how to use a protractor. Plus, Sal doesn t have the smaller lines to show ones."
    },
    {
        "video_name": "uaPm3Tpuxbc",
        "Q": "At 1:20 why doesn't -14 + -5 = 19 because two negatives equal a positive right? Please explain..",
        "A": "You are confusing the rules for multiplication with the rules for addition. When multiplying, a negative * a negative = a postive. When adding and the 2 numbers have the same sign, you keep the common sign. Another way to think of it. If you owe me $14, then you have -14 dollars. If you then borrow 5 more dollars from me, you much do you now owe me? You owe me $19. So, you have -19 dollars. Hope this helps."
    },
    {
        "video_name": "uaPm3Tpuxbc",
        "Q": "At 1:09, Is -2*7+-5 consiered to be equal to -2*f(-6)+g(1)",
        "A": "YES! since f(-6) = 7 and since g(1) = -5, then given -2 * f(-6) + g(1), substitute the values of each of the functions you get -2 * 7 + -5. Good one!"
    },
    {
        "video_name": "uaPm3Tpuxbc",
        "Q": "I have a question at 0:26 and at 0:46 is it that if it's a negative X it goes up and if it's a positive X it goes down?",
        "A": "No, the direction you go is based upon the function you need to use. The top graph is for function f(x). So, to find f(-6), you have to go up to the f(x) line. You could also find f(5) which means you would go up to f(x). Similarly, when you need to find g(1), you need to use the function g(x) which is the lower graph. Hope this helps."
    },
    {
        "video_name": "uaPm3Tpuxbc",
        "Q": "at 0:41 how do you know that its that one and not the other one",
        "A": "Your question is unclear. Do you mean how does he know to use the bottom graph (pink) rather than the top graph (blue)? If yes, the reason he is using the bottom graph is that g(1) means use the function g(x) to find Y when X=1. The bottom graph is the one labeled g(x). If you are looking for something else, comment back and I ll try to help."
    },
    {
        "video_name": "uaPm3Tpuxbc",
        "Q": "At 1:09, Is -2*7+-5 consiered to be equal to -2*f(-6)+g(1)? Because I got confused. Or did I think it wrong?\nThanks..!",
        "A": "Yes, these two expressions are indeed equivalent, as inputting something into a function is like inputting an x value to get a y value out. (To elaborate further, f(x) is equivalent to y , but you must input something in for x in order to get a specified result for y ). Sal goes through how to find inputs and outputs ( x and f(x) ) in the video, but to answer your question, yes, the two expressions are indeed equivalent."
    },
    {
        "video_name": "Nq88IHw4Y9E",
        "Q": "3:40 explain this part please",
        "A": "You cannot measure \u00e2\u0080\u009c geometry. Geometry. Is a very broad. Subject that includes area. Volume. Perimeter and other things. You. Can however . Measure things like the side of a. Geometrical shape in different units if that helps"
    },
    {
        "video_name": "v5SAMuRanGM",
        "Q": "At 7:22, isn't it supposed to be phi+1 to 1 instead of phi+1 to phi?",
        "A": "Yes. The mistake has already been reported, and dealt with. If you watch the video out of full screen, then at around 7:22 , a message should pop up in the video correcting the mistake. Nice catch though."
    },
    {
        "video_name": "v5SAMuRanGM",
        "Q": "The entire base ratio, CD:BC, is equal to phi:1. The entire altitude ratio, AC:AQ, is equal to phi + 1:1. If you wanted to find the ratio of, AB:AP, in terms of phi, how would you know which value or ratio of phi to use. Would it be phi:1 or phi+1:1. What determines which to use? Thank You.",
        "A": "I think because AC:AQ refers to the two different triangles, that is the ratio we use as we know the triangles are similar already. whereas CD:BC is just the ratio of the base parts of the larger triangle..."
    },
    {
        "video_name": "QOtam19NQcQ",
        "Q": "it is hard to under stand in 0:01 when it just starts out",
        "A": "because you dont get more stickers after watching a video once"
    },
    {
        "video_name": "QOtam19NQcQ",
        "Q": "At 0:05, are there more methods than just grouping the number with place value?",
        "A": "if you don t finish khan you will be sorry"
    },
    {
        "video_name": "QOtam19NQcQ",
        "Q": "At 1:17 , what is barging.",
        "A": "Borrowing is a tool that you use when adding or subtracting numbers with multiple digits."
    },
    {
        "video_name": "dXHqO955HxE",
        "Q": "Time 1:30\nYes, but what's confusing to me is what the fraction is of two or more circles.\nFor example, if there are 2 Circles, each with 3 part's, then the combined number of part's is 6.\nIf only 5 of the parts (Pizza Slices) are shaded red, then wouldn't the fraction be 5/6?",
        "A": "I was having trouble with this too. I guess they want you to compared the shaded portion to one circle. Technically you are correct, though."
    },
    {
        "video_name": "dXHqO955HxE",
        "Q": "at 6:08 I want to now how to solve a real fraction equation",
        "A": "They are on other videos not this one.This is for this specific problem"
    },
    {
        "video_name": "CDmJL-VNlaM",
        "Q": "At 6:10 could the function have been simplified a little more to f(e + 0.1) = 2 + 1/5e ?",
        "A": "Yes, it could have. Since that was outside the scope of the exercise, Sal didn t take too long in making the equation as pretty as it could be."
    },
    {
        "video_name": "CDmJL-VNlaM",
        "Q": "at 5:05 why do you write (ln(e^2))+(2/e)*(x-e) when we determined that f'(x)=2/x? Why does it change from 2/x to 2/e?",
        "A": "it s really confusing that he said for a = e , when there was no variable a established in the entire question. for L(x) when a = e? when x = e, not a. x."
    },
    {
        "video_name": "pANZpI9sOYg",
        "Q": "I followed all the steps but at 5:00 of the video I still had three variables to solve for, how do I solve this?",
        "A": "It happens, just solve for another variable and/or one of the other equations."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "I am so confused with everything... But to start what is the definition of a parametrization?\n\nAlso at 13:28 why is it \"b - a\"? Why not \"a - b\"? And how do we know it is b minus a not a minus be?",
        "A": "Parameterization is the method by which we set all variables to be functions of one single parameter. As for your second question, it could be a-b. Either b-a or a-b will work."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "At 14:33, if t(b-a) is a vector it should not only originate from (0,0), perhaps from (1,1), (2,2), or any points in R2. So shift it to (0,0) will go through these two points, but what if it starts from (1,1) ?",
        "A": "Yes, vectors can be drawn anywhere, but we re looking for a mathematical representation of a specific set, so we need to represent it in terms that start from standard position. If we draw it somewhere else, the mathematical representation starting from standard position becomes t(b-a) plus wherever we draw it from. That s where he gets the next part of L = {b + t(b-a)}."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "hello 21:22 i am not so clear what happen line equation if i am p2- p1 instead of p1-p2 ? and two 90 degree parallel vector with different length meet a line so how to fine this line?",
        "A": "two arrow up from the ground with different slightly high and one arrow come from side through touch the top of both arrow top and slide to the ground"
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "So far I've seen several different representations of vectors:\n\n1. A boldfaced letter\n2. A letter with a sort of carrot above it that I saw in the \"Adding Vectors\" exercise\n3. A letter with an arrow above it.\n4. At 7:37, somewhat of a half arrow that Sal puts over his V in '-2V'\n\nAre all of these acceptable formats, or are their use dependent on the circumstances? Are there specific names to refer to the characters above the letters?",
        "A": "If the notation is just to signify a vector then these are interchangeable, and depend on the author and the fashion at the time of writing or the subject. Sometimes a hat above the letter means not just a vector, but a vector with length 1."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "why the slope is 2? I thought it should be one half. (at around 4:45)",
        "A": "No Sal had mistaken it. It is gonna be 1/2"
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "at 14:03 why are we multiplying (b-a) by a scalar? Is that to bring it into standard position?",
        "A": "Multiplying by a scalar lengthens or shortens (b-a). The set of all such expansions and contractions forms a line that is collinear with (b-a). If (b-a) is in standard position, the line will contain the origin. If (b-a) is not in standard position, the line will not contain the origin."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "At about 10:33, Sal talks about the equations of lines in higher dimensions, but could you not just say that for n dimensions, the equation of your line is as follows: y = n1x1 + n2x2 + n3x3. . . + nnxn + b, where n1 through nn are constants and x1 through xn are the independent variables? As messy as that might get, it still seems more straight forward than Sal's equation.",
        "A": "That would not give you a line. Since there are n independent variables, that would mean that the thing this maps to is n-dimensional. A line must be 1-dimensional. What Sal did, by creating the parameter t was make sure there only be 1 independent variable."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "At 3:57 Sal mentions that we are working with a cartesian plane, are polar coordinates used often when when dealing with vectors? It seems like they would be much easier to use than cartesian coordinates when dealing with vectors. Thanks",
        "A": "Polar coordinates are not actually easier. Take plain old vector addition for example. What is the resultant vector if you add 4 at 46\u00cb\u009a and 76 at 0.23\u00cb\u009a ? To solve this you either need to delve into a big geometry problem, or break it up into components. But the components are the xy form of the vector anyway. The basic operations of vectors are much much easier to do in xy notation. Feel free to ask more questions if you need more help"
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "At 24:00, is it right for me to say that \"t\" is a parameter which determines the magnitude of the line? And am I right to say that the parametric equation describes the direction of the vector L? Thank you.",
        "A": "for second question, no. the direction depends on t values. for t=1 you can get some direction but otherwise you can t. first question s answer is no,too. values can do that but t is parameter."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "What does the little upside down triangle at 1:24 mean?",
        "A": "it s a vector because of the right arrow over it."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "somewhere at 15:45, Why a-b is a vector in a standard form? but b+t(a-b) is not a vector in standard form?",
        "A": "Actually, b+t(a-b) IS in standard form. Its tail is the origin and its head is a point on the line defined by t(a-b) that s been shifted away from the origin by the addition of vector b."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "Khan said at 4:37 that the line that S = cV forms has a slope of 2. Doesn't it really have a slope of 1/2?",
        "A": "Yes"
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "At 9:02, wouldn't all the vectors in set L have arrows that point to every point on the blue line, not actually being part of the blue line? The arrows start at the origin, they don't start on the point (2,4).",
        "A": "The vector arrowhead tips are what defines the line. The arrowhead tips all lie on the blue line because all of the vectors in set L are in standard position."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "Is that a left handed coordinate system in 22:00?!? I'm having a hard time with telling the difference",
        "A": "If positive y is going into the screen (i.e. away from viewer), it s a right handed coordinate system, else left. Get your right hand, stick your thumb along positive x, your index finger along positive y, your middle finger (once you make it perpendicular to the digits already discussed) will be pointing along positive z for right handed coordinate systems."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "at 24:02 Sal states that x + y + z is a plane. Isn't that a point instead of a plane???\n\nIf I follow the discussion, to refer to a plane, wouldn't x, y and z be expresed in terms of t, as a parametric expression?\n\nI am just learning linear algebra so maybe I missed the point.\n\nThanks for the great video!!!",
        "A": "x + y + z = k is not a point, because, holding z constant, there is an infinite number if solutions, and since there is also an infinite amount of values for z, it expands in 2 dimensions, making it a plane."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "At about 13:36, Salman Draws the vector a-*b*. Wouldn't the vector point the other way because [2, 1] - [0, 3] = [2, -2]?",
        "A": "Sal draws the vector b - a, which goes in the correct direction. a - b would go in the opposite direction, yes."
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "Hi, i dont understand the part where you show the traditional parametric definition of the line (min 18:53). I think that the green L is wrong because vector b+(b-a) is not equal to the line L. The answer is (-2,5) and this point is not on the line L. It is maybe b-(b-a)? Maybe i am thinking wrong...",
        "A": "What did you get for your answer"
    },
    {
        "video_name": "hWhs2cIj7Cw",
        "Q": "[At time 8:15] I went through the previous questions but I wasn't too clear about one concept. When you add v +x where x is a single point rather than a vector , I understand why the vector v is shifted to the blue line (basically, you're shifting the origin). However, if x is a vector, shouldn't v + x result in a vector which is 4i + 5j?",
        "A": "v + x, would be <4, 5> That is correct. At time 8:15, Sal is defining the set of all vectors that are x + t v. There is nothing wrong with that."
    },
    {
        "video_name": "apWQoMKVmW4",
        "Q": "Is there a better or (at least) another way to see if the vector ds is pointing in the right direction other than the right-hand rule Pr. Khan begins to mention at 3:30?",
        "A": "A vector that points toward increasing x crossed with a vector that points toward increasing y will always result in a vector that points toward increasing z."
    },
    {
        "video_name": "igJdDN-DPgA",
        "Q": "Can someone explain to me about the graph at 5:15?",
        "A": "He said f(x) is greater or equal to g(x) and f(x) is less than or equal to h(x). At x=a, g(a) and h(a) are the same and f(a) must be between g(a) and h(a) which means f(a) is equal to g(a) and h(a). The graph may look strange but forget about the look and understand that f(x) is always between g(x) and h(x) in this case."
    },
    {
        "video_name": "igJdDN-DPgA",
        "Q": "Can someone explain to me the part around 5:00?",
        "A": "Your function f is squeezed between 2 functions, h above and g below. If h and g come together and meet (or at least, if their limits are the same), then f is still squeezed in-between them. But if f is in-between h and g, and they both approach L, then f must have limit L, since L is the only number that is in-between L and L."
    },
    {
        "video_name": "igJdDN-DPgA",
        "Q": "On the graph visible at 7:07, Why must f(x) approach the necking point? If it had a run of multiple points that were discontinuous in that area of the graph, isn't it quite possible, under certain conditions, that the graph would not approach that necking point, and therefore give a value other than L?",
        "A": "No. f(x) is defined as, at any given point, less than h(x) and more than g(x). Therefore, even a point that is discontinuous must still be in between the values of h(x) and g(x) at that point. If h(x) and g(x) are not defined at that point, then f(x) must also be undefined, as there is no such thing as more or less than undefined."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at 7:19 in the video, couldn't x=2 also make it undefined?",
        "A": "yes, but since you can still see that when you look at the simplified form, you do not need to state that explicitly."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 15:10, can x =/= to 3 and 1/2? Because if you multiply 1/2 to 2x-1 that'll cause the expression to be zero right?",
        "A": "yes you are correct"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 7:31, Sal says that the answer is x+5/x-2, and x cannot equal -1.\n\nWhy is 2 not included as a number that x cannot equal? If I plug in 2 to the original equation it is undefined.",
        "A": "Because the (x - 2) is not eliminated/canceled like the (x + 1)/(x + 1) terms. When you eliminate a term via division simplification, it is at that point you must declare the constraint on the remaining expression where the eliminated term could be made = to zero in order to mitigate an undifined result in the initial unreduced expression."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at 10:59, couldn't Sal have factored 3x-6, further (the common factor being 2) ?",
        "A": "No, because 3 can t be divided by 2 without a remainder. If you did, you d get 2(1.5x-3), which is not correctly factored"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at about 5:00 he starts adding y= into the beginning of the expression. why is that expression equal to y? where did he get the y from?",
        "A": "He is giving an example"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at 7:29 wouldn't you have the condition that x cannot equal 2 as well? since the denominator would also equal zero then?",
        "A": "Looking at the simplified answer, we can see that x-2 is in the denominator. So it is already clear that x cannot equal 2. But, because the x+1 factor has already been divided out, we cannot tell just by looking that it used to be in the denominator. That s why it requires a separate statement that x cannot equal -1."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "Im confused by the whole factoring thing, like at 3:03. How would you factor something like 6n-20? or would you even factor something like that?",
        "A": "You could factor that to: 2(3n - 10)"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "Why can't be x = -3 in the 2nd equation? at 3:47",
        "A": "@Jan Kaye Apa How many times are you going to post the same question ?"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "why didnt he just simply divide the solution in 6:30 part",
        "A": "There are many different ways of finding the solution to this problem."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 3:02, isn't (x+3)(x+3) equal to x^2+6x+9",
        "A": "Yes, but he s writing (x + 3)(x - 3), not (x + 3)^2."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 7:29, Sal has the simplified expression (x + 5) / (x - 2). He says that x cannot be equal to -1, but that was it. The denominator in the simplified case can't have +2 as x. Why didn't he mention that?",
        "A": "It is understood that x \u00e2\u0089\u00a0 2, as you mentioned by looking at it, but by looking at the simplified expression alone we wouldn t know that -1 is restricted originally. So we add the restriction of the original expression to say that while this expression is defined at x = -1, but it is originally cannot be -1, so don t evaluate it at x = -1"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "how can you get x not equal to -1 at 8:00",
        "A": "The factor that was fully divided out of both the denominator and numerator was ( x + 1 ). When that factor equals zero, x = -1. Because that factor no longer shows up in the final answer, as part of the simplified answer we have to say that x cannot equal 1 ."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 15:07, why does Sal not exclude x=1/2 from the domain of the rational expression?",
        "A": "This is because the final product still contained the (2x - 1) in the denominator. It was implied that x \u00e2\u0089\u00a0 1/2 in the final statement but since (x + 3) was cancelled, it was impossible to deduce from the final statement that x \u00e2\u0089\u00a0 -3 which is why Sal made a special note of -3 and not 1/2. Both ARE excluded from the domain, but 1/2 can be deduced so it is not necessary to take special note of that."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 7:28, can't you also have the condition that x cannot = -2 as well? They are in the denominator, and if you do put -2 for x in the original, you will get undefined.\n\n(2^2)-2-2\n4 - 2 - 2\n= 0",
        "A": "I think you meant that x cannot equal 2. And yes, you can have that condition, but it is unnecessary since x-2 is still in the denominator. The reason the condition is necessary is because, if you look at the equation after it is simplified, you would not know that x=1 cannot be a solution."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 7:00 what about x=2? Wouldn't that be undefined too?",
        "A": "Yes, but since we re not canceling the (x -2), it s not necessary to include that as a condition."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 3:25, I don't understand why x squared minus nine is (x+3)(x-3)? I greatly appreciate your support and help.",
        "A": "OK, well we have to multiply it out. (x + 3) (x - 3) x(x - 3) + 3(x - 3) x*x - 3*x + 3*x - 3*3 x\u00c2\u00b2 - 3x + 3x - 9 x\u00c2\u00b2 - 9 I broke it up into more than the normal number of steps. I hope that helps. This is a common pattern in maths, the (x + a) (x - a), so that you get x\u00c2\u00b2 - a\u00c2\u00b2. Don t worry if it doesn t feel natural at first; just practice. Remember, don t practice what you are good at, practice what feels uncomfortable . Soon it will become natural."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 11:09, Sal writes down 2x^2+5x+3 but in the actual problem, it is written as 2x^2+5x-3. Is it wrong in the video or am I mistaken?",
        "A": "Yes, at that time there is an annotation on the video, which reads: Sal wrote 2x^2+5x+3 but meant 2x^2+5x-3 . He notices this mistake at 12:30."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "Okay so with the condition x cannot equal negative 1 @ 7:25. Why do we not add the condition x cannot equal 2. Is it because we only add the condition for something we've cancelled out? ;D\n\nThank you",
        "A": "Yes, you are absolutely right. When we cancel the (x+1), we lose the fact that (x+1) was in the denominator of the original expression. Since we still have (x-2) in the denominator of the final expression, we do not have to include that restriction."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "The video is very well explained, however i have a question on the grouping method @8:14 , wouldn't it be easier to divide the whole thing by 3, thus getting rid of the 3x^2 and more importantly simplifying the equation?",
        "A": "You can t divide by 3 that would lose the 3. But, you could factor out a 3 first, then factor the new trinomial. 3x^2+3x-18 = 3(x^2+x-6) = 3(x-2)(x+3). This is the same as what Sal got as 3(x-2) = 3x-6 (one of Sal s factors). Sal s version is just not completely factored."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 8:00 shouldn't you factor 3 out of the whole problem first?",
        "A": "I think it s probably a good idea to always factor completely, though."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at 15:20 why didn't sal write x is not equal to 1/2 as well or is that just understood.\nthanks.",
        "A": "You re right. The 2x-1 in the denominator makes it clear that x can not be 1/2. He says what else x can not be equal to because after canceling out the factor that was on the top and bottom of the fraction, there is no other evidence that x can not be -3 unless you state that it is not part of the domain."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at 8:15 in the video, why didn't you take out a factor of 3 from the numerator to rewrite the factoring (after some steps) as 3(x+3)(x-2) ?\n\nthank you so much for building this resource!",
        "A": "He could have done that. It is just a different method that I (and many other people) think is more complex and not as easy as taking out the 3 first."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 4:57 you say x cannot equal to -1/3 where did you get -1/3 from?",
        "A": "x cannot equal -1/3 because if you plug it in to 4(3x+1) you get: 4(3[-1/3]+1) 4(-1+1) 4(0) 0 If x=-1/3 the denominator will equal zero and you can t divide by zero."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 6:00 Saul says \"so what two numbers when multiplied together equal 5, but added together equal 6\". How/why does this work?\n\nIt seems random to me. Wouldn't this not work for most examples? For example x^2 - 2X +7. After all, what two numbers multiplied together equal 7, but added together equal -2? Confused...",
        "A": "If it doesn t work, then it can not be factored and you would need to use another technique to solve the problem. Of course in most cases if you are being asked to factor a trinomial, it can be factored."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "Why can't be x = -3 in the 2nd equation? at 3:47",
        "A": "The original fraction (x^2-9)/(5x+15) is undefined if x = -3 Even after we reduce the fraction, we want to maintain the integrity of the original fraction. Since the original fraction was undefined if x = -3, we restrict the new fraction in the same way. Hope this helps."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "When Sal writes -6 at 13:08, shouldn't that be -3?",
        "A": "No. The equation is 2x^2 + 5x -3 He s factoring by grouping. There is a 2 on the x term. Hence A*B = 2* -3 (which is -6) and, A+B has to equal 5."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 7:06 how did you get that x does not equal to -1 i dont understand where the -1 came from.",
        "A": "When x = -1, x+1 = 0, so you would have division by 0."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at 6:27 when he writes that the numerator is (x+5)(x+1) couldn't it have also have been (x+2)(x+3)?",
        "A": "No, because 3 x 2 does not equal five and 3+2 does not equal six. You are trying to find two numbers that when multiplied they equal five, and when added they equal 6. Thus, 5 and 1 are the only two numbers which satisfy what I just described. Do you understand what I just explained?"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at 0:54 how does it get down to 1/3 does he just simplify?",
        "A": "yes he just simplify s."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 7:04, why does x not equal -1? I thought x cannot equal 2 because in the denominator it said x-2, which then makes the equation undefined.",
        "A": "We need to be true to the original fraction. Notice, when the fraction is 1st written in factored form, the denominator contains both (x-2) and (x+1). The x+1 tells us that x cannot = -1 either. So, the original equation is not defined for either x=2 or -1. Sal writes in the x not equal -1 to maintain the value of the original fraction."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 11:10 you made a mistake in the video. You mention that you should have done the video over again. I agree with this statement and question why you did not do this. It makes it hard to follow the steps....",
        "A": "Yeah, he made a few mistakes in this video. As tomm said, it must have been a long day for Sal."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At the 3rd example (7:20), shouldn't the final answer's domain also be x != 2 and not just x!=-1?",
        "A": "The x not = -1 has to be stated because it can no longer be determined from the simplified expression. The x not = 2 can still be determined from the expression, so it isn t required that it be stated."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "In the example in 2:45, why did x^2-9 turn into (x+3)(x-3)? Shouldn't both of them be x-3's?",
        "A": "No, because (x-3)(x-3) would be x^2 - 6x + 9. Remember, two negative numbers multiplied together yields a positive number. And in the case of the expression x^2 - 9, the last term has to be negative (and is shown through 3 * -3)."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "Also, I don't understand the equation at around 5:30...why is it simplified as x+5 over x+2? I thought for sure that the simplified answer is 5x + 3? Surely 'FOIL' needs to be taken into account when expanding the original equation into brackets...? if we're multiplying the outer numbers in both sets of brackets as the denominator (x and POSITIVE 1), how did we get to x-2??",
        "A": "x-2 came out of the denominator. x^2+6x+5=(x+5)(x+1). x^2-x-2=(x-2)(x+1). The you factor out the (x+1)"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at 7:21 -ish, Sal is saying that the (x+1) in the numerator & the denominator cancel each other out, leaving (x+5) over (x-2). He then says we must add the condition that x cannot = -1. Why doesn't Sal also include a condition that x cannot = 2?",
        "A": "That is already implied by having \u00f0\u009d\u0091\u00a5 - 2 in the denominator. However, the \u00f0\u009d\u0091\u00a5 + 1 was canceled out so in the simplified expression there was no way to deduce the domain restriction of \u00f0\u009d\u0091\u00a5 \u00e2\u0089\u00a0 -1. Granted, he could have added \u00f0\u009d\u0091\u00a5 \u00e2\u0089\u00a0 2 to reduce confusion."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 8:00, is there a reason Sal does not take out the GCF? It seems like that would make things much easier.",
        "A": "I agree with you. Sal may simply have overlooked doing that. Eventually, though, he does factor out the three."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 10:22, he has a binomial factor of 3x - 6. I notice that this can be simplified by factoring out 3, but Sal doesn't do that. Why?",
        "A": "You can factor it further to 3(x-2). I don t know why he didn t or chose not to. We only need to simplify the original expression, not factor it completely, you can leave it at 3x-6."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "5:0 Why -1/3 instead of something else?",
        "A": "Because when x=-1/3, it makes the denominator 12x+4 = 0. When the denominator is 0, it is undefined, therefore x cannot be -1/3. While anything else doesn t make 12x+4 = 0, therefore anything else would be defined."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at 14:46 how did 2x(x+3)-1(x+3) not factor to (2x-1)(x+3)^2? Why doesn't the (x+3) have to be squared?",
        "A": "Think the distributive property and FOIL. When simplifying the expression (2x-1)(x+3), the 2x is multiplied to both x and 3, 2x(x+3), and the -1 is multiplied to both x and 3, -1(x+3). From this, the condensed form only has one factor of x+3. Hope this helps!"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 7:36, why is the condition x cannot equal 2 not listed? Because plugging that number in also makes the expression undefined. Thanks!",
        "A": "The fact that it cannot be 2 is obvious from looking at the denominator, which was why he did not list it. The fact that x cannot be -1 is not so obvious, however, since he canceled those factors, so it s important to explicitly state that restriction to ensure the functions are still the same after simplification."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at 7:24, why cant a condition also: x not equal to 2, since it would make it undefined?",
        "A": "It is obvious from the new function definition that x cannot be equal to 2. What is not obvious is that x cannot equal -1 as well since, when we factored the polynomials and were able to cancel the x+1 term, we visually lost that constraint."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 8:44, can't you also use a GCF for the numerator: 3?\n3(x^2 + x - 6)? Then you can factor the expression in the parenthesis and it makes it easier as well.",
        "A": "Yes, you could have. The final solution is the same though."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 7:13 when the excluded variable is mentioned, why is -1 excluded but 2 isn't? Wouldn't positive 2 also make the denominator undefined?",
        "A": "I agree I think you re right. the domain should be all real numbers such that x =/= 2, -1."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "I am lost at 11:15, wasn't the Quadratic: 2x^2 + 5x - 3 and not 2x^2 + 5x + 3? Is -3 a typo?",
        "A": "Yes, it seems to be a careless mistake. Factoring the right expression would give: 2x^2+ 6x - x - 3 = 2x (x + 3) -1 (x + 3) = (2x - 1) (x + 3)"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "Why isn't x cannot be equal to 2 also a condition? You said that about (x+1) at 7:02, but not (x-2). What excludes it?",
        "A": "The reason is, that is still true for the simplified expression. In other words, that still applies to (x+5)/(x-2). So you are correct, and 2 is not part of the domain, except you don t need to write a note about it, like you do the (-1). The reason we need to say x can t be -1 , is we reduced that part of the equation, so it needs a note, so that it matches the domain of the original function. I hope this helps!"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "I don't think Sal should say ( at 9:12) that he \"split\" the 3x into -6x and 9x because those terms do not \"fit\" into 3x.",
        "A": "I don t know what you mean by they don t fit into 3x. -6x + 9x = 3x So, it is perfectly acceptable to replace 3x with the 2 new terms because they are equal. Algebraic substitution lets us replace anything with a value that is equal to the original value."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "Sorry if this has already been asked, but (3:47) when you say x cannot be equal to -3, shouldn't that be a positive 3 because if it were, then the top part would equal zero and we can't have that?\nAnd At 10:09, I understand where the 3 is coming from, but why is it positive now instead of staying negative?",
        "A": "No - having a zero in the top part is not a problem. A zero in the top just means that the number is a zero. It is having a zero in the bottom that is a problem, since division by zero is not defined. That is why the restriction is a -3 and not a +3. As for your second question, I am going to assume you mean the purple 3 in the parenthesis. It becomes positive because you factored out a negative, which changes the sign of everything in the parenthesis. You can think about it like division -18/-6 = +3"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "How come at 7:20 when he adds the NPVs he doesn't add +2? Wouldn't you have to add that because the other binomial is X-2?",
        "A": "X not equal to 2 doesn t need to be stated again because it is already clear from just looking at the denominator that x cannot be 2. The -1 had to be stated because the simplified fraction no longer included the (x+1) in the denominator."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "8:39 Great video, very helpful but why don't you mention the box and diamond method for factoring?",
        "A": "I m sure there are various ways for factoring. I personally think that factoring by grouping is the most accurate and efficient way to factor."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 7:30, shouldn't the other condition also be that x != 2?",
        "A": "You are right that x=2 also cannot be a solution, but you dont have to write it as an extra condition when the formula was simplified. that x !=-1 was only written because you couldnot see anymore in the formula x+5/x-2 that x cannot be -1. you still can see in the formula x+5/x-2 that x cannot be 2."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 6:38, he starts to right out the denominator that will supposedly equal x\u00c2\u00b2-x-2. He writes out (x-2)(x+1). Doesn't this equal x\u00c2\u00b2-2 and not x\u00c2\u00b2-x-2?",
        "A": "If in doubt when factoring, try multiply it out to see if it is correct. It s what I \u00c3\u00a4lways do to double check to make sure I factor it correctly. Try using FOIL to multiply (x-2)(x+1) and see what you get."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at 7:26, wouldn't there also be a restriction of x != 2? because of the x - 2",
        "A": "When simplifying an expression, you only need to make a note of what would set the denominator to 0 if you are canceling it (because canceling removes all evidence of that factor from the denominator). Since he only canceled (x+1)/(x+1), that denominator of x+1 is the only place he needs to worry about what value of x would lead to a zero in the denominator."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "How did Sal get that x \u00e2\u0089\u00a0 -1/3 at around 4:42?",
        "A": "The denominator had a factor of (3x+1) When x=-1/3 the (3x+1) =0 and 0 cannot be a factor of the denominator of because anything divided by 0 is undefined. I hope that helps!"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 8:30, how is Sal doing the grouping? I'm really confused lol. Why does he do\na * b = -54 and\na + b = 3x",
        "A": "Unfortunately, Sal made a poor selection for variables. Everyone confuses his variables with the a and b in the quadratic, like ax^2 + bx + c. Sal is using a and b to represent 2 unknown values. Relabel them to use different variables like m and n . It might make more sense. You basically need to find 2 numbers that multiply to create -54: m*n = -54 And, the same 2 numbers need to add to 3: m+n = 3 The 2 numbers are: 9 and -6. Hope this helps."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "time approx 12:10 and before.... (3x-6)(x+3) does not expand into 3x squared + 3x - 18. it expands into 3x squared -3x -18,.... does it not\u00c3\u0089",
        "A": "(3x-6)(x+3) = 3x(x+3) - 6(x+3) = 3x*x + 3x*3 - 6*x - 6*3 = 3x^2 + 9x - 6x - 18 = 3x^2 + (9x - 6x) - 18 = 3x^2 + 3x - 18"
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 8:42, Sal says to use grouping to find what two numbers multiply to get -54 and add up to get 3. Is it more efficient to factor 3x^2 + 3x - 18 using the grouping method or taking out a 3 first?",
        "A": "It is usually much easier if the leading coefficient is 1 without leaving any fractions elsewhere, so typically you would want to start by factoring out the 3."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 3:48, the simplified expression (x - 3) / 5 cannot have x as -3. But if you plug that in, it won't have an undefined number. Why, then, is -3 not an option to replace x?",
        "A": "x \u00e2\u0089\u00a0 -3 because of the original expression where it was undefined. You have to carry the restriction from the original expression when you write the new simplified expression, because when you write the new expression, it is defined at -3, but remember, it didn t define at -3 originally."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 4:38, what do you mean by the horizontal line when y=3/4?",
        "A": "If you had an x/y graph and plotted every point where y was equal to 3/4 you will in up with a horizontal line running across the chart at y=3/4. Get some graph paper, draw up a x/y chart and plot a few points."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "at 6:18, shouldn't it be (x+5) and (x+1)? Not (x-1)?",
        "A": "Yes, I think he made a mistake."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "how do you know where to put which a or b first @ 9:50?",
        "A": "You can put them either way, the point is to notice which of the terms you get have a common factor to undistribute . Putting 9x first is only useful in that it makes it a little more obvious which terms do have a common factor, but if you want to write them in the right order , that means you need to have that step figured out in your head before you even write it down, so there s no real advantage."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "Is there a way to factor other than what Sal did at 7:33?",
        "A": "You can initially factor out a 3 from the numerator, and that would make the quadratic easier to factor. I.E. you wouldn t have to use factoring by grouping, since the coefficient of the x squared term would be 1 then."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 10:51, why didn't he factor out a three from the 3x-6?",
        "A": "Good question, and my only guess would be that he wasn t looking to factor anymore and with only a binomial it was simple enough as it stood so taking out the GCF wouldn t help much? Just a guess though haha."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At around 7:29, would of 2 be another condition?",
        "A": "Said by another user JaniceHolz When simplifying an expression, you only need to make a note of what would set the denominator to 0 if you are canceling it (because canceling removes all evidence of that factor from the denominator). Since he only canceled (x+1)/(x+1), that denominator of x+1 is the only place he needs to worry about what value of x would lead to a zero in the denominator."
    },
    {
        "video_name": "7Uos1ED3KHI",
        "Q": "At 2:56 how do you factor the expression?",
        "A": "It answers your question right after that time"
    },
    {
        "video_name": "St3MU5i3chc",
        "Q": "At 6:38, what is a non degenerate triangle? (And, if it exists, what is a degenerate triangle?)",
        "A": "I looked this up on Google. A degenerate triangle is a triangle formed by 3 collinear points. It doesn t look like a triangle. It looks like a line segment. In the video, if b were zero, side c would collapse back onto side a. So, bearing this in mind, I guess that a non-degenerate triangle is the usual type of triangle that we see. It would be formed when not all 3 points were collinear. Hope this has been helpful!"
    },
    {
        "video_name": "St3MU5i3chc",
        "Q": "At 4:50, did Sal mean to say, \"Assume: Segment connecting a point off line and closest point ON line is not perpendicular to the line,\" instead of \"Assume: Segment connecting a point off line and closest point TO line is not perpendicular to the line\"?",
        "A": "He corrects himself when he says what he meant."
    },
    {
        "video_name": "MRAIgJmRmag",
        "Q": "at 3:48 why did he divide by -2? arent you supposed to divide by positve 2 to make sure the 2s are canceled out??",
        "A": "He took a shortcut. He could have divided both sides by 2 to get: -y = 4 and then divided by -1 to get: y = -4 He just divided by -2 instead, and combined both steps into one step. The two different methods get the same result."
    },
    {
        "video_name": "IXRMVcoqRRQ",
        "Q": "So at the end of example 3 of understanding the notation of functions 2:05 of 2:12 mins he's talking about the volume of the water balloon and comes to 36 inches squared times pi but doesn't go any further why? Pi has a number value 3.14 so why didn't he multiply pi times 36 isn't the problem in compete without doing that?",
        "A": "Yes, to find the volume of the balloon you would multiply pi by 36 to get about 113 inches cubed. But, pi is irrational, which means the digits never repeat or come to an end. Multiplying by pi would output a number that has to be rounded."
    },
    {
        "video_name": "IXRMVcoqRRQ",
        "Q": "I cant understand what you're doing after 1:48 you're going to fast!",
        "A": "He is going fast! You can slow down any Khan Academy video by clicking options at the bottom of the video and then selecting (1/4). You can also speed it up! To answer your question, Sal is combining (4/3) * pi * 27 by multiplying the (4/3) by 27 to get 36 so he can just write 36pi."
    },
    {
        "video_name": "IXRMVcoqRRQ",
        "Q": "2:00 pm: in this video sals answer was 36 pie inches to be more accurate can i put the value of pie=22/7 or 3.14 to get answer more accurate",
        "A": "It is more accurate to leave it as pi, because pi is an irrational number and neither 22/7 or 3.14 are precisely equivalent."
    },
    {
        "video_name": "IXRMVcoqRRQ",
        "Q": "Why was pi not multiplied In 1:44",
        "A": "Unless told to give a numeric answer, you generally want to leave irrational numbers in their exact state rather than calculate an approximation. So that means leaving things like \u00cf\u0080 or \u00e2\u0088\u009a2 as such rather than punching them into your calculator."
    },
    {
        "video_name": "IXRMVcoqRRQ",
        "Q": "At 1:53 don't you multiply each of the terms by three to get 4 as a whole number, not divide?",
        "A": "If you multiplied by 3 you would have to multiply the left side V(r) by 3 as well, which would give the wrong answer (unless you divided the result by 3). Since all the terms are being multiplied by 4/3, due to the associative property of multiplication which says you can multiply or divide whichever numbers you want first, you could just view the equation as (4\u00cf\u0080r^3)/3 or (4\u00cf\u0080r^3)*(1/3)--> (4\u00cf\u008027)/3 or 4\u00cf\u0080*(27*1/3)--> (4\u00cf\u0080)*(27/3) --> 4\u00cf\u0080*9 --> 36\u00cf\u0080."
    },
    {
        "video_name": "eTSVTTg_QZ4",
        "Q": "at 1:17 can you do the same by just dividing by 1 half?",
        "A": "Yes, but that is the exact same thing as multiplying by two."
    },
    {
        "video_name": "eTSVTTg_QZ4",
        "Q": "At 2:10, why didnt he subtract b from both sides?",
        "A": "You have to use division for that, not subtraction. H is being multiplied by B, and if you subtracted B, your answer would be wrong."
    },
    {
        "video_name": "eTSVTTg_QZ4",
        "Q": "Hi, at 1:14 why do you have to get rid of the 1/2. Isn't the 1/2 in the equation for a reason? Thanks",
        "A": "You have to get rid of it so bh is alone, which will make it easier to get h alone."
    },
    {
        "video_name": "GWZKz4F9hWM",
        "Q": "At 0:47,how did he know which equation to box?",
        "A": "When you solve with substitution, you want to first solve one of the equations for a single variable (meaning putting it alone on one side). That was already done in this case, with the term y = 2x - 5."
    },
    {
        "video_name": "GWZKz4F9hWM",
        "Q": "Video time: 4:21\nWhen working out this problem I did the PEMDAS (order of operations). So I did not distribute the 4 to the numbers within the parentheses. I think this where I went wrong. But I don't understand why you don't do the PEMDAS here? How do you know when to distribute factors versus PEMDAS method. Please advise! Thank you!",
        "A": "The video is only 4:20 minutes long. So, time stamp of 4:21 doesn t make sense. Are you referring to the distribution of -4 at 1:48 in the video? If you are, the video is following PEMDAS. Sal needs to deal with the parentheses. He can t add/subtract 2x-5 because these are unlike terms. So, to eliminate the parentheses, he uses the distributive property. Hope this helps."
    },
    {
        "video_name": "xH_GllPuymc",
        "Q": "At 1:23, why does put x+7 twice? Thanks",
        "A": "Because a squared number is the number times itself. (x+7)\u00c2\u00b2 means (x+7) squared. Therefore (x+7)\u00c2\u00b2=(x+7)(x+7)"
    },
    {
        "video_name": "xH_GllPuymc",
        "Q": "@2:50 what is the difference between: (x^2+2ax+a^2) and (a^2+2ab+b^2)",
        "A": "Both versions represent the standard pattern for squaring a binomial. The only difference is the binomials used to start with. In the video, Sal is using (x+a)^2 and it creates x^2+2ax+a^2 The pattern is often shown as: (a+b)^2 = a^2+2ab+b^2. Hope this helps."
    },
    {
        "video_name": "xH_GllPuymc",
        "Q": "3:37 makes no sense how can you get a squared 2 times when origianly you had two a's\nthe original problem said (x+a) (x+a) so our answer is now\nx2 + 2ax + a2\nand isnt 2ax the same as a squared? i'm absolutly confused right now",
        "A": "x^2 is short hand for x times x whereas 2ax means: 2 (a) (x). Let s try an example that replaces a with a number. It might be clearer. Note: this is shown in exreme detail. (x + 4) (x + 4) = x(x) + 4(x) + 4(x) + 4(4) = x^2 + 4x + 4x + 16. Add 4x + 4x (x + 4) (x + 4) = x^2 + 8x + 16 Now compare this to the formula. Remember, a = 4 in this example. (x + 4) (x + 4) = x^2 + 2(4)x = 4^2 Simplify and you get the same result: x^2 + 8x + 16"
    },
    {
        "video_name": "xH_GllPuymc",
        "Q": "At 2:55, how is it b^2?",
        "A": "At 2:55, it is NOT b^2. It is ab. Sal made a mistake. It would be b^2 if he was multiplying (a+b)(a+b). Sorry that that was confusing for you."
    },
    {
        "video_name": "xH_GllPuymc",
        "Q": "At 2:55, Isn't the binomial (x+a)(x+b) = x^2 + (a+b)x +ab? Thanks.",
        "A": "You are correct. Sal made a mistake."
    },
    {
        "video_name": "3tRc6mUumFk",
        "Q": "at 3:52 why would would you add $20 to the 8$?The $20 is your monthly pay isnt it",
        "A": "It is $20 to be a member for a month, plus $8 per session. In the table, he has 20 for the month with 0 sessions. He doesn t have to pay the 8 dollars if he doesn t have a session. Next in the table, he has 28 dollars for 1 session. He pays 20 just to be a member, then adds 8 dollars for the session. = 28 dollars"
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 12:41 why didn't he say 39=3(13)?",
        "A": "because those are not perfect squares"
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 12:40, can't 39 be factored into 3 and 13?",
        "A": "Yes it can, but he is looking for perfect squares and neither 3 nor 13 are perfect squares."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "I'm confused, why did he divide at 13:27? When I learned about this awhile ago, our class used graphing calculators once we subbed into the formula so I don't really know how radicals work by hand.",
        "A": "Dividing all terms in the equation would eliminate the 2 in front of the sqrt(39), and makes it easier to solve the problem overall."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 4:11, why can't I simplify -4 in the numerator and 2 in the denominator? I did that and my final answer was different.",
        "A": "You can t do that because the -4 is being added to the plus or minus 10, not multiplied by the plus or minus 10. If it were being multiplied like this: (-4 * \u00c2\u00b1 10) / 2 Then you could simplify that like this: -2 * \u00c2\u00b1 10 But you can t do that if the -4 is being added. In that case, you have to add first and then divide. When you re dealing with fractions like this, it s best to just pretend that there are parentheses around the numerator, and do that stuff first before dividing. Anyways, I hope this helps."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "Why is it possible to change the signs at 4:15. Sorry, I haven't dealt with positive-negative signs a lot.",
        "A": "The positive-negative sign indicates that there are actually two equations, one with a positive sign and one with a negative sign. It occurs when taking the non principle even root. This occurs in the quadratic formula because the principle square root sign is included, but you should be taking the non principle version. (If you take the square root of 100, there is two answers, 10 and -10 {10*10 = -10*-10}.) You are not really changing the signs, but you are solving both versions of the equation."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 13:33, how did he get -6 +or-root 39 over -3? How did he get the -6 and -3?",
        "A": "At about 13:24, Sal explained We can divide the numerator and the denominator, maybe, by two. Two was a common multiple of the straight-forward numbers in the numerator and the denominator as well as the coefficient on the square root, so he divided the entire fraction by two. He was just simplifying... I hope this is helpful! ;)"
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "7:09 - How can it be a function? Don't functions look like this- f(x)=x+ 1 ?",
        "A": "Linear functions look like f(x) = mx + b Quadratic functions look like f(x) = ax^2 + bx + c Exponential functions look like f(x) = a (b)^x There are a wide variety of functions that still meet the rule of each x has only one y."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 5:52, you cannot cancel the 4 and 2 with 10?",
        "A": "You could do that. It would become -2 \u00c2\u00b1 5 , which works fine. Nice observation."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 11:27, why did Sal write -4 for multiplying a and c? Doesn't the formula say 4ac?",
        "A": "Because it is in the formula? x = [-b \u00c2\u00b1 \u00e2\u0088\u009a(b\u00c2\u00b2 - 4ac)]/2a"
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "With the quadratic equation at 2:20, why don't I find the correct x values when I try to solve it with the perfect square approach? I get only 7 for x. Thanks",
        "A": "If you completed the square, you would have taken the square root of both sides of the equation. Remember, the side of the equation with the constant needs to have 2 signs in front: +/- the square root of the constant. If sounds like you lost one of these signs which would then give you only one answer. Hope this helps."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "10:10 why do you need to graph it?",
        "A": "you don t necessarily need to.. Sal did to prove that there are, in fact no real solutions to the equation, and also to show everyone a visual representation that the quadratic formula works... Now for you, you would only need to graph it if; A) you get a question that asks you to graph the equation. or, B) you just really wanted to check your work."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "@ 13:34 , why does sal remove the two from the \"2sqr root of 39?\"",
        "A": "everything was divided by 2. 12/2=6, -6/2=-3, and 2/2=1 when you have 1 on the outside of the radical, it means 1xsqrt39=sqrt39 so the 1 is just left out."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "at 13:35 Sal simplifies the equation from a -12\u00c2\u00b12\u00e2\u0088\u009a39/-6\n(over -6) to -6\u00e2\u0088\u009a39/-3 (over -3) Why did he divide the two from the top to both the -12 on the top and -6 on the bottom of the equation?",
        "A": "Because the denominator applies to both terms, not just one. And, when we reduce fractions we cancel common factors. We do not cancel terms. We need to factor the numerator, then reduce. (-12 \u00c2\u00b1 2\u00e2\u0088\u009a39)/(-6) = 2 (-6 \u00c2\u00b1 \u00e2\u0088\u009a39) / (-6) = (-6 \u00c2\u00b1 \u00e2\u0088\u009a39) / (-3) Hope this helps."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 14:04, can I write x = 2+- (sqrt 39)/3 as x = 2+- (sqrt 13/3)? The second result is what I got from completing the square and factoring. Is it any more simplified than the first equation?",
        "A": "Your version is less simplified than the 1st version. You need to simplify your radical. When the radical is simplified, you should not have a fraction inside the radical. And, you should not have a radical in the denominator. If you complete those steps, you will get the 1st version."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 12:41, 39 can be further simplified into 13 and 3.\nIt's prime factorization.\nRight???",
        "A": "It can be divided by them, but there is no point because they aren t perfect squares and so they don t help you to continue to solve the equation and they just complicate things, so he just left them out."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 13:30 where did the 2 go ?",
        "A": "He has the equation (-12 \u00c2\u00b1 2 \u00e2\u0080\u00a2 \u00e2\u0088\u009a39) / -6 He simplifies both the numerator and denominator by 2 to simplify it (-12/2 \u00c2\u00b1 2/2 \u00e2\u0080\u00a2 \u00e2\u0088\u009a39) / (-6/2) Since 2/2 is equal to 1, he could have left it s value in there and it would have looked like (-6 \u00c2\u00b1 1 \u00e2\u0080\u00a2 \u00e2\u0088\u009a39) / -3. But, since 1 \u00e2\u0080\u00a2 \u00e2\u0088\u009a39 is the same as \u00e2\u0088\u009a39, Sal just removed it completely from the equation."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "Why didn't Sal reduce the answer of the second question at 8:55?",
        "A": "Once he gets a negative number in the radical sign, there is no real purposes of going on until you start learning about imaginary numbers. As he graphs it, he shows that it does not cross the x axis, so there are no real solutions. If you understand imaginary numbers, then you can go on and reduce."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "12:29\nCould someone run that by me again? I didn't get it.",
        "A": "Sal is prime factoring 156 in order to simplify \u00e2\u0088\u009a156. \u00e2\u0088\u009a156 = \u00e2\u0088\u009a(2 \u00e2\u0088\u0099 2 \u00e2\u0088\u0099 3 \u00e2\u0088\u0099 13) = \u00e2\u0088\u009a(2\u00c2\u00b2 \u00e2\u0088\u0099 39) = \u00e2\u0088\u009a2\u00c2\u00b2 \u00e2\u0088\u0099 \u00e2\u0088\u009a39 = 2\u00e2\u0088\u009a39."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "at 09:04 could you simplify all by the denominator 6? even tho the result wouldn't change",
        "A": "In some cases this could work, but not in this case. The expression on screen at 9:04 has a 6 in the denominator, but to simplify you have to divide the 6 out of every term in the numerator. The first term, -6, divides by 6 nicely. However sqrt(-84) does not, so the expression would actually get more complicated."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 11:48 Sal said that -4 * (-3) cancels out and then you turn both of their signs from negative to positive, whereas in a separate problem (7:54) when multiplying the 4 by variable a and variable c, Sal does not make the minus sign a part of the 4. Can someone explain why he would ignore the minus sign and then in another problem make it a part of the 4?",
        "A": "Yes he did. The words minus and negative are interchangeable. They have the same meaning. When you have two minus signs you end up adding. When you have two negative numbers you end up with a positive. The same is true for one of each. A positive times a negative remains negative, therefore the operation is subtraction and not addition."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 8:36, can I simplify x= (-6+- sqrt (-84))/6 to x= -1+- sqrt (-14)?",
        "A": "No you can t. You cancelled out sqrt(6) with the 6 in the denominator. These are not the same value. The only way to simplify the rational expression is to: 1) use complex numbers to 1st simplify the radical sqrt(-84) = sqrt(-1)sqrt(4)(sqrt(21) = 2i sqrt(21) This turns the answers into: x= (-6+- 2i sqrt (21))/6 2) then, you can cancel out a common factor of 2: x= (-3+- i sqrt (21))/3 Hope this helps."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "Considering the way quadratic equations are written, when adding 10 to the left side at 5:10, if I were to replace 3x^2 with 10, make the b term 3x^2 and make the c term 6x, would that affect the solution of the answer?",
        "A": "I assume that you are saying you want to write it a 10 + 3x^2 + 6x = 0 instead of 3x^2 + 6x +10 = 0. First off, since this is not standard form, you cannot just arbitrarily say a = 10, b = 3, and c = 6. That does not work because no matter what order you have it in, the coefficient of the x^2 is always a, the coefficient of the x term is always b, and the constant is always c, so merely changing the order of the terms does not change what a, b, and c are and thus would not affect the solution."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "14:17 is 2 plus or minus the square root of 26 an ok answer?",
        "A": "Yes! It s often better to leave your answer like that than convert it to a decimal, since that s an exact answer and when we write it as a decimal we lose accuracy."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "why did 100 = 10 @ 4:16?",
        "A": "The square root of 100 is 10."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "Hi, at 13:22 how did he knew that he had to divide the numerator and denominator by 2 and why did he do it? Thanks",
        "A": "He saw that the denominator was divisible by 2 and that both terms in the numerator were divisible by 2. He did it to simplify the fraction."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At time 5:00, Sal indicates the solution is 3 or -7.\n\nEvery time I practice quadratics in Khan Academy I get the answer wrong unless I order my solutions from lowest to highest in quantity (i. e. the solution for 5:00 should be -7 or 3).\n\nDoes anyone know if this a rule - in which case the video has got it wrong - or are the practice exercises forcing an unnecessary convention?",
        "A": "The practice exercises instruct you to enter the answers from lowest to highest. I think this helps simplify the programming behind the website as they don t have to code for all possible ways that the number could be entered."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 12:36 he writes 156 as (rad sign)2 * 2 * (rad sign)39. Why does he leave 39 under the radical sign instead of rewrite it as (rad sign)3 * 13?",
        "A": "Because the three wont simplify out of the equation"
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "12:19 Sal uses prime factorisation to simplify 156, 78 goes into 156 twice, 39 goes into 78 twice but then he stops there at 2*39. Why doesn't he simplify it further as 3*13= 39 as both are prime numbers?",
        "A": "He didn t further reduce 39 because he knew that it contained no factors that were perfect squares. For simplifying a square root you need only find the factors that are perfect squares -- after factoring out all the perfect square factors, the square root can no longer be simplified. Thus, if you know for sure that a number contains no perfect square factors, you need not continue factoring it for the purposes of simplifying a square root."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "At 14:49, why does Sal write the square root of 156 as the square root of 2 * 2 times the square root of 39? How does he decide how to split the numbers, and then where to divide them up?\n\nWould it have affected the answer if he had done it differently?",
        "A": "No. No matter how you simplify a square root, if you do it correctly the answer will not change. Ex: sqrt 156 = sqrt 6 times 26 = sqrt 3 times 2 times 2 times 13 = sqrt 39 times 4 =2 root 39"
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "I don't quite understand how, at 4:36, be brought (-4 +/- 10) / 2 into -2 +/- 5.",
        "A": "Sal just divided both of the terms in the numerator by the denominator. -4 / 2 = -2 and +/-10 /2 = +/-5 so he was left with x = -2 +/- 5"
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "at 4:18\nhow did \u00e2\u0080\u00944 +- 100\nbecome \u00e2\u0080\u00944 +- 10?",
        "A": "the 100 was under a square root and then he took the square root of 100 to get 10"
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "About 13:20 in, he says \"now we can divide the numerator and denominator by 2\".\nI'm not sure why we do this step. Is there a rule that I'm missing?",
        "A": "To further simplify your answer, you can divide the numbers by two. That is the only way to make it completely simplified and able to be solved."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "Can someone else who has already learned imaginary numbers help me figure out what the solution(s) to the equation introduced around 6:20 is? I think it would be -1 +/- i 2\u00e2\u0088\u009a21. Is this correct?",
        "A": "Close! You forgot to divide the i 2\u00e2\u0088\u009a21 term by 6 as well. Remember (a + b)/c = a/c + b/c. So the answer is -1 +/- i\u00e2\u0088\u009a21/3 Nice One! Keep Studying!"
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "at 13:30 how does sal simplify the equation from-12 + or - 2 to -6 before the radical he seems to have skipped how he got to the -6 out of that mess",
        "A": "it is because he divided the whole problem by 2"
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "where did the 2 go when dividing the numerator and the denominator of 12+- 2 square root 39? on 13:23",
        "A": "When you divide the numerator by the denominator, the 2 cancels out, so you are left with square root 39."
    },
    {
        "video_name": "i7idZfS8t8w",
        "Q": "Starting at 13:23 when you divide the numerator and denominator by 2 you get -6 so when you re wrote it you got -6 plus/minus sqrt of 39... but what happened to the 2 in front of the sqrt of 39? please help me understand this part.",
        "A": "Sal is reducing the fraction. To reduce fractions, we cancel a common factor. Sal skipped a couple of steps. I ll write it out in more detail. We start with: [-12 +/- 2sqrt(39)] / -6 Factor out a 2 from the numerator: 2 [-6 +/- sqrt(39)} / -6 Cancel out the common fractor of 2 from numerator & denominator: [-6 +/- sqrt(39)} / -3 Hope this helps."
    },
    {
        "video_name": "JVDrlTdzxiI",
        "Q": "I still don't have the intuition of why pivot entries must be the first entry in a row. It could be a logical consequence of its definition but womehow I'm missing the point.\n\nFor instance, in the example Sal put forward at 10:00, why should the variable X1 (row 1, col.1) be the pivot entry and not X2 (row 1, col. 1)?",
        "A": "If you get a 1 you are saying the coefficient on the variable is 1. So you know x1 = 3 instead of, say, 3*x1=9. For square matrices you get something like: 1 2 2 | 20 0 1 2 | 16 0 0 1 | 7 Then you know instantly that x3 = 7 and you can do back substitution easily to find the other variables. It also makes things easier for more advanced concepts in linear algebra."
    },
    {
        "video_name": "JVDrlTdzxiI",
        "Q": "Sal mentioned at 3:27 that, \"pivot entries which are always going to have the coefficient 1, or the entry 1, it should be the only non-zero term in my row.\"\nIs it not the non-zero term in column?",
        "A": "It should be column, but for cases where there is a unique solution, it will be the only non-zero term in the row as well. Most practice problems have a unique solution, so it s a natural slip of the tongue."
    },
    {
        "video_name": "JVDrlTdzxiI",
        "Q": "At 10:14 Sal shows that if you have the same amount of pivots as columns you have a unique solution. I was wondering if on the last pivot (last row), could d =0 such that 1 =0 on the last row?",
        "A": "You may have any of those numbers on the right side of the partition (that vertical line in the augmented matrix) as zero. This simply means that your unknowns you are solving for are zero (eg. x=0, y=0, z=0). This is perfectly permissible."
    },
    {
        "video_name": "JVDrlTdzxiI",
        "Q": "at around 3:26 Sal says \"The pivot entry should be my only non-zero entry in my row\". Does he mean column instead of row? If so, this is a repeating mistake throughout this video series",
        "A": "Yes, Sal meant column."
    },
    {
        "video_name": "xPb6HLM3xEQ",
        "Q": "At 7:50, Sal drew the x and y axes slanted. I don't understand why he did this.",
        "A": "He s drawing a three-dimensional solid. If you were to draw the x- and y-axes on the ground, that s what the function would look like from a location off to the side. As for the shape of it, each cross-section of the solid is the same shape as the functions he s been talking about for the last couple of videos. The basic shape is outlined at 9:10 or so, but Sal never (not even in the next few videos) draws the actual solid in its entirety."
    },
    {
        "video_name": "T0IOrRETWhI",
        "Q": "At 1:58, why would the square root of 74 be 74?",
        "A": "Hi there. We are not just taking the square root here, but we are squaring the square root. Squaring and taking the square root are inverse operations in a sense. They cancel each other out. It s like doing this: 1 + 2 -2, leaving you with 1. So the (sqrt74)^2 is just 74."
    },
    {
        "video_name": "T0IOrRETWhI",
        "Q": "At 1:29 he said \"longerst\"",
        "A": "He probably had an accent or a lisp."
    },
    {
        "video_name": "h0SNEO25vIw",
        "Q": "At about 5:00 in, could the answer also be simplified to (7radical(3))/(34)?",
        "A": "No, because the fraction on the left is 15/34, not 15sqrt(3)/34. Just like 15 - 8x does not equal 7x; it s equal to 15 - 8x, and you can t simplify it further."
    },
    {
        "video_name": "h0SNEO25vIw",
        "Q": "At 3:10, when finding the cosine of 60 degrees, and at 3:54. when finding the sine of 60 degrees, why do you use the example triangle that you drew? Why not just use the triangle given with the problem?",
        "A": "The triangle given in the problem doesn t have a 60 degree angle."
    },
    {
        "video_name": "h0SNEO25vIw",
        "Q": "At 4:45 how did you determine that 8/17 multiplied by square root of 3 equaled 4 square root of 3 over 17?",
        "A": "In multiplication you can divide the factors with each other. I can t really show you since I m limited by a keyboard, but I ll try: 8/17* sqrt 3/2 <=> 8/2*sqrt 3/17 and 8/2 is 4, which he multiplied with sqrt 3/17. He s not doing anything fancy, just writing a fraction directly in simplest form. Excuse my bad English, I m doing my best to explain."
    },
    {
        "video_name": "h0SNEO25vIw",
        "Q": "5:05 should it be (last step) 15/34-4sqrt3/34, not 17? tell me if I am wrong. thanks",
        "A": "hi danish12--i think sal sort of simplified the previous step in his head, cancelling out a 2. so: 8/17 * sqrt3/2 became: 4/17 * sqrt3/1. so final answer became: 5/34 - 4sqrt3/17 hope that helps."
    },
    {
        "video_name": "6Ozz3J-LRrY",
        "Q": "At 4:34, Sal gets volume by multiplying the curcumference by its width - dx. In the previous section he duscussed that Volume = pi * height * ( (outer radius)^2 - (inner radius)^2 ). Now he simply replaced the last multiple with dx with no explanation. Is that because dx is infinitely small and should not be accounted for? Or is that a mistake?",
        "A": "That is the washer method; this is the shell method. They re different."
    },
    {
        "video_name": "8Yl_u_Otcjg",
        "Q": "At 3:45 , Why will this arcsin gives value between -90 to + 90 ?\nAnd how do we know that cos(theta) between this range will always be positive as said by sal at 3:54 ?",
        "A": "As Sal said, arcsin is usually defined to give values between -pi/2 and pi/2 radians. Cosine is positive in the first and fourth quadrants, because it s defined as the x-coordinate of the corresponding point on an unit circle."
    },
    {
        "video_name": "8Yl_u_Otcjg",
        "Q": "at 1:23\nHow does x=2sin(theta)? I dont understand how Sal got that value",
        "A": "From the form a^2 - x^2 he said x could be written as x=asin(theta) Then you got that 8 - 2x^2 could be rewritten as 2(2^2 - x^2). There you have your a on the thing you got now inside the parenthesis, and it is equal to 2. So if you replace it on what you had as x=asin(theta) you get x=2sin(theta)"
    },
    {
        "video_name": "8Yl_u_Otcjg",
        "Q": "Could someone explain the step at 4:21 which jumps from \u00cf\u0080/2 \u00e2\u0088\u00ab d\u00c3\u00b8 to \u00cf\u0080/2\u00e2\u0088\u00ab\u00c3\u00b8 please?",
        "A": "The integral is the antiderivative. The derivative of x is 1, so the antiderivative of 1 is x. (Or in this case, \u00e2\u0088\u00ab d\u00c3\u00b8 = \u00c3\u00b8.) Note that the second expression in the video no longer has the integral sign."
    },
    {
        "video_name": "8Yl_u_Otcjg",
        "Q": "At 00:17, why is it possible to assume that X = asin(theta)?\nIs that a trigonometric identity? Where does it come from?\nThank you in advance.",
        "A": "No, it s not a trig identity. This method is similar to the u-substitution method you learned earlier in Integral Calculus. The important thing to keep in mind with substitution is you can try any substitution you like. You re creating a new variable so x = a sin t because you said so. Of course, some substitutions will lead you nowhere so you need to be careful. But this substitution has been known to work and so that s why Sal is using it in this video."
    },
    {
        "video_name": "JaScdH47PYg",
        "Q": "At 1:58 why are we multiplying 4?",
        "A": "Sal has an equation: 0.25 * number = 150 Remember, 0.25 is the same as 1/4 The 0.25 (or 1/4) needs to be moved to the other side of the equation. There are different ways to do this and accomplish the same result. Sal is multiplying by 4 because 1/4 * 4/1 = 1 -- You could also divide both sides by 0.25 or by 1/4 Hope this helps."
    },
    {
        "video_name": "JaScdH47PYg",
        "Q": "hey at 2:20 sal multiplied by 4 but why?i do also have the same question as jack but not satisfied with the answer",
        "A": "Hello..I understand that 4 is equivlent to .25% but how do arrive at 4? For this question it was easy, but lets say you re asked 175 is 37% of what number? how would you work that out?"
    },
    {
        "video_name": "JaScdH47PYg",
        "Q": "At 1:47, how did he get 4?",
        "A": "0.25= 25/100= 5/20 = 1/4 so instead of writing 0.25 I could write 1/4, that is now the equation going to be: 1/4.NUMBER= 150 So to get the NUMBER we need to multiply by 4 both sides or you could say to simplify we have to multiply by 4 both sides"
    },
    {
        "video_name": "JaScdH47PYg",
        "Q": "At 2:01, where did you get the 4 from?",
        "A": "to make computing the answer easier, sal wants to turn that .25 ( or 25%) into a whole number. he knows that 4 x .25 will = 1.00. so he decided to use 4 in this situation. (think of how four quarters = one dollar... same as 4 x .25 = 1.00)"
    },
    {
        "video_name": "qSE1Z5F_vrU",
        "Q": "In 0:14 it says 'till whats the definition of 'till?",
        "A": "Till is until, so it s 6 minutes TILL 4:00 etc"
    },
    {
        "video_name": "qSE1Z5F_vrU",
        "Q": "Varsan\nThink all times as american time\n\n\nI was flying from London to Sydney with no stop-overs. I left London at 2:20 p.m. and arrived in Sydney at 6:42.\n\n\nWhen did I arrive",
        "A": "So, this is like elapsed time. So how long does it take you to get there? Well add 4 hours and 22 minutes so it took you 4:22 for you to get there because 2:20 + 4:22 = 6:42."
    },
    {
        "video_name": "pi3WWQ0q6Lc",
        "Q": "At 4:30 is it possible that in an exam they can tell you to switch -21/20 to a proper fraction and how.",
        "A": "-21/20 is improper. It can become a mixed number (divide). It = -1 1/20."
    },
    {
        "video_name": "pi3WWQ0q6Lc",
        "Q": "At 3:13, is negative times negative always positive?",
        "A": "Yes because only negative times positive is negative."
    },
    {
        "video_name": "pi3WWQ0q6Lc",
        "Q": "At 1:16, why is ONLY the numerator negative?",
        "A": "Because if both the numerator and the denominator had negative signs in front of them, the negative signs would cancel out"
    },
    {
        "video_name": "pi3WWQ0q6Lc",
        "Q": "so at 2:54 the answer to the problem is 1/9 if so, does that mean if there both negative it would be the same as both are positive?",
        "A": "Yes. Positive number can be get by multiplying both positive or both negative numbers."
    },
    {
        "video_name": "pi3WWQ0q6Lc",
        "Q": "At 4:31, Sal. says that we cannot simplify 21/20 anymore. Can't we turn it into a mixed number, say 1 1/20?",
        "A": "Good question! Yes, we can turn into a mixed number, but it is simplified as a fraction. numerator and denominator do not have any common factors other than 1."
    },
    {
        "video_name": "pi3WWQ0q6Lc",
        "Q": "someone explain this to me pls i couldnt hear anything he said in this video because my computer is dumb so it be a help if anyone told me from 0:01 to 4:31",
        "A": "Look under options and turn on interactive transcript. You can pause each step of the way and read what he is saying."
    },
    {
        "video_name": "pi3WWQ0q6Lc",
        "Q": "At 0:49, why is multiplying a negative and a positive a negative? Is it always?",
        "A": "One way to think of it is that when multiplying by a negative you are repeatedly subtracting your number (or more precisely, you are repeatedly adding a negative number but of course we know those are the same thing). So if you have -3 * 4 that is like -3 - 3 - 3 - 3 = -12. So yes, it will always be negative."
    },
    {
        "video_name": "DPuK6ZgBGmE",
        "Q": "What does MAD mean at 1:41",
        "A": "the mean absolute deviation works like this, you find the mean then you take any number higher or lower than the mean and see how far away it is than you divide by how many numbers there are. example: 5,6,3,4,7 7+3+4+6+5=25 divided by 5 equals 5 not enough room look at comments"
    },
    {
        "video_name": "DPuK6ZgBGmE",
        "Q": "At 3:03 how would Tara deviate from the mean?She has less.",
        "A": "The deviation is the difference, positive or negative, between the data point and the mean. That s why, when you do see the formula, it has the absolute value signs. |1 - 4| = 3"
    },
    {
        "video_name": "DPuK6ZgBGmE",
        "Q": "4:41 Why do you have to do absolute value? Can you just flip it? For example: 3-4 to 4-3.",
        "A": "It s just the same thing."
    },
    {
        "video_name": "PPOIlLhsT6s",
        "Q": "At 4:21 Sal refutes the second option that: \"The 2nd column represents the price of toothpaste in the two cities.\" He says that that the second column is refering to the objects coming out of NYC.\nAs Sal mentioned in the previous video, and in this video, the horizontal row is called a 'row' and a vertical row is called a 'column'. If so, why wouldn't the 2nd column be referring to the second row going vertically, which actually would be a representation of toothpaste?",
        "A": "2nd column does mean 2nd vertical row . If you look carefully, the 2nd column is represents both toilet paper and toothpaste in NYC. The 2nd option says that the 2nd column represents the price of toothpaste in two cities which is false. The 2nd column represents NYC not toothpaste. The correct statement would be the 2nd row represents the price of toothpaste in both cities . Hope that helps!"
    },
    {
        "video_name": "7_QPAdHILzw",
        "Q": "at 1:46 Sal said \"We now have 1, 2, 3, 4, 5, 6, 7, 8, 9. We now have 9 ones\".",
        "A": "he said 1,2,3....for counting them.. there were 9 objects hence 9 and as it is a one digit number(ones place value),it has 9 ones.."
    },
    {
        "video_name": "uFZvWYPfOmw",
        "Q": "At 3:00 he says that C only has two real roots so it's excluded. But how can we say that for sure? Couldn't the graph curve back down off to the left or the right side somewhere that we don't see?",
        "A": "As always, there are some unspoken assumptions. In this type of problem, the implication is that all of the interesting features of the graph are visible. In practice, if you saw a curve of this nature and did not know the formula, you could say that it is at least locally approximated by a curve with only two zeroes."
    },
    {
        "video_name": "uFZvWYPfOmw",
        "Q": "At 2:11, Sal says that there are 3 roots. How was this determined? Is it because that is how many different x terms there are in the equation?",
        "A": "The degree of the polynomial (the highest exponent on the variable) tells you how many roots. You can sometimes get duplicates that reduce the number."
    },
    {
        "video_name": "rcLw4BlxaRs",
        "Q": "at 1:47 why do you need a ruler and compass?",
        "A": "so that you can make the diagram more neat and clear."
    },
    {
        "video_name": "rcLw4BlxaRs",
        "Q": "At 0:58, the guy said just set it right ontop of those six. But he said there were 8 3D circles. what does he mean?",
        "A": "in total their were 8 3D circles: the huge one, the big one, and the 6 small spheres. in the second diagram, he only used 2 of the small spheres to solve the problem."
    },
    {
        "video_name": "rcLw4BlxaRs",
        "Q": "What does tangent mean? (for example he said it in 0:42 and it's in the problem)",
        "A": "In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that just touches the curve at that point. - Wikipedia. I personally think tangent is just a line which touches a circle or ellipse at just one point in geometry, while it is one of the trigonometry functions. In a right triangle, the tangent of an angle is the opposite side over the adjacent side."
    },
    {
        "video_name": "rcLw4BlxaRs",
        "Q": "Why are they using tangent? isn't that trig?\n\n(well, except at 4:02 when he draws the right triangle.)",
        "A": "Tangent doesn t necessarily only mean sinx/cosx. A tangent has uses in Geometry and Calculus where it s defined as a line that intersects a curve or other surface at exactly one point. A secant is a line that touches a curve or surface at exactly two points. I d look up on google images some pictures of a tangent and secant line."
    },
    {
        "video_name": "rcLw4BlxaRs",
        "Q": "At 3:20, when he is making the second cross section where he cuts the first diagram in half, is he taking the outer two most small spheres? If so when he is drawing the 8th circle how does he know to use those two circles as the external points, and how big the 8th circle had to be. I understand the math I'm just having trouble visualizing it. I appreciate any help and feel free to let me know if I've just failed to understand the visuals.",
        "A": "the 8th sphere is touching the two smaller circles external points and the huge circle s inner points because it s tangent to all of them. All the 6th circles that line up on the hexagon are all touching the huge circle, and these circles are lined up on the half way line in the sphere. That s how he was able to take half the sphere with the two small circles on the sides and the bigger circle on the top. Sorry if made it too confusing.... bad at explaining."
    },
    {
        "video_name": "MZl6Mna0leQ",
        "Q": "At about 1:20, he says that x^2 is the greatest common x degree in all of them, but one of the x degrees is x^3. How is x^3 a degree of x^2?",
        "A": "Because x^2 times x equals x^2 :)"
    },
    {
        "video_name": "MZl6Mna0leQ",
        "Q": "Why did he put x^2 on 1:26",
        "A": "He factored out the value 2x^2 from the polynomial."
    },
    {
        "video_name": "MZl6Mna0leQ",
        "Q": "How does x squared go into x cubed? 1:20",
        "A": "x^3 is equal to x \u00e2\u0080\u00a2 x \u00e2\u0080\u00a2 x x^2 is equal to x \u00e2\u0080\u00a2 x So we can say that x^3 is equal to x^2 \u00e2\u0080\u00a2 x. Thus, x squared can go into x cubed. Or more mathematically said, x squared is a factor of x cubed."
    },
    {
        "video_name": "MZl6Mna0leQ",
        "Q": "Why is Sal multiplying and dividing at 2:53?",
        "A": "so that he can simplify."
    },
    {
        "video_name": "MZl6Mna0leQ",
        "Q": "At 3:35 why is x^3 \u00c3\u00b7 x^2 just x ?",
        "A": "This can be demonstrated by expanding the powers x^3/x^2 = x*x*x/x*x now pair one x on top with one x on bottom - note that the last x on top does not have a pair so it is by itself. x*x*x/x*x = (x/x)*(x/x)*x and use the fact that any non-zero divided by itself equals one (x/x)*(x/x)*x = 1*1*x = x and you should note that this simplification is only true if x does not equal zero"
    },
    {
        "video_name": "MZl6Mna0leQ",
        "Q": "At 1:36 the y just dissappears?",
        "A": "Sal just stopped writing for a little bit. The y didn t disappear. It comes back as he continues the problem."
    },
    {
        "video_name": "MZl6Mna0leQ",
        "Q": "The only thing I'm having trouble understanding is at 1:23. how is x^2 the common factor when there is an x^3? X^4 is a factor of x^2, but 2 is not a factor of 3, so I'm confused. Are there different rules for exponents? Thanks.",
        "A": "You re trying to divide the exponents on the X s. x^3 is divisible by x^2. You divide the x s, not the exponents on the x s. X^3 / x^2 = x^(3-2) = x Hope this helps."
    },
    {
        "video_name": "MZl6Mna0leQ",
        "Q": "I get confused at 2:06 . Im a younger student taking algebra 1 and this chapter is one of my weaknesses; I was wondering if anyone could make sense of what Sal is saying at 2:06 ? Thank you.",
        "A": "At 2:06 Sal is taking the first term (4x^4 y) and dividing it by Greatest Common Factor (2x^2). Does that make sense?"
    },
    {
        "video_name": "MZl6Mna0leQ",
        "Q": "How can x^4, x^3, and x^2 all be divisible by x^2? (1:19)",
        "A": "To be divisible by x^2, the division should not create a fraction or remainder. Let s divide each item by x^2 and see what happens. x^4 / x^2 = x^(4-2) = x^2 We have no fraction. So, x^4 is divisible by x^2 x^3 / x^2 = x(3-2) = x^1 = x. We have no fraction. So, x^3 is divisible by x^2 x^2 / x^2 = 1. Any number divided by itself = 1. Again, no fraction. So x^2 is divisible by x^2. Hope this helps."
    },
    {
        "video_name": "MZl6Mna0leQ",
        "Q": "at 3:22, why does he include the y and where does he get one",
        "A": "at 3:22, he gets the y from the polynomial he s working with. He is just working on the polynomial to change it."
    },
    {
        "video_name": "MZl6Mna0leQ",
        "Q": "At 2:02, how do you multiply 2x squared times 4x to the fourth y over 2x squared?",
        "A": "You multiply them in the same way you multiply numeric fractions. Multiply numerator to numerator; and denominator to denominator. Make sure you reduce your final answer (cancel out any common factors)."
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "\"14:15\", {V1,V2,...Vn} do these components same as S and T sets he discussed earlier? And together they for Basis for U subspace?",
        "A": "The vectors V1 until Vn are indeed the basis for the U subspace. S and T are EXAMPLES of bases (that s the plural form of basis) for the U subspace. For example, we have the basis T which consists of: V1 = (1,0) and V2= (0,1) (Note that n=2, so V2 is your Vn). Those two vectors span the subspace R2 and are linearly independent, therefore, they are (one of many examples of) a basis for U."
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "Sal said in the last video at 8:40, the zero vector is a valid subspace, and in this video at 0:30, he said that a basis must be linearly independent. However, any set containing the zero vector is linearly dependent because the constant multiplied by the zero vector can be non-zero, and the linear combination results in the zero vector. So, what is the basis of the zero vector if not the zero vector itself?",
        "A": "There would be no basis of the zero vector since it is Lin. Dep. and therefore doesn t qualify. Same for scalar multiples or any set containing the zero vector, scalar multiples, or sets with free variables ( basically anything Lin dependent)."
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "At 17:00 when Sal says (C1-D1) and (C2-D2) must each be equal to 0. why is that ? why is that the defenition of linearly independant?",
        "A": "Recall that a set of n vectors v1, v2, \u00e2\u0080\u00a6, vn is said to be linearly independent if and only if the following is true: if k1, k2, ..., kn are scalars such that k1 v1 + k2 v2 + \u00e2\u0080\u00a6 + kn vn = 0, then k1 = k2 = \u00e2\u0080\u00a6 = kn = 0. Since 0 = (c1 - d1) v1 + (c2 - d2) v2 + \u00e2\u0080\u00a6 + (cn - dn) vn, it follows that all the scalars ci - di must be 0, i.e., c1 - d1 = 0, c2 - d2 = 0, \u00e2\u0080\u00a6, cn - dn = 0."
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "At the very end of the video at 18:30, he talks about how adding a third vector will make the set linearly dependent and no longer a basis. Is this always true? What I mean is, when you have more than n-vectors in a subspace Rn, will it always be linearly dependent and thus not a basis?",
        "A": "Yes, for a space in \u00e2\u0084\u009c\u00e2\u0081\u00bf, a base must have exactly n elements, any less and you won t be able to map the whole space, any more and they won t be able to be independent."
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "I'm probably missing something here, but how come, at 03:54, T is clearly not a linearly independent set?",
        "A": "That s an important thing to ask about if you don t see it yourself. :) T is not a linearly independent set because v_1 + v_2 - v_s = 0 by the definition of v_s. Since we have a non-trivial linear combination of the set vectors equaling the zero vector, T is not l.i.d."
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "At 13:09 why can't we just put an arrow like we would put over any other vector over our unit vector? It is a vector too. I know unit vectors are very useful but are they really THAT special?",
        "A": "You could just put an arrow. There are not a set of rules about notation, it is just convention. As long as you are consistent it is fine. The hat is just a common way, and has the useful benefit of making it clear that it is a unit vector."
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "At 18:00 you show that you can't add another vector to the basis for R2 and still have it be linearly independent. Is it ever possible to have a basis of any subspace with more vectors than dimensions in the vector? E.g., a basis consisting of four 3-element vectors? Or is it by definition impossible for them to be linearly independent?",
        "A": "What about the opposite, spanning more dimensions than the number of vectors?"
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "At about 4:34, instead of saying {v1,v2...vn} is the basis for V, can we also say {v2,v3...vn,vs} is the basis for V? because v1 can be obtained by vs-v2",
        "A": "Your basis is the minimum set of vectors that spans the subspace. So if you repeat one of the vectors (as vs is v1-v2, thus repeating v1 and v2), there is an excess of vectors. It s like someone asking you what type of ingredients are needed to bake a cake and you say: Butter, egg, sugar, flour, milk vs Butter, egg, egg, egg, egg, sugar, sugar, flour, milk Basis is asking for the minimum amount of information to answer the question adequately."
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "at 4:02, I thought that a subspace could not have any vectors that were linear combinations of other vectors, so how does Span (T)=V?",
        "A": "I thought that a subspace could not have any vectors that were linear combinations of other vectors nope"
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "Since the c's have to equal the d's around 17:00, does that mean that there is only one combination?",
        "A": "Yep"
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "at minute 12:58, Sal says we've shown S is a Basis for R^2. but Sal said earlier, \"Basis is the minimum set of vectors that spans the Subspace.\" we used the definition there, S is a Basis for V. is R^2 also a Subspace? i thought a Linear Subspace was a subset of R^2.",
        "A": "R2 is a subspace of R2. Every set is its own subset. When the subset has fewer elements than the the set it is called a proper subset. When the subset is equal to the set it is called an improper subset. If S is a basis for R2 and it is also a basis for V then V is equal to R2 and is thus an improper subset of R2"
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "From 19:56, you gave an example of (1,1) to span in R^2?\n\nMy understanding was that you need two linearly independent vectors that live in R^2 to span R^2. So (1,1) can be a subspace of R^2 but it can't span R^2..... is that right??\n\nThank you! :)",
        "A": "Yes, you are correct that the vector (1,1) alone does not span all of R^2, since two linearly independent vectors are needed to span all of R^2. The span of the vector (1,1) alone would be a subspace of R^2, namely the line y = x. Have a blessed, wonderful day!"
    },
    {
        "video_name": "zntNi3-ybfQ",
        "Q": "2:03 why does not being able to represent one of the vectors by a combination of the others mean it is linearly independent?",
        "A": "What do you mean by that? How is that a reason?"
    },
    {
        "video_name": "ja4WRZ4DrAw",
        "Q": "In the first graph that Sal made, y=2^x+3, at 2:20. Why is the y intercept 5? I thought that if x is 0 then 2 would be to the power of 3 which is 8, right? or is it a error?",
        "A": "As drawn by Sal, the y-intercept is 6. However, you are correct that it should be 8. The difference comes because Sal s drawing is only an approximate sketch. Further on in the video at about 4:11, there is a pop-up note that also mentions another inaccuracy of Sal s sketch."
    },
    {
        "video_name": "eR8vEdJTvd0",
        "Q": "At 10:35, is Professor Sal defined a pivot column in a matrix to be a column that has only 1 non-zero number. If so, would the second column of the matrix A be considered to be a pivot column?",
        "A": "In reduced row echelon form, a pivot column is any column containing the leading 1 of a row (which will be the only non-zero value in the column). Here are the 3 criteria for RREF|: Rows with more leading 0 s must be below rows with fewer The first non-zero value in any row must be 1 The leading 1 of each row must be the only non-zero value in its column Any column described by the 3rd criteria is a pivot column."
    },
    {
        "video_name": "eR8vEdJTvd0",
        "Q": "At 17:32, Sal says that for T to be ONTO, rref(A) needs to have a pivot in each column, but couldn't a matrix just have the same number of pivots as the dimension of the codomain? At 21:30, he says the matrix must have m pivot entries, so couldn't a matrix A have more than m columns as long as it has m pivot entries?",
        "A": "First, that should be a mistake, rref(A) needs to have a pivot in every row for the transformation T to be onto. When rref(A) has a pivot in every column, the transformation T is one-to-one. Secondly, yes, it just needs at least as many pivots as the dimension of the codomain. Sometimes, this is impossible. It is impossible on for example a 4x3 matrix. In general, if m>n, it is impossible for an mxn matrix. In order to be onto, rref(A) needs to be an mxn matrix where m<=n, and rref(A) has m pivots."
    },
    {
        "video_name": "eR8vEdJTvd0",
        "Q": "Hi Sal, around 11:22 you say that a free variable leads to many solution. Isnt it the case that with a free variable you have infinite solutions, because you can use any number of the set of R for the free variable?",
        "A": "Yes, whenever you have a free variable, there are infinitely many solutions."
    },
    {
        "video_name": "DY9Q3qNmZnw",
        "Q": "At 3:27 Sal says that you multiply -0.99 to get positive 0.99 but when I multiplied them together I got .980. I don't know if this matters or if I'm just being overreactive. Can someone please explain it to me?",
        "A": "Notice that the terms have increasing exponents. What he actually says is that you get the next term in the series which in this case is positive 0.99 squared which is in fact 0.9801 as you found."
    },
    {
        "video_name": "DY9Q3qNmZnw",
        "Q": "at 4:54, for the third example, why do we assume the a sub 1 is the first term? How do we know that this is in fact the first term and not the second term following a sub 0.",
        "A": "We know that a-sub-1 is the first term in the series that Sal gives us because that s the way he gives it: he s the one defining the series, so he gets to decide, and we are obliged to follow along (assuming we want to follow!-). He could have called it a-sub-0, but there would be no reason to do that in this case. In other cases, a-sub-0 is a good choice for the first term."
    },
    {
        "video_name": "DY9Q3qNmZnw",
        "Q": "At 1:34, what happened to the first 1. Did Khan multiply it into the expression? It seems to have come back at 1:49",
        "A": "He indeed multiplied the expression (1 - (10/11)^50 by 1 at 1:34. However, it does not come back at 1:49; there, he multiplied the whole expression by 11 to get rid of the 1/11 in the denominator."
    },
    {
        "video_name": "DY9Q3qNmZnw",
        "Q": "At 1:54 I did not know how to do the arithmetic part of 11 ( 1 - ( 10/11 ) ^50 ). I got a very huge number because it is to the power of fifty!",
        "A": "Actually it will not be a very large number. (10/11) is less than 1. When you take a number times a number smaller than one, you will get a smaller number. (10/11)^50 is approximately 0.008518551 The rest should be easy, giving you an answer of approximately 10.90629594"
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "At 4:10 when he is talking about multiplying the reciprocal of 'c', why is he multiplying b-a/ab * 1/c and not b-a/ab * c/1? Or just multiplying b-a/ab * c? Wouldn't the reciprocal of b-a/ab/c just be c?",
        "A": "The fraction at first threw me off, but when you write it out, (b-a/ab) \u00c3\u00b7 c, you see that it is not 1/c, but just c. Looking at it this way, you can see that in fact, the reciprocal is 1/c. I hope this helps!"
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "3:22 why ar we subtracting the factions first is that not ignoring the order of opperations (PEMDAS)?",
        "A": "Sal followed the Order of Operations (PEMDAS) rules. Because the subtraction of the fractions is in the numerator of a fraction, the division bar in the overall fraction acts as a grouping symbol (parentheses). You have to simplify the numerator before you do anything else in that fraction. Hope this helps."
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "in 4:33 why is the reciprocal of c \" 1/c\" ? isn't it \"c/1\" ?",
        "A": "No, and here s why: If c were any whole number, then its fractional equivalent would be said number over 1. For example, if c were 2, then its fractional equivalent would be 2/1, since 2 divided by 1 is 2. Therefore, the reciprocal is the fraction reversed, which is 1 OVER said number, or 1/2. That s why the reciprocal of c is 1/c, because it s the opposite of c/1, aka c divided by 1, aka c."
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "i do not get what he did at 5:00, how did he even multiply it out?",
        "A": "Just before the 5:00 mark, Sal has 3 fractions being multiplied (b-a)/ab * 1/c * d/1 To multiply fractions, you multiply numerators, then denominators. Numerators: (b-a) * 1 * d = d(b-a) Now, distribute the d: d(b-a) = db - ad Denominators: ab * c * 1 = abc Put the fraction back together: (db-ad) / abc Hope this helps."
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "At 0:47 does Sal write the wrong color? It was the same color as 'a' instead of the previous magenta 'b' right? or is it just me?",
        "A": "Hey Stone Sentinel, Nice Catch! He did use the wrong color. I am pretty sure it did not confuse the concept or anything up though. . . - JK"
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "at 4:50 couldn't the denominator ABC be C(A)(B)?",
        "A": "It could. It is just nicer to list them in alphabetical order."
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "At 3:41 1/a - 1/b then becomes b/ba - a/ba why did the a and b get reversed? shouldnt it be a/ba - b/ba still? sorry, this video is the first on the site that has confused me. thanks",
        "A": "We had: (1/\u00f0\u009d\u0091\u008e) - (1/\u00f0\u009d\u0091\u008f) Multiplying the first fraction by \u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008f: (\u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) - (1/\u00f0\u009d\u0091\u008f) Multiplying the second fraction by \u00f0\u009d\u0091\u008e/\u00f0\u009d\u0091\u008e: (\u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) - (\u00f0\u009d\u0091\u008e/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) Therefore: (1/\u00f0\u009d\u0091\u008e) - (1/\u00f0\u009d\u0091\u008f) = (\u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) - (\u00f0\u009d\u0091\u008e/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) Comment if you have questions."
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "3:25 mins in... \"we can multiply 1 over a times b over b\" this is were I got lost, unfortunately I do not know how to tackle that section of the problem. Any explanation or direction to an explanation would be helpful. Thanks",
        "A": "To add/subtract fractions you need a common denominator. For example: 1/2 + 1/3 -- LCD = 6 -- To convert any fraction to the LCD, you multiply both the numerator and denominator by a common value. This creates an equivalent fraction 1/2 (3/3) = 3/6 1/3 (2/2) = 2/6 -- Then add: 3/6 + 2/6 = 5/6 When Sal says we can multiply 1 over a times b over b , he is using the same process. The LCD = ab . Convert to the LCD by multiplying 1/a (b/b) = b/(ab) Hope this helps."
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "I don't really understand how Sal got a-b/ab at around 4:00, could anyone explain it to me?",
        "A": "Sal skipped a few steps in his work on the upper right. We need to subtract: (1/\u00f0\u009d\u0091\u008e) - (1/\u00f0\u009d\u0091\u008f) 1) Common denominator = ab 2) Next convert each fraction to the common denominator -- Multiplying the first fraction by \u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008f: (1/\u00f0\u009d\u0091\u008e)*(\u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008f) - (1/\u00f0\u009d\u0091\u008f) = (\u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) - (1/\u00f0\u009d\u0091\u008f) -- Multiplying the second fraction by \u00f0\u009d\u0091\u008e/\u00f0\u009d\u0091\u008e: (\u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) - (1/\u00f0\u009d\u0091\u008f)*(\u00f0\u009d\u0091\u008e/\u00f0\u009d\u0091\u008e) = (\u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) - (\u00f0\u009d\u0091\u008e/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) 3) Subtract the fractions: (\u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) - (\u00f0\u009d\u0091\u008e/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) = (\u00f0\u009d\u0091\u008f-\u00f0\u009d\u0091\u008e) / (\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) Therefore: (1/\u00f0\u009d\u0091\u008e) - (1/\u00f0\u009d\u0091\u008f) = (\u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) - (\u00f0\u009d\u0091\u008e/\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f) Hope this helps."
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "I really do not understand how Sal got the 1/C starting at 4:04. Why is it 1/C and not just C? I thought that was the reciprocal of fractions.",
        "A": "In his example he s got C underneath everything right? That (1/a - 1/b) is all divided by C. That is the same as multiplying (1/a-1/b) by the reciprocal of C which is 1/C."
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "1:37\nd(eep) was purple\nwas this what you were thinking when you chuckled sal?\n:)",
        "A": "i didnt see it it would probably help if I watched the video i guess.."
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "At around 3:22, Sal did the subtraction first. With Pemdas, you do the division first. You could of wrote the equation as 1/a-1/b divided by c. Doesn't this mean that you do 1/b divided by c first instead of 1/a=1/b?",
        "A": "It s because that bit of subtraction is actually in parentheses. When an expression is all over another number then it really means that you should resolve the top before dealing with the rest. So you could also write that part as (1/a - 1/b)/c"
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "At 3:30, Sal made 1/a into b/ba, but he \"didn't change it's value.\" How is that?",
        "A": "Sal took the LCD (least common denominator) of the fraction, just like turning a 1/2 into 2/4 . The values didn t change."
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "At 4:10, why does Sal multiply by the reciprocal of c instead of just dividing by c? If you divide by c, then the answer is (cdb-cda)/ab. Is this not the right format? Like does it need to be simplified more?",
        "A": "If you divide by c the answer is the same (db - da)/abc Remember, you can write 10\u00c3\u00b75 as 10/5 which is the same as (10/1)*(1/5). Multiplying by the reciprocal of a number is the same as dividing by the number. This is a very handy equivalence since often it is way easier to multiply by the reciprocal than to do a division. Which do you prefer, to multiply of divide?"
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "is'nt the reciprocal of c c/1 not 1/c in 4:12",
        "A": "No. c/1 is the same thing as c because when you divide a number by one you end up with that number. The reciprocal would be flipping the fraction. Since c is equal to c/1, the reciprocal is 1/c."
    },
    {
        "video_name": "_BFaxpf35sY",
        "Q": "at 4:19, wouldn't the reciprocal of C be 1/a - 1/b?\nbecause in the original problem C was on the bottom...",
        "A": "Your question isn t clear, however the reason why Sal multiplied by 1/c is because originally we re dividing by c, which is actually c/1. So if you re DIVIDING by c/1, the reciprocal you would use to MULTIPLY, is its opposite: 1/c."
    },
    {
        "video_name": "Qwet4cIpnCM",
        "Q": "7:46 I'm sure I'm mis-applying or not remembering an exponent rule correctly, but I don't see how (1/2h)^2 becomes h^2/4. Can someone enlighten me? Thanks!",
        "A": "Well, there s a 1/2 and a h both squared, which is (1/2)^2 and h^2 which is 1/(2^2) and h^2 which back together is (h^2)/4."
    },
    {
        "video_name": "Qwet4cIpnCM",
        "Q": "At 7:45 why does half(H) squared equal (H)sq over 4? please explain because i dont really get it",
        "A": "Sal was able to do this by: (1/2*h)sq 1/4*h sq h sq/4 Hope this helps"
    },
    {
        "video_name": "Qwet4cIpnCM",
        "Q": "from 7:50 to 8:08 how you multiplied.what is logic behind it ? i am not familiar with the steps you did, Plz explain, it could be helpful further",
        "A": "He is using the pythagorean theorem to find side a. That theorem say a^2 + b^2 = c^2, where c is the hypotenuse. In this example, that translates to A^2 + (h/2)^2 = h^2. So he squares the terms to get: a^2 + (h^2)/4 = h^2 and then solves for A."
    },
    {
        "video_name": "Qwet4cIpnCM",
        "Q": "O.k just to clarify at 7:50 where he says we subtract h^2 from both sides, does he in fact mean we subtract h^2/4 from both sides?\nThis is to my understanding what actually happens:\n\nH^2/4 + A^2 = H^2\n\nSubtract H^2/4 from both sides:\n(-H^2/4) H^2/4 + A^2 = H^2 (-H^2/4)\n\nLeft side cancels out, we are left with:\nA^2 = H^2 - H^2/4\n\nIs this correct?\n\nAlso I don't understand the next steps, but I realize there is already a question about this so I'm watching the factoring vids.",
        "A": "Yes, that is correct. Also in the next steps, h^2 - (h^2)/4... h^2 is a common factor so put that in front and put a 1 where it used to be -> h^2(1 - 1/4) 1-1/4 = 3/4 -> so you get h^2(3/4)"
    },
    {
        "video_name": "Qwet4cIpnCM",
        "Q": "at 8:02 I don't understand how sal went from h^2 - h^2/4 to h^2(1-1/4)",
        "A": "Thanks"
    },
    {
        "video_name": "84NkvxfMPtM",
        "Q": "When he says that if you orient your head in the direction of the normal vector and you were to walk along the path, the surface would be on your left (1:15-1:26), does that imply that you are outside of the surface?",
        "A": "Yes, it is like you are walking around a small hill in a counterclockwise direction, with the hill always to your left."
    },
    {
        "video_name": "R8YKuGJ0plI",
        "Q": "At 4:28 when Sal's talking about putting the same numerator above different denominators, and uses 4/9 and 4/7, he could've put it as 5/9 and 5/7 and it would've been the same answer?",
        "A": "Yes, it would have been the same answer since 5/7 > 5/9."
    },
    {
        "video_name": "YwNX9uWszQY",
        "Q": "1:57 in question not mentioned equal parts but answer only equal parts only selecting please reply",
        "A": "When you divide a number, it gets broken down into equal parts. He is only selecting the one because when you divide something into thirds, the 3 parts have to be equal to each other. If you were dividing 15 people into 3 groups, each group must have 5 people. All parts must be the same size."
    },
    {
        "video_name": "YwNX9uWszQY",
        "Q": "at 0:48, how is it that the first three options are right and the bottom one is wrong. Shouldn't they all be wrong?",
        "A": "You mean the yellow circles or the green squares? It s hard to tell from the timestamp. Yellow circles: The first 3 options are divided into 4 equal sections. They are right. Green squares: The first 3 are not right. The 2nd one is wrong."
    },
    {
        "video_name": "d8De3xcVmnw",
        "Q": "At 1:44, if the integers are 3, 5, 7, and 9, why do you only do [x + x + 2 + x + 4]\nand not [x + x + 2 + x + 4 + x + 6]? Why doesn't the [x + 6] come in?",
        "A": "I wonder about that too. He should have four integers, not three!"
    },
    {
        "video_name": "d8De3xcVmnw",
        "Q": "At 2:10, I would have added 6 to both sides of the equation rather than subtracted 6 because that would get 3x + 12 on the left-hand side. Then, you can divide both sides by twelve to get x + 4, which is the largest integer.\nAlso, I may have set up the original equation as x+(x-2)+(x-4)=231, because then x= the largest integer and it will be easier to answer the question. Do you agree with my methods? Thanks!",
        "A": "Yes. Very clever but even easier is divide by three to get middle one."
    },
    {
        "video_name": "d8De3xcVmnw",
        "Q": "hi um i don't really know why but my video froze at 3:50 in the video so if you dont mind can someone just sum it up for me thank you!! ( :>*",
        "A": "He just checks his answer by adding 75, 77, and 79. He gets 231 so that s correct. Nothing important that you missed really."
    },
    {
        "video_name": "EAa3J_nDkoI",
        "Q": "I'm really confused at 3:09 when he puts the undefined circle at the point (2,5) why 5?",
        "A": "f(x)=x+3, so f(2)=2+3, which is equal to 5. so the coordinates are (2,5). You just have to look at the function as if it didn t have a discontinuity."
    },
    {
        "video_name": "EAa3J_nDkoI",
        "Q": "At 1:00 when he says \"If it was defined, the limit would be what ever the function is\" does that mean the limit would have been 2 IF it was defined, or would it have been the solution? (If there had been a solution, which there was not since it was 0/0)",
        "A": "To be clear, a limit equals whatever the function is at the limiting value, IF and ONLY IF both of these conditions are met: 1. The function is defined at the limiting value. 2. The function is continuous at the limiting value. If either of these conditions is not met, then the limit is not what the the function is at that point and the limit might not even exist."
    },
    {
        "video_name": "EAa3J_nDkoI",
        "Q": "At 2:59.. How did Sal find the slope as 1?",
        "A": "For f(x) = x+3, the graph of the function is 1. It is in the form y = mx+b, where m is the slope of the function, and b is the y-intercept. If this is difficult for you, you might want to go back to algebra to review the slope of a line."
    },
    {
        "video_name": "EAa3J_nDkoI",
        "Q": "at 2:50 how come you are considering it a y intercept......and if so why in other sums you do not do the same",
        "A": "A y-intercept is whatever value the function takes on, if it exists, when x=0. I haven t recently watched this video, could you specify which sum you thought ought to be a y-intercept? Note, the value or values that the function takes on whenever y=0, if there is such a value, is known as the x-intercept, not y-intercept. You can have any number of x-intercept, but a single-valued function can have at most 1 y-intercept."
    },
    {
        "video_name": "EAa3J_nDkoI",
        "Q": "At 2:53, why is the slope equal to 1? Is that just a value that you picked, or is there a reason?",
        "A": "He did not just pick that value, no. If you simplify the expression, for all values except x=2 the result is y=x+3, right? So, we have the format y=mx+b, where m (slope) is 1, and b (y intercept) is 3."
    },
    {
        "video_name": "EAa3J_nDkoI",
        "Q": "For the equation at 0:15, couldn't I plug in 1.9999 and 2.0001 directly into the original equation? Why would I need to factor the equation first?",
        "A": "Plugging in these values gives you an approximation for the limit. If you want the exact value then you have to factor."
    },
    {
        "video_name": "EAa3J_nDkoI",
        "Q": "At 2:37, Sal says that y=f(x). I never thought of this before, but why is it so called?",
        "A": "Notice on the grid, the horizontal or x is the independent variable and the vertical or y is the dependent variable. Normally our function is in the form y = x. But our function in the video is in the form f(x) = x, where f(x) is the dependent variable that corresponding to y. Hope that helps."
    },
    {
        "video_name": "EAa3J_nDkoI",
        "Q": "At 2:58 how'd you get the slope is equal to one?",
        "A": "f(x)=x+3 AKA y=mx+b, m=slope; b=y intercept"
    },
    {
        "video_name": "EAa3J_nDkoI",
        "Q": "When you factored it at about 1:40 did you mean to say \"sum is negative 3\"?",
        "A": "it s a misstake, he must say sum is positive 1"
    },
    {
        "video_name": "EAa3J_nDkoI",
        "Q": "At 5:42 do you really have to test from the left and the right of 2 to see that it goes to 5? Could you just plug in 2 into x+3 since you were able to cross out the (x-2) in the numerator and the denominator and get 5?",
        "A": "I think that was just to show you that it actually does approaches 5 from left and right. It s not necessary."
    },
    {
        "video_name": "EAa3J_nDkoI",
        "Q": "he says at 0:51 that the function f(2)=0/0 is not defined, but since 0/0 has been proved to equal 2, wouldn't f(2) = 2?",
        "A": "Proof that 0/0 = 3: 0/0 = (0*3)/(0*1) = 3/1 = 3 Proof that 0/0 = i: 0/0 = (0*i)/(0*1) = i/1 = i 0/0 is still undefined. the error stems from =(10+10)(10-10)/10(10-10) =(10+10)/10 When you cancel out the (10-10) terms, it s the same thing as canceling zeroes, as I did above. You can t do that, because it leads to inconsistencies."
    },
    {
        "video_name": "CCysGY2cXUo",
        "Q": "At 3:40 I did not understand what he meant",
        "A": "He is saying that the marbles, when placed in , displaced some water, and when taken out, the water level dropped."
    },
    {
        "video_name": "CCysGY2cXUo",
        "Q": "At 2:36 how does the water drop to 5.9 cm?",
        "A": "When you put something in water, it will always move out of the way it s own weight in water. This is called displacement and was discovered by Archimedes."
    },
    {
        "video_name": "MQtsRYPx3v0",
        "Q": "At 1:43, Sal divides each term by 5. If he didn't do this, and instead just went on to factor it, would that be wrong? Because I thought you didn't have to simplify it first, and when I tried not simplifying I got 5(x-3)(x-1)",
        "A": "Your 1st step should be to deal with the common factor of 5. You can either factor it out. Or, you can do as Sal chose and divide the equation by 5. Both will create the same results."
    },
    {
        "video_name": "MQtsRYPx3v0",
        "Q": "at 3:41 where does he get the 2 from? How does he know what the x-value was?",
        "A": "The 2 is the axis of symmetry, which is the point between the two x intercepts and is basically a line of where the vertex is going to be on. The middle of 1 and 3 is 2. To find the axis of symmetry, you add the x intercepts together (1+3) and divide by 2, which is 2."
    },
    {
        "video_name": "MQtsRYPx3v0",
        "Q": "At 3:42 where did you get the two from?",
        "A": "well the Vertex is ALWAYS on the middle of the 2 other points so what Number is between 1 and 3? 2 so our X-coordinate is (2.__) Hope This Helps!"
    },
    {
        "video_name": "MQtsRYPx3v0",
        "Q": "Lol; At 2:35, you guys here him say -3+1 is negative 4.",
        "A": "This mistake has been found and corrected already, if you look at the bottom right corner at this point, it has the correction box that say what Sal meant to say."
    },
    {
        "video_name": "MQtsRYPx3v0",
        "Q": "At 1:50, Sal divided the whole equation by five and got x^2-4x+3=0.\n\nIs it better to factor it out and get 5(x^2-4x+3)=0?\n\nThank you.",
        "A": "This is an equation not an expression. When you factor out a five, you can then divide by 5. Since 0/5 = 0, both equations are equivalent (in terms of their solutions). Having the extra 5 is unnecessary in final factored form, and it s a lot easier to deal with a simplified equation."
    },
    {
        "video_name": "MQtsRYPx3v0",
        "Q": "At 3:32 what do you do if you have an uneven space between the two intercepts, like -1 and 2?",
        "A": "You want the midpoint, in another word the average of the two points. (-1 +2)/2= 0.5"
    },
    {
        "video_name": "MQtsRYPx3v0",
        "Q": "At 3:40, how do you know that x is 2?",
        "A": "The axis of symmetry is in between 1 and 3 which makes x=2 Hope this helps!"
    },
    {
        "video_name": "MQtsRYPx3v0",
        "Q": "at 1:37, what if all of your terms aren't divisible by the coefficient? what happens then?",
        "A": "Numbers can always be divided by one another, but it might not be as neat as it is here. 5/14 is still dividing 5 by 14, but it s not able to be simplified."
    },
    {
        "video_name": "MQtsRYPx3v0",
        "Q": "What if the all the variables in the equation are not divisible by the same number?\nAs seen in: 1:30",
        "A": "If they are not divisible by the same number, that just means that you cannot factor out a GCF to simplify. You can still factor the expression, it will just be a little more challenging. He s mostly demonstrating that an expression MAY be simplified if there does happen to be a GCF."
    },
    {
        "video_name": "1BH2TNzAAik",
        "Q": "At 1:40, due to my poor understanding of how to deal with fractions, I am lost when teacher writes the lower half of the equation as 18x20xpi. I re-watched several times and can't understand why the 20 was added here. It's like a triple decker fraction or something and I can't make sense of it.",
        "A": "He got the 20pi in the denominator because the 20pi which is the circumference of the entire circle goes in the denominator."
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "at 9:23 how did he change (2RadY)2 into 4y?",
        "A": "Awesome, Thank you"
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "Why would you only need to rationalize the denominator? Does it make a problem easier to leave a numerator nonrational? How would you solve that problem, let's say at 2:06?",
        "A": "Yes. Rationalising the denominator makes problems easier and is a standard mathematical rule. Hope I helped!"
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "At 6:00, when he does 2+\u00e2\u0088\u009a5, is it the opposite sign or is it always positive?",
        "A": "It s the opposite sign, because the result should be a difference of two squares . If you wouldn t do a difference of two squares, you would end up with a radical in the denominator again."
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "At 7:00 Sal stated the answer was -24-12 x square root of 5.\n\nCan it also be -36 x square root of 5?\n\nThanks.",
        "A": "No, that would be like subtracting 12x from -24, you could only subtract 12x from -24x. You can t subtract/add a radical with a non-radical number. For example: (3)(squareroot(4)) - 2 =/= (1)(sqaureroot(4)) because we know squareroot(4) = 2 ... 3(2) - 2 =/= (1)(2) 4 =/= 2 hope that helps"
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "At 9:48, while he was expressing different ways to write 10y(sqrt)y, is it possible to write it as 10(sqrt)y^3?",
        "A": "I assume you mean the y^3 is inside the radical: 10 * sqrt(y^3) You could write it this way, but it would not be considered simplified. So, if you left your answer in this form, it would be incomplete."
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "At 4:10, Can't Sal just cancel out the square root of five from both the top and the bottom to get\n12/2-5 instead.",
        "A": "No, because a square root of 5 would have to be canceled out of the 5 as well. If the terms in the denominator were multiplied, then yes, you would be able to cancel a square root of 5."
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "Stupid question time: at 6:54, why - 24 - 12 sqrt5? Why not - 24 + 12 sqrt5? The minus signal should be distributed by all the terms?",
        "A": "Sal s final answer was: - 24 - 12 sqrt5."
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "At 4:20, when you discuss rationalizing denominators in the form 2+\u00e2\u0088\u009a5 by multiplying 2-\u00e2\u0088\u009a5, is that the same thing as saying \"multiply both the numerator and the denominator by its discriminant\"? I'm learning multiple things at once and trying to understand them all.",
        "A": "Actually, 2-\u00e2\u0088\u009a5 is called the conjugate of 2+\u00e2\u0088\u009a5. The word discriminant is used to refer to the B^2 - 4AC (the contents of the radical) within the quadratic formula."
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "At 6:50, Why does the negative 1 carry to both terms in the numerator?",
        "A": "After rationalizing the denominator, it became -1. Sal placed the minus in front of the fraction making the fraction which is the same as saying: -1[24 + 12 sqrt(5)]. You need to distribute the minus across the 2 terms. Another way to look at this is if Sal hadn t distributed the 12 already, he would have had 12[2 + sqrt(5)] / (-1). 12 divided by -1 = -12. So, he would have then distributed -12 across the [2 + sqrt(5)] creating the same answer of -24 - 12 sqrt(5)."
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "8:30- difference of squares?",
        "A": "Anything that looks like this format: (a - b)(a + b) is factored form of a difference of squares a^2 - b^2 In this problem, it is less obvious: (2\u00e2\u0088\u009ay \u00e2\u0080\u0093 5)(2\u00e2\u0088\u009ay + 5) Think of it this way a = 2\u00e2\u0088\u009ay b = 5 (a-b)(a+b) = (2\u00e2\u0088\u009ay - 5)(2\u00e2\u0088\u009ay + 5) So, it makes: a^2 - b^2 = (2\u00e2\u0088\u009ay)^2 - (5)^2 = 4y - 25"
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "Why at 6:56 did he write -24 - 12(sqrt(5))? should it have been plus?",
        "A": "Sal distributed the minus sign. Re watch that part to remember."
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "3:30 what is a binomial",
        "A": "is a polynomial with 2 terms...how do you you know how many terms there are? well, each term is separated by a + or a -. Examples of binomials: 3xp + 2nmyp, m - n , etc"
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "At 6:56, Sal wrote the answer as negative 24-12 square root 5. But doesn't the sign in between 24 and 12 square root 5 be '+'? I don't know whether Sal is correct or I am correct... Please revert. Thanks",
        "A": "The fraction was (24 + 12*sqrt5) / -1, so since the division by -1 affects both terms in the numerator, it is Sal who is correct."
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "At 4:13, when he has the 12 and square root 5, over 2 and square root 5, why wont it work? Why cant he just simplify it?",
        "A": "At 4:13 he had 12\u00e2\u0088\u009a5/(2\u00e2\u0088\u009a5-5) Because of the -5 in the denominator, this could not be simplified any better that the original 12/(2-\u00e2\u0088\u009a5) . That s why took a new direction and multiplied the 12/(2-\u00e2\u0088\u009a5) by (2+\u00e2\u0088\u009a5)(2+\u00e2\u0088\u009a5) which resulted in 4-5 = -1 in the denominator and 24+12\u00e2\u0088\u009a5 in the numerator (24+12\u00e2\u0088\u009a5)/(-1) resulting in your answer of -24-12\u00e2\u0088\u009a5. I hope that helps."
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "I don't understand why he is adding the five at 8:10. What happens to the minus?",
        "A": "He does that to cancel out the -5"
    },
    {
        "video_name": "gY5TvlHg4Vk",
        "Q": "At 7:00, why did he put -24-12 times the square root of 5? Isn't it -24+12 times the square root of 5?? If I am wrong, please explain. Thanks!!!!",
        "A": "It s because the whole term is negative, not just the 24. If the minus sign was close to the 24, then you would be right. Because the minus sign is next to the division sign, the minus sign has to be applied to everything, making it -24 and -12(square root of 5). Wow, this is hard to explain in words. Did that make any sense?"
    },
    {
        "video_name": "f-gCYOCFMYc",
        "Q": "At 00:47 he says the area is the length times the length. Shouldn't it be the length times the width?",
        "A": "In theory, yes; but since this is a square the length equals the width."
    },
    {
        "video_name": "GmD7Czmol0k",
        "Q": "At 1:32 why is he multiplying .25 by 8? Shouldn't it be -8?",
        "A": "It is because it was 7j+5-8(0.25). Multiplication comes before subtraction, so you first multiply and then subtract the answer. So in a way, he did do -8x 0.25 because it has the same answer. I hope I didn t confuse you more"
    },
    {
        "video_name": "GmD7Czmol0k",
        "Q": "at 1:17 why did he divide 7 in half to make 3.5",
        "A": "The 3.5 comes from multiplying 7 * 0.5. 7 * 5 = 35, then you need one decimal place, so you get: 3.5"
    },
    {
        "video_name": "GmD7Czmol0k",
        "Q": "At 1:40 in the video he got 2 from 8k so does that mean for that part of the problem I have to divide the denominator {I/4} to the 8?",
        "A": "yes, just think about what multiplies by 4 would equal 8, that number would be 1/4 of 8, going back to the definition of a fraction. the denominator is how many pieces are the whole thing, using a candy bar as an example, if you had a candy bar with 4 pieces in it all together, how many candy bars would you need for 8 pieces."
    },
    {
        "video_name": "GmD7Czmol0k",
        "Q": "3:12 you say that you can cancel out the three....How does that work? Why not just solving it?",
        "A": "Whenever you ve got a string of addition like that it s easier to just ignore the ones that cancel instead of solving, since 3-3=0. By solving it I think you mean doing 8+3, then subtracting 3 from that. This is more steps than just ignoring the 3 s altogether. Consider this example- 425+637-425 The answer is 637. You could have done 425+637, then subtracted 425 from that, but we can save ourselves the trouble and notice that 425-425=0, which give us the answer immediately without having to do any computation."
    },
    {
        "video_name": "GmD7Czmol0k",
        "Q": "@ 1:58 why does he chuckle?",
        "A": "I don t know why i m answering this He laughs because a few seconds earlier he says try and work through it by yourself and right after that he starts to go through the problem."
    },
    {
        "video_name": "GmD7Czmol0k",
        "Q": "At 3:20, how is it that the answer is 8? Don't we have to go by order of operations when solving this? I'm pretty sure addition is before subtracting and I got the answer 9...Little explanation would be nice on how this problem.",
        "A": "When adding and subtracting it doesn t matter in which order we perform the operations. 8 \u00e2\u0088\u0092 3 is just a shorter way of writing 8 + (\u00e2\u0088\u00923), which is the same thing as (\u00e2\u0088\u00923) + 8. So, Sal didn t do anything wrong because: 3 + 8 \u00e2\u0088\u0092 3 = 3 + 8 + (\u00e2\u0088\u00923) = 3 + (\u00e2\u0088\u00923) + 8 = 3 \u00e2\u0088\u0092 3 + 8 = 0 + 8 = 8 And, your way works as well: 3 + 8 \u00e2\u0088\u0092 3 = 11 \u00e2\u0088\u0092 3 = 8"
    },
    {
        "video_name": "GmD7Czmol0k",
        "Q": "He does this so fast it's hard to keep track can someone break down the problem at 2:45 for me please.",
        "A": "If he s talking too fast, I would recommend slowing the video speed down to 0.5 and turning the captions on. That way you can read along as he talks."
    },
    {
        "video_name": "GmD7Czmol0k",
        "Q": "at 1:03 in the video he says if we get half of 7 we get 3.5 but how does he get the that?",
        "A": "The decimal representation of 1/2 is 0.5, which is what he uses in the video. He wrote 0.5 and says one half , since they a representations of the same thing. Hope that helped."
    },
    {
        "video_name": "GmD7Czmol0k",
        "Q": "at 0:57,why didn't he just multiply the 8(.25) just the way it was?\ndo I really have to convert the decimal to a fraction?\nI'm asking because I solved the problem without the conversion, and a got 28.70 as an answer.",
        "A": "No, you do not have to change 0.25 into a decimal. But, somewhere you have a math error. You should have got the same answer as Sal. 7(0.5) + 5 - 8(0.25) = 3.5 + 5 - 2 = 8.5 - 2 = 6.5 See if you can find your error use the work I showed. If you have questions, comment back."
    },
    {
        "video_name": "GH8-URjRQpQ",
        "Q": "How does the e^x lose the exponential at 3:03?",
        "A": "he s substituting X with 1, e^1 = e. He didn t lose the X, it just got substituted"
    },
    {
        "video_name": "GH8-URjRQpQ",
        "Q": "at 1:00, why is the product rule used before the chain rule? How do you decide that?",
        "A": "You are starting with a product, so apply the product rule first. In the course of applying the product rule in this case, we needed to apply the chain rule."
    },
    {
        "video_name": "QkepM8Vv3kw",
        "Q": "In min 4:00 Sal builds a matrix, the matrix seems to be wrong.\n-1 1\n0 1\n1 1\n2 1",
        "A": "The matrix you wrote is the matrix Sal wrote at 3:12, and I believe that it is correct."
    },
    {
        "video_name": "D6mivA_8L8U",
        "Q": "How did Sal get 50a^3 in this video? Around 1:07 into the video he gets 50a^3 for his answer. Did he add the variable a and three together?",
        "A": "To multiply terms with variables in them (for instance 10a times 5a^2), you have to : 1. Multiply the numerical coefficients : in this case, 10 times 5 = 50 2. Look for the same variable : in this case, a times a^2. 3. Write the variable with an exponent that is the sum of the exponents : in this case, 1 + 2, giving a^3. 4. So the answer to 10a times 5a^2 is 50 a^3. Hope this has been helpful."
    },
    {
        "video_name": "_jRXtJz3CYc",
        "Q": "At 2:10 when she makes the straight angle she makes it vertical. Does it have to be vertical or can it be horizontal?? Thanks,",
        "A": "Because her word problem is based on the oven knob, she has to start at the top Off position. That s the only reason she s starting there. In real life, a 180 degree angle (also known as a straight line) can be vertical, horizontal, or anywhere in between."
    },
    {
        "video_name": "z1hz8-Kri1E",
        "Q": "At 3:15 why do we subtract one equation from the other? And could you reverse the method? Also, at 5:11, how are you simplifying, doesn't that change the answer? I think I missed an important lesson, do you have a recommended video?",
        "A": "Ok, thank-you. I think I need to work on these types of problems some more."
    },
    {
        "video_name": "z1hz8-Kri1E",
        "Q": "While Sal writes out the equation during 1:45 how does he know which one should be on top and the other on the bottom? does it matter?",
        "A": "commutative property of addition=it doesn t matter."
    },
    {
        "video_name": "z1hz8-Kri1E",
        "Q": "At 2:47 in the video, can you multiply -1 to the top equation instead of the bottom or does it matter?",
        "A": "No, it doesn t matter. Sal just did that for simplicity. If you have 2 equations, say 5x+y=6, and 4x+y=7, it doesn t matter which one you multiply by negative one."
    },
    {
        "video_name": "z1hz8-Kri1E",
        "Q": "At 3:15, how can we multiply by negative one? Why is that fundamental change to the equation allowed without also changing the second equation?",
        "A": "You can multiply an equation by anything you want without changing its meaning, provided you multiply each term by the same factor."
    },
    {
        "video_name": "z1hz8-Kri1E",
        "Q": "2:51 Why do you have to multiply the equation by -1? Can't you just subtract the 2nd from the first equation? It's much easier that way than multiplying the 2nd equation by -1 and adding it to the first equation.\n\nSubtracting is the same thing as adding something negative.",
        "A": "Yes that is completely true. You can definitely do that. But you see, the reason most people don t do this however is because it s very easy to forget which parts of the term should be negative or not; it s very easy to forget if a + or - sign should be assigned(it really depends on the person and how they do it). That s why people multiply by *-1* and then add, but you can definitely resort to just subtracting."
    },
    {
        "video_name": "z1hz8-Kri1E",
        "Q": "At 2:01 Khan says to multiply both sides of the bottom equation by -1 so they cancel out when you add the two equations together, but why can't you just subract the bottom equation from the top equation and skip a multiplication step?",
        "A": "Someone just asked that question somewhere on this page just look around for it. It has already been answered."
    },
    {
        "video_name": "SfxULALs_u8",
        "Q": "At 1:00 why do you write the problem before you do the equation.",
        "A": "how do you mean? how are you supposed to write the equation without answering the question?"
    },
    {
        "video_name": "SfxULALs_u8",
        "Q": "at 1:48 Why don't we first add the carry and then multiply with the number 6?",
        "A": "Think back at the order of operations. PEMDAS. Parenthesis, Exponents, Multiplication and Division, and lastly, Addition and Subtraction. When a number has pending addition and multiplication you do the multiplication first then add."
    },
    {
        "video_name": "SfxULALs_u8",
        "Q": "At 01:31, Sal says, \"And we carry the 4 in 42 up to the tens place.\" Does he mean that he'd like to put the digit 4 of the number 42 up to the tens column?",
        "A": "this means that you leave the 2 in the ones column, and then take the 4 (that you got from multiplication, in which the answer was 42) and put it in the tens column so that you can add it to the result of the tens multiplication. this is called carrying."
    },
    {
        "video_name": "SfxULALs_u8",
        "Q": "At 0:10 he wrote the problem from left to right. Is it possible to do the problem that way?",
        "A": "Yes. In multiplication and addition, it doesn t matter what order you do the equation. 12 x 3 is the same product as 3 x 12. Hope this helped :)"
    },
    {
        "video_name": "yj4oS-27Q3k",
        "Q": "When Sal talks about the two line segments being equal at 7:11, does the = sign with the ~ over it mean congruent? If so, what is the difference between \"congruent\" and \"equal?\" Sal pointed out that they are basically the same thing, however, from the way each of the words is used, they aren't really. Please explain!",
        "A": "It is really just context. Congruent is used more for geometric things like line segments and shape, while equal is more of algebra sense, like integers and such. In the end is is really just a subtle difference, as most of this is for geometric proofs so you can use certain theorems/postulates"
    },
    {
        "video_name": "yj4oS-27Q3k",
        "Q": "08:56\nSo the RSH postulate is basically the same thing as the Pythagorean Theorem?",
        "A": "That s a good way of thinking about it. The Pythagorean Theorem is another way of saying that RSH and SSS are equivalent in right triangles."
    },
    {
        "video_name": "yj4oS-27Q3k",
        "Q": "4:18 Sal is saying equal distant right not some other term I'm unfamiliar with? it sounds like he's say equadistance like it's a word?",
        "A": "It is equidistant but i still means equal distant its just one word"
    },
    {
        "video_name": "v-pyuaThp-c",
        "Q": "At 3:56 what tool is Vi using? I can't tell because it is sped up.",
        "A": "a compass. usually used in geometry"
    },
    {
        "video_name": "v-pyuaThp-c",
        "Q": "I have watched this video so many times (ever since she posted it onto YouTube) and the very beginning has always confused me. At 0:06 she says, \"But what he doesn't even realize is that what he's teaching isn't even Algebra.\"\nHow are parabolas NOT Algebra? They are one of the major lessons in Algebra!",
        "A": "Although everything learned in an algebra class can probably be called algebra, the way algebra is currently taught in most schools is more about procedures and less about developing mathematical thought processes. While having a procedural knowledge of equations might be useful for many applications, it has little relevance to pure mathematics/pure algebra."
    },
    {
        "video_name": "v-pyuaThp-c",
        "Q": "at 4:31, How did she draw that?",
        "A": "She made 12 or so points in a circle, and then made each point the edge of one circle."
    },
    {
        "video_name": "v-pyuaThp-c",
        "Q": "At 1:53, how does she do that star?",
        "A": "she practiced alot"
    },
    {
        "video_name": "v-pyuaThp-c",
        "Q": "What is that orange thing that she has at 5:43? I know it's not the same ribbon from a few seconds before, because it's stiffer.",
        "A": "It s a pipe cleaner, so that she can stand it up without it collapsing."
    },
    {
        "video_name": "v-pyuaThp-c",
        "Q": "At 1:44 how does she get to the star shape?",
        "A": "At 1:14 she explains how to to perpendicular lines can form a parabola and she makes a star made out of lines that then joinded form the star you see in the video"
    },
    {
        "video_name": "v-pyuaThp-c",
        "Q": "At 3:45 how does she do that spinny thing?",
        "A": "You draw dots in a circle shape. You get a compass. you name one dot the side. you make the rest centers. you then draw the circles with the center on each of the dots and the pencil on the dot that you call the side"
    },
    {
        "video_name": "v-pyuaThp-c",
        "Q": "what is a porrabala as vi says in 0:20",
        "A": "A parabola is a graph that is plotted by various forms of the equation: y=x^2. It looks somewhat like the letter U."
    },
    {
        "video_name": "v-pyuaThp-c",
        "Q": "at 4:29 how did she draw the heart?",
        "A": "She drew twenty dots in a circle evenly spaced. Then she picked one point to be on the edge of every later circle that she would draw, in this case the top point. So she drew nineteen circles, all of which pass through the one point that she picked. The center of each of the nineteen circles were the nineteen other dots."
    },
    {
        "video_name": "v-pyuaThp-c",
        "Q": "At 3:18, Vi mentions that it's better to teach people HOW to think rather than WHAT to think.\n\nDoes anyone have any good ideas on how to do that?",
        "A": "wow thats my motto sort of. i happen to play minecraft and there is soemthign in the game that adds computer logic. if i teach sombody, i always make them not understand WHAT they are and what they do but also WHY they work. my exact motto is the what is good, but they why is at least 200 times better. i THINK thats what it means."
    },
    {
        "video_name": "v-pyuaThp-c",
        "Q": "how did she get that figure at 4:50",
        "A": "She explains at 4:38"
    },
    {
        "video_name": "v-pyuaThp-c",
        "Q": "At 3:33, Vi was talking about tautology. What is that?",
        "A": "teagan please stop being so strange"
    },
    {
        "video_name": "wt6XqG59t5U",
        "Q": "at 1:44 how did 12w came in the equation\nw.sq+12w+36\ncan u plz answer",
        "A": "( 6 + w ) ^2 = ( 6 + w ) ( 6 + w ) Then use FOIL to multiply the two binomials: First terms : 6*6 = 36 Inner terms : +6w Outer terms : + 6w Last terms : w*w = w^2"
    },
    {
        "video_name": "wt6XqG59t5U",
        "Q": "At 1:40 why did Sal write w2+12w+36?",
        "A": "Do you have to do it this way?"
    },
    {
        "video_name": "wt6XqG59t5U",
        "Q": "Sal, At 1:40 how did you get 12?",
        "A": "He was squaring (6+w). Squaring a binomial follows the pattern (a + b)^2 = a^2 + 2ab + b^2, so he got 6^2 + 12w + w^2, but he reversed it at the same time so it would be in correct polynomial order. So 12w in this case is the middle term 2ab because it s 2*6*w. If you don t remember the pattern (a+b)^2 = a^2 + 2ab + b^2, you can do it yourself using FOIL by multiplying (a+b)(a+b)."
    },
    {
        "video_name": "wt6XqG59t5U",
        "Q": "At 1:40 you multiplied 6x2. Where did the 2 come from to end up with W sq +12",
        "A": "He was squaring (6+w). Squaring a binomial follows the pattern (a + b)^2 = a^2 + 2ab + b^2, so he got 6^2 + 12w + w^2, but he reversed it at the same time so it would be in correct polynomial order. If you don t remember the pattern (a+b)^2 = a^2 + 2ab + b^2, you can do it yourself using FOIL by multiplying (a+b)(a+b)."
    },
    {
        "video_name": "ios3QL9t9LQ",
        "Q": "@ about 1:38 i wasn't sure how to or what to do can anyone help?",
        "A": "What was the problem there, KimberlyPK? Tell us so we can answer."
    },
    {
        "video_name": "ios3QL9t9LQ",
        "Q": "At 1:34 Sal says that 10 to the -2 divided by 10 is 10 to the -3. I don't quite understand that. Sal doesn't really explain how he got to that answer and I don't get how that makes sense. When I think about it 10 to the -2 is 100. then divided by ten is 10.. but i know that's incorrect. Can someone help me understand this? thanks, Oliver",
        "A": "10\u00c2\u00b2 = 100 10\u00e2\u0081\u00bb\u00c2\u00b2 = 1/100 10\u00e2\u0081\u00bb\u00c2\u00b2/10 = (1/100) / 10 = 1 / (100 \u00c3\u0097 10) = 1/1000 = 10\u00e2\u0081\u00bb\u00c2\u00b3"
    },
    {
        "video_name": "lEGS5ECgFxE",
        "Q": "At 4:35\nI don't see how Sal got -6, I thought that it would be positive 6 because the difference between 9 and 3 is 6?",
        "A": "He is using -6 because he is figuring out what to add to the coordinates. If you start from 9 and want to get to 3 you do 9 + (-6) = 3. Otherwise we would be going in the wrong direction."
    },
    {
        "video_name": "lEGS5ECgFxE",
        "Q": "I need a video to explain how to find a collinear point that is between two given points, This video explains how to find a collinear point that is PAST two given points. I do not understand how to make the new ratio (It was 3:1, but since we are finding a point between the given collinear points, we change the ratio to 3:4). I do not understand why/how to change that ratio.\nThanks,\nMayda",
        "A": "The video previous to this one (Dividing line segments: graphical) is the video you are looking for. I had the same problem like 20 min ago. You can basically figure out the problem he gives you at 3:36. Hope this helps :)"
    },
    {
        "video_name": "lEGS5ECgFxE",
        "Q": "I get the 1:3 is = to 4 parts so why has the exercise following have 1/3 as the answer i.e. three parts instead of 4 parts hence I got the answer wrong! Who is wrong Sal or the person who worked out the answer of the exercise?",
        "A": "same thing happened to me and i think it s a mistake"
    },
    {
        "video_name": "lEGS5ECgFxE",
        "Q": "At 2:05 could somebody better explain why parallel line components(x, y length) have the same ratio as the line lengths?",
        "A": "Try drawing the changes in x and y so that you have two overlapping right triangles (with AB and AC as the hypotenuses). You ll see that the triangles are similar (by angle-angle-angle), so the corresponding sides have the same ratio."
    },
    {
        "video_name": "lEGS5ECgFxE",
        "Q": "I learn better by example, unfortunately there isn't a video showing how I derive a fraction from something like \"The ratio of AB to BC is 1:4.\", the only thing I can find on the internet is to turn it into 1/4, but I know this is completely wrong. I'm finding this really frustrating as I should of easily mastered this subject by now.",
        "A": "How is that wrong? AB : BC = 1 : 4, hence we can simply rewrite this as a fraction (which demonstrates the same thing): AB/BC = 1/4"
    },
    {
        "video_name": "lEGS5ECgFxE",
        "Q": "At 3:24 why would you add the 10 and the -6?",
        "A": "Because in order to find the C X coordinate, you have to add the X (which is 10) to whatever X coordinates you already have (-6). Basically just fining the length of that X line that he drew so that you can get the X coordinates."
    },
    {
        "video_name": "lEGS5ECgFxE",
        "Q": "At 10:32 what is the formula for finding ratio between two points?",
        "A": "The formula is known as scale factor, or as you ll commonly find to be referred to as _k_ . Scale factor: x2 y2 -------- x1 y1 The dashed line is the fraction divisible. Hope this helps."
    },
    {
        "video_name": "lEGS5ECgFxE",
        "Q": "Why does he reciprocate at 4:45? Or at the earlier time? How does it help, or is it needed?",
        "A": "You can just cross multiply, which is another way"
    },
    {
        "video_name": "lEGS5ECgFxE",
        "Q": "by 3:20 why does sal use A and not B to minus the 10?",
        "A": "The ratio is AB:AC is 2:5. We use the coordinates of A and B at ratio 2, to calculate the change from B to C at ratio 5, then the reference point will be B and not A, yet he uses the reference as A, so that would give the ratio of AB:AC at 2:5, and not the original AB:BC."
    },
    {
        "video_name": "lEGS5ECgFxE",
        "Q": "in one of the exercises (on this same topic) I\u00c2\u00b4ve found this statement: because the ratio of AB to BC is 1:3 the ratio of AB to AC must be 1:4.\n\nbut I couldn\u00c2\u00b4t understand the relationship... please someone could explain it to me",
        "A": "It is easiest if you draw a line segment AC and put B about 1/4 of the way between A and C. Then you can label AB as 1 unit and BC as 3 units long. By adding these two numbers together, you get the total length of AC which is 4 units, and maybe that will make sense to you."
    },
    {
        "video_name": "gHzLHknEk1M",
        "Q": "at 5:00 wouldn't derivative of 8 equal to 0 ?",
        "A": "Using the constant rule d/dx af(x) = a[d/dx f(x)] d/dx [8*3^x] = 8 [d/dx 3^x] So you don t differentiate 8 in this case. Had it been d/dx 8+3^x then you would use the sum rule, d/dx f(x) + g(x) = d/dx f(x) + d/dx g(x). d/dx 8 + 3^x = d/dx 8 + d/dx 3^x = 0 + ln(3) *3^x"
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "At 1:40, how did Sal get 1/3?",
        "A": "Here s an algebraic answer to your question. 8^x is the same as (2^3)^x which is the same as 2^3x. So. given that 8^x = 2, then 2^3x also equals 2 which can be written as 2^1. So, 2^3x = 2^1. The bases are the same, so the exponents must be equal. 3x = 1 x = 1/3"
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "at 4:13 could we put just -3 as the answer? Instead of -1/3 ?",
        "A": "No, we can t. They re very different things. 8^(-3) would be 1 / 8^3 which equals 1 / 512. 8^(-1/3) is 1 / 8^(1/3) which says 1 divided by the cube root of 8 and therefore equals 1 / 2. Since the problem is asking for the power of 8 that equals 1/2, the correct answer is -1 / 3."
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "A 1:50 how can you show that 8^1/3 is equal to 2? I know the reason why its one-third because 2^3=8 but I can't understand how you would like actually solve 8^1/3 and get 2. Can anybody show me? Thx!",
        "A": "I have edited my answer, hope u understand this time:) (2\u00c2\u00b3)^1/3 = 2^(3\u00c3\u00971/3) = 2\u00c2\u00b9 = 2 also, (2\u00c2\u00b3)^1/3 = 8^1/3 [as 2\u00c2\u00b3 = 8] as u can see, (2\u00c2\u00b3)^1/3 is equal to 2 as well as 8^1/3 so, 8^1/3 = 2"
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "At 1:29, how do we identify what sort of fractional power does log 2 to the base 8 stand for? Well, Sal here tells us that log 2 to the base 8 is 1/3 but how do we solve other questions?",
        "A": "log 2 to the base 8 is 1/3 because any fractional power is a root. Such as 25^0.5 = sqrt(25) or 5."
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "At 1:40 did you make a mistake as surely 8 to the power of 1/3 is 2.6666666 and 8 to the power of 1/4 is 2 ??",
        "A": "I m sorry, but you re wrong. You do not multiply the base by the exponent: 8^(1/3) = \u00e2\u0088\u009b8 = 2 8^(1/4) = \u00e2\u0088\u009c8, which is the simplest form it is approximately 1.68179283..."
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "At 1:31 why would log base 2 (8) be equal to 1/3? Wouldn't it be equal to 1/4?",
        "A": "Be careful --- log_2(8) = 3, not 1/3, since 2 has to be raised to the 3rd power in order to get 8. At 1:31, the problem is different (Sal has changed the base) --- it s log_8(2). So we have to ask ourselves, what power of 8 will give us 2, or mathematically, 8^x = 2 and then solve for x. Since 8^x is the same as 2^3x , we then have the equation 2^3x = 2^1. The bases are the same, so the exponents must be equal. 3x = 1, so x = 1/3."
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "At time stamp 1:32 isn't the answer 1/4 because Sal was saying that it was equal to 1/3. But I did the math with a calculator and got 1/4. I am so confused which is right? 1/4 or 1/3?\nThanks",
        "A": "It is 1/3. It is 8 raised to the power of x not 8x. We are dealing with exponent and not multiplication."
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "At 3:57, why the answer is 1/2?",
        "A": "Keep in mind that the top part remains the same (1), so the question is really why the bottom parts is: 8^(1/3) = 2. This is the equivalent of taking the cube root of 8, or alternatively asking what number raised to the 3rd power is equal to 8? The answer to this question is 2. Note: taking something to the power of 1/2 is the same as taking the square root. Taking something to the power of 1/3 is taking the cube root. 1/4 is the fourth root, etc."
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "At 2:38 would the log also be equal to 1/16?",
        "A": "1/16 would be the simpler form for 2^(-4) , not 2^(-3) being what Sal s tried to calculate from."
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "At 1:35, wouldn't it actually 8 to the 1/4 power? Because 8 to the 1/3 power actually is a decimal, and 8 to the 1/4 power is 2 even. I'm sort of confused.",
        "A": "8 = 2^3 so 8^(1/3) = 2^(3/3) = 2"
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "Right about 2:25, I paused the video and tried to solve log2(1/8) by myself. I came up with log2(1/8)=(1/16). 2(1/16) is equal to 1/8. Why isn't the answer log2(1/8)=(1/16)? What am I missing?",
        "A": "with logs,the way you re trying to do it, the 2 would be raised to the 1/16 not multiplied by it, so 2^(1/16) would give you 1.044... when you do this problem youre asking yourself 2 raised to the what will give me 1/8. The answer is -3 because 2^(-3) equals 1/(2^3) which equals 1/8"
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "At 4:14, can you write negative 3 instead of negative 1/3 as the answer? Or must it be in fractional form?",
        "A": "-3 is -3/1, it is not the same as -1/3 If it was -3/1, then of course you can write it as -3."
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "1:50 So it's practically reversed?",
        "A": "Pretty much, yes."
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "in 2:37, i thought 2 to the negitive third power was negitive eight, how does that turn into one eighth",
        "A": "if the exponent is a negative, instead of the 2 being in the numerator it goes to the denominator, then the exponent is applied to the number."
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "at 2:50 how come log8^2 is 1/3 and log2^1/8 is -3.\nIsn't also the first one in negative??",
        "A": "when you do a log, you find the exponent. Here, log base (power) = exponent so, base^exponent = power Lets do log8^2 = 1/3 => 8^(1/3) = 2 ------ just reconvert them into the above form to make sense of it. Lets also do log 2 (1/8) = -3 => 2^(-3) = 1/8 ------- Likewise Hope this has helped you!"
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "at 3:38, I thought that 8^1/3 is 2.666666667, not 2.",
        "A": "Looks like you did 8 * 1/3 = 8/3 = 2.666... 8 to the exponent of 1/3 or 8^(1/3) = cube root(8) Cube roots are asking you to find the number that was cubed (raised to the power of 3) that = 8. 2 x 2 x 2 = 2^3 = 8 So, 8^(1/3) = cuberoot (8) = 2."
    },
    {
        "video_name": "eTWCARmrzJ0",
        "Q": "Can somebody please explain what Sal said starting from 0:40? It confuses me.",
        "A": "First: Are you aware that a root, such as a square root or a cube root, is the same thing as a fractional exponent? Specifically, \u00e2\u0088\u009ax = x^(\u00c2\u00bd) \u00e2\u0088\u009bx = x^(\u00e2\u0085\u0093) \u00e2\u0088\u009cx = x^(\u00c2\u00bc) And so on... Thus log\u00e2\u0082\u0088 (2) = x, can be rearranged to 8\u00cb\u00a3 = 2 So, to find what x is, you have to know what power you have to put 8 to to get 2. Since 2\u00c2\u00b3 = 8, then \u00e2\u0088\u009b8 = 2 But \u00e2\u0088\u009b8 is the same thing as 8^(\u00e2\u0085\u0093). So, the power you have to raise 8 to in order to get 2 is \u00e2\u0085\u0093. Thus, x=\u00e2\u0085\u0093 whether you write the problem in the log form or in the exponential form."
    },
    {
        "video_name": "pLCmwHsDYqo",
        "Q": "Wait, how did Sal cancel out the denominator? 5:29",
        "A": "He distributed the x^2-x+1 to the fractional term, which has a denominator of x^2-x+1. Thus they cancel! It would be like multiplying 3(x + y/3)=3(x)+3(y/3)=3x+y. Does that help?"
    },
    {
        "video_name": "pLCmwHsDYqo",
        "Q": "Can't you factor a 5 out of 5x - 5 at 4:26? Also, you could factor x^2 - x +1 to get (x - 1)(x - 1). So, you would have 5(x - 1) divided by (x - 1)(x - 1). This would result in 5 divided by x - 1. So, shouldn't the final answer be x + 1 + 5/x-1?",
        "A": "Actually you made a small mistake there. x^2-x+1 doesn t simplify to (x-1) (x-1). If you multiply -1 to itself it s true that is equal to 1 which is the same as the constant C from the original equation (which is in the form of Ax^2+Bx+C), so that checks out. However if you add -1 and -1 together you will get -2 which is not the same as the coefficient B of x. The original coefficient B of x is -1 not -2. That s why Sal just divided the equation from the start because you can t factor it perfectly."
    },
    {
        "video_name": "eSjbEWb4Qp4",
        "Q": "At 1:18 Sal puts his compass down but it jumps to the wrong point, he then uses it to bisect a line perpendicularly, is that right?",
        "A": "Kind of. While it is not exactly precise, like he stated, the system is merely recognizing the general placement of lines and compasses to see that you attempted to find the correct circle. The result would be something similar to the right circle. So you would get it right on here."
    },
    {
        "video_name": "eSjbEWb4Qp4",
        "Q": "What does circumcenter mean at 0:22?",
        "A": "It is where the perpendicular bisectors of a triangle meet. It is also the center of the circle that contains the vertices of the triangle."
    },
    {
        "video_name": "eSjbEWb4Qp4",
        "Q": "why does the perpendicular bisector have to be half way between the two points, at 0:51",
        "A": "In order for the angles to be equal, the bisector has to cut the angle in half so it can fulfill it."
    },
    {
        "video_name": "nGCW5teACC0",
        "Q": "There is a jump in the explanation of g(-1). How did Sal get x=1? This falls at 5:23 in the video.",
        "A": "I didn t hear Sal mention x = 1 in the video. He does mention g(-1) = 11 when he says the point is (-1, 11) g (-1) = -4 (-1) + 7 = 11 And the slope of y = -4x + 7 is - 4"
    },
    {
        "video_name": "nGCW5teACC0",
        "Q": "In the video @2:00, while defining the limit as x approaches -1, Sal does g(x) - g(-1) for the y coordinate of that limit. It's the same with the denominator, x+1. It seems to me that he should be doing it vice versa, because he should be going down the y ( or f(x)) axis. Please tell me if I'm looking at this the wrong way.",
        "A": "It is perhaps more conventional to calculate slope as you suggest, but you get the same result working in either direction, because (a - b)/(c - d) is the same as (b - a)/(d - c). In effect, when you change the direction you re multiplying the top and bottom of the fraction by -1, which leaves the result unchanged."
    },
    {
        "video_name": "nGCW5teACC0",
        "Q": "how does the negative 11 @4:12 come in? I mean I get everything else except tht",
        "A": "At that point Sal is evaluating an expression that requires us to subtract g(-1), as shown in the expression written just above this one. In other words, we need to subtract the outcome of plugging -1 into the function g(x). We re given (in the original problem) that g(x) = -4x + 7, so we get g(-1) equals -4 times -1 plus 7, which is 4 + 7 = 11, and because we re subtracting g(-1) we get -11."
    },
    {
        "video_name": "nGCW5teACC0",
        "Q": "At 0:46 ...how did sal construct the slope (the light green coloured slope)\n.Im sorry for the stupid question..",
        "A": "He was given the slope in the question, g(x) = -4x+7. This is called slope-intercept form of an equation of a line. It is more commonly written y = mx + b and the slope of the line is m. What it says is that you can construct the line by taking the equation, substituting an x value and the equation will produce a y value on the line."
    },
    {
        "video_name": "nGCW5teACC0",
        "Q": "At 3:25, how do we know that the slope is equal to the value of the limit?\nHowever, thank you this video it is very helpful. :)",
        "A": "In a line (which is a function where the slope is always constant), the secant will always be equal to the slope (a line that pass through the same two points as some other equal in size line will always be the same line). Since we use secant to find limits and in a line the secant is the same as the slope, finding the limit in a line is using the slope."
    },
    {
        "video_name": "nGCW5teACC0",
        "Q": "Maybe I've been awake too long but I'm having an aneurism on this one but at 4:40, we're going from 4x + 7 - 11 / x + 1 to -4 ( x + 1 ) / x + 1. How do we get from substituting 4x + -4 in the numerator to -4(x + 1). Is it that \"x\" in the numerator is substituted by the denominator \" x + 1 \"?",
        "A": "You re missing a minus sign at the beginning of the numerator, which was -4x + 7 - 11. Combining the two constants we get -4x - 4 in the numerator, and then factoring out -4 we get -4(x + 1)."
    },
    {
        "video_name": "nGCW5teACC0",
        "Q": "4:44 min mark, how did sal get the last part in the numerator\n(x+1)?\n\ncheers",
        "A": "He s moving kind of fast there so it may be hard to see, but he found that the numerator was -4x - 4, and then factored out -4 to get -4(x - 1) (actually -4(x + 1), as Josh pointed out below -- thanks for the correction)."
    },
    {
        "video_name": "nGCW5teACC0",
        "Q": "At 1:27 of the video, Sal says \"an arbitrary point\". What does that mean?",
        "A": "That means he just picked a point at random."
    },
    {
        "video_name": "nGCW5teACC0",
        "Q": "At 4:12 why do you not plug in (-1) for x?",
        "A": "You cannot plug the -1 in too early . Keep in mind that the function is undefined at -1, so if you started by just putting it in you d have g(-1) - g(-1) / (-1) + 1 which gives 0 / 0. We are looking for what happens near -1, not at -1, since we are finding the limit. You have to algebraic ally manipulate the function basically until it is defined at that point before plugging it in. Sal kind of half plugged it in this video too"
    },
    {
        "video_name": "1F7LAJEVp-U",
        "Q": "At 0:20, why does Sal have to use another program just to write on the question? Did they take off that scratch paper thingy.",
        "A": "He has to because his methods and organization work better with the program he uses most of the time, and so his personal preference is to use that program rather than the scratchpad option."
    },
    {
        "video_name": "1F7LAJEVp-U",
        "Q": "At 2:16, does the little triangle mean change in?",
        "A": "Yes,change in amount."
    },
    {
        "video_name": "1F7LAJEVp-U",
        "Q": "At 0:32, Sal says \"y is going to be equal to some constant times x\". Sal wrote a K when he said constant. Does K mean constant?",
        "A": "No the K is just the variable"
    },
    {
        "video_name": "aeyFb2eVH1c",
        "Q": "At 2:30, why is the range of y always grater or equal to 1? If i solve the equation for -2 I get 0..",
        "A": "The function is f(x)=(x+2)^2+1, so if you solve f(-2) you get: f(-2) = [(-2)+2]^2+1 f(-2) = [0]^2+1 f(-2) = 1 You may be looking at when Sal writes the equation solved for the exponential expression as y-1=(x+2)^2 and forgetting to finish solving for y. y-1 = (x+2)^2 y-1 = [(-2)+2]^2 y-1 = 0^2 y-1 = 0 :But we haven t solved for y yet! y=1"
    },
    {
        "video_name": "aeyFb2eVH1c",
        "Q": "At 1:09, isn't this whole subtracting and 2rooting process the same as just switching x and y in their respective places in the initial equation. I believe this is what i was taught... just wondering/suggesting?",
        "A": "Seems like it is the same. But only if you continue on and solve for the y after you switch them. I can t think of any reason why switching them first would be a problem."
    },
    {
        "video_name": "aeyFb2eVH1c",
        "Q": "At 5:55 do we have to rename or is Sal just trying to make it more clear?",
        "A": "Because it is the inverse of f(x) it makes more sense and is more clear to write it as f-1(x)"
    },
    {
        "video_name": "aeyFb2eVH1c",
        "Q": "At 6:43 wouldn't the inverse be \u00c2\u00b1\u00e2\u0088\u009ax-1)-2 or would it be the way he had it, because my teacher says \u00c2\u00b1\u00e2\u0088\u009a of something or if there is a even root there has to be the plus/minus sign in front?\nThank you.",
        "A": "In this case, it would be the way he had it. The inverse of a full parabola is not a function, hence the limitations on the domain of x from the start of the problem. Sal talks about this at 1:12 or so in the video. Chuck Towle also has a good answer to a related question on this page about 5 months ago."
    },
    {
        "video_name": "aeyFb2eVH1c",
        "Q": "At 4:52, what is the point of renaming the transformed function in terms of y to again x. If its the same why change it? Also why would it be the same if x and y will have different values.",
        "A": "we re looking for the inverse of x, and since we replaced f(x) with y in the beginning, and used it to find the inverse, we have to change it back to f^-1 (x)"
    },
    {
        "video_name": "aeyFb2eVH1c",
        "Q": "Where is the video with the answer to why we had to constrain it to x <= -2 as you aforementioned at \"0:17\"",
        "A": "to visualize, the inverse of a full parabola would be look like a u shape with the opening towards right hand side. this implies (except for the vertex) a particular x value can produce TWO y values. This is not allowed in function."
    },
    {
        "video_name": "aeyFb2eVH1c",
        "Q": "At 1:17 I still don't understand why Sal took the positive square root. Even though the domain of f is will make the expression (x+2) greater than or equal to 0, I don't see why Sal couldn't have taken the negative square root. It would have made logical sense right?",
        "A": "Yes but how would you be able to plot the points on the graph with the negative square root"
    },
    {
        "video_name": "JEHejQphIYc",
        "Q": "At 1:43 in the video it says how the 3 numbers after all the decimals is how many spaces you go to the left to place the decimal. Why can you add all the places after the decimal together if they are not in the same alignment as the other decimal? Please help.",
        "A": "You add them together because both are decimals none the less. And also, if you do not do this you will get the wrong answer. I know this is kinda restricting, but it is the only way to do it."
    },
    {
        "video_name": "JEHejQphIYc",
        "Q": "Is the hight dot the same thing as a multiplacation sign? 0:10",
        "A": "Yes, the dot is a multiplacation symbol in Algebra."
    },
    {
        "video_name": "JEHejQphIYc",
        "Q": "at 1:54 ,sal is talking about the decimals place and i kinda need help understanding it,plz help me if you can.thnx!XD",
        "A": "thnx!that helped alot!"
    },
    {
        "video_name": "JEHejQphIYc",
        "Q": "Why do you move the decimal to the left when you're done? Why don't you move it to the right ?Watch 2:00",
        "A": "He moved the decimal to the left. Are you sure you know your right form left?"
    },
    {
        "video_name": "JEHejQphIYc",
        "Q": "I don't get it. In Sal's other videos, he said that you have to line up the decimals. That's what my teachers, my mom, and my dad say too. But at 0:40, Sal says that you do not need to do that because it doesn't really matter. So then which theory is correct?",
        "A": "For addition and subtraction, you have to line up the decimals for place value purposes. For multiplication, both methods are correct. If you line up the decimals for this example you d have 32.12 x 0.50 ------------ Any number multiplied by 0 is 0, so the 0 in the hundredths place does not affect the answer. You d get the same result by not lining them up, multiplying by 5, then counting the decimal places."
    },
    {
        "video_name": "JEHejQphIYc",
        "Q": "At 2:03, Sal shows us the decimals have to move up 3 times. what if the numbers behind the decimal are all the same?",
        "A": "Then you just keep multypling them even if they are the same."
    },
    {
        "video_name": "JEHejQphIYc",
        "Q": "I don't understand what he is doing at 1:07 if some one could help",
        "A": "He is doing multiplication of numbers. You ignore the decimal point at first, Since there are 3 places after the decimal point between the two numbers that means there will be 3 decimal values to the right of the point in the result."
    },
    {
        "video_name": "JEHejQphIYc",
        "Q": "At 0:06, when he said \"12 hundredths\", didn't he mean \"12 tenths\"?",
        "A": "0.12 it means that tihs means a decimal"
    },
    {
        "video_name": "xXIG8ouHcsc",
        "Q": "At 0:58,how do you take the number like that?",
        "A": "When you have a number less than the dividend, put that under it, subtract it, and then bring the next number down."
    },
    {
        "video_name": "xXIG8ouHcsc",
        "Q": "@2:06-7 you say that \" 7 times 1 is 28 plus 1 is 29\".",
        "A": "If at 2:11 you look in the bottom right corner, a pop up box says Sal said `7 times 1 is 28 but meant 7 times 4 is 28"
    },
    {
        "video_name": "xXIG8ouHcsc",
        "Q": "what does he mean by \"roughly\" at 2:40",
        "A": "Roughly, usually means it is an estimate."
    },
    {
        "video_name": "xXIG8ouHcsc",
        "Q": "Am I the only person who finds 0:14 / 0:13 funny?",
        "A": "The dog howling in the background is kinda funny. : )"
    },
    {
        "video_name": "AmN0YyaTD60",
        "Q": "I have a question: What does Vi mean at 2:29? She says it went through 4-D space. What does she mean?",
        "A": "Chirality is impossible to change unless it is reflected. However, a reflection could just be called a rotation through an extra dimension."
    },
    {
        "video_name": "AmN0YyaTD60",
        "Q": "Did anyone notice that at 1:01 when Vi tries to cut a hexaflexagon on down the middle it doesn't work? I'm pretty sure this is because a hexaflexagon is a M\u00c3\u00b6bius strip of some sort. Does anybody know more?",
        "A": "Yes. All hexaflexagons are M\u00c3\u00b6bius strips. In fact, it is the properties of the M\u00c3\u00b6bius strip that allow hexaflexagons to do what they do."
    },
    {
        "video_name": "AmN0YyaTD60",
        "Q": "at around 0:40 vi mentions how to take apart a hexaflexagon but she never mentions how to make one in any of her videos, does anyone know how to make one?",
        "A": "if you google hexaflexagon you will find hundreds of webpages that have instructions."
    },
    {
        "video_name": "AmN0YyaTD60",
        "Q": "At 3:55 I like how the hexaflexagons are tied in with the food. Very Creative isn't it?",
        "A": "Yes,yes it is."
    },
    {
        "video_name": "AmN0YyaTD60",
        "Q": "to my understanding, with a trihexaflexagon, one side has two combinations when flexed. the picture becomes reversed. but as shown on 3:05 to 3:09 it is shown that on a hexahexaflexagon one side has THREE combinations. is that right? and if so how can you tell where the different parts line up to make three pictures out of one design? and if im wrong, is it just a trihexaflexagon with a clever design? thank you :)",
        "A": "I had this same question, and I watched it again--when she shows the snake hexaflexagon again, she mentions that is is actually a hexahexaflexagon. (thanks for finding the time stamp)"
    },
    {
        "video_name": "AmN0YyaTD60",
        "Q": "At 0:32, why are the fingernails on Vi's right arm painted yellow and the ones on her other hand are not, and why is the sleeve on her left arm red and the other one is gray.",
        "A": "I don t know, but it sure looks cool!"
    },
    {
        "video_name": "v-H_7o7EMoU",
        "Q": "At 5:25, why is Vf not subtracted from Vg?",
        "A": "Because it s revolving around Y=1, so it s like the functions are flipped, with g being below f"
    },
    {
        "video_name": "v-H_7o7EMoU",
        "Q": "3:90\nDoes it matter whether the exponent (2), is in the parentheses or outside the parentheses?",
        "A": "YES! Overwhelmingly, yes. When you do this, you ll use [R(x)^2 - r(x)^2]. If you put it like R(x^2), you re subbing in every x for x^2, which is wrong; if you write [R(x) - r(x)]^2, you ll actually have [R(x)^2 - 2R(x)r(x) + r(x)^2], which is not proper either. You must square each function individually!"
    },
    {
        "video_name": "VMEV__2wW3E",
        "Q": "at 0:22 why does he mention the squaring I thought the squaring was finished",
        "A": "He s squaring both sides of the equation to get rid of the square root sign on the left. The effect of that shows up on the right side, where the (x+2) quantity is squared."
    },
    {
        "video_name": "VMEV__2wW3E",
        "Q": "I didn't understand what Sal said at 3:33 that by taking the negative square root we would have satisfied the equation with -5.\nBut how is that possible since the Square Root of a Negative number does not exist, it's an imaginary number!",
        "A": "He is talking about the negative square root not the square root of a negative; there is a difference. He is talking about taking the square root of the positive number, then flipping the sign so the answer is negative. You are right to be confused though, because -(sqrt)() looks a lot like (sqrt(-)."
    },
    {
        "video_name": "Jkr4FSrNEVY",
        "Q": "OK, so around 3:00, you say that n is a moment in time. Why is it then the limit as n approaches infinity? Shouldn't it be approaching 0 because it becomes a smaller and smaller interval?\nFor example, a minute is 1/60 and a second is 1/3600.\nThat's the one thing that doesn't make sense to me. O.o",
        "A": "n is the number of of trials. If you think of 1 hour allowing 24 trials in a day. you can then extrapolate from that that if we could do an infinite number of trials in one hour, we would have to make the length of time a trial could take shorter and shorter. The idea is to get to the point where a trial represents an instant in time."
    },
    {
        "video_name": "Jkr4FSrNEVY",
        "Q": "Hi Sal... at 12:00 you simplified lim<n->\u00e2\u0088\u009e> (1-\u00ce\u00bb/n)^n as e^-\u00ce\u00bb and lim(n->\u00e2\u0088\u009e> (1-\u00ce\u00bb/n)^-k as 1. Question is when you could simplify (1-\u00ce\u00bb/n) as (1-0) as n approaches \u00e2\u0088\u009e in lim(n->\u00e2\u0088\u009e)(1-\u00ce\u00bb/n)^-k.. why did not you simplify the lim(n->\u00e2\u0088\u009e)(1-\u00ce\u00bb/n)^n as (1-0)^n which would also be 1 as n approaches \u00e2\u0088\u009e. which would have left us with further simplified value of (\u00ce\u00bb^k/k!) ?",
        "A": "I wondered about this too. If we only look inside the parentheses, they are both the same, and as n approaches infinity, the contents of the parentheses approach 1. And 1 raised to any power, including infinity, is still 1. So the contents of the parentheses must not be exactly 1, but something infinitely close to 1. Then when you raise it to any non-infinite power it is still infinitely close to 1. But if you raise it to an infinite power, it can come to something other than one... I am guessing."
    },
    {
        "video_name": "Jkr4FSrNEVY",
        "Q": "At 9:00, is he using the l'hopital's rule?",
        "A": "You could deduce that using L H\u00c3\u00b4pital s Rule, but what Sal did was actually simpler. As n approaches infinity, the lower order terms would become negligible in comparison with n^k. To be a little more formal, you could divide the numerator and the denominator by n^k, and you would get: (1 + a/n + b/n^2 + c/n^3 + ...)/1, where a, b, c are constants that we need not to know, because if we take the limit as n approaches infinity, it ll become: (1 + 0 + 0 + 0 + ...)/1 = 1"
    },
    {
        "video_name": "Jkr4FSrNEVY",
        "Q": "I got a little confused around about \"7:30\", does n^k make it's way into the numerator because there are 'k' terms, and since it's a limit to infinity, terms like (n-1)(n-2)...(n-k+1) become effectively equal to (n)(n)...(n), which is n^k?\n\n(i.e. if n is infinity, then n is approximately equal to (n-1)?)",
        "A": "to jbh212. Lets say k=3 when n=10 10*9*8/10*10*10 = .72 (10*10*10 is also 10^3 or n^k) when n=100 100*99*98/100*100*100 = .9702 when n=1000 1000*999*998/1000*1000*1000 = .997002 therefore as n approaches infinity n(n-1)(n-2)....(n-k+1)/n^k will approach 1. hope this helps"
    },
    {
        "video_name": "FYMn61HLw1k",
        "Q": "at 01:45, Sal said \"derivative\" instead of \"vector\", Its very minor issue, but an annotation would be much appreciated. Thanks",
        "A": "You help create annotations yourself by clicking on Report a mistake above, and writing out what the mistake is and give a time stamp. Then, if it is approved by the guardians, it becomes an annotation."
    },
    {
        "video_name": "FYMn61HLw1k",
        "Q": "Why is the multiplication by a differential \"dt\" considered to be hand wavy (like Sal does at 2:19)? What would be a more rigorous way of showing this? I use/see this trick so often it makes me worry that I don't know the answer to this question.",
        "A": "Exactly! It s just that dr/dt is nowadays not even thought of as a concrete fraction, dr is the change in the output and dt is the change in time. It s introduced as notation, so Sal has to accommodate for those guys too so he can t make it seem as simple as it actually is to multiply out dts."
    },
    {
        "video_name": "FKraGDm2fUY",
        "Q": "At 0:57, Sal says d/dx [y(u(x))] = (dy/du) * (du/dx). Shouldn't this be (d [y(u(x))]/du) * (du/dx)? Because Sal is implying that d/dx [f(g(x))] = (d [f(x)]/d [g(x)]) * (d [g(x)])/dx.",
        "A": "So you wrote: Shouldn t this be (d [y(u(x))]/du) * (du/dx)? Try this instead: ( d [y(u(x)]/d[u(x)] ) * ( d[u(x)] / dx ) Notice I included a the whole u(x) in place of the lone u that you sometimes wrote. By writing just u , you just used a short hand notation for u(x) . This is what Sal is doing doing by writing dy/du, instead of: ( d[y(u(x))] / d[u(x)] ) * ( d[u(x)] / dx ). Either way is fine, if you know that y is short for y(u(x))."
    },
    {
        "video_name": "FKraGDm2fUY",
        "Q": "At 4:26 and thereafter, Sal uses the notion from his previous video about the change in continuos function being 0. But, in that video, u was the dependent variable and x was the independent variable, so my question is that how can he take that same logic here, that as delta x increases delta u also increases, whereas in this case u and x have no relation at all?",
        "A": "At the beginning of this video, he s trying to take d/dx (y(u(x)). What this means is that u is a function of x. And y is a function of u(x). x is the independent variable, while u is the dependent that depends on x. And y is a dependent that depends on u. Hope that makes sense."
    },
    {
        "video_name": "CGZZINHT0I8",
        "Q": "i am confused at 2:21. why does the -1/2 mean?",
        "A": "As x increases by 1, y decreases by 2. It doesn t matter if the rate of change is -1/2 or 1/-2. They are both the same value."
    },
    {
        "video_name": "CGZZINHT0I8",
        "Q": "At 1:10 you say that they already give the x-intercept. How do you find the x- intercept on a table that doesn't have a zero?",
        "A": "If there are at least two points, you can first determine the equation of the line, which you can then use to find the x-intercept."
    },
    {
        "video_name": "yDwIfYjKEeo",
        "Q": "At 4:48, in order to prove that the length of a vector does not change when represented with respect to orthogonal basis, shouldn't we start by finding length of C^-1*X rather than C*X because X represented in basis B is equal to C^-1*X ?",
        "A": "He s using C here to represent any transformation with matrices whose columns are orthonormal basis vectors. Its not important here that it can transform from some basis B to standard basis. We know that the matrix C that transforms from an orthonormal non standard basis B to standard coordinates is orthonormal, because its column vectors are the vectors of B. But since C^-1 = C^t, we don t yet know if C^-1 is orthonormal. All we know is that its r o w vectors are an orthonormal set."
    },
    {
        "video_name": "yDwIfYjKEeo",
        "Q": "06:28\n\nSo matrix x matrix isn't commutative but matrix x vector is?",
        "A": "Ax = b is the product of mxn and nx1 matrices equal to an mx1 matrix. So matrix x vector is also matrix x matrix, so it s not commutative either."
    },
    {
        "video_name": "8Ft5iHhauJ0",
        "Q": "at 4:11 cant you just write R",
        "A": "Yes. A common notation when you have a remainder is to write R and then the number remaining."
    },
    {
        "video_name": "8Ft5iHhauJ0",
        "Q": "3:04 Three times nothing is not 23! Anything times 0 is 0!",
        "A": "Yes it has to be"
    },
    {
        "video_name": "8Ft5iHhauJ0",
        "Q": "at 4:34 my computer skipped so what didi the guy say",
        "A": "lets do division with bigger numbers"
    },
    {
        "video_name": "8Ft5iHhauJ0",
        "Q": "At 10:52 i don't understand why 0 became 10 and 6 became 5 please help.",
        "A": "it is called borrowing"
    },
    {
        "video_name": "BYTfCnR9Sl0",
        "Q": "I totally get how he is using the graph of the derivative to predict the slope of the exponential function at 5:15 but can the same math be applied in a reverse manner to achieve the anti-derivitive graph?",
        "A": "You can piece together AN anti-derivative graph, but not THE anti-derivative graph. The derivative graph tells you the SLOPE of the anti-derivative graph for any point x, but you cannot tell from the slope where the graph actually is. So you can construct an infinite number of possible graphs from the information of slope, but you have to add an arbitrary bit of data to figure out exactly where to start the graphing."
    },
    {
        "video_name": "I9efKVtLCf4",
        "Q": "1:38 could he have multiplied the three numbers in any order?",
        "A": "Yes, the numbers could be multiplied in any order"
    },
    {
        "video_name": "I9efKVtLCf4",
        "Q": "at 1:27 how does he get the block to rotate without touching it",
        "A": "It rotated automatically. Clicking on the box would make it stop."
    },
    {
        "video_name": "I9efKVtLCf4",
        "Q": "at 1:30 you said width, which side is the width?",
        "A": "The width is taken as the shorter dimension of the perimeter."
    },
    {
        "video_name": "Tm98lnrlbMA",
        "Q": "At 2:34, wouldn't it be x -> x+1?",
        "A": "Ok thank you now I understand"
    },
    {
        "video_name": "Tm98lnrlbMA",
        "Q": "Why does Sal use 'x' (0:40) instead of some other letter like 'k' or 'a'?",
        "A": "Because the Europeans who started learning about algebra had no letter or sound for it. They borrowed the character x from another culture. when it was translated to Latin, it became X."
    },
    {
        "video_name": "Tm98lnrlbMA",
        "Q": "why do people usually use x as their variable and not any other letter? 00:39 seconds in video",
        "A": "x and y have been used traditionally as co-ordinates in geometry with the x-axis typically representing the independent variable. So it gets used often in functions/equations"
    },
    {
        "video_name": "Tm98lnrlbMA",
        "Q": "At 1:00 he says that we could use any symbol,would people still under stand what we were talking about?",
        "A": "in a math course you will see all sorts of letters and symbols used to represent a variable. theta, x, y, z are some of the most common"
    },
    {
        "video_name": "Tm98lnrlbMA",
        "Q": "at 1:06, why does Sal mention smiley face plus three? isn't it just seven plus three??",
        "A": "Yes, but he s assuming we don t know that x is actually seven. It is considered an unknown. His point is that we can represent that unknown with an x, or a smile face or any type of symbol."
    },
    {
        "video_name": "Tm98lnrlbMA",
        "Q": "At 2:45 sal says \"This is just a convention that kind of comes to use from history.\" but what is the history behind this. I have always wondered. Why can't I use a amiley face, heart, star etc. in an equation or expression.",
        "A": "Because the teacher would not understand it:)"
    },
    {
        "video_name": "Tm98lnrlbMA",
        "Q": "Why can't we use just symbols, as Sal mentioned at 2:48 in a equation, instead of x and y? Why can't use smiley faces and stars and swirls in 6th grade math class, just to spice up our equations?",
        "A": "Technically you can, but you don t because in your future classes you will use x and y, so it is simpler to learn with x and y when they are what you will use in future classes."
    },
    {
        "video_name": "Tm98lnrlbMA",
        "Q": "At 0:53, what did x stand for? I don't really understand.",
        "A": "x would be the thing you have to figure out, like a puzzle. x+3=10 So, what +3 equals 10 7 7+3=10 You will get it, then it will get hard.......then you will get it....so on and so fourth........ till you learn it all!"
    },
    {
        "video_name": "Tm98lnrlbMA",
        "Q": "At 1:04 you put a smiley face i understand that you could use any letter, but why shapes or symbols and not letters?",
        "A": "Sal just used shapes and symbols as examples. You can use anything even letters (there is no rule but letters is the one we use)."
    },
    {
        "video_name": "Tm98lnrlbMA",
        "Q": "At 1:39, the smiley face and the underscore are not even letters!",
        "A": "those smiley and underscores are just for examples for showing that anything can be used instead of x"
    },
    {
        "video_name": "Tm98lnrlbMA",
        "Q": "at 2:14, isn't x and y axis'?",
        "A": "x and y can be variables too. But that type of equation is directly related to x and y axes. The y is for our y value on a graph, and our x is for the x value on the graph. So in that equation y = x + 1, you can have 3 for x and get 4 for y. So our point would be (3, 5) on the graph."
    },
    {
        "video_name": "Fvi9A_tEmXQ",
        "Q": "On a more philosophical note (\"7:07\"): in reality rain does tend to fall in a super exact quantity, however, we just don't measure it super exact.",
        "A": "Indeed - how could anyone get it to the nearest ton?"
    },
    {
        "video_name": "Fvi9A_tEmXQ",
        "Q": "Isn't the part you mentioned from 4:40 that use calculus to find out the probability as 1.9<Y<2.1 should be CDF(cumulative distribution function) instead of PDF?or the PDF includs CDF??\nplease answer me the question so i can find out whether i got wrong\u00ef\u00bc\u0081thx",
        "A": "A CDF function, such as F(x), is the integral of the PDF f(x) up to x. That is, the probability of getting a value x or smaller P(Y <= x) = F(x). So if you want to find the probability of rain between 1.9 < Y < 2.1 you can use F(2.1) - F(1.9), which is equal to integrating f(x) from x = 1.9 to 2.1."
    },
    {
        "video_name": "Fvi9A_tEmXQ",
        "Q": "At 4:30, Sal is saying that the probability of exactly 2 inches is zero. But how can the probability be zero. Isn't there even at least a one in a million chance that it could be exactly 2 inches? So couldn't there be like a 0.0000000001 chance?",
        "A": "Well, to achieve a 0.0000000000...1 percent chance, you would have to have an infinite number of zeroes. This means you would never even reach the 1."
    },
    {
        "video_name": "Fvi9A_tEmXQ",
        "Q": "Can someone explain to me how at 5:00 P(lY-2l<.1) is the same as P(1.9<Y<2.1)? If Y=5, than lY-2l=3, which is outside that range. I think that since Y is a continuous random variable, maybe saying Y=5 isn't possible, or has no meaning, but that doesn't answer my question and just confuses me even more. Thanks!",
        "A": "Absolute value means distance from some number. So lY-2l is the distance of Y from 2. And lY-2l<0.1 means that Y is less than 0.1 away from 2 on the number line. That means Y has to be less than 2.1 on the right side of 2 and greater than 1.9 on the left of 2. Or combining both results, 1.9<Y<2.1. And finally P(lY-2l<0.1) is asking what is the probability Y is a number greater than 1.9 and less than 2.1?"
    },
    {
        "video_name": "Fvi9A_tEmXQ",
        "Q": "@0:20 Sal says that a discrete random variable is finite but according to my Stats book but he is wrong. Say you have a random variable X and you say X=1,2,3,... where 1,2,3.. represent the first flip that a head appears on a coin toss. Now technically X is a discrete value with an infinite amount of discrete possibilities (albeit many are very unlikely).",
        "A": "Think of the poision distrubution."
    },
    {
        "video_name": "Fvi9A_tEmXQ",
        "Q": "7:08 According to this there will not be rain because if we consider each specific amount of rain then is zero possibility of any rain. for e.g. you say what is the chance of between 0 and 1 inch of rain. I say look (in theory) the chance of every possible amount of rain falling. you then get every chance as 0 thus there 0 chance of rain. How is this possible?",
        "A": "A very important thing to wrap your mind around in probability theory is that an event with probability 0 can occur. Imagine throwing a dart at a dartboard. The probability that it hit the infinitely precise spot that it did is 0, and yet it did. These are the kinds of counter-intuitive things that can happen when your event space is infinite -- if you divide 1 into an infinite number of equally likely outcomes, there s no way for any of them to be non-zero even though they are all possible."
    },
    {
        "video_name": "Fvi9A_tEmXQ",
        "Q": "Isnt the stuff drawing at 9:55 and Probability mass function? Im just curious?",
        "A": "Since at 9:55 Sal is referring to a continuous random variable (and not a discrete one), so his picture is of a probability density function. Now, if it was a discrete random variable, then it would technically be called a probability mass function (or often just probability function )."
    },
    {
        "video_name": "Fvi9A_tEmXQ",
        "Q": "At 7:47 Sal says \"Hopefully that is not an infinite number.\" Isn't integral of f(x) on the interval where it is defined (in our case from 0 to inf) always 1? In other words, the probability that any possible event will happen is 1 (or the area under f(x) is 1).",
        "A": "I think that Sal was speaking in generality. The integral of every arbitrary function will not necessarily converge. If it does, then we can scale it appropriately to be a pdf, but if it doesn t converge, we might be out of luck."
    },
    {
        "video_name": "RweAgQwLdMs",
        "Q": "at 1:40 he talks about a positive and negative version of something being squared to give the same result. A positive or negative x value of 6 can be squared to equal 36. Is that the same thing as saying a positive or negative square root of 36 is equal to x?\n\nWhy don't we write( positive or negative x) = (positive or negative 6)? Why make only one side positive or negative when you're taking the square root of both?",
        "A": "Don t think of -x as being a numbered value, think of it as the opposite value of x. From that, you see that the value of -x can be deduced from the value of x. If x=10 then -x=-10, if x=-10 then -x=10. Also, for that matter, the negative value of x can t be -10 or 10, it has to be one of the two depending on the value of x used. Hopefully that makes sense."
    },
    {
        "video_name": "xjkbR7Gjgjs",
        "Q": "At 2:36 does it matter if \"a\" comes first in \"ab\" or can you also write \"ba\"? Since in multiplication, the order shouldn't matter?",
        "A": "No, it doesn t matter - as you say, ba is equivalent to ab. The convention (I think) for ease of reference is to list variables in alphabetical order, so ab is generally preferred."
    },
    {
        "video_name": "xjkbR7Gjgjs",
        "Q": "between 2:40 and 3:00 captions have wrong spelling.",
        "A": "For me it is Spelled Correctly.I think they Fixed it :)"
    },
    {
        "video_name": "AOxMJRtoR2A",
        "Q": "Sal said that not all equations can be graphed as a line so can all LINEAR equations be graphed as a line? 6:11",
        "A": "yes because you can always graph a STRAIGHT line, which is linear, but you can t graph a nonlinear equation because it is not a straight line it would have a curve like an oval or a circle"
    },
    {
        "video_name": "AOxMJRtoR2A",
        "Q": "At 7:28 Sal mentions a constant. Isn't the formula for constants\nK= y/x ?",
        "A": "Constants are values that do not change. Notice: when Sal uses the word constant , he is referring to the 12 and the -3. These don t change. The variables in these equations represent the values that can change."
    },
    {
        "video_name": "AOxMJRtoR2A",
        "Q": "At 0:29, what does all the xy pairs that satisfy this equation mean?",
        "A": "Linear equations are called linear equations because when you graph them in a coordinate system, they create a straight line. The line is made up of an infinite set of points. Each point can be defined by an ordered pair of the form (x, y). So, the phrase you are asking about is referring to all the points on the line. Each point will satisfy the equation (i.e., make the equation be true). Hope this helps."
    },
    {
        "video_name": "WvxKwRcHGHg",
        "Q": "Around 5:50 Sal says that the functions don't need to be defined at c only when approaching c? How come this is the case? Don't the functions need to be defined at c as well to properly 'squeeze' the middle function?",
        "A": "They don t need to be defined at c because the squeeze theorem is used in reference to a limit, which looks at values as they get very close to a number, rather than at the number itself."
    },
    {
        "video_name": "WvxKwRcHGHg",
        "Q": "at 5:34, what if c was not at the point where all three functions met. The limit of them would then not be equal, correct?",
        "A": "That s correct, Maral! The example Sal gave us is to explicitly show us the point at which the limits of each of the graphs could all be equal to each other. However, (in terms of the graph given) if the question was what is the limit of g(x) as it x approaches c/2, then g(x) would not be equal to the limits of f(x) or h(x)."
    },
    {
        "video_name": "l5zJvZKfMYE",
        "Q": "\u00c3\u00ae\u00c2\u00b7\u00c3\u00ae=1 ?\nIs this expression true?\n\nThat is, when one has the x-unit-vector and the dot-product with another x-unit-vector (as at 2:50), does that always equal one? Because it always seems to disappear when this happens.",
        "A": "yes it is true. \u00c3\u00ae\u00c2\u00b7\u00c3\u00ae=|i|*|i|*cos0=1 the dot product is evaluated as the product of the magnitude of each vector times the cosine of the angle between them."
    },
    {
        "video_name": "l5zJvZKfMYE",
        "Q": "At 1:31, what does dr mean? Also is there any video that Sal mentioned that explains about dr?",
        "A": "The dr simply means the derivative with respect to r. I am sure that he mentions this in other videos. This also exists when you have to do triple integrals in spherical or cylindrical coordinates."
    },
    {
        "video_name": "l5zJvZKfMYE",
        "Q": "At 3:45 he says that he considers the minimum and maximum values of x and states that he can split this into two function of y on x. Why is this legal to do? The way he draws it I think it's legal on this example, but in general those half loops may not be functions of x at all (such as when the curve takes multiple y values for a given x value).",
        "A": "That is because you are not interested in the vector field as a whole, but only in the components that affect your path. So if your path is not random and follows one or more functions of x (or functions of y) the points you choose to calculate your vector field can also be written as a function of x or of y (Ex: the vector field is acting on the line y=x^2, so if F(x,y)= x+y is your vector field and the points you are considering are in the line then: F(x,y)=F(x,y(x))= x+x^2)"
    },
    {
        "video_name": "l5zJvZKfMYE",
        "Q": "So where did the surface that Sal integrates the region R upto at 11:40, that forms the volume, come from again? I can't understand the relationship between partial P wrt y and the magenta surface drawn in teh diagram.",
        "A": "The partial derivative of P with respect to y is a scalar function that, in general, depends on x and y. Therefore, it can be represented as a surface over the xy plane."
    },
    {
        "video_name": "_HFTRQRpzFM",
        "Q": "at 2:08 sal says \"conjugate pairs\". what does it mean?",
        "A": "For a nonreal number, you can write it in the form of a + bi To form its conjugate, just change the sign of the imaginary portion: a \u00e2\u0088\u0092 bi"
    },
    {
        "video_name": "SgKBBUFaGb4",
        "Q": "@ 0:22 , he says that p=-1, how does he know?",
        "A": "Answer this question..."
    },
    {
        "video_name": "SgKBBUFaGb4",
        "Q": "At 4:06 How does -3 change to -4?",
        "A": "3p-p= -4p just like -3-1= -4 if you add variable p to all the values it will be 3p-1p= -4p"
    },
    {
        "video_name": "SgKBBUFaGb4",
        "Q": "At 0:37 why is it -4p and not -3p?",
        "A": "If you have negative three of something (-3p), and you take away one more of that something (-1p), you are going further into the negatives. You will have even less. -3p - 1p = -4p"
    },
    {
        "video_name": "SgKBBUFaGb4",
        "Q": "at 3:11, where did the -3 come from?",
        "A": "He already found the solution set, and that is that p is greater than -4. Now, he s plugging in a number for p to see if it works. The number he picked is -3, which is greater than -4, so it should work."
    },
    {
        "video_name": "SgKBBUFaGb4",
        "Q": "at 0:01, is one plus one 2? or is it 11. or 21. Please help.",
        "A": "One plus one is 2. 11 is 1 plus 10, and 21 is 1 plus 20. Helpful?"
    },
    {
        "video_name": "SgKBBUFaGb4",
        "Q": "at 0:45, I don't get why you would make it -4p-7 < 9.",
        "A": "Because the p is positive ( -3p-7 < p+9 ) and to get rid of it, you must subtarct it from both sides of the equation. As to say if the p was negative then the result of that problem would be ( -2p-7 < 9 ) .. Understand?"
    },
    {
        "video_name": "SgKBBUFaGb4",
        "Q": "From 1:40 to 1:45, why does Sal call the inequality an ''Equation''?\n\nI know that ''Equations'' and ''Inequalities'' are processed the same way, but still, why does Sal use the term ''Equation'' when we are dealing with ''Inequalities''?\n\nThank you!",
        "A": "An inequality is processed the same way as an equation, and yes, he should have been saying inequality. He probably made an honest mistake, like all of us do at some point in time."
    },
    {
        "video_name": "SgKBBUFaGb4",
        "Q": "An 2:34, isn't -3.9999... actually considered to be EQUAL to -4? (Just type it on google and you'll see).\n\nReference: Converting repeating decimals to fractions.",
        "A": "-3.999999999999... will go on repeating forever and never be exactly equal to -4, though it is very close."
    },
    {
        "video_name": "SgKBBUFaGb4",
        "Q": "so at 2:17 its say to switch the sign is it only when dividing a negative number or just dividing any number?",
        "A": "Just by dividing by a negative number. For instance: 10>4. If you wanted to divide by -2, it wouldn t be -5>-2, you would have to flip the sign to make it be -5<-2."
    },
    {
        "video_name": "SgKBBUFaGb4",
        "Q": "1:17 why is it still less then",
        "A": "If \u00f0\u009d\u0091\u008e < \u00f0\u009d\u0091\u008f, then \u00f0\u009d\u0091\u008e + \u00f0\u009d\u0091\u0090 < \u00f0\u009d\u0091\u008f + \u00f0\u009d\u0091\u0090, because we ve moved \u00f0\u009d\u0091\u008e and \u00f0\u009d\u0091\u008f the same amount, and in the same direction, along the number line."
    },
    {
        "video_name": "rLdoKZ7w0xI",
        "Q": "@ 8:57 Sal states that E(-Y)^2 = -E(Y)^2. Did I miss something before? What is the proof or reasoning behind that? Does that follow from the previous identity for the substraction of random variables? (i.e. follows from E(X-Y) = E(X)-E(Y)?) Thanks!",
        "A": "If I were to ask you what the expected value of Y is, it would just be the mean of Y, lets call it x. Then, if I asked you what the mean of -Y is, it would just be the mean of Y multiplied by -1, so -x. So, E(-Y) = - E(Y), and therefore E(-Y)= -x. You may also want to review linear transformations of a distribution , most common stats textbooks discuss this."
    },
    {
        "video_name": "rLdoKZ7w0xI",
        "Q": "I have a question regarding your Var formula of summation at 5:18.\nMy formula handbook states Var[X+Y] = Var[X] + Var[Y] + 2Cov[X,Y].\nIs this not the same as Var[Z] where Z = X + Y?\n\nThanks for your great videos!",
        "A": "The covariance of two independent random variables is 0, so var[x+y] = var[x] + var[y] because cov[x,y] = 0 in this case. The regression line between two independent variables has a slope of 0 (has maximum square error). And from the slope formula of the regression line m = cov(x,y) / var(x) => 0 = cov(x,y) / var(x). Since var(x) can t be 0 (division by 0 is not allowed) => cov(x,y) = 0"
    },
    {
        "video_name": "FWEqB0J6mgA",
        "Q": "At 1:43, why did Sal write, 46-4= tens+ones. Don't understand",
        "A": "Not sure what you really asking but it maybe this: 46 is 4 tens and 6 ones, 4 is 4 ones.so when you are adding or subtracting make sure your place values are lined up correctly under each other. 46 -4=42 If you lined them up wrong you might get 06 for a wrong answer. Hope this helps."
    },
    {
        "video_name": "lf4BBZK1Vzs",
        "Q": "at 2:27,do we draw 4 or 5 cans or do we draw cans equal to 45?",
        "A": "We draw 5 cans because we are given that she had 5 cans."
    },
    {
        "video_name": "w2M5CzTFYfI",
        "Q": "On 2:20 Sal mentioned 46 as 50 because because it is 6 more than 40 and 4 less than 50.",
        "A": "it s called rounding up"
    },
    {
        "video_name": "JVrkLIcA2qw",
        "Q": "Can this be an example of a 3:4:5 right triangle?",
        "A": "yes it can be. 3 x 5 =15, 4 x 5=20 and 5 x 5=25. Those are equal to the the lengths of the right triangle. and they all have to be multiplied by 5 to get 15 20 and 25 so yes it can be an example of sides in ratio of 3:4:5. Hope this helped you!!"
    },
    {
        "video_name": "JVrkLIcA2qw",
        "Q": "2:24, does this only work with right angle triangles",
        "A": "at 2:24 yes it only works with right triangles. He states that in the video..."
    },
    {
        "video_name": "JVrkLIcA2qw",
        "Q": "In 3:33 Sal starts to figure out what the squares are for each side but in the end the final step he got 25^2 equals r, which he would have gotten if he had just added the 15 and 20 together. Will this work for all pythagoreon theorem eqations?",
        "A": "No. If he had added both sides together, he d get 35^2 instead of 25^2. You ve got to add the squares together and take the square root of the result to get the length of the hypotenuse. There s a reason it s a^2 + b^2 = c^2 and not a + b = c :)"
    },
    {
        "video_name": "JVrkLIcA2qw",
        "Q": "At 2:53, why did Sal square 15, and 20?",
        "A": "The Pythagorean theorem states that for any right triangle, a^2 + b^2 = c^2 where a and b are the two shorter sides, and c is the longer side (the hypotenuse). In the video, 15 and 20 were a and b , the two shorter sides. Sal wanted to find the longer side, c . So he used the equation, plugging in 15 and 20 for a and b and then solving for c . It looked like this: a^2 + b^2 = c^2 became 15^2 + 20^2 = c^2 So then he just had to solve for c , and that s what he did. I hope this helps."
    },
    {
        "video_name": "JVrkLIcA2qw",
        "Q": "At 4:05, Sal says, \"The principal root\". Is principle root the positive square root?",
        "A": "Yes, the principal root of a number is positive. If you re using the principal root, an expected negative result will have the minus before the root like so -\u00e2\u0088\u009a9."
    },
    {
        "video_name": "JVrkLIcA2qw",
        "Q": "Why did Sal,at 3:11, call the line that he drew over the hypotenuse green?\nIt was more of a yellow.",
        "A": "Color doesn t matter ;)"
    },
    {
        "video_name": "JVrkLIcA2qw",
        "Q": "At about 4:03 Sal talks about taking the principle root. What does it exactly mean?? Is it always the positive root??",
        "A": "Yes, the principal square root is always the positive root."
    },
    {
        "video_name": "m-pRnFIeoNA",
        "Q": "what happens if you switch the equation in 1:04",
        "A": "If you mean switching as if to make 7/3 divided by 2/5, then it would go like this. You flip 2/5 into 5/2 so that it will be multiplication. First you multiply the denominators. 3 times two is 6. Then you multiply the numerators. 7 times 5 is 35. We get 35/6. We divide 35 by 6 and get 5 remainder 5. We keep the denominator, so the answer will be 5 5/6. I hope I answered your question!"
    },
    {
        "video_name": "Yw3EoxC_GXU",
        "Q": "i don't understand how this works does the dot such as in : 4'5 ( the ' means dot . look at 1:11 to 3:30 ) does that dot system work fo more than 1 digit numbers ??",
        "A": "Yes, it works anywhere the x would work or anywhere that you can substitute in the word times . Which way you label the fact that you are multiplying is up to you. One important note.......the dot will be in the air .......dots on the ground are decimals. I would not use the dot to multiply when I m multiplying decimals until I am very comfortable with multiplication and decimals! Good luck!!"
    },
    {
        "video_name": "Yw3EoxC_GXU",
        "Q": "he spelt times like tmes in the video\n0:48",
        "A": "yeah gusse your write opps spelt guess wrong XD"
    },
    {
        "video_name": "Yw3EoxC_GXU",
        "Q": "At 1:48 what does (7) (5) mean",
        "A": "That means the same thing as 7x5. (7)(5) = 7x5"
    },
    {
        "video_name": "kdMep5GUOBw",
        "Q": "7:07, Is the f(a,b) also being divided by h or is it outside of the fraction?",
        "A": "The f(a,b) is divided by h. This formula is based on what was covered in 1:24 to 2:46, and is very similar to the df/dx formula he derived. If you remember, the concept of this general formula is centered around dy/dx. dx, of course is h, while dy is equivalent to the numerator in the formula."
    },
    {
        "video_name": "BXHNzUaIRR0",
        "Q": "at 1:42 fomr 49 - 40 - 8 = 1 how is the awnser 1?",
        "A": "49-40=9, 9-8=1. 1:21"
    },
    {
        "video_name": "BXHNzUaIRR0",
        "Q": "At 1:14 Sal does the order of operations, why?",
        "A": "Once you re introduced into PEMDAS, you pretty much use it through your whole math life."
    },
    {
        "video_name": "BXHNzUaIRR0",
        "Q": "I'm only nine years old but i kind of get this. But i have two questions all the videos i have watched already has not helped me on algebra fractions and in 1:00 how does 72 equal 49 (sorry about the 2 i don't know how to make a small two)",
        "A": "It means 7 squared which is 49. Squared means the number times itself."
    },
    {
        "video_name": "BXHNzUaIRR0",
        "Q": "at 1:15 in video why is the answer to 10(-4)= -40 wouldn't it be 40 instead",
        "A": "When you multiply a positive number with a negative number, the result will be a negative. When you multiply a positive number by a positive number, the result will be a positive. When you multiply a negative number by a negative number, the result will be a positive. When you multiply a negative number by a positive number, the result it negative. This works the same for division."
    },
    {
        "video_name": "BXHNzUaIRR0",
        "Q": "at 0:47 and many other examples I noticed sal automatically adds parenthesis. how or why does he do that is there a rule there? he does that in almost all the videos dealing with the algebra",
        "A": "Parentheses signify multiplication. For instance 4(1 + 79) you do inside the parentheses first and end up with 4(80) which is the same as 4 x 80. Hope that helps. (Yes I am aware that I am using an x instead of a dot)"
    },
    {
        "video_name": "BXHNzUaIRR0",
        "Q": "0:11 sal says that it is 4, would it not be -4?",
        "A": "Yes, but just a few seconds later he corrected himself when doing the substitution so his verbal mistake did not affect the math involved. Try producing 3000+ videos and see if you can do it without the occasional minor slip of the tongue/mistake."
    },
    {
        "video_name": "BXHNzUaIRR0",
        "Q": "Did anyone notice that at 0:11, Sal said that b was 4, but at 0:18 he says that b equals -4?",
        "A": "It s hard to always say the right thing, but most of the time Sal is good about correcting his errors."
    },
    {
        "video_name": "BXHNzUaIRR0",
        "Q": "at 0:38 was strange how did he mad those equations?",
        "A": "he walked through it piece by piece and he replaced his unknown with what they are (displayed on the left) and then used order of operations to do the rest."
    },
    {
        "video_name": "BXHNzUaIRR0",
        "Q": "bracket means to multiply right ? but at 1:16 how can it be 49+(-40) and the result should be a negative. right or wrong?",
        "A": "The bracket does not mean to multiply. it means to do the equation inside the bracket first. But since its just 40 you ignore the brackets. Once your in Algebra 1 and combining like terms using the distributive property you will start multiply."
    },
    {
        "video_name": "ANnuQZyEKVo",
        "Q": "Isn't it a parallelogram because a rhombus is a parallelogram? I got this from 6:35.",
        "A": "A QUADRILATERAL is any shape with with 4 sides (or corners); a parallelogram must have 2 pairs of parallel sides; and true, a rhombus is a parallelogram with all 4 sides congruent (equal)."
    },
    {
        "video_name": "ANnuQZyEKVo",
        "Q": "At 0:40, how did he decide that the line would pass zero?",
        "A": "Try it out: x = 0; y = x/2; y = 0/2; y = 0; Therefore the line would pass zero."
    },
    {
        "video_name": "b76wePnIBdU",
        "Q": "what did sal say at 4:22?",
        "A": "The derivative of e^u is e^u; the antiderivative of e^u is e^u."
    },
    {
        "video_name": "b76wePnIBdU",
        "Q": "At 4:14,\n\nWhat is the antiderivative? Is that just the integral?",
        "A": "Yes: antiderivative \u00e2\u0087\u0094 integral"
    },
    {
        "video_name": "b76wePnIBdU",
        "Q": "At 3:01 how could he rewrite like that? I thought we couldn't bring variables outside of the integration expression.",
        "A": "Take as closer look, that did not happen, everything is after the integral symbol."
    },
    {
        "video_name": "b76wePnIBdU",
        "Q": "at 4:19 Why is e^u*du anti derivative e^u?",
        "A": "The function e^x is defined to be that the changing rate of x is equal to the function itself: e^x. The number: e actually did the trick. By the way, don t for get the constant when taking anti derivative."
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "How is it it that at 9:41, X is greater or equal to two? Shouldn't have the sign flipped after eight was divided by four, leaving the X all alone on the left side? Overall however, this really helped me study for my upcoming EOC.",
        "A": "The sign of an inequality ONLY switches when you multiply or divide by a negative number. If Sal had been dividing by -4 instead, the sign would have flipped."
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "Isn't what sal describes at 1:10 know as the Intersection of two functions? Just like 8:40 is the Union?",
        "A": "yes what he describes is commonly known as intersection"
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "3:34 does this whole process provide for different looking problems?",
        "A": "If your talking about the inequality then yes you use the same process to isolate the variable, but if you mean the number line you have to keep in mind that the direction the arrow goes depends on the sign used in the inequality."
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "Sal mentions at 11:30 that the words or and and affect the inequality greatly. What is the reason that these words will change the possible solutions sets to the inequalities?",
        "A": "If both conditions must be true, for ex. x < 3 and x > 1, then the solution is restricted to the intersection of both cases. The solution would be 2. If either condition can be true, for ex. x < 3 or x > 1, then the solution includes the union of both cases. The solution would include every number less than 3 and every number greater than 1. In this situation, there is an infinite number of solutions. So just by changing from AND to OR, the solution set changes drastically."
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "At 2:28, why does Sal put the x back into the 2 answers?",
        "A": "Because it saves up space and also makes it more comprehensible. There s nothing wrong with saying, for example, \u00f0\u009d\u0091\u00a5 < 5 and \u00f0\u009d\u0091\u00a5 > 0, but it s easier to write it as 0 < \u00f0\u009d\u0091\u00a5 < 5, or even \u00f0\u009d\u0091\u00a5 \u00e2\u0088\u0088 (0, 5)."
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "at 8:31, why does he draw an open circle? why can't it be filled in like the others?",
        "A": "An open circle means that the point does not count (in the cases of > or <), and an closed circle would mean the point counts (in the cases of \u00e2\u0089\u00a5 or \u00e2\u0089\u00a4)."
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "At 5:33 on the video how do you know when to divide",
        "A": "Thanks For the help"
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "At 10:30\nis it helpful to plot the inequality on a number line??",
        "A": "Yes it is helpful because it shows you what is higher/lower."
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "At 1:05, Is -5=< x-4 equation or inequality?",
        "A": "It is an inequality. Equations never have a less than (<) or greater than (>) symbol. They will only have an equals symbol."
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "At about 10:32,wouldn't you switch the inequality's sign to 'x is less than or equal to 2'?",
        "A": "no you would not swap the sign because you are not multiplying or dividing be a negative."
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "at 5:43 why does the sign flip",
        "A": "You only switch the inequality sign when you multiply or divide by a negative number. Example: 1 < 2 if you multiply both sides by -1, the inequality now becomes -1 < -2 which is incorrect! So you flip the sign to make it correct: -1 > -2. Hope you found that useful!!"
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "At 10:49, Is there some way to write both results as an interval? Or should it be separately?\n\n(-\u00e2\u0088\u009e, 2/3); [2, \u00e2\u0088\u009e)",
        "A": "You would use U , which means union. (-\u00e2\u0088\u009e, 2/3) U [2, \u00e2\u0088\u009e)"
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "At 3:10, you shaded the line to the right. My teacher tells me that if it has the less than (<) sign, it goes to the left, because that is the way that it points. This apparently isn't the case here, could you maybe explain?",
        "A": "He shaded to the right of -1 because he was shading for: x >= -1 When he is shading by x<= 17, he shades to the left of 17."
    },
    {
        "video_name": "A3xPhzs-KBI",
        "Q": "At 2:06 you said 13 + 14, but the real equation says 13 + 4. Is this a mistake?",
        "A": "Its a dramatic mistake, like dramatic as in the dramatic song. It s just super dramatic."
    },
    {
        "video_name": "STyoP3rCmb0",
        "Q": "0:00-3:49 it not clear can some one help me?plz ;(",
        "A": "If I do .5 x.5 I get .25. Its just like regular multiplying just add a decimal in front of it."
    },
    {
        "video_name": "STyoP3rCmb0",
        "Q": "0.6 is the same as 6 divided by 10 (video time is 0:34)\nI don't understand how its the same thing. can you please explain. Thank you",
        "A": "When dividing by ten, you can just move the decimal point one digit to the left. Conversely, if multiplying by ten, you can move the decimal point one digit to the right."
    },
    {
        "video_name": "STyoP3rCmb0",
        "Q": "so at 1:39 all you have to do is multiply this and that, then divide it by decimal? then the remainder is the decimal?",
        "A": "It is very straight forward just like Sal explained in the video. The way he explained seems to be longer than you think. When you actually do the work, you can eliminate the small steps and make it shorter and more simple too. I suggest you to watch the video over and over again. Use pause and replay as many times as possible, and try to understand little by little, then you should be able to get it. That is what I have been doing for myself too. Sincerely."
    },
    {
        "video_name": "STyoP3rCmb0",
        "Q": "at 3:25 khan said divide 54 by 10 but how do you do that",
        "A": "54/10 is not a whole number, but it is possible. The answer would be 5.4"
    },
    {
        "video_name": "STyoP3rCmb0",
        "Q": "At 3:21, i don't understand. HELP!",
        "A": "What Sal does at 3:21 is shows that he divided one of the two numbers (whose product we are calculating) by 10, and the product also ends up divided by 10. 9x0.6=9x(6/10)=(9x6)/10. Does that make sense?"
    },
    {
        "video_name": "STyoP3rCmb0",
        "Q": "(2:25) So what he is saying is that when there is a zero at the end of a number you move the decimal over that many times?",
        "A": "I couldn t watch the video, sorry. Did you mean when you multiply say 10 and 2.1? If you ignore the decimal for just a moment and multiply the two numbers, you get 210. But since there was a decimal with just one number to the right of it, you d move the decimal of the answer one number to the right. The final answer would be 21.0 I hope this helped, and if you have any more questions, I ll be perfectly happy to help!"
    },
    {
        "video_name": "STyoP3rCmb0",
        "Q": "Hi everybody who is watching the video with me right now, please chat with me if you are. it if 9/22/2017. Please say something before 11:12 strikes",
        "A": "Khan Academy question sections are used SOLELY for questions, and answers to those question. It does not allow for socialization. There are other ways to do that."
    },
    {
        "video_name": "mzOBlH32qdk",
        "Q": "I think the power rule fails with constant functions. As Sal says at 3:00 the derivative of a constant is always 0, but according to the Power Rule it's 1, because every constant has an exponent of 1, so the derivative of A^1 = 1 * A^0 = 1 * 1 = 1.\nAnything wrong in this reasoning?",
        "A": "When you find the derivative of a function you find the derivative with respect to some variable. In most cases that variable is x. Constants have no x variables, so they stay the same no matter what x value. So when you find the derivative of an equation with constants the constants become zero because they will not change when x changes. Tell me if I just confused you even more."
    },
    {
        "video_name": "mzOBlH32qdk",
        "Q": "At 2:30 if y=constant, then how can a tangent line exist based on the definition of a tangent line. The only way I can picture a line touching at one pint would be a vertical line with an undefined slope.",
        "A": "I think what you are asking is if y=constant and is graphed as a perfectly horizontal line, then how can you possibly draw a tangent line with a slope of zero that only touches y=constant at ONE POINT ONLY. According to the definition, you are right. It can t be the tangent line because it touches at every single point possible. Visually it is wrong, but mathematically, it works out"
    },
    {
        "video_name": "mzOBlH32qdk",
        "Q": "At 7:47 x^-4=-4x^-5\nIs it really right? Why its not 1/5x^4?",
        "A": "Power Rule: d/dx(a\u00e2\u0080\u00a2x^n) \u00e2\u0086\u0092 a\u00e2\u0080\u00a2n\u00e2\u0080\u00a2x^(n - 1) Example: f(x) = x^-4 f (x) = d/dx(x^-4) Apply power rule d/dx(a\u00e2\u0080\u00a2x^n) \u00e2\u0086\u0092 a\u00e2\u0080\u00a2n\u00e2\u0080\u00a2x^(n - 1), where a = 1, n = -4: f (x) = 1\u00e2\u0080\u00a2-4\u00e2\u0080\u00a2x^(-4 - 1)\u00e2\u0080\u00a2d/dx(x) f (x) = 1\u00e2\u0080\u00a2-4\u00e2\u0080\u00a2x^(-4 - 1)\u00e2\u0080\u00a2dx/dx f (x) = 1\u00e2\u0080\u00a2-4\u00e2\u0080\u00a2x^(-4 - 1) f (x) = 1\u00e2\u0080\u00a2-4\u00e2\u0080\u00a2x^(-5) f (x) = [-4\u00e2\u0080\u00a2x^(-5) or -4/x^5]"
    },
    {
        "video_name": "mzOBlH32qdk",
        "Q": "When Sal takes out the constant from the derivative brackets at around 5:00, doesn't the constant still need to be differentiated? I mean, if the constant where in fact, a function, then it would be. But how come the constant is exempted in this case?",
        "A": "You can factor out constants because they are not affected by the differentiation process. You may need review the rules of limits. One of the basic rules of limits is: lim x\u00e2\u0086\u0092c { a \u00e2\u0088\u0099 f(x) } = a { lim x\u00e2\u0086\u0092c f(x) } (where a is a constant)"
    },
    {
        "video_name": "mzOBlH32qdk",
        "Q": "At 8:17 could you take 3x^2 - 4x^-5 a step further and put the negative exponent on the bottom to get: 3x^2 over 4x^5?",
        "A": "No. That would be done as 3x\u00c2\u00b2 - 4/x\u00e2\u0081\u00b5. The 4 is not raised to the -5, only the x is. Furthermore, they are subtracted, not multiplied."
    },
    {
        "video_name": "mzOBlH32qdk",
        "Q": "At 3:28, why can you do that?",
        "A": "The difference quotient is: [f(x+h) - f(x)] / h as h approaches 0 right? Let s say we have d/dx(A(f(x)) we get: lim h-> 0 [A * f(x+h) - A(f(x))] / h This is the same as: lim h->0 A * [f(x+h) - f(x)] / h We know from limit properties that lim x->c Af(x) = A * lim x->c f(x) So we have: A * d/dx(f(x)) which is what we wanted to prove, i.e. that: d/dx [ Af(x) ] = A * d/dx[ f(x) ]"
    },
    {
        "video_name": "mzOBlH32qdk",
        "Q": "At 4:40, why is it beneficial to remove the constant from the derivative? The sample polynomial yields the same result and seems to make the equation more tedious and complex than simply leaving the constant in the derivative.",
        "A": "The derivative of a constant is 0."
    },
    {
        "video_name": "mzOBlH32qdk",
        "Q": "At 3:47, because one may move the constant to the front of the derivative of f(x), is this a case of the distributive property?",
        "A": "I guess, kind-of, but not really. You need to prove it. For example, d(c*f(x))/dx = lim h->0 (c*f(x+h)-c*f(x))/h=lim h->0 c*(f(x+h)-f(x))/h = c * (lim h->0 (f(x+h)-f(x))/h)=c* df(x)/dx The distributive property is used there, but, I d call the statement different than the distributive property. Note that this property relies on the property that you can take a constant factor out of a limit."
    },
    {
        "video_name": "Ncv6gt2Luew",
        "Q": "At 0:43, how do I know it is h'(1.8)? It makes sense when Sal says it, but on a real AP test, how do I interpret the written question correctly to assume they are asking for the derivative (h'(x))?\nHope this question makes sense, and thanks so much in advance!",
        "A": "The question asked for the rate at which h(x) changes at x = 1.8. When it asks for the RATE at which something is changing, it is asking for the derivative. Did this answer your question?"
    },
    {
        "video_name": "Ncv6gt2Luew",
        "Q": "At 0:43, how do I know it is h'(1.8)? It makes sense when Sal says it, but on a real AP test, how do I interpret the written question correctly to assume they are asking for the derivative (h'(x))?\nHope this question makes sense, and thanks so much in advance!",
        "A": "Well if you look at the wording it says find the rate so immediately you know it s the derivative with respect to X = 1.8. Always remember rate = derivative"
    },
    {
        "video_name": "9VZsMY15xeU",
        "Q": "At 4:43, why would you multiply 4*84? I don't understand why you multiply 4.",
        "A": "Because he s using the sum of the scores. The sum of the scores for all 5 exams is 4 times the average for the first 4 + the score for the 5th = 4*84 + x"
    },
    {
        "video_name": "9VZsMY15xeU",
        "Q": "Why Sal multiples 5.88 (min 5:26)? Thanks.",
        "A": "Using the properties of equality, Sal is multiplying both sides of the equation by 5. By doing this, he eliminates the fraction. Hope this helps."
    },
    {
        "video_name": "9VZsMY15xeU",
        "Q": "At 3:00, does he mean\n80\n81\n87\n+ 88",
        "A": "He means 80 + 81 + 87 + 88."
    },
    {
        "video_name": "9VZsMY15xeU",
        "Q": "At 4:44, why does he multiply 4 by 84? And how is that going to get the sum of the first four exams? I also do not get the concept of why he did 4x84+x/5",
        "A": "Remember that the he added the 4 exams then divided by 4, so if he multiplies by 4 what is he going to get? only the 4 exams added without it being divided by 4 (cause you multiplied) so the equation is the sum of the 4 exams which is 4*84 plus the fifth exam which is x divided by the total exams there are which is 5. and you solve for x which gives you the grades you need on the fifth exam to have that certain average on your grade ticket."
    },
    {
        "video_name": "9VZsMY15xeU",
        "Q": "During 1:40 of the how did he get 7.25",
        "A": "He added the 4 numbers, divided it by however many numbers there were (4) then found 7.25. Doing the division he ran into 7 remainder of 1 so he added a decimal and two invisible zeros."
    },
    {
        "video_name": "9VZsMY15xeU",
        "Q": "At 4:50, I am not sure why we are dividing the average score (84%) by the number of exams taken. Wouldn't this only be accurate if the student had gotten a score of 84% on each of those 4 exams? I would really appreciate any clarification on this! Thanks.",
        "A": "that s the problem with averaging each individual is unique but if you have to describe them as a group it s a way to do it average is figured the way he shows to make it so that for every high there s an equal low, or two that are half as low maybe, compensating reciprocally. overall it balances out pretty well but super-simplifies the picture you get"
    },
    {
        "video_name": "9nH3A4riVxs",
        "Q": "Find A:B:C,if A:B=7:4and B:C=8:9",
        "A": "A:B:C=14:8:9 Is it right? Help me"
    },
    {
        "video_name": "4vNloi-zNjU",
        "Q": "At 0:10, if the x value was negative, would we move the other way?",
        "A": "Yes. -x ----(-3,0)-----(0,0)-----(+3,0)---- +x"
    },
    {
        "video_name": "MaMk6-f3T9k",
        "Q": "In the first problem, couldn't you just simplify 24:40 to 3:5 and then find equivalent fractions of that?",
        "A": "Yes, but there already is a ratio with 3 but missing the 5."
    },
    {
        "video_name": "MaMk6-f3T9k",
        "Q": "what is 29:32 simplified?",
        "A": "It can t be simplified. It is simplified in its current form."
    },
    {
        "video_name": "MaMk6-f3T9k",
        "Q": "Can you explain why the 1:1, 2:4, 3:9 is not a correct ratio?",
        "A": "You can look at it as a fraction- 1:1, 2:4, and 3:9 would equal 1/1, 2/4, and 3/9. 1/1 is not equal to 2/4, nor 3/9, so therefor, they are not equivalent."
    },
    {
        "video_name": "MaMk6-f3T9k",
        "Q": "How is 12:3 Not equal to 11:2? Can you not add? Someone help.",
        "A": "You can simplify 12:3 to 4:1 by dividing the 12 by 3 and and 3 by 3. 12:3 12 divided by 3 = 4 3 divided by 3 = 1 4:1"
    },
    {
        "video_name": "MaMk6-f3T9k",
        "Q": "At 4:18, Sal says that table 2 is not legitimate. I am not able to get it clearly. Can somebody please explain it to me?",
        "A": "Means that it does not fit your needs. So he needs a table that has equivalent ratios but table 2 does not have equal ratios"
    },
    {
        "video_name": "MaMk6-f3T9k",
        "Q": "Filling missing measurements\nScale 1:5,size drawn 250m is?\nGive actual size",
        "A": "250 times 5 is 1250m in reality."
    },
    {
        "video_name": "MaMk6-f3T9k",
        "Q": "how would you work out, 'given K:m = 1:5 and m:n=8:1, what would k:m:n be'?",
        "A": "I like that answer!! :)"
    },
    {
        "video_name": "UvDcEvDC4vg",
        "Q": "at 2:11 did he say million or billion?",
        "A": "he said a million"
    },
    {
        "video_name": "UvDcEvDC4vg",
        "Q": "At 4:28, Sal writes his 2 like a z and I'm not sure which one it is.",
        "A": "its an 2, since z wouldn t make sense in this context"
    },
    {
        "video_name": "UvDcEvDC4vg",
        "Q": "what does x70 mean, 2:13, and what does y70 mean",
        "A": "In both cases, the words and voice clearly say greater than, so it is x > 0 and y > 0."
    },
    {
        "video_name": "2MYA8Ba2PvM",
        "Q": "At 0:22 we are considering a 6-sided dice... but what if we were doing... a 100 sided dice? How would that change the answer?",
        "A": "Even with 100 sides, half of them are even, right?"
    },
    {
        "video_name": "2CZrkdtgeNU",
        "Q": "Anyone else solve as he does in the vid, like at 1:16 ? I am so used to writing each step one at a time.",
        "A": "I d do it a bit like that, but I d subtract 7x and 194 in my head, going straight to 2x = -12. You should do it any way that is easiest for you, and makes you less likely to make a mistake. Writing out each step one at a time is good - less chance of errors, and in a quiz, you should get credit for showing all your working, even if you make an accidental error."
    },
    {
        "video_name": "2CZrkdtgeNU",
        "Q": "at 1:24, Khan says that we have to subtract 196 from both sides to maintain the inequality. But shouldn't he just subtract 182 from both sides instead?",
        "A": "When solving an equation, we try to end up with the variable terms on one side of the equation and the constant term on the other. If Sal had subtracted 182 from each side, then the equation he was left with would have been 2x + 12 = 0 and in order to solve this he would then have had to add -12 to each side. The answer would be the same in each case, but Sal s video method was a quicker way to solve the equation."
    },
    {
        "video_name": "2CZrkdtgeNU",
        "Q": "I've got a question. And either I' really dumb or just didn't see it.\nWhy does Sal divide by 2 at 1:50? I'm just really confused. Please help meh.",
        "A": "When you have 2x = 12, you have to divide the side of the equation with the variable into the other side of the equation, but leaving the x behind. (my method) 2x = 12 Move the 2 to the other side of equation to get x = 12/2 x = 6 The universal method: 2x = 12 Divide both sides by 2: 2x/2 = 12/2 You now have x = 6 I usually use my method because that is how I learned algebra. I think both ways are equally as simple."
    },
    {
        "video_name": "ondmopWLiEg",
        "Q": "At 2:07, the matrix multiplication he performs does not make sense to me. It lo0ks like at first he's treating v1, v2, v3... as the column vectors of matrix A, which would have dimension 1xm (causing it to have the expected mxn dimensions, as there are n vectors) , but then he multiplies them by the x vector, which is an nx1 matrix. You cannot perform matrix multiplication between a 1xm and an nx1 matrix. Am I overlooking something?",
        "A": "A has n vectors, which are each m x 1. So you can t multiply them by x as a vector (as x is n x 1), but that is not what is happening. He is multiplying them by the elements of x, so x1, x2 to xn, and then summing the result. Each element of x is just a scalar, which obviously can multiply the vector columns of A. This is just another way to go through the mechanics of multiplying.. using the elements of x as coefficients of the vectors of A, and it gives the same answer as doing it say the dot product way."
    },
    {
        "video_name": "ondmopWLiEg",
        "Q": "7:06 Shouldn't Sal have a vector arrow over the x?",
        "A": "I think he did forget to put one, but as long as you know that x is a vector, the notation isn t really important."
    },
    {
        "video_name": "ondmopWLiEg",
        "Q": "On 1:08, why did Sal insist on writing a bold A?\nI thought only vectors were bolded?",
        "A": "A[ v ] is a symbol that references a vector (since it is a member of R^m), so it makes sense to make it bold. But, as Sal suggests, he forgets to do it often and the world doesn t collapse. Indeed, if it were up to me, he would have made T bold as well, since it is a vector function, but meh."
    },
    {
        "video_name": "ondmopWLiEg",
        "Q": "At 2:04 - 2:33, how did Sal find out the members of Rm?",
        "A": "The sum is generated by multiplying the matrix A by vector x. If you visualize matrix multiplication, you can see that for every row in A, the first element of the row will be multiplied by x_1. The sum of all of those multiplications will be v_1*x_1, as v_1 is the first column of A and it s elements are always the first of the row. The rest of the elements in the sum generating Rm are found similarly."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "at 10:18 how did x^2 - 3x +9/4 become (x-3/2) squared",
        "A": "Remember that (a+b)^2 = a^2 + 2ab + b^2. So if you substitute x and -3/2 for a and b you get (x - 3/2)^2 = x^2+2(-3/2)x+(-3/2)^2 = x^2 - 3x + 9/4"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "at 8:17 why do you have to take the 4/5 and move it to the other side?",
        "A": "You want to do this so you can make the left side of the equation a perfect square by adding on (b/2)^2 (the left side of the equation is in the form of ax^2+bx). If you wanted to add on to the -4/5 so it becomes a perfect square (you would have to add on 4/5 more), you could, but this way seems easier."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "In 1:46 of the video, how did he find the 2ax?",
        "A": "It comes for using FOIL to multiply: (x-a)(x-a) = x^2 - ax - ax +a^2 Combine the 2 middle terms, you get -2ax. Hope this helps."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 3:40, Sal factors a quadratic equation that isn't equal to zero. Don't they always have to be equal to zero?",
        "A": "They don t necessarily have to be equal to 0, but in general that is how quadratics are written because it is much easier to solve."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "at 12:30, sal says - 0. 246, i thought you cant have a negative zero?",
        "A": "It s not a negative zero because after the zero there are decimal numbers so it s just a decimal number."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 8:28, where does Sal get the 2? As in, \"what number times 2 is equal to -3?\" Why the 2?",
        "A": "He s using the fact that (a - b)^2 = a^2 - 2ab + b^2 In the case at hand, he has a=x and 2ab = 3x. Solving for b is fairly easy: 2ab = 3x 2bx = 3x 2b = 3 b = 3/2"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "sir? the example you explained at 12:44 cant we just divide the whole equation by 10 or first we have to divide it by 2 and then 5?",
        "A": "Yes, you could just divide by 10. I don t know why Sal divided by 2 and then the 5."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 8:35 why do you divide 3 by 2? I understand that you square it because that's what you do when you distribute the perfect square. Does that work for every single square completion?",
        "A": "When you complete the square, you need to understand what happens when you square a binomial. (x+n)^2 = x^2 + 2nx + n^2 Notice the middle term: It is 2 times the number that is squared to create the last term. Thus, when we are completing the square, we need to divide the middle term by 2 to find the number that needs to be squared. Hope this makes sense."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 5:37 the video completes the traditional factoring method to get the answere of (x-5)(x+1)=0. This is then stated as the final answere of x = 5 or x = -1. Why does the negative (-5) become positive and the postive (1) become negative? I noticied this was the case for the other examples in the previous video as well. If you just go with (x-5)(x+1) your asnwere would not mtach the \"completing the square\" method.",
        "A": "If (x-5)(x+1) = 0 then one of the factors has to equal zero. Either x-5 = 0 or x+1 = 0. If x-5 = 0, then x=5 If x+1 = 0 then x = -1 The sign changes when we move the number to the other side of the equation."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "Shouldn't at 2:30 onwards, a be 2 and not -2?",
        "A": "Half the coefficient of the x term is - 4 / 2. The sign must always be included. That s where Sal gets - 2."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "So what do you do when after you find b2/2 (like at 2:55) and the right side of the equation is not a complete square? I keep getting to that kind of equation when i work these out.",
        "A": "Two possibilities that I can think of without seeing your work: 1. Make sure that you are dividing b by 2 and then squaring that result, (b/2)^2, and not this: (b^2)/2. 2. Be sure to add (b/2)^2 to both sides. Can you show an example of what you are doing?"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 2:00 what exactly does \"a\" stand for? I don't understand why he wrote the 2ax underneath the -4x. There is no \"a\" in the original problem.",
        "A": "The a means the unknown digit."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 2:00 how did he get 2ax? I understand how he got x^2 and a^2 or did he do (x-a)(x-a)?",
        "A": "Because they are the same"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 2:05, I think Sal made a mistake that kind of confused me. (x-a) squared is x squared -2ax+ a squared. This means that -4x should be equal to -2ax(not 2ax as Sal wrote it) which implies that a is 2. If a was -2, then (x-a) squared will become (x+2) squared. Please tell me whether I am wrong or is Sal wrong.",
        "A": "Sal was forming a generic model, so the actual \u00c2\u00b1 relation to the middle term isn t necessary at that point."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 2:08, shouldn't a=2? Not -2?",
        "A": "I guess it was because that the perfect square was (x-2)^2 so that it was -2a, then it should be 2. But yeah, I understand."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 1:12, he says \"there has to be some number here that if I have my number squared I get that number, and then if I have two times my number I get negative 4.\" What on earth does that mean? I have watched all of the multiplying and factoring section, as well as all of the previous quadratic videos with little to no confusion, but I have no clue what he meant. I watched the rest of the video hooping my question would be answered, and I still do not know what he meant, or why that strategy works.",
        "A": "there has to be some number here (reference to -2) that if I have my number (-2) squared I get that number (+4) , and then if I have two times my number (-2) I get negative 4. It might sound a little obscured but think about it and it ll make sense, I kinda had the same question."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 3:45, why to find a number that when multiplied by itself gives 4 and when added to itself gives\n-4?",
        "A": "Well, we can turn this into two equations. Your first condition can be written: a\u00c2\u00b2 = 4 And your second condition: 2a = -4 In this particular problem, we can throw away the first condition and look at the second. If 2a = -4, then a = -4/2 = -2. We can check this with our first equation: (-2)\u00c2\u00b2 = 4. Done!"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 4:06 how do you take the square root and get x-2= +/- 3? I don't get the how you can take the square root of (x-2)^2 and get x-2 or how you take the square root of 9 and get that it's equal to -/+ three. Could someone explain that more in detail please?",
        "A": "Whenever you have a letter that s squared, you have to take the square root of it in order to solve for the letter. Take for example an easier equation: x\u00c2\u00b2 = 9. It s quite obvious without even solving the equation that the solutions are x = 3 and x = -3. When you take the square root of something you don t want to lose solutions, so we add a \u00c2\u00b1 to either side of the equation. Note that x = \u00c2\u00b19 is the same thing as \u00c2\u00b1x = 9 (just multiply both sides by -1 if you don t believe me)."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "at 3:35, in the equation x^2 - 4x + 4 = 9, is the + 4 part of the equation? If yes, how is x^2 - 4x + 4 = (x - 2)(x - 2)",
        "A": "Yes, +4 is part of the equation. For your 2nd question, you ll need to know your algebraic identity: (x - a)(x - a) = x^2 - 2ax + a^2 and set a = 2."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 2:08 Sal gets that a is -2, but I think it should be +2 - because -4x is supposed to be -2ax not 2ax- if you look In the formula..",
        "A": "It is -2ax only when you square (x-a)^2 = x^2 - 2ax + a^2, it is +2ax when you square (x+a)^2 to get x^2+ 2ax +a^2. So you always divide the middle term with an x by two and that is always what goes into the completed square. Thus, -4/2 = -2 which is correct."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 4:10, is it always going to be plus or minus 3? Or is it going to be another number?",
        "A": "The reason it is a 3 is because he s squaring the numbers so yes you can get another number"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "Sir, in the video \"Completing the square\" at time: 3:54, you wrote (x-2)^2 as (x-2) (x-2), sir isn't that supposed to be (x+2) (x-2)",
        "A": "No. Remember, if you have a number squared, you are multiplying that number with itself. The same applies to binomials. (x-2)^2 tells you to multiply (x-2) with itself. So, you get (x-2)(x-2)"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "at 3:40 sal khan says that a = -2 which is also the factorization of x^2 - 4x+4=9.\nis it allways like that?\n( a = factorizition)",
        "A": "In this situation, yes. Because it makes a perfect squared binomial. (x - 2)^2 So this only applies to completing the square, where it goes like (x - a)^2. Other quadratics won t use a like how Sal uses it in this video."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "by 10:15 sal says that x^2-3x+9/4=61/20 is the same as (x-3/2)^2=61/20. how did he factor the first equation to get the second? i distributed (x-3/2)^2 and i see that it IS equal to x^2-3x+9/4=61/20 but i don't see how to figure that out because it's very complicated to factor. can someone please explain how to factor it?",
        "A": "The quadratic form (a+b)^2 always equals a^2+2ab+b^2. If we take the square root of 9/4, we get 3/2, and that doubled is 3. Therefore, it can be factored into the form (a+b)^2"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "What if the \"b\" number is an odd number? how do you divide it in half to find the missing number at 2:00?",
        "A": "I remember answering this question already, so I\u00c2\u00b4m very confused. Anyways, if b is odd, then you will either keep it as a fraction or make a decimal and continue solving as you normally would. I hope this helps. :)"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 2:02 where did the -2ax come from?",
        "A": "Use FOIL to expand: (x - a)^2 (x-a)(x-a) = x^2 - ax -ax + a^2 Combine the middle 2 terms: x^2 - 2ax + a^2 That is where the -2ax comes from. Hope this helps."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At around 7:30, when Sal shows why it is hard to use factoring sometimes couldn't he have used grouping to solve the equation 5x^2 - 15x - 4 = 0?",
        "A": "No... this quadratic is not factorable. If you use grouping, you would multiply 5(-4) = -20 Then, you would need to find 2 numbers that multiply to -20, and also add to -15. There are no factors of -20 that will work. FYI... If it was factorable, I would do the factoring before completing the square. it would likely have been easier. But, it wasn t an option for this quadratic. Hope this helps."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "at 8:01 why is there a fraction instead of a decimal?",
        "A": "It is possible to solve the equation using decimals or fractions. I m guessing Mr. Khan decided to use fractions to show the process used to complete the square."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "Why would you need to complete the square if you can factor it? for example 1 at 1:44",
        "A": "After about 5:12, Sal explains the advantages of Completing the Square over Factoring, and uses the rest of the video to explain and elaborate on this. Hope this helps!"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "Two questions:\n1). At 1:37, how did he come up with (x-a)^2?\n2). At 4:10, how come x-2= plus or minus 3? How did it go from just 9 (on the right side of the equation) to plus or minus 3?",
        "A": "1.) It s the rule. It s a notable product, and when you expand it you got the thing that he put at the video. The real rule is (a-b)^2 = a^2 -2ab - b^2. He trade a per x and b per a, just for better understanding. 2.) He take the square root at the two sides of the equation. When you got the square root of a number, it s always plus or minus, because when you elevate a number to it s square, it s going to have the same result if you take that number in negative form."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 7:50, he said he'll explain 2 ways of completing the square for that question but I don't think he explained the second way. The first way he explained was dividing 5 on both sides and moving 4/5 to the right side of the equation. Is the other way to factor 5 out of the equation first so it becomes 5(x^2 -3x -4/5) = 0?",
        "A": "I believe you are correct, but many people do not include the constant and just do 5(x^2-3x) -4 = 0"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "What do you do if once youre at 4:04 in the problem where you use the square root of both sides, for example (x+4)^2 = 18, which you then have x + 4 = the square root of 18, what if there isnt an even square root, such as in this problem, how do you solve it?",
        "A": "You need to simplify the radical by finding and removing all perfect square factors. 18 = 9 * 2. Thus sqrt(18) = sqrt(9) * sqrt(2) = 3 sqrt(2) Thus you have: x = -4 +/- 3 sqrt(2) Hope this helps."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "why did you factor out the 5 first at 6:45? can't you just move the -4 over by adding 4 to both sides before factoring the 5 out?",
        "A": "You could indeed have done that (and I think that s what I would have done), but both ways get you the correct answer."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "Wait, how did he get 9/4 at 9:04?",
        "A": "To complete the square, you use the following formula: (b/2)^2 when a is 1. So in this case, b is -3. So, using the formula, you would get: (-3/2)^2 = 9/4."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 3:49, when he says, \"Well, we already answered that question.\" Is he referring to the number you square and at to both sides?",
        "A": "He is referring the number solved for on the right side of the screen."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "I am confused around 14:00 he uses the grapghing calculator and gets the answer 1E-14. I did the same equation on my graphing calculator, and I got 1.4915E-10. I do not understand.",
        "A": "His calculator just gives an average. DId you use the negative answer that he used or did you get the exact amount of digits that he got in his negative answer?"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 1:32 how did you get \"a\"?",
        "A": "a is the unknown number in the problem"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 6:45, why did he have to divide everything by 5 and not just the first number\nexample, couldn't he just do this?\n( 5x^2 )/5 -15x-4 = 0/5?",
        "A": "One of the rules in math is when you divide one number in the equation in order for the equation to work you must divide all the numbers in the equation. 4y+2x=8 if you just divide the first number by 2 you would get 2y+2x=8 which is not the same as if you divided the whole equation by 2 which is 2y+x=4"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "I don't understand. How did he get 9/4 at 9:00? I mean I understand he reduced his quadratic equation simply by reducing it without making it a struggle, but why fractions?",
        "A": "The middle term has a coefficient of -3. To complete the square, 1) you divide the -3 by 2 (Sal showed this as a = -3/2. 2) you then need to square a . (-3/2)^2 = 9/4 That is where the 9/4 came from. Hope this helps."
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "At 2:59 ish he adds 4 to the 5 in the equation but don't you have to add the opposite??",
        "A": "No, because he added four to the left side of the equation already. If he subtracted 4 from the right side, it would be like adding 8 to the left side. Here is an example: If we have x^2 +b =a and we add +4 to the left side, then we must add +4 to the right side, or else the equation is imbalanced. Does that make sense? This can be very confusing!"
    },
    {
        "video_name": "bNQY0z76M5A",
        "Q": "at 10:10, wouldn't it be better to change the fraction into a mixed fraction - in this case, 3 1/20?",
        "A": "I wouldn t change it to 3 1/20. The reason is that you will very soon be apply a square root. You need the improper fraction to do the square root. You can t do the square root of a mixed number. Hope this helps."
    },
    {
        "video_name": "4h54s7BBPpA",
        "Q": "At around 0:49, why can't it be defined if it is negative?",
        "A": "Unless you are using imaginary numbers, you cannot take the square root of a negative number. The square root means which number multiplied by itself gives this result? Since a positive * positive = positive and negative * negative = positive, square roots of negatives are not real numbers."
    },
    {
        "video_name": "4h54s7BBPpA",
        "Q": "How did Sal get rid of the square root sign? For example, at 0:30 he gets rid of them obtaining the inequality 2x-8<or= to 0? How did he do this i know about square roots and squares but i am stumped to how he got rid of the square root sign??",
        "A": "He didn t get rid of the radical. He only needs to evaluate what s inside the radical so that he avoids a square root of a negative."
    },
    {
        "video_name": "4h54s7BBPpA",
        "Q": "At 0:35, how does Sal know what inequality sign to write? He wrote the sign that is \"greater than or equal to.\" How does he know if he should write <,>, or less than or equal to? THis is all very confusing.",
        "A": "With square roots, the content of the radical has to be zero or above (which is >=) to be defined within the real number system. This is something you learn as you work with radicals."
    },
    {
        "video_name": "4h54s7BBPpA",
        "Q": "When he says that there are more different root symbols. what does he mean? 00:43",
        "A": "He is referring to the set of complex numbers."
    },
    {
        "video_name": "4h54s7BBPpA",
        "Q": "At 0:39 , How do you determine which inequality sign to use?",
        "A": "Relating to the video, you can t take a square root of a negative number, but you can take square root of 0, and all positive numbers. Therefore it must be equal or greater than 0."
    },
    {
        "video_name": "4h54s7BBPpA",
        "Q": "At 0:40 why will it be undefined if the number is negative? Don't understand.",
        "A": "As you would then be square rooting a negative number which doesn t give any real solutions."
    },
    {
        "video_name": "4h54s7BBPpA",
        "Q": "0 is a positive number? 0:40\nIsn't a positive number a number that is greater that 0?",
        "A": "Radicals need to contain a non-negative number. Sal mentions this at about 0:26 in this video. This would include zero. We can find the square root of zero. It = zero. Hope this helps."
    },
    {
        "video_name": "4h54s7BBPpA",
        "Q": "At 0:06 what does he mean by principal square root?",
        "A": "That just means the positive square root. For example: \u00e2\u0088\u009a36 = \u00c2\u00b16 However the principal root is just 6."
    },
    {
        "video_name": "4h54s7BBPpA",
        "Q": "At 0:03 and later in the video, Sal calls it the \"principal square root\". Can someone tell me what this is? Is it just another way of saying the square root?",
        "A": "The unique nonnegative square root of a nonnegative real number. For example, the principal square root of 9 is 3, although both and 3 are square roots of 9. The concept of principal square root cannot be extended to real negative numbers since the two square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point and can then be distinguished. Since either choice is possible, there is no ambiguity in defining as the square root of ."
    },
    {
        "video_name": "pSSqUF_mmtM",
        "Q": "So what would h prime of (-1) would be at around 6:25 sec in the video? Given that g prime of -1 is again -1 as explained in the video? Any thoughts?",
        "A": "Well, h(x) = x^2, so h (x) = 2x, and it looks from the graph like g(-1) = -1. Therefore h (g(-1)) = h (-1) = 2(-1) = -2. Does that help?"
    },
    {
        "video_name": "sh-MP-dVhD4",
        "Q": "At around 1:15. Sal completed the square with the 16x, making it (x+8)^2-55. Couldn't you also complete the square with the 9 at the end? So like substituting a^2 as 9. This would give the final equation as (x+3)^2+10x. Wouldn't this work as well?",
        "A": "It wouldn t help you. The term outside the perfect square should be a constant (i.e., not x in that term. You want to use the x^2 and the entire x terms when you complete the square. Hope this helps."
    },
    {
        "video_name": "sh-MP-dVhD4",
        "Q": "At 1:27, how did he geat a=8?",
        "A": "He tried to find what(a) times 2 = 16x. So you could simple multiplication to get 8*2=16. you could also do 16/2=8 (easier for bigger numbers). Hope this helps :)"
    },
    {
        "video_name": "ayRpoJgph0E",
        "Q": "At 12:08 so the formula would be B= A*3+1",
        "A": "yes that is right"
    },
    {
        "video_name": "ayRpoJgph0E",
        "Q": "At 0:33 couldn't you just go through and add 3 once you got the first 2 patterns?",
        "A": "Yes the pattern is an arithmetic increase of 3, but you have to follow the directions first in order to notice that pattern."
    },
    {
        "video_name": "YjT3QYfoy4Q",
        "Q": "At 0:52 why did you go over four up one? I was taught that it was over 2 and up 1 and it stays like that throughout the whole line.",
        "A": "The slope is always constant. Sal did a couple of points where he went up 4 over 2. Note: 4/2 = 2/1. So, he is using the same slope. Hope this helps."
    },
    {
        "video_name": "YjT3QYfoy4Q",
        "Q": "Why did Sal move up 4 spaces at 0:39?",
        "A": "He decided to skip writing it up 2 spaces so he went right up to 4."
    },
    {
        "video_name": "YjT3QYfoy4Q",
        "Q": "at about 0:44 why do you move up 4",
        "A": "The slope is 2. Written as a fraction it is 2/1 which tells you to move up 2 spaces every time you move right one space. He decided to double both of them. Since he moved right 2 spaces, he needed to move up 4 spaces. This is the same thing as moving right 1 and up 2, and then moving right 1 and up 2 again."
    },
    {
        "video_name": "YjT3QYfoy4Q",
        "Q": "At 3:08, how did you know to fill it in?",
        "A": "To know where to shade, imagine the slope of the line is as a slope of a hill that you would walk on. If y is greater (or greater or equal) you would shade the upper side of the slope, the side that you walk on. The opposite side if y is less than (or less than or equal to). In a system the overlapping shades is the solutions of the systems. If they don t overlap, there is no solution."
    },
    {
        "video_name": "YjT3QYfoy4Q",
        "Q": "At 3:08, how did you know to shade in the right of x>1?",
        "A": "since x is greater than 1 we want to shade the section of the graph that has x increasing into positive numbers... if we went to the left, we would be getting into negative x values..."
    },
    {
        "video_name": "swFohliPgmQ",
        "Q": "2:12 How does he get the 2x?",
        "A": "(x+1)\u00c2\u00b2 = (x+1)(x+1) Use FOIL to get (x)(x) + 1x + 1x + (1)(1) = x\u00c2\u00b2 + 2x + 1"
    },
    {
        "video_name": "MT1-bqKpal8",
        "Q": "I know this is a dumb question but at 3:15, how did he know u(x) was sin(x) and not sin(x) +1. Thanks in advance!",
        "A": "He could have, but then v(x) = cos(x) and u(x) = sin(x) +1 and we would get the same result, mathematical equivalent of potato potAto"
    },
    {
        "video_name": "-nlMXVrgtjw",
        "Q": "at 0:32 sal says that x+4 goes into 3x^3 3x^2 times. I'd understand if the divisor was just x but what does he do with the 4?",
        "A": "When you divide by polinomial you basically divide by the first term and then estimate how much less or more you should have in the quotient based on whether your denominator is bigger or smaller than the one (first term) you originally divided by."
    },
    {
        "video_name": "JK-8XNIoAkI",
        "Q": "At 6:08, Sal asks if the transformation is linear. I have noticed he asks the same question often in other videos. What is the significance of something being linear transformation? What would be the issue if it wasn't linear transformation? Thanks.",
        "A": "A linear transformation can be defined using a single matrix and has other useful properties. A non-linear transformation is more difficult to define and often lacks those useful properties. Intuitively, you can think of linear transformations as taking a picture and spinning it, skewing it, and stretching/compressing it. Anytime you want those types of effects, linear transformations are wonderful."
    },
    {
        "video_name": "JK-8XNIoAkI",
        "Q": "12:44 I notice that the determinant of A is 0. Is this just a coincidence or does it mean something?",
        "A": "Yes, the determinant is 0. That actually shows that the image of the transformation isn t all of \u00e2\u0084\u009d\u00c2\u00b2, which we already knew from the fact that the image will always be a line."
    },
    {
        "video_name": "U1lQbh21y1k",
        "Q": "Around 0:50 what does he mean by \"with respect to\" ?",
        "A": "With respect to something means in regards to or relating to ."
    },
    {
        "video_name": "6dIMIBO_2mc",
        "Q": "At 0:43 , what does it mean to put the ~ over the = sign?",
        "A": "This is the standard notation which basically means that both the figures are congruent to each other. That s what the \u00cb\u009cover the = sign means. Hope this helps."
    },
    {
        "video_name": "6dIMIBO_2mc",
        "Q": "At 0:43 , what does it mean to put the ~ over the = sign?",
        "A": "It means congruent."
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "In this video , at 7:30 , how the coordinates of that point on the cartesian plane came ?\nPlease help\nThank you",
        "A": "That point comes from the Unit circle which is a circle of radius 1 and by using trigonometric identities and pythagorean theorm you can get any point on the circle."
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "I feel so frustrated! I don't understand. At 4:15, why do you have to think of y as a function of x? And what does that even mean? What does respect to x or respect to y mean? At 3:30, why is he suddenly doing respect to y instead of respect to x? What does implicitly even mean? PLEASE HELP! Thanks!",
        "A": "I would be happy to explain this material to you, but it would help if you would mention what your previous level of mathematical study was. It may be that you re jumping too far ahead. But, in brief, y is a function of x because it varies with x -- y is not a constant. Differentiating with respect to a particular variable means that is the variable we are treating as the independent variable."
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "At 1:06, Sal says \"derivative of the slope of the tangent line\". Isn't this a double-negative, since derivative can be thought of as the slope of the tangent line?",
        "A": "Sal meant to say the derivative of the function or the slope of the tangent line . The derivative of the slope of a tangent line is always zero (since the derivative of a constant is always 0)."
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "Hmm... okay, so how come at 3:13, we can't just substitute y^2 for (1-x^2) and take the derivative?\n(edit: oops, made a mistake in my original question! thanks, newbarker!)",
        "A": "Well think about it. If you substitute y^2 for (1-x^2), then you get: x^2 + (1-x^2) = 1 1=1 Which is true, but tells you nothing. You cannot substitute your function INTO that same function, because all the variables will cancel out, and it will leave you with a true equation involving constants, but not helpful. Also, when taking the derivative, we WANT a y in the equation. Otherwise we couldn t solve for the derivative of y."
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "At 3:01, why do you only use the chain rule for the y term and not for the x term? Please help, thank you!",
        "A": "Remember you are differentiating over x. Therefore, x = 1. If you use the chain rule with x all you are going to have is your equation times 1, which doesn t change the overall result. So using the chain rule with x or not makes no difference in this case. Since you are differentiating over x and not y, you need to explicitly write y , as y might be something other than 1."
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "At 4:21 why does it appear that x is being multiplied by y? I understand the Y is a function of X but shouldn't it be X+Y? or is that the function that y(x) is stating?",
        "A": "It could be taken as multiplication but that is functional notation, y of x, that is y(x) (just like f(x)). At that point we are taking the derivative of (y(x))\u00c2\u00b2 (y to the x all squared). This part of the demonstration is to show why you need to apply the chain rule to d/dx[y\u00c2\u00b2], since d/dx[y\u00c2\u00b2] is equivalent to d/dx[y(x)\u00c2\u00b2]."
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "At 3:10 to 5:10, how did he get the derivative of d/dx y^2. Can somebody please give me an explanation.",
        "A": "A way to think of it is that y is usually equal to a bunch of stuff with x s in it. Because y is a function, we have to use the chain rule. Since we don t know what all the x stuff is, we can just put in the place-holder for the derivative of y: d/dy or y (I personally like y , it s cleaner). Derivative of the outside with the inside left alone (2(y)) times the derivative of the inside (y ). d/dx y^2 = 2y*y"
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "can you explain how we are applying the chain rule when finding the d/dx of (y2) at 3:16? What would be considered the inner and outer function?",
        "A": "The outside function is the squaring function. (---)^2. The inside function is y. Remember, we are treating y as a function of x. If you knew what y was, it would be easy. For example, if you found the derivative: d/dx (3x-5)^2 = 2*(3x - 5)(3). Where did the 3 come from? The derivative of the inside of course. That is why: d/dx(y^2) = 2(y)y Where did the y come from? The derivative of the inside!"
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "At 4:25 when Sal is explaining the derivative of y^2 with respect to x, can I just think of it this way:\ny'(x) is the same thing as dy/dx.\nSo I can think of y as being y(x) just to make sense of this implicit differentiation concept.\nSo instead of writing d/dx[y^2], can I think of it as just getting the derivative of (y(x))^2?\nThank you.",
        "A": "Yep. As long as you don t forget to treat _y_ as a function, and hence apply the Chain Rule, it can be viewed in that way."
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "I'm not really sure how implicit and explicit differentiation differ. What are their concrete definitions? What I am basically trying to ask is what makes implicit differentiation different from explicit differentiation? Also, how are the two functions he talks about at approximately 0:50 similar to the function he derives by the end. Thank you very much in advance.",
        "A": "The difference comes in handy when you can t solve for y and so cant use explicit differentiation."
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "i got lost after 4:49..How did you get 2y X (Dy/Dx)",
        "A": "He started with d(y\u00c2\u00b2)/dy * dy/dx. Since d(y\u00c2\u00b2)/dy = 2y that means the original equation becomes 2y * dy/dx"
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "When Sal does d/dx (y^2) at 3:09 and applies the chain rule, is he in effect saying that the \"y\" that we are taking a derivative of is some function of x we just chose to represent as \"y\"? Meaning, y could equal x+6 or x^7 or x^2+6.....?",
        "A": "Correct. And he mentions it briefly at 3:45 we re assuming that y does change with respect to x. y is not some type of constant... That s the key with these problems - to recognize the variable your taking the derivative of (x), dependent variable (y) and any constants."
    },
    {
        "video_name": "mSVrqKZDRF4",
        "Q": "At 1:23, Sal said that the first step is to take derivatives on both sides of the equation. However, I'm not completely convinced about that. Since d/dx (1)=0, does that imply that d/dx [x^2+y^2] is also 0? I'm getting confused. If anyone could bother explain it to me, please leave your answer. Thanks!",
        "A": "It does imply that it equals zero, but there are many different values that x and y could take that satisfy this answer. You are trying to solve for dy/dx, so it would counter productive to immediately substitute it with a 0. d/dx [x^2+y^2]=d/dx [1] 2x+2y(dy/dx)=0"
    },
    {
        "video_name": "rHonltF77zI",
        "Q": "At the beginning of the video it is stated that v1 and v2 are basis vectors for a subspace.\nIf they are basis vectors then shouldn't they be orthogonal to each other.\nIf they are orthogonal to each other how can they have a projection onto each other at 8:38 in the video.",
        "A": "Basis vectors don t have to be orthogonal, they just have to be linearly independent."
    },
    {
        "video_name": "rHonltF77zI",
        "Q": "So when he says \"u1\" again, at 2:15, does he mean the LENGTH of u1? Because we've just defined it to be ( v1 / || v1 || ) so are we normalizing a second time?",
        "A": "Yes, it s mentioned in bottom right of the video."
    },
    {
        "video_name": "rHonltF77zI",
        "Q": "At \"2:30\", you forgot to put the || signs. You wrote u_1 instead of ||u_1||",
        "A": "Here the letter u is meant to stand for unit vector , so you don t need to write ||u1|| ."
    },
    {
        "video_name": "rHonltF77zI",
        "Q": "At 10:18, we dot product v2 with u1 and than multiply with u1 again, to write the projection of v2 on v1. Why we multiply with u1 again? Isn't it by definition of dot product v2 dot u1 is the projection of v2 on u1 (and here therefore v1)? Thanks.",
        "A": "No, it is not. *v\u00e2\u0082\u0082* \u00c2\u00b7 *u\u00e2\u0082\u0081* is just a scalar. It is the amount that *v\u00e2\u0082\u0082* is going in the direction of *v\u00e2\u0082\u0081*. You then need to multiply by *u\u00e2\u0082\u0081* in order to actually create a vector that is going in that direction."
    },
    {
        "video_name": "rHonltF77zI",
        "Q": "Is there an error at 16:00?\nHe copies the text, but should the second term he copies also be negative along with the first?",
        "A": "yes, you are right. should be y_3 = v_3 - [projv_3], and the negative sign will be distributed to all proj terms"
    },
    {
        "video_name": "HCC96awA-FM",
        "Q": "at 0:58 sal talked about decimals why do we use decmals now and in the future!?!?",
        "A": "The most common place is when you re dealing with money. Personally, I prefer fractions, but there are times when the fractions would be so large and cumbersome it s better to use decimals. Not only that, you will not be able to use calculators without understanding decimals, and you will be stuck doing everything by hand. Decimals are a huge pain in the everything, but they re worth knowing. Cheers"
    },
    {
        "video_name": "RPhaidW0dmY",
        "Q": "At 2:11-2:29, how do you simplify a numerator on one of the fraction and not the denominator of that fraction too?",
        "A": "you divid the numerator by a no. then do the same with the denominator with the same number or make it into a mixed number."
    },
    {
        "video_name": "RPhaidW0dmY",
        "Q": "On 1:30 why would you plus the to 35?",
        "A": "I m eleven too. :)"
    },
    {
        "video_name": "RPhaidW0dmY",
        "Q": "At 2:13 Sal divide the numerator and denominator by 4 my quation why he only divide once he only divide the 36 and the 4 why not the 5 and the 7 too ?? i see that in addition we can divided every things",
        "A": "5 and 7 have nothing in common besides 1 unlike 36 and 4."
    },
    {
        "video_name": "RPhaidW0dmY",
        "Q": "At 2:18, it is said they are doing cross-multiplication or cross-cancelling. Why does this work? What is the logic behind it?",
        "A": "Since Multiplication is commutative and associative, you can rearrange the factors however you want and group them however you want. In this case you have 7/4 * 36/5 but we can break that up to be 7 * 1/4 * 36 * 1/5 then rearrange and regroup it to be (36 * 1/4) * (7 * 1/5) then we have 36/4 * 7/5 and of course we can simplify 36/4 and get 9 * 7/5."
    },
    {
        "video_name": "RPhaidW0dmY",
        "Q": "at 4:50, how would you multiply a huge amount of numbers like 7*36",
        "A": "7*36 is the same as 7*30 + 7*6. 7*30 is the same as 7*3*10 7*3 is 21, so 7*30 is 210. 7*6 is 42. So 7*36=210+42=252."
    },
    {
        "video_name": "RPhaidW0dmY",
        "Q": "in 2:27 is it important to do the step where you have to simplify the fractions!?",
        "A": "When simplifying fractions it is much easier for others to analyze your answers in other problems. Also when you put that simplified fraction and add it with a fraction you wouldn t have to simplify it again at the end. So it is a lot easier and important to simplify. When you take state tests or so you should make sure the answer is simplified. Hope this helps! :)"
    },
    {
        "video_name": "RPhaidW0dmY",
        "Q": "at 2:18, why does he divide the numerator of one fraction and the denominator of the other",
        "A": "As you know the rule that we have to divide first before multiplication.here in this fraction we are allowed to divide first,the numerator of one fraction with the denominator of the another fraction if they are possible to divide by same number, as we are dividing 4 and 36 by 4........... i hope it will work"
    },
    {
        "video_name": "k5EbijWu-Ss",
        "Q": "what is with the subscripts at 7:25",
        "A": "Different subscripts are given to identify different data points in the sample."
    },
    {
        "video_name": "k5EbijWu-Ss",
        "Q": "At 0:11 Sal says he wants to differentiate between a sample and a population. what does that mean?",
        "A": "Well, he answers that question pretty well between 0:12 and 6:42."
    },
    {
        "video_name": "k5EbijWu-Ss",
        "Q": "At 4:34,why does that x with a bar over it mean ''sample mean''?",
        "A": "That s just the notation selected for this context."
    },
    {
        "video_name": "k5EbijWu-Ss",
        "Q": "at 00:40, Sal says there is a geometric mean. What is that?",
        "A": "geometric mean of a and b would be sqrt(a*b)... hope this helps."
    },
    {
        "video_name": "k5EbijWu-Ss",
        "Q": "Why on 4:34 did you put a bar over the x? Thanks Sal!",
        "A": "The bar, technically called vinculum, is simply used to indicate the mean. Sal used x-bar (as you refer to it) to indicate that he was going to find the mean of certain terms. The usage of x is preferential, AFAIK, since most of the time, values along the lines of xsub1,xsub2...xsubn are used as examples in statistical formulas. You can probably use y-bar and get away with it in casual company, but you d probably have to explain your usage in that case. In comparison, x-bar is well known."
    },
    {
        "video_name": "k5EbijWu-Ss",
        "Q": "I meant 2:42 what does \"macho statistics\" mean not mean median and mode but look it up in a dictionary. THX p.s. PLEASE ANSWER",
        "A": "please stop asking questions and go to hell"
    },
    {
        "video_name": "k5EbijWu-Ss",
        "Q": "I'm a little confused on what the i stands for, or why we use \"x sub i\" at 7:35.",
        "A": "In math i stands for initial value (physics) or a value of a row/col of an array of values / matrix (linear algebra). In the case of the latter it refers to the first column values."
    },
    {
        "video_name": "k5EbijWu-Ss",
        "Q": "At 4:33, why does Sal put a bar over x? I have learned that the bar over a certain number means repeating so I don't get why a bar is put over x.",
        "A": "In statistics when we put a bar over a letter, it means we are averaging something. If we call our data x, then the x with a line over it (pronounced: X-bar) is the symbol we use for the sample mean, the average of all the values."
    },
    {
        "video_name": "k5EbijWu-Ss",
        "Q": "At 7:02, What do you mean by subscripts?",
        "A": "The subscript is the little number after the x that tells you which x you re talking about."
    },
    {
        "video_name": "pAlq9fFwtus",
        "Q": "At 1:21, Sal says that the x axis is often called other things. What else is it called?",
        "A": "vertical line"
    },
    {
        "video_name": "pAlq9fFwtus",
        "Q": "at 1:30 he says y axis what does he mean",
        "A": "The y axis is the line on the graph that goes up and down. It tells you how far up or down your point should go."
    },
    {
        "video_name": "XkRD9lv_y44",
        "Q": "Using the problem at 2:35, would the variable become -(-7) or just -7? I didn't get what Sal did to answer that question. Please help!\nThanks!",
        "A": "-(-7) becomes 7 because there are two negatives, which cancel out into a positive."
    },
    {
        "video_name": "XkRD9lv_y44",
        "Q": "Sal did something wrong at 2:42, the question states h= - 7. Then substituting the value of h in -h= (--7)= +7. Therefore u don't subtract u add.",
        "A": "THX FOR correction"
    },
    {
        "video_name": "XkRD9lv_y44",
        "Q": "At 2:45 during the video, Sal says \"instead of adding an negative H, you can subtract a regular H.\" What I did not get is if suppose I wanted to add a negative H, would the H be equal to -7 or 7 because the 2 negatives become a positive ( -H = -(-7)?",
        "A": "7 because the two minus signs next to each other turn into a positive"
    },
    {
        "video_name": "n17q8CBiMtQ",
        "Q": "At 4:43 divide zero by what??",
        "A": "Sal says divide by zero and not divide zero by as you seem to have understood. That means he is looking for any numbers that make the denominator equal to zero."
    },
    {
        "video_name": "n17q8CBiMtQ",
        "Q": "Where does the -6 come from and how, in around 3:30?",
        "A": "Absolute value x is smaller than 6. Based on the root,the denominator can not be zero or less than 0. Because of the absolute value, all of the negative numbers are positive. if x= -7 ,the denominator less than zero in the root. Undefined."
    },
    {
        "video_name": "n17q8CBiMtQ",
        "Q": "At 2:07, he says x can't be equal to zero, but if x is zero, then wouldn't it be 6-0 under the root? which would be root 6 and then it would be defined? I'm confused.",
        "A": "What Sal said is the value of the expression 6-|x| cannot equal 0, because the square root of 0 is 0, then we would have 0 in the denominator, and division by zero is undefined."
    },
    {
        "video_name": "n17q8CBiMtQ",
        "Q": "At 4:50. Why would one divide by 0 when x=9 or when x= -10?",
        "A": "If x=9, then the 1st factor (x-9) would become zero. And, zero times the other 2 factors is still zero. The same thing happens if x = -10. The factor of (x+10) would become zero causing the whole denominator to = zero. Hope this helps."
    },
    {
        "video_name": "n17q8CBiMtQ",
        "Q": "About the example Sal is working on from min. 3:35 aprox., if we asume that x=-10, then both terms of the fraction will be 0.\nAnd now my question is:\nWhat is 0/0?\nCould it be 0 instead of undefined?\nBecause if it is 0, then 10 would be as well in the function domain.\nI would appreciate a clarification about this point.\nThanks.",
        "A": "Mr. Khan stated toward the end of the video that 0/0 is undefined. This means that this function has no solution at x = -10. I cannot emphasize enough that it is never possible to divide by zero."
    },
    {
        "video_name": "n17q8CBiMtQ",
        "Q": "OK so at 3:30 and on I have no idea what he's even talking about or doing anymore can someone explain it to me?",
        "A": "He s making a piecewise function. This is a function that you evaluate differently depending what your input is. For the example he gave, if the input is 5, the output is pi. For all other inputs, you evaluate it using the top part. If your input is -10 or 9, the denominator will be 0, so the function is undefined for those values."
    },
    {
        "video_name": "n17q8CBiMtQ",
        "Q": "At 4:48 why does he divide by zero and how did he get that number",
        "A": "The denominator of the fraction is ( x+10)(x-9)(x-5) if any one of these are zero, then multiplying two other things times zero will still give you zero, If x+10 = 0 gives x = -10, x - 9 = 0 gives x = 9, and x-5 = 0 gives x = 5. The third case he eliminates by limitations, With any of the other two choices, the fraction is divided by 0 and becomes undefined. This answers his earlier question of what makes that function undefined."
    },
    {
        "video_name": "n17q8CBiMtQ",
        "Q": "At around 6:05 , Sal says that we can't cancel the terms in the numerator and the denominator ( ( x-10 ) ) because it would \"change the function\" ... What did he mean by that? And why would it change the function? Please answer quick .",
        "A": "If it would change the function it must mean that they would not be equivalent, which would mean they would not have the same domain. However, the reason of why it would change the function is not clear to me, If someone could explain that, it would be awesome."
    },
    {
        "video_name": "n17q8CBiMtQ",
        "Q": "2:49 Isn't the absoulute value always positive? Sal wrote -6",
        "A": "The output of the absolute value will always be positive. But, the values inside the absolute value can be negative. Sal is defining what values X can have when inside the absolute value. Hope this helps."
    },
    {
        "video_name": "n17q8CBiMtQ",
        "Q": "At 4:39. How do you determine the range and domain by looking at a graph?",
        "A": "The domain is the lowest and highest part of x, range is the distance"
    },
    {
        "video_name": "n17q8CBiMtQ",
        "Q": "in 1:56 why did we remove the equal to sign when we put 6-|x| > 0",
        "A": "6-|x| cannot be equal to zero because it is in the denominator . if you take 6-|x| to be equal to zero , you would get the value for 1 upon square root of (6-|x| ) as 1 upon square root of 0 which would be 1 upon 0 which in turn is undefined"
    },
    {
        "video_name": "n17q8CBiMtQ",
        "Q": "1:11 What's radical?",
        "A": "A radical is the symbol used on square roots, cube roots, 4th roots, etc. So, these are called radicals or radical expressions."
    },
    {
        "video_name": "v9NLtiVt3XY",
        "Q": "Time ~0:15: \"Assume that one person cannot hold more than one office\". What if a person could hold more than one position? An example would be to change P, VP, S into something like Most Popular, Most Valuable, and Most Likely to Succeed. Choose all possible combinations with 9 people if each person can hold more than one title. How does this change the example?",
        "A": "Instead of removing 1 candidate when counting how many people could fill in the next position, you use the same number. So 9*8*7 becomes 9*9*9. You can t really call it a combination in the mathematical sense though, since the ordering matters (being VP and S is not the same as P and S)."
    },
    {
        "video_name": "QmfoIvgIVlE",
        "Q": "At 0:12, it was said that a protractor can be used to construct an angle, but how we construct angles WITHOUT a protractor? My teacher says a pair compasses and a ruler can be used but how exactly?",
        "A": "The protractor is the easiest way to construct an angle. I m not sure what your teacher was referring to though. A compass is used to make circles in mathematics."
    },
    {
        "video_name": "QmfoIvgIVlE",
        "Q": "at 2:34, how come he saw and read the question(make a 155 degree angle) but got it wrong?",
        "A": "he probobly forgot"
    },
    {
        "video_name": "QmfoIvgIVlE",
        "Q": "I don't think I did the reference right at 1:35, it is not responding to my click, what am I doing wrong?",
        "A": "I just clicked it and it brought me to that time in the video. You did nothing wrong."
    },
    {
        "video_name": "LqyA96oYtwE",
        "Q": "At 1:34 Sal states that 2 to the 0th power is 1, why? Shouldn't it be 0 because anything times 0 = 0? Plus 2 times itself 0 times...I'm confused. :l\nAlready answered.",
        "A": "The easiest way to explore is this: Anything divided by itself is 1. [Except 0, 0/0 is undefined] So (n^a)/(n^a)=1 What happens when divide bases to powers? We subtract them. ( n ^ a ) / ( n ^ a ) = n ^ ( a - a ) . a-a=0. Therefore n ^ 0 = n ^ ( a - a ) = ( n ^ a ) / ( n ^ a ) = 1 n ^ 0 = 1. However, since 0/0 is undefined, 0^0 is also undefined. So it is a MOSTLY true statement. One exception, however, isn t bad when you realize losing that exception breaks mathematics."
    },
    {
        "video_name": "LqyA96oYtwE",
        "Q": "At 5:37, why is it log base 2 (x+2) and not (x-2)? The blue graph is shifted 2 points to the left and not the right.",
        "A": "In graphing, replacing x by x-h in the equation of a function shifts the graph h units horizontally. In this case replacing x by x+2 = x-(-2) shifts the graph -2 units horizontally, i.e. 2 units left, since h = -2"
    },
    {
        "video_name": "cqK3uRoPtk0",
        "Q": "At 2:45 how can P(X=2) = 3/8. when there are also 2 H in HHH? Why the HHH is not considered for P(X=2)?",
        "A": "When we say X=2, we mean exactly 2. If we wanted to include HHH, we d say P( X \u00e2\u0089\u00a5 2 )."
    },
    {
        "video_name": "cqK3uRoPtk0",
        "Q": "At 3:31 Sal says 'You can have probability larger than 1\", how is that possible?",
        "A": "If you check the transcript, he is actually saying You can t have a probability larger than 1 . Your intuition was correct, the largest probability comes when every element of the sample space meets the criterion, and that s a probability of 1."
    },
    {
        "video_name": "3FManXv4mZM",
        "Q": "at 2:14 in the video why do you multiply 1/2 im still cunfuzzled and ive watchd the vid about 5 times",
        "A": "At 2:14 I do not see him multiply by 1/2. But the two times that he does multiply by 1/2 are: the first time is because in finding the area of triangle ABC the formula is (1/2)bh. The second time is when he points out that BE is half of BD so by substitution he is trying to get the full length diagonal (BD) into the formula to prove that the area of rhombus ABCD is (1/2) (diagonal1)*(diagonal2)"
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At 3:05, why does Sal say that k is >= (greater than or equal to) 0? If k were 0, the whole exponent would be zero and the answer would be 1 right?\n\nThanks!\nThe #1 Tau Proponent",
        "A": "In this example, all Sal wants is for k to not be negative. That is the same as k being greater than or equal to 0. When i is taken to the power of any multiple of 4 (i.e. 4,8,12...), it will yield a result of 1. The any multiple of 4 part is expressed with 4k , with k being a positive integer or 0."
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "Couldn't one alternatively memorize the pattern 1, i, -1, -i starting from zero exponent and apply that to the last digit of a non-variable exponent?\ne.g. At 2:15, i^501 since the last digit is 1, it would equal i.",
        "A": "You have a good idea, but it doesn t exactly work that way...it s the remainder when dividing by 4. (i.e. i^501...501/4 = 125 r 1, since the remainder is 1 (one), i^501 = i)"
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At 4:16, I did not get how i^7340 gets^converted to 1 ? A more elaborate way would have been useful.",
        "A": "i^1 = i i^2 = -1 i^3 = -i i^4 = 1 i^5 = i i^6 = -1 i^7 = -i i^8 = 1 As you can see, the cycle repeats. So i^n = i^(n+4) . Hence, i^7340 = i^7336 = i^7332 = ... = i^8 = i^4 = i^0 = 1 (Divide your exponent by 4. The remainder is what you raise i to, to find the answer.)"
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At 5:12, Sal solves the problem i^99. I paused the video and used a different approach. (i^99) = (i^100 * i^-1) = (1/i)\nis this answer correct? If yes/no then why",
        "A": "Yes but we generally prefer the answer to be the form of i instead of 1/i. i\u00e2\u0081\u00b9\u00e2\u0081\u00b9=i\u00c2\u00b3 i\u00c2\u00b3=i\u00c2\u00b2*i=-1*i=-i 1/i * 1= 1/i * i/i = i/i\u00c2\u00b2 = i/-1 = -i or 1/i * 1 = 1/i * i\u00e2\u0081\u00b4 = i\u00c2\u00b3"
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At 3:03 why is k >= 0?",
        "A": "The main reason behind this move is to make sure that k is positive or 0. Raising i to a negative power would not help Sal s current demonstration, so he uses k >=0 to limit k s possible values to being non-negative"
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "Sal accidentally wrote i to the i power at 4:09, but I'm just wondering what that would equal.",
        "A": "We need Euler s equation e^(ix)=cos(x)+isin(x) Set x=\u00cf\u0080/2 to get e^(i\u00cf\u0080/2)=cos(\u00cf\u0080/2)+isin(\u00cf\u0080/2) e^(i\u00cf\u0080/2)=i (simplified the trig functions) Raise both sides to the i power to get [e^(i\u00cf\u0080/2)]^i=i^i e^(i\u00e2\u0080\u00a2i\u00cf\u0080/2)=i^i e^(-\u00cf\u0080/2)=i^i So, oddly enough, i^i is a real number."
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "Starting at 0:35 it says 100 hundred is a factor of I to the 4th so it's 1 but it's also a factor of I to the 5th and that has a different I value. and 0 is not a factor of 4 but it has the same I value so please someone help!",
        "A": "Don t worry! You just made a simple misunderstanding. The video says that 100 is a factor of four. This means that you can manipulate the expression that is given like so: i^100 = (i^4)^25 = 1^25 Since 1 to any power equals 1, this expression equals 1. Therefore, if you have i to the power of any multiple of four, you get 1. Does this help?"
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At 5:06, Sal figured out intuitively the highest multiple of 4 less than 99. How?",
        "A": "Since we know 100 is divisible by 4 then 96 must be too."
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At 2:33, why does i^4 times twenty-five simplify to 1?",
        "A": "1 to the power of any real number is always 1. For example: 1^-4/3= 1/(1^4/3)= 1 to the power of four thirds is 1 cube root times one cube root times one cube root times one cube root which really just equals one and 1/1 is yet again one."
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At 1:00, we can see that if i's exponent is a multiple of four, then it equals one. In other words, can we use this rule as shortcut when solving larger powers of i? What are some other shortcuts?",
        "A": "Yes, one of the rules of i is that its powers are cyclic. i^1=i, i^2=i*i=-1, i^3=-1*i=-i, and i^4=-i*i=1. We see that if we multiply i^4 by i again, we ll get i, which starts us back at the beginning of the cycle. You can simplify i to any power by taking the power mod 4. That is, i^58 is the same as i^(58 mod 4)=i^2=-1. Just remember to keep in mind, when dealing with negative powers, that you add remainders, so -5 mod 4 is 3, not 1."
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At 0:01 why is i^2= -1?",
        "A": "By definition, i=sqrt(-1), so we can conclude that i^2 is equal to -1."
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At 1:51, how did we get from i^501 to i^500?",
        "A": "i\u00e2\u0081\u00b5\u00e2\u0081\u00b0\u00c2\u00b9=i\u00e2\u0081\u00b5\u00e2\u0081\u00b0\u00e2\u0081\u00b0\u00e2\u0080\u00a2i\u00c2\u00b9 by exponent properties, (a^m)(a^n)=a^(m+n) so i\u00e2\u0081\u00b5\u00e2\u0081\u00b0\u00e2\u0081\u00b0\u00e2\u0080\u00a2i\u00c2\u00b9 = i\u00e2\u0081\u00b5\u00e2\u0081\u00b0\u00e2\u0081\u00b0\u00e2\u0081\u00ba\u00c2\u00b9 = i\u00e2\u0081\u00b5\u00e2\u0081\u00b0\u00c2\u00b9"
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "5:11 couldn't I do i^100*i^(-1)",
        "A": "2012396, Sure, you could do it that way i^100*i^(-1) i^100 = 1 so you get i^(-1) and i^(-1) = 1/i And to get the i out of the denominator, you could multiply by i/i 1/i * i/i = i/(i*i) = i/(-1) = -i So, yes it can be done, but it seemed easier to do i^96 * i^3"
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "what if i do i^100 = (i^2)^50 then the answer will be -1 (at about 1:15)",
        "A": "-1 is also cyclic when elevated, but instead of 4 posibilities (as \u00f0\u009d\u0092\u00be) it only has 2: (-1)\u00e2\u0081\u00b0 = 1 (-1)\u00c2\u00b9 = -1 (-1)\u00c2\u00b2 = 1 (-1)\u00c2\u00b3 = -1 ... As you can see any even power gives 1, while any odd power gives -1. So when you do: \u00f0\u009d\u0092\u00be\u00c2\u00b9\u00e2\u0081\u00b0\u00e2\u0081\u00b0 = (\u00f0\u009d\u0092\u00be\u00c2\u00b2)\u00e2\u0081\u00b5\u00e2\u0081\u00b0 = (-1)\u00e2\u0081\u00b5\u00e2\u0081\u00b0 = 1 Because you are elevating -1 to an even power."
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At 3:00, Sal restricted K to being not negative, but was that really necessary?\nNegative exponents mean you take 1 over the value. So i^-4, for example, would just be 1/i^4, which would be the same as 1/1 (because i^4 = 1), so i^-4 should still equal 1. The same would go for i^-8 (because 1/i^8 = 1/1 = 1), and also i^-12, and so on.\nSo, was the constraint k>=0 really necessary? Or did Sal just throw it in to keep things from getting too complicated for now?",
        "A": "It s Necessary! Because without it , the problem would be incomplete."
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "Do I always have to use 4 to solve i^k? At 1:43 he does i^501 then uses 4. Could I use 3? 501 is divisible by 3.",
        "A": "No, divisibility by three doesn t really help here. It s s specifically four because i^4 = 1."
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At 3:18, why did he say \"...to the Kth power\"?",
        "A": "Sal just picked a random variable which if you try to solve for k then you will actually get infinite solutions.\u00f0\u009f\u0098\u0083"
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "at 2:42 i^1 * i^1 is i^2 and that is -1, but he said jsut 1",
        "A": "You misunderstood. Sal turned i^501 into i^500*i^1. Then, he made the i^500 into i^4^125. i^4=1, so then 1^125 is still 1 and i^500 is therefore 1. 1 times i^1 is i."
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "at 4:22 Sal said i to the i power, what would that number be",
        "A": "Here is a proof using Euler s formula... e^(ix) = cos(x) + i*sin(x) e^(i\u00cf\u0080/2) = cos(\u00cf\u0080/2) + i*sin(\u00cf\u0080/2) = i i^i = (e^(i\u00cf\u0080/2))^i = e^(i^2*\u00cf\u0080/2) = e^(-\u00cf\u0080/2)"
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At 3:28, what does he mean by \"i to the 4k plus 2\"?",
        "A": "Sal is raising i to the 4*k plus 2 power, where k is a positive integer. Raising i to the 4th power yields 1, and i to the power of any multiple of 4 also yields 1. So, raising i to the 4*k plus 2 power equals (1)*(i^2), or -1, for any positive integer value of k."
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "At one 1:45 Sal multiplies i^501 as (i^4)^125 and so on. Couldn't he have gotten the same result by multiplying i^501 as (i^3)^167? ( (i^3)^167 = -i^167... so on.)",
        "A": "As with many things, there are several ways to do this and get the correct answer. The reason he wrote this as a power of i^4 is that this value is one, which is the multiplicative identity so all you really have to look at is what is left over to determine the value."
    },
    {
        "video_name": "QiwfF83NWNA",
        "Q": "In the i^100 example ( at 1:00) Sal said i^100 was i^ 4x25. Why could we not just say i^ 2x50.",
        "A": "Because then you d get different values. i ^ 4 is 1, so i ^ 4 * 25 = 25. i ^ 2 is -1, so i ^ 2 * 50 = -50. It works because the powers of I wrap around after i^4 (i.e, i ^ 4 is 1, i ^ 5 is i, i ^ 6 is -1, etc.)"
    },
    {
        "video_name": "BUmLw5m6F9s",
        "Q": "At 1:38 how does he knows \"less than or equal\"?",
        "A": "Luis, as mentioned in the problem, has a $25 gift card. He cannot spend more than $25, so the total cost of everything needs to be $25 or less, hence the less than or equal to . Hope that helps. (I guess they re not working out the tax or shipping...)"
    },
    {
        "video_name": "BUmLw5m6F9s",
        "Q": "At 2:24, couldn't he flip it around and instead have songs as the x-axiz and games as the y-axis? It doens't matter if he did. Because I think it will be the same result. Just wondering?",
        "A": "Indeed. The labeling of each axis on this graph is rather arbitrary and can, indeed, be switched around at your leisure."
    },
    {
        "video_name": "BUmLw5m6F9s",
        "Q": "When Sal is drawing the coordinate system around 3:00, he takes a guess at which resolution to use on the axes. This is not a good way to do it. Better is to wait until the step he does around 7:00, that is, figure out the maximum values that g and s can take while still satisfying the second inequality. Had he done that, he would not have had an unused half of the coordinate space.\n\nActually, better is to figure out the values for both inequalities first, and then figure out how to best draw them.",
        "A": "His stated goal is to solve by graphing, so it would be cheating for him to use algebra to find the answer and then solve by graphing! Now if he wanted to publish his results somewhere, he would probably readjust his scales and use a graphing utility to give straight lines and perfect intercepts."
    },
    {
        "video_name": "BUmLw5m6F9s",
        "Q": "At 9:23, he shaded the overlapping portion. Would this be the range?",
        "A": "The range is all the results that a function can produce. E.g. in f(x), f maps x to any result. So if the system of inequalities would be a function, then the overlapping portion would be the range."
    },
    {
        "video_name": "BUmLw5m6F9s",
        "Q": "At 2:24, couldn't he flip it around and instead have songs as the x-axiz and games as the y-axis? It doens't matter if he did. Because I think it will be the same result. Just wondering?Indeed. The labeling of each axis on this graph is rather arbitrary and can, indeed, be switched around at your leisure.",
        "A": "Yes, though it would result in a reciprocal when you checked both of them. They are indeed reversible, though we must first look for the dependent and dependent variable (y and x), and if there is none, choose."
    },
    {
        "video_name": "BUmLw5m6F9s",
        "Q": "8 games and 16 songs doesnot equal 15? At 5:14",
        "A": "No, 8+16 is 24. However, the equation shows that s + g \u00e2\u0089\u00a5 15, so as long as it is exactly 15 or more than 15, it is a solution to that equation."
    },
    {
        "video_name": "BUmLw5m6F9s",
        "Q": "How do you know at 5:45 to \"shade\" one side of the line as apposed to the other side?",
        "A": "My general rule of thumb is to shade on top of the line for greater than and under the line for less than. If it s unclear, plug in a coordinate pair on the side you think should be shaded. If it satisfies the inequality, you re good, if not, try a coordinate pair from the other side. Hope this helps!"
    },
    {
        "video_name": "BUmLw5m6F9s",
        "Q": "At 4:15 in the video why do you connect the two dots?\nP.S. I know you have to I just want to know why.",
        "A": "you have to connect them so that you can solve the system, if you didn t connect the dots then it would be difficult to find the solution because of no clear borders in the graph"
    },
    {
        "video_name": "_JpS8k1a9yE",
        "Q": "at 4:16, what happens to dN? I understand that 1/N goes to ln N. But why does dt go to just t while dN just disappear?",
        "A": "Once you take the integral of something, the dN and the dT will go away. That s just the way it is. For example, if you take the integral of x^2 dx, the answer would be x^3/3 and the dx goes away."
    },
    {
        "video_name": "_JpS8k1a9yE",
        "Q": "At 4:19, why is the anti-derivative of 1/N equal to the natural log of the absolute value of N?",
        "A": "Logarithm can t have negative inputs, geometric proof is called looking at the graph."
    },
    {
        "video_name": "OLO64d4Y1qI",
        "Q": "At 2:12, where does the du go? Why is there only one Ln IxI after this?",
        "A": "The du means that you integrate with respect to u . Just as \u00e2\u0088\u00ab 1/x dx = ln|x|, \u00e2\u0088\u00ab 1/u du = ln|u| . It s the same thing, just with different variable."
    },
    {
        "video_name": "OLO64d4Y1qI",
        "Q": "AT 2:15 why isn't the derivative of 1/u not rewritten as u^-1 giving us -u^-2?",
        "A": "At 2:15 Sal is taking the antiderivative not the derivative. So he gets ln | u |"
    },
    {
        "video_name": "tFhBAeZVTMw",
        "Q": "why at 1:16 did he add three lines?",
        "A": "at 1:16, he wrote three lines because that line is congruent to (equal to) itself."
    },
    {
        "video_name": "tFhBAeZVTMw",
        "Q": "At 0:11 Mr.Sal mentioned a video on finding the area of rhombi, but there is no video on that!Why isn't there one?",
        "A": "Because rombi is another way of saying an parallelogram"
    },
    {
        "video_name": "tFhBAeZVTMw",
        "Q": "At around 4:30, is DC the base and if so why?",
        "A": "Yes, because it is the bottom side of the shape."
    },
    {
        "video_name": "tFhBAeZVTMw",
        "Q": "0:50\n\nSo essentially to find out the area of a parallelogram you can split the shape into two triangles and find out the area from there?",
        "A": "That is a way you can do it, but you can just multiply base length by height because a parallelogram is incredibly similar to a rectangle. The slants don t matter because the added area cancels them out"
    },
    {
        "video_name": "tFhBAeZVTMw",
        "Q": "at 2:31, did Sal really mean [ADC] or [ABC]?",
        "A": "Sal meant that both triangles had the same area, so you only had to use one triangle to find out the entire area of the parallelogram. Hope this helps!"
    },
    {
        "video_name": "tFhBAeZVTMw",
        "Q": "At 1:25, Sal writes a symbol that looks like an equal sign (=) and half of an approximation symbol. Does this sign mean congruent?",
        "A": "Yes that is the sign for congruent."
    },
    {
        "video_name": "tFhBAeZVTMw",
        "Q": "At 3:00 why do you times the inside hight of the parallelogram not the side?",
        "A": "Aren t they the same measurements?"
    },
    {
        "video_name": "tFhBAeZVTMw",
        "Q": "Does the triangle thing at 0:55 work on all parallelograms?",
        "A": "Yes, by definition, you can always split a parallelogram into two triangles"
    },
    {
        "video_name": "tFhBAeZVTMw",
        "Q": "At 0:48, Sal said diagonal, are diagonals and traversals the same thing or am I confused?",
        "A": "A diagonal is a line segment passing through non adjacent vertexes"
    },
    {
        "video_name": "tFhBAeZVTMw",
        "Q": "At 2:21, Sal uses brackets to represent area. Are brackets common symbols used to represent area?",
        "A": "Yes they are."
    },
    {
        "video_name": "OhISsmqv4_8",
        "Q": "Why was the bounds of the integral changed on 3:40? Would keeping the original bounds still work when you replace u back after taking the integral of u with respect to the original bounds?",
        "A": "The original bounds were in terms of x. Once we did the u-sub, we are now in terms of u, so to find the new bounds in terms of u we did the process you referenced. In answer to your second question, YES, if you want to convert back to terms of x after you do the integration in terms of u, then you can use the original bounds. Do what works best for you."
    },
    {
        "video_name": "ZIqW_sXymrM",
        "Q": "2:06 I don't understand how he factored out the p.\nI guess I can understand how the exponent for the first p drops to one, but how does the second term turn into p-2??",
        "A": "The 2 terms we re factoring are p^2 and -2p. They each have a common factor of p, so we remove the common factor to be outside the parentheses and leave the other factors of p and - 2 inside the parentheses. to get the expression p ( p - 2 ). (As a check, do the multiplication to be sure we still have what we started with.)"
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "At 2:22 the equation didn't have to be two stepped, it could've been ( 2 - 14) / 3",
        "A": "Ya. I think he was just trying to make it clear."
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "At 0:23 why does khan show a weight?",
        "A": "He s saying that equations are similar to the balance of weights, in this case the weights are perfectly balanced so the weight on each side is equal; 14 yellows on the right, 2 yellows and 3 blues (X) on the left, if you assume the yellows are 1 , the equation would be: 3x + 2 = 14 From that you can figure out the weight (or simply value) of X: 3x = 12 x = 4"
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "At 4:21 in your video you left six boxes on the right.",
        "A": "No, he has lines thru all but 4 boxes on the right. Watch him as he draws the lines. Some of the lines are pretty light, but they are there."
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "At 3:54, I don't get why multiplying each side by 1/3 is the same a dividing each side by 3...",
        "A": "1/3 means 1 divided by 3. If you were to multiply a number by x * 1 % 3. (% is used as a division symbol) order of operations says x % 3 * 1 is equivalent to the above expression. x*1/3 any number times 1 is itself x/3 so multiplying a number by 1/3 is the same as dividing by 3"
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "At 4:20, Sal missed 2 blocks. Did he do that on purpose or did he just didn't see it? Because I would be surprised if he did do it on purpose.",
        "A": "You think he messed up?"
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "At 4:24, he left 6 block not 4.",
        "A": "This is a known error in the video. A box does pop up at about 4:20 to tell you this."
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "Did anyone notice how Sal Khan forgot about the 2 1kg boxes on bottom right hand side at 4:37s and below!",
        "A": "No he doesn t. He crosses out (or removes) 2 1-lb boxes from each side at 2:25 in the video."
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "At 3:48+ was it really necessary to do everything that you did with 1/3? I saw much easier ways to do that.......",
        "A": "Multiplying by 1/3 is the same as dividing by 3, so you could write it either way."
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "3:50-3:58, do you always have to multiply by the reciprocal? or could you have just divided it by 3 on both sides?",
        "A": "You could also divide by 3. You will get the same result."
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "At 0:22 Sal says a mathematical equation but would it not be an expression because he only ask for the formula of one side the scale and it does not ask for a solution yet?",
        "A": "Equations are made up of 2 expressions. The entire scale pictured in the video represents the equation. Each side of the scale represents one side of the equation. By the end of the video, Sal has created the entire equation: 3x+2 = 14"
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "2:57.\n\"X bloxes\" haha, love it.",
        "A": "I think that was an accident, but it was pretty funny haha!"
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "At 4:05 khan multiples by 1/3. Couldn't you divide by 3 as well?",
        "A": "It s really the same operation, because division and multiplication are inversely related. So dividing by 3 is the same exact thing as multiplying by 1/3. It ll get you the same answer. So yes, you could do either."
    },
    {
        "video_name": "_y_Q3_B2Vh8",
        "Q": "at 3:07 used th concept of algebra",
        "A": "Yes it did..? This whole video is supposed to be about algebra."
    },
    {
        "video_name": "HyThzLRuqXo",
        "Q": "At 3:11, Sal said that we have to take DE as negative then why didn't he took BC as negative on LHS? Isn't it also dropping?",
        "A": "He is saying DE is negative while AD is positive so that the slope of AE is negative, so he is using this fact for a purpose, not just to say that it is negative for no reason. On the other hand, if you start from A, you are correct that BC is negative, but AB would also be negative, so when you talk about the slope of AC, when you divide a negative by a negative, you would get a positive. Hope this helps"
    },
    {
        "video_name": "2axaRJQkfVk",
        "Q": "at 3:50 when it's the integral of the derivative why isn't it the same as in part c? I thought it was the same thing where it's H'(10)'H'(0) in c so that part d is just B'(10) -B'(0) ? Why does he actually integrate it?",
        "A": "From my understanding of the information given this is because in the table they give you values for H(t) because they don t give you the equation for h (t), so you can t integrate it yourself. However they do give you the equation for b (t) and you have to integrate it to find the B(t) function to find out the values at the given times"
    },
    {
        "video_name": "2axaRJQkfVk",
        "Q": "At 3:48, why do you divide by -.173?",
        "A": "F(t) = \u00e2\u0088\u00ab( -13.84\u00e2\u0080\u00a2e^(-0.173\u00e2\u0080\u00a2t) )dt F(t) = -13.84\u00e2\u0080\u00a2\u00e2\u0088\u00ab( e^(-0.173\u00e2\u0080\u00a2t) )dt u = -0.173\u00e2\u0080\u00a2t ( d/dt(-0.173\u00e2\u0080\u00a2t) )dt = du ( -0.173\u00e2\u0080\u00a2d/dt(t) )dt = du ( -0.173\u00e2\u0080\u00a2dt/dt )dt = du ( -0.173 )dt = du dt = ( 1/-0.173 )du dt = ( -1/0.173 )du F(u) = -13.84\u00e2\u0080\u00a2\u00e2\u0088\u00ab( e^(u)\u00e2\u0080\u00a2-1/0.173 )du F(u) = -13.84\u00e2\u0080\u00a2-1/0.173\u00e2\u0088\u00ab( e^(u) )du F(u) = 13.84/0.173\u00e2\u0088\u00ab( e^(u) )du F(u) = 80\u00e2\u0080\u00a2\u00e2\u0088\u00ab( e^(u) )du F(u) = 80\u00e2\u0080\u00a2e^(u) u = -0.173\u00e2\u0080\u00a2t F(t) = 80\u00e2\u0080\u00a2e^(-0.173\u00e2\u0080\u00a2t) F(t) = [80\u00e2\u0080\u00a2e^(-0.173\u00e2\u0080\u00a2t) + C]"
    },
    {
        "video_name": "2axaRJQkfVk",
        "Q": "At around 6:39: Wherefore do you subtract 65.817247 from 100?",
        "A": "B(t) = integral B (t) from 0 to t + y-intercept It s said that when t = 0, B(0) = 100 You can think of it as.. if x = 0, then y = 100 y-intercept is 100 B(0) = integral B (t) from 0 to 0 + y-intercept B(0) = 0 + y-intercept B(0) = y-intercept B(0) = 100 so when actually solving the problem where t = 10... B(10) = integral B (t) from 0 to 10 + 100 B(10) = -65.8172 + 100 does that help clarify?"
    },
    {
        "video_name": "oqAgaGeBOAU",
        "Q": "Is there a video or a simple explanation of how I can find out what to do with this - Divide 306 into the ratio 9:5:3?",
        "A": "is this a homework problem or something? can you tell me the entire problem word for word"
    },
    {
        "video_name": "oqAgaGeBOAU",
        "Q": "at 1:16 what was the reason for multiplying the fraction by 10?",
        "A": "By multiply the numerator and denominator by 10, Sal changed the fraction from a complex fraction (it s complex because the fraction contains a decimal) to a simpler fraction (no decimals within the fraction). Hope this helped."
    },
    {
        "video_name": "wblW_M_HVQ8",
        "Q": "At 7:37 why we need k+2 to be factored out into [(k+1)+1]? I don't get it",
        "A": "Because it is easier to recognize that we have proven that it also works for k+1, the following number after k. But it is redundant, just to make it easy for us to see it."
    },
    {
        "video_name": "wblW_M_HVQ8",
        "Q": "at 4:00 in the basic step we take n=1, what if we take some other number. Will the answer be correct?",
        "A": "By starting with 1 and proving that k+1 is true if k is true, we prove it for all n > 1. If we had started with 10 and proved that k+1 is true if k is true, we d have proved it for all n > 10. So, usually lower is better."
    },
    {
        "video_name": "wblW_M_HVQ8",
        "Q": "At 6:35 Sal starts factoring out k(k+1)+2(k+1), I really don't get this. How does he get from k(k+1)+2(k+1) to (k+1)(k+2)?",
        "A": "Well.., I guess I ll just answer myself, LOL! Turns out (k+1) is the GCF so you just take it out in the front and you re left with k+2. Watched the Factoring and the Distributive Property 3 and everything became crystal clear. ;)"
    },
    {
        "video_name": "wblW_M_HVQ8",
        "Q": "At 6:35 Sal starts factoring out k(k+1)+2(k+1), I really don't get this. How does he get from k(k+1)+2(k+1) to (k+1)(k+2)? Help please:[",
        "A": "Sal is just using the distributive property to factor out a common factor of (k+1). Consider: kx + 2x Hopefully, you see there is a common factor of x . If you factor it out you get: x(k+2) Sal did exactly the same thing... except, instead of x , the common factor is (k+1) Hope this helps."
    },
    {
        "video_name": "wblW_M_HVQ8",
        "Q": "At approximately 7:35, how does he get ((k+1)+1) from k+2? Thank you!",
        "A": "because 2 = 1+1. He separated it out to suit what he was doing."
    },
    {
        "video_name": "wblW_M_HVQ8",
        "Q": "At 2:36, he says that in the induction step, we assume this is true for s(k)... where'd k come from? I thought the function was s(n)?",
        "A": "Sal uses S(n) = n(n+1)/2 for the value of the sum for some general n, and S(k) = k(k+1)/2 for the value of S(n) at some particular k. It s like saying f(x) = 2x + 3, and f(2) = 2*2+3 = 7."
    },
    {
        "video_name": "GBtcGO44e-A",
        "Q": "At 2:16, you say that 15 - 8 = 17, little slip of the tongue :)",
        "A": "my name is dawidklon"
    },
    {
        "video_name": "kkrF9X6Fycg",
        "Q": "At 0:30, Sal writing the -10x^3 under both, but shouldnt it be -10x^3 / -10x^3 AND (F) / -10x^3 ? Thanks",
        "A": "No because these two terms are being multiplied together rather than added. He did not have to use parentheses to separate them because they are all one term, but it is easier to see. If they were added instead of multiplied, then the opposite would be to subtract and that is a different problem."
    },
    {
        "video_name": "GwjiR2_7A7Y",
        "Q": "2:37",
        "A": "I meant only mean, not meanly,sorry"
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "At 6:00, how does multiplying 1/(sqrt3) by (sqrt3) equal (sqrt3)/3? How do the ratios stay consistent?",
        "A": "He is multiplying times \u00e2\u0088\u009a3/\u00e2\u0088\u009a3"
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "At 6:31, Sal said the final answer was (4+\u00e2\u0088\u009a3)/(3) + 2. However, you could simplify this by changing 2 into 6/3, which would turn it into (6+4\u00e2\u0088\u009a3)/(3). Which one is a more acceptable answer?",
        "A": "Remember to add parentheses to get (6 + 4 \u00e2\u0088\u009a3)/3 in this representation. I do not think that one or the other is anymore acceptable than the other, but when we do the quadratic formula, we more often would see it somewhat like your form because that is how it starts."
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "at 5:39 i was wondering why you don't add 1 over the square root of 3 because just before it is a - and i thought you would have to add it but you subtracted it, I'm really confused!!",
        "A": "The length of ED is sqrt3 - 1/sqrt3 And to find the perimeter he added ED to BE to BD, which is 2+ (sqrt3 - 1/sqrt3) + 2/sqrt3. Hope this is the answer you were looking for !"
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "I can\u00c2\u00b4t understand the way he took since the 5:54 to the end.\nHe says that will multiply 1/\u00e2\u0088\u009a3 +\u00e2\u0088\u009a3 +2 by \u00e2\u0088\u009a3. So \u00e2\u0088\u009a3 (1/\u00e2\u0088\u009a3 +\u00e2\u0088\u009a3 +2).\nWhy he gets \u00e2\u0088\u009a3/3+3\u00e2\u0088\u009a3/3+2?\n\nWhen we get \u00e2\u0088\u009a3 (1/\u00e2\u0088\u009a3 +\u00e2\u0088\u009a3 +2) we can rewrite as =\u00e2\u0088\u009a3 (1/\u00e2\u0088\u009a3 )+\u00e2\u0088\u009a3 (\u00e2\u0088\u009a3 )+\u00e2\u0088\u009a3 (2), so we will get:\n\u00e2\u0088\u009a3 /1(1/\u00e2\u0088\u009a3 )= \u00e2\u0088\u009a3 /\u00e2\u0088\u009a3 =1\n\u00e2\u0088\u009a3 (\u00e2\u0088\u009a3 )=\u00e2\u0088\u009a9=3\n\u00e2\u0088\u009a3 (2 )= 2\u00e2\u0088\u009a3\n\n1+3+2\u00e2\u0088\u009a3 = 4+2\u00e2\u0088\u009a3\n\nAm i wrong? Why? Please someone explain me. Thanks",
        "A": "He had: (1/\u00e2\u0088\u009a3) + \u00e2\u0088\u009a3 + 2 He did not multiply this entire sum by \u00e2\u0088\u009a3. He simply multiplies the first term by \u00e2\u0088\u009a3/\u00e2\u0088\u009a3 to rationalize its denominator: (1/\u00e2\u0088\u009a3)(\u00e2\u0088\u009a3/\u00e2\u0088\u009a3) = \u00e2\u0088\u009a3/3 And for the second term to have a common denominator with the new first term, he multiplies the second term by 3/3: (\u00e2\u0088\u009a3)3/3 = (3\u00e2\u0088\u009a3)/3 So now he adds the first and second terms: (\u00e2\u0088\u009a3/3) + (3\u00e2\u0088\u009a3)/3 = (4\u00e2\u0088\u009a3)/3 Plus the additional 2 he had as the third term."
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "at 3:41 can somone help , i am in chaos, i understand the video but at 3:41 SAL said mulitipled 1/sqrt3*3 get 1 which is 30degree angle. if shortest length opposite to 30 is 1/sqrt 3 then opposite 60 must be double to opposite of 30, but sal said it is 1/sqrt3.\nhelp...!",
        "A": "I think you are mistaken. Sal says that it is 2/sqrt(3)."
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "At 2:15, Sal says that the 60-degree side is going to be \u00e2\u0088\u009a3 times 1, but I don't understand how he got \u00e2\u0088\u009a3.",
        "A": "The 30-60-90 ratio states that if the side across from 30* angle is x, then the side across from 60 will be x*\u00e2\u0088\u009a3 and the one across from the 90* will be 2x. Therefore, if x is one, then the side across from 60 will be 1*\u00e2\u0088\u009a3 = \u00e2\u0088\u009a3"
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "at 2:18 how does he get that line BC is sq. root of 3?",
        "A": "The side across from the 60 degree angle is always sqrt 3 times the side across from the 30 degree angle in a 30-60-90 triangle."
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "Why did Sal give the side opposite the 30 degrees triangle a 1/sqrt3 at 4:30 in the video?",
        "A": "what are you talking about"
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "how long can a 30:60:90 triangle be?",
        "A": "What do you mean? The lengths of a triangle can be as long or short as you want, as long as they actually connect and make a triangle."
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "Beginning at 3:10, Sal explains how we can figure out the side opposite of angle ABE (30 degree) given the length of side AB (which is 1). I'm having a hard time following how the side opposite of angle ABE (or side AE) is 1/sqr.3. I understand that this side is originally the hypotenuse/2 so I just don't see how we get 1/sqrt.3. I hope this is clear enough for someone to take a stab at it. I may just need to re-watch previous videos. Thanks.",
        "A": "Another way to look at it is that AE=BE/2. We know AB=\u00e2\u0088\u009a3*BE/2. We know that AB is 1 so 1=\u00e2\u0088\u009a3*BE/2. This gives BE=2/\u00e2\u0088\u009a3, or to rationalize it = 2*\u00e2\u0088\u009a3/3. We know that AE = BE/2 so this gives AE=( 2*\u00e2\u0088\u009a3/3) / 2. Or for easier math AE= (2*\u00e2\u0088\u009a3/3) * 1/2 which simplifies down to = \u00e2\u0088\u009a3/3. Sal writes it as 1/\u00e2\u0088\u009a3 and when you rationalize this you get \u00e2\u0088\u009a3/3. Hope this made sense."
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "At 2:20, how does he get square root of 3?",
        "A": "You times the number by the square root of 3 across from the side of 60 degrees."
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "At 6:36, why couldn't have Sal have just written (4 * sq. r.t (3) + 6)/3 ?",
        "A": "That s a possible answer, however, mathematicians enjoy simplified answers and in this case, 6/3 can be simplified."
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "Can someone please explain what rationalizing means. Sal mentions it at 5:50.",
        "A": "Rationalizing is just converting a fraction to make sure the square root (radical) is on the numerator. Here is would be (3^0.5)/(3^1)=3^(0.5-1)=3^(-0.5)=1/(3^0.5). 1/sqrt3 is same as sqrt3/3"
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "4:05 I am confused when I do 1/ sqrt 3 I get sqrt 3/ 3 and not 1 / sqrt 3",
        "A": "Those are the same thing. (3^0.5)/(3^1)=3^(0.5-1)=3^(-0.5)=1/(3^0.5)"
    },
    {
        "video_name": "dgHksfBFbjk",
        "Q": "At 2:20, How did you get Square Root of 3. I understand but not completely.",
        "A": "1 : sqrt(3) : 2 is the ratio used here. For the 30 side, the length is one. Apply the ratio, and the 60 side is sqrt(3). I hope this helps!"
    },
    {
        "video_name": "Ei54NnQ0FKs",
        "Q": "At 5:11, do we always try to simplify the fraction?",
        "A": "A calculator will do that for you. If you put 15/24 into your calculator and press enter, you will get 5/8, which is the simplified form of 15/24."
    },
    {
        "video_name": "Ei54NnQ0FKs",
        "Q": "At 4:05, why didn't he just subtract 6000 from 6100, then subtract 100 from each side?",
        "A": "He couldn t do that because the 6000 was not a stand-alone term. The 6000 was part of the term -6000 cos(theta)."
    },
    {
        "video_name": "Ei54NnQ0FKs",
        "Q": "@ 3:55 he has it set to 400 = 6100 - 6000 cosine Theta, and from there he subtracts it from both sides to isolate theta, should he have taken the square root of 6100 and 6000 and 400 as well, then isolated cosine theta?",
        "A": "No, you only need to take the square root when you re using the cosine rule to calculate a side. If you re calculating an angle it won t matter that the sides have been squared because the same thing was done to all of them so their relation to each other (the angles between them) will be unaffected. Its the principle of similar triangles , which you ll probably remember studying in earlier trig, look up a few a check out the angles :)"
    },
    {
        "video_name": "Ei54NnQ0FKs",
        "Q": "How come at 3:59, Sal didn't subtract 6,100 - 6,000?",
        "A": "It s because the 6000 part is being multiplied to cosine of theta. It is not standing in the equation by itself. It s like this: 560 - 560(4). You wouldn t do the subtraction first ; you d do the multiplication first in accordance to PEMDAS."
    },
    {
        "video_name": "Ei54NnQ0FKs",
        "Q": "3:57 he had 400=6100-6000*cos(theta)\nwhy didn't he just do the subtraction?\nit would be:\n400=100*cos(theta)\n400/100=4=cos(theta)\ncos^-1(4)=theta\nis there some reason he didn't do that if so I would like to now.",
        "A": "Because of order of operations. It is required to do 6000*cos(theta) before adding to 6100. Since theta is the variable, you have to subtract 6100 from 400 first to solve for theta. If you had a simpler equation, hypothetically, 10 = 2 + 4x, I think you would agree readily that we should subtract 2 from 10 first to solve this equation. If you see it there, now look back at your original one. It s the same format."
    },
    {
        "video_name": "Ei54NnQ0FKs",
        "Q": "At 5:53, could you have simplified 19/20 to 0.95 or would that mess up the math when multiplying by cos^-1 ?",
        "A": "Fractions and decimals will always give the same result if they are equal. Fractions are most commonly used because ratios are easier to work with (in proportions) rather than decimals. So yes, you can used 19/20 or 0.95."
    },
    {
        "video_name": "Ei54NnQ0FKs",
        "Q": "At 2:12, isn't it easier to rewrite the rule as cosA=b^2+c^2-a^2/2bc ?",
        "A": "it can be cos A = (b^2+c^2-2bc)/a^2."
    },
    {
        "video_name": "Ei54NnQ0FKs",
        "Q": "At 0:38, why does he say angle theta instead of something else?",
        "A": "Because theta is the standard variable for an unknown angle measure."
    },
    {
        "video_name": "Ei54NnQ0FKs",
        "Q": "At 6:06 why do we do inverse cosine?",
        "A": "you use inverse when ever you are doing a angle"
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "At 3:07, how do you know whether something is positive or negative in a problem like this?",
        "A": "pos+pos=pos neg+neg=neg pos+neg=take sign of larger..pos*pos=pos neg*neg=pos pos*neg=neg. in this video at the end he had 21/-21. only one -sign makes the answer also neg. had there been 2 neg signs -21/-21 they cancel each other out making the answer pos."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "Where did he get the -3? 1:47 of the video",
        "A": "On the left side of the equation (-9-9x+6) he just combines -9 and 6. -9+6 or +6-9 = -3"
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "I dont get the combination of -9 and 6 at 1:24. Please explain.",
        "A": "what you guys are basicly saying is that -9 and 6 are like terms so you can add those together. not like terms is -9 subtracting -9x. this cant be done because they do not share the same variable."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "From 2:20 to about 2:23, Sal subtracts 12 from the right and left sides...was that a rule (necessary) or could he have subtracted, I mean, added 9? Thanks.",
        "A": "Yes. You could have added 9x instead of subtracting 12x. If you add 9x, then your next step would have been to subtract 18 from both sides,"
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "1:47\n\nWhy does -9 go in front of the -3 when we combine like terms. Why could it not be the other way around?",
        "A": "Sal is using -1 for representation purpose for better compreshension , you can use 1 also before 4x ."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "at 2:34 could you get rid of the -9 ?",
        "A": "Assuming you re referencing the -9x and not just the -9, it s combined with the like term of -12x to form a greater term -21x. This is only done to remove the -12x from the other side of the equation. Hope that makes sense."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "At 1:48 when Sal combined -9 and 6 he got -3 so it became -9x-3=12x+18, but could you have done -3-9x=12x+18 instead or would that have changed the answer? I always get confused when solving equations when you combine 2 terms that are on the same side.",
        "A": "Either version will work. Just make sure you keep the right signs with the right numbers. As long as you do, then the order of the terms will not matter. This is because the commutative property of addition applies."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "At 5:16, how is negative 9- (-15) the sames as -9 + 15? I cant exactly remember.",
        "A": "It s the same as 9+15, not -9+15. You should review adding and subtracting negative numbers in the pre-algebra playlist, that should help clear up everything"
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "At 0:20 Sal replaces the \"-\" sign with \"-1\" but why?\nHow is minus the same as negative one?",
        "A": "A negative sign is the same as multiplying a term by -1."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "At 1:26 why do we group -9 and 6 together? Please explain.",
        "A": "-9 and 6 are like terms. They can be grouped together to make the equation simpler. If you were to reorganize the left hand side of the equation, it would be -9x - 9 + 6 You can go right ahead and calculate -9 + 6 to make it one term. -9 + 6 = -3. So then, the equation will be simplified and shortened to -9x - 3"
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "At 1:57 how is -9x-3 showing that 3 is negative ??",
        "A": "The answer is the same either way. You can either subtract 3 from -9x, or you can add a -3 to -9x; both ways are mathematically correct. In Algebra, however, you don t say subtract , refer to adding a negative . So the problem -9x - 3 should, technically be read as negative 9x plus negative 3. It doesn t really change anything, it s just proper Algebraic grammar , if you will. So when he says that the problem you gave says that 3 is negative his is correct."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "At 0:29, why does he put a one there? He explains it, but I still don't get it.",
        "A": "He puts the 1 in there to make it clear that distributing a - is the same as distributing -1."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "On 0:35 I didn't quite understand it. You see if you had a question such as\n31-11n=8-12(8n-9) what would happen to the 8? and what would the problem look like then? is the twelve a negative because of the minus? so confuzzled please halp :D",
        "A": "I ll just solve it, it should be easier since I can t explain well in words. 31-11n=8-(12)(8n)-(12)(-9) 31-11n=8-96n+108 The 8 still remains after the 12 is distributed in, and yes, the 12 is negative because of the subtraction symbol in front of it."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "At 2:07-2:14, Sal says \"to get all x terms on the left side of equation, we need to get rid of 12x by subtracting it form both sides of the equation.\" Won't it be much easier if we transfer the 12x to the left hand side (which makes 12x negative) and then continue to simplify?",
        "A": "What you are doing is the same thing as what Sal said. You just aren t writing in the -12x on both sides. The way the 12x disappears on the right sides is to subtract it from both sides. You just skip the step of writing out 12x - 12x on that side. But, you are doing it. Hope this helps."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "how does the negative one appear at 0:34? If you didn't multiply with that, what'd happen?",
        "A": "The equation wouln t give you the right answer at the end if you switch signs. What happened is this: the parentheses also mean a multiplication in algebra. But since, in this case, the numbers inside the parentheses are multiplied by minus, we consider that it s -1. It was Sal s way to explain how things in the ( ) were changing sign."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "At 1:42, when he combines his like terms -9 and +6, he puts it to the right of -9x. So, -9 -9x +6 becomes -9x -3. This might just be a bad memory, but I recall people combining like terms and putting their new number in some other spot then the second number they combine. What's the rule about where to put a newly combined number? Where do you put it if you're combining more than one number?",
        "A": "It doesn t matter where you place the like terms. It ll be the same equation. He could have made it 6-9x and it would still be correct."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "0:21 - 0:24 Would the same thing apply if it was a positive sign?",
        "A": "Yes you still could add a 1 in front, but multiplying a positive 1 times anything does not change it, so with a positive, the parentheses can just be dropped. With a negative, everything inside has a sign change."
    },
    {
        "video_name": "YZBStgZGyDY",
        "Q": "two numbers are in the ratio 5:3. If they differ by 18,what are the numbers",
        "A": "First number is 5n Second number is 3n 5n : 3n = 5:3, so the ratio remains true. 5n - 3n = 18, this is the difference. Solve for n then figure out the numbers."
    },
    {
        "video_name": "mbc3_e5lWw0",
        "Q": "at 2:15, how do you find prime roots",
        "A": "What do you mean by prime roots . At 2:15 in the video, Sal is doing sqrt(25). 25 is not a prime number. Please clarify your question."
    },
    {
        "video_name": "mbc3_e5lWw0",
        "Q": "At 4:43, when he says that the square root of 9 = x and x^2 = 9, he says that the x's are almost equal, but I'm not sure if he's telling the truth. For the first equation, x has to be 3, but for the next equation, he says x can be 3 or -3. If x = -3, -3^2 doesn't equal 9 because -3^2 = -(3*3) = -9, so x for both equations has to be 3. If he made the second equation (x)^2 = 9, then he will be right.\nDoes anyone have the same question or disagree? if yes, please reply.",
        "A": "But it you did it correctly, (-3)^2 is 9 or (-3)(-3) = 9. The way you show is just the negative value of 3 squared, not -3 squared. Your use of parentheses would not really help, because using your logic, it would be (-3^2) which still gives you - 9."
    },
    {
        "video_name": "mbc3_e5lWw0",
        "Q": "Did you know at 5:23 in the video, right before it ends in the bottom left corner there is triangle?",
        "A": "yea i noticed that to"
    },
    {
        "video_name": "mbc3_e5lWw0",
        "Q": "at 5:09, it says that 3^2 is 3 and -3. I don't get this.",
        "A": "Rewatch that section. It does not say 3^2 is 3 and -3 It tells you that if you have an equation: x^2 = 9, then x can = 3 or -3. This is because 3^2 = 9 and also (-3)^2 = 9. Hope this helps."
    },
    {
        "video_name": "mbc3_e5lWw0",
        "Q": "I don't get the X = -3 part in 4:48 ?",
        "A": "Whenever you are finding square roots of a number, remember to consider the possible cases in which, when squared will give you this number. -3^2= -3*-3= +9 and 3^2= 3*= +9"
    },
    {
        "video_name": "mbc3_e5lWw0",
        "Q": "2:55 when learning this in class my teacher told me that the principal root is the positive square root of a number and that the square root sign is called a radical. But in this video the person teaching the lesson is saying that the radical (square root sign) is called a principal root. What is the name of the square root sign?",
        "A": "The positive number is the principal square root and the square root sign is called the radical. But when the radical sign is by itself it is usually just asking for the principal square root. Usually when someone wants both answers they will put a plus sign over the top of a minus sign in front of the radical."
    },
    {
        "video_name": "mbc3_e5lWw0",
        "Q": "At 9:05, what is a cube root and what is a square root? Aren't they the same?",
        "A": "A cube root of a number is a number multiplied by itself three times to produce that number. So the cube root of 27 would be 3, because 3 \u00c3\u0097 3 \u00c3\u0097 3 = 27. A square root of a number is a number multiplied by itself two times to produce that number. So the square root of 144 would be 12 ."
    },
    {
        "video_name": "mbc3_e5lWw0",
        "Q": "I don't get how at 2:31, you get 9 for (-3)^2. Does someone know?",
        "A": "Well, (-3)^2 is the same thing (-3) squared. Which is the same thing as (-3) x (-3) which equals 9. A negative number times a negative number is always positive. I hope this helps"
    },
    {
        "video_name": "mbc3_e5lWw0",
        "Q": "At 3:40 Sal says that any negative squared becomes a positive. So does that mean that any negative to the x power (x being ANY number) is equal to a positive?",
        "A": "No. If your power if an EVEN number, then your answer will be positive. If you power if an ODD number, then your answer will be negative. Like this: (-3) ^2 = -3 * -3 = 9 (-3) ^3 = -3 * -3 * -3 = -27 (-3) ^4 = -3 * -3 * -3 * -3 = 81"
    },
    {
        "video_name": "Vm7H0VTlIco",
        "Q": "3:41 What is 9x^0? How is 9x^0 the same as positive 9? x^0 cannot be zero because 9*0 would be zero and not nine. Is x^0 equal to one? (9* (x^0 = 1) ) = 1?",
        "A": "X no matter what number (unless it is negative) when raised to 0 will be 1 so 1 times 9 is 9"
    },
    {
        "video_name": "Vm7H0VTlIco",
        "Q": "Did Sal make a mistake at 5:26 ?",
        "A": "He meant that the ones that are polynomials are polynomials."
    },
    {
        "video_name": "Vm7H0VTlIco",
        "Q": "I still do not get why a monomial is the same thing as a polynomial. Isn't a polynomial made up of separate monomials? I am referring to 1:20. Please help! Thanks :)",
        "A": "A polynomial is the sum of 1 or more terms (items being added/subtracted) where all the variables have positive exponents and there are no variables in the denominators. A monomial is the simplest polynomial. It has a single term."
    },
    {
        "video_name": "Vm7H0VTlIco",
        "Q": "At 3:51, Sal said the term has to finite number. Then why is Pi considered an finite number if it goes on forever.",
        "A": "Infinity (either positive or negative) is unbounded. It is larger (or, if negative, smaller) than any number that can be named. On the other hand, we know that \u00cf\u0080 is less than 4 and more than 3. It is only its decimal part that goes on forever."
    },
    {
        "video_name": "Vm7H0VTlIco",
        "Q": "@5:20 Sal states that 9a^a is not a polynomial because \"a\" could be negative. If he added that a \u00e2\u0089\u00a5 0, would that satisfy the definition of a polynomial?",
        "A": "No it wouldn t. The crucial thing here is that a is a variable. Regardless of any constraints you might put on them (such as a \u00e2\u0089\u00a5 0) the exponents can NOT be variables. They have to be non-negative integer constants."
    },
    {
        "video_name": "Vm7H0VTlIco",
        "Q": "At 8:05 Sal mentions that the highest degree will be the degree of the polynomial. What is there was a polynomial that had the same degree for all of its terms, what would the degree be?",
        "A": "The degree of the entire polynomial would just be that degree"
    },
    {
        "video_name": "Vm7H0VTlIco",
        "Q": "At 6:10, all the monomials have a coefficient, variable and a power. Do all polynomials need to have powers/exponents?",
        "A": "well, if you are talking about the terms in the polynomial, yes. Even if they do not look like it, every number has an exponent and variable. For example, if there is a term like 4, it can be rewritten as 4x to the 1 power."
    },
    {
        "video_name": "Vm7H0VTlIco",
        "Q": "At 9:35 Sal says that standard form is ordering the terms in degree order. What if two terms have the same degree? Does the one with the bigger coefficient lead? Or does it matter?",
        "A": "If you had two terms of the same degree then you would combine them. So, for example, \u00f0\u009d\u0091\u00a5 + 3\u00f0\u009d\u0091\u00a5\u00c2\u00b2 + 4\u00f0\u009d\u0091\u00a5\u00c2\u00b3 - 5\u00f0\u009d\u0091\u00a5\u00c2\u00b2 + 2 would become 4\u00f0\u009d\u0091\u00a5\u00c2\u00b3 - 2\u00f0\u009d\u0091\u00a5\u00c2\u00b2 + \u00f0\u009d\u0091\u00a5 + 2"
    },
    {
        "video_name": "Vm7H0VTlIco",
        "Q": "At 7:25, what if the polynomial does not have an exponent integer? For example 2x+1; would the degree be a 1st degree term? Would it have a degree at all?",
        "A": "The x in 2x+1 does have an exponent. It is x^1 . So, it is considered a 1st degree polynomial. A polynomial of degree 0 would be a constant, like the 1 in 2x+1"
    },
    {
        "video_name": "Vm7H0VTlIco",
        "Q": "At 4:26, when he says a \"non-negative\" power, couldn't he just say positive? Or is there some crazy in-between-negative-and-positive term that I haven't learned about yet?",
        "A": "There is a power of zero. This applies to polynomials which is why the definition uses non-negative powers."
    },
    {
        "video_name": "wl2iQAuQl7Y",
        "Q": "at 2:10 what are those symbols for?",
        "A": "They are capital deltas from the greek alphabet, and they mean, the change in"
    },
    {
        "video_name": "wl2iQAuQl7Y",
        "Q": "At 2:08 what are the triangles used for?",
        "A": "The triangles are the greek symbol for delta, which in this case represents the change"
    },
    {
        "video_name": "INqaHHZGt8w",
        "Q": "At 5:31, what does he mean by infinity?",
        "A": "Infinity is a concept that states that numbers go on forever in both directions, negative and positive."
    },
    {
        "video_name": "INqaHHZGt8w",
        "Q": "At 2:34, he said (-5,6). but he was supposed to say (6,-5). Am I wrong?",
        "A": "There s been a correction added on this, so this question can be safely deleted."
    },
    {
        "video_name": "INqaHHZGt8w",
        "Q": "how come at about 2:48 we didn't do anything with the z axis? just wondering. Thanks!",
        "A": "It is not a 3-D plane, but a 2-D plane. There is only a z-axis in a 3-D plane."
    },
    {
        "video_name": "4Eax48gDg9g",
        "Q": "Why do you subtract \"b\" when have it under the division sign? 0:45 Is about where it takes place.",
        "A": "i have no idea"
    },
    {
        "video_name": "xGmef7lFc5w",
        "Q": "how come x and y are particularly placed there at the top left corners? why not the bottom right ones? (0:35)",
        "A": "just to give emphasis on the positive direction of the y axis and x axis"
    },
    {
        "video_name": "xGmef7lFc5w",
        "Q": "At 4:00 how do you know that the line goes straight and it doesn't curve or something like that? Or are you just amusing it goes straight, or did you double check that it goes straight?",
        "A": "This is a very good question! You could try to convince yourself that s it s straight by choosing values of x, finding their y values and then plotting more and more points. Or you could just accept that as long as the exponents of x and y are not higher than 1, then the equation represents a straight line. Essentially, that is what the generalised equation y = mx + b indicates."
    },
    {
        "video_name": "xGmef7lFc5w",
        "Q": "by 2:07, i still didnt get it, plz help",
        "A": "Hey, So all he saying is that if you have an equation and want to solve for the x-intercept you would substitute a 0 for y and then it would the other way around if for the y-intercept."
    },
    {
        "video_name": "xGmef7lFc5w",
        "Q": "Must you plug in the values of x and y separately, or is it possible to tackle both coordinates at once in the same equation? (Starting at 1:31 in the video.)",
        "A": "Unless you re checking if a coordinate set works (e.g. (0,5)), you ll need to find each intercept separately by plugging in 0 for the respective values."
    },
    {
        "video_name": "xGmef7lFc5w",
        "Q": "At 2:09, how did he get x= -4 by crossing out 4 in the first place and then divide by negative 5?! I cannot understand. Can anyone help?",
        "A": "In the equation -5x + 4y = 20 Sal made y = 0 so it then became -5x + 4*0(y) = 20. Well, 4*0 is 0, therefore, we end up with -5x + 0 =20, or simply -5x = 20. We can now find out the value of x by dividing both sides of this equation by -5, i.e., -5x/-5 = 20/-5. This results in x = -4."
    },
    {
        "video_name": "xGmef7lFc5w",
        "Q": "at 1:50, what if the x intercept doesn't go into the answer equally?",
        "A": "Then your variable is equal to a fraction"
    },
    {
        "video_name": "kDZlMFkKhWA",
        "Q": "At 07:10. The population's size is the half-way between the initial population(P) and maximum population(a). The initial population is 700, not 0. Shouldn't the middle point be shifted by 350?",
        "A": "The initial population is 700, but this is where t=0. What Sal did was finding the vertex of dP/dt, which is a function of P, not t. The vertex is halfway between the points where dP/dt is equal to 0. dP/dt is equal to 0 when P is equal to 0. This is not the same thing as the initial population. Therefore the middle point is not shifted over."
    },
    {
        "video_name": "hIAdCTNi1S8",
        "Q": "At 4:39, why did you multiply 5 and 3,600?",
        "A": "Because we went 5 meters per second. So to find the distance traveled in 3,600 seconds u would multiply 5 meters by 3,600 seconds..."
    },
    {
        "video_name": "hIAdCTNi1S8",
        "Q": "5:28 Wouldn't it be easier to divide 18000 by 1000?",
        "A": "Yes, but Sal is showing you another way."
    },
    {
        "video_name": "CkQOCnLWPUA",
        "Q": "At 5:50, did Sal mean to write the 0 vector and not just 0?\nx1a1+x2a2+x3a3+x4a4+x5a5=0 vector?",
        "A": "Yes, it should be the 0 vector. Sal forgot to write the 0 as a vector."
    },
    {
        "video_name": "CkQOCnLWPUA",
        "Q": "01:07 Sal said ''Column span'' but he probably meant ''Column space'' correct me if I'm wrong",
        "A": "Column span and column space are the same thing. The column span would mean the span of the column vectors, which is exactly what the column space is."
    },
    {
        "video_name": "CkQOCnLWPUA",
        "Q": "At roughly 3:45 did you mean to say that C(A) = span (a1,a2,a3,a4,a5) = {a1,a2,a4}?",
        "A": "What Sal said was correct. C(A) = span(a\u00e2\u0082\u0081,a\u00e2\u0082\u0082,a\u00e2\u0082\u0083,a\u00e2\u0082\u0084,a\u00e2\u0082\u0085) = span(a\u00e2\u0082\u0081,a\u00e2\u0082\u0082,a\u00e2\u0082\u0084) The span of a set of vectors gives you a subspace and the columnspace needs to be a subspace. {a\u00e2\u0082\u0081,a\u00e2\u0082\u0082,a\u00e2\u0082\u0084} is just a set of vectors, it s not a subspace."
    },
    {
        "video_name": "Xc3oHzKXVh8",
        "Q": "on 11:14, what if i said triangle BED is congruent to triangle CFD and i gave my reason as ASA, because the line is in between the two angles, would i get it wrong if i reasoned it as ASA?",
        "A": "In ASA, the side is between the two angles. You would be correct if you said that. Good job on finding that track of reasoning!"
    },
    {
        "video_name": "Xc3oHzKXVh8",
        "Q": "At 5:30, how can ED=EF?",
        "A": "We need to prove that, which could be false, or could be true."
    },
    {
        "video_name": "Xc3oHzKXVh8",
        "Q": "At 2:30 , around, how do you identify SSS, SAS, ASA, HL, and other triangle congruencies? I don't understand.",
        "A": "If there is a right angle, and you know the triangles are congruent, it is most likely HL. SSS is applied if all three sides are congruent, SAS is used if there are two sides and one angle and ASA is used if there is one side and two angles."
    },
    {
        "video_name": "Xc3oHzKXVh8",
        "Q": "at 4:55,how is BF=CE\n\nIs it because congruency of Traingle BAF=CAE is proved?",
        "A": "Yes. If congruency of two triangles is proved, it means that each corresponding angle and side of the triangles are equal to each other. Thus BF=CE, BA=CA, and FA=EA."
    },
    {
        "video_name": "Xc3oHzKXVh8",
        "Q": "At 8:40 you have to first establish that F is the midpoint of CA and E is the midpoint of BA, right?",
        "A": "No, you don t have to show that it is the midpoint, meaning that the point F splits CA into two perfect sections of equal measure. We don t have enough information for that."
    },
    {
        "video_name": "Xc3oHzKXVh8",
        "Q": "At 2:27, instead of \"Reflexive property\" can you write \"Given\"?",
        "A": "Given is used only on what is stated as fact to start the problem, all other parts of a proof have to be based on something else either a property, axiom, previously proven theorem, definition, etc. Since angle A being congruent to itself is not stated, we need to use the reflexive property as reason for this step."
    },
    {
        "video_name": "Xc3oHzKXVh8",
        "Q": "At 1:52, Sal says that angles ABF and ACE share A as a vertex. However, isn't B the vertex of ABF, and isn't C the vertex of ACE?\nThen, from 2:00, he talks about angle BAF instead of ABF. The same for 2:10, he talks about angle CAE instead of ACE. I must be missing something, I think.\nAny help?",
        "A": "At 1:47 Sal says TRIANGLES ABF and ACE, not angles."
    },
    {
        "video_name": "Xc3oHzKXVh8",
        "Q": "At somewhere around 8:00 why can't you just use CPCTC",
        "A": "We can t claim that ED and DF are parts of congruent triangles until we show that \u00e2\u0088\u0086BDE \u00e2\u0089\u0085 \u00e2\u0088\u0086CDF. Notice that ED is not a side of \u00e2\u0088\u0086ACE and DF is not a side of \u00e2\u0088\u0086ABF. Both of these line segments are just parts of sides of those triangles. Sal finds two triangles that are congruent and have ED and DF as sides. Once we know these things, then C(orresponding) P(arts) of C(ongruent) T(riangles) are C(ongruent) applies."
    },
    {
        "video_name": "Xc3oHzKXVh8",
        "Q": "At 2:47, I thought you couldn't make angles reflexive to itself, only line segments... right?\nAlso at 7:08, can there be corresponding sides because I thought it was only corresponding angles making the side corresponding?",
        "A": "At 2:47, it is not the matter of reflection. Clearly, angle BAF and CAE have the same value. At 7:08, yes, there are corresponding angles as well as corresponding sites"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "@ 3:25 , on the second problem ; why does the order of operation not apply ? I thought in these types of problems you work out what is in the parenthesis first. Apparently I am incorrect , but it helps my brain digest this stuff when I ask about anything I am unsure about .",
        "A": "No it does apply, and what he did was apply it. You cannot simpluify the terms inside the bracket (that is 3x+2). So you eliminate the brackets, by multiplying them by the -1 in the front. Note: -(3x+2) is the same as -1(3x+2)"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "I'm also wondering how you know when to stop for the answer. For example at 4:44 time in the video, the answer is 3/2 but why dont you keep going and divide the two into the 3 which ends up as 15? How do I know when to leave it a fraction or divide the fraction..? Thanks ! :)",
        "A": "As the equations are presented as fractions, not decimal, it s accepted that the solutions should be in the same format. Also, 3/2 = 1.5, not 15. :)"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "Why would we divide both the top and bottom by 2? For @6:48?",
        "A": "golden rule: what you do to one side you do to the other"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "7:36 Wheres the 1? 1 over 1 is equal to 1? where did it go?",
        "A": "You don t have to write the 1 since 1 multiplied by a number is equal to that number."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 5:05 Sal changes the 3s/8 into  (s) 3/8.\ns( 3 \u00c3\u00b7 8 ) is the same as 3(s) \u00c3\u00b7 8?",
        "A": "That s correct. Another way to look at it is 3s/8 = (s/1)*(3/8)"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "Can someone help me understand why he changed the operation sign at 3:35? What did he mean at 3:37?",
        "A": "Well, I would assume so. I think that you are correct young ma am/sir."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 4:55\n\nI think I understand the equation, but I'm not sure.\nCan you tell me if my method is correct?\n\ns - 3s/8 = 5/6 Equation\ns - 3/8s = 5/6 Moved second s to end of the first fraction\n1 - 3/8s = 5/6 Changed first s into a 1\n8/8 - 3/8s = 5/6 Made 1 into a fraction\n5/8s = 5/6 Subtracted 3/8 from 8/8\ns = 4/3 Multiplied reciprocal of the first fraction on both sides",
        "A": "he didnt change the s into a 1. The step he did here was, s - 3/8s = 5/6 1s- 3/8s = 5/6 (note that any letters with no numbers in front actually have a 1) then he factored out the s s(1-3/8) = 5/6 however instead of putting the s in front, he put it in the end. This doesnt make any difference. (1-3/8) s = 5/6 and you have it right"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "I don't get what Sal did at 5:25- HELP!!!!!!!!!!",
        "A": "you mean like changing 1/1 to 8/8? oh, he is just trying to find a common denominator by using equalivilant franctions"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 3:43 how did Sal change the addition sign to a subtraction sign?",
        "A": "He moves it to the opposite side which would inverse the type it was: x-12=50 ---------> x=12+50"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 3:27 sal said he wanted to distribute the -1. Where did he get into the negatives? How did he work the problem into 5x-3x-2=1",
        "A": "Hi Kieran. The expression was 5x - (3x + 2) = 1 Now imagine it this way: 5x -1(3x + 2) = 1 Sal is using the distributive property to apply that invisible -1 to both terms (both the 3x as well as the positive 2) When you perform that operation you will get: 5x - 3x - 2"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "(at approximately 1:00min)\nim confused...when Sal wrote:\n1.3x - 0.7x = 12 and then said to merge the two x terms\n(l.3 - 0.7)x =12 <----- woudnt there be two x's after the parentheses like (1.3 - 0.7)2x /",
        "A": "If you use the distributive property, you multiply both numbers inside the parenthesis by x. Expanded, it would still be 1.3x-0.7x=12"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "How did sal get 35 at 9:55?",
        "A": "He s simplifying 5(q-7) . He s doing this by multiplying 5 with q and then with -7. 5*q = 5q 5*-7=-35"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 3:26 where did Sal get the negative 1 from? I didn't see it in the equation.",
        "A": "The equation 5x - (3x + 2) = 1 could also be written as 5x (-1(3x + 2)) = 1. In other words, the minus before the (3x + 2) is basically a -1 times (3x + 2). Therefore at 3:26 Sal distributes or multiplies the -1 by the 3x and the 2 to get -3x - 2. Hope this makes sense!"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 2:19, Sal kept 0.6 as a decimal. Could you also make that into a fraction (6/10) and then multiply both sides by 10 (getting 6x = 120), then dividing both sides by 6, getting x = 20? It seems much easier that way.",
        "A": "Gauri, the way you described is a completely correct way of doing it, and I agree: it is easier. Usually, if you can covert decimals to fractions because fractions are easier to work with, that s a good way to do it. Hope that helps!"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 5:30, why did sal choose to use 8/8 as an equivalent to 1? How would you know what number to use?",
        "A": "He had to subtract 3/8 from 1 i.e 1 - 3/8. In order to match the denominator of 3/8 i.e 8 to 1, he rewrote 1 as 8/8 so that 8/8-3/8= 5/8 Its basically LCM."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 5:23, did he need that 1?",
        "A": "Yes, because it makes the distributing process easier. If not, you probably would have a lot of people asking where the 1 came from. Hope this is helpful!"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "at 6:54 Sal simplified\n(5(q-7))/12\n\nto\n(5/12)(q-7)\n\nI already reassociated the 5 but I expected\n(5/12)((q-7)/12)\n\nWhy is q-7 no longer over 12? Should the diviser only ever affect one factor of the dividend?",
        "A": "When you have 5(q-7)/12 you can change it to 5/12(q-7) because you are just reordering the terms, You are stilldoing the same thing, multiplying. Hope this helps!"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 6:28 how was Sal able to divide the fives of the numerator and denominator by 5 and the 8 and 6 by 2 if they weren't each part of the same fraction?",
        "A": "Christina, When you multiply fractions you multiply the numerators and multiply the denominators so (5/6) / (8/5) = (5*8)/(6*5) And multiplication is commutative so (5*8)/(6*5) = (5*8)/(5*6) which is also (5/5)*(8/6) and now you can see that the 5/5 = 1 and the 8/6 = 4/3. That is why when fractions are multiplied together, you can cancel out or reduce any of the numerators against any of the denominators that have common factors. I hope that helps make it click for you."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "at 8:13 why does he turn the seven into 7/35",
        "A": "He doesn t turn it to 7/35, he turns it to 35/5. He changes it so it has an equal denominator as the 8/5 so he can add it."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 3:40 why did he write the sum as 5x-3x-2 in stead of 5x-3x+2?",
        "A": "Because the 2 is now negative. He took -1, multiplied it by the 3x to get -3x and the 2 to get -2."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "at 4:45 could you write x as a decimal (1.5) instead of a fraction?",
        "A": "Yes, you can, but fractions are easier to deal with in general."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "Hi! So at 3:26 you said that we had to factor a -1, but I couldn't see that number in the equation. Then the expression 5x-(3x+2) became 5x-(3x-2.) Could anyone explain how this works? I didn't understand it. Thank you :)",
        "A": "ok, thank you so much!"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "Can someone please explain to me how and why Sal rearranged the equation at 6:56?\n\nI don't understand the 5/12 part and this relates back to the original equation in the question.\n\nThank you in advance.",
        "A": "How: I believe the original equation was 5(q-7)/12, no? Well, basically, he is un-multiplying. (q-7) by itself can easily be read as (q-7)/1. 5/12 multiplied by (q-7)/1 is the original problem. So undoing the multiplication allows him to end up with 5/12 times (q-7). Why: It allows for the easy cancelling of 5/12, as shown in the video."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "at 8:17 how can 7 be written as 35?",
        "A": "He wrote 35 is because 5 is the common denominator and since we are dealing with 7, we can multiply 7 by 5 to get 35 and write 35/7. The reason why this fraction works because 35 divided by 7 equals 5."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 7:28, why does he multiply each side by the inverse operation? Wouldn't you have to do the parentheses first?",
        "A": "No, you don t have to do that. PEDMAS or the order of operations is just a tool that beginners are given until they learn all the properties of algebraic operations. Once you know those properties, you can do the operations in any convenient order -- provided you know what you are doing."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "at 3:37, I don't understand how he distrubuted the numbers. can someone please explain?",
        "A": "Thank You! That helped a lot."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "What is a coefficient as mentioned at 1:36 ?",
        "A": "A coefficient is the number being multiplied with the variable. In the term 6x, the coefficient is 6."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At around 13:24, isn't 0.75m/0.75 = 1?",
        "A": "0.75m/0.75 = 1 x m =m"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 5:07, how come he can say 3/8 s when it was initially written as 3s/8, wouldn't that end up meaning something else?",
        "A": "The unit expressed is the numerator unit. In this case 3/8 s = .375s And 3s/8 is also .375 s"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 5:14 why couldn't you divide both sides by 3/8? Thanks",
        "A": "I don t think that s right, because s-s would equal zero, not 5/6 * 8/3."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "10:32 Eastern Time\nHow did -2 come about in 5x-(3x+2)=1. I refuse to go on until I understand. I see how -1 comes from -3x+2. but I cannot comprehend how -2 comes in the equation.",
        "A": "5x - (3x + 2) = 1 5x + -(3x + 2) = 1 5x + -1 * (3x + 2) = 1 5x + -1 * 3x + -1 * 2 = 1 5x + -3x + -2 = 1 5x - 3x + -2 = 1 5x - 3x - 2 = 1 2x - 2 = 1 2x = 1 + 2 2x = 3 x = 3/2"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "Can someone help me understand why he added a \"-1\" into the equation @ 3:25 when there was no \"-1\" in the equation at all? I've read the other answers but none of them make sense to me.",
        "A": "We have 5x-(3x+2) = 1. This is the same as 5x+(-1)(3x+2) = 1. You can then distribute the -1 into 3x + 2, giving you 5x-3x-2=1."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "Why 35/5 at 8:24?",
        "A": "Think of it this way, if Mr.Khan had used 7/1 instead of 35/5, would it be easier to add 8/5 to 7/1, or would it be easier to add 8/5 to 35/5? 35/5, right? Yeah! That s why he chose to use 35/5, instead of another fraction equal to 7. It would be possible if you used a different fraction, but this fraction, 35/5, would be more convenient and take less work. Got it?"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "i still don't understand why at 7:27 Sal uses 12/5",
        "A": "Sal is using 12/5 because it is the reciprocal of 5/12. It accomplishes 2 things: 1) it enables you to cancel out the existing denominators and 2) it moves the 5 to the other side. An alternate approach would be to distribute the 5: (5q - 35)/12 = 2/3 Multiply by 12 to cancel out the denominators: 5q - 35 = 8 Add 35: 5q = 43 Divide by 5: q= 43/5"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At about 3:26 Sal says he has to distribute the negative 1. There is no negative 1. What does he mean by that?",
        "A": "The equation was as follows: 5x - (3x + 2) = 1. When Sal distributed the -1, he was distributing the minus sign right in front of the parenthesis. Since you are subtracting everything inside of the parenthesis, you must distribute the -1. You just have to remember that, just as x is the same as 1x, that subtracting something in parenthesis is the same as multiplying everything inside by -1. I hope this helps!"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "at 3:47 why did 5x-(3x+2) turn into 5x-3x-2?",
        "A": "This is because the quantity -(3x+2) is the same thing as -1*(3x+2). And multiplication is distributive over addition which is just a fancy way of saying that: -1*(3x+2) is the same thing as: -1* 3x + -1 * 2 Which is the same as: -3x -2"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "Why is it it negative 3x? This is at 3:49.",
        "A": "It is negative 3x because your are multiplying the negative sign by the two numbers in the parenthesis, thus changing their signs. I hope this helps!"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 5:26 how did you factor out one of the \"s\"?",
        "A": "s(1-3/8) is the same thing as s-3/8s. Multiply s(1-3/8s) and you should get the answer s-3/8s"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 5:34 why did you or how did you rewrite it to 8/8?",
        "A": "He did that to make the sum more simpler, The number one can be written as a fraction where the numerator and denominator are the same number, for example 1= 2/2=3/3; Because if you make the divison is going to be equal to one"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "I'm confused about solving the equation 5(q-7)/12=2/3 which shows up at about 6:50 in the video. Sal comes up with the answer 43/5. I see how he did that but it makes no sense in the equation. If you rewrite the equation, substituting x for 5(q-7), it looks like this: x/12 = 2/3. So, x=5(q-7), which means x should be 8. (8/12 = 2/3) If you simplify Sal's answer {43/5} you get 8 3/5, which isn't REALLY 8 unless you round it off but why would you do that?",
        "A": "If you substitute 5(q-7) for x, you don t get the value of q, you get the value of 5(q-7). See, if x=5(q-7) and x=8, then 5(q-7)=8. Solving it for q: 5(q-7)=8 => q-7=8/5 => q=8/5+7 => q=(8+35)/5 => q=43/5 Hope you get it"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "Can someone explain why he changed the Equation on 3:38 on the video to a negative( - ) when it was positive?",
        "A": "When you are removing parentheses, you have to distribute using the sign in front of it."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "at 7:55 where did he get the 8/5ths? it looked like it would be 4/5ths! please help!",
        "A": "he takes 2/1 times 4/5 to get the left side solved and when timesing two fractions you need to do 2 times 4/1 times 5"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 4:25 he adds 2 to the 2x and the 1. Why do you add 2 to the answer and not the other 2?",
        "A": "well....... ............. ok i dont know this is sooooooooo hard"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 5:28 he left out the other s did it just disappear or is there some reason?",
        "A": "He basically simplified the equation."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 5:00, how does the factoring of s work? How is one s removed?",
        "A": "Sal is using the distributive property to factor out the S. Look for videos on Factoring out the Greatest Common Factor. An alternate approach would be: 1) Multiply equation by LCD for 8 & 6 = 24: 24(S) - 24(3S/8) = 24(5/6) It creates: 24S - 3(3S) = 4(5); OR 24S - 9S = 20 2) Combine like terms: 15S = 20 3) Divide by 15: S = 20/15 = 4/3"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "at 12:37 how did he get 7.7",
        "A": "He used the number line to figure it out"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 11:27, Sal translates the words \"75 cents PER mile\" into the formula \"C = 2.35 + 0.75 m\". I think I get the solution using common sense, but why does \"PER\" translate into a multiplication and not into a division (like in \"per cent\"?)",
        "A": "because a cent is a hundredth of a dollar but because he had other whole numbers in the equation the 75 cents is the 0.7m in the equation hope i answered your question :-)"
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At time 10:33 in the video, why did he finish the problem by multiplying both sides by 1/5th? Why didn't he divide by 5 because with 5q=43 i would have thought that because it's multiplication with the 5 TIMES Q you do the OPPOSITE operation which then is division. I would have done this to continue isolating the Q and get the answer on the right side of the equation. Then, however, I end up with 43 divided by 5 which ends up as 86 and def. not right. Thanks :)",
        "A": "First, multiplying by 1/5 always gives the same result as dividing by 5, because 1/5 and 5 are RECIPROCALS. So either way is okay. Second, 43 divided by 5 is not 86, but rather 8.6. This could be written other ways, such as 43/5 or 8 3/5."
    },
    {
        "video_name": "tuVd355R-OQ",
        "Q": "At 5:48 how did he get 5/8? why 8. why not some other number? or is it because there's 8 in the denominator next to that?",
        "A": "No Sal got 5/8 s because in the parentheses he got the equation 8/8 - 3/8 which is equal to 5/8.Hope that helps! :)"
    },
    {
        "video_name": "FP2arCfAfBY",
        "Q": "5:05 What rule allows Sal to say that n = l+m?",
        "A": "Imagine you have 10^3 = 10^x; the bases are the same and therefore 3 = x since both sides must equal 1000. Now what if you have 10^3 = 10^(x+y)? The bases are still the same, therefore 3 = x+y. In the video, Sal has x^n = x^(l+m). The bases are again the same (they re both x), therefore n must equal l+m for both sides to balance."
    },
    {
        "video_name": "FP2arCfAfBY",
        "Q": "So around 3:10 Sal states that x^n = A*B. Where did he get this from? It seems as if he just stated that for the purpose of the proof.",
        "A": "There are 2 issues I like to mention in the answer for your question: 1- x^n = A*B, is just 1 case of logarithm to be study. 2- what he spoke right there is an error, and the system admit the error by having a small pop-up window at 3:25. He should say: x^n = A*B, that means log base x of A*B = n"
    },
    {
        "video_name": "FP2arCfAfBY",
        "Q": "From around 3:00 to 3:10, Sal is writing some really crazy equations. Where is he getting those varibales from? You can't just use any variable randomly and say \"Oh let this equal \"that\" and etc.\". That is completely random and doesn't follow the rules of math. Could someone explain this to me? Thanks.",
        "A": "Actually, you can use variables randomly. As long as you have not already defined a variable, you can make it whatever you want. It s a variable."
    },
    {
        "video_name": "FP2arCfAfBY",
        "Q": "At 5:18 ,How to get the same base , powers should be the same ?",
        "A": "The base has to be the same from the start or you cannot directly use the logarithm properties. There is, however, a method for converting between logarithm bases, but it isn t of much use nowadays. But if you ever need it here it is: logx(a) = logy(a) / logy(x)"
    },
    {
        "video_name": "FP2arCfAfBY",
        "Q": "At 3:15, technically isn't Khan only able to use 256^3 (16777216) colours, as only that many is perceivable by the human eye?",
        "A": "actually i think you re right about that--"
    },
    {
        "video_name": "BSKtQcLQLWU",
        "Q": "at 5:56, when finding the gradient of g, why do we write the partial derivative of g with respect to s when the equation has no s in it?",
        "A": "The function g(h,s) is 20*h + 2,000*s. (I can understand some confusion happening here because the s does look like a g ). So, writing it out, it s 20 times however many [hours], and 2,000 times however much [steel]."
    },
    {
        "video_name": "ieC5LgoGJVg",
        "Q": "At 3:21 what was all about the constant terms",
        "A": "The constant terms are the ones that don t have any variables. If you do something we call the Box method , they are the ones you should get in the bottom right. Again, they don t have a variable. No letter A-Z. I hope this helps you out!"
    },
    {
        "video_name": "0pv0QtOi5l8",
        "Q": "At 4:56, where is Sal getting that y=x^2 (or x=sqrt(y)). Is this just intuition based upon looking at the 3-D graph from the z-axis, or is there a specific method used to solve this and set it as a bound?",
        "A": "In the previous video, Sal starts of by saying that we re finding the volume under z = xy\u00c2\u00b2 that s bound by y = x\u00c2\u00b2 and the x-axis. So he figure it out from anywhere, it s part of the initial problem."
    },
    {
        "video_name": "0pv0QtOi5l8",
        "Q": "I understand why he uses a function as the upper bound on his integral (7:00), very cool. But I was wondering if there was an alternate way to find this volume:\nCould you simply integrate the function y=x^2 from 0 to 1, and THEN integrate this area along the y-axis? If this method is possible, would it be done without the use of a function as part of the integral boundary? Thank you.",
        "A": "That won t work. Keep in mind you are trying to find the volume under z=xy^2. The area you would get by integrating y=x^2 would just be the projection into the xy-plane."
    },
    {
        "video_name": "KGSvigZQKZY",
        "Q": "At 3:32, does 2^4 = 4^2?",
        "A": "Yes. 2\u00e2\u0081\u00b4=2*2*2*2 =4*4 =4\u00c2\u00b2 This is just a special case. Don t be misunderstood that the base-exponent equal to its reverse. For example: 2\u00e2\u0081\u00b5\u00e2\u0089\u00a05\u00c2\u00b2 2\u00e2\u0081\u00b5=2*2*2*2*2=32 5\u00c2\u00b2=5*5=25"
    },
    {
        "video_name": "KGSvigZQKZY",
        "Q": "At 1:54, what dos he mean by computationally intensive?",
        "A": "A computation is just a mathematical calculation, so to say something is computationally intensive meaning that its hard to calculate. For instance, 1+1 is very easy to compute, so easy in fact you probably did it in your head without even thinking. Something like 128^4, or 53619 x 79863 would be considered hard to compute (without a calculator of course (unless perhaps you re a genius))"
    },
    {
        "video_name": "KzaPBzFFLRM",
        "Q": "@6:05 the result of f^-1(f^-1(13)) was found to be 9. From f^-1(13)=5 and then f^-1(5) = 9. I am confused because I thought that it would have been 13 again, just as the inverse of 7 example. I am having a hard time reconciling this issue.",
        "A": "f^-1 maps the function from input to output. When you said that it would be 13 again, you were assuming that it would go output to input after the first functions. That is incorrect because the question is asking for input to output again after the first function."
    },
    {
        "video_name": "KzaPBzFFLRM",
        "Q": "@ 4:30 I don't understand how you got the inverse of 7.",
        "A": "All Sal really did was look for an input (an x value ) that would give him a function value of 7. If you look at the orange/brown table you can see that a function value of 7 occurs when the input is -7."
    },
    {
        "video_name": "EINpkcphsPQ",
        "Q": "at around 2:45 Sal says that the angle of the opposing sides of the 90 degree are 90 times theta. What is theta?",
        "A": "Theta is the 8th Greek letter. But that doesn t really matter. All you really need to know is that Sal uses it as a variable. Something that would replace an unknown number. So next time he uses it, just know that he is using it as an unknown/variable."
    },
    {
        "video_name": "EINpkcphsPQ",
        "Q": "AT 8:00 how did he get a^2 + 2ab + b^2",
        "A": "We know that (a+b)^2 can be re-written as (a+b)(a+b). To expand this expression use FOIL (i.e. first, outer, inner, last). First multiple the inner terms of each parentheses (so a^2), then multiple the outer terms (ab), then the inner terms (ab) and finally the last terms (b^2). When you add all of these together you get a^2 + 2ab + b^2."
    },
    {
        "video_name": "EINpkcphsPQ",
        "Q": "At 7:38 when expanding the equation Sal get a^2+2ab+b^2. my question is how did he get 2ab? isn't (a+b)^2 the same as a^2+b^2?",
        "A": "You cannot just square the parts if there is a plus or a minus in the parenthesis. You have to write it as (a+b)(a+b) then multiply the binomials."
    },
    {
        "video_name": "EINpkcphsPQ",
        "Q": "I don't understand what is happening at 2:28. Could somebody explain that to me?",
        "A": "You should watch 3:00"
    },
    {
        "video_name": "EINpkcphsPQ",
        "Q": "At 7:44 why is the equation, \"a square + 2ab\"? I don't get why its 2ab.",
        "A": "(a+b)\u00c2\u00b2=(a+b)(a+b) =a\u00c2\u00b2+ab+ab+b\u00c2\u00b2 =a\u00c2\u00b2+2ab+b\u00c2\u00b2"
    },
    {
        "video_name": "EINpkcphsPQ",
        "Q": "at 7:38 from where did 2ab come? is it a different property",
        "A": "when you have a squared binomial like: (a+b)^2 there is a formula for its solution, which is: a^2+2ab+b^2. If you have the time you can apply the distributive property to (a+b)(a+b), it works out."
    },
    {
        "video_name": "EINpkcphsPQ",
        "Q": "@4:30 how did he get \"looking for the area of a trapezoid\" from 3 trangles?",
        "A": "Because you can form a trapezoid out of three triangles, and the b x h thing is used because of the legs of the triangles, I think. If you look at the diagram it has the three triangles assimilated into the trapezoid shape, and that s where the formula for a trapezoid comes from."
    },
    {
        "video_name": "HXg_a9oJ5nA",
        "Q": "at 0:54 i got confused can someone help",
        "A": "It s 13 because it s between 12 and 14 on the x axis."
    },
    {
        "video_name": "HXg_a9oJ5nA",
        "Q": "AT 2:52 how did you get a 7?",
        "A": "He ended up with a function that gives 2x - 7, so if he put 4 in for x, there is 2(4) - 7 = 1 and for x = 7, then 2(7) - 7 = 7 which is where the 7 comes from. He continues to x=10, so 2(10) - 7 =13 ..."
    },
    {
        "video_name": "FJo18AwLfuI",
        "Q": "at 1:04 when Sal says to evaluate for infinity, does it matter if it comes out as infinity/infinity or pos. infinity/ neg. infinity in order for l'hopitals rule to still apply?",
        "A": "L Hopital s rule can be applied in all cases of indeterminate forms which include 0/0, infinity / infinity, (infinity) /(-infinity) and (-infinity) / (infinity)."
    },
    {
        "video_name": "FJo18AwLfuI",
        "Q": "At 2:02 why isn't the denominator 6x^(-3)?",
        "A": "The exponent was 2, not -2. Via the power rule, the derivative of 3x^2 is 2*3x^1 or 6x; then there s the minus sign in front to make it -6x. Remember, x^n turns in to n*x^(x-1). Had the exponent been -2 (so the denominator was 1 - 3x^(-2)), the derivative would ve been (0 - (-2)*3x^(-3) or -6x^(-3)."
    },
    {
        "video_name": "FJo18AwLfuI",
        "Q": "at 2:47 Sal says that \"It doesn't matter what value this [limit] is approaching.\" I'm not sure I understand how that could be? The fraction would not be there at any point besides infinity, because otherwise it wouldn't be indeterminate, to my thinking. Or am I just reading too much into it? Thanks.",
        "A": "At that point in the analysis, we re looking at the limit of a constant as x goes to infinity, and Sal is making the point that a constant will always have the same value no matter what limit we re approaching. He isn t saying that the specific limit (infinity) is irrelevant to the overall problem, but just noting that once you ve got a constant you re done and don t have to worry about the limit any more."
    },
    {
        "video_name": "FJo18AwLfuI",
        "Q": "Sal used L'Hopital's Rule twice. I solved the question by using L'Hopital's Rule once.\n\nWhen I apply the limit (as x approaches infinity) to the derivative (8x-5) / (-6x) at 2:03, I know that \"-5\" is negligible when x reaches infinitely large values. That leaves (8x) / (-6x), which is scaling at the rate of 8 / -6, or 4 / -3. Hence, the answer to the question asked.\n\nCan somebody please confirm if my approach is correct? Thank you!",
        "A": "It is correct. You did what Sal said at the begining of the vidieo about already knowing how to get the limit. While your way is correct, Sal only continued taking L Hopital s rule one step further to get the same result."
    },
    {
        "video_name": "Ot0FxL8Vl7I",
        "Q": "At 2:43, how he took out the 'a'?",
        "A": "This is because he did the chain rule (but didn t show work). To find the derivative of sqrt(4-ax): f(x) = sqrt(4-ax) = (4-ax) ^(1/2) f(x)= u ^(1/2) u = 4-ax Chain rule time: f (x)=1/2 * u ^(-1/2) * d/dx[u] f (x)=1/2 * (4-ax) ^(-1/2) * d/dx[4-ax] f (x)=1/2 * (4-ax) ^(-1/2) * (-a) f (x)= -a * 1/2 * (4-ax) ^(-1/2) Hope this helps"
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "At 8:25 she says 2 is binary. What does that mean?",
        "A": "2=bi. Base 2 is binary even though binary doesn t use 2."
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "At 3:44, what's a Mersenne prime?",
        "A": "A Mersenne prime is a prime number of the form 2^p - 1."
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "At 10:15, why did Vi write merry cristmath instead of merry cristmas?",
        "A": "Everything is about math..."
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "Unary? How would someone write anything in unary (base 1)? (8:30)",
        "A": "The number of digits would be one greater than the number itself. For instance: 0\u00e2\u0082\u0081 = 0 00\u00e2\u0082\u0081 = 1 000\u00e2\u0082\u0081 = 2 Etc."
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "Vi Hart is amazing!\nDid anyone notice her say a right equilateral triangle at 8:44?\nIs that even possible?",
        "A": "I think she made a mistake. But it is possible in spherical geometry. If your on the surface of a sphere, say the equater of the earth, and you walk 1/4 of the way around earth then turn 90 deg (so now your pointed towards a pole) and walk 1/4 of the way around again (now your at a pole) now turn 90 deg and walk another 1/4 of the circumfrence and guess what, you just traced out an equalateral right triangle and your back where you started even though you only made 90 degree turns! On flat\u00e2\u0080\u00a6 (more)"
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "At 0:48 Vi said that 2 is the only even prime number. What about -2?",
        "A": "The definition of a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. -2 is clearly not greater than 1, and it does have a positive divisor 2 that is not 1 or itself."
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "Anybody else find it funny when Vi makes a mistake and uses it humorously, like in 00:30?",
        "A": "Lots and lots of practice. I should know --- I ve been playing for oh, around... 8 years or so."
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "At about 3:41, Vi talked about Mersenne Primes. What does she mean?",
        "A": "A Mersenne Prime is any prime that follows the pattern 2^n - 1 where n itself is prime. They are some of the easiest primes to find."
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "At 2:24, I heard that there was no such thing as \"year zero\". Was this an error?",
        "A": "No year 0 is 0 B.C. B.c. years run backwards toward zero so year zero mark the birth of Christ and the turn to A.D"
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "Why does it say a Mobious strip has one side at 3:07",
        "A": "try it cut a piece of paper into a strip and give it a half twist and tape it together then get a pen or pencil and draw a line on it then take the tape off and you will have drawn a line on both sides of the paper! don t take my word for it try it yourself!"
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "At 8:44 Vi Hart said there was a right equilateral triangle. Was that supposed to say isosceles?",
        "A": "Yes, she did mean to say right isosceles. There is actually no such thing as a right equilateral triangle, because an equilateral triangle has three 60 degree angles, so it can t have a right angle."
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "i think that at 7:57 she kinda ran out of ideas, (not surprising given how long the song is,) but at that point she started doing all the bases from 1-10, and i noticed she gave up on the 12th day completely: anyone have an idea why? O.o",
        "A": "She was trying to be funny. Remember how in the beginning she mentioned she didn t like to be so long?"
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "At 3:06, wouldn't a mobius strip have two sides, its inside and its outside?",
        "A": "no! one side twists into the other."
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "At 8:34, how does Unary System work?",
        "A": "Since the unary number system only has one digit, it basically works like tally marks. Each number is represented by that number of 1 s (although I guess it really does not have to be 1 s). Therefore: 1 = 1 2 = 11 3 = 111 4 = 1111 5 = 11111"
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "For the last one, 12, at 10:08, what do you think she should put?",
        "A": "Good point... Thanks, Tejas."
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "At 8:33 what are the digits in \"unary\"?",
        "A": "Actually , u use 0."
    },
    {
        "video_name": "sxnX5_LbBDU",
        "Q": "At 7:22 what if it is an odd number?",
        "A": "at 7:22 , an example has been given of even numbers. If it is an odd number it wont work. The simplest thing to do would be to use the fomula of the sum of n natural numbers which is equal to n(n+1)/2. This can be used for sum of any number of natural number."
    },
    {
        "video_name": "gs9O255gdEY",
        "Q": "If the series at 0:44 started at n=0, would the value at n=0 be the first term or does the first term only exist at n=0?",
        "A": "IF the series started at n = 0, we would have a real problem: the first term would be undefined, and therefore the sum would be undefined. \u00c2\u00b9\u00e2\u0081\u0084\u00e2\u0082\u0080 is a pretty effective showstopper without even getting into the (-1) multiplier"
    },
    {
        "video_name": "405boztgZig",
        "Q": "On 1:03 how does 1/3 x equal to 12?",
        "A": "The equation is 2y + 1/3x = 12 and you want to find the x-intercept. On a graph, the x-intercept point is where y = 0, so all you have to do is sub y = 0 into the equation. Then you will get 1/3x = 12. This means x = 36."
    },
    {
        "video_name": "405boztgZig",
        "Q": "I'm just curious around 1:35 when Sal multiplies both sides by the reciprocal of 1/3, can you only do that when the fraction has a 1 in it? Or could you do it if the fraction was like 3/5?",
        "A": "You can do it with any fraction."
    },
    {
        "video_name": "GdIkEngwGNU",
        "Q": "What does he mean at 2:00 that \"It's a pretty\" ?",
        "A": "He was about to complete the sentence, probably about to say something like its a pretty good visual to represent these data values"
    },
    {
        "video_name": "GdIkEngwGNU",
        "Q": "At 9:00, this formula (if it is a formula) for finding MAD is interesting. Who invented this MAD?",
        "A": "it wasnt actually invented by anyone it is undefined who invented it but we can say that it was discovered :)"
    },
    {
        "video_name": "c-_yrA-GUow",
        "Q": "After explaining the relationship between multiplication and division, at 1:16 Sal says \"this is generally true\". What might be an example where this is not true for multiplication or division?",
        "A": "There are probably nuances in more advanced math, but for what is being taught in this video this will always be true except for one instance. The one instance is that 0 * 0 = 0, but 0 / 0 does not equal 0 (it is undefined because you cannot divide any number by zero)."
    },
    {
        "video_name": "8dMLK2Wueaw",
        "Q": "Does the interactive script stop at 1:46 for you?",
        "A": "No, it goes through the whole video for me."
    },
    {
        "video_name": "fnuIT7EhAvs",
        "Q": "At 0:36, why do we have to subtract instead of dividing it by 2w?\nIn other equations (which I can't remember where) you divide it to get rid of the variable on the right side.\nIs there an indication as to what operation to use when you put the denominator on both sides?",
        "A": "You do. What you want is the required unknown alone on one side of the equal to sign. To do this, you can do different operations. If you got 2 unknown on one side,separated by an addition or subtraction sign then you use subtraction or addition (whichever is right). x+y=3 x=3-y P=2l+2w P-2w=2l l= (P-2w)/2 If you got 2 unknowns separated by a multiplication or division sign, you just multiply or divide (as appropriate) x*y= 4 (which can also be written as xy=4) x = 4/y w/r = 2 w =2r Hope this helps!"
    },
    {
        "video_name": "fnuIT7EhAvs",
        "Q": "At the time 1:07 the 2 is cut with 2l on the right hand side why can't we cut the 2 with 2w on the left hand side? Is it because it is attached with P and makes a whole equation so we can't cut it?",
        "A": "I assume by cut you mean divide. (p - 2w) / 2 is the same as p/2 -2w/2 or simplified it becomes p/2 - w The division by 2 applies to the entire side, not just a single term. Hope this helps."
    },
    {
        "video_name": "fnuIT7EhAvs",
        "Q": "At the 0:38min of the video, why are we subtracting from both with \"2w\" and not multiplying with \"2w\" like in other examples",
        "A": "If you have y = x + 2w and you want to isolate the x, the only way to do it is to subtract 2w from both sides. Dividing both sides by 2w doesn t help much at all - you d get y/2w = x/2w + 1 which is even messier for x than before."
    },
    {
        "video_name": "fnuIT7EhAvs",
        "Q": "at 1:13 why did he divide by 2",
        "A": "He divided by 2 because he wanted to isolate the l variable from the rest of the other variables and wanted it as just one variable without any coefficients."
    },
    {
        "video_name": "fnuIT7EhAvs",
        "Q": "At 0:40, could you also divide 2l, 2w, and P by 2? And then subtract w from the right side after?",
        "A": "Yes you could, and if that way seems better for you, go right ahead."
    },
    {
        "video_name": "fnuIT7EhAvs",
        "Q": "at 0:34 why cant you just distribute it, so that it goes like P=2(L+W)?",
        "A": "That doesn t help us, in fact it takes us in a direction away from solving the problem. Since we want to solve for L, that means we need to isolate the L."
    },
    {
        "video_name": "fnuIT7EhAvs",
        "Q": "why at 1:00",
        "A": "You have 2 times L. You want L by itself, so to get rid of the 2 you divide both sides by it. You should watch previous videos if you re unsure of why."
    },
    {
        "video_name": "fnuIT7EhAvs",
        "Q": "at 1:17 could u make the answer this as well thanks\nl = (p-2w)/2?",
        "A": "Yes... your version is equivalent to Sal s answer."
    },
    {
        "video_name": "N_ZRcLheNv0",
        "Q": "At 6:30. it is said that, directional derivative is real no., or it is not a vector, but nabla operator should be a vector. isn't is ?",
        "A": "Nabla is vector. But maybe your question is concerning product with nabla, dot product is skalar."
    },
    {
        "video_name": "ZS1OZj_oWao",
        "Q": "At 3:14 in the video, how, when 120 is 3/4 of the remaining distance, how is a fourth of it 1/3 times 120?",
        "A": "1/3 of 3/4 = 1/4 Imagine I slice a pizza in 4 equal pieces, then I take 3 of them. If I divide what I ve got by 3, then I ll have a single slice, which is 1/4 of the pizza."
    },
    {
        "video_name": "ZS1OZj_oWao",
        "Q": "Sorry, he lost me at 2:48... how did he get forty for each of the fourths?",
        "A": "So Judy walked 120 yards, correct? But that length was just 3/4 of half the actual distance. Since the remaining amount is 1/4 of half the actual distance, you have to find 1/3 of 120 or 120 divided by 3 in order to find the length of half of the total distance. 120 divided by 3 is 40, so each fourth equals 40 yards. I really hope this helped Rashad : )"
    },
    {
        "video_name": "ZS1OZj_oWao",
        "Q": "At 2:05, why does the number line show that she walked so much farther than she skipped? The problem says that she ran 1/2, skipped 1/4, and walked the rest. Wouldn't the amount of distance she walked be 1/4, because that is all that is remaining?",
        "A": "She ran 1/2 and of the remaining half she skipped 1/4. 1/4 of 1/2 = 1/8. If they would have said she skipped 1/4 of the whole distance you would be right. :)"
    },
    {
        "video_name": "EwEbZI57P1o",
        "Q": "At 6:25, what would the graph be if the equation were the other way: y = (-3)^x?",
        "A": "x is a logarithm, and y is its antilog. So for real x you can t have a negative base (or a base of 0 or 1, for which a^x is a straight line y=0 or y=1)."
    },
    {
        "video_name": "EwEbZI57P1o",
        "Q": "at 7:03 why does y= (-3)^x graph go through the y axais at one?",
        "A": "It s -(3^x) and the relevant graph is the utmost left one. The graph of -(3^x) should go through (0, -1). Every number on the zero power is equal to one and the negative of that is -1. Therefore the graph goes through the point (0, -1)"
    },
    {
        "video_name": "NU1v-8VRirU",
        "Q": "At 6:03, Sal interpreted N_0 as \"N sub naught.\" Is his English pronunciation of N_0 incorrect?",
        "A": "N sub naught and N naught are both correct. So would be N sub zero and N zero . I think naught is more common than zero ."
    },
    {
        "video_name": "Zp5z0wa0kgo",
        "Q": "At the very end (3:57 ish), could the absolute value signs be dropped, since x^4+7 will always be positive?",
        "A": "yes, they could be dropped"
    },
    {
        "video_name": "Zp5z0wa0kgo",
        "Q": "how is du/u=1/u at 2:59",
        "A": "If you have a/b, you can rewrite it as (1/b)*a. Sal does the same thing here, rewriting du/u as (1/u)*du (but without the multiplication sign)."
    },
    {
        "video_name": "Zp5z0wa0kgo",
        "Q": "At around 3:05 he drops the du, what happens to it?",
        "A": "du means with respect to u . After you integrate with respect to u, you ve dealt with it. Consider where it comes from. If y is some function of x, when you take the derivative you replace y with dy/dx. The dx seems to come out of nowhere, so that s where it goes when you integrate. (It still means the same, as you take the derivative of y with respect to x.)"
    },
    {
        "video_name": "Zp5z0wa0kgo",
        "Q": "At around 3:30 why does he eliminate the du from the equation? arent you supposed to plug back in the value for du just as we do for u?",
        "A": "When he eliminates du from the equation he gets the indefinite integral."
    },
    {
        "video_name": "Zp5z0wa0kgo",
        "Q": "When Sal rewrites the integral around 2:07, where does the original dx go? Shouldn't it be (integral symbol)( (4x^3dx)/(x^4 + 7)dx)?",
        "A": "He just stuck it in the numerator. Your statement would seemingly have dx in two different places, which does not work. It certainly won t multiply into the denominator. Does that help?"
    },
    {
        "video_name": "Zp5z0wa0kgo",
        "Q": "9 At 3:59, could we ignore the absolute value (or mod) because for any value of x the number inside the 'brackets' would be positive...........?",
        "A": "You can, but it is recommended you don t, as the problem is concerned with finding out the value of the integral, the general value of the integral of 1/u is log|u| + constant, it shows you know the integral, which is essential."
    },
    {
        "video_name": "Zp5z0wa0kgo",
        "Q": "At 3:04, I'm confused as to why it becomes 1/u du..",
        "A": "it s (1/u)* du yup it s the same thing."
    },
    {
        "video_name": "79lagkER9qY",
        "Q": "at 6:39 he said he multiplied both sides of the equation by AB. how did he do that?",
        "A": "You can do what you want to an equation as long as you do the same thing to both sides. He started with slopeBC = -1/slopeAB, and multiplied both sides by slopeAB. On the left we get (slopeAB)(slopeBC) and on the right we get (-1 * slopeAB)/slopeAB. The slopeABs cancel to leave -1. The equation is thus changed to (slopeAB)(slopeBC) = -1"
    },
    {
        "video_name": "E4HAYd0QnRc",
        "Q": "At about 9:49, what is an arbitrary number?",
        "A": "Any number you would like to choose."
    },
    {
        "video_name": "E4HAYd0QnRc",
        "Q": "8:45 when adding the numbers do i have to add the zero or can i just leave it out?",
        "A": "No you don t have to add it in because it doesn t change the value. However, you still have to count it when it comes to the division step because it is a value that was returned to you. So you still divide by 5, even though one of those values was 0."
    },
    {
        "video_name": "E4HAYd0QnRc",
        "Q": "At 5:04 Sal talks about how sigma is the greek letter for variance. I thought the greek lettre looks more like an E, such as \u00ce\u00a3, not \u00cf\u0083. Could someone please clarify?",
        "A": "\u00ce\u00a3 is a capital letter, while \u00cf\u0083 is a lowercase letter."
    },
    {
        "video_name": "E4HAYd0QnRc",
        "Q": "In what case will either Variance or Standard Deviation be preferred over each other ?\nAt around 9:55 Sal says that Variance has an odd set of units so is standard deviation better as it has the same units as the data itself ?",
        "A": "In practical settings, the standard deviation is probably almost always preferred. the variance is more often used in the background, deriving this or that, or used in the theory of something."
    },
    {
        "video_name": "E4HAYd0QnRc",
        "Q": "At 12: 31 or 12:32, he says that it is used to measure the distance from the mean. Isn't mean the average so doesn't he mean the centre or the median?",
        "A": "the median is the value located on the center of the data, the values of the data do not alter the median. The mean is the average of the data, so if you change the values you will change the mean."
    },
    {
        "video_name": "E4HAYd0QnRc",
        "Q": "At 10:44 how did Sal get 10 square root 2? He says \"what's the square root of 200? Well...\" and then I don't understand the next bit at all. Can someone explain the maths? Thanks!",
        "A": "You can break down the number into its factors. The square root of 200 is the same as the square root of 100 times the sqr (short for square root) of 2. sqr(100) is simply 10; hence you can say it is 10 times sqr(2). Hope this helps."
    },
    {
        "video_name": "E4HAYd0QnRc",
        "Q": "At 3:44 how does 30 minus -10 equal 40 because I don't understand that?",
        "A": "At 3:44 to get the Range of the data, Sal does 30 - (-10) = 30 + 10 = 40 I hope this is clear."
    },
    {
        "video_name": "E4HAYd0QnRc",
        "Q": "At 10:45, why doesn't he just take the square root of 200? That is what I expected so I calculated the square root of 200 and got the answer 14.142. He instead solves it as 10*the square root of 2. How do you determine whether you actually have to solve the square root or not?",
        "A": "The question is not whether you have to solve the square root or not, the question is better asked do you want an exact answer 10 \u00e2\u0088\u009a2 (pure math). or do you want an approximate answer so that you can use it for some other purpose (applied math)?"
    },
    {
        "video_name": "E4HAYd0QnRc",
        "Q": "When you were at 5:11 and calculating the variance would it be possible to use the \"\u00ce\u00a3\" instead of \"\u00cf\u0083\u00c2\u00b2\"",
        "A": "Yes and no. In Statistics, \u00cf\u0083\u00c2\u00b2 is the symbol for the variance of a population. The uppercase \u00ce\u00a3 gets used for something called the variance-covariance matrix . If we have several variables that we are collecting data on, then each variable will have a variance - \u00cf\u0083\u00c2\u00b2 - and each pair of variables will have something called a covariance , which is related to the correlation. All these variances and covariances can get put together into the variance-covariance matrix, which gets denoted \u00ce\u00a3"
    },
    {
        "video_name": "hbJ2o9EUmJ0",
        "Q": "at 8:32 , it shouldn't be -34 instead of +34?",
        "A": "Yes. It s a mistake. It should be A = -5/17 and B = 3/17"
    },
    {
        "video_name": "hbJ2o9EUmJ0",
        "Q": "at 10:01 when I put A= -5/17 into (-5A+3B=2) i got B=7/51 , which is different than what sal found",
        "A": "You made some algebraic mistake, because the result is the same as Sal got: -5A + 3B = 2 -5(-5/17) + 3B = 2 25/17 + 3B = 2 3B = 2 - 25/17 3B = 34/17 - 25/17 3B = 9/17 B = 3/17"
    },
    {
        "video_name": "hbJ2o9EUmJ0",
        "Q": "At 5:11 shouldn't -4Asinx and -4Bcosx both be positive?",
        "A": "No. Sal s guess solution is y=Asinx+Bcosx and the differential equation is y -3y -4y=2sinx so he calculates -4y which is -4(Asinx+Bcosx) if we remove the brackets by multiplying we get -4Asinx-4Bcosx Hope that helps"
    },
    {
        "video_name": "hbJ2o9EUmJ0",
        "Q": "02:07 so finding the particular solution in undetermined coefficient equations is all guess work ? idk seems wrong like no factual formula to determine the solution.",
        "A": "In this case, yes, but the good news is, all the guess work is in finding the form of the equation needed to get the particular solution you need, and the calculations follow finding the coefficients. As long as you have first semester calculus conceptualized, it should be relatively straightforward."
    },
    {
        "video_name": "hbJ2o9EUmJ0",
        "Q": "at 2:22 if the function equaled 2cosh(x) what what it be then?",
        "A": "Since cosh(x) = 1/2\u00c2\u00b7(e\u00cb\u00a3 + e\u00e2\u0081\u00bb\u00cb\u00a3), a good starting guess to find the particular solution would be to propose y(x) = Ae\u00cb\u00a3 + Be\u00e2\u0081\u00bb\u00cb\u00a3"
    },
    {
        "video_name": "Cn0skMJ2F3c",
        "Q": "What is 'the inth data point' that he's talking about at 2:11 ?",
        "A": "The n-th data point is the last data point in the data set. When we have n=9 data points in our sample set, this would be the 9th data point."
    },
    {
        "video_name": "Cn0skMJ2F3c",
        "Q": "at 3:45 the video is telling ' when you have a smaller sample size you are more likely to get a sample mean that give a bad estimate of the population \"\nBut the matter is instead of dividing 'n' we are dividing it by 'n-1' ie., we are deducting the number of data point we have and hence smaller the size of the sample . Than the facts here are contradictory! is'nt it!?",
        "A": "You are not deducting a data point when you divide by n-1. The variance of all the data points are included in the top of the fraction, it is just the denominator that is changed, from n to n-1."
    },
    {
        "video_name": "Cn0skMJ2F3c",
        "Q": "At 5:42, Why do we have to multiply by n/(n-1)",
        "A": "Because we d like to get the most precise value of \u00cf\u0083\u00c2\u00b2. In order to do that, we have to multiply both side of the equation with n/(n-1) to cancel out the (n-1)/n on the right hand side, or in other words to get the value of \u00cf\u0083\u00c2\u00b2."
    },
    {
        "video_name": "arY-EUZDNfk",
        "Q": "At 1:10 couldn't you do 4+5*3 instead of 3(4+5)",
        "A": "no because of PEMDAS your equation: 4+5*3 4+15 19 Sal s equation: 3(4+5) 3(9) 27"
    },
    {
        "video_name": "arY-EUZDNfk",
        "Q": "at 1:10 what does calculate mean",
        "A": "Calculate means to solve something mathematically. I hope this helps!"
    },
    {
        "video_name": "arY-EUZDNfk",
        "Q": "At 0:59 , can you write the 3 at the end of the expression? If you can't, why not?",
        "A": "You can write three at the end of the expression but make sure to not include it inside the brackets"
    },
    {
        "video_name": "N0Y8ia57C24",
        "Q": "@5:35 if we go more than 4pie say 6 or 7pie and so on, does this mean x^3=1 has infinitely many roots, which seems to be absurd to me as a cubic can have only three solutions.",
        "A": "If you went more then 3 times the results would start repet itself. For example: 4 -> (e^(6pi*i))^(1/3) = e^(6*1/3*pi*i) = e^(2pi*i) = 1 so the same solution as for e^(0pi*i)"
    },
    {
        "video_name": "N0Y8ia57C24",
        "Q": "At 10:43, isn t -i to the fouth -1 ? Sal said 1.",
        "A": "No, Sal is correct. Think about it. (-i)^2=i^2= -1. Therefore, (-i)^4=i^4=((-i)^2)^2=(i^2)^2=(-1)^2=1."
    },
    {
        "video_name": "N0Y8ia57C24",
        "Q": "At 9:10, How can Khan say its \u00e2\u0088\u009a3/2 ? Plz HELP ME!",
        "A": "Simple. Let s just call the triangle he drew with a hypotenuse of a right triangle, which it is. Let s call the side opposite the 60 degree angle x. Since Sine=opposite/hypotenuse, sin(60)=x/1. From there, sin(60)=\u00e2\u0088\u009a3/2. So \u00e2\u0088\u009a3/2=x/1, or just x=1"
    },
    {
        "video_name": "N0Y8ia57C24",
        "Q": "I know this is a really stupid question and against the point of this video, but why couldn't you just take the cube root of both sides at \"0:18\"?",
        "A": "Well, think about this: x^2=1 If you take the square root of both sides, you get x=1. But x=-1 is also valid. Because you re taking the principal square root to get x=1. Same in this case, you would be taking the principal cube root if you would be x=1. but if you think about the non-principal cube roots, either you use the method of this video or you use factorisation."
    },
    {
        "video_name": "N0Y8ia57C24",
        "Q": "At 9:10 - 9:20 I'm a little unclear how he found the values of cos (2theta/3) + isin (2theta/3) ? Thanks in advance for your help here.",
        "A": "The cosine is opposite over adjacent"
    },
    {
        "video_name": "N0Y8ia57C24",
        "Q": "Ok I get the whole video riiiight up until 08:35 where he puts the answers into the form a + bi . I am familiar with that form but I don't understand where cos and sin come from. and how he decided to use them. Is it to do with a triangle that I have to figure out each time or is it a rule with a complicated proof\n\nThanks in advance,\nHelp is much appreciated",
        "A": "rcostheta= hyp(adj/hyp)= adjacent = a, rcostheta= hyp(opp/hyp)= opposite= b Make right triangles, with the hypotenuse being z (r)."
    },
    {
        "video_name": "r9aTLTN16V4",
        "Q": "What does he say at 1:11 ? Row etch-a-long form? What is \"row etch-a-long form\"?\n\nThanks,",
        "A": "He says row echelon form. Put any lines of all zeros on the bottom of the matrix, make the first entry in each row a one, and each entry in each row to the right of the first entry in the row above it. In form it is easy to see the solution to the equation."
    },
    {
        "video_name": "-DTMakGDZAw",
        "Q": "At 4:45 Sal says y is greater than or equal to 6, shouldn't it be y>6 because otherwise 0 will also have to take part in the function?",
        "A": "Sal s domain is correct. Y can be = 0. If y = 6, then the function f(y) = \u00e2\u0088\u009a(y-6) becomes \u00e2\u0088\u009a(6-6) = \u00e2\u0088\u009a(0) = 0."
    },
    {
        "video_name": "-DTMakGDZAw",
        "Q": "At 6:07 why the domain of h(x) is defined in curly brackets? In earlier vedio of interval notation, sal showed to use square brackets in closed interval. In here we only have two defined values 0 and 1, so why not use square brackets ?",
        "A": "Writing [0, 1] means the set of real numbers between 0 and 1, including 0 and 1 Writing {0, 1} means the set containing 0 and 1 Writing (0, 1) means the set of real numbers between 0 and 1, excluding 0 and 1 Writing [0, 1) means the set of real numbers between 0 and 1, including 0, but not 1 Writing (0, 1] means the set of real numbers between 0 and 1, including 1 but not 0 The second set contains exactly two elements. All the others have infinitely many."
    },
    {
        "video_name": "-DTMakGDZAw",
        "Q": "At 6:06 didnot we said 0 is undefined?",
        "A": "0 is the output not the input. 0 can be the output but in certain cases cnnot be an input. Hope this resolves your issue"
    },
    {
        "video_name": "-DTMakGDZAw",
        "Q": "At 3:30 instructor said: domain cannot be equal to zero, then at 4:32 he said y-x is greater than or equal to zero. Is there any contradiction or am I missing something? Please explain this. Thanks.",
        "A": "I see what you mean. 3:30 and 4:32 are different examples. Usually, the instructor gives new examples in a different colour or he colour-codes the writing to make it clearer and easier to follow. But yes, they re different examples. No worries. :-)"
    },
    {
        "video_name": "-DTMakGDZAw",
        "Q": "Is the example at 5:31 a matrix? I have not completely learned about them yet, but it looks like one.",
        "A": "No, although I think I see the similarity you are seeing -- namely, large brackets enclosing pairs of things -- this sort function notation is not directly related to matrices in any way that I am aware of."
    },
    {
        "video_name": "-DTMakGDZAw",
        "Q": "At 3:31 he is talking about how all real numbers besides 0 can go into that function but what I am confused about is that I thought 0 wasn't a \"real number\"",
        "A": "Real numbers are every number you can find on a number line. Any point on the number line is a real number. Irrational ones, natural ones, integers. All of them! 0 is on the number line, so it s a real number."
    },
    {
        "video_name": "-DTMakGDZAw",
        "Q": "Hey, i have a question, at 6:00, if h(pi) = 1 is h(1)=pi ?",
        "A": "No. The inverse of h at x = 1 is pi. The x and y values are interchanged by the inverse of h."
    },
    {
        "video_name": "-DTMakGDZAw",
        "Q": "At around 4:18, Sal said that y-6 cannot equal a negative number. How is it not possible to find the square root of a negative number? Why is the square root of a negative number considered undefined?",
        "A": "The square root reverses the operation of raising a number to the power of 2 (or squaring the number). When you square any real number, it becomes positive. For example: 3^2 = 9 and (-3)^2 = 9 So, if you are asked to find the sqrt(-9), it can t be done using real numbers. Note: If you continue with your math studies, you will learn about imaginary & complex numbers that let us deal with negative numbers in the square root. These exist outside the real number system. Hope this helps."
    },
    {
        "video_name": "-DTMakGDZAw",
        "Q": "at 3:52, g(y) is not a function because there are two possible outputs for an input.",
        "A": "No. g(y) is a function. For example: g(10) = sqrt(10-6) = sqrt(4) = 2. You get one solution, not two. Possibly you are thinking that radicals can have 2 roots. That is true. But the problem tells you which root to use. There is no minus in front of the radical, so it just wants the positive root. If the minus was in front of the radical, it would want the negative root. In either case, you get one solution for each input value. Hope this helps."
    },
    {
        "video_name": "-DTMakGDZAw",
        "Q": "At 4:50, could the domain be 6 is less than or equal x which is less than or equal to infinity?",
        "A": "Nearly Yes for the first part. 6 <= y (the equation uses y - not x ). Yes for positive infinity. For example 1,000,006. Subtracting 6 from it gives -1,000,000, which is a very positive number if you will that we ll take the square root of (giving 1000). But not for negative infinity! For example -1,000,006. Subtracting 6 from it gives -1,000,012, which is a very negative number if you will we won t try to take the square root of it."
    },
    {
        "video_name": "hq1bUM2tyg0",
        "Q": "0:25-0:28 big pause",
        "A": "Not too big, it was only like, 2 seconds long."
    },
    {
        "video_name": "hq1bUM2tyg0",
        "Q": "At 0:40, is Sal solving for s or C, or is s equal to 3?",
        "A": "He s solving for C, knowing that the number of sessions is 3 (s=3). So if the full equation is C=20+8s, then the exact problem Sal is solving is simply 20+8*3. In other words the monthly cost is $20 (monthly fee for basic plan ) plus the $8 per session fee times 3 sessions."
    },
    {
        "video_name": "hm17lVaor0Q",
        "Q": "At 2:19, Sal says that we multiply the base and the height to get the area of a parallelogram, like rectangles. But that is not possible, because parallelograms do not have 90 degree angles. And most of the time, we do not solve equations on the computer; we normally solve it with a pen and paper. And therefore we cannot cut-copy-paste items a lot of the time. So then how do you find the area of parallelograms?\nThanks!",
        "A": "If you don t understand, you may check out the next thing, Area of Parallelograms. It explains to you and you can interact with the parallelogram to form it into a rectangle. I hope that helps you!"
    },
    {
        "video_name": "81SseQCpGws",
        "Q": "At 1:20, why couldn't Sal start at a different point? for example, point (3,1). If you wanted to find the slope between that point and (0,2) wouldn't the slope be 1/3?",
        "A": "No, because there is always a negative and a positive so you get-1/3"
    },
    {
        "video_name": "81SseQCpGws",
        "Q": "How would you write the rise over run with a ratio? because all my problems say to write a ratio. Sal doesnt show how to do that....but im assuming like if the rise was 2 and the run was 3 you would write it like 2:3 ?\n~aubrey",
        "A": "I believe this is correct. After all, you can write a ration such as 2 to 3 as 2/3, so it should work."
    },
    {
        "video_name": "81SseQCpGws",
        "Q": "at 0:58 Sal says delta. Where does this come from?",
        "A": "Delta is the fourth letter of the Greek alphabet which is usually used for denoting change between two values. For e.g. \u00ce\u0094y means the change or difference between the two values of the y co-ordinates."
    },
    {
        "video_name": "RI874OSJp1U",
        "Q": "Is the unfair coin on 1:32 real?:-/",
        "A": "I think it is possible to make one side of the coin heavier than the other side, but ive never seen one."
    },
    {
        "video_name": "RI874OSJp1U",
        "Q": "I've got a mind blank. At 5:02, it shows P(H,H2). does the H2 thing mean 2nd flip?",
        "A": "Yes, Sal says heads on our first flip and heads on our second flip, and as he says this, he writes: H1 and H2; so H2 means exactly heads on our second flip. If you look closely, you ll see (I think :) that there is no comma between H1 and H2."
    },
    {
        "video_name": "RI874OSJp1U",
        "Q": "At 1:30 he first introduces the unfair coin. How likely is it that there would even be an unfair coin? It doesn't quite make sense to me. Which is why it was difficult for me to calculate the probability, because I can't picture an unfair coin...",
        "A": "The unfair coin is just an example for a probability, where not every possibility is equally likely. By learning to calculate the possibilities for an unfair coin, you learn how to cope with all situations, where the chance of getting one result is not equal to the chance of getting another result.For example, when producing things, 1% of these things may be defect. So P( defect ) is 1% and P( Not defect ) is 99%. Bingo, the same as an unfair coin: P( tails)=1%, P( heads )=99%."
    },
    {
        "video_name": "RI874OSJp1U",
        "Q": "At 4:33, shouldn't it be \"four over five\", not \"four over fifths\"?",
        "A": "Yeah, that s what I meant. Sal said two over fifths , but shouldn t it be two or five ?"
    },
    {
        "video_name": "RI874OSJp1U",
        "Q": "At 4:47, Sal said the probability of getting heads is 40%. He meant the probability of getting TAILS is 40%, right?",
        "A": "This is a known error, and a pop-up box appears in the lower right-hand corner of the video to say so."
    },
    {
        "video_name": "RI874OSJp1U",
        "Q": "At 2:21, he says that the probability of getting heads is 60%. How does he get the 60%? Or is it just an example?",
        "A": "It s just an example of an unfair coin. He could have easily made it 51% or 70% or 12% - anything except 50%, which is a fair coin."
    },
    {
        "video_name": "RI874OSJp1U",
        "Q": "At 5:41 why do we multiply P(A) by P(B)",
        "A": "Probability of event A times probability of event B. Since the probability of head is 0.6, the probability of two heads in a row is equal 0.6 * 0.6 = 0.36"
    },
    {
        "video_name": "RI874OSJp1U",
        "Q": "At 4:46, how exactly do you know its 60% and 40% not 70% and 30% or anything else.",
        "A": "it s just an example he is using, on a bias coin you determine the probabilities when making it or if it is given to you, the probabilities are given."
    },
    {
        "video_name": "RI874OSJp1U",
        "Q": "a gazillion is not a number at 3:20 and at 4:40",
        "A": "Sal was using a gazillion to represent a very large number. He wasn t suggesting that it is a specific number."
    },
    {
        "video_name": "XdzcU5JbVcA",
        "Q": "If the equation was y = lnx instead of y = \u00e2\u0088\u009ax, and I was rotating around of y = -1 instead of y = 1, I presume that each washer would equate to \u00cf\u0080(lnx + 1) instead of - 1. Is that presumption accurate? See 2:25.",
        "A": "That s very close. Remember, we are finding the volume, so we will need to find the area of the circle. Since the radius is ln(x)+1, the area is \u00cf\u0080(ln(x) + 1)^2. Also, to find the volume, we will need to multiply the area of the circle by the width of each washer, which is dx. The dx is important. With dx, you will summing up an infinite number of finite values, which is infinite, but with dx, you will be summing up an infinite number of infinitesimal values, which is finite. I hope this helps!"
    },
    {
        "video_name": "XdzcU5JbVcA",
        "Q": "At 3:45, how do we know that the interval is from 1 to 4?",
        "A": "The interval comes from the setup of the problem in the first 45 seconds of the video. We re working with a figure that begins at the point where the curve y = sqrt x crosses the line y = 1, and that occurs when x = 1. The figure ends at x = 4, which is just an arbitrary ending point that Sal states early in the video."
    },
    {
        "video_name": "MMv-027KEqU",
        "Q": "At 2:43, Sal says you put a dx around the function you are trying to take the antiderivative of. is that to show the opposite of a derivative as there you seem to be dividing by dx (though you don't) whereas here you seem to be multiplying it?",
        "A": "In a sense, dx acts as a multiplier in integration, as you ll see when using integrals to find areas and volumes. Note that we don t put dx around the function, but rather put the integral symbol on one side and dx on the other, so that in a sense the two act as a single symbol that goes around the function being integrated."
    },
    {
        "video_name": "MMv-027KEqU",
        "Q": "At 3:30, you talk write a dx after 2x; why do we have to put that there? What is the meaning of that dx? Does it stand for a value?",
        "A": "dx means an infinitesimal change in x so for dy/dx it means an infinitesimal change in y for an infinitesimal change in x which is basically just slope. In the case of integrals and anti derivatives the dx tells us what we are taking the integral with respect to in this case x. Additionally it can be seen almost as a measurement telling us what intervals to go on (infinitely small ones). Hope this helps"
    },
    {
        "video_name": "MMv-027KEqU",
        "Q": "at 0:35 how is pi and 1 zero?",
        "A": "The derivative of ANY constant is 0. This can be shown quite easily from the definition of a derivative. So, it doesn t matter if the constant is \u00cf\u0080. 1, -10 billion, or e or any other constant. The derivative is always 0. Here s the proof f(x) = c (where c is any constant you like). f (x) = lim h\u00e2\u0086\u0092 0 [ f(x+h) - f(x) ] / h = lim h\u00e2\u0086\u0092 0 (c - c) / h = lim h\u00e2\u0086\u0092 0 0/h = lim h\u00e2\u0086\u0092 0 0 = 0"
    },
    {
        "video_name": "YZD5ifHZILE",
        "Q": "At 1:18pm, so if we had 7/10 + 30/1000 would you convert the 10 into 1000 to find the answer.",
        "A": "Yes, which would give you 700/1000 + 30/1000 = 730/1000, which equals 73/100. Or you could first write 30/1000 as 3/100, and then convert 7/10 to 70/100 and add them together. 70/100 + 3/100 = 73/100, which is the same answer. Hope this makes sense! Let me know if it doesn t."
    },
    {
        "video_name": "ZN48Ub3k8uE",
        "Q": "1:56 why cant it be one because if you spit it into half one will be heavier than the other if you take one off at a side and look at sides and if you took them off one side was still heavier the heavy ball is still on but when it evens out the then the one you took off on the side that was heavier is the heavy ball",
        "A": "but 0.5+0.5=1 so it is the same"
    },
    {
        "video_name": "ZN48Ub3k8uE",
        "Q": "i don't understand : i know weigh one versus one. om the minutes 3:52",
        "A": "The balls are originally broken into groups of 3 balls. Group A is 1-3. Group B is 4-6. Group C is 7-9. After weighing Group A vs. Group B you will know which of the three groups is heaviest (if even, Group C has the heavy ball). Step 2, which you reference, requires you to weigh two balls out of the heavy group. Say Group C (balls 7, 8, 9) was heaviest, you would weigh ball 7 vs. ball 8. If they are even, 9 is heaviest. If they are not even the scale will tell you whether 7 is heavier than 8."
    },
    {
        "video_name": "ZN48Ub3k8uE",
        "Q": "this is one of my favorite brain teasers, especially at 4:05 when it all starts to come together",
        "A": "I like this brain teaser too, though I originally saw it in coin form, where the heavier one is a counterfeit."
    },
    {
        "video_name": "ZN48Ub3k8uE",
        "Q": "at 0:46 I have an answer! Couldn't you just pick it up with your hand and figure out? So mine is 0",
        "A": "Sal did say that the heavier ball is only slightly heavier. You may find the balls weight so close that you would not be able to use your hands to tell the difference."
    },
    {
        "video_name": "ZN48Ub3k8uE",
        "Q": "At 2:00 I paused so I guess put one ball down and on the other side put one down measure and then replace the ball and do it til the scale tips?",
        "A": "That would give an expected number of weightings to be 2 2/3, so on average, you would need 2 2/3 weightings to figure out the heavier ball. However, it can be improved upon. There is a more efficient method."
    },
    {
        "video_name": "ZN48Ub3k8uE",
        "Q": "I was getting really intrested by the 0:15 mark so you know it must be going somewhere!",
        "A": "Yes! Sal is teaching us a logic problem."
    },
    {
        "video_name": "iKy-d5_erhI",
        "Q": "at 4:59 you found the number of ways in which you could arrange 3 people how would you find it for x people?",
        "A": "At 4:59 Sal is using the formula n! / (n-k)! OR 6! / (6-3)! which is 120 / 3! or just 120/6. In this case k = 3 or to answer your question k would be equal to x."
    },
    {
        "video_name": "iKy-d5_erhI",
        "Q": "I don't understand, at 4:50 Salman says, to find the number of ways to arrange three people from the six, the equation is 120 (total number of permutations) divided by 6 (number of ways to arrange the letters in a set). Why wouldn't the equation to find all arrangements of three people be the same as finding the total number of possibilities in a set? (3 letters*3chairs = 9 different arrangements)",
        "A": "There are 3 people who can sit in chair one. Then there are only two of those three left for chair two and then one for chair three. 3*2*1 equals 6"
    },
    {
        "video_name": "iKy-d5_erhI",
        "Q": "If 6C3 is the notation for combination, what is the notation for the corresponding permutation? See 5:18",
        "A": "5P3. If the corresponding means, everything is same. (But, the result will be different.)"
    },
    {
        "video_name": "iKy-d5_erhI",
        "Q": "At about 5:30, Sal writes \"6 C 3\" with the 6 and 3 as subscript numbers, before and after the C. I've seen the notation \"6 C 3\" before, but only with the 6 as a superscript before the C. Are both notations in common use?",
        "A": "They are equivalent expressions."
    },
    {
        "video_name": "iKy-d5_erhI",
        "Q": "1:27 'So you have a total of 30 scenarios where you have seated six people in the first two chairs'\n\nHow do six people fit in two chairs? This furthers my confusion, Or did I just read it wrong?",
        "A": ".It might be easier to relate it to something that makes sense, lets say we have 6 people, and they can run for either president or vice-president. That means two people will hold an office and 4 will not. The same with 2 chairs (like on a bus maybe with just two seats left). That means 2 of the six will be seated and 4 will have to keep standing."
    },
    {
        "video_name": "tOd2T72eJME",
        "Q": "A:B= 1:4 and B:C = 2:3 then what is A:C",
        "A": "1:3 is the answer."
    },
    {
        "video_name": "tOd2T72eJME",
        "Q": "help this is confusing what does he mean by that @ 1:00",
        "A": "to get from 10 to 60, you multiply by 6... so to get from 7 to 42, you multiply by 6.... Thus: 7/10=42/60"
    },
    {
        "video_name": "tOd2T72eJME",
        "Q": "mr.khan i have a question>The ratio between the length sides of a triangle is 2:3:4.Ig the perimeter of the triangle is 54cm,find the length of each side of triangle",
        "A": "Answer: 2:3:4 equates to 12:18:24 = 54cm"
    },
    {
        "video_name": "tOd2T72eJME",
        "Q": "Ok, can someone please explain the different between this video and the one before=?\nIn the first video, with students, the girl to boy ratio is 5 to 8 ( 5:8, 5/8). Let's say 5 is g and 8 is b and totalt is t. Then we get: Ratio g:b = 5:8 --> g+b=t 5+8=13\n\nBut in this video example we can't do that. 7cm : 10km can't be written as a:b = 7 : 10 and a+b=t --> 7 + 10^6=total.\nWhat is the different between the ratio problem of girls and boys and this problem of scale?",
        "A": "This problem is not asking for a total number of boys and girls. This problem is asking you to convert between two measurements (cm & km). You cannot ask a total question when you have two different measurements."
    },
    {
        "video_name": "tOd2T72eJME",
        "Q": "Hey Sal, what is the answer to the ratio 4:28",
        "A": "I think you mean what is the ratio 4 to 28, or 4:28 . It is not actually a time in the video =0. The answer would be 1 to 7, 1:7 , or 1/7. Hope this answers your question :)"
    },
    {
        "video_name": "fa__960kzVE",
        "Q": "At 5:55 to 6:00, Sal is talkin about associative property. Could someone brief me about it?",
        "A": "Definition: The associative property states that you can add or multiply regardless of how the numbers are grouped. By grouped we mean how you use parenthesis . In other words, if you are adding or multiplying it does not matter where you put the parenthesis. Add some parenthesis any where you like ! Please Vote up If you like my answer . Thank You !"
    },
    {
        "video_name": "sBhEi4L91Sg",
        "Q": "What is the video he mentions at 9:53?",
        "A": "\u00c3\u0097 Logarithmic Scale Understanding how logarithmic scale is different from linear scale and why it could be useful"
    },
    {
        "video_name": "sBhEi4L91Sg",
        "Q": "At 2:26, did Sal start it at one because you can't use zero in logarithms?",
        "A": "if he would have had taken zero then the scale was not possible because he multiplies by 10 and again and again,so if there was zero then 0*10=0 hope that helps"
    },
    {
        "video_name": "sBhEi4L91Sg",
        "Q": "At 2:20, I understand why you didn't start with zero, but is it possible to have a log-scaled graph that does include zero? For example, if I needed a graph that has both positive and negative y values, can I still use a logarithmic scale on the y axis?",
        "A": "No. Can you take a positive number to a power and make it negative? or take any number except 0 to a power and make it 0?"
    },
    {
        "video_name": "86nb02Bx_5w",
        "Q": "at 3:11 sal said there will be chance of 9/20 chance for sunfish. I think in that case there will be chance of 9/19. Am I correct?",
        "A": "Yes, he made a mistake or misspoke, but does not affect his solution of the problem posed (since he was discussing a how the problem would differ if they were not replacing the fish after catching them)."
    },
    {
        "video_name": "1tn0vd2835k",
        "Q": "At 2:49, wouldn't any number over three be a negative number? Since evaluating the expression with three gets zero, any number more than three would end in a negative fraction, like -1/-1, which is impossible. Please satisfy my grade 5 mind",
        "A": "A negative number divided by a negative number is a positive number. Why do you think -1/-1 is impossible?"
    },
    {
        "video_name": "dD71fFNwVJw",
        "Q": "At 2:24 how do the seconds cancel out?",
        "A": "When you have the same units on the top and bottom of a fraction, you can divide the top and the bottom by that unit to simplify the fraction, so it cancels out. This is just like if you have 6/8 and you divide the top and the bottom by 2 to get 3/4."
    },
    {
        "video_name": "dD71fFNwVJw",
        "Q": "At 4:00 til the end how did you get an extra 10 to make it 10 to the 11 power? And then why are you dividing when its a multiplication symbol? Is it because you have to divide it to make it qualify for scientific notation?",
        "A": "Yes, you are correct. When you have a number that doesn t qualify for scientific notation, like 23.45 * 10^3, you have to make it qualify as a scientific notation number. To move one place to the left, we have to divide, even though the * means to multiply. Once we have divided one place to the left, the number becomes 2.345*10^4."
    },
    {
        "video_name": "dD71fFNwVJw",
        "Q": "At 3:30, if the decimal is moving one to the left, then how are you going to multiply it by 10^1, because that would make 150, not 1.5. Shouldn't it be 15(10^-1)? I thought that when you move the decimal to the left, you use negative integers.",
        "A": "It is 10^1 because when you take 1.5 * 10 it equals 15. If you took 1.5 * 10^-1 you would get .15 You are multiplying 15 by 10^1, not the 1.5."
    },
    {
        "video_name": "vbGwcvXgDlg",
        "Q": "At 3:18 in the video it seemed that doing the area model was harder than just doing the multiplication. Were you doing just to explain the concept or did you actually just want to teach us how to do it?",
        "A": "explain the concept"
    },
    {
        "video_name": "vbGwcvXgDlg",
        "Q": "At 7:50, isn't it 20x6? Please answer!",
        "A": "yes he just said it wrong"
    },
    {
        "video_name": "tedzsRH0Jas",
        "Q": "At 1:50\nSal, says that to represent the 1st interval, -9 can't be included, but why is it -9<x why not -9>x?",
        "A": "-9>x means -9 is larger than X, or X is smaller than -9. This means the line would be to the left of -9. It isn t! The line extends from the -9 to -5. This means X is greater than -9 (thus, -9 < x), but less than or equal to -5."
    },
    {
        "video_name": "tedzsRH0Jas",
        "Q": "At 0:22, what is a piecewise function?",
        "A": "This video tells you what a piecewise function is. Obviously, after only 22 seconds into the video, you have basely started the lesson. Did you watch the entire video? By the end, you should know that a piecewise function is a function where the value of the input (x) determines what part of the function you perform. Not all values of X go thru the same operations.."
    },
    {
        "video_name": "tedzsRH0Jas",
        "Q": "f(x)=sqrt(x) is not a function....Then, why did Sal say so? At 0:07",
        "A": "f(x)=\u00e2\u0088\u009a(x) is a function because we are only taking the principal square root of x. In other words, only the positive square root. If he said f(x)=\u00c2\u00b1\u00e2\u0088\u009a(x) then it would not be a function because this includes both positive and negative square roots."
    },
    {
        "video_name": "tedzsRH0Jas",
        "Q": "At 0:49, Sal puts calls the y axis f(x). What does this represent?",
        "A": "Y and f(x) represent basically the same thing. So, sometimes the y-axis is refereed to as f(x). f(x) just means what is the y value at a certain x value . The equation y = mx + b is the same as f(x) = mx + b. Function notation (which means using f(x)) will be used a lot more in upper level math."
    },
    {
        "video_name": "tedzsRH0Jas",
        "Q": "When defining the boundaries of the piecewise function near 2:38, why does Sal write the y coordinate first? (Eg: 6, -5<x<=-1 not 6, -5<x<=-1)?",
        "A": "After thinking about your q, the reason Sal writes the y coordinate first is because f_ ( _x ) is the _y value! It is the output of the function. Therefore the first thing to write is the y value. Then he writes the qualifying statements (only if ...). This is my guess but it feels right."
    },
    {
        "video_name": "2N62v_63SBo",
        "Q": "At about 15:20, what was Sal referring to about \"flipping the sign\"? I didn't see any sign flipping in the section of the chart he was talking about.",
        "A": "If you calculate f(x) for x = -4.1, -3.9, 3.9, and 4.1, you will see that f(x) changes sign as you cross -4 and 4, the 0 s in the denominator."
    },
    {
        "video_name": "2N62v_63SBo",
        "Q": "at 8:25 does he mean\n\nnegative 4?\n\npositive 4?\n\nboth?",
        "A": "both its positive and negative"
    },
    {
        "video_name": "2N62v_63SBo",
        "Q": "At 3:05, I don't understand how to find the horizontal asymptote.",
        "A": "Just let x keep getting larger. In this problem, f(x) = 1/-15, 100/84, 10000/9984, 1,000,000/999,984, and so on."
    },
    {
        "video_name": "2N62v_63SBo",
        "Q": "At 16:32, what does sal mean when he says \"these 2 terms dominate?\"",
        "A": "It means that those two terms determine the value of the output for the most part. The other terms will only make relatively small adjustments. So we can look at the two terms to determine the general behavior of the function."
    },
    {
        "video_name": "2N62v_63SBo",
        "Q": "At 18:09 you have explained that if there are no common factors between the numerator and the denominator then the graph would have an vertical asymptote. Could you please explain it further. For eg: (x-2) / (x^2+x+6) , will be simplified to (x-2)/ (x-2) (x+3). After removing the common factors from the numerator and denominator the equation will be simplified to 1/(x+3). Can't this equation have -3 as vertical asymptote? Please help.",
        "A": "For your example, there is a vertical asymptote at x = -3. Since the (x-2) canceled out from the numerator and denominator, there is a removable discontinuity, or hole, at x = 2. If you were to graph the function f(x) = (x - 2)/(x\u00c2\u00b2 + x + 6), it will look like 1/(x + 3), but there will be a hole in the graph at x = 2, where the f(x) is undefined."
    },
    {
        "video_name": "2N62v_63SBo",
        "Q": "why does y approach 1 at 03:35 can anyone give another example?",
        "A": "Sal is trying out numbers later on in the video. The bigger the number is the less important any values of lesser order. In x^3-x^2+x-1, the x^3 is most important, as it is of the highest order. If you have (x^2 + 1)/ x^3 the function would approach 0 because no matter how big x^2 gets, x^3 will get a lot bigger than that. In Sal s example x^2 / (x^2 - 16) is approximately the same as x^2/x^2 = 1"
    },
    {
        "video_name": "vmS3cTaxKdA",
        "Q": "So at 1:14 is the bottom radius of the taken disk not bigger than the top radius? Is it not more something like a section of a cone rather than a cylinder? Or is it taken like a cylinder because of the limit?",
        "A": "The radii are different, but as the height becomes infinitely small, so does the difference in radius. It s much like the problem with Riemann sums, where the edges of the rectangles stick out to one side or the other of the function graph. As the rectangles get narrower and narrower, the error disappears. So yes, the limit makes it a cylinder."
    },
    {
        "video_name": "7BkcNLOf56w",
        "Q": "why doesn't he just say 0.5 to the 4th at 3:00?",
        "A": "He expands it so that it is clearer for us to understand what he is doing. He also does that to help the people that did not learn about powers yet."
    },
    {
        "video_name": "7BkcNLOf56w",
        "Q": "at 4:57, why are the probabilities added and not multiplied? when should probabilities be added and when multiplied??",
        "A": "Probabilities are generally multiplied when something is done over and over in a row ( ex. flipping a coin 4 times and getting heads 4 times in a row is .5^4 or .5 * .5 * .5 *.5 ). Probabilities are generally added together when you are trying to find a total probability between two independent probabilities on a probability tree ( ex. the whole video shows a probability chart. )"
    },
    {
        "video_name": "7BkcNLOf56w",
        "Q": "At 5:57 why does he add the results of the two but not multiply them?",
        "A": "Picking an unfair coin is a 3/4 chance. Picking a fair coin is a 1/4 chance. Adding them together gives us the 100% chance."
    },
    {
        "video_name": "ROIfbUQrSY4",
        "Q": "At 0:12, what does he mean by positive square root? Is there a negative square root? If so, what does the symbol look like?",
        "A": "Positive square roots are when it is above zero. Negative square roots are below zero."
    },
    {
        "video_name": "ROIfbUQrSY4",
        "Q": "At 0:18 sal says that every number has two square roots. I know that the sqare roots of 25 are -5 and 5. so the sqrt of 25 is 5 and the sqrt of 25 is -5. the two eqautions stat that sqrt 25=-5 and sqrt 25 =5. therefore sqrt 25=-5=5, if I eliminate the 25, it says the 5=-5. i know that this is wrong, so could someone please explain this to me.",
        "A": "Sal doesn t say every number has two square roots. Only positive numbers have two square roots. As Sal says at 0:14, that symbol means the positive square root, so in your example, when you say \u00e2\u0088\u009a25 = 5, that s correct. When you said \u00e2\u0088\u009a25 = -5 that is not correct as we already know that \u00e2\u0088\u009a means the positive square root."
    },
    {
        "video_name": "t4xOkpP8FgE",
        "Q": "At around 3:45, I think it was a little too early for subtracting negatives and positives because that subtracting segment is *AFTER* this segment. (It's January 2017 just so you know)",
        "A": "(yep, it s 2017 awesome, right?). It is :)"
    },
    {
        "video_name": "t4xOkpP8FgE",
        "Q": "at 4:42, it said the answer to 3 - (-2) = 5. But I really don't get it because doesn't 3 - (-2) = 1? Or am I wrong and 3 + -2 is 1?",
        "A": "When you have a minus sign and a negative sign, you end up adding. So 3 - (-2) becomes 3 + 2."
    },
    {
        "video_name": "t4xOkpP8FgE",
        "Q": "at 10:41, why is the greater number always on the right side?",
        "A": "Where the heck did you get a 10:41? This video is 5:29 minutes long."
    },
    {
        "video_name": "t4xOkpP8FgE",
        "Q": "4:36- Why did you place the two in parenthesis? I never can clearly understand that and it happens a lot. It's holding me back from understanding a few things and I just really need help on that.",
        "A": "There is a convention in Math that we never write 2 math operations side by side. For example: 3 - - 2 is not done. Instead, we wrtie: 3 - (-2)"
    },
    {
        "video_name": "t4xOkpP8FgE",
        "Q": "Just wondering, At 3:53, why does -5 equal 5? I don't get that. But then at 4:47 5 equals 5 which is obvious, but why does -5 equal 5?",
        "A": "Let s say that you are at the point 0. Then if you went 5 steps to the east, or 5 steps to the west, the distance from you to each place wouldn t change. If east is negative and west is positive, the distance you went to west is not bigger than the distance you went toward east. They equal each other, and the - sign or + sign is only a sign telling you which way is which."
    },
    {
        "video_name": "t4xOkpP8FgE",
        "Q": "Did anyone else notice that Sal's voice cracks up at 2:23?",
        "A": "Nope, I never knew so thanks!"
    },
    {
        "video_name": "t4xOkpP8FgE",
        "Q": "I am having Trouble picking up to the video's pace.\nCan You Please Help Me With Absolute value as distance between numbers ?\nMostly in 2:17 - 5:19 - See In Transcript.",
        "A": "Basically the absolute value of a number is just the positive of a number. |-5| = 5 |9| = 9"
    },
    {
        "video_name": "t4xOkpP8FgE",
        "Q": "At 2:00, I tried an experiment to see if that was true. I did a=-5 and b=6. I don't think I got it. Would you use absolute value on your answer?",
        "A": "To find the distance between the numbers but you didn t know which was greater, yes, you would use absolute value. so if you did 5-6, you ll get -1, which isn t the distance between them. |5-6| is equal to 1, because the absolute value makes anything positive. I hope that clears things up for you."
    },
    {
        "video_name": "RomnHMWSLoE",
        "Q": "Why did he include just the Discriminant part in the case of the hyper bola at 3:43 ?? Where as in case of the circle he considered the whole formula....??",
        "A": "he used the discriminant part in the previous video too just re-watch the 2nd part and you will understand that he uses the whole formula in the later part of the video when he is solving for b in terms of m , from the equation he got by equating the discriminant"
    },
    {
        "video_name": "PErhLkQcpZ8",
        "Q": "At 11:20 you do an example from R^2 to R^3. Could you do an example from R^2 to R^1? I have a problem that has stumped me. It says T: R^2 -> R^1 is a linear transformation, such that T(1,1) = 2 and T(1,0) = 3. What is T(x,y). The solution is given in the book, but I don't understand how it was found: 2y +3(x-y) which equals 3x-y. I understand that the Identity matrix was used and the property T(av + bw) = aT(v) + bT(w), but I get lost when I try to work it out.",
        "A": "You don t even need the identity matrix for this problem. A simple solution would be to treat this as a system of equations. T(1,1) can be written as [1,1]vertical times [x,y]horizontal equals 2. Solve for x: you get x=2-y. Do the same with T(1,0): you get x=3. You insert x=3 into the first equation and find out that y=-1. There is your solution: T(x,y)=(3)x+(-1)y = 3x-y."
    },
    {
        "video_name": "PErhLkQcpZ8",
        "Q": "At 11:00 and forward, what was the purpose of the last example? I don't think I understand what it's supposed to prove.",
        "A": "It uses a specific transformation to transform the columns of the R^2 I matrix, then the sum of products these transformed column vectors with their corresponding components of x is Ax where A is a matrix of the transformed column vectors, and is equivalent to the original transformation by the definition of a linear transformation."
    },
    {
        "video_name": "PErhLkQcpZ8",
        "Q": "5:00 - 6:30 If the x vector contains elements equal to zero, is it still correct to say that I*x still spans Rn? How can a vector span Rn if it doesn't protrude into one of its dimensions?",
        "A": "A single vector can never span R^n, unless n=1, i.e. the vector space is one-dimensional. But you can certainly have a set of one-dimensional vectors (which have elements equal to zero) that span a multi-dimensional space -- in three dimensions, the vectors (1,0,0), (0,1,0) and (0,0,1) are an example."
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "At 1:30 Sal is showing how to zero out the equations. But what if they arent't the same? What if it was x+y+z=3, 2x+2y+3z=2, and 4x+y+2z=1?",
        "A": "What do you mean by zero out? None of the rows were reduced to all zeros. None of the original equations are the same."
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "What is \"pivot entry\" at 0:58",
        "A": "This may not be an exact definition, but it should help you to understand what Sal means by a pivot entry. When solving the matrix, we want to end up with 1 s going down the diagonal and with zeroes everywhere else (I m of course talking about the entries to the left of the dividing line). Those 1 s are in the pivot positions."
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "at 2:20 Why we had to come back to R1 again to make row 1, column 2 = 0?\n\nlike what rows, column numbers we have to make zero by row-echelon form?",
        "A": "RREF has 3 criteria: Rows with more leading 0 s are below rows with fewer leading 0 s The first non-zero element in each row must be 1 The leading 1 in each row must be the only non-zero element in that column"
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "at 3:36 why does Sal replace the third row with the 3rd row minus 2times the second row...?",
        "A": "You can replace any row with: 1) Its multiple 2) Itself minus (or plus) any other, or a multiple of any other, row)."
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "What's a pivot entry? At around 0:53.",
        "A": "This may not be an exact definition, but it should help you to understand what Sal means by a pivot entry. When solving the matrix, we want to end up with 1 s going down the diagonal and with zeroes everywhere else (I m of course talking about the entries to the left of the dividing line). Those 1 s are in the pivot positions."
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "What does echelon form mean at 2:20?",
        "A": "Reduced row echelon form has 3 criteria: 1) Rows with more leading 0 s must be below rows with fewer 2) The first non-zero element in each row must be 1 3) The leading 1 in each row must be the only non-zero element in its column Here s an example of a matrix in reduced row echelon form: \u00e2\u008e\u00a11 0 2 0 0\u00e2\u008e\u00a4 \u00e2\u008e\u00a20 1 1 0 3\u00e2\u008e\u00a5 \u00e2\u008e\u00a20 0 0 1 6\u00e2\u008e\u00a5 \u00e2\u008e\u00a30 0 0 0 0\u00e2\u008e\u00a6"
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "At 1:15, it says how to zero out the 1 in the second row, but how do you come up with the equation? It says \"Replace the 2nd row with the 1st row minus the second row\" - where did that come from? How do you figure out what you have to do?",
        "A": "You can replace any row with: 1) Its multiple 2) Itself minus (or plus) any other, or a multiple of any other, row). If it says Replace the 2nd row with the 1st row minus the second row I also find that confusing, but actually it s equivalent to replacing row 2 with row 2 - row 1 (that is, r2-r1), because: r1-r2 = -(r2-r1) is equivalent to r2-r1 ; because: if r1 is a=b and r2 is c=d, then r1-r2, or a-c = b-d is the same as -(a-c) = -(b-d) is the same as c-a = d-b , or r2-r1."
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "At 2:36, he Sal says, \"The Augmented part of it.\" In piano the augmented chord is an extended part of the chord. If augmented means extended, then how is it an extended part of the matix, if the dividing line means equal to? If you have the same question, can you please give this one a vote? Thanks.",
        "A": "The augmented matrix represents Ax = b , where b is the added column vector, and it s components are constants that are not coefficients as are the components of the other column vectors, and are obviously manipulated and interpreted differently. Matrices that aren t augmented have importance of their own, as you will see as you continue."
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "At 5:00, why couldn't've Sal done substitution?",
        "A": "He could have, but then he would have been diverting from the basis of this video. Which is the RREF, as it is abbreviated to in most text."
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "at 3:47 and 3:52 why is he using 2 times?",
        "A": "So that he can eliminate the 2 in the 3rd row. (r3 - 2*r2)."
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "Fort the situation occurring at the point in time 5:42,\nCould we also take the 2nd row minus negative 2 times the 3rd row in order to progress this matrix into reduced row echelon form?",
        "A": "You can replace any row with: 1) Its multiple 2) Itself minus (or plus) any other, or a multiple of any other, row)."
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "Is there any real difference in the second equation - first equation or the first-equation minus the second? Sal said around 1:38 that we want to have the rest of second row zeroed out. I am a little confused about this concept of eliminating equations within the matrix.",
        "A": "r2-r1 is equivalent to r2-r1, because if r1 is a=b and r2 is c=d, a-c = b-d <-> -(a-c)= -(b-d) <-> c-a = d-b If one equation is a multiple of another, they are the same, and only 1 of them is allowed."
    },
    {
        "video_name": "lP1DGtZ8Wys",
        "Q": "im wondering if it would be no solution scince x=5 but 1,0,and 0 dont equal to 5\nits about 7:01 in the video",
        "A": "1, 0, and 0 are probably the coefficients of x, y and z."
    },
    {
        "video_name": "WkspBxrzuZo",
        "Q": "Im confused at 1:25-1:28\n\nWhat does he mean, I saw him do it but what is the final Delta Y and Delta X? Is the -7 or -2?",
        "A": "Check out 3:23 ish to see that it can also be Delta X=7 Delta Y=14 The slope is still -2."
    },
    {
        "video_name": "WkspBxrzuZo",
        "Q": "At 4:23 in the video how do they get -14\n\nplease help me i dont get this at all please answer me",
        "A": "I like to Sing-a! About the Moon-a and the June-a and the Spring-a! I like to Sing-a!"
    },
    {
        "video_name": "WkspBxrzuZo",
        "Q": "at 2:15 do you have to write change of x = -3 - 4",
        "A": "No, he just writes it that way to fit the slope equation, (the yellow equation in the video), because x1 is equal to -3 and x2 is equal to 4, So x1 - x2 = -3-(4)"
    },
    {
        "video_name": "WkspBxrzuZo",
        "Q": "2:26 how can the positive 4 turn to negative 4? and also i am studying for this SAT test and i would really appreciate if you could answer soon",
        "A": "the formula is y2-y1, so it did not turn to a negative 4, the formula provided the minus sign"
    },
    {
        "video_name": "WkspBxrzuZo",
        "Q": "Do you have to use the method that Mr. Sal talked about at around 2:02 ish, or can you find the points on a graph and then find the slope from there?",
        "A": "Using a graph would work. But, you should memorize the formula to calculate the slope. It will come in handy. And, it is very likely that your teacher will expect you to know it."
    },
    {
        "video_name": "WkspBxrzuZo",
        "Q": "At 2:05 in the video, I don't understand why 4 to -3 is -7.\nI thought that it would be 1, like 4-3=1?\n\nCan somebody help? :(",
        "A": "4 - 3 is certainly 1. But 4 - (-3) is 7. And if you start at 4 and wanted to arrive at -3 you would have to subtract 7. Thus the distance from 4 to -3 is -7."
    },
    {
        "video_name": "WkspBxrzuZo",
        "Q": "at 0:36 the variable m is introduced for slope, why a 'm'. When was m first used to represent slope? Is slope in a different language a word that begins with a m? Who came up with m and not an s?",
        "A": "I ve seen: y = mx + c y = mx + b y = ax + b The beauty of variables is that you can choose any symbol (Remember Sal using a star as a variable once). So if the slope hasn t been assigned a symbol yet, and you want to use s for the slope, no one can stop you."
    },
    {
        "video_name": "WkspBxrzuZo",
        "Q": "At 4:36, why did Sal say that there is a negative of -7? --7 doesn't make any sense!",
        "A": "No, not really. But in math, the negative symbol actually means the opposite of , like -2 meaning the opposite of two . With this in mind, you can place negative symbols however you wish to denote opposites. So, a negative negative number is the opposite of the negative number, so it would be the positive number. Gargamel"
    },
    {
        "video_name": "WkspBxrzuZo",
        "Q": "0:35 why m and not s",
        "A": "It stands for modulus, which could mean the gradient/slope."
    },
    {
        "video_name": "WkspBxrzuZo",
        "Q": "So at 3:31 is negative two the slope?",
        "A": "Yes, -2 is the slope"
    },
    {
        "video_name": "nLY2bzRfQyo",
        "Q": "At 1:55 i saw that the total perimeter is less then 60, it's 32. does the perimeter always have to be less then the area?",
        "A": "In general the total perimeter is less than the area. However, it is not always the case. For example, if you have \u00c3\u00a0 rectangle 7 by 1. You will have a perimeter of 16 (7+7+1+1) and an area of 7 (7x1)"
    },
    {
        "video_name": "nLY2bzRfQyo",
        "Q": "at 2:53 through 3:12 Sal said,the area of the second rectangle is 17 but how is that true?",
        "A": "its not because a traiangle has three sides."
    },
    {
        "video_name": "lfZGtjSWcQs",
        "Q": "At 3:05, why is it (-1)^(n+1)? Is that the same as (1)^(n-1)?\nThanks",
        "A": "(-1)^(n-1) does equal (-1)^(n+1) for every n value. Both equations oscillate every other N, every (-1)^ to an even number equal(-1)^ to a different even number. You can change the (n-1) to any odd number [for example (n+67) or (n-7)] and get the same answer."
    },
    {
        "video_name": "lfZGtjSWcQs",
        "Q": "At 2:39, how do you find the formula for the nth term?",
        "A": "Well, he tells you in about the next minute of the video! Are you asking for more clarification? The formula for the nth term will be the definition stated in terms of n such that it will produce the sequence that he has given."
    },
    {
        "video_name": "qVn_Lfec-Ac",
        "Q": "At ~1:30, when Sal is setting up the integral, could you not calculate the area using half the value of the cross-product of two vectors separated by theta, taking r to be a vector-valued function? And how might this look? I'm thinking something like A \u00e2\u0089\u0088 (1/2) Sum ( r1 x r2 ). Any ideas how to move this forward?",
        "A": "Remember r1xr2 = ||r1|| ||r2|| sin(theta). When theta is very small (that is, as d(theta) approaches zero), sin(theta) \u00e2\u0089\u0088 theta, and ||r1|| \u00e2\u0089\u0088 ||r2||, so you ve got dA = 1/2 r^2 d(theta), which is exactly what Sal derived above."
    },
    {
        "video_name": "2Zk6u7Uk5ow",
        "Q": "At 0:43 why did you draw to only -4 but there is a 5?",
        "A": "Yes, a number line extends forever, those numbers are just the numbers he needs in the video. Just remember there is no end to numbers, so there is no end to a number line (the arrows at the ends of the number line are supposed to show that the number line keeps going)."
    },
    {
        "video_name": "6vRgqB0PSXE",
        "Q": "At 0:48, why does Sal say APC instead of CPA?",
        "A": "As long as the vertex of the angle is in the middle, you can switch the order of the other two points. So angle APC is the same as angle CPA."
    },
    {
        "video_name": "1Vb8t7Y-pI0",
        "Q": "what if at 7:00 instead of 2x15=30 you get 5x3=15",
        "A": "It doesn t matter because if you see, after doing 2*15, Sal got 5*3 so if you get 5*3 before, all it would do is save you a step. Just remember, prime factors of a number never change so no matter what prime factors you get during prime factorization, it ll be correct. Hope this helped :)"
    },
    {
        "video_name": "1Vb8t7Y-pI0",
        "Q": "7:33 what did u mean",
        "A": "Sal is representing the pencils by the yellow circles, and the binders by the pink circles. Creating the maximum number of identical sets out of 21 binders and 30 pencils."
    },
    {
        "video_name": "1Vb8t7Y-pI0",
        "Q": "At 3:09 , you are doing the sum through prime factorization method . I think that is not the method . It is factor tree method . Why would you give wrong informations like this ?",
        "A": "No... Sal initially shows you how to find the LCM by finding the multiples of each number (the sums). He then repeats the problem using the prime factorization method. He is showing how to do the same problem 2 different ways."
    },
    {
        "video_name": "1Vb8t7Y-pI0",
        "Q": "At 6:25, could LCM and GCF be used interchangeably?",
        "A": "No. They are completely different. LCM (Lowest common multiple) is the lowest number that both numbers can go into. Ex: The LCM of 5 and 6 is 30. GCF (Greatest common factor) is the highest number that can go into both numbers. Ex: The GCF of 36 and 60 is 12."
    },
    {
        "video_name": "1Vb8t7Y-pI0",
        "Q": "at 7:22 if the gcf is 3. Than what is the lcm is it bigger than 3 or less?",
        "A": "the LCM of 25 and 30 is greater than 3 because the lcm(25,30)= 210"
    },
    {
        "video_name": "daCT_24RnIY",
        "Q": "im confude does he mean 7 cubes on 6:18 please comment",
        "A": "Do you mean the volume? Or the area of one side? If you are asking any of these, then 7 cubes doesn t match anything. The area of one side is 3*4 (since you are trying to find area, and you do know the formula for area, right?), and 3*4= 12. If you are trying to find the volume, it is 3*4*2, or 12*2=24."
    },
    {
        "video_name": "daCT_24RnIY",
        "Q": "The answer is 24 at 3:01",
        "A": "Well to be precise it s 24 centimeters cubed. You can t forget the unit."
    },
    {
        "video_name": "DzJvR56Suss",
        "Q": "At 0:57 what do those things mean (3+2)?",
        "A": "It s just a group of numbers. 5 is the same as 3+2, the ( ) just separates the numbers from the rest of the equation"
    },
    {
        "video_name": "fvtv2uYjo_E",
        "Q": "At 3:09 how does he get the numerator a ten?",
        "A": "Basically, he multiplied both the numerator and the denominator by 2, therefore, from 5/6, he got 5*2/6*2 = 10/12"
    },
    {
        "video_name": "fvtv2uYjo_E",
        "Q": "at 3:37 where did you find the number 5",
        "A": "you reduce, 2 goes into 10 5 times and it goes into 12 6 times,giving you the reduced fraction 5/6"
    },
    {
        "video_name": "fvtv2uYjo_E",
        "Q": "I have a question, at 0:38, he talks about denominator, what is a denominator?",
        "A": "A denominator is the number or equation on the bottom of a fraction."
    },
    {
        "video_name": "Q1vMNyIP4Us",
        "Q": "At 0:34, how come u don't write the answer to the problem",
        "A": "He does. We are unable to make a value out of the expression because x is unknown and we have no way to define the value of x in this expression."
    },
    {
        "video_name": "Q1vMNyIP4Us",
        "Q": "at 1:28 -1x doesnt equalx",
        "A": "The negative sign in -1 equals -x"
    },
    {
        "video_name": "RX0DY9eRp8g",
        "Q": "7:59 what does \"shifting\" in this case mean?",
        "A": "It means moving the entire graph horizontally to right or left."
    },
    {
        "video_name": "RX0DY9eRp8g",
        "Q": "At 5:17 Sal says that you could model it with either but what would the cosine version look like?",
        "A": "Here are the 4 equations I got (all from the original graph with t=0 at 10am (That s all you need.) 7.5sin((pi/12)(t+14))+10.5; (sin(t*pi/12 + 7pi/6)).(t=0 (midline) for sine is 10am.) 7.5sin((pi/12)(t-10))+10.5; (sin(t*pi/12 - 5pi/6)). 7.5cos((pi/12)(u+8))+10.5; (cos(u*pi/12 + 2pi/3)). (u=0 (max) for cosine is 4pm.) 7.5cos((pi/12)(u-16))+10.5; (cos(u*pi/12 - 4pi/3)). (If you extend the graph out another 24 hrs and relabel the x axis with 4pm = 0, you get the cosine graph and its proper 0 point)."
    },
    {
        "video_name": "RX0DY9eRp8g",
        "Q": "At 9:23 , When Sal substitutes -10 isn't the time then going to be 8 p.m ?\nThe question asks for temp after midnight which is after 12 a.m, shouldn't it be -14 then? if he is going to shift it to the right at 12 a.m.\nI don't understand can someone please explain?",
        "A": "But by substituting -10 that means he s moving right 10 hours. From 10 a.m. or 10 p.m., when you go right by 10, you end up at 8."
    },
    {
        "video_name": "T3nVKbxEQdo",
        "Q": "At 2:33, I don't understand how did you get the result of (t+8)(t-3).",
        "A": "He just factorized it. t^2+5t-24 =t^2+8t-3t-24 = t(t+8)-3(t+8) = (t+8)(t-3)...."
    },
    {
        "video_name": "9UhChwcVWq8",
        "Q": "2:08 While Sal wasn't wrong in part a per se, he made the problem much more difficult than he had to. Nowhere in the problem does it say the answer must be in slope-intercept form. He could have simply used point-slope form.",
        "A": "thats true, but it s hardly much more difficult"
    },
    {
        "video_name": "9UhChwcVWq8",
        "Q": "@3:46 Why are we evaluating/integrating from 0 to 1/2 instead of 0 to 1?",
        "A": "Because we are integrating with respect to x, and the x values in consideration range from 0 to 1/2. We could change the integration to be with respect to y, in which case you would integrate from 0 to 1."
    },
    {
        "video_name": "XAzFGx3Ruig",
        "Q": "why is the multiplication sign a dot at 2:45?",
        "A": "A dot is just another way to write the multiplication sign. In pre-algebra and algebra, the variable x is used a lot. People may confuse the x for a variable or a multiplication sign, so the dot is a simpler, easily recognized way that is used in higher levels of math. Another way that you may see the multiplication sign is as an asterisk (*). Here s an example of three ways to write a multiplication sign: 5\u00c3\u00977=35 5\u00c2\u00b77=35 5*7=35 Hope this helps!"
    },
    {
        "video_name": "XAzFGx3Ruig",
        "Q": "At 1:45, isn't he doing scientific notation or something like that?",
        "A": "No, he is not doing scientific notation in this video. Instead, he is just breaking up the number 2943 into its digits to help clarify the 9 Divisibility Rule. Scientific notation would be saying that the number 2.943*10^3"
    },
    {
        "video_name": "XAzFGx3Ruig",
        "Q": "At 2:27 9x9x9 does not make sense\n9 times 9 times 99.......\nhow could nine divide into 99?",
        "A": "9 times 11 equals 99, so nine is divisible 99. i hope i answered the question you were asking, if not just comment please."
    },
    {
        "video_name": "f3ySpxX9oeM",
        "Q": "Why does Sal turn the fraction upside down and change the operation at about 2:00",
        "A": "because he s using the reciprocal"
    },
    {
        "video_name": "f3ySpxX9oeM",
        "Q": "In 4:07 he mentions something called a reciprocal. What is a reciprocal? And how does it relate to the problem?",
        "A": "M is totally right. I d just like to add that you can also have a reciprocal of a whole number (not a fraction) like 4. Because you could also write 4 as 4/1, its reciprocal would be 1/4."
    },
    {
        "video_name": "f3ySpxX9oeM",
        "Q": "At 1:33 it says that it could be 8, but wouldn't it be a fraction, not a whole number?",
        "A": "Every whole number can be represented by a fraction, 8 = 8/1 or 16/2 or 24/3 and so on. So, essentially the result can be a fraction or a whole number. But not be confused, not all fractions can be represented as a whole number, for example 1/3 can t be represented like a whole number because it s not a whole number. I hope this helps."
    },
    {
        "video_name": "f3ySpxX9oeM",
        "Q": "By the way, what is a reciprocal? It is at 4:04.",
        "A": "I can t explain a reciprocal in words but I can with numbers. Example:the reciprocal of 2 is 1/2. The reciprocal of 8 is 1/8. So it is like multiplying a number to power of negative 1. So the reciprocal of 2/3 is 3/2 as (2/3)^-1=3/2. Hope this helps :)"
    },
    {
        "video_name": "f3ySpxX9oeM",
        "Q": "I'm still a little confused. At 1:39 when he says, \"So we could view 8/3 divided by 1/3 as being equal to 8\", I got a little confused, because 8/3 as a mixed number is 2 2/3, and 2 2/3 divided by 1/3 is 0.88888888888. That is less than 1, so it definitely is not 8. Anyone got an explanation for this?? I know he tries to explain it, but i'm still confused... \u00f0\u009f\u0098\u0095 \u00f0\u009f\u0098\u0095 \u00f0\u009f\u0098\u0095 Also, at 2:28, he says 8*3 equals 8, and he puts 8 as the numerator, when it's really 24. I'm SO confused!",
        "A": "To get to an answer of 0.888888..., you had to have multiplied the 2 numbers. 8/3 * 1/3 = 8/9 = 0.88888.... You didn t divide which is why you got a different answer."
    },
    {
        "video_name": "f3ySpxX9oeM",
        "Q": "At 2:16, Sal begins to write about multiplying the fractions together. It is like multiplying the reciprocal of 1/3...?\nIf I am not right, could you explain why. Many thanks",
        "A": "That is correct. To divide fraction, you change the division to multiplication by using the reciprocal of the next fraction."
    },
    {
        "video_name": "g8yk6zQDAek",
        "Q": "Hey! At 3:23 the answer should be 3 r 5, not 5 r 3?",
        "A": "Either way works. For example, the remainder is the same as .83333..... .83333 + 5 is still = to 5 + .833333. Therefor, it doesn t matter what way you say it, kind of like communicative property."
    },
    {
        "video_name": "g8yk6zQDAek",
        "Q": "Sal made a mistake! At 3:13 the answer is 3 r5 ! He probobly (sorry , mistake) didn't change it because I saw a question by Anonymous THREE YEARS AGO ! And I'm in 2016!",
        "A": "No he did t I think you made a mistake in your calculation"
    },
    {
        "video_name": "olrXr3J5xq4",
        "Q": "at 1:50, why does he say that in different contexts, we call it different things?",
        "A": "Because some people, in their work, find it more useful to say the limit goes to infinity , while others are fine with leaving it at the limit does not exist ."
    },
    {
        "video_name": "HwSszh3L358",
        "Q": "At 1:20 what do you mean",
        "A": "i guess it means no matter in what order the problem it is, the answer will be the same. i hope this helps"
    },
    {
        "video_name": "HwSszh3L358",
        "Q": "at 0:40 when will this kind of problem be delt with?",
        "A": "well on tests and sometimes in real life but you do need to know how to deal w/ it anyways(you-know for tests and SATs and MAPs and SOLs)"
    },
    {
        "video_name": "wPZIa3SjPF0",
        "Q": "At \"1:17\" he says that not every parallelogram is a rhombus but every rhombus is a parallelogram.\nI don't get that, and I don't know how to define it.\n\nCan anyone explain it for me?",
        "A": "rhombus has to have 4 equal sides and parallelogram has 4 sides, but only 2 pairs are equal just like a square and rectangle"
    },
    {
        "video_name": "wPZIa3SjPF0",
        "Q": "From 1:15 to 1:25, I am confused because Sal says, \"not every parallelogram is a rhombus, but, every rhombus is a parallelogram.\" What does Sal mean by this?",
        "A": "so what he means is that rhombuses are parallelograms but a different shape parallelograms are not rhombuses"
    },
    {
        "video_name": "wPZIa3SjPF0",
        "Q": "How does a trapezoid have parallel lines?\nDoesn't the line have different lengths at 2:09.",
        "A": "Lines do not need to be the same length to be parallel. Parallel lines are lines that have the same slope. The top and bottom sides of the trapezoid at 2:09 are parallel. The other 2 sides are not parallel (the left side would have a negative slope and the right side looks like it has a very steep positive slope. So, the slopes can not be the same). Hope this helps."
    },
    {
        "video_name": "wPZIa3SjPF0",
        "Q": "What does he mean at 3:15",
        "A": "There isn t a 3:15 bro, sorry. Try another timestamp, I ll be happy to answer any questions you have!"
    },
    {
        "video_name": "wPZIa3SjPF0",
        "Q": "At 4:09,why is it good to know math",
        "A": "is every square a rectangle ?"
    },
    {
        "video_name": "wPZIa3SjPF0",
        "Q": "At 0:16, what is meant by \"congruent side\"?",
        "A": "Congruent sides are sides of equal length. For example, an isosceles triangle is thus named for it s two sides being of the same length (also for the corresponding congruent angles connected the line between the congruent sides)"
    },
    {
        "video_name": "wPZIa3SjPF0",
        "Q": "How do you know at 1:42 that the quadrilateral has one pair of parallel sides? It doesn't tell you they're parallel.",
        "A": "At 1:42 the two sides Sal said were parallel were parallel, meaning they could go on forever in the same direction without ever crossing. To find out if a shape has parallel sides, look at the angles. If you add the angles opposite each other at the sides endpoint s, they should equal 180 if it is parallel. If they don t, they are not parallel to each other."
    },
    {
        "video_name": "6FrPLJY0rqM",
        "Q": "shouldnt be -4 in 10:50????",
        "A": "-2C = -8 /-2 /-2 C = 4 If you re dividing by negative number, it s the same as multiplying by it s inverse: so -8 / -2 is the same thing as -8 * -1/2 and you know negative * negative = positive :)."
    },
    {
        "video_name": "6FrPLJY0rqM",
        "Q": "At 2:20, He says that jf we make x equal to 2 then thats a factor, but if x is 2 doesn't that make something/0 and is devision by 0 forbidden?",
        "A": "dividing by zero is not forbidden, but it is undefined."
    },
    {
        "video_name": "6FrPLJY0rqM",
        "Q": "I must be missing something about the \"long way\". At 8:30 I see that the left side has a 10x^2 while the right side has only Ax^2. I then figured that 10=A. Why is this incorrect? I re-watched Partial Fraction Expansion 1 but it only did the fast way.",
        "A": "I see now that the last part of the problem is (Bx+C)(x-2), thus we do have a Bx^2 as well. So it is 10=A+B for the x^2 portion."
    },
    {
        "video_name": "6FrPLJY0rqM",
        "Q": "When we are setting our equations equal to each other (at 5:57), what do we do if part of our factored denominator has a degree greater than 2?",
        "A": "in this situation we can decompose further, because x^3 can be negative and equation can be 0. But i have now a question, what we do if in denominator we have x^4+5x^2+10 it can t be 0, but what we will have in numirator"
    },
    {
        "video_name": "6FrPLJY0rqM",
        "Q": "5:50 why is it Bx + C ? why cant it be just B ?",
        "A": "Because you need the degree of the numerator to be one less than the denominator, so if the denominator is x^2, the numerator is x."
    },
    {
        "video_name": "6FrPLJY0rqM",
        "Q": "At 1:30, he says that 2 is a zero, or 2 makes the expression 0. What does that mean?",
        "A": "If you want to factor a polynomial, such as x^3 - 8 , one way is to ask the question, What value of x will make the polynomial equal to zero? If you find an x-value that satisfies this, then you know that x minus that number is also zero and hence is a factor of that polynomial. Sal then goes on and uses polynomial long division to find the other factor. These two factors are then used as the denominators of the partial fractions."
    },
    {
        "video_name": "6FrPLJY0rqM",
        "Q": "I don't understand how he gets the Bx+C at 5:50.. even after listening to the explanation several times. Can anybody explain that to me please?",
        "A": "What if equation did not have (x-2) in denominator? How would we decompose non-fractionable denominator then?"
    },
    {
        "video_name": "6FrPLJY0rqM",
        "Q": "At 8:50, Sal \"gets rid of\" the (Bx+C)(x-2) term by making x equal 2. But the original fraction is not defined when x is 2. Are we allowed to substitute values of x for which the original fraction is undefined?",
        "A": "That s great that you were able to answer your own question!"
    },
    {
        "video_name": "6FrPLJY0rqM",
        "Q": "At 2:15 Sal says that since 2 sets x^3-8 to 0, therefore (x-2) must be a factor. Does this mean that every time you can set an expression to equal 0 and solve for x, then (x-[the solution]) must be a factor?",
        "A": "Kind of... If the leading coefficient is one, then this is true. If not, then you could end up with a zero of... say 1/2, but the factor would be (2x-1)"
    },
    {
        "video_name": "6FrPLJY0rqM",
        "Q": "At 3:38, Sal puts a negative sign in front of the x cubed in the long division. Why does the negative 2x squared become positive?",
        "A": "Sal is subtracting the binomial. Any time you subtract a polynomial, the minus sign must be distributed: -(x^3 - 2x^2) = -x^3 + 2x^2 Distributing the minus causes all the signs to change."
    },
    {
        "video_name": "Yhlv5Aeuo_k",
        "Q": "To 2:14 to 2:30, what was that a picture of and what does \"Stanislaw Ulam\" mean?",
        "A": "That was a picture of the Ulam Spiral, which is where you start the natural numbers spiraling out, and then shade over the primes. Stanislaw Ulam is the name of the man who came up with the Ulam Spiral."
    },
    {
        "video_name": "Yhlv5Aeuo_k",
        "Q": "at 3:25, what happens if you do that but with even and not odd?",
        "A": "Try it and find out! It s probably really cool and they ll name it the (enter someones name here) s triangle ."
    },
    {
        "video_name": "Yhlv5Aeuo_k",
        "Q": "Anyone knows if this \"spiral\" number square (0:15) has a special name, so I can find out more about it? Thanks!\nGreat video BTW!",
        "A": "it s a pattern to get an Ulam Spiral , if you make it big, and zoom out, that s what it is. It looks a bit like a fancy QR code."
    },
    {
        "video_name": "Yhlv5Aeuo_k",
        "Q": "At 2:41, what is the image called again?",
        "A": "The Ulam Spiral. Black dots represent the prime numbers. The image was courtesy of Wikipedia.org."
    },
    {
        "video_name": "Yhlv5Aeuo_k",
        "Q": "at 4:41, the \"rules\" thing looks like that special triangle! check it out! did she do that on purpose?",
        "A": "Vi said it was Sierpinski s Triangle, didn t you hear? :)"
    },
    {
        "video_name": "Yhlv5Aeuo_k",
        "Q": "at 5:20, Vi mentioned Floyd's triangle. Exactly how does that work?",
        "A": "i think it works the same as pascals. just drawn differently"
    },
    {
        "video_name": "Yhlv5Aeuo_k",
        "Q": "At 5:00, What happens when you highlight all the prime numbers on a Pascal's Triangle?",
        "A": "Also, the prime numbers will only occur on the edges. Not the edge that s all 1 s, but the next one in-the one that just counts by 1. This isn t super intuitive, but kind of makes sense if you understand Pascal s triangle as being a list of combinations."
    },
    {
        "video_name": "Yhlv5Aeuo_k",
        "Q": "What doe the alien mean when he responds at 2:10 \"I dunno. Whats ur Base?\"",
        "A": "There are different bases with which you can add things together. We use base 10 most of the time probably because we have 10 fingers or for some other reason I don t know. Probably one of the other most well-known bases is base 2, or binary. Lewis Carroll uses base 18 in Alice in Wonderland, but I can t think of that many other real-life examples. It s pretty insane!"
    },
    {
        "video_name": "Yhlv5Aeuo_k",
        "Q": "At 0:07 is Stanislaw Ulam a real person?",
        "A": "Yes. Ulam was one of the core mathematicians in the Manhattan Project that designed the first nuclear weapon, and he had a number of keep insights into the ways in which computers could be used to attack previously unsolvable problems."
    },
    {
        "video_name": "Yhlv5Aeuo_k",
        "Q": "The triangle at 3:28 is a triforce",
        "A": "which triforce? courage, power, or wisdom?"
    },
    {
        "video_name": "Yhlv5Aeuo_k",
        "Q": "At 2:40, is it just nonsense that she wrote \"nice ribs\"?",
        "A": "When she was making the spiral of numbers, she put lines through the primes, then mentioned they resembled bones and called them Prime Ribs. So saying Nice Ribs was i think intended to make people laugh."
    },
    {
        "video_name": "SRn3WhhS6vs",
        "Q": "at 4:20, you make the equation by which you solve the problem. How do you make it with what you have and how do you know how to make it.",
        "A": "By knowing 14m-2w=40 and w=7m-8, he converted the 2w into 2(7m-8)=14m-16, then cancels out the two 14ms, therefore solving the equation as false."
    },
    {
        "video_name": "3SO1BQQ9_1E",
        "Q": "At around 2:10 in this video, Sal draws a vertical line down the graph. In school, I learned that it's called the vertical line test (VLT). Is this right? If not, is there a specific name for this?",
        "A": "You can distinguish if something is a function or a relation with the Vertical line Test. A vertical line has a consistent x value but ever-changing y values, making it a relation rather than a function. Just to be clear, a vertical line has an undefined slope and a horizontal line has a slope of zero. I hope this was helpful."
    },
    {
        "video_name": "d8vvVjfTbYY",
        "Q": "At 2:24 you were multiplying and I got confused because you were just dividing those fraction. So my question is am I supposed to divide or multiply the fractions?",
        "A": "Division by a fraction is multiplication by the reciprocal."
    },
    {
        "video_name": "d8vvVjfTbYY",
        "Q": "I dont understand 1:45",
        "A": "Slope is rise/run. Rise was (4/7) , Run was (5/7) Therefore the slope is (4/7)/(5/7) and then to simplify division by a fraction you would need to multiply by the reciprocal (4/7)*(7/5). The 7(s) cancel out and you re left with (4/5). Hope that helps."
    },
    {
        "video_name": "d8vvVjfTbYY",
        "Q": "At 0:20, isn't the plural of octopus octopuses?",
        "A": "Yes, but it is also octopi or octopodes."
    },
    {
        "video_name": "d8vvVjfTbYY",
        "Q": "what does reciprocal mean?\n(time - 2:26 - 2:29)",
        "A": "A reciprocal is like flipping over a number. If the number is a whole number, you divide it by one. For example, the reciprocal of 5 is 1/5. If the number is a fraction, the numerator and denominator switch places. For example, the reciprocal of 3/7 is 7/3."
    },
    {
        "video_name": "y2G03Lumhe0",
        "Q": "at 0:08, you said \"size\". Did you mean to?",
        "A": "At 0:08, Sal said 6 free throws . The caption is incorrect it says size ."
    },
    {
        "video_name": "7IkufOBIw5g",
        "Q": "At 6:39, why is A equal to 1 minus B?",
        "A": "Short answer: because A + B = 1. Why does A + B = 1? We know that (x-4)/(x+1)(x-1) = [(A+B)x + (B-A)]/(x+1)(x-1) for all x. So we must have (A+B)x + (B-A) = x-4 for all x, and that means A+B = 1, B-A = 4."
    },
    {
        "video_name": "7IkufOBIw5g",
        "Q": "5:45 why must A+B always be equal to that coefficient and B-A always equal to the constant? I see that for any example I do it's true, but I don't understand why.",
        "A": "Sorry it s late, just got here. However, remember x+y=x+y, therefore 1x=(A+B)x and -4=(B-A); since your x terms are specified in the equation, hence A+B=1."
    },
    {
        "video_name": "7IkufOBIw5g",
        "Q": "For the second part of the new expression of the integral we get at 7:50, I distributed 5/2 into 1/(x+1) so that I got 5/(2(x+1))=5/(2x+2). Then I took the integral:\n\u00e2\u0088\u00ab5/(2x+2)dx = (5/2)\u00e2\u0088\u00ab2/(2x+2)dx. With u-substitution, I get (5/2)ln(2x+2)+C instead of Sal's (5/2)ln(x+1)+C. Where did I go wrong?",
        "A": "You re not mathematically wrong, though you could argue the (2x+2) bit is not in its simplest form. ln(2x+2) = ln(2(x+1)), which by the law of logarithms = ln(2) + ln(x+1) And, since ln(2) is a constant it can be rolled up with C. After multiplying by 5/2, obviously. In other words the results are the same."
    },
    {
        "video_name": "7IkufOBIw5g",
        "Q": "Did anyone else hear the ringtone at 4:48 :)",
        "A": "YeahhXD and I replayed the video just to confirm that I wasn t dreaming it up XD"
    },
    {
        "video_name": "AiW7syKXfJM",
        "Q": "At 1:58 I don't get it why does he want to include 0, as 0 candy bars can't be bought, sounds illogical. Buying 0 candy bars is equivalent to buying none, for which the function isn't defined. The Question says \"Purchased\" unless and until it is 1 it can't be purchased.",
        "A": "You are correct. If the definition of making a purchase is the exchange of good(s) for money then both of these things have to happen in order for a purchase to occur. And that is exactly what purchased no candy bars or purchased 0 candy bars means. The 0 and no candy bars negates the act of purchasing, stating a purchase hasn t occured. This has to be included because it is a possible state."
    },
    {
        "video_name": "mvye6X_0upA",
        "Q": "at 2:47 What do we do when it is less than 30?",
        "A": "Use a t table."
    },
    {
        "video_name": "TD1zuENbEdk",
        "Q": "AT 12:00 at night is the sun like somewhere else",
        "A": "It is on the other side of the earth."
    },
    {
        "video_name": "5jwXThH6fg4",
        "Q": "At 1:26, shouldn't the i to the right of the sigma be i+1?",
        "A": "No, it has to be i. We re adding the numbers from 1 to 10. The sigma notation says we re going to add up the results of applying the rule to the right of the symbol to each of the i s from 1 to ten. In this case we re applying a rule that does nothing, just gives back i. So the first result in the addition is 1, then 2 and so on up to 10. If we had i+1 to the right of the symbol, the first result in the addition would be 2, and we would end up adding the numbers from 2 to 11."
    },
    {
        "video_name": "5jwXThH6fg4",
        "Q": "At 3:44 I don't understand how this example is working. Sal seems to be saying that the index is equal to 0 and so shouldn't each number increase by 0? Because when i=1, each number increased by 1, shouldn't that same rule be applied here? Or does 'i' just mean that is where the index starts, at 0 or 1 to use Sal's example.",
        "A": "the i=0 or i=1 or i=127 under the sigma notation is just the beginning of the index. The index is always increasing by 1 to the n number written above the sigma."
    },
    {
        "video_name": "5jwXThH6fg4",
        "Q": "At 2:18, what are all of the i's?",
        "A": "The letter i is a common choice for the indexing variable. Other common choices include j , k and n , but you could use any symbol you like."
    },
    {
        "video_name": "5jwXThH6fg4",
        "Q": "At 1:36, when you say \"Increment the index\", how do you know what value to increment the index with?",
        "A": "By definition in Sigma notation, the index is incremented by 1 each time, always."
    },
    {
        "video_name": "5jwXThH6fg4",
        "Q": "At 1:13, when Sal says lets start at the i=1, would that be the same as n=1?",
        "A": "Yes, i is only used as a variable, it can be any letter."
    },
    {
        "video_name": "5jwXThH6fg4",
        "Q": "At 2:30, the index is i=0. What I remember learning is that the number on top of the Sigma (10 in the first example, 100 in the second, and 50 in the third) represents the number of terms in the sum, so in the third example if i=0, it should only go to 49, not 50. To make it go to 50, the number at the top should be 51, so then there will be 51 terms (0+1+2+3...49+50). If I got this wrong, please explain how the index equaling to 1 is different from it equaling to 0.",
        "A": "The number on top of the sigma is NOT the number of terms in the sum. It is the upper limit. And the lower number (the i=0) is the lower limit. So 5 \u00ce\u00a3 i = 0 + 1 + 2 + 3 + 4 + 5 i=0 (six terms) 6 \u00ce\u00a3 i = 3 + 4 + 5 + 6 i=3 (four terms). I don t know if you do any computer programming, but you can think of i as working like the variable in a for-next loop"
    },
    {
        "video_name": "5jwXThH6fg4",
        "Q": "At 1:43 Sal says increment the index. Are you incrementing the index to the value of the index? Like if i=1/2 would you increment by one half?",
        "A": "The index must be a whole number, so you cannot increment by one half."
    },
    {
        "video_name": "Bq9cq9FZuNM",
        "Q": "At about 2:50, it says that x is going to be negative. But if x is less than or equal to one, it could then just simply be 1. So how did Sal know is would be negative?\n\nThanks.",
        "A": "Thanks so much!"
    },
    {
        "video_name": "Bq9cq9FZuNM",
        "Q": "how come you graph half of a parabola, and not the other half? at 0:15 or so",
        "A": "Because the function has an important condition for x <= 1 . With that condition, only the x values less than or equal to 1 show up in the graph."
    },
    {
        "video_name": "Bq9cq9FZuNM",
        "Q": "I did not understand how on 6:14 Sal starts to graph the equation.\nI generally do not understand how to graph points from an equation given.\nIt would help if someone guided me here.",
        "A": "You would just want to plug in numbers for x that would work out to be whole numbers for Y."
    },
    {
        "video_name": "Bq9cq9FZuNM",
        "Q": "at 2:57 he says you have to use the negative square root thingy. How do you know when to use the negative square root vs. the normal square root? Would you use the negative root if the value was positive? Do you always use it?",
        "A": "If the value that is getting squared is negative, you take the negative square root. Otherwise, you take the positive one. This is done so we get the inverse of the function and not some other function altogether."
    },
    {
        "video_name": "Bq9cq9FZuNM",
        "Q": "around 2:28\nx <= 1 right?\nOn the third line, it's y+2 = square root of(x-1)^2\n\nso what happens if x=1 ?\nwhy is the expression x-1 negative then?\n\nyou can't take the negative square root coz (x-1)= (1-1) = 0\n?",
        "A": "Find f-1 (x) f(x) = \u00e2\u0088\u009b(x-8)"
    },
    {
        "video_name": "Bq9cq9FZuNM",
        "Q": "Sal please explain in a more clear way how the negative square root of (x-1)^2 = x-1 . Shouldn't the x-1 come from under the radical and then the negative gets applied to the x becoming -x-1?? Sal, please give a clearer explanation! This is done at 2:25 in the video.",
        "A": "yes, but the way it is done is to put some number s in for X -> if X=-2, -\u00e2\u0088\u009a(-2-1)\u00c2\u00b2= -\u00e2\u0088\u009a(-3)\u00c2\u00b2= -\u00e2\u0088\u009a9=-3 which is the same as -\u00e2\u0088\u009a(-2-1)\u00c2\u00b2 but without the negative square root and the squaredf sign which is the same as X-1 if you put back x for 3."
    },
    {
        "video_name": "Bq9cq9FZuNM",
        "Q": "at 5:02 , why doesn't Sal includes x <= 1 along with y=> -2 in the functions domain? where did x<= 1 go? :\\",
        "A": "thanks, but i do understand why domain restrections exist in generally... my question was, however, why does at the begining of the exercise there were two restrections : x<=1 and y => -2, but further along - at 5:02, for some reason x <= 1 got dropped from the restrection list, leaving us with only y => -2... is it a simple mishappen, or the x restrection was cancelled for a reason?"
    },
    {
        "video_name": "Bq9cq9FZuNM",
        "Q": "at 2:45 it says it's negative, what about if x is 1?",
        "A": "by negative he actually means non positive which also includes zero. Hope this helps!!!"
    },
    {
        "video_name": "wtLTb_VaI-k",
        "Q": "@4:26, and other places in the video, Sal refers to the unit vectors as \"i\" and \"j\". Shouldn't he be saying \"i hat\" and \"j hat\"?",
        "A": "Sure, he should do that to insure clarity. Note that he did state that he that he was using engineering notation. @1:00 he even calls out the \u00c3\u00ae as the i unit vector and continues when writing \u00c4\u00b5 , the j unit vector."
    },
    {
        "video_name": "8r8Vp_1iB4k",
        "Q": "5:43 Here, the secant line didn't fit well with the curve. In this case, how can we say that this gives the average rate of change between those two points (x, and x2) in the curve? Is it just an approximation?",
        "A": "It does give the average rate of change between the given points, and it gives it precisely. When it doesn t fit the curve well, it means that simply looking at the average over the interval doesn t give a good sense of what s going on in the interval. If we want a better picture, we need other techniques."
    },
    {
        "video_name": "8r8Vp_1iB4k",
        "Q": "At 6:36, what is the difference between a secant line and a tangent line?",
        "A": "A secant line intersects a curve at two points; a tangent line only meets a curve at a single point, and its slope at that point can be thought of as equal to the slope or instantaneous rate of change of the curve at that point."
    },
    {
        "video_name": "8r8Vp_1iB4k",
        "Q": "At 4:46, why does sal uses X1 and X2 instead of X0 and X1, is there a change in concept after switching to a curve line?",
        "A": "No, it does not matter. You can use any subscript you like."
    },
    {
        "video_name": "4BNIGTHUTTM",
        "Q": "at 2:59 it's obvious that 324 - 100 is gonna be 224",
        "A": "It is not that hard ,it is so easy. Do you think it is hard?"
    },
    {
        "video_name": "DMyhUb1pZT0",
        "Q": "At 1:31, how can we know if that is a -4 or a minus 4. An how is the product a positive 4? Thanks",
        "A": "At 1:24 Sal explains that there is a given number in between the subtraction sign and the parenthesis. In fact at 1:27 he uses a pink font to draw the 1 into the expression. It is understood that there is an invisible 1 when interpreting negative operations. It is with the use of this understood 1 that we evaluate the expression for simplification. When distributed, -1*-4 equals positive 4."
    },
    {
        "video_name": "DMyhUb1pZT0",
        "Q": "why at 1:25 did you add a negative one? and can you do that with all negative or subtraction sings in that order you solved them in?",
        "A": "Imagine there is a 1 next to all parentheses that have no number as with the middle section- +(-2x^2+X-2) he just drops it down as in that parentheses its all times by +1 which is the same numbers, but in the case of the last section its -1 so you times each one by -1. I really hope this helps :)"
    },
    {
        "video_name": "DMyhUb1pZT0",
        "Q": "at 1:27 there is a negative 1 in front of the back parenthesis why aint there a negative one in front of the middle parenthesis",
        "A": "The person in the tutorial had - (3x-4) so he put a -1 to show that we must distribute to get -3x+4 there is no - in front of ( -2x^2+x-2) because there Is no negative in front of this equation"
    },
    {
        "video_name": "DMyhUb1pZT0",
        "Q": "I meant at 0:12 it said divided by 10 when it means times ten.",
        "A": "Yoo who this cuz all yaall stink ."
    },
    {
        "video_name": "DMyhUb1pZT0",
        "Q": "2:12 I thought adding two negative numbers equaled a positive? Why is -6 + -2 = -8? Shouldn't it be positive 8?",
        "A": "Multiplying two negative numbers gives a positive. But say, for example, you owe me $6. So you have -$6. I buy you something for $2. So you now owe me an extra $2. Your debt is now $8, so you have -$8."
    },
    {
        "video_name": "DMyhUb1pZT0",
        "Q": "At 2:01 Sal cancels out the 3x and -3x and leaves it with just x\nBut doesn't the whole term cancel out so the final answer would be x^3 - 2x^2 - 4?",
        "A": "There are three terms containing x 3x + x - 3x Yes, 3x and -3x cancel out to 0, but that still leaves the x term Hope this helps."
    },
    {
        "video_name": "gkifo46--JA",
        "Q": "At 2:05, isn't the triangle right below it slanted a bit? If it is, this would make the whole problem wrong.",
        "A": "we don t care about the right angle right now, we cut that angle in half to make the bottom triangle s angle"
    },
    {
        "video_name": "vcwvzUVLPw0",
        "Q": "At the end (11:30 and beyond) shouldn't the velocity vector for r2 be the EXACT SAME as the velocity vector in R1... since 2(1/2)i + 4(1/2)j = 1+2j?",
        "A": "Michel - Don t forget that the time t=1/2 is being used in r2 (t). r2 (t) is light blue in the upper right of the screen at 11:30. r2 (t) = 2i + 8tj substituting 1/2 for t gives: r2 (1/2) = 2i + 8(1/2)j ---> r2 (1/2) = 2i + 4j"
    },
    {
        "video_name": "vcwvzUVLPw0",
        "Q": "at 6:26 , Sal says that the path is the same. Yes, the paths appears to look the same, but how can they truly be the same if the graphs use different scalings (in x axis). If the graph on the right were scaled the same way as the graph on the left (each \"notch\" on the x and y axis represents 1 unit), the graph on the right would be much steeper and thus the \"path\" would -not- be the same.",
        "A": "He is not using a different scalar multiple. Think of the vector r as (x, (y)^2). Now what he did was multiply r by 2 (2*r) which would multiply x and y in the same fashion (2*x, (2*y)^2), However (2*y)^2 = 4*y^2. Remember order of operation is parenthesis then exponents."
    },
    {
        "video_name": "cfOcOgr0E7U",
        "Q": "At 1:14, what's the difference between f(x) as x->a, and f(a)?",
        "A": "One is a limit, the other is an evaluation of the function. If the function is continuous and defined at (in your example), a, then they re equivalent. But you can get some very interesting results if the function is not continuous or not defined. The limit is basically saying what the function seems to be going to as x gets closer to closer to a, but the function may not be defined at that point."
    },
    {
        "video_name": "S-dcMvJlMJs",
        "Q": "Around 5:26 the answer to the second example is that g'(4) > g'(6). I guess it makes sense as g'(4) is less negative than g'(6)... but I find this confusing because the instantaneous rate of change at g'(6) is greater than at g'(4), and the example would make it seem like those instantaneous rates of change are what's being compared. Am I off base here?",
        "A": "it s kind of similar to comparing velocity and speed(absolute value of velocity). g (6) has more speed than g (4), but g (4) and g (6) are both negative, so g (4) has a higher velocity than g (6) because it s less negative. i hope i haven t confused you."
    },
    {
        "video_name": "S-dcMvJlMJs",
        "Q": "At 5:23 we know slope in terms of Tan-Tetha, so g'(4) would be negative as it thetha goes to second quadrant, tan-thetha is negative in second quadrant while g'(6) would be positive as it goes to first quadrant. why g'(4)>g'(6)?\nit should be g'(4)<g'(6)...",
        "A": "What is your reasoning for g (6) to be in the QI? If we are to construct the equation for that graph it would be g(x) = 5sin(x/2). The graph has a period of 4\u00cf\u0080. If you divide that into 4, each quadrant is one \u00cf\u0080 (we ll use 3.14 approximation). We can see that the QI is from 0 to 3.14, QII is from 3.14 to 6.28. Both 4 and 6 are in (3.14, 6.28) which mean both are in QII. g(x) = 5sin(x/2) g (x) = 2.5cos(x/2) g (4) = 2.5 cos(4/2) \u00e2\u0089\u0088 -1 g (6) = 2.5 cos(6/2) \u00e2\u0089\u0088 -2.5"
    },
    {
        "video_name": "S-dcMvJlMJs",
        "Q": "So is it okay to draw any arbitrary tangent line at 3:40? I'm confused.",
        "A": "There s only one tangent line that can be drawn at that point. The slope is defined, since it s the same as that of the graph, and a point is defined, since we re drawing it at that point on the function. So the line is well-defined, since it has a fixed point and slope."
    },
    {
        "video_name": "S-dcMvJlMJs",
        "Q": "can i know where is the increasing function in the graph 2 at 5:03",
        "A": "By taking the derivative, you can know where the graph is increasing or decreasing. If the derivative is < 0 then the graph is decreasing. If the derivative is > 0 then the graph is increasing. Where the derivative = 0, the graph is transitioning from increasing to decreasing or vise versa, which means there is a local maximum (increasing to decreasing) or a local minimum (decreasing to increasing)."
    },
    {
        "video_name": "YJgIGwTysk0",
        "Q": "at 5:29, Mrs. Thompson is buying 1 3/4 pounds of turkey and 3/4 as much cheese as turkey at a deli, is she buying less or more turkey than cheese?",
        "A": "Turn 1 3/4 into an improper fraction by multiplying 1 by the denominator (4) then add the nunerator (3) which equals 7. Then put 7 over the original denominator giving you 7/4. 7/4 is bigger than 3/4 so more turkey"
    },
    {
        "video_name": "YJgIGwTysk0",
        "Q": "at 0:10, sal says \"How Much Oats should you use?\" isn't \"How much oats\" bad grammar?",
        "A": "No. It s correct grammer."
    },
    {
        "video_name": "YJgIGwTysk0",
        "Q": "At 0:27, what does the interpunct ( \u00c2\u00b7 ) mean?",
        "A": "It means to multiply"
    },
    {
        "video_name": "YJgIGwTysk0",
        "Q": "at 5:29, Mrs. Thompson is buying 1 3/4 pounds of turkey and 3/4 as much cheese as turkey at a deli, is she buying less or more turkey than cheese?",
        "A": "less because 3/4 is less than 1"
    },
    {
        "video_name": "YJgIGwTysk0",
        "Q": "at 5:29, Mrs. Thompson is buying 1 3/4 pounds of turkey and 3/4 as much cheese as turkey at a deli, is she buying less or more turkey than cheese?",
        "A": "less because the amount of turkey is 3/4 of the amount of cheese"
    },
    {
        "video_name": "iEpqcdaJNTQ",
        "Q": "At 8:39 We have this expression:\n\nd\u00ce\u00a8/dx = d\u00ce\u00a8/dx + d\u00ce\u00a8/\u00ce\u00b4y * dy/dx\nA = B + C * D\n\nSo the A = B then this wouldn't be infinitely recursive?",
        "A": "You transcribed it wrong, using d (lower-case D, which means derivative) instead of \u00ce\u00b4 (lower-case delta, which means partial derivative with respect to either x or y). It should be: d\u00ce\u00a8/dx = \u00ce\u00b4\u00ce\u00a8/\u00ce\u00b4x + \u00ce\u00b4\u00ce\u00a8/\u00ce\u00b4y * dy/dx"
    },
    {
        "video_name": "iEpqcdaJNTQ",
        "Q": "@ 4:10 , when sal takes the derivative w.r.t x , isn't g(y) a constant ?Why not ?",
        "A": "He did the multivariable chain rule. So he is saying that d/dx ( f(x)g(y) ) is equal to \u00e2\u0088\u0082f/\u00e2\u0088\u0082x*g + f*\u00e2\u0088\u0082g/\u00e2\u0088\u0082y*dy/dx. So he is taking the derivative of g w.r.t. y, not x."
    },
    {
        "video_name": "iEpqcdaJNTQ",
        "Q": "at 2:52, why are there n terms in the f(x)g(x). Its an extremely confusing example",
        "A": "well sal is trying to show you a pattern and he extended the terms to infinity to generalize it to show you what is happening when he rearranges the data . he is showing you are that with groping we can say that the derivative of phi is just taking partial derivatives ."
    },
    {
        "video_name": "iEpqcdaJNTQ",
        "Q": "If you get confused by this example, you should skip to 8:43 and listen carefully; 'the second variable of the function is also a function of x, so the chain rule is this...'\nThen, just watch the video again and it will be clear :)",
        "A": "If you re asking how to take the derivative of x*v(x): d/dx (x*v(x)) = x * v(x) + x * v (x) would be correct as you say, following the product rule."
    },
    {
        "video_name": "iEpqcdaJNTQ",
        "Q": "at about 0:43 seconds. The Greek character is psi, not xi, I believe?",
        "A": "Yes, it surely is psi. I guess that is just his pronunciation of psi , like in psychology / psychiatrist ."
    },
    {
        "video_name": "iEpqcdaJNTQ",
        "Q": "Looking at ~2:40, is the right side of the equation (dPsi/dx + dPsi/dy) * (dy/dx) OR (dPsi/dx) + ((dPsi/dy) * (dy/dx)) ? If so, can the dy / dx terms cancel out like any other term? eg (dPsi/dy) * (dy/dx) = dPsi/dx ?",
        "A": "Not exactly. If you look at the symbols for the Psi and y multiplied with (dy/dx), you ll notice they are actually partial derivatives. That phrase is actually the partial derivative of Psi with respect to y * (dy/dx). Earlier in the video he actually states you can t cancel the partial y and dy out, but you can kinda in order to make some intuitive sense. They are different and hence don t cancel out like that."
    },
    {
        "video_name": "owKAHXf1y1A",
        "Q": "Beginning at 7:37, Sal defines a parameter t that determines how much we have rotated about the z-axis. Where do we measure t from? Do we measure it counterclockwise from the x-axis like we measure theta in the unit circle?",
        "A": "The choice of initial position and direction of rotation is completely arbitrary, but in this video it looks like t is measured counterclockwise from the y-axis (assuming you re looking down from above the x-y plane)."
    },
    {
        "video_name": "owKAHXf1y1A",
        "Q": "At 5:00 Sal says \"the angle which the radius makes with the xz plane\". shouldn't it be the yz plane?",
        "A": "or the angle in the yz plane that the radius makes with the xy plane, like szinte55 said (since it won t always be over the y axis)"
    },
    {
        "video_name": "owKAHXf1y1A",
        "Q": "I think it should be \"3Pi/2\" instead of \"3Pi/4\" at 8:55 and 9:20.",
        "A": "Yes. Sal made a mistake there. 3Pi/2 corresponds to 270 deg whereas 3Pi/4 corresponds to 135deg. But if you are able to correctly visualize the torus with respect to its orientation around the three axis, it shouldn t be a problem. :)"
    },
    {
        "video_name": "owKAHXf1y1A",
        "Q": "Is it just me or do the areas he shades in at 18:00 not match? He is right in that from 0 - pi/2 for S and T only sketches 1/4th of the outer surface for 1/4th of the circumference, or 1/8th of the total torus surface. But on the right diagram he starts shading this parametrization on the y-axis while the left he starts from the x-axis. It is important to maintain the same starting point so parameters and bounds align correctly",
        "A": "Good catch, based on how he defined t, the diagram on the right is correctly shaded, but the upper left (top view) is not ... I ve put your observation into the Tips&Feedback, I think he is more likely to see those!"
    },
    {
        "video_name": "WAPZihVUmzE",
        "Q": "At about 3:20 you mention that it is okay to do this because it's in the first quadrant. Does this have anything to do with the symmetry rule. I have seen some people use the symmetry rule and I don't understand why they are doing it. For example for x = y^2, x = 1 - y^2, about x=3; I get 20pi/(3 times sqrt 2) but others get (10sqrt2 times pi)/3 and they say they've done by symmetry.",
        "A": "well yea, it s the same value. as long as the logic and math are correct, the answer is right. that s the cool thing about math."
    },
    {
        "video_name": "WAPZihVUmzE",
        "Q": "At 7:40, does the order of the functions matter? For example, we could write y^2 - 2 as the outer radius and sqrt(y) - 2 as the inner radius. Since it is being squared, their quantity is still the same.",
        "A": "Because we re squaring these quantities, your formulas would produce the same result as those in the video, but your formulas wouldn t correspond to the problem, because y^2 - 2 is a negative number in this range. I hope you ll agree that it s better to stick with formulas that correspond to what s happening in the problem you re solving."
    },
    {
        "video_name": "VXrn5HOQmHQ",
        "Q": "At 1:40, Sal says something about the blue group and the purple group. Which one is the blue group and which one is the purple group? Because the groups aren't labeled. Just confused a bit.",
        "A": "It shows you the color of the groups"
    },
    {
        "video_name": "VXrn5HOQmHQ",
        "Q": "In 4:38 also could be 5x(4x6)",
        "A": "Yes it could be if you apply both the commutative property & associative property. The commutative property changes the position of the 4 and 5 (4 x 5) x 6 = (5 x 4) x 6 Then the associative property moves the parentheses (5 x 4) x 6 = 5 x (4 x 6) Great observation."
    },
    {
        "video_name": "yWGeFDXgQvg",
        "Q": "Sal at 0:34 that's not a circle.",
        "A": "Its called sketching... It wont always be exact"
    },
    {
        "video_name": "oqCfqIcbE10",
        "Q": "Sorry if this seems silly, but I still do not quite understand where the 1/7 comes from at 1:47 instead of just 7?",
        "A": "Unfortunately, we can t simply make up a constant to bring in. To remedy this, he effectively multiplies the primary equation by 1, which we can do, since 1 multiplied by any number is itself. 1/7*7=7/7=1, so by using this method, he is able to bring in a constant without changing the primary equation to something it is not."
    },
    {
        "video_name": "oqCfqIcbE10",
        "Q": "If you have the integral of 2(1+2x)^4, what would you do with the exponent of 4? i understand that at 1:13, you can change the way the integral is looked at, can you do the same with an exponent?",
        "A": "u = 2x + 1 du/dx = 2 => 2dx = du \u00e2\u0088\u00ab 2(1 + 2x)^4 dx = \u00e2\u0088\u00ab u^4 du = 0.2u^5 = 0.2(1 + 2x)^5"
    },
    {
        "video_name": "oqCfqIcbE10",
        "Q": "At around 2:07, how did you know you had to use 1/7 and 7 in front of the the radical? Your response would be greatly appreciated. Thanks for creating a website like this! You make us understand calculus in a more easier and simpler way!",
        "A": "He used the 1/7 and 7, because he was thinking about how the u-substitution would work out. It takes practice to recognize those sorts of tricks. So, basically, just put that trick in your brain if you are stuck on a u-substitution question sometime. It might help you work it out."
    },
    {
        "video_name": "oqCfqIcbE10",
        "Q": "At 3:49, you say to multiply the new thing by its reciprocal. Why would we have to multiply by the reciprocal? And will it always have to be multiplied by the reciprocal if it is to a fraction power?",
        "A": "The why pertaining to your first question is if you don t take the reciprocal, and you were to attempt to take the derivative if your final result, you would not end up with your original function. It applies to all exponent powers."
    },
    {
        "video_name": "oqCfqIcbE10",
        "Q": "AT 4:04, the u^3/2. I've always learned that the same fraction is put in front, meaning it would be 3/2 u^2/3, instead of 2/3 u^3/2. Which is the correct way to do it?",
        "A": "The derivative of u^(3/2) is 3/2u^(1/2). The integral of u^(1/2) is 2/3u^(3/2)."
    },
    {
        "video_name": "oqCfqIcbE10",
        "Q": "At 3:52, when Khan says the integral of '''u^1/2 = 2/3 u^3/2''', shouldn't that be '''u^1/2 = -1/2 u^3/2'''?\n\nThanks in advance!",
        "A": "Nope. If you take the derivative of -1/2 u^(3/2), you don r get u^1/2."
    },
    {
        "video_name": "oqCfqIcbE10",
        "Q": "At about 2:00, why can't you simplify 1/7*7... to just ... by canceling the 1/7 with 7?",
        "A": "You can do so. But, because the inside of the radical has a 7x + (constant that I forget), we will need a constant 7 for u-subs. That is, we let u = 7x + (const), then du = 7dx. now the integral becomes sqrt(u)du. (The 7 that we did not cancel gets lumped into the du.)"
    },
    {
        "video_name": "oqCfqIcbE10",
        "Q": "at 4:35, when that 1/7 is distributed to the rest of the problem, wouldn't the result end up with a 1/7 C?",
        "A": "Any constant multiplied by a scalar is still just some constant. Right after that Sal talks about how you could call them C_1 and C_2 to show that they are different constants. However, this precision is not normally required and is often left out entirely since it doesn t change the result. If you wanted to be thorough, you would call them C_1 and C_2 and also state that C_1 = 1/7 C_2."
    },
    {
        "video_name": "oqCfqIcbE10",
        "Q": "Towards the end when Sal distributed the 1/7 at 4:26, why doesn't the C become C/7?",
        "A": "C stands for any constant . You can multiply, divide, add or subtract some constant to any constant and still have any constant (so long as you don t multiply or divide by 0). So, we just leave the representation as C . Remember that without some additional information, C represents every constant. Thus, an indefinite integral has infinitely many solutions and we summarize them by replacing the infinite number of possible constants with C ."
    },
    {
        "video_name": "eJxSDYO82c4",
        "Q": "at 3:31, you say it is -4 over 2 for the slope. It is confusing because you use the marks on the graph instead of the actual numbers. It's still the same though, right?",
        "A": "Yes. The scale is 2, so the slope is -2/1 = -2"
    },
    {
        "video_name": "eJxSDYO82c4",
        "Q": "At 1:39 he writes the derivative as g'(x) but shouldn't it be g(1) since you are differentiating what is inside the bracket?",
        "A": "I don t see where you see a 1 in the problem. With the chain rule you need to differentiate the inner function and so g (x) is correct. And then evaluated at x = 4 is g (4)."
    },
    {
        "video_name": "LKzpw_HUKNQ",
        "Q": "When you raise x to the fourth at a little before 6:00 and the result is 81, wouldn't that negative sign become positive? ie Yielding positive 81 in lieu of negative 81??",
        "A": "It would become positive if we had (-x)^4, because then we d be multiplying -x by itself four times, But in this video we have -x^4 instead. The way to evaluate that expression is to multiply x by itself four times and then applying the minus sign. So when x = 3 we get -(3*3*3*3) which is -81."
    },
    {
        "video_name": "Hrjr5f5pZ84",
        "Q": "at 0:52 how andis it diffrent to write it as a fraction?\nPlease Answer",
        "A": "It s pretty much the same"
    },
    {
        "video_name": "Hrjr5f5pZ84",
        "Q": "It's supposed to be 30.24/0.42 at 0:50 seconds.",
        "A": "There is a correct box that pops up on the video and says that Sal made a mistake. He wrote 30.42 when it should have been 30.24."
    },
    {
        "video_name": "Hrjr5f5pZ84",
        "Q": "Please tell why at 0:45 you changed .24 and .42!",
        "A": "Sal made a mistake"
    },
    {
        "video_name": "Hrjr5f5pZ84",
        "Q": "At 0:42, Sal writes 30.42 instead of 30.24. ? And then he says the answer is 72, which is actually the answer to the real problem. ?",
        "A": "At least on my computer, there s an annotation mentioning Sal s mistake. However, while he explained the moving on the decimals on the right-hand side, he didn t change the numbers of the original problem in the left-hand side and only shifted the decimals the same way he did on the R.H.S. Thus, the answer is still correct."
    },
    {
        "video_name": "Hrjr5f5pZ84",
        "Q": "at 8:30pm khan put 30.42 and0.42 into a fraction instead of 30.24 and 0.42",
        "A": "wrong time there"
    },
    {
        "video_name": "Hrjr5f5pZ84",
        "Q": "At 1:54 Sal changes the division into a fraction. In any video does he then simplify the fraction to make the division easier? An example is: 3024/42 are both even so it's divisible by 2/2 equaling 1512/21. Then that's divisible by 3/3 equaling 504/7. Finally, divide by 7/7 to equal 72. It may take more steps, but each division is by a single digit which is easier to do.",
        "A": "he is trying to teach us the more challenging way so we can be prepared"
    },
    {
        "video_name": "Hrjr5f5pZ84",
        "Q": "at 1:10 how come it say 30.42 instead of 30.24",
        "A": "I m pretty confident that Sal meant to write 30.24, he just made a mistake and wrote 30.42 on the right. The fraction 30.24/.42 does indeed equal 72."
    },
    {
        "video_name": "Hrjr5f5pZ84",
        "Q": "at 0:13 how are we suppose to figure it out are self's if we're watching this to find out how too?",
        "A": "Just go to the end of the video."
    },
    {
        "video_name": "Hrjr5f5pZ84",
        "Q": "what did he mean at 0:34",
        "A": "you ll get the same answer when you multiply or divide both numbers with same things. If I multiply both the denominator and the nominator with the same number. I still get the same answer. For example: 12/3 = 4 If I divide both the denominator and the nominator by 2. (12\u00c3\u00b72) / (3\u00c3\u00b72) 6 / 1.5 I still get the answer is 4."
    },
    {
        "video_name": "R48KimqkYTA",
        "Q": "at 1:53 how come 80 + 80 = 160",
        "A": "Accent your posts with bold or italics and format code if (x < 0) { return; } return; 8+8=16 16*10=160 } `"
    },
    {
        "video_name": "CDvPPsB3nEM",
        "Q": "At 4:18 how is 720 meters the correct answer?",
        "A": "Presumably you are asking why 0.72 kilometres is equal to 720 metres? Well, kilo means 1000 . So a kilometre is 1000 metres, and therefore 0.72 kilometres is 0.72 x 1000 metres, which is 720 metres."
    },
    {
        "video_name": "Aig1hkq3OsU",
        "Q": "At 0:39, when Sal draws one side of the rectangle as the width and one as the length, doesn't he point out the wrong sides? To elaborate more, isn't the longer side the length, not the width?",
        "A": "Yes, that s the convention, but it doesn t really matter since they re just labels and as long as you understand what they represent then it doesn t matter which sides you define as length and width."
    },
    {
        "video_name": "Aig1hkq3OsU",
        "Q": "When Sal talks about an alternate way to solve the example problem, he says at 4:24 that \"we can factor out a 2\". What does this mean????? Help!!!!!!\n\nP.S. Vote up if you think this is a good question!",
        "A": "divide by 2"
    },
    {
        "video_name": "Aig1hkq3OsU",
        "Q": "At 2:07 , Sal said to subtract 2l . Why do you subtract 2l and not divide it by 2l so it would equal\nP/2l = 2w",
        "A": "Because the relationship of the 4 sides to the perimeter is not direct proportion. The perimeter is the sum of the 4 sides."
    },
    {
        "video_name": "ZMLFfTX615w",
        "Q": "At 4:18, I Do not Understand Distributive Property",
        "A": "(5x - 7)(3x + 2) What we want to do is distribute each term to the other binomial. In other words we will take the 5x and multiply it by 3x and 2 and we will take -7 and multiply that be 3x and 2: (5x - 7)(3x + 2) = 15x\u00c2\u00b2 + 10x - 21x - 14 = 15x\u00c2\u00b2 - 11x - 14"
    },
    {
        "video_name": "ZMLFfTX615w",
        "Q": "At 4:12 why does the equation in the brackets with the minus sign: (5x-7) need to go in front of the equation that has the plus sign?",
        "A": "It doesn t need to go in front of the first equation, Sal just showed it like that for demonstration purposes to show the easier way to do these equations."
    },
    {
        "video_name": "ZMLFfTX615w",
        "Q": "At 1:04 what does foil mean?",
        "A": "It s a method of expanding two linear terms multiplied by each other. I m not entirely sure what each letter stands for individually since I wasn t taught it like that at school, but it s where you multiply the inside terms, the outside terms, the two terms on the left side of each bracket, and the two terms on the right side of each bracket. I wouldn t worry about it, chances are you just learnt it a different way."
    },
    {
        "video_name": "Ojtwh8XDF8M",
        "Q": "I thought it wasn\u00e2\u0080\u0099t possible to separate a division. Look at 2:20.",
        "A": "Look carefully at what Sal has written at 2:32. He hasn t separated a division, he has merely written the multiplication in the numerator and denominator more clearly. (sinx)(sinx)/(x)(1+cosx) = (sinx)/(x)*(sinx)/(1+cosx)"
    },
    {
        "video_name": "TIwGXn4NalM",
        "Q": "At 0:35 a+b+c=7\nand 5a+5b+5c=?\ncouldnt you just do 5*7 b/c all the terms are being multiplied by 5",
        "A": "Yes... that s the way I would do it."
    },
    {
        "video_name": "TIwGXn4NalM",
        "Q": "at 4:37, why is the answer -40? shouldn't it be 58 as -1/-9=9\n-7/7=-49\n9 - -49= 58??",
        "A": "The expression was ( -9 ) ( -1 ) - 7 ( 7 ). = + 9 - 49 (there s only one negative sign in the second term ( - 7 * 7 ), so its result is negative) = - 40 [Note: the way you ve written the terms ( using / ) indicates division, not multiplication.]"
    },
    {
        "video_name": "TIwGXn4NalM",
        "Q": "At 2:50, why do A, B, and C when multiplied by 9= -9?",
        "A": "A+B+C=-1, so multiplying that sum by 9 = -9."
    },
    {
        "video_name": "TIwGXn4NalM",
        "Q": "At 3:18 why does he say -9(a+b+c)? shouldn't it be -9(a-b-c)?",
        "A": "Two negatives make a positive."
    },
    {
        "video_name": "ppBJWf_Wdmc",
        "Q": "At around 1:41, what you're saying is that speed is equal to the absolute value of the velocity?",
        "A": "yeah, speed is the absolute value (magnitude) of the velocity, whether it be one dimensional or a two dimensional vector or a three dimensional vector"
    },
    {
        "video_name": "ppBJWf_Wdmc",
        "Q": "At the beginning of the video (0:35), Sal talks as if speed has some significance in the process for finding the answer. I understand the difference between velocity and speed by definition, but what significance does speed play in solving part a.)?",
        "A": "if you watch until the end, he shows that in this case, the fact that they asked whether the SPEED, not the velocity, was increasing or decreasing does indeed matter in part a."
    },
    {
        "video_name": "ppBJWf_Wdmc",
        "Q": "At 7:01: Why is the speed increasing?",
        "A": "So unlike velocity, speed is a scalar quantity. That is, it only has a magnitude (not direction). So, at 7:01 when Sal references the particle s speed and says that it is increasing , he s doing so with the knowledge that the particle s velocity vector is moving in the negative direction. Try picking a negative number and moving in the negative direction, the magnitude (or absolute value) of the number increases, doesn t it? It is confusing, but this is all he s saying here. I hope this helps."
    },
    {
        "video_name": "ppBJWf_Wdmc",
        "Q": "At 7:12, isn't speed and velocity the same thing? how come velocity is negative but speed is increasing?",
        "A": "Speed is the magnitude of velocity. Velocity is a vector, with direction and magnitude."
    },
    {
        "video_name": "Vc09LURoMQ8",
        "Q": "AT 7:44 why does sal say he is going to distribute the negative of x(a+2b)?",
        "A": "It just makes it easier to simplify at the end. You can try to see what happens if you don t distribute that negative side. Just makes it harder to solve."
    },
    {
        "video_name": "Vc09LURoMQ8",
        "Q": "At around 2:00, is the midpoint just the average of X & average Y values of the two points? Thanks",
        "A": "Yes, they are. Great job on finding that out! I was a little confused when Sal was talking about that being the midpoint... I hope this helps! Please reply to this if you have any further questions!"
    },
    {
        "video_name": "Vc09LURoMQ8",
        "Q": "+Sal At 2:58, what does he mean by change in Y? And in 3:06 when he says change in X? Isn't slope m= y2-y1 over x2-x1? O.0",
        "A": "Sal is using the slope formula, but y2-y1 is basically the change in y. If your first coordinate is (1,1) and your next one is (1,2), y2- y1 is 1 unit, or the change in y. He s just stating how much the coordinate moved vertically."
    },
    {
        "video_name": "Vc09LURoMQ8",
        "Q": "Can someone tell me how he got the equation of the line at 4:05 or the link to the video. I seem to forgotten most of my algebra =/",
        "A": "Forgot your algebra? not good bro, not good... But ill elaborate anyway, Okay so slope is rise /I\\ over run -> so he subtracted b from A/2 (because thats how much farther right b is than a/2, or how much the line RUNS) and then he subtracted 0 from C/2 (because thats how much higher that C/2 is than 0, or how much it RISES) and then sal divides them. rise/run, so C/2 divided by A/2 - - b. multiply the whole bunch by 2 and you get C divided by A+2b. Because minus a negative is positive."
    },
    {
        "video_name": "osIFa9zcI-w",
        "Q": "At 1:09, Sal made 5 circles in orange on top of the pink ones. there are 4 more pink ones left. Isn't that the answer?",
        "A": "Yes, you re correct. The answer is 4 and that s what Sal says. He then continues by showing how to represent the problem with number instead of marbles. While you can easily look at the drawing of marbles to find the answer, as we progress in math it is more convenient to work with number than physical objects."
    },
    {
        "video_name": "6NBkb98nFOM",
        "Q": "At 1:29, he says in the future he will show us faster ways, why can't he show them now?",
        "A": "Sal likes to keep his videos shorter. Also, the faster way are often also more advanced."
    },
    {
        "video_name": "CEPuNmVKnZM",
        "Q": "Why are we converting Polar to Cartesian form ( 3:08 ) ? Y can't we just use ds (arc length) = rd\u00ce\u00b8 and integrate it?",
        "A": "Since r=f(\u00ce\u00b8), the length of the arc is not given by r*\u00ce\u00b8, (that is only valid on segments of circles)."
    },
    {
        "video_name": "xR9r38mZjK4",
        "Q": "At 1:30, Sal states: \"it might not be a line; it might be a curve.\" regarding the demand/ price linear graph. If the graph were a curve, and y values were always decreasing as x increased, could it still be described as a linear relationship? If not, what sort of relationship would this be?",
        "A": "That would be a nonlinear relationship"
    },
    {
        "video_name": "0-wa7voc0uM",
        "Q": "At 2:10, Sal meant plus -50. He corrects this later in the video on another calculation, but please flag it for viewers to avoid confusion.",
        "A": "2044-(-50)= 2044+50=2094 The parentheses of -50 means multiplication. It multiplies the negative signs to make it positive."
    },
    {
        "video_name": "eGRZKkmI_fo",
        "Q": "I'm very lost at 3:45, where does he get x(a+b-t) from?",
        "A": "choosing the quantity (a+b-t) is a nice trick that satisfies * when t=a, (a+b-t)=(a+b-a)=b * when t=b, (a+b-t)=(a+b-b)=a That is, (t) and (a+b-t) reverse values at a and at b"
    },
    {
        "video_name": "eGRZKkmI_fo",
        "Q": "During 4:10. Shouldn't it be b<= t <= a? not a<= t <= b? I mean I understand it is a <= t <= b if you think that the line starts at a, but when I think of it graphically I mean t increases as we move on the positive x-axis direction I think its b<= t <= a? I'm confused which way to think about it, but for now I'll think of it that t strats at the top being t=a and we move down to t=b.",
        "A": "I think the key thing to realize is that T is what does not change between the two integrals. it s starting value and ending value are the same in both cases. Therefore, in order to make the integral move in the reverse direction you have to change how the integral interprets the end points (if that makes sense). Now, if you went and messed with the value of T directly, you couldn t make a direct comparison between the integrals without substituting in and getting back the A+B-T."
    },
    {
        "video_name": "RM644gFKo_g",
        "Q": "At 6:36 didn't he use the same technique that Carl Frederich Gauss used many years ago to calculate \"n=100 sigma index=1\" or 1+2+3+4+5+6+7+8+9+10+11+12+13...+100.",
        "A": "Yeah. Gauss was an annoying kid that got a bunch of work as punishment, which he finished in only a few minutes..."
    },
    {
        "video_name": "MIAmN5kgp3k",
        "Q": "Hello! How do you figure out geometrically that you need to subtract vector x0 from x1, and not x1 from x0, at 2:14? Thanks!",
        "A": "Tracing the vector head and tail, you ll see pictorially that x1 = x0 + (x1 - x0) Thus (x1 - x0) = x1 - x0 algebraically, these two equations are also the same"
    },
    {
        "video_name": "MIAmN5kgp3k",
        "Q": "Why does Sal use (1 2, -1 0) matrix at 6:21 ?",
        "A": "It s just an example. He could have used any matrix with the right dimensions."
    },
    {
        "video_name": "7N5orPxUoGo",
        "Q": "At 0:54 what do you do if they don't give the height to you",
        "A": "if they don t give you the height, that means that it is a length times width also means area."
    },
    {
        "video_name": "7N5orPxUoGo",
        "Q": "What if you don't have the base ( or the height ). How would you find the area? ( Like at 0:53 when Sal said they don't always give it to you. )",
        "A": "You would have to find the area by the Points of the triangle. So if you had a triangle and it only told you the base. you would split it in half then you would see 2 triangles. You will get the base easily cause you just divide the number in half. Then all the sides should be the same length. then you just add it all together."
    },
    {
        "video_name": "S_OX3ByvBSc",
        "Q": "At 1:58. I thought according to PEMDAS addition came before subtraction. I ended up getting -5. I wanted to know what I did wrong.",
        "A": "Addition and subtraction are equal just do it left to right, same for multiplication and division."
    },
    {
        "video_name": "S_OX3ByvBSc",
        "Q": "At 0:30, Sal says to plug the variables into the equation. But, what if you don't know one of the variables, or you can't solve for one?",
        "A": "In this type of problem, you would be given the value of the variable to use."
    },
    {
        "video_name": "S_OX3ByvBSc",
        "Q": "at 2:01 why did he do 6-2 instead of 2+9 since addition is before subtraction and do 6-11",
        "A": "The PEMDAS rules specify that you Add & Subtract from left to right. It does not say do all the addition, then do all the subtraction. This is why the 6-2 was done ahead of the other steps."
    },
    {
        "video_name": "9_3OxtdqmqE",
        "Q": "At 0:13, Sal says \"...if we were to draw a transversal...\" What is a Transversal?",
        "A": "(of a line) intersecting a system of lines."
    },
    {
        "video_name": "9_3OxtdqmqE",
        "Q": "At 3:54 how can an angle be 0 degrees you can see that its more than that?!",
        "A": "If it is 12 noon, the angles formed by the two hands of the clock for a zero degree angle. Even though the sides have length, there is no distance between the hands of the clock and the angle is zero."
    },
    {
        "video_name": "9_3OxtdqmqE",
        "Q": "What does he mean by contradiction in 0:56?",
        "A": "z ended up with 0 degrees.. as sal said we can concluded by two possibilities.. 1) they are overlapping each other.. OR 2) they do not intersect at all.. hence, its a contradiction.."
    },
    {
        "video_name": "9_3OxtdqmqE",
        "Q": "At 4:35, what is contradiction?",
        "A": "a combination of statements, ideas, or features of a situation that are opposed to one another."
    },
    {
        "video_name": "9_3OxtdqmqE",
        "Q": "From 3:51 to the end, how can it really be a triangle if there is a 0 degree angle? How can Z be equal to Y if they are representing a different number/angle? How can Z be 0?",
        "A": "If a triangle has, or it seems as if it has a zero degree angle, then the triangle cannot exist. Therefore you have a contradiction."
    },
    {
        "video_name": "9_3OxtdqmqE",
        "Q": "I did not understand anything at 1:14",
        "A": "he is proving it by contradiction."
    },
    {
        "video_name": "9_3OxtdqmqE",
        "Q": "At 7:36 pm. PLEASE HELP. You don't show how to do a question like this in any of your videos. Angles A and B are corresponding angles. if m Angle A =4x and angle B = 3x +7 find the value of x. ( I don't need to know the answer, just show me how to do it.)",
        "A": "Corresponding angles are equal. To find x, set angle A = angle B. 4x=3x+7 Then solve for x."
    },
    {
        "video_name": "9_3OxtdqmqE",
        "Q": "I don't get how Z= 0 at 3:31",
        "A": "Z is = to zero because when you have x+180-x+z=180 you can cancel out the +x and -x leaving you with 180+z=180 if you subtract 180 from both sides you get z=0"
    },
    {
        "video_name": "9_3OxtdqmqE",
        "Q": "i did not understand the explanation at 1:41",
        "A": "Okay, so you assume that l is not parallel to m. so of course you d draw two lines intersecting eachother. Also, if you want to prove that two lines are either parallel or not, you have to have a transversal(pink line). And so to find x and y, y plus the angle next to it (or SUPPLEMENTARY) will equal 180. you get 180 - y. SO.... 180-y + x (the angle that sal gives you) plus z(the other angle) = 180. If you solve it, you get z = 0."
    },
    {
        "video_name": "-EG10aI0rt0",
        "Q": "At about 0:58, Sal is applying the chain rule to take the derivative of both sides. Why, when he takes the derivative of 5x-3y , doesn't he take the derivative of 5x? He gets 5-3 dy/dx. Shouldn't it be 5 dx/dy - 3 dy/dx?",
        "A": "Sal did take the derivative of 5x. It is 5. There is no need to put the dx / dx, since the variable involved is the one that s being differentiated in respect to. (You can think of this as reducing to 1 if you like!). The derivative of -3y WITH RESPECT TO X, is - 3 dy/dx because the variable involved is different to the one that s being differentiated in respect to."
    },
    {
        "video_name": "-EG10aI0rt0",
        "Q": "Why do you distribute the 5 in this video at 1:50?",
        "A": "Sal isn t distributing 5 here. He s distributing that hairy thing after the equals sign: -sin(5x-3y) He has to distribute this in order to break apart the factor (5-3(dy/dx)), so that the part relating to dy/dx can be moved to the left side of the equation, isolating all the dy/dx terms on that side."
    },
    {
        "video_name": "-EG10aI0rt0",
        "Q": "In this video at 3:10, When we subtract 3sin(5x-3y)dy/dx from both sides, shouldn't the dy/dx cancel out because there is dy/dx on the left, and you are subtracting dy/dx from dy/dx...?\nI am very confused. Thank you.",
        "A": "if you factorise x - ax, you get (1-a)(x). Here you factorise (dy/dx) - 3sin(5x-3y)(dy/dx). You get ( 1 - 3sin(5x-3y) ) (dy/dx)."
    },
    {
        "video_name": "-EG10aI0rt0",
        "Q": "At 2:48, why is it that the dy/dx can be reduced to 1?\nThank you",
        "A": "ppl have so much trouble with this. it is the distributive property. get out of the calc videos and back into pre algebra and the distributive property videos."
    },
    {
        "video_name": "-EG10aI0rt0",
        "Q": "At 2:33 he substracted 3sin(5x-3y) from 1 and at the end he got -5sin(5x-3y) / (1-3sin(5x-3y)). However if we substract the other way and get 3sin(5x-3y)-1 we can get 5sin(5x-3y) / (3sin(5x-3y)-1) . Would it not be right as well?",
        "A": "-5sin(5x - 3y)/(1 - 3sin(5x-3y)) = 5sin(5x - 3y)/(3sin(5x - 3y) - 1). We re just multiplying by -1 on top and bottom, and a/b = ka/kb for k \u00e2\u0089\u00a0 0."
    },
    {
        "video_name": "-EG10aI0rt0",
        "Q": "i will never understand the step at 2:46! where does the one come from?!",
        "A": "Anything can be multiplied by 1 without changing its value. Thus, we normally do not explicitly write out a coefficient of 1 unless we have a good reason to do so. Thus, y = x + 7 is the same thing as 1y = 1x + 7 The same goes for a differential dy/dx = 2x dy/dx + 37y is the same thing as 1 dy/dx = 2x dy/dx + 37y"
    },
    {
        "video_name": "-EG10aI0rt0",
        "Q": "In this video at 2:40 as well as in the last video, sal subtracts something from both sides that contains dy/dx, in order to get all dy/dx to the same side. But I don't get how one of the dy/dx becomes 1. In this video it results in (1-3sin(5x-3y))dy/dx. How does dy/dx-3sin(5x-3y)*dy/dx= (1-3sin(5x-3y))*dy/dx",
        "A": "He factored the dy/dx out. For example, in the expression x+5x we can factor out the x to get (1+5)x. The 1 is left behind to show that we had an x there before we factored. Another way to see this is to try multiplying the dy/dx back into the expression. If there was no 1 sitting there than we would have magically lost the dy/dx term in the original expression."
    },
    {
        "video_name": "boH4l1SgJbM",
        "Q": "@ 6:02 - 6:39 , I don't list the choices that way. I typically list them like:\nAB, AC, AD\nBC, BD\nCD\nIs this still correct?",
        "A": "Yes, it is still correct."
    },
    {
        "video_name": "jgWqSjgMAtw",
        "Q": "Would the the \"true\" fraction of the triangle at 3:46 be 1/8? Or is there too little information to tell?",
        "A": "there is too little information. you would have to divide it in a different way to tell."
    },
    {
        "video_name": "jgWqSjgMAtw",
        "Q": "At 2:54, The first shape is 1/3, the second shape is 1/5, and the last shape is 1/4,but the last shape that is 1/4 is not in equal parts,so it is not 1/4",
        "A": "awsome people"
    },
    {
        "video_name": "jgWqSjgMAtw",
        "Q": "In 4:01 what do you call a fraction that has a few equal parts and a few that aren't ?",
        "A": "He was pointing out that it wouldn t make a fraction at all..... Because the parts are not equal."
    },
    {
        "video_name": "jgWqSjgMAtw",
        "Q": "Oh, never mind, at 3:57, he points out that it's unequal.",
        "A": "This is not a question."
    },
    {
        "video_name": "jgWqSjgMAtw",
        "Q": "when i watched the video at 5:31 the moon is large to block the sun but the sun is supposed to be soo much larger?",
        "A": "the moon is closer to the earth then the sun. Every once in a while, the moon is in such an exact place that it appears to cover the sun since the moon is closer."
    },
    {
        "video_name": "jgWqSjgMAtw",
        "Q": "I didn't quite get what he meant at 1:16. The whole divided into four parts would still be four parts, so viewing in that way doesn't make sense.\n\nAlso, at 4:13, how would we represent the triangle's shaded part in fraction? Is it possible or fractions MUST have equal parts? Is this what Sal was trying to tell us? (I assume they do, since you can only add x/8 + 2x/8 rather than x/8 + x/13?)",
        "A": "With every fraction, in this case 1/4, you can either look at it is a whole divided into four equal parts or as one of four equal parts that would make up the whole; either way you do still have four parts that are equal. You are correct, you cannot represent that triangle s shaded part as a fraction. In order for something to be written as a fraction each part must be equal. As this triangle does not have equal parts, you cannot write it as a fraction."
    },
    {
        "video_name": "jgWqSjgMAtw",
        "Q": "At 4:10, when you say the triangle is not 1/4 because it is not equal, what is it then?",
        "A": "1/8. If you draw a few more lines inside the big triangle to make all the pieces congruent to the red triangle, you get 8 of the small triangles like the one he highlighted in red. That example is to show that all the pieces need to be equal to be able to figure out the fraction by counting the number of pieces made. Sorry if this is confusing!"
    },
    {
        "video_name": "jgWqSjgMAtw",
        "Q": "at 4:05 do people get tricked bt that",
        "A": "Yes, people can be easily tricked by that if they don t pay attention to if the shape is divided equally."
    },
    {
        "video_name": "wiVWafc3JBM",
        "Q": "Now I know that dS = r*\u00e2\u0088\u009a(2)\nBut at 12:04 don't we use \"r dr d\u00c3\u00b8\", instead of just dr d\u00c3\u00b8.\nIf not, why?",
        "A": "You re thinking of area, but dS is arc length. dA = r dr d\u00ce\u00b8 dS = dr d\u00ce\u00b8"
    },
    {
        "video_name": "xmVBv7vk3Gg",
        "Q": "For part b at 6:08, Shouldn't the answer be multiplied by \u00cf\u0080 for Volume?",
        "A": "NO! this is a square not a circle."
    },
    {
        "video_name": "nDXFgexOM5c",
        "Q": "At 5:00 when you've simplified the limit to 100/sqrt(100+x)+sqrt(x), would it be possible to further simplify? My brain instantly wants to pull the 100 out of the bottom sqrt and end up with 100/10sqrt(x) + sqrt(x) then simplify to 10/sqrt(x) + sqrt(x) then 10/2sqrt(x) then finally 5/sqrt(x). I feel like I'm making a mathematical mistake, but I don't know where.",
        "A": "If the problem were sqrt(100*x) you could do that, but having an addition in there is what makes it not work."
    },
    {
        "video_name": "toKu2-qzJeM",
        "Q": "at 0:10, I like how half her notebook is filled! My question, is why is the smaller snowflake on the right of the mirror? Don't mirrors make things appear larger?",
        "A": "Planar mirrors make images appear neither smaller nor larger, but what Vi showed was not a mirror at all. She drew a line down the center of the page, on the left she showed snowflakes with mirror symmetry, and on the right she showed snowflakes with rotational symmetry."
    },
    {
        "video_name": "toKu2-qzJeM",
        "Q": "At around 2:33, Vi unfolds a golden shape. How is it a sphereflake?",
        "A": "It is a sphere -- a cut-up beach ball -- and Vi cut it up to have snowflake patterns. Sphere + snowflake = awesome sphereflake! --Blue Leaf"
    },
    {
        "video_name": "toKu2-qzJeM",
        "Q": "At 0:05, Vi Hart says: \"Finite symmetry groups on the Euclidean plane are fun and all...\" What does \"Finite symmetry groups on the Euclidean plane\" mean?",
        "A": "It basically means 2D x-fold symmetry paper snowflakes, even though paper snowflakes aren t actually 2D, they are 3D, like Vi said at 1:14."
    },
    {
        "video_name": "TJaHNxJk5-4",
        "Q": "in 3:216, why does Sal -1(a+b+c=180). Could he not just cancel out the b's from (a+b+c=180)\n(-4a+b=-50)",
        "A": "You can only cancel out when you re dividing (and even then it is factors, not terms, that you can cancel out). You re not dividing here, so you cannot cancel out. So, no, you cannot cancel out here."
    },
    {
        "video_name": "9P80OLC6wKY",
        "Q": "At 5:48, how can you convert rational numbers to fraction form.\n0.219...",
        "A": "0.219 = 219 / 1000 And that is already in the simplest form"
    },
    {
        "video_name": "AR1uqNbjM5s",
        "Q": "At 6:42, why does (a*1)^3 = a^3? wouldn't it be a^4? since previously when he took a^3-a^2 its = to a^1/a",
        "A": "(a*1)^3 is equal to a^3 because simply put, a*1 just equals a , and therefore (a)^3 = a^3. I think you were confusing this operation with multiplying terms like these. For example this equation would equal a^4 if it was a^3*a^1 , but it does not in this case because it is (a*1)^3 Hope this helps and feel free to contact me if you still do not understand."
    },
    {
        "video_name": "AR1uqNbjM5s",
        "Q": "at 4:05 why did sal have one in the numerator? we got nothing",
        "A": "Because 3*3*3*3 = 3*3*3*3*1, not 3*3*3*3*0 which equals 0. So if everything cancels, there is still the invisible 1 that was multiplied by it in the first place."
    },
    {
        "video_name": "AR1uqNbjM5s",
        "Q": "At 7:00 could it also be (ab) to the 9 power ?",
        "A": "The 3 outside the parentheses doesn t affect a\u00c2\u00b3, it belongs to b\u00c2\u00b3 since it s sticking to those brackets. So to answer your question, no it s not the same. However, if the parentheses includes a\u00c2\u00b3 such as (a\u00c2\u00b3b\u00c2\u00b3)\u00c2\u00b3, then it certainly would be (ab) to the 9th power."
    },
    {
        "video_name": "AR1uqNbjM5s",
        "Q": "6:20 Why Sal first divide and only then raise to a power?",
        "A": "To make it easier. You could raise every term to the power of 3 but why not save yourself the trouble? Remember that (2*2*2*2 / 2*2*2)^3 = (2)^3 is the same thing as (2^4 / 2^3)^3 = (2^1)^3 and if you have ( (2^4 / 2^3 )^3 you could just simplify it as 2^3 like above, instead of doing your way: (2^4*3) / (2^3*3) (2^12 / 2^9), and now it s 2^(12-9) = 2^3 So its easier, hope it helped."
    },
    {
        "video_name": "AR1uqNbjM5s",
        "Q": "@4:07 where Sal is getting 1 to the nominator, coz there is no numbers left? thx ;)",
        "A": "Each time Sal cancels out a pair of 3 s, he is doing 3 divided by 3. This = 1. Thus, the numerator = 1 * 1 * 1 * 1 = 1. Hope this helps."
    },
    {
        "video_name": "AR1uqNbjM5s",
        "Q": "7:45 why do use letters its just confusing",
        "A": "the letters represent numbers or variables as they are called"
    },
    {
        "video_name": "AR1uqNbjM5s",
        "Q": "At 8:15, wouldn't 25/20 be 5, since the x's cancel out like that? Or is division for those numbers like simplifying, not subtracting?",
        "A": "The rules described in the video only apply to exponents. 25/20 isn t working with exponents, it s just a simple fraction. So, you would simplify by dividing both the numerator and denominator by the greatest common fraction (5 in this case)."
    },
    {
        "video_name": "AR1uqNbjM5s",
        "Q": "At 1:19 why is he subtracting if this is a division problem?",
        "A": "1 / (2^2) is the same as 2^(-2) so 2^4 / 2^2 is the same as 2^(4-2)"
    },
    {
        "video_name": "AR1uqNbjM5s",
        "Q": "I can't understand how to simplify this:\n128:4^3/16",
        "A": "To do this,firstly change the number 128:4 into 128/4.Simplify this, we get 32.Now, simplify 3/16 we get 0.1875.Now, you will need help of a calculator to do this."
    },
    {
        "video_name": "AR1uqNbjM5s",
        "Q": "In 3:23, Sal said \"Segway\". what does \"Segway\" or whatever that is, means?",
        "A": "A Segway is a sentence where you transition into another subject gracefully."
    },
    {
        "video_name": "_X_QwpXsdOs",
        "Q": "At \"6:00\", I think it should be F(x)-F(0) rather than F(x) as F(0) is not 0 if F(x)=cos x and f(x)=sin x.",
        "A": "F(x) there is not standing there for it being a definite integral, it stands for an indefinite integral. It is just for us to know that anti-derivative of the function f(x) is equal to F(x)."
    },
    {
        "video_name": "Z3X6QqezqUg",
        "Q": "I am confused when he does the derivative for f(3.5) at 2:56. My textbook tells me that you use two secant lines(that are close to the point whose slope you are trying to estimate) to approximate the derivative, is there something I am missing here?",
        "A": "You can do that if you have a formula, but if you only have a table of values to draw from, you might be limited on how many slopes you can calculate. Sal is using a version of the difference quotient that uses an input value on either side of the input you are looking for. This is sometimes called the balanced difference quotient and is a bit more efficient than the standard formulation. The formula for the balanced difference quotient is lim h->0 of (f(x+h)-f(x-h))/(2h)."
    },
    {
        "video_name": "Gx5D09s5X6U",
        "Q": "at 2:08 the snakes head falls off. does anyone think its funny",
        "A": "It s kind of funny"
    },
    {
        "video_name": "Gx5D09s5X6U",
        "Q": "how did she get all of those snakes? 0:05",
        "A": "They are probably available at orientaltading.com, where you can buy TONS of cheap stuff, like snakes. You can also get them on amazon."
    },
    {
        "video_name": "Gx5D09s5X6U",
        "Q": "At 1:41, what does Vi mean by the binary expansion of pi?",
        "A": "11.001 and stuff"
    },
    {
        "video_name": "Gx5D09s5X6U",
        "Q": "How come the food at 2:07 teleports form the corner right to the snake?",
        "A": "Vi took a picture, and moved the snake. Then she repeated. After, when the snake gets to the food, she adds it to the snake, and gets another segment."
    },
    {
        "video_name": "Gx5D09s5X6U",
        "Q": "How do you make that snakestache at 0:39?",
        "A": "Looks like it is a head part, a body part and a tail part, joined together."
    },
    {
        "video_name": "Gx5D09s5X6U",
        "Q": "at 1:42 she could be hideing a secret message hmmmmmmmmmmmmmmmmmmmmmmmmmm?",
        "A": "Then you could throw it at someone and freak him/her out and then she yells at you and then notices the paper inside, opens it, and sees 100 s of ants on a small puddle of honey inside."
    },
    {
        "video_name": "Gx5D09s5X6U",
        "Q": "At 1:42, how do you find the digits of the binary expansion of pi? Thanks! :D",
        "A": "The way Vi found them was by simply searching binary pi on the internet."
    },
    {
        "video_name": "Gx5D09s5X6U",
        "Q": "In 0:32 isn't the \"space-filling fractal\" the hilbert fractal?",
        "A": "yes, that s one example of a space filling fractal curvy thing."
    },
    {
        "video_name": "Gx5D09s5X6U",
        "Q": "At 0:39, is that really Vi !",
        "A": "Yeah. I guess so!(IuI)"
    },
    {
        "video_name": "Gx5D09s5X6U",
        "Q": "How can she say snakes so fast 1:24 ?",
        "A": "Sorry to spoil the magic but most likely she record herself saying snakesnakesnake... as fast as she can and then speeds up the tape for the vid. :)"
    },
    {
        "video_name": "3Ayt7mOd_To",
        "Q": "At 0:38 Sal says that Y is always zero, but wouldn't it be two?",
        "A": "This is because the slope means how much you move in order to get to the next point. Since, the number remains as 2 no matter how much the x-value changes, it would be 0. However, if it was actually 2, the y-coordinates would change 2 units to the right for each change in the x-intercept"
    },
    {
        "video_name": "3Ayt7mOd_To",
        "Q": "At 3:56, why was -2/3 times 7 suddenly become -14/3? He just says it then doesn't explain how that happened :/",
        "A": "-2x7 is 14, and 3x1 is 3, making it -14/3"
    },
    {
        "video_name": "RhzXX5PbsuQ",
        "Q": "Isnt X over both 25 and y at 0:10",
        "A": "If we wanted to make the exponent of x applicable to both the denominator and the numerator, it would be expressed as (25/y)^x, since is wasn t expressed in this manner, it is presumed that the exponent of x is only applicable to the numerator."
    },
    {
        "video_name": "RhzXX5PbsuQ",
        "Q": "(beginning around 1:06, through about 1:32): In the original problem (separating out the question of \"y,\" for a moment), is the exponent \"x\" for 25 or is it applied to the log base 5 of 25? I'm trying to understand the logic of multiplying that x times the log base 5 of 25 (moving it out front).",
        "A": "This is how I see it: 5log(x) = log(x) + log(x) + log(x) + log(x) + log(x) = log(x * x * x * x * x) = log(x^5)"
    },
    {
        "video_name": "p5e5mf_G3FI",
        "Q": "Is there a particular order to +/-/x/\u00c3\u00b7 on an equation?\nIf I at 00:37 instead multiplied both sides by 4 first then took of 3 then x would = -66 instead of -72.....",
        "A": "Always do things in parentheses first Then do exponents After that: multiplication and division, going left to right after that: addition and subtraction, going left to right"
    },
    {
        "video_name": "p5e5mf_G3FI",
        "Q": "at 1:55 why do we multiply",
        "A": "Multiplying is the inverse operation of division. To get rid of a number being divided in an equation, you multiply that number to each side of the equation to get rid of it."
    },
    {
        "video_name": "p5e5mf_G3FI",
        "Q": "At 1:32, I got lost for what to do with the x/4?",
        "A": "We have -16 = x/4 + 2. We want to isolate the x. Step 1, Remove the 2 by subtracting 2 from both sides -16 - 2 = x/4 + 2 - 2 which gives -18=x/4 The key thing to remember here is that x/4 can be written as (x)(1/4). It should now be easier to see that if we multiply both sides by 4 we can get rid of the 1/4: (-18)(4) = (x)(1/4)(4) = (x)(4/4) = (x)(1) = x so now we have (-18)(4)=x. Now just multiply (-18)(4) and your done."
    },
    {
        "video_name": "jYOfMszfzAQ",
        "Q": "4:38\nWhat is odd number?",
        "A": "See answer posted on your question for even numbers. The answer also described odd numbers."
    },
    {
        "video_name": "jYOfMszfzAQ",
        "Q": "5:00\nWhat is even number?",
        "A": "An even number is any integer that ends in 0, 2, 4, 6, or 8. For example, these are all even numbers: 210, 0, -4, -112, 16, -58 An odd number is any integer that ends in 1, 3, 5, 7, or 9. For example, these are all odd numbers: 1, 25, -39, -207, 4183 Hope this helps."
    },
    {
        "video_name": "jYOfMszfzAQ",
        "Q": "At 2:41 does the same aply with other negitive numbers like 1 2 3 4 ...............",
        "A": "When I do the multiplication I come up with 81. The answer is -81 but why? Why am I not getting -81"
    },
    {
        "video_name": "jYOfMszfzAQ",
        "Q": "at 2:36, why are there parenthesis on a negative number?",
        "A": "To help you read it better. if it looked like this: ---1 instead of this: ( - { - [ - ( -1 ) ] } ) you wouldn t be able to read it as well."
    },
    {
        "video_name": "jYOfMszfzAQ",
        "Q": "At 0:43, 1 raised to 8 is just 1*1*1*1*1*1*1*1, right?",
        "A": "Yes, that s right"
    },
    {
        "video_name": "OWPVZoxNe-U",
        "Q": "At 2:21 he calls the y-intercept =b\nI have always learned it as being =c\nDoes it matter?",
        "A": "I don t think it does. As long as variables have the same meaning / value, then you shouldn t worry."
    },
    {
        "video_name": "6_9xNMtwnfs",
        "Q": "At 2:05, it is mentioned that the orange line has to have a slope greater than the slope of the blue line. I thought if a line's slope needs to be greater, it would be more negative, or have a steeper slope, shown at 2:15. However, the video makes the blue line's slope steeper than the orange line's. Why and how?\nThanks!",
        "A": "I originally understood it like you did, and got some questions wrong in practice. I think by saying less negative he means greater than. Like how -5, while negative, is still greater than -6. I do think he should have been clearer though."
    },
    {
        "video_name": "wRxzDOloS3o",
        "Q": "5:16 there are actually 3 ways: diagonally",
        "A": "He said at 1:06 that you can only move forward, down, or right. Note that that list does not include diagonally."
    },
    {
        "video_name": "wRxzDOloS3o",
        "Q": "In 11:08, there should be a video that does go into complicated graphing stuff. Does anyone know where to find one or if it exists?",
        "A": "In this specific website,I don t think Sal added any more.In 11:08,he showed different possibilities,of the cube.They might be more in other websites."
    },
    {
        "video_name": "Z96vkuybvjE",
        "Q": "On the second problem, how would -a=4? Isn't -a a negative number, thus 4 not being an option. (Around 3:55)",
        "A": "a is just a variable that hold some value, that value can be negative or positive, so it doesn t necessarily mean that -a should be a negative value for example: a = -3 so -a = - (-3) the negatives cancel out and become a positive and you get -a = 3"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "I found a mistaken at 10:53 x suppose to be positive 8 but sal put negative 8",
        "A": "Mirza, x = -8 is correct. When the equation was factored to (x+8)(x+10)=0 then the equation is solved when either x+8 = 0 or x+ 10 = 0 And x must = -8 for (x+8) to equal zero. I hope that helps!"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 2:34, why did root 4x2 becomes 2 root 2?",
        "A": "Jonathan, \u00e2\u0088\u009a8 = \u00e2\u0088\u009a(4*2) \u00e2\u0088\u009a(4*2) = \u00e2\u0088\u009a4 * \u00e2\u0088\u009a2 And because \u00e2\u0088\u009a4 = 2 then \u00e2\u0088\u009a4 * \u00e2\u0088\u009a2 = 2 * \u00e2\u0088\u009a2 Which is 2\u00e2\u0088\u009a2 So \u00e2\u0088\u009a8 = 2\u00e2\u0088\u009a2 I hope that helps."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "at 2:23 how did it became 2\u00e2\u0088\u009a2 in \u00e2\u0088\u009a4.\u00e2\u0088\u009a2? factoring that out that would be \u00e2\u0088\u009a2.\u00e2\u0088\u009a2.\u00e2\u0088\u009a2",
        "A": "When simplifying radicals, you want to actually do the square root where possible. \u00e2\u0088\u009a4 is doable. It equals 2. So, \u00e2\u0088\u009a4.\u00e2\u0088\u009a2 = 2 \u00e2\u0088\u009a2 (because you simplify the \u00e2\u0088\u009a4 into 2)."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "Why can't you just do the equation normally? For example,at 6:09 couldn't you just do it like this:\nx^2-10x+25=9\nx^2-10x+25-25=9-25\nx^2-10x=-16\nx^2-10x+10x=-16+10x\nx^2=-16+10x\nx=\u00e2\u0088\u009a(-16+10x)\nWouldn't that work?",
        "A": "There is nothing wrong with your math (except that x^2 = y becomes x=+/-\u00e2\u0088\u009ay), but the goal is to solve for x. You still have an x term on the left, so the equation x=\u00e2\u0088\u009a(-16+10x) does not make it clear what values of x are the solutions."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "at 6:48, how did he get x-5?",
        "A": "HoOaMeYe, Sal had x\u00c2\u00b2-10x+25 -5*-5 = -25 -5+-5 = -10 So this factors to (x-5)(x-5) which is (x-5)\u00c2\u00b2 I hope that is of some help."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "and at 11:00, why is it x= -8 or x= -10 although the equation was (x+8)(x+10), if we just see from the traditional ways??",
        "A": "as u see (x+8)(X+10) =0 will be possible either if x+8=0 or x+10=0 so then solving the equations we can get that either x= -8 or x= -10"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 4:47 I was thinking \"Wow, that's a lot easier than I thought!\" But does anyone know when they teach this kind of stuff in school?",
        "A": "I m taking Algebra ll and learning it right now."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "how does he solve the (2\u00e2\u0088\u009a2)^2 to be 4 * 2 ?? 5:28",
        "A": "He separately squares both 2 and sqrt(2). If you square 2, you get 4. If you square sqrt(2), you get 2."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "I don't understand what happens at 5:26, how does Sal solve that?",
        "A": "What part do you not understand?"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "at 1:52 why does he change it to 4*2 instead of just 8?",
        "A": "Square root of 8 is equal to square root of 4 times square root of 2. Since square root of 4 equals to 2, the expression can be simplified."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 2:27, how does Sal change the square root of 8 into 2 times the square root of 2?",
        "A": "Since 8 = 4*2 sqrt(8) = sqrt(4*2) =sqrt(4)*sqrt(2) =sqrt(2*2)*sqrt(2) =2*sqrt(2)"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "How did you get the 4x + 1 = (positive or negative) 2 (square root of) 2 \"2:17 on video\"\nWhere did the twos come from and why isn't 4x+1 changed?",
        "A": "Ah. So in this step he did some algebraic manipulation and simplification. So he had (4x+1)^2 on one side and 8 on the other side. To get rid of the exponent, he had to square both sides. It then became 4x+1 on one side and the square root of 8 on the other side Whenever you take the square root of something, you have to account for both possibilities (positive and negative) and he did that with +/-. The square root of 8 simplifies to 2\u00e2\u0088\u009a2 That is where he got the two s from."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 2:17, how did he got 2 square root of 2??",
        "A": "Kahleah, First you need to know that \u00e2\u0088\u009a(a*b) = \u00e2\u0088\u009aa*\u00e2\u0088\u009ab That is, if you have the square root of something times anotherthing that is the saem as the squareroot of something times the squareroot of anotherthing. And 8 = 2*4 so \u00e2\u0088\u009a8 = \u00e2\u0088\u009a(2*4) = \u00e2\u0088\u009a4*\u00e2\u0088\u009a2 And the \u00e2\u0088\u009a4 = 2 so \u00e2\u0088\u009a8 = \u00e2\u0088\u009a4*\u00e2\u0088\u009a2 = 2*\u00e2\u0088\u009a2 I hope that makes it click for you."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 3:22, why does 1 have to be subtracted before the radical to get -1 +/- 2 radical 2. Why can't it be +/- 2 radical 2 - 1?",
        "A": "you can write it the second way, but I try to put things that are not under the radical in front just so there is no confusion about what is under the radical sign and what is not"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 8:30, how does Sal get x={2, 8}?",
        "A": "if you have x^2 - 10x + 16 = 0 (x - 8)(x-2) = 0 [if you times this out, you will get the equation above] x^2 - 2x -8x + 16 = 0 x^2 - 10x + 16 = 0 Hope that helps."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "5:48 In the video, the equation would equal to 4*2-8 which is zero but don't you have to square 2*the root of 2? If I'm right, then in that case it would be 16-8. What did I not get?",
        "A": "(2\u00e2\u0088\u009a2) squared is equal to 2 squared times \u00e2\u0088\u009a2 squared which as we know equals 4 times 2 or eight."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "at around 3:30 he said that +1 minus 1 is equal to -1 am i missing something? can someone explain how he got that -1\nThanks",
        "A": "when you subtract -1 from both sides of 4x+1=2* squareroot2, (4x+1)-1: 1-1 is zero 2 * squareroot2 -1: one cannot be subtracted from 2 * squareroot2. When i subtract, i think of it as 0+ 2 * squareroot2, so when you subtract one, you get -1. so the equation is now: 4x= -1 +/- 2 * squareroot2"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "can somebody explain to me about 1:41. i don't get it",
        "A": "What he s doing here is taking the square root of both sides of the equation to eliminate the exponent of 2. When he takes the square root of the left side, the square root and the exponent cancel each other out. This helps him solve the problem."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "Between 1:41 and 1:46 (approximately) why did Sal take the negative and positive square root of 8... I don't understand the reason!:(",
        "A": "It is positive and negative because if you have 3^2 then the answer will be 9. But in the same case if it is (-3)^2 it will still be 9. That is why when you square root a number like 9 it can either be -3 or positive 3 that is why plus and minus are used at the same time"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 7:07, how did Sal figure out that:\nx^2 - 10x + 25 = 9 is equivalent to (x-5)^2 = 9?",
        "A": "Sal used the binomial square, which states that a^2+2ab+b^2 = (a + b)^2. An easy way to know if something can be factored in this way is to take the square root of the part that as an x^2 and the square root of the part that doesn t have any x, and then multiply them together and then double that. If you get the middle part (the part with x) then you can put it in that form, if it doesn t then you can t."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 8:05, Sal subtracts the square of nine so that it's positive/negative three. How does he get positive eight and positive two? Shouldn't it be negative?",
        "A": "He did not subtract, he square rooted both sides."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 4:00, why does Sal not divide the radical by 4 along with the -1 + or - ?",
        "A": "Desiboy, At 4:50 Sal was able to cancel out the /4 so it turned out to be simpler never to change it to (1/4)\u00c2\u00b11/2\u00e2\u0088\u009a2"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "at 7:14 you said a dead end would that be the answer or \"no solution?\"",
        "A": "Leonie, What Sal means by that would be a dead end is that this method can t be used to solve the quadratic equation-- but that doesn t mean that there is no solution. It only means that we can t find the solution using the method he is describing in the video (the method of Square Roots). You might still be able to find the solution using the factoring method. And you will ALWAYS be able to find the solution using the quadratic equation."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "for problem one, at 4:45, approx, can you simplify it anymore? if you divided by four, you would get\nx= -1/4 +or- 1/2 \u00e2\u0088\u009a2\ncorrect?\nor am i wrong?",
        "A": "Yep, that s correct."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 6:30 how does that work?",
        "A": "This is a basic factoring problem you are talking about. If: (x-5)(x-5) is done out, you have x^2-5x-5x+25. The two middle terms are like terms so you add them. Then, you do indeed have x^2-10x+25. Therefore, it is a perfect square of binomials."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "at 8:54, the problem is x^2 - 10x + 25 = 9. sal factors it as (x-5)(x-5) = 9. if there were three x's at the beggining, how would you end up with just 2?",
        "A": "If you multiply it out, you would get x^2 - 10x +25 = 9."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "AT 1:44, isn't the square root of 8, 2? Where did 2 square root of two come from?",
        "A": "No the square root of 8 is 2 the square root of 2 because when you get the square root of 4 times two, four is a perfect square, so it comes out as two. The other two is not a perfect square, so it stays in the square root. The square root of eight is not a perfect square."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 5:55 Sal mentions the quadratic formula is derived from this. Could someone please explain what the quadratic formula is?",
        "A": "If you have an equation in the form of: ax^2 + bx + c = 0 where a, b, and c are numbers, you can find the 2 values of x by using the quadratic formula, thus: x = (-b +- sqrt(b^2 - 4ac)) / (2a) If the part inside the square root is positive, you will get two real roots. If it is zero, you will get a double root. If it is negative, you will get two complex (real + imaginary) roots."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 7:43 9 gets rooted into a negative and positive number \u00e2\u0088\u00933 , but if you were to root -9, it would make an imaginary number.\nHow is this avoided here i am not understanding please",
        "A": "We are looking for a number such that when you mulitply it by itself, it gives you 9. (-3)*(-3) = 9 because the product of TWO negatives is a positive. It would not equal to -9. When you are doing this with a calculator you must be careful. If you do not put parenthesis, the calculator gives -3^2= -9, because it is only squaring the 3. It does not include the negative. To square a negative in a calculator you must type (-3)^2. Hope that helped"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 7:00 how did you get (x-5)(x-5)",
        "A": "If u FOIL it out again x*x= x^2 x*-5 +x*-5= -10x 5*5=25 So u get the original equation of x^2-10x+25"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 5:30 where did you get the 4(2) from?",
        "A": "you have to distribute the exponent. 2 squared is 4 and the root of 2 squared cancels out the root sign and you re left with 2, so you have 4 times 2"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "at 6:50 Sal says that X^2-10X+25 is the same thing as (X-5)(X-5).\nHow does that work?",
        "A": "If you break it apart and foil (x-5)(x-5) you get that. It is simple math, you find two numbers that multiply to ac, and add to b, in this case it is -5 and -5"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 4:44, Sal said \"expressions\" but meant \"equations\"",
        "A": "There s a Report a mistake button off to the side of the video. You can use that"
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "8:37 how does he go from (x-8)(x-2)=0 to x=8 and x=2? you cannot simplify it just like that",
        "A": "Sal applies the Zero Product Rule. If says: if ab = 0, then a=0 and/or b=0. This allows Sal to split the factors and set each one = to zero. x-8 = 0 creates x = 8 x-2 = 0 creates x = 2 Hope this helps."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "where did the squared go in 5:27",
        "A": "the square root and the radical cancels out because (radical 2) squared = radical 4 and the square root of 4 is 2. So instead of going through this whole process, you can just cancel out the radical and the square root."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "At 3:40 why does the -1 go to the front of the equation?",
        "A": "He subtracted 1 from both sides of the equation to simplify."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "I don't get it. At 2:18 he takes the square root of both sides and says the square root of 8 is + or - 2 times root 2 .How can square root be negative? .And how is it + and - at the same time?\nI'm sorry but i'm completely new to the topic.",
        "A": "It s positive and negative because the concept of the square root is a number when multiplied by itself results in the number under the square root. For example, the square root of 4 would be 2 and -2 because 2 multiplied by 2 makes 4, and -2 multiplied by -2 makes 4. So both of those numbers satisfy that condition, meaning when both of those numbers are multiplied by themselves they result in the number under the square root, that number being 4."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "3:24, would it be the same thing if he put\n+or- 2\u00e2\u0088\u009a2 -1?\nDoes it make a difference in the solving of the problem?",
        "A": "No, that would not make a difference. That s actually what he did. He subtracted 1 from each side, yielding 4x=+or- 2sqrt(2)-1."
    },
    {
        "video_name": "55G8037gsKY",
        "Q": "why does the 4 cancel with the other 4 when you are plugging the 1st equation in\ntime - 5:04",
        "A": "If they are on both sides, they do not matter to the equation."
    },
    {
        "video_name": "PR-A3UAO7_0",
        "Q": "at 0:35 what is that box thing Sal drew",
        "A": "It does not have any meaning, it s just there for emphasis."
    },
    {
        "video_name": "JyArK4jw3XU",
        "Q": "at 1:52, Sal says \"a positive c greater than zero\". but shouldn't it be \"greater than one\" since this is a vertical stretch (not a compression, from the way it was drawn)? thanks!",
        "A": "Sal said he chose to draw it with c = 3, just because he had to use some value. But what he showed will work for any value of c > 0. He says the line may scaled up or down depending on c."
    },
    {
        "video_name": "PlY3e_-9JUA",
        "Q": "At 0:40 couldnt he also write 3/6 = 2/x or 6/3 = x/2? Thanks",
        "A": "Yes, you could. switching the denominator and the numerator on both sides of an equation has no effect on the result. An example: If you have 3/6 = 3/6. Switch the denominator and numerator, and get 6/3 = 6/3. That is the same thing with x. 6/3 = x/2 can be 3/6 = 2/x. Now, when using the Angle Bisector theorem, you can also use what you just did. You will get the same result! Hope this answers your question."
    },
    {
        "video_name": "PlY3e_-9JUA",
        "Q": "At 2:20,why did they combine like terms,the 5x and the 7x on opposite sides of the equal sign?",
        "A": "Sal added 5x to both sides so he could then divide both sides by 12 to find out about the value of x. For example, how would you solve x + x + x = 6 x + x + x = 6 3x = 6 x = 6 \u00c3\u00b7 3 x = 2 It s the same idea."
    },
    {
        "video_name": "PlY3e_-9JUA",
        "Q": "1:09 A what triangle?....",
        "A": "the triangle is isosceles. which means that 2 sides of the triangle are equal"
    },
    {
        "video_name": "PlY3e_-9JUA",
        "Q": "At 1:05 in the video when he first started talking about \"A\" and \"B\", where did \"X\" come from?",
        "A": "Not sure which of the two examples you mean, but in each case, x was marked as the length we needed to determine given the other information."
    },
    {
        "video_name": "XDOH9Ul2OOM",
        "Q": "I have a question: 2 complementary angles are in the ratio of 3:2, how do u find the (reduced) ratio of their supplementary angles?",
        "A": "according to me the answer is 36*2=72 SOLUTION Let the no. be x so the values are 2x+3x=180 (180 is the sum of supplementary angles) 5x=180 x=180/5 x=36 Therefore 2x=72 Hope I am correct....."
    },
    {
        "video_name": "XDOH9Ul2OOM",
        "Q": "At 1:21 can we simplify \"2x + 46 + 3x - 6 = 90\" to 5x +40=90??",
        "A": "You are able to simplify 2x+46+3x-6=90. You start by adding the x s together. 2+3=5 so it is 5x. then you have 46 + (-6), which is equal to 46-6, or 40. put the 2 terms together, and add the =90, you have your equation 5x+40=90"
    },
    {
        "video_name": "XDOH9Ul2OOM",
        "Q": "At 0:42, what do the lines mean that are above OL+ON?",
        "A": "An angle is formed by two rays. The symbol for a ray is what you see above each. They are not lines because they have an endpoint O (the vertex of the angle)."
    },
    {
        "video_name": "XDOH9Ul2OOM",
        "Q": "What does X stand for in the movie titled \"Example using algebra to find measure of complementary angles\". It was 1:12. It would be very helpful if someone could give me an answer.",
        "A": "x stands for one of the numbers that add up to the measurement of the angle"
    },
    {
        "video_name": "XDOH9Ul2OOM",
        "Q": "in 1:32 you substract 46-6 so in equations do u always substract the integers that are by the x ?",
        "A": "It depends on the equation. The only reason he subtracted 46-6 was because he had to simplify the equation. The equation was 2x+46+3x-6=90 46 and -6 are common. He subtracted 6 from 46 because the 6 was negative. However he added the integers with the x because they were both positive. Like I said before it depends on the equation."
    },
    {
        "video_name": "nb3m_Q7rfT4",
        "Q": "Would like to know , why would we need ti find the net gain at first if we can just come up with the \"equation\" N=10X-15 in the first place. If I came up with that eqt in the first place then I can just skip to 2:50 and just come up with the answers ( as the original mean and SD were given)",
        "A": "You don t need it. Sal does it in the video only for illustration purposes."
    },
    {
        "video_name": "b8q6i_XPyhk",
        "Q": "@3:35 cm2, not the cm3",
        "A": "Sal made a mistake there. What he ment to say was cm2."
    },
    {
        "video_name": "b8q6i_XPyhk",
        "Q": "Why is salmon-pink referred to as orange at ~2:35?",
        "A": "haha. probably not. He is pretty accurate with his other colors... :D"
    },
    {
        "video_name": "b8q6i_XPyhk",
        "Q": "what is that orange thing that jumped up right in the corner on 3:15",
        "A": "A Skype thing. I have not heard of Skype a lot, but it says Skype, if I m correct."
    },
    {
        "video_name": "b8q6i_XPyhk",
        "Q": "At 3:10 why do you half it?",
        "A": "To create triangles."
    },
    {
        "video_name": "b8q6i_XPyhk",
        "Q": "im not sure i understand... how do you know those sides are equivalent? i forgot how you determine it. at 0:50",
        "A": "The definition is kite is a quadrilateral with two sets of consecutive sides that are congruent."
    },
    {
        "video_name": "qBfc57x_RSg",
        "Q": "Between 16:20 and 17:45 Sal explains how A^tx=0 is equivalent to x^TA=0^t. This sounds very confuse because the left part of the equivalence creates a 2x1 matrix and the second part creates a 1x3 matrix. The left part i just need to equal to zero, but what should I do with the right part?\nAlso, just to be sure, a transpose of zero matrix is zero right?",
        "A": "A is a n\u00e2\u00a8\u0089m matrix, so A\u00e1\u00b5\u0080x\u00e2\u0083\u0097 is a m\u00e2\u00a8\u00891 column vector. (since A\u00e1\u00b5\u0080 is a m\u00e2\u00a8\u0089n matrix and x\u00e2\u0083\u0097 is a m\u00e2\u00a8\u00891 column vector.) Therefore in the equation A\u00e1\u00b5\u0080x\u00e2\u0083\u0097 = 0\u00e2\u0083\u0097, the 0\u00e2\u0083\u0097 is the zero m\u00e2\u00a8\u00891 column vector. Now, the equation x\u00e2\u0083\u0097\u00e1\u00b5\u0080A is a 1\u00e2\u00a8\u0089m row vector (since x\u00e2\u0083\u0097\u00e1\u00b5\u0080 is a 1xm row vector), and therefore 0\u00e2\u0083\u0097\u00e1\u00b5\u0080 is also the zero 1\u00e2\u00a8\u0089m row vector."
    },
    {
        "video_name": "qBfc57x_RSg",
        "Q": "At about 6:50, Sal says that x1 is the pivot entry. But couldn't we have solved for any of the other x's also, and then expressed them in the same way?",
        "A": "Yes, he could. Depending on which row you put on top, and which column comes first and second, and how you do the elimination, you could make any of the 3 variables the pivot variable. Also notice that in this problem the span of any of the column vectors is the same as span of any other."
    },
    {
        "video_name": "qBfc57x_RSg",
        "Q": "At 14:45, does Sal mean \"column space of the R transpose matrix\"?\nif not, what is \"column span\"?\nHe sometimes says \"column span\", and it confused me.",
        "A": "I think by that he can only mean the span of all the column vectors which is the column space. What is the definition of column space? It s the span of all the column vectors."
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "at 6:02 if 4x^2/4 is same as that of X^2, i dont understand from where 4 come from.\ni didnt understand what is the idea behind 4x^2/4, why SAL making complicated with number 4",
        "A": "Turtle, Sal had BD\u00c2\u00b2 + x\u00c2\u00b2/4 = x\u00c2\u00b2 I ll fill in some steps Sal left out To solve for BD\u00c2\u00b2 you need to subtract x\u00c2\u00b2/4 from both sides. BD\u00c2\u00b2 + x\u00c2\u00b2/4 - x\u00c2\u00b2/4 = x\u00c2\u00b2 - x\u00c2\u00b2/4 The x\u00c2\u00b2/4-x\u00c2\u00b2/4 becomes zero and disappears BD\u00c2\u00b2 = x\u00c2\u00b2 - x\u00c2\u00b2/4 To add or subtract fractions we need to have a common denominator. and 4/4 = 1 and we can multiply anything by 1 without changing it, so we multiply the first x\u00c2\u00b2 by 4/4 BD\u00c2\u00b2 = 4x\u00c2\u00b2/4 -1x\u00c2\u00b2/4 Now we can subtract the fractions. BD\u00c2\u00b2 = 3x\u00c2\u00b2/4 I hope that helps make it click for you."
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "I'm so confused. At 5:46, How does he go from (BD)squared + (X/2)squared = Xsquared to BDsquared + X/4 squared = 4/4Xsquared?",
        "A": "Did he just find the square of 1/2 to get 4? Why is x considered 1 ?"
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "ok starting at 0:27 Sal says that the 30-60-90 triangles sides are hypotenuse = x and the short leg = x over 2 (x/2) and the long leg = square root 3 x over 2 (root 3 x/2) where I was taught that it is short leg would = x the long leg would = x root 3 and the hypotenuse would = 2x. am I wrong? or is it just a different way of doing it? have I been doing it wrong?",
        "A": "They are equivalents. It is a different way of doing it because they are all just divided by 2. I guess his way has a little more meaning on the unit circle - do not know if you have dealt with this topic yet Lets do it Mathematically with the Pythagorean Theorem: x^2 + (x SQRT(3))^2 = (2x)^2 or x^2 +3x^2 = 4 x^2 which is true for you (x/2)^2 + (xSQRT(3))/2)^2 = x^2, so x^2/4 + 3x^2/4 = x^2, then 4 x^2 /4 = x^2, and dividing by 4, x^2 = x^2 which is also true"
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "At 5:50, isn't (BD)^2 equivalent to B^2 D^2 rather than BD^2? Because I read BD^2 as B(D^2), not B times D the quantity squared. When you take the square root of B(D^2) you get the sqrt. of B times D. I know it's a bit persnickety of me to point out a minor notation issue but that confused me for a second.",
        "A": "He meant to keep the parentheses in, but he didn t, probably to save time. It can be inferred that there are still supposed to be parentheses around these variables. However, since he treats the problem as the parentheses are still there, not writing them doesn t affect anything."
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "At 6:48, how do we supposed to find b in the 30-60-90 triangle",
        "A": "in 30, 60, 90 its s, s times the square root of three, and 2s Respectively"
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "At 0:55, where did Sal get square root of 3 times x divided by 2?",
        "A": "In a 30-60-90 triangle, the side adjacent to the 30 degree angle would always be the measure of the side adjacent to the 60 degree angle times \u00e2\u0088\u009a3. In the video, the side adjacent to the 60 degree angle is x/2. The side adjacent to the 30 degree angle would be x/2*\u00e2\u0088\u009a3 would give you \u00e2\u0088\u009a3x/2. Hope this helped."
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "Got another question: At around 5:51, after Sal squares everything, shouldn't BD squared be B squared D squared? If I am wrong, can someone please explain to me why?",
        "A": "BD refers to the length of segment BD which means that it is a single value. You misinterpreted BD as being B * D. If that were the case, then you would be correct."
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "At 0:56, he says that the hypotenuse is x, the shortest side is x/2, and the side opposite the 60-degree angle is (\u00e2\u0088\u009a3)x /2. My teacher calls the smallest side s, the side opposite the 60-degree angle s\u00e2\u0088\u009a3, and the hypotenuse 2s. These both mean the same thing, right?",
        "A": "That s correct if you change (\u00e2\u0088\u009a3)x/2 to \u00e2\u0088\u009a3(x/2)."
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "at 3:35 he says \"congruent\".What does congruent mean?",
        "A": "Congruent means that two objects have the same size and shape. So if you were to move one object, you could make it overlap another exactly. For triangles, we often want to know when they are congruent. The various posutlates/theorems (SSS, ASA, AAS, ASA) tell you when you have enough information to know if two triangles are congruent."
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "how do we know that it is x/2 at 4:21\n\nI haven't taken geometry, but I am taking trig.",
        "A": "We know that segment AC equals x , triangle ABC is equilateral, and triangle ABC has a median starting at point B going through segment AC. Any median of an equilateral triangle will bisect the opposite side, or the side it goes through. So, if segment AC is the side the median is bisecting, and x divided by 2 equals x/2, we can conclude that each part of segment AC will be x/2."
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "are all of them 30-60-90? 4:40",
        "A": "Yes and no. The equilateral triangle by whole is not 30-60-90 it s 60-60-60. But when you slice it in half, we ll end up with two 30-60-90 triangles."
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "I don't understand where the 4 comes from at 5:57? any help would be great.",
        "A": "The four comes from working out (x/2)^2 = (x^2)/(2^2) = (x^2)/4."
    },
    {
        "video_name": "SFL4stapeUs",
        "Q": "5:54: where does that 4 come from?",
        "A": "You need to distribute the exponent 2 to x and to 2. When you do that, you get x^2/2^2. When you do 2^2, you get 4. Hope this helps you out."
    },
    {
        "video_name": "m5Tf6vgoJtQ",
        "Q": "On the first problem (1:15), why did Sal change the percent difference of 96.5% to a decimal --> 0.965, but then later (3:20) used it in the equation and called it percentages? I thought you would just leave it as a percent or just solve for the decimal ?",
        "A": "The two forms are equivalent. You should be able to fluidly go back and forth between forms of the same number, including referencing a context for the number. For example, if you had the ratio 3:4, and we used it in a problem as the fraction 3/4, I hope you would not get upset with me when I called 3/4 a ratio."
    },
    {
        "video_name": "m5Tf6vgoJtQ",
        "Q": "at 0:58 how did he come up with .965?",
        "A": "since decay is 3.5 % when you move the decimal twice, you get .035 1-.035 = .965"
    },
    {
        "video_name": "m5Tf6vgoJtQ",
        "Q": "at 5:20, I don't understand why you use 1.08, and at 1:27, why did you use 0.965 and not 0.035?",
        "A": "at 1:27 The substance decays by 3.5%. This means you take away 3.5% from the original. The original = 100% 100% - 3.5% = 96.5% or 0.965 at 5:20 The number of stores increases 8% each year. This means you are adding 8% to the original. The original = 100% 100% + 8% = 108% or 1.08 Hope this helps."
    },
    {
        "video_name": "m5Tf6vgoJtQ",
        "Q": "Do you have to make a chart/table like Sal did at 0:39?",
        "A": "No, but it can make visualizing the question and the exponential values easier."
    },
    {
        "video_name": "m5Tf6vgoJtQ",
        "Q": "Can anybody explain why did he use the number 1 at 4:48 plz?",
        "A": "The number 1 (Assuming you mean in the table) represents the number of years after 1999, so one year after 1999 would be 2000. Hope this helped! :)"
    },
    {
        "video_name": "m5Tf6vgoJtQ",
        "Q": "At 1:50, why do you do multiply 0.965 by 0.965? I understand that 0.035 is taken away from each hour so wouldn't you just do 0.965-0.035= 0.93? I also clearly understand how you multiply by 100 to make a percent.",
        "A": "Each hour, you take away 3.5% of what is left. If you subtract 3.5% each hour, you are assuming that the base is still 100% But, after the 1st hour, you no longer have 100%. You have 96.5%. Thus, the new starting point, is 96.5%, not 100%. This is why we multiply 0.965*0.965 It calculates 3% of the new base. hope this helps."
    },
    {
        "video_name": "m5Tf6vgoJtQ",
        "Q": "At 4:53, isn't 8% the same as 0.08, not 1.08? If not, why? And at 5:57, they say it should by 1.8. Aren't BOTH of those numbers wrong?",
        "A": "at 4:53 : 8% increase, that makes the total number of store is 1.08 * 200 (nothing wrong withat statement. at 5:57 : the speaker said 1.8. It is an error, and I am sure he meant 1.08 (not 1.8). You may see this type of error in many videos. Follow the procedure to report those errors. So the system will correct them for future use."
    },
    {
        "video_name": "m5Tf6vgoJtQ",
        "Q": "@5:25 I still don't understand why Sal adds the \"1\" and get 1.08. Could anypne explain this to me clearer.",
        "A": "they do this so your answer wont be this long decimal. that way when you multiply it, it makes your answer"
    },
    {
        "video_name": "m5Tf6vgoJtQ",
        "Q": "Where did he get 1.08 for the Nadia problem? (It starts coming up at 4:50)",
        "A": "It increases every year by 8%. So if you set the first year to 100% you have 108% in the second. Percent means per hundred so 100% means 100/100 or 1.00. 108% = 108/100 = 1.08. So you can either multiply the 200 stores by 1.08 or by 108/100. See also the videos about percents somewhere around here."
    },
    {
        "video_name": "m5Tf6vgoJtQ",
        "Q": "at 4:44, where did he get that 1 in 1.08 please help ??\nthanks",
        "A": "The idea is that, if there s exponential GROWTH, when the stores grow by 8%, that s 8% more than the original amount, or 108% total. To represent this as a decimal, we use 1.08"
    },
    {
        "video_name": "1UQ5IbihJNI",
        "Q": "At 0:11, why did Sal say \"4 units wide\" instead of \"2 units wide?\"",
        "A": "Ther is a small comment on the bottom right of the video that says Sal said 4 units wide but meant 2 units wide ."
    },
    {
        "video_name": "1UQ5IbihJNI",
        "Q": "0:19 whats a right pyramid",
        "A": "A three dimensional right triangle."
    },
    {
        "video_name": "qbMe4f2yvKs",
        "Q": "it's 1:16 and does 9 have to be boxed in red?",
        "A": "beaches its a different place value"
    },
    {
        "video_name": "qbMe4f2yvKs",
        "Q": "0:14...Sal said to write 12.98 as 12+ 0.98 why not just .98?",
        "A": "You can write .98 and it would be correct. However, many people write a 0 before the decimal point, to call attention to the fact that there is a decimal point. Since whole numbers don t have a zero in front, then when you see the leading zero you are more likely to notice the decimal point."
    },
    {
        "video_name": "qbMe4f2yvKs",
        "Q": "in 0:53 how is the common denominator like 9 over 10 turns into 90 over 100?",
        "A": "Sal has multiplied both the numerator and denominator by 10."
    },
    {
        "video_name": "qbMe4f2yvKs",
        "Q": "At 1:36 he says we have to simplify the fraction. It it considered wrong if we don't simplify it?",
        "A": "No. Simplified fractions are exactly the same as the fraction you have, such as 6/9 and 2/3. Simplified just means that you can t get a fraction equal to it that is in smaller numbers."
    },
    {
        "video_name": "4IWfJ7-CYfE",
        "Q": "at 1:20 why can't 3.2=32 divided by 10 be 32 divided by 100?",
        "A": "because 32/100 is 0.32 since we move two times to the left what we basicly do when we divide by 10 or 100 or any number of a 1 and zeros we move to the left a number of times equal to the number of zeroes for example 32/1000000 is 0.000032 wo moved six times to the left"
    },
    {
        "video_name": "4IWfJ7-CYfE",
        "Q": "I'm sure he was just seeing if everyone was listening at 3:54, but 6 + 3 is not 13.\nHe meant to say 6 + 7 is 13.",
        "A": "yes , 6+7 = 13 and not 6+3 = 13 the latter is completely wrong"
    },
    {
        "video_name": "4IWfJ7-CYfE",
        "Q": "3:54 he said 6 plus 3 equals 13.",
        "A": "I agree with Nick"
    },
    {
        "video_name": "4IWfJ7-CYfE",
        "Q": "what does aggeregate mean at 5:38",
        "A": "To over react"
    },
    {
        "video_name": "yPZ7In2tdOQ",
        "Q": "At 0:37 where did the other x^2 go? because i thought it was an x^4... ??",
        "A": "He factor it out like (x^2 + 7) (x^2 + 1), because the x^4 is a common factor, so if you multiply the parentheses one by the other you get the original polynomial. Hope it makes sense."
    },
    {
        "video_name": "yPZ7In2tdOQ",
        "Q": "At 1:18 why is it x^2*x^2. Why is it not x^3*x",
        "A": "Good point! However, x^2*x^2 is actually equivalent to x^3*x. Since x can be written as x^1 and when multiplying exponents with the same base (x) we add the powers, x^2*x^2 = x^4 and x^3*x^1 = x^4"
    },
    {
        "video_name": "poGjjX_VNbM",
        "Q": "At 5:40 how come you don't subtract the denomenator",
        "A": "Because the denominator is the thing your adding"
    },
    {
        "video_name": "poGjjX_VNbM",
        "Q": "why did we let 5/8 like that only in 4:19",
        "A": "Can you please explain your question better?"
    },
    {
        "video_name": "Yh4TXMVq9eg",
        "Q": "5:27 how do you do inequalities with varibles",
        "A": "Do you realize you just linked to 5:27 when this video is only 5 minutes and 9 seconds long?"
    },
    {
        "video_name": "Yh4TXMVq9eg",
        "Q": "At 4:11 , why is 4<4 correct but 10>10 incorrect?",
        "A": "Look closely. It is not 4 < 4, it is 4 \u00e2\u0089\u00a4 4. It IS the case that 4 is less than or equal to 4, that is why the inequality holds. For the other, it is impossible that 10 > 10, so the inequality here does not hold."
    },
    {
        "video_name": "Yh4TXMVq9eg",
        "Q": "People at 2:19-2:20 why does 2 not satisfy the right hand side inequality and 0 does?",
        "A": "On the right hand inequality, if you substitute 0 in for X, you get 4 > 0. This is a true statement. 4 is greater than zero, always. So, if X=0, the inequality is true. If you substitute 2 for X in the same inequality, you get 10>10. This is a false statement. 10 is not greater than 10. If the inequality had been 10 >= 10, then it would be true. A value is determined to be a solution if it creates an inequality that is true. A value is not a solution is it makes the inequality false."
    },
    {
        "video_name": "Yh4TXMVq9eg",
        "Q": "at 2:51 how would i get to that particular part",
        "A": "He plugged in the x and y values of a point given to see if it satisfied both inequalities. The check mark means it did satisfy it."
    },
    {
        "video_name": "XstL_x4ucm4",
        "Q": "I don't really understand what he said at 5:01, can someone please help.",
        "A": "Do you mean the word scalar? A scalar is a quantity or magnitude that doesn t have any other parameters associated with it, such as direction. For example 4 feet is a _scalar _ quantity, and 4 feet north, is what is called a vector because it has a direction with it. A scalar is just a number without a direction."
    },
    {
        "video_name": "XstL_x4ucm4",
        "Q": "At 5:00, Sal said that the solution of the system is not going to change as long you as multiply both sides of the equation by a scaler. Would dividing both sides of the equation give the same result?",
        "A": "Yes, as long as you divide the entire equation by the same value."
    },
    {
        "video_name": "XstL_x4ucm4",
        "Q": "What is a scalar? 5:02",
        "A": "A scalar is just a number. It simply has a magnitude. Like 5 seconds, or 3 miles etc. This contrasts with vectors that specify both a magnitude and a direction. For example 3 miles to the west."
    },
    {
        "video_name": "evmDZkDvlNw",
        "Q": "Does it matter the order of the denominator towards the end of the problem (as shown in 3:03 of the video)? For example, would it matter if I did (3x+1)(8x+7), rather than (8x+7)(3x+1)?",
        "A": "The order does not matter (8x+7)(3x+1) is the same as (3x+1)(8x+7) based upon the commutative property of multiplication. Hope this helps."
    },
    {
        "video_name": "evmDZkDvlNw",
        "Q": "3:23 I'm a little confused on the (-6x^3) distributing to (8x+7). I didn't know you could do that. I thought a number is not supposed to have parhenthesis around it in order to multiply numbers in parhenthesis.",
        "A": "The parenthesis around -6x^3 is to show that it is negative. Otherwise it would be (8x+7)-6x^3."
    },
    {
        "video_name": "evmDZkDvlNw",
        "Q": "At 2:05, why did we change - to + and then changed 6x cube as -6x cube?",
        "A": "we had to distribute the negative"
    },
    {
        "video_name": "R-NeYKSEqns",
        "Q": "2:14: the P(A/B) = 1/2?\nthe probability of A given that B has happened.....\n\nwhat is the probability that B happens:? : 1/3\n\nwhat is the probability that A happens once B has happened? 1/2 out of that initial 1/3\n\nso P(A/givenB) should be 1/6",
        "A": "P(A | B) means (roughly) the probability that A happens once B has already happened, or 1/2. What you describe is the probability of A and B: P(A & B) = P(B) * P(A | B) = 1/3 * 1/2 = 1/6"
    },
    {
        "video_name": "C3HFAyigqoY",
        "Q": "Wouldn't the slope of the terminal ray (around 0:55) be 1 if it's a unit circle?",
        "A": "No, the length of the segment would be 1, but the slope would be different due to it s having a different angle."
    },
    {
        "video_name": "1JT9oODsClE",
        "Q": "At 6:20 , how np > 5 come ?",
        "A": "Thank you Marie"
    },
    {
        "video_name": "1JT9oODsClE",
        "Q": "at 5:09, shouldn't the standard deviation be the square root of 150(0.3)(0.7) instead of just square root of (0.3)(0.7)? I vaguely recall that the variance of a binomial distribution is n(1 - p)(p)",
        "A": "The distribution is a Bernouli distribution not a Binomial distribution. Please understand this. It s very essential that you do. In case of a Bernouli distribution, the mean is p and the variance is (p)*(1-p). Thanks for asking this question, as I myself was confused and referred to the concept."
    },
    {
        "video_name": "1JT9oODsClE",
        "Q": "Since this is a sample (around 5:45), why do we not divide the sample proportions by n-1?",
        "A": "Because you are assuming the null hypothesis is true, and 0.3 represents the true population proportion, you are actually calculating the true standard deviation of the population and not estimating it with the sample standard deviation. Therefore, you divide by n"
    },
    {
        "video_name": "b9H22F0Qbgw",
        "Q": "at 5:59, how does this skill apply in real life?",
        "A": "Well, of course it would apply to testing, but it also applies to certain jobs. For example, coordinate planes are used in graphing and mapping. Hope this helps!"
    },
    {
        "video_name": "E3OfSGjpDoI",
        "Q": "on 2:45 i know why because the last digit is in the 100ths place",
        "A": "Wonderful! Now use this knowledge to answer the questions and rule the world!"
    },
    {
        "video_name": "E3OfSGjpDoI",
        "Q": "at 3:00 after 35 hundreds there should be 2 zeros",
        "A": "He s saying hundredths, not hundreds. Adding two zeroes at the end of the equation would add nothing, and as a rule unnecessary 0000s at the end of decimals are omitted."
    },
    {
        "video_name": "E3OfSGjpDoI",
        "Q": "From 0:09 to 0:11, Sal mentions to re-write 21/60 so that 100 is the denominator. Isn't it the same thing as saying to re-write it as a percentage?",
        "A": "Yep, you could say it that way, they are just different ways of phrasing one thing."
    },
    {
        "video_name": "cRboY08YG8g",
        "Q": "2:45 But what are the odds that \"12x^2 - 200x + 600\" just happens to be in the correct format to use it in the quadratic formula? What if it were: \"-200x + 600 = 0\" or \"12x^3 - 200x + 600 = 0\"?",
        "A": "Hi Steven. That is an interesting question. Actually it happens to be a quadratic by chance, or more precisely, because the people who invented that exercise wanted the solver to combine his knowledge in calculus and the quadratic formula. In many other cases it is not so simple to get an exact answer, and you would have to use the graphing method Sal explained or make some numerical approximations (either to maximize the function or to set the derivative equal to zero)."
    },
    {
        "video_name": "cRboY08YG8g",
        "Q": "At 5:39, how will the x's overlap with each other if x is greater than 10?",
        "A": "Sal explains that the value greater than 10 yields an answer which has no meaning. Don the waste your time with answers which have no meaning. Learn the optimization methods instead."
    },
    {
        "video_name": "xoXYirs2Mzw",
        "Q": "What does Sal mean at 0:30 when he said you would divide by that amount per group?? Please help",
        "A": "She wants (1/5) of a can in each group, so that is the amount she wants in each group or it is the amount per group that she wants."
    },
    {
        "video_name": "xoXYirs2Mzw",
        "Q": "in 0:00 \u00e2\u0088\u0092 2:31 what does potpourri mean?",
        "A": "Definition of Potpourri : A mixture of dried petals and spices placed in a bowl to perfume a room."
    },
    {
        "video_name": "xoXYirs2Mzw",
        "Q": "At 2:33, Sal said \"single bonds\" but meant \"covalent bonds.\"",
        "A": "dude there is noot even 2:33 in this video it goes up to 2:31"
    },
    {
        "video_name": "xoXYirs2Mzw",
        "Q": "At 2:18, what is the reciprocal?",
        "A": "The reciprocal would be when you switch the numerator and denominator. When you multiply the reciprocal and your former fraction, you should get a product of 1. For example, let s say you have the fraction 5/9. The reciprocal of it is 9/5, so 5/9 (your fraction) x 9/5 (reciprocal) = 1. Hope this answered your question! Happy New Year!"
    },
    {
        "video_name": "WsZj5Rb6do8",
        "Q": "At 2:42, is that Desmos Graphing Calculator?",
        "A": "Yes, it certainly seems like it."
    },
    {
        "video_name": "WsZj5Rb6do8",
        "Q": "At 2:41 in the video we see that the contour lines for z=1, not intersecting at point (0,0) in the X-Y plane.\n\nBut at 3:46 in the video, Grant marks a contour line as z=1 which intersects at pint (0,0) in the X-Y plane.\n\nCan please someone explain this difference?",
        "A": "I believe those are not the same graphs, or its some kind of an error."
    },
    {
        "video_name": "KFgvOQtH0Z0",
        "Q": "At 0:29, isn't that more than or equal to?",
        "A": "What Sal said at that time is correct. He just made an example to find the probability of getting a number equals or less than 2."
    },
    {
        "video_name": "KFgvOQtH0Z0",
        "Q": "At 6:30, Sal talks about the chance of the sun rising. Is that theoretical or experimental probability?",
        "A": "its theoretical because you would hope that sun rises tomorrow morning but it might not happen"
    },
    {
        "video_name": "njb6xYF0GZ0",
        "Q": "Why do you need more info at 4:30?",
        "A": "Because the broccoli and spinach both decreased by two thirds. Since the ratio is the same, we can t determine the difference in quantity, and therefore can t solve the problem."
    },
    {
        "video_name": "EbmgLiSVACU",
        "Q": "At 2:09, Sal said \"5 times 2 point 3\" but it looks like 5 times 2 times 3.\nIs there any other method (besides * and the dot) to show multiplication?",
        "A": "Another way is to use no symbol but to have parentheses around the second factor. It would look like this: 5(2.3) Of course if you have no variables you can use the multiplication sign, \u00c3\u0097 When a dot is printed in a book, it is usually very clear that it is the multiplication sign or decimal depending on the size (multiplication sign is bigger) and position (multiplication dot is higher). When writing it is better to use the star or parentheses to avoid confusion with decimals."
    },
    {
        "video_name": "EbmgLiSVACU",
        "Q": "At 3:23 the denominator comes as 11.50000*10 to power 2! Here can i not move the decimal as per the rule and make it 1.150000*10 to power 3 ? I went ahead and did it the ans was 1.2500*10 to power 1 !! CAN I DO LIKE THIS ? IS IT CORRECT ??",
        "A": "Hi. Yeah, you shall!! And the reason he didn`t do this is because we are still in the middle of the problem and it is easier to work with 11.5, rather than with 1.15. However, either you do this with 1.15 or 11.5, the final answer should be the same [8*10^2, in this case]. As you have got a different answer, let me do this again with 1.15. 9.2 * 10^5/ 11.5 * 10^2 ==> 9.2 * 10^5/ 1.15 * 10^3 ==> 9.2/1.15 * 10^5/10^3 ==> 8 * 10^2. Hope this helps."
    },
    {
        "video_name": "EbmgLiSVACU",
        "Q": "At 3:32 shouldn't he have converted 11.5 to scientific notation first for consistency's sake?",
        "A": "Yes he could have. I believe he did not do that for the sake of working with easier numbers."
    },
    {
        "video_name": "EbmgLiSVACU",
        "Q": "At 2:06 there is a 4.6 in the numerator and a 2.3 in the denominator. Can this be simplified to a 2 in the numerator prior to performing the multiplication so that the division isn't quite as cumbersome?",
        "A": "Good eye! Yes, that would be a great example of simplifying ahead of your calculations to make the problem much simpler. If Sal would have seen that, he could have easily done 4 divided by 5 in his head to find 0.8."
    },
    {
        "video_name": "gHTH6PKfpMc",
        "Q": "At 3:32 sal added a 0 to his answer when do you add a zero to your answer when dividing.",
        "A": "You do that when you run out of numbers for the problem. If you don t have any numbers left in the dividend that already haven t been used then you add a 0 so you could figure out the rest of the problem."
    },
    {
        "video_name": "gHTH6PKfpMc",
        "Q": "At 0:48 sal says 6 is more than 25",
        "A": "I don t think he did but if he did, it was probably a mistake."
    },
    {
        "video_name": "gHTH6PKfpMc",
        "Q": "At 3:27 why does Sal add a zero to the top of his answer ,and when do you do this?",
        "A": "If you write the decimal point in the 6250, you have 6250. You need to have digits all the way up to the decimal point. He still had a blank space between his answer and the decimal point, so it got filled in with a zero."
    },
    {
        "video_name": "gHTH6PKfpMc",
        "Q": "at 7:53 he says \"plus 2\", he means \"plus 5\"",
        "A": "you re right Dandanfrank!"
    },
    {
        "video_name": "gHTH6PKfpMc",
        "Q": "at 8:15, Sal says the number below 597 has to be smaller, but what do i do if its bigger?",
        "A": "yes thats true"
    },
    {
        "video_name": "gHTH6PKfpMc",
        "Q": "DIVISION LEVEL 4 IS EASY WHEN I LEARN IN 3RD GRADE. Like 4 digit number divide by 2 digit ; for example: 6250 divided by 2 in 0:44.How about 4000 divided by 20?",
        "A": "that is a lot more simple 4 divided by two is two and the two zeros cancel out so 200 is the answer"
    },
    {
        "video_name": "gHTH6PKfpMc",
        "Q": "3:07 why 5 time...??",
        "A": "Sal knew that multiplying by 5 would give him the result he wanted. If you don t know to do times 5, you need to guess and check. You might guess 4 and see that it is a little low. You might guess 6 and see that it is too much. Finally, you settle on 5."
    },
    {
        "video_name": "gHTH6PKfpMc",
        "Q": "At 7:59 he said +2 instead of +5. Why?",
        "A": "He made a verbal typo. No big deal."
    },
    {
        "video_name": "gHTH6PKfpMc",
        "Q": "What do you mean 6 is more than 25 0:43\ndid you mean less?",
        "A": "He meant 6000 is more than 25 at 0:43"
    },
    {
        "video_name": "gHTH6PKfpMc",
        "Q": "At 3:37 is that a 8 or 0",
        "A": "It is an 0, Sal just put a 0 under it to subtract them since 25 goes into 0 zero times."
    },
    {
        "video_name": "TrQTK-B4bzM",
        "Q": "At 3:35, Sal ruled out the second possible answer, but I think it may not be right. What if the function looks like a parabola curve crossing over the origin, i.e. reaching a zero value for f(x). This would make the second answer possible right?",
        "A": "f(c) could be equal to zero, but it is not guaranteed by the intermediate value theorem. The only thing that you could say that must be true based on the intermediate value theorem is that the function must touch all possible values between 3 and 6 on that interval."
    },
    {
        "video_name": "eytyVWA5ZQs",
        "Q": "At 6:55, how Sal came up with the limit expression intuitively?",
        "A": "Because Sal wanted to prove lim x\u00e2\u0086\u0092c (f(x)) = f(c) if f(x) is continuous at c, he simply reduced f(c) from both sides to get [lim x\u00e2\u0086\u0092c f(x)] - f(c) = 0 But f(c) is a constant, so we can put it inside the limit to get lim x\u00e2\u0086\u0092c (f(x) - f(x) = 0 After this, Sal has to show that this is true whenever the function is continuous at c."
    },
    {
        "video_name": "eytyVWA5ZQs",
        "Q": "At 8:58, Sal said that the limit of X minus C as X is approaching C is 0. Why is this?",
        "A": "Let s use an example with a value for c. let c=5. as x gets closer to 5, the difference between x and 5 gets smaller and smaller. x can get as close to 5 as you want and x - c, or as we have decided in this example, 5 - 5 = 0 So can you see that as x gets closer and closer to any value c the difference between x and c gets closer and closer to zero. In math speak we describe that situation by saying the limit of x-c as x approaches c is zero ."
    },
    {
        "video_name": "KSrnZMAfwTM",
        "Q": "At 03:33 Sal writes +5 but don't you have to add 4? Besides, 18+5 does not equal 22.",
        "A": "That was an error that he made. He was supposed to write +4 instead."
    },
    {
        "video_name": "KSrnZMAfwTM",
        "Q": "At 3:32, why did Sal write +5 instead of +4?",
        "A": "It was an error, there is a little note of the mistake in the bottom right hand corner, during the video, at that time."
    },
    {
        "video_name": "BOIA9wsM4ok",
        "Q": "Just to clarify, could you do that problem in a different approach? Like at 2:50, couldn't you multiply both sides by 5, and then subtract by 0.6? Because I found out it gives you the same answer. Am I right, or him? (Probably him), but please help.",
        "A": "At 2:50, YES, you could multiply both sides by 5, but remember that you have to multiply EVERY term by 5, so on the left side you have n/5 times 5 plus 0.6 times 5. On the right side you have 2 times 5. That gives you n + 3 = 10. Then, you subtract 3 (not 0.6) from both sides to get n = 7, which is the correct answer that Sal got. Hope this helps. Good Luck."
    },
    {
        "video_name": "BOIA9wsM4ok",
        "Q": "At 6:02 , he just gets rid of the 0.5 on the left hand side. But doesn't he need to divide the (r+2.75) by 0.5 too? I am so confused.",
        "A": "He does not just get rid of the 0.5. What he does is anything divided by itself is 1. He divided 0.5/0.5 which gives him 1."
    },
    {
        "video_name": "BOIA9wsM4ok",
        "Q": "at 4:00 why does he multiply by 5",
        "A": "People cancel n because is a variable."
    },
    {
        "video_name": "LGnL17ESs3Q",
        "Q": "at 1:32, Sal says that in previous videos, it is shown that the measure of an inscribed angle is half the measure of the arc it intercepts. I can't find any video on Khan Academy showing this? All I can find are videos proving that the measure of an inscribed angle is half the measure of a central angle that subtends the same arc, which seems related to, but not quite the same as what he is getting at here. Could anyone point me in the direction of the video/ explain the connection between these two ideas?",
        "A": "I am having the same problem. There actually aren t any videos. I had to google it up..."
    },
    {
        "video_name": "LGnL17ESs3Q",
        "Q": "At 3:24, why does Sal do it as 1/2(360-2x)?",
        "A": "The inscribed angle theorem which was shown in the video before this one. An inscribed angle ( one that touches the circle) is related to the angle it subtends by a scale factor of two, If I know the angle, I double it to get the arc (which he did at the beginning to get 2x), and if I know the arc, I cut it in half to get the angle which is where the expression you asked about comes from. I assume you are okay with the 360-2x to be all of the circle not part of the 2x."
    },
    {
        "video_name": "LGnL17ESs3Q",
        "Q": "At 2:16, if we are given the measurement of 3 arcs out of the 4 arcs formed by a quadrilateral how can we solve the missing arc using 2 angle measurements that add up to 180 in the quadrilateral?",
        "A": "I am not sure what you are talking about 3 out of 4 arcs. Sal is showing the proof that opposite angles of inscribed quadrilateral are supplementary by showing only two arcs, one yellow and one blue which must add up to 360 because they go around the whole circle. Are you asking a question unrelated to the video because he is unconcerned about specific values?"
    },
    {
        "video_name": "vkhYFml0w6c",
        "Q": "at 6:11 why do you divide",
        "A": "Because he had 2 y s, but he needed it to be only one. In math we like to know what one of something equals. If he had not divided then it would be like someone asking what your age is and you saying two times my age is 30. Which, while maybe true, is not really helpful. At 6:42 he tells use that one y is equal to 2, which is as simple as an equation can get."
    },
    {
        "video_name": "vkhYFml0w6c",
        "Q": "at 2:09, what would you do if your equation is -3x+4y-5=-14?",
        "A": "you would have to do subsitution etc. (I forgot how to do it so don t ask) :("
    },
    {
        "video_name": "vkhYFml0w6c",
        "Q": "At 1:18, why does it matter if you put the y first on the right side instead of the 7; for example: why can it not be 3y +3=7+y? Why does it have to be 3y+3=y+7?",
        "A": "You don t, but in algebra it is common to write the variable first if on the other side of the equation, the variable is first. Just makes it easier to isolate y."
    },
    {
        "video_name": "vkhYFml0w6c",
        "Q": "At 2:15, why would you subtract 3 from 7 and 3? Could you subtract 7 instead? And does it have to be the lowest number?",
        "A": "Okay this is the equation, 3y+3=y+7 take away three from both sides so it equals 3y=y+4. Then subtract y from both sides, that should equal 2y=4. Then divide two from both sides so your left with y=2. You re trying to get the numbers to the right side of the equation and the letter on the left side of the equation."
    },
    {
        "video_name": "vkhYFml0w6c",
        "Q": "couldn't he have subtracted 7 from both sides instead of doing y first and still got the same awnser? 2:36",
        "A": "It is easier to do get the y s on one side first"
    },
    {
        "video_name": "vkhYFml0w6c",
        "Q": "@2:25 does it matter what side of the equation the variable is on?",
        "A": "No, it does not, x=1 and 1=x is the same thing. just like 1+1=2+0 and 2+0=1+1 are the same thing."
    },
    {
        "video_name": "5a6zpfl50go",
        "Q": "At 1:12 he says that every point on the line represents the equation. Is there a chance that a point can NOt represent the equation?",
        "A": "With such a simple equation, y is dependent on x and x is allowed to take on any value from negative infinity to positive infinity. Now since it s a straight line, it s trivial to find a straight edge to draw through a handful of points and extend the line as far as you want. Those points on the line will still be a solution to the equation. (Assuming that the line was drawn correctly). The points not on the line though do not represent the equation."
    },
    {
        "video_name": "5a6zpfl50go",
        "Q": "at 1:25, how did he get the slope as 1????",
        "A": "slope=m=change in y/change in x"
    },
    {
        "video_name": "5a6zpfl50go",
        "Q": "Toward the end of the video around 7:39 when you wrote: \"3+3(3)=6. How did that equal nine? I am confused on those types of problems.",
        "A": "I m afraid you must have misheard and not seen it clearly for Sal wrote 3=3(3)-6, which is indeed true. What he said was that 3x3 equals 9 and 9 minus 6 equals 3. Well, I hope this helped clear up any confusion you had! :D"
    },
    {
        "video_name": "5a6zpfl50go",
        "Q": "At 0:35, how did you get the slope from the equation? Also, I understand that x would be 1/1, but what if it was -5x? Then what would the x point be?",
        "A": "if it was -5x as a slope, since slope is y/x, and -5x is the same as -5/1x, the -5 is y. In other words, for every time that y goes down 5 points, x goes to the right 1."
    },
    {
        "video_name": "5a6zpfl50go",
        "Q": "At 2:00, why isn't the the point on -1? Why is it on the zero?",
        "A": "(0, 3) is the only point that fits both equations. That means it is the only value of x and value of y that makes the two equations true at the same time. A point involving -1 will not do that."
    },
    {
        "video_name": "5a6zpfl50go",
        "Q": "At 6:18 How did he get a negivive slope? Didn't he say it was positive?",
        "A": "A lines slope on a graph works the same weather it is positive or negative."
    },
    {
        "video_name": "5a6zpfl50go",
        "Q": "At 0:47, what does it exactly mean to have a slope of 1?",
        "A": "It means that for if there is a change in x of 1 , then there is a change in y of 1. Example: __________ ----x----|----y---- ----1----|-----1--- ---2----|-----2--- ---3-----|-----3---- ...and so on."
    },
    {
        "video_name": "5a6zpfl50go",
        "Q": "At 2:07, Sal just switched the direction of the line since x was negative, are you able to do this in any situation if the x is negative, but the rest of the equation is the same?",
        "A": "The principle behind this is that if the equation says y = -x + 3, that means the slope is -1 (-1 multiplied by x equals -x). The slope is negative. But if y = x + 3, the slope is 1, positive 1. So just remember, if the equation y = mx + b and m is negative (ie: -4, -1, etc.) the slope is negative. If m is positive, the slope is positive."
    },
    {
        "video_name": "5a6zpfl50go",
        "Q": "At 2:07, why does it move down and right simply because it is negative ?",
        "A": "So if you have a negative number attached to your x you will go down but if there s no sign it s an implied positive so you move up"
    },
    {
        "video_name": "5a6zpfl50go",
        "Q": "GUys i dont get what he did on 6:13 to 6 :30 i dont know he was talking about pls help",
        "A": "At 6:13 to 6:30 the teacher is plotting the second equation y = -x + 6 on the graph"
    },
    {
        "video_name": "5a6zpfl50go",
        "Q": "At 5:47, why do you go up twice with one? For example, the slope is three, so you rise 3 and run 1. I don't see why you would rise 3, run 1, then rise 3 and run 1 again.",
        "A": "It s not wrong that he did that. In a way, by plotting more points, he (Sal) can graph a more accurate line than by just eye-balling the line and drawing it. As you can see, when he did rise 3, run 1, and then again and again, he still drew a line that satisfied the equation y= 3x-6."
    },
    {
        "video_name": "qGTYSAeLTOE",
        "Q": "2:30 is the same as 2/30 right or am i wrong",
        "A": "it really depends on what website you are on but i know that 2/30 is the same as 2 divided by 30 and 0.0666666667"
    },
    {
        "video_name": "qGTYSAeLTOE",
        "Q": "At 3:00, could you write 35 to 60, because there's 60 minutes in an hour or the correct way to write is 35 to 1 ?",
        "A": "It depends on what unit you are using. If you are using hours, then you can not because you were using minutes, not hours. If you are saying, I am going 35 miles per sixty minutes it is completely different from I am going 35 miles per hour ."
    },
    {
        "video_name": "qGTYSAeLTOE",
        "Q": "Can you do it in fractions what I mean is....... 1/2:8 the 8 is the people per serve. 1/2:8....8 times 8 which is 64. you use that 8 and multiply to 1/2",
        "A": "No, you don t use them together, also that example makes the same answer without the ratio in it"
    },
    {
        "video_name": "EFdlFoHI_0I",
        "Q": "At around 6:26, I don't understand why he's referring to the degrees of freedom as m*n. I'm only familiar with df as \"the number of values in the final calculation of a statistic that are free to vary\" (wikipedia) and that it is used in virtually every hypothesis test",
        "A": "He is using m*n as the number of values horizontally times the number vertically to get the total number of data points in the set. He s actually equating df to (m*n)-1. If you don t understand why df=(m*n)-1 take a look at the question above this one."
    },
    {
        "video_name": "R0JY8oWsuOY",
        "Q": "At 17:05 why did sal choose the unit step function instead of the direct delta function when doing the inverse laplace transform because isn't the equation for the laplace transform of the unit step function e^(-as)/s , not just e^(-as). a=alpha",
        "A": "He s getting the unit step function from the definition he has chosen of the Laplace transform of a shift in time (see 3:03). It makes it clear that prior to t=2pi, that whole term is equal to zero. I have a text where a shift in time is written as L{f(t-a)}=e^(-as) L{f(t)}. Where it is understood that there is nothing before t=0 (because of the definition of the Laplace transform). So the unit step function is unnecessary. It seems like he is writing it as if you wanted to input it into a computer program."
    },
    {
        "video_name": "Upw1b0EohIM",
        "Q": "May seem off-topic, but why doe Mr. Khan repeat himself so much? Especially at 0:37 - 0:45?",
        "A": "It could be because he is restating what he said earlier while writing it, just to make sure if you missed what he said the first time, you hear it again. He might also record his videos without editing or in one take, and (from personal experience) if you are talking for a while, you tend to develop a habit of repeating phrases, especially while thinking about what to say next."
    },
    {
        "video_name": "Hur3v1hrX3U",
        "Q": "at 0:31, it said you should go into counterclockwise direction. Why cant we go in clockwise direction?\nAlso, when you are dealing with shapes irregularly put in position, how would you know if it is 90 degrees?",
        "A": "Because the angle is positive, you rotate it counterclockwise."
    },
    {
        "video_name": "sjUhr0HkLUg",
        "Q": "At 1:15, Sal says \"a bunch of datas.\" Is this correct? I thought \"datum\" was singular and \"data\" was plural. If so, \"datas\" would be pluralizing an already-plural word. Wouldn't that be wrong?",
        "A": "You are correct. However, you have to remember that Sal is doing these videos pretty much without a script and in an informal way. I think he d be more careful in his usage if he were writing a paper, for example. The important thing to learn from these videos is the mathematics, not the grammar."
    },
    {
        "video_name": "sjUhr0HkLUg",
        "Q": "Why pi/2 is 90, isn't it 1.5707....? 2:15",
        "A": "Pi / 2 does equal 90 DEGREES. It also equals 1.5707..... RADIANS."
    },
    {
        "video_name": "sjUhr0HkLUg",
        "Q": "At 9:20 how do we know when to use brackets or parentheses?",
        "A": "Brackets vs. parentheses is dictated by whether or not you want to include the border number. So if the inequality has a greater than or EQUAL sign then you use brackets, if the inequality is strictly greater than (or less than) then you use parentheses as you do not want to include the border number. Example: x < 1 you would use parentheses as you do not want to include x=1, so (-inf, 1) x =< 1 you would use brackets at 1, so (-inf, 1]"
    },
    {
        "video_name": "sjUhr0HkLUg",
        "Q": "In 6:54, Sal starts talking about going into the \"negative direction\", but how is it possible to have negative theta? How can it be possible to have a negative angle measure?",
        "A": "A negative angle simple means to measure going clockwise instead of counterclockwise. So -pi/2 = 3pi/2."
    },
    {
        "video_name": "sjUhr0HkLUg",
        "Q": "At 5:45, why does he have to curve the tips of the graph before going back down? I thought he mentioned just \"connecting the dots\".",
        "A": "I guess its because if you actually do a table of values for all of the values in between pi/2 and pi, or 3pi/2 and 2pi, or any two points on his graph,we get that shape."
    },
    {
        "video_name": "sjUhr0HkLUg",
        "Q": "At 2:24, why is pi/2=90 Degrees and pi=180 Degrees?",
        "A": "How many diameters in a circumference? radii? How many radii in half a circumference? How many radians are subtended by (in) half a circumference? A quarter of a circumference?"
    },
    {
        "video_name": "sjUhr0HkLUg",
        "Q": "How did he know where to put pi/2 at 4:48?",
        "A": "He knows sin(pi/2) is 1, sin0 is 0, and sin(pi) is 0; so he locates pi/2 where he wants a maximum value of sin(theta)."
    },
    {
        "video_name": "sjUhr0HkLUg",
        "Q": "At 8:10, Sal states that the domain can be any real number but on the table and graph the line only goes from -1 to 1, so wouldn't the domain be -1 to 1 or am I forgetting something?",
        "A": "The input, or domain of the sine function is all real numbers, that is (-\u00e2\u0088\u009e,\u00e2\u0088\u009e) BUT, the output, or range of the sine function is from -1 to 1, that is [-1,1]."
    },
    {
        "video_name": "sjUhr0HkLUg",
        "Q": "At 2:18 and 2:19 Sal said \"pi/2 is the same thing as 90 degrees\". How had he got 90 degrees from pi/2 ?\nPLEASE EXPLAIN !",
        "A": "In radians 360 degrees = 2pi, 180 degrees = pi. If we divide pi by 2 we get 90 degrees, if we divide 2pi by 2 we get 180 degrees, just to confuse you (hope not) if we divide 2pi by 4 we get what? 90 degrees, get it?"
    },
    {
        "video_name": "sjUhr0HkLUg",
        "Q": "At 8:04, can't we say that domain of the sin(thetha) = (-infinity, +infinity) ??",
        "A": "You can do that. Sal wrote the domain in words. You have written it in interval notation. They mean the same thing."
    },
    {
        "video_name": "sjUhr0HkLUg",
        "Q": "Can we say that the domain of a sine function is minus infinity to plus infinity? Is that the same as saying the domain is 'all Real numbers' (at 8:11)?",
        "A": "Yes, the domain of a sine function is from negative to positive infinity. The domain is written as (-\u00e2\u0088\u009e,\u00e2\u0088\u009e) in interval notion. It it the same thing as the domain is all real numbers."
    },
    {
        "video_name": "sjUhr0HkLUg",
        "Q": "At 1:33,Is it possible to use all the radians on the unit circle?",
        "A": "Yes, Sal is just using the most common radians, but you can use all of them."
    },
    {
        "video_name": "waWYIKAfkWw",
        "Q": "At 2:07, the graph appears to be of a relation than a function? (Vertical Line Test)",
        "A": "No way, sin(x) is a function (trigonometric function); x^2+1 is a function too. The ratio between two fuctions should be also a function. I ve already made the vertical line test on that function and a I didn t find nothing. This is because every value of x takes only a value of y. Every x gives only a y=sin(x). Every x gives only a x^2+1. Every x gives only a x^2+1/sin(x)"
    },
    {
        "video_name": "K_PiPfYxtao",
        "Q": "At 2:17-2:24, Sal talks about the horizontal asymptote. Do exponential functions have vertical asymptotes?",
        "A": "Exponential-horizontal Logarithmic-vertical"
    },
    {
        "video_name": "EqNzr56h1Ic",
        "Q": "5:31, and at 1:00 i don't get it im confused",
        "A": "What are you confused about? Maybe rewatch the video."
    },
    {
        "video_name": "q19UnHEofOA",
        "Q": "what does quantity mean. haerd it on 2:02",
        "A": "It s similar an amount or count of something. So if I have 3 baskets, each with 10 apples, my total quantity of apples is 3 x 10, which is 30 apples."
    },
    {
        "video_name": "q19UnHEofOA",
        "Q": "At 0:25 why did you have to put parenthesis?",
        "A": "Sal just did that to show you where in the problem he was focusing. All he did was break up the whole number and the fraction. Another way you could have done it would be to multiply the denominator and the whole number and add it to the numerator to get just a fraction without it being a mixed number anymore!"
    },
    {
        "video_name": "q19UnHEofOA",
        "Q": "00:23 Do you always have to put a parentheses around a mixed number if there is a minus in front of it? I still don't get this. Why is he not viewing 11 12/16 - 5 8/16 as 11 + 12/16 - 5 + 8/16 like he normally would if that minus was a plus, or in other words if all terms were positive? What's going on here?\n\nI see that he is viewing the mixed number as a whole, as if there was a parentheses around the whole thing. But that's just it - there is no parentheses around it!?",
        "A": "When the negative has to apply to both 5 and 8/16, you can either write it as - 5 - 8/16, or as - ( 5 + 8/16 ). If you write - 5 + 8/16 (with no parentheses), that implies that only number 5 is negative and 8/16 is positive. (Which is not the case in this video)"
    },
    {
        "video_name": "q19UnHEofOA",
        "Q": "At 1:19, I don't get that. I don't really understand that, because it just confuses me. I have to do 4 3/4- 2 1/4, 5 3/8- 3 7/8, 2 3/10- 1 7/10, 6 5/12- 2 11/12 on this sheet and clearly I cannot do that.",
        "A": "Why can t you? Convert them all into improper fractions and do the math like any other subtraction problem. :)"
    },
    {
        "video_name": "q19UnHEofOA",
        "Q": "at 5:18 can you go back",
        "A": "Sorry, what exactly is your question?"
    },
    {
        "video_name": "sZMezOCZr40",
        "Q": "The first triangle he draws at 0:27 is it a right angled scalene because it has a right angle or is it just a scalene",
        "A": "You cannot say that it is a right scalene triangle because it does not have the right angle symbol (_|). If it did, you would have clear evidence that it is a right scalene triangle"
    },
    {
        "video_name": "sZMezOCZr40",
        "Q": "At 0:57, Sal said And in most CIRCLES you could also say this is isosceles because isosceles would be at least 2 sides being equal. The word circle doesn't make sense in this case. Was this a mistake or not?",
        "A": "Likely a mistake."
    },
    {
        "video_name": "sZMezOCZr40",
        "Q": "At 2:17, if you change the angles that are written to 45 degrees, wouldn't that make a right equilateral triangle?",
        "A": "Equalateral is 60\u00c2\u00ba,60\u00c2\u00ba,60\u00c2\u00ba. Iscolese right is 90\u00c2\u00ba,45\u00c2\u00ba,45\u00c2\u00ba."
    },
    {
        "video_name": "sZMezOCZr40",
        "Q": "At 3:09, wouldn't the first triangle be called a right scalene triangle, since it has an angle of exactly 90 degrees? Just checking! Thanks!",
        "A": "It can be called a right scalene triangle, but that is a type of scalene triangle, so it is easier just to call it a scalene triangle. Hope that helped! ;)"
    },
    {
        "video_name": "sZMezOCZr40",
        "Q": "At 0:17, would the scalene be an isosceles?",
        "A": "No. Isosceles triangle has two or more sides the same length. Scalene triangles have no sides similar."
    },
    {
        "video_name": "sZMezOCZr40",
        "Q": "2:23 Sal said interior angles is the in angles that are inside or outside of the triangle",
        "A": "interior angles are inside the shape"
    },
    {
        "video_name": "lEhf75ma7Ww",
        "Q": "but what if you have like, 1,1 and 1,2. what do you do? i'm confused at 0:15. what do you do?",
        "A": "Basically, if you input one x value you will get only one y . Ex. for getting one y: f(x) = 2x x = 3, then y is 6 Ex. for getting two y s: f(x) = sprt of x x = 9, y can equal 3 OR -3. Hope this helps"
    },
    {
        "video_name": "7Ig6kVZaWoU",
        "Q": "At 2:15 why do we have to take the log of both sides. Why can't it just be put into log form and then solved using graphics calculator?",
        "A": "That would work, but you can t always be guaranteed to have a graphing calculator with you."
    },
    {
        "video_name": "7Ig6kVZaWoU",
        "Q": "at 4:09, Sal mentions that Ln is Log base e.\nive heard of this, but dont understand why.\nwhats the fuss about log base e?",
        "A": "Unfortunately, if you ve been watching the videos in order, ln and e have not been introduced yet. They are not introduced until the Continuous compounding and e section, many videos later. When you get there, you ll see what all the fuss is about. :)"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "what does he mean by\"rational form\"\n01:24",
        "A": "It is improper to leave a root in the denominator of a fraction. Rationalizing means to rearrange the fraction so that the root is on top. In this case multiplying 1/sqrt(2) by sqrt(2)/sqrt(2) would do the trick. It works because sqrt(2)/sqrt(2) is equal to 1 and multiplying anything by one doesn t change it, however it does allow the denominator to be sqrt(2)^2 which = 2, so in the end you get sqrt(2)/2."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "at around 4:13, how does adding 360 degrees to the angle and 2\u00cf\u0080 to the length make all of the solutions part of the arcsine? I tried plugging it into my calculator but I just think I don't understand the concept",
        "A": "Add 360 degrees (or in terms of radians, 2\u00cf\u0080) to the angle theta has the same trig function value because one unit circle is (by definition) 360 degrees (or 2\u00cf\u0080). An analogy could be: The number 4, when divided by 3, leaves a remainder of 1. If you add multiples of 3 to 4, these numbers will still leave a remainder of 1. Hopefully, you can find the similarity between these situations."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 07:07 why is it that the angle is in the fourth quadrant and not also the third quadrant when there is a negative square root of three over two in it as well.",
        "A": "Technically you are right and later in Trig this will be explained. But for now Sal is talking about the principle answer. It s like square roots. Take the square root of say 4 for example. You wouldn t answer -2 although this is correct. You would answer 2, because this is the principle square root. Hope this helped"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "at \"8:12\" how does he know if the triangle is 30 60 90 triangle\nand how does he know ir the adjasent side's length is 1/2?\nsmb tell me plz!!!!!!",
        "A": "he knows because the side opposite to the 60 degree is 3*sqrt/2 of the hypotenuse and since the hypotenuse is 1 and the side the side opposite to the 60 degree is 3*sqrt/2 we know that is a 30 60 90 triangle watch the video on 30 60 90 triangles if u dont understand"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "Where is the video for this convention that Sal talks about at 5:50? I remember watching it but cannot find it.",
        "A": "The convention was first mentioned in this video when Sal explained it at 5:50. The reason it is used is to keep arcsin a function."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "If my current instructor were to give us the second problem in the video (begins at 6:50), arcsin (-rad3/2), she would want us to give what radians is in quadrant IV. Meaning, -60degrees in quadrant IV is 300degress, and in radians is 5pi/3. Is what's given in the video the same thing? The -pi/3 answer?",
        "A": "The correct answer would be -pi/3 because the range of arcsin() is from -pi/2 to pi/2 However, arcsin(-sqrt(3)/2) = - pi/3 rad = 5pi/3 rad Also both -60 degrees and 300 degrees lie in the IV quadrant and are equal"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "0:26 - 0:28 How! No seriously, how. \u00cf\u0080/4 = 45\u00c2\u00b0. I am confused. I really am.",
        "A": "\u00cf\u0080 radians is equal 180 degrees. Therefore, \u00cf\u0080/4 radians is equal 180/4 degrees. Do the math and then try it with other numbers such as 5\u00cf\u0080/4, and you will start to understand it."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 8:55, why do we multiply 60 degrees to pi rad over 180 degrees? I'm so confused",
        "A": "When you got to covert angle measured in degree to its corresponding radian measure , then you use that formula. Radian measure = Degree measure * ( pi / 180)"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 1:23 he multiplies it by the square root of 2 over the square root of two. Can somebody explain this to me?",
        "A": "it would make no difference if he didnt since it is the same thing as 1 he does it because it is the convention not to have a square root in the denominator"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 5:50, why is arcsine restricted to the first and fourth quadrants?",
        "A": "Take any point on a unit circle. This point is (x , y) where x = cos (\u00ce\u00b8) and y = sin (\u00ce\u00b8). Suppose the point is in second quadrant then sin(135) = sqrt( 2)/2 which is same as sin (45) = sqrt(2)/2. That is why the sin(\u00ce\u00b8) range is restricted to First Quadrant and Fourth Quadrant. The values are repeating itself. y = f(x) cannot take two values where -1 <= x <= 1."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 1:21, how did he get x= 1/\u00e2\u0088\u009a2?",
        "A": "We know that \u00f0\u009d\u009c\u008b (radians) = 180\u00c2\u00b0 \u00e2\u0087\u0094 \u00f0\u009d\u009c\u008b \u00e2\u0088\u0095 4 = 180\u00c2\u00b0 \u00e2\u0088\u0095 4 = 45\u00c2\u00b0 We also know that in a 45-45-90 triangle, both legs are of equal length. Let s call that length \u00f0\u009d\u0091\u00a5. Soh-Cah-Toa tells us that Sine = Opposite \u00e2\u0088\u0095 Hypotenuse. If the hypotenuse is 1, we get sin(\u00f0\u009d\u009c\u008b \u00e2\u0088\u0095 4) = \u00f0\u009d\u0091\u00a5 \u00e2\u0088\u0095 1 = \u00f0\u009d\u0091\u00a5. The Pythagorean Theorem tells us: \u00f0\u009d\u0091\u00a5\u00c2\u00b2 + \u00f0\u009d\u0091\u00a5\u00c2\u00b2 = 1\u00c2\u00b2 \u00e2\u0087\u0094 2\u00f0\u009d\u0091\u00a5\u00c2\u00b2 = 1 \u00e2\u0087\u0094 \u00f0\u009d\u0091\u00a5\u00c2\u00b2 = 1 \u00e2\u0088\u0095 2 Since we know that \u00f0\u009d\u0091\u00a5 \u00e2\u0089\u00a5 0, we can safely take the square root of both sides, giving us: \u00f0\u009d\u0091\u00a5 = \u00e2\u0088\u009a(1 \u00e2\u0088\u0095 2) = \u00e2\u0088\u009a1 \u00e2\u0088\u0095 \u00e2\u0088\u009a2 = 1 \u00e2\u0088\u0095 \u00e2\u0088\u009a2"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 7:10, why Sal is taking the value -sqrt(3)/2 as the y - value in the co-ordinate system?",
        "A": "He s solving for arcsine(-sqrt(3)/2)."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 4:06, what did Sal meant when he said \" But I could just keep adding 360 degrees or 2 pi and all of those would get to to same point of the unit circle\". I couldn't understand what he meant to say. Please help!",
        "A": "He was just saying that if a full rotation of 360 degrees was added to the angle, the terminal ray would arrive back at where it started from and therefore that larger angle would have the same sine as the angle we started with. He then goes on to discuss the domain of arcsine ( -1<= x <=1 ) and mention how the range of arcsine is restricted so that everyone knows which angle is being used."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 6:00, how can the range just be restricted between -Pi/2 and Pi/2?",
        "A": "Sine function is many to one function. It means that two or more elements of domain is connected to one element of range. This means that it s inverse mapping would be one to many relationship which would not satisfy one range value requirement to be a function. To fix this problem, we restrict the range of inverse sine function between -pi/2 to pi/2."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 06:49\nWhy does arcsin hav to be in quadrant I or IV?",
        "A": "you have to travel along the y axis +sine 1st quad, - in 4th."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 1:13, what happened to the 2 on the other side of the equation? The one that was the exponent of 1.",
        "A": "Sal applied the exponent to simplify: 1^2 = 1 * 1 = 1 Hope this helps."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "In 1:21 you said that the square root of 1/2 equals 1 over the square root of 2. Why is that? Is it like that with all fractions that you need to find the square root for?",
        "A": "To take the square root of any fraction, you do the square root of the numerator and the square root of the denominator. For example: sqrt(4/9) = sqrt(4) / sqrt(9) = 2/3 Hope this helps."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 0:58, how is Sal automatically able to determine that the hypotenuse is 1 on the enlarged triangle as concluded from the unit circle?",
        "A": "The enlarged triangle s hypotenuse is a radius of the unit circle. By definition all the radii of the unit circle are equal to 1. so the hypotenuse of the enlarged triangle is equal to 1."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 3:30, Sal asks what is the arcsin of root 2/2. He says the answer is pie/2. could he have also said 45 degrees or no since we're talking about radians?",
        "A": "The answer he gives is actually pi/4 (radians), which is the same as 45 degrees (as you say). He could have used degrees, instead, but it s customary to use radians in contexts such this one, and of course we must always be systematic: we cannot switch units in the middle of calculations (at least: not without making other changes in our results, to compensate for the change in units)."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 6:00, why is the domain restricted to between -1 and 1? And why is it the convention to only use the first and fourth quadrants?",
        "A": "and if the domain is restricted to -1 and 1, how can the answer be ~ -1.04?"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 6:10, I get the first quadrant being pi/2, but how is the fourth quadrant -pi/2? Or why?",
        "A": "The output of the sine function ranges from -1 to 1. All values of the sine function can be found in quadrants I and IV, with no duplications."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 8:06,wouldn't the y-length be square root of 3 and the x-length be 2, since sin= y/r?",
        "A": "No, you could label the y-length as sqrt(3) and the radius as 2, but in that case the x-length ought to be labeled as 1."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "Just wondering why the range of the arcsin is restricted to Quadrants I and IV if the domain is between -1 and 1? (6:01)",
        "A": "because if u go to the V quadrant or above 360 degrees then for each ratio there will be more than one angle, so the function can t exist. hope it helps."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 1:52, how did Sal calculate the sin of pi/4? I don't quite know how to calculate sin, cos, and tan. (And I didn't understand the radians part.)",
        "A": "I suggest you watch trigonometry videos on KA if you don t know how to compute trig functions for special angles. All those values are derived from the special right triangles. Watch radian videos as well if you are not sure how to use it. You can search these topics in the search bar."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "at 3:50, what is an arcsine?",
        "A": "The SINE of an angle is a RATIO of 2 sides of a right triangle ( opposite/hypotenuse). The ARCSINE is the ANGLE itself. So, for example, sin 30\u00c2\u00b0 = 1/2 but arcsine (1/2) = 30\u00c2\u00b0 ( In words, the angle whose sine is 1/2 equals 30\u00c2\u00b0.)"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 9:14, he says that the arcsin of negative root 3 over 2 is negative pi over 3, are there not two other answers within the 0 to 2 pi range, and how does he determine the restrictions for both examples",
        "A": "He s using the restriction that negative pi/2 is less than or equal to theta, which is less than or equal to pi/2, thus providing only one answer"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "Why is the Domain restricted to -1<= x <= 1 for arcsinx?\nWhy the restriction of the 1st and 4th quadrants. at 5:14 onwards",
        "A": "The definition of sine is y/r. r will always be greater than or equal to y, so it is impossible to get anything larger than 1."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 3:35, Sal says that arcsin sqrt(2)/2 = \u00cf\u0080/4, but doesn't it also equal 3\u00cf\u0080/4? On my calculator, it tells me \u00cf\u0080/4, but my unit circle tells me both. Thanks!",
        "A": "It s right You know:Sin(\u00cf\u0080-a)=sin a"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 2:33, Sal asked the question that for what angle x , the inverse sine of which will give root 2 over 2. But can't the same equation mean what is the cosec of root 2 over 2 degree angle and get the required magnitude (though it will give the ratio, but if i am just taking for magnitude) ?",
        "A": "When Sal says inverse , he means arcsine, not 1/sine. Inverse , not reciprocal . Mathematically, he s saying: sin(x) = \u00e2\u0088\u009a2/2 Solve for x. So, when you see sin\u00e2\u0081\u00bb\u00c2\u00b9(x), it means arcsine, not sin(x)\u00e2\u0081\u00bb\u00c2\u00b9. I know it s confusing, the notation is stupid."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "? Did I miss something? Sal doesn't explain at 1:00 1, x and another x in the triangle why? and at 1:09 to 1:30. Most of the video after I don't get it lol",
        "A": "Sal doesn t explain that the hypotenuse has length 1 because we are supposed to know he s talking about a triangle in the unit circle (the little diagram he draws first, at around 0:20). He uses x as his variable, and gives this length to both legs of the triangle because this is an isosceles triangle."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 1:22, how did you simplify x^2 = 1/2 into x = 1/\u00e2\u0088\u009a2, wouldn't the identity be x =\u00e2\u0088\u009a(a/b) or x = \u00e2\u0088\u009a(a)/ \u00e2\u0088\u009a(b)? Thanks",
        "A": "\u00e2\u0088\u009a(1/2) = (\u00e2\u0088\u009a1)/(\u00e2\u0088\u009a2) = 1/(\u00e2\u0088\u009a2) because \u00e2\u0088\u009a1 = 1"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "For 9:18, why does the answer to the problem have to be -pi/3 or can it be 5pi/3 too?",
        "A": "It is because we restricted it to -\u00cf\u0080/2 \u00e2\u0089\u00a4 \u00ce\u00b8 \u00e2\u0089\u00a4 \u00cf\u0080/2 -\u00cf\u0080/3 = -\u00cf\u0080/3 + n*2\u00cf\u0080 where n is an integer. So yes, it can be written as 5\u00cf\u0080/3"
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "what was that thing sal said at 4:00 about adding 360 and getting the value.",
        "A": "360 degree is a full turn. If you turn 360 degree you end up at the same spot. This is also true for multiples of 360 degrees like 720 degree, 1080 degrees and so on."
    },
    {
        "video_name": "JGU74wbZMLg",
        "Q": "At 0:30, why does Sal start from the horizontal line and not the vertical?",
        "A": "When we re talking about angles on the co-ordinate axes, their initial ray is always the positive X-axis."
    },
    {
        "video_name": "-YI6UC4qVEY",
        "Q": "at 3:57 how is 18*9*7*2=36*9*7?",
        "A": "Commutative Property of Multiplication states that A*B=B*A. Basically just means you can swap any terms inside multiplication around, as they will evaluate to the same value. So; 18*9*7*2=18*2*9*7, if you process the first step on the right side of the = sign (18*2, or 36), the equation becomes 18*9*7*2=36*9*7."
    },
    {
        "video_name": "-YI6UC4qVEY",
        "Q": "Whoa! I completely lost Sal at 3:46. Why did he multiply 2 * 18 and not 7 by 9? He also starting writing them in individual square roots and confused me even more.",
        "A": "He factored out those other square-root-able numbers (a.k.a. Square Numbers)."
    },
    {
        "video_name": "-YI6UC4qVEY",
        "Q": "At 4:26 how did he get 18\u00e2\u0088\u009a7? I understand the \u00e2\u0088\u009a7 part, because it was left over from the formula since there isn't a whole number \u00e2\u0088\u009a7, I just don't understand how he got the 18?",
        "A": "sqrt(18 * 9 * 7 * 2) = sqrt(36 * 9 * 7) because 18 * 2 = 36. This separates out the perfect square factors. So, sqrt(36) = 6 and sqrt(9) = 3 -> 6 * 3 = 18 The sqrt(7) is the remainder so the answer is 18 * sqrt(7)."
    },
    {
        "video_name": "-YI6UC4qVEY",
        "Q": "At 4:19, what does Sal mean by saying negative sq roots? I don't seem to see any in there.",
        "A": "Negative square roots are not used in normal calculations."
    },
    {
        "video_name": "-YI6UC4qVEY",
        "Q": "Ok, I understood Heron's theorem up to where he was getting the area. At the 3:45 mark, he gets 36, 9, and 7. Why does he multiply 18 by 2 and not by 9 and 7 as well? If anyone could help I'd appreciate it. Thanks!",
        "A": "He could have. It would have been the same answer. But he s looking for intermediate numbers which he can squareroot easily. 18x2 = 36. sqrt(36) = 6. sqrt(9) = 3. Because of the commutative property of multiplication (A*B = B*A or in this case a*b*c*d = d*a*b*c) he can swap any of the four numbers being multiplied around. He can also do 18*9*7*2 = 18*18*7 because 9*2 = 18. sqrt(18*18*7) is like sqrt(18*18) * sqrt(7) which is 18*sqrt(7). 18*sqrt(7) = sqrt(2268)"
    },
    {
        "video_name": "-YI6UC4qVEY",
        "Q": "I'm a little confused. During 4:10, why can't you just do the square root of 36*9*7 ? Why do the square root of 36 times the square root of 9 times the square root of 7 ? Does it even make a difference?",
        "A": "Techinically it doesn t matter, but you will notice that both the 36 and 9 have a perfect root. Being Sal, he noticed this already so he would already have the whole numbers pulled out. If you were to multiply the three together square root of 2268 which also simplifies back down to 12root7 hope this helps"
    },
    {
        "video_name": "-YI6UC4qVEY",
        "Q": "at 04:40 of Herons formula it shows 18 square 7. How do I find the actual area of the space?\nI found the answer to be 47.62352359916263 from another site but not sure how that answer was found.",
        "A": "18\u00e2\u0088\u009a7 is irrational meaning that the numbers go on forever without repetition in its decimal form. Your answer 47.62352359916263 is an approximation of 18\u00e2\u0088\u009a7. Since we cannot write an exact answer in decimal form, we keep it as a radical (square root). There are algorithms for approximating square roots much like long division. But these algorithms are often complex. If you want to learn how to evaluate square roots by hand, you can search how to calculate square roots by hand on Google."
    },
    {
        "video_name": "-YI6UC4qVEY",
        "Q": "At 1:35 S is a variable but shouldn't it be T (triangle) or A (area) what does S have to do with this?",
        "A": "S stands for semiperimeter. It is 1/2 of the perimeter of the triangle."
    },
    {
        "video_name": "-YI6UC4qVEY",
        "Q": "So I understand untill 4:00. Why can you just do 18 times 2, then suddenly you take the square root out of the remaining numbers then you multiply 9 by two again and leave the 7 to be squared. I'm not sure how to formulate my question, but this makes absolutely no sense to me. Both the squareroots of 2 and 7 are irrational numbers, so why would 18 times 2 be good and not 18 times 7?",
        "A": "18 times 2 gives you a perfect square, 36, but 18 times 7 does not give you a perfect square. He wants to group numbers into perfect squares so he can take the square root of them. Also, it s easier to multiply 18 and 2 than it is to multiply 18 and 7."
    },
    {
        "video_name": "DxSmZ0MDSxM",
        "Q": "at 0:33 if we assume that y is larger than x then, by calculation, solution is (23, 47). Is this valid?",
        "A": "actually yes"
    },
    {
        "video_name": "gR8-vRg6Yp0",
        "Q": "at 6:43 Sal says d^-t is approaching 0 as t approches infinity, but if t is approaching infinity then it means our 1/t value is gonna be smaller and smaller, which gonna make the \"t\" approach 0, and everything to the 0th power is 1, so it's it true to say that d^-t is approaching 1?",
        "A": "You are interpreting the expression incorrectly. And, thus your conclusion is incorrect. d^(-t) does not equal d^(1/t) The correct interpretation is: d^(-t) = 1 / d^t As t gets larger, then d also gets larger. And 1 divided by a very large number will get smaller and smaller, thus it approaches 0. Hope this helps."
    },
    {
        "video_name": "gR8-vRg6Yp0",
        "Q": "Isn't (at 7:12) -d^-t an asymptote?",
        "A": "It has an asymptote of y=0. (It s always negative.)"
    },
    {
        "video_name": "gR8-vRg6Yp0",
        "Q": "At 2:20, how do we get C^2^2 - 1^2 / c^2-1 equal to (c^2+1) - (c^2-1) / c^2+1 ? Where did the 1^2 pop up at?",
        "A": "From the 1 . (c^4 - 1)/(c^2 - 1) = ((c^2)^2 - 1^2)/(c^2 - 1) = ((c^2 - 1)(c^2 + 1))/(c^2 + 1) = c^2 - 1. This is factoring the difference of two squares, then cancelling common factors. Have there been videos yet on that?"
    },
    {
        "video_name": "MamrTJ7V_Vg",
        "Q": "AT 5:30 it's really confusing",
        "A": "Then rewatch it."
    },
    {
        "video_name": "MamrTJ7V_Vg",
        "Q": "What does he mean at 3:59 when he says the solution?",
        "A": "Solution is just a synonym for answer."
    },
    {
        "video_name": "MamrTJ7V_Vg",
        "Q": "At 3:34 in the video sal says it is square cm, he writes cube cm. Is that correct?",
        "A": "If it is a 2-D object you use units^2 and if it is 3-D you use units^3, so it depends on the shape if it is 2-D or 3-D, you decide."
    },
    {
        "video_name": "MamrTJ7V_Vg",
        "Q": "When he mentions the pink trapezoid, at 3:09, how would you be able to calculate the extra portion of the shape if it looks similar to the green shape, even though at the end of the video he says that the red shape is the most similar to the green one? Would you just use the formula for the area of a triangle?",
        "A": "You can tell that the extra portion is half the 5 square units sticking out at the end, so you could measure that area and divide by 2, or.... Use the area of a triangle formula like you said!"
    },
    {
        "video_name": "rYG1D5lUE4I",
        "Q": "at 1:54 when sal takes sqrt(1) he says that the answer is wrong. He is only looking at the principal square root. sqrt(1)=1 but it is also equal to -1 [ (-1)*(-1) =1 ]\nSo he got the answer",
        "A": "No, the error lies in assuming the distributive property of roots for complex numbers. In other words, you can t assume that sqrt(z*w) = sqrt(x)*sqrt(w) (you can t assume it holds for any root)."
    },
    {
        "video_name": "rYG1D5lUE4I",
        "Q": "3:52 Sal mentioned that the radical sign always implied the positive answer. However, in my experience, this hasn't been true... Especially in Calculus, square roots of a positive would always produce 2 answers, one root positive and the other negative (as if the +/- were generally implied). Is this not universally true in math, or is it just a mistake in the video?",
        "A": "The root square principle only holds true for any REAL number utilised in the number inside the root. An example of this would be as you indicated above where an answer can have two numbers. In this case, sqrt (4) = +-2 in this case. However, with regards to imaginary numbers, this is a little different as i or i^2=1. And theoretically, there are no two REAL numbers that make up -1. Hence one of these numbers must be an imaginary number of i."
    },
    {
        "video_name": "rYG1D5lUE4I",
        "Q": "0:10 what does principal square root mean?",
        "A": "When taking square roots of a positive number you always get two answers. For example, the square root of 4 can be either 2 or -2. This is because the square of either one is 4: (2)(2) = 4 = (-2)(-2). So if you want to specify that you mean the positive answer then you would say the principal square root . So...the principal square root of 4 is 2. The other one is -2."
    },
    {
        "video_name": "rYG1D5lUE4I",
        "Q": "what if a is negative but b is positive at 2:28",
        "A": "The property holds if a is negative but b is positive. As long as one or the other or both are positive, the property holds."
    },
    {
        "video_name": "rYG1D5lUE4I",
        "Q": "My math teacher taught me that root a * root b is the same as root ab always. But at 2:32 Sal says that isn't true if both numbers are negative. Does this mean that she taught it to me incorrectly?",
        "A": "Chuck is correct: the property is different if you have negatives. Here is how to do it correctly. So we don t get confused, let a and b be positive, and we have: (\u00e2\u0088\u009a-a)(\u00e2\u0088\u009a-b) = (i\u00e2\u0088\u009aa)(i\u00e2\u0088\u009ab) =i\u00c2\u00b2\u00e2\u0088\u009a(ab) =-\u00e2\u0088\u009a(ab) Thus, the product of the square roots of two negative numbers equals the negative of the square root of the product of the absolute value of those two numbers: \u00e2\u0088\u009a-7 * \u00e2\u0088\u009a-2 = - \u00e2\u0088\u009a14"
    },
    {
        "video_name": "rYG1D5lUE4I",
        "Q": "At 2:15, you said using the property when both numbers are negative is wrong. However, I tried applying the property to the square roots of positive numbers (and then factoring these to have both negative factors) and after working it out, I got the negative version of the answer. eg. for root 56, I got -2 times root 14. So my question is, what makes that property to be considered wrong, yet it provides an acceptable answer?",
        "A": "The principal root of a square root is always the default. The principle root is the positive root. You got the negative root as your default, which is what makes it incorrect."
    },
    {
        "video_name": "rYG1D5lUE4I",
        "Q": "At 3:55, if the radical symbol means principal square root or 'positive square root', how would you represent 'negative square root'?",
        "A": "To represent the negative square root, you could simply do -sqrt(x). Same notation as for the positive square root, just with a negative sign in front of it so it returns the negative answer."
    },
    {
        "video_name": "rYG1D5lUE4I",
        "Q": "In the beginning at 2:28 when Sal said a and b both can't be negative did he mean that not even one of them can be negative or is it that at least one of them is negative?",
        "A": "One of the numbers, either a or b, can be negative. The property does not work for both a and b<0, but it works if only one <0."
    },
    {
        "video_name": "rYG1D5lUE4I",
        "Q": "at 4:23, what is the pn that you are writing? What does it stand for?",
        "A": "At that time Sal was saying principal square root and he used pn to indicate principal ."
    },
    {
        "video_name": "hjigR_rHKDI",
        "Q": "5:24\n\nI understand the whole video right until that point. Why is he doing -2(4) -2 and -8 -2 = -10 ? Where is he grabbing those numbers from?",
        "A": "The numbers are the coordinates of the point that both equations meet. He is substituting them for the x and the y to see if he graphed them correctly. If he substituted them and they didn t work then that means that he graphed the equation wrong."
    },
    {
        "video_name": "hjigR_rHKDI",
        "Q": "At 0:33, he mentions an odd way for me to find the maximum. I usually use -b/2a, but it would equal 0/-2 in this case, which obviously is incorrect. Am I doing something wrong here?",
        "A": "You can use x = -b/(2a) to find the x-coordinate of the vertex. And, your value for x is ok. x = 0/(-2) = 0 If you substitute x=0 back into the equation, you will get y=6 So, the vertex is at (0, 6) which is the same value that Sal calculated."
    },
    {
        "video_name": "hjigR_rHKDI",
        "Q": "At 3:34 when Sal says if move back 2 we move up 2 in the y direction did he mean to say 4. Because the equation says a slope of -2/1",
        "A": "A slope of -2/1 is equivalent to -4/2 or -6/3. As long as use a fraction that is equal to the original slope, you are ok. Sal used the equivalent fraction of -4/2 (reduced it is = to -2/1)."
    },
    {
        "video_name": "hjigR_rHKDI",
        "Q": "what is a linear system at 4:02",
        "A": "It s a a system of linear equations. For example: x+y=2 x-y=0"
    },
    {
        "video_name": "UUyAoEQtToI",
        "Q": "(3:37) How can h be differentiable at 3 and 7? Aren't they the endpoints in the interval?",
        "A": "\u00e2\u0084\u008e(\u00f0\u009d\u0091\u00a5) might very well be differentiable over some interval (\u00f0\u009d\u0091\u008e, \u00f0\u009d\u0091\u008f), such that \u00f0\u009d\u0091\u008e < 3 and \u00f0\u009d\u0091\u008f > 7, and then \u00e2\u0084\u008e(\u00f0\u009d\u0091\u00a5) is of course differentiable at \u00f0\u009d\u0091\u00a5 = 3 and at \u00f0\u009d\u0091\u00a5 = 7."
    },
    {
        "video_name": "MHtTP6vc4RU",
        "Q": "At 1:24, Sal says that P(x) / x-a has the remainder of P(a). If this is the case, does that mean that if it was something like x+3, the remainder would be P(-3)? Thanks in advance!",
        "A": "Yes, if the divisor was (x+3), then the remainder = P(-3)"
    },
    {
        "video_name": "MHtTP6vc4RU",
        "Q": "At 1:45, I don't understand how he got -3x8-4x4+20-7? Was it that he just distributed the 2, but then how did he get rid of the x's in the equation?",
        "A": "he is talking about the reminder theorem. you can do it through long division P(x)/(x-a) and get the reminder or use r=P(a) to get the reminder which is an easier way to do it. the function was p(x)= -3x^3-4x^2+10x-7, what is the reminder of this equation if it was divided by (x-2). reminder = p(a), in this case x=2. you plugin 2 in to the Xs then you get -27 which is the reminder. so to answer your question, he did not distribute the 2s but plug it in to P(x) to get the reminder."
    },
    {
        "video_name": "MHtTP6vc4RU",
        "Q": "At 1:56, where does 20 come from ?",
        "A": "from evaluating the + 10x term where x=2."
    },
    {
        "video_name": "MHtTP6vc4RU",
        "Q": "At 3:18, when Sal says it's essentially to the zero degree, wouldn't that be incorrect? Since anything to the 0th power is 1 - which is a constant, I know - Sal saying that it being to the zero degree would be wrong because the constant could be any number.",
        "A": "In this case, he is referring to \u00e2\u0080\u009cx\u00e2\u0080\u009d. He means \u00e2\u0080\u009cx\u00e2\u0080\u009d to the 0th power, rather than the coefficient. Hope this helps!"
    },
    {
        "video_name": "pMFv6liWK4M",
        "Q": "From 14:20 and forward: Having a span defined by three vectors...\nI don't see why it's closed under multiplication. Couldn't you just multiply any of the three vectors with a negative number resulting a vector outside of the span/subset?\n\nEdit: I understand how it works on a span with one vector. I don't see how it works with multiple vectors.",
        "A": "As Sal says the span of the vectors is all linear combinations of them. This includes linear combinations with negative coefficients. So in the video those c_1, c_2,...,c_6 could be negative. Does this help?"
    },
    {
        "video_name": "pMFv6liWK4M",
        "Q": "At 13:15, should he have said \"falls out of our subset\" instead of subspace? Because he just proved that it wasn't a subset?",
        "A": "Yes. It falls out of Sal s proposed (but impossible) subspace, which is a subset of R^2."
    },
    {
        "video_name": "pMFv6liWK4M",
        "Q": "23:11, Sal states that the the vector 1 1 is a valid subspace.\n\nWould this be incorrect since since 1 1 is a valid subset instead of a subspace?\n\nIs my logic correct: All subspaces are subsets but not all subsets are subspaces. Thank you.",
        "A": "He did not say that the vector (1, 1) was a valid subspace, he said that the vector (1, 1) spans a subspace. Otherwise, your logic is correct; it is indeed so that every subspace is a subset, but not every subset is a subspace."
    },
    {
        "video_name": "pMFv6liWK4M",
        "Q": "At 14:00, If X1 were only equal to 0, then could S be an subspace?\n\nI mean, if x1=0 , so when you multiply the scalar A to x1, x1 would be 0. It makes S be an subspace?\n\nI hope you can understand my question, sorry about my english. I am from Brazil.",
        "A": "the zero vector space containing only (0) as a point is a vector space"
    },
    {
        "video_name": "pMFv6liWK4M",
        "Q": "when a specific vector is not \"closed under scalar addition and/or multiplication\", does that mean that the vector is outside of the subspace? (reference at 14:00)",
        "A": "At 14:00 Sal found a vector that is not a member of the mentioned subset. But as this vector is found by multiplying a vector of the subset by -1 the set is not closed under multiplication. Therefore the subset is not a subspace of R^n. Put another way: If a vector of a subset is multiplied with a scalar and the resulting vector is not a member of the set, than the subset cannot be a subspace of R^n. So you should replace the word vector with the word subset in your sentence."
    },
    {
        "video_name": "pMFv6liWK4M",
        "Q": "In roughly 10:19, Sal refers to the x-axis as x1 and y-axis as x2. Can anyone explain why this is?",
        "A": "It s a form of generalising. When dealing with only 2 or 3 dimensions it s easy to use the letter x, y and z, but this formulas apply to vectors on spaces of any size, so if you re dealing with vectors in 30 dimensions there would not be enough letters to label each of them, so it s common to use numbers instead."
    },
    {
        "video_name": "pMFv6liWK4M",
        "Q": "@ 22:55 of the video, we conclude that linear subspaces are closed under multiplication and closed under addition. I was wondering is there something like closure under division or subtraction. Since division is multiplication with an inversed number, and subtraction just addition of a negative value of some number, does something like closed under division or subtraction exist, and when do we use it ?",
        "A": "We would never note something as closed under subtraction and division because they are covered under addition and multiplication. As you say, subtraction is addition and division is multiplication so to reference them on their own would be redundant."
    },
    {
        "video_name": "pMFv6liWK4M",
        "Q": "How can a set of vectors, U, ever not have the zero vector? (16:33) Just multiply every vector by zero and you have the zero vector, right? :O",
        "A": "You re right. We can ALWAYS construct a linear combination from ANY set of vectors to get the zero vector."
    },
    {
        "video_name": "SBqnRja4CW4",
        "Q": "At 4:07 you use 2pi/3 when instead you could use tau/3. Why did you use pi?",
        "A": "Historical tradition. Historically \u00cf\u0080 has been used much more than \u00f0\u009d\u009c\u008f. So much in fact, that a lot of people in the world would not understand the meaning of \u00f0\u009d\u009c\u008f. That is why in almost every math book or course you will see only \u00cf\u0080 and almost never \u00f0\u009d\u009c\u008f. But it s good that you notice that it can be used, just be careful when you use it, since a lot of people might not understand the meaning of \u00f0\u009d\u009c\u008f, while all will always understand the meaning of \u00cf\u0080."
    },
    {
        "video_name": "SBqnRja4CW4",
        "Q": "When Sal says the amplitude of a sinusoid is half the difference between the minimum and maximum values at 0:17, does he mean the absolute value of this difference, since the number could be negative?",
        "A": "The absolute value of the difference is correct. The reason being is that the amplitude describes a distance, not a coordinate, and distances cannot be negative. Like Sal explained in the video, changing the sign ( - , or +) of the amplitude merely flips the function over it s midline; all of it s maximum points would become it s minimum points and vice versa."
    },
    {
        "video_name": "SBqnRja4CW4",
        "Q": "At 2:00 Sal explains why the amplitude of the equation is equal to the amount we factor our cosine, I didn't really get why it's only 1/2, y is determined also by the cos3x, why taking only the 1/2?",
        "A": "Remember that \u00f0\u009d\u0091\u00a6 = cos(\u00f0\u009d\u009c\u0083) has the range \u00e2\u0088\u00921 \u00e2\u0089\u00a4 \u00f0\u009d\u0091\u00a6 \u00e2\u0089\u00a4 1 In this case \u00f0\u009d\u009c\u0083 = 3\u00f0\u009d\u0091\u00a5, which doesn t affect the range of \u00f0\u009d\u0091\u00a6."
    },
    {
        "video_name": "SBqnRja4CW4",
        "Q": "At 6:40, he says that cos 0 is 1. Why is this?",
        "A": "If you look at a unit circle, the point at 0 degrees/0 radians doesn t has an opposite length 0, so the radius/hypotenuse is the same length as the adjacent side. That means they re both 1. Cosine = Adjacent/Hypotenuse = 1/1 = 1. Also, looking at the graph, at 0pi, the cosine graph starts at 1. (Remember cosine is just the x of the unit circle!) I hope that helped."
    },
    {
        "video_name": "SBqnRja4CW4",
        "Q": "at 4:08, how does he know to use radians for the x axis? I thought of it using degrees, so my thought was that the period would be 120 (360/3). Is it a convention not to use degrees for plotting trig functions?",
        "A": "You might say this is a convention, but it s also a matter of handling problems in the way that makes the most sense. Calling a circle 360 degrees is an arbitrary convention with no connection to the inner workings of math. Radian measure comes from the characteristics of a circle, so it s more organic or natural, even though it doesn t seem natural at first after a lifetime of thinking in degrees. Radian measure is ultimately more revealing, and allows proofs to proceed in a more direct path."
    },
    {
        "video_name": "SBqnRja4CW4",
        "Q": "at 7:40 you said that we completed another cycle. Wouldn't that have been the first cycle because the direction of the line wasn't the same at 2pi/4 as it was at the beginning thus altering the slope?",
        "A": "Sketch a graph of y = cos(3x) (don t worry about -1/2, and mark off your x axis in intervals of pi/6.) and find out where all the periods start and end. Then if you don t get the answer ask more questions."
    },
    {
        "video_name": "SBqnRja4CW4",
        "Q": "at 4:05 when Sal defines period since cos is being multiplied by 3 and it takes 2 pie to get a full circle is it true that he inverts the equation to get the cosine?\nis the cosine equal to the period?",
        "A": "3 is the angular frequency of this sinusoid, which means that it completes 3 cycles over a 2\u00cf\u0080 interval. So, to find its period, which is the interval that contains one cycle of the sinusoid, we divide 2\u00cf\u0080/3."
    },
    {
        "video_name": "SBqnRja4CW4",
        "Q": "at 1:42, Sal mentions why the amplitude is 0.5 . I didn't quite get it. could somebody please explain it more simply?",
        "A": "Cos(v) (and sin(v)) always vary between 1 and -1, because they treat v like an angle (either radians or degrees), so that 1 and -1 are the largest possible magnitudes of cos and sin, which vary continually between these two values. So what are the largest possible magnitudes of (1/2)cosv? -(1/2)cosv? What are their amplitudes? (Sketch their graphs if necessary. The amplitude is the absolute value of half the difference between max and min.)."
    },
    {
        "video_name": "SBqnRja4CW4",
        "Q": "At 4:02 Sal says the period is 2Pi / 3. I had guessed it would be 360 / 3? This got me thinking. Nowhere does it say anything about degrees or radians. It's just an equation with an x variable and a y variable, neither of which is an angle.\n\nHow do we know when to use degrees or radians? I get different y values for this equation depending on what I set my calculator to.",
        "A": "it often does not matter whether you use one or the other, as long as you are consistent. But if someone else has made the assumption for you, then you can usually tell by looking at the units on the x axis and the units of x in sin(x)"
    },
    {
        "video_name": "SBqnRja4CW4",
        "Q": "At 6:39 why is 0 of cosine 1?",
        "A": "0 is not cosine of 1. He said that cosine of 1 is 0. This is because when a line on the unit circle is at 0 degrees (or radians), the x value of the segment in the unit circle becomes 1 and the y value becomes 0. The cosine value of the angle on the unit circle refers to the x value of that point that it intersects the unit circle."
    },
    {
        "video_name": "SBqnRja4CW4",
        "Q": "At 4:07, why is sinx and cosx have 2\u00cf\u0080 as their period?",
        "A": "He answers that specifically, starting at about 4:40."
    },
    {
        "video_name": "SBqnRja4CW4",
        "Q": "At around 4:53, can you substitute 2pi fot Tau?",
        "A": "true. this is very true."
    },
    {
        "video_name": "wSiamij_i_k",
        "Q": "At about 2:00, Sal says that the functions -y+4 and -x+4 are the same thing. But, if we didn't change the y to an x in the first equation, would the graph be the inverse of the original equation?",
        "A": "In other words, if y stays in the equation, then the new equation would not be an inverse of the original, it would just be a solving for x instead of y. Right?"
    },
    {
        "video_name": "wSiamij_i_k",
        "Q": "At 0:32, shouldn't the opposite of f(x)=x+4 be -x-4 instead of -x+4?",
        "A": "If you are doing the same problem as in the video, it is f(x) = -x + 4 To find the inverse function, you solve y = -x + 4 for x. Add x to both sides: x + y = 4 Subtract y from both sides: x = -y + 4 So, inverse function is: f^-1(x) = -x + 4"
    },
    {
        "video_name": "wSiamij_i_k",
        "Q": "At 1:19, why does he change the Y to an X ?",
        "A": "That s how you get the inverse function."
    },
    {
        "video_name": "wSiamij_i_k",
        "Q": "At 2:21 how does Sal know that -x+4 is perpendicular to the line y=x. Is it because only lines perpendicular to y=x have the same inverse function or is there any other way of knowing?",
        "A": "There is another way of knowing, just take the negative reciprocal of the slope of the function and you have a function perpendicular to the original function. In this case we have f(x) = x and g(x)= -1x +4 we look for the slope (the number that is multiplied by x). In the case of f(x) the x is multiplied by 1 and in the case of g(x) the x is multiplied by -1. Guess what the negative reciprocal of 1 is... it is -1 :D (Reciprocal means switching denominator and numerator)"
    },
    {
        "video_name": "wSiamij_i_k",
        "Q": "At 4:50, where does the 1/2 come from?",
        "A": "Dividing both sides of the equation by -2"
    },
    {
        "video_name": "pYaIjA4CsKo",
        "Q": "what would 40:000.000.000.000.000.000.000.000.000.000.000.000.000.000. be?",
        "A": "4x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10 or around that"
    },
    {
        "video_name": "pYaIjA4CsKo",
        "Q": "At 2:37, am I wrong or is the answer really 5 instead of 6,700,000? The question is 'how many zeros...'? I can see 5 zeros..",
        "A": "yes you re right. The answer is 5"
    },
    {
        "video_name": "pYaIjA4CsKo",
        "Q": "The time \"3:45\" i was a bit confused , can someone explain?",
        "A": "Basically, when you multiply by ten you put another ten after the original number right? So when you divide by ten you take away a zero."
    },
    {
        "video_name": "pYaIjA4CsKo",
        "Q": "At 0:12, does 10^5 have to be 1*10*10*10*10*10? Can it just be 10*10*10*10*10? If so, why did he say 10^5 is 1*10*10*10*10*10 instead of 10*10*10*10*10 if it's the same thing?",
        "A": "It can be just 10 * 10 * 10 * 10 * 10 because anything times 1 is equal to the same thing that it was before it was multiplied! So adding a 1 to it is completely unnecessary."
    },
    {
        "video_name": "pYaIjA4CsKo",
        "Q": "2:35 Sal did not answer the question, instead he wrote the number in the standard form which is the question.",
        "A": "Well, the purpose of the videos are not to answer the question but to help you to figure it out yourself."
    },
    {
        "video_name": "pYaIjA4CsKo",
        "Q": "At 6:51 when you divided by a power of ten do you always get a decimal",
        "A": "You do in that example, but that is not always the case. If you divide by 10^1, that is like dividing by 10. 1 / 10 = 0.1 and that is a decimal, but 1000/10 is 100. That is not a decimal What about 10^2? (10^2 = 100) 1 / 10^2 is the same as 1 / 100 which equals 0.01. That s a decimal. What about 1000 / 10^2. That s the same as 1000 / 100 which equals 10. That s not a decimal."
    },
    {
        "video_name": "s9PvbFG2yJA",
        "Q": "at 0:48 would making the fraction bars into 6 equal pieces or even 12 equal pieces would that change the decimal and make it impossible?",
        "A": "At 0:56, it teaches decimals with a 2nd grid."
    },
    {
        "video_name": "ZjeMdXV0QMg",
        "Q": "6:17 where did you get that \"-3\"",
        "A": "One of his solutions is X < -2. The -3 would be in that solution set. Sal just picked a number from the solution set to see if it would work."
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "At around 7:30, how does x + 3 < 0 become y = -(x+3)?",
        "A": "Sal gives two options, x + 3 < 0 or >0. Less than zero , like you asked, means it s negative while greater than 0 id positive."
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "At 10:17, does the graph of an absolute value equation just look like the V or is the graph both lines?",
        "A": "I was also confused as to how he got the v, I watched the video and paused it several times. The graph of both lines form a v. y=-x-3. It forms a downward sloping line, or it has a negative slope.It s y intercept is at -3 and it s slope is -1. When y=x+3. It s y intercept is at 3 and it s slope is 1. It forms a upward sloping line, it has a positive slope.They will intersect at the same point, -3, and their lines will form a v shape."
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "At 6:20 how come Sal writes it as inequalities and not the equal sign?",
        "A": "The inequalities were used in order to graph the lines for the original function y = |x + 3|. It s to help sort out which lines need to be drawn for correct representation."
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "At 3:02 in the video it shows you how to do a problem with the equals sign, what do you do when there isnt one in the problem",
        "A": "You can t do anything except for simplifying, if possible."
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "on 8:58, i do not understand the graph. how does the graph work?",
        "A": "Do you know how the graphs of functions work? If you don t, I highly recommend that you watch some other videos on this site about them. But if you do, then remember this: With absolute value equations, you sort of get two functions once you solve them and graph them. Does that help out?"
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "Two related questions:\n1. At about 6:12, when Sal is discussing graphing y = the absolute value of x-3, he says there are two scenarios: x-3 is less than zero and x-3 is greater than zero. But couldn't x-3 equal zero?\n2. On the graph, he includes -3 on the x axis. But x is never equal to -3. It's either greater than or less than -3. So shouldn't the dot on -3 be hollow to show that the graph doesn't include -3 (unless x can equal - 3, which goes back to my first question)?",
        "A": "On number one X could equal zero I think..well..... yea....I guess"
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "at 4:59, why does |4x-1|=19, they never explicitly explained how they got 19 out of the equation",
        "A": "I guess I don t understand your question... the 19 is given as part of the original problem. It wasn t created by the equation."
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "at 6:36 why x+3 can be > 0 or < 0 instead of x-3=-y or x-3=y ?? i didnt understand the relationship between an absolute value of a function",
        "A": "ummm i don t know"
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "At 2:07 Khan says both would satisfy the equation? Does that mean 2 right answers or do you have to just guess which one?",
        "A": "both answers will work"
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "At 2:04, why is x - 5 = 10 the same as x - 5 = -10?\nThanks!",
        "A": "The problem starts at: |x - 5| = 10 What number could be inside the absolute value and create a 10? There are 2 numbers: 10 or -10 |10| = 10 and |-10| = 10 So to solve the absolute value equation, you have to deal with both scenarios. You need a version where the x-5 = 10 (the 1st scenarnio); and a version where x-5 = -10 (the 2nd scenario). Hope this helps."
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "At the time 2:14 I did not quite understand how x-5 could be either 10 or -10.",
        "A": "Because x-5 is in the absolute value. Absolute value means that any number placed inside of it will result in a positive result. This means that |10| = 10 and |-10|=10. Therefore how do we know if the input was the positive or negative number? That is why we cover our bases and check both options."
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "At 10:50, if the pink line represents where x is greater or less than -3, should the x intercept be an open circle? It's exactly -3.",
        "A": "No, the original function is an equation. X can = -3. It creates the vertex. with the inequalities, Sal is trying to show you how to determine where the rest of the graph will go. One side controls the graph when X> -3 and the other one controls the graph when X <-3. These are not meant to say that X = -3 is not a valid input to the function."
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "At 3:36 how did you get x to be -8?",
        "A": "He had x + 2 = -6 Now subtract 2 from both sides to get x + 2 - 2 = -6 - 2 => x = -8"
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "At 2:42 which one did he say was the solution? Also, why did he put -10 at the end? Wasn't the equation 10?",
        "A": "They are both the solution. Absolute value equations have two solutions, since you can have a x-value that makes the equation negative and another one that will make the equation positive, and both will be true since when you take the absolute value they both end up being positive. Sal sets the equation to -10 since we need to figure out which value will give us that negative answer that the absolute value then transforms to being positive."
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "In 2:04, why does both of them satisfy the equation?",
        "A": "The equation is saying that the absolute value of x-5 equals 10. The absolute value of a number is the same as the distance from 0 on a number line. -10 and 10 are both 10 away from 0, so they have the same absolute value even though they are different numbers. That s why there are two answers that satisfy the equation. If it were any other number (besides 0) then there would also be 2 answers."
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "at 2:30 - do you get to choose whether it's positive or negative? how do you know which one it is? thnx",
        "A": "You don t. :O MATH ISN T ALWAYS PERFECT!"
    },
    {
        "video_name": "u6zDpUL5RkU",
        "Q": "at 3:47, can you put parentheses around -2 when you are using an absolute value sign?",
        "A": "FunGuy, Yes, you could use parentheses. | x - (-2)| = 6 is perfectly acceptable."
    },
    {
        "video_name": "upO6Mh862PI",
        "Q": "At 5:32, Sal wrote a -1/pi. Why did he do this?",
        "A": "He wrote -1/pi to cancel the -pi in -pi*sin(pi*x) , since (-1/pi)*(-pi) = 1. That way he has the original function, sin(pi*x), because anything times 1 is itself."
    },
    {
        "video_name": "upO6Mh862PI",
        "Q": "isnt function derivative of sin of x equal to -cos of x? 4:18 he writes d/dxcospix and forgot the minus signal? :D",
        "A": "The derivative of sinx is positive cosx"
    },
    {
        "video_name": "upO6Mh862PI",
        "Q": "Maybe I'm mistaken, but did Sal forget to include a pi when he did the derivative of cos(pi*x) at 5:00?\n\nAlso, I don't understand why Sal multiplied -1/pi to -pisinpix when he rewrote the antiderivative of sin(pix). at 6:20 he says he an just carry it over becuase its a constant but how did he just mulitply it randomly?",
        "A": "He multiplied it intentionally, as a counter operation to multiplying by pi. This way he is able to complete the antiderivative as pi cosine pi x without changing the overall integral expression."
    },
    {
        "video_name": "NEaLgGi4Vh4",
        "Q": "@ 1:00 what is the n=5 for if he already has n=1?",
        "A": "He s just giving another example."
    },
    {
        "video_name": "NEaLgGi4Vh4",
        "Q": "At 1:30, he was explaining how x to the zeroth power is 1. He said any number except for zero follows this rule. What is zero to the zero power then?",
        "A": "0^0 is considered by most mathematicians to be 1. Some say it is undefined. It does not involve dividing by zero."
    },
    {
        "video_name": "NEaLgGi4Vh4",
        "Q": "at 2:22\nwhy is it expressed as 4 times 5^1 -2 times 5^0? why isnt it just 4 times 5 - 2 times 5?",
        "A": "If you look at the original expression, you ll see that it s 4 times n^1 - 2 times n^0 To evaluate the expression where n=5, you put the number 5 wherever n is. So, 4 times n^1 - 2 times n^0 becomes 4 times 5^1 - 2 times n^0 It s not just 4 times 5 - 2 times 5 because the original expression tells us that exponents are needed."
    },
    {
        "video_name": "NEaLgGi4Vh4",
        "Q": "At 1:15, isn't 0^0 still one? 0 to the power of 0? That's one, right?",
        "A": "When x\u00e2\u0089\u00a00, the follwing are both true. x^0 = 1 :any number to the power of 0 equal 1 and 0^x = 0 :zero to any power equal 0. But both cannot be true when x=0 if 0^0 = 1 (because any number to the power of zero is 1) and 0^0 = 0 (because 0 to any power is 0) then 0 = 0^0 = 1 so 0=1. This contradiction simply cannot be. So which is it? 0^0 = 0 or 0^0=1 ? There are some good arguments to say 0^0=1, and it can be useful in several situations, but it is not always true. So we generally say 0^0 is undefined. 0^0 is indeterminate."
    },
    {
        "video_name": "NEaLgGi4Vh4",
        "Q": "wait.... that would be = to -18, right? not 18.\nat 2:55, i mean.",
        "A": "ohhhh. thanks! i thought it was because it was -20 + 2 but i get it now. thanks alot!"
    },
    {
        "video_name": "NEaLgGi4Vh4",
        "Q": "From 2:46 to the end i do not get it. how 5 to the power 1 equal to 1?",
        "A": "Here are some rules about exponents. I m afraid you just have to memorize them: Any number (other than 0) to the power of 0 = 1 0 to the power of 0 is a meaningless expression and cannot be solved. 0 to any other power than 0 = 0 The equation was 4n - 2n^0 with n=5 Since we have a problem where we have a number (other than zero) raised to the power of zero, we know that equals 1. So, 5^0 = 1 Now we can solve our problem 4n - 2n^0 with n=5 4(5) - 2(1) = 20 - 2 = 18"
    },
    {
        "video_name": "NEaLgGi4Vh4",
        "Q": "at 0:14 why do n=1 first? I am a 4th grader.",
        "A": "so that n doesn t trip you up later on in the equation, where you see a whole bunch of numbers and letters, but have no idea what it means. Hope that helped!"
    },
    {
        "video_name": "NEaLgGi4Vh4",
        "Q": "At 0:45 why does he use that equation?",
        "A": "An example to make the equation easier to understand."
    },
    {
        "video_name": "pzQY-9Nmtws",
        "Q": "I didn't get the part at 4:50\nif we subtract a negative from another negative negative it is going to be positive>\nCan anyone please explain me the whole concept there?",
        "A": "It s not about subtracting a negative from a negative that turns it positive. It s the sign that changes. 2 = (=2) is the same as 2 + 2 which equals 4. In the video, it was =(=2) = (=2) = (=2) which is equal to +2 + 2 + 2 which is equal to 6. The way I do it is when I see a minus sign and a negative beside each other (e.g. *=(=*2)), I directly change it into a plus sign (e.g. =(=2) -> 2)."
    },
    {
        "video_name": "pzQY-9Nmtws",
        "Q": "at the equation 5:30 why does it start with -(-2)-...... instead of -2-(-2)-(-2)? Cuz i understand the ones behind but since the number at the beginning is -2x-3 ?",
        "A": "The 3 bracketed underneath is positive, but in the question, it is negative. So each term above (all three -2s) need to have a negative in front."
    },
    {
        "video_name": "pzQY-9Nmtws",
        "Q": "During 1:09 what is that symbol underneath the 3+3. and what does it mean and how do you use it ?",
        "A": "It is a bracket, and the end of the bracket is leading you to something like a group of something, or an answer to something."
    },
    {
        "video_name": "pzQY-9Nmtws",
        "Q": "I still didn't get why neg x neg is pos! at 12:10pm",
        "A": "they are the same sign so it is + not -"
    },
    {
        "video_name": "pzQY-9Nmtws",
        "Q": "I think you are hand-waving at the middle example (around 3:30). No explanation is given to why 2 x -3 suddenly becomes -2 -2 -2 You are kind of relying on the first example and just switching the signs as a graphical analogy. Can't you explain it logically? You are trying to do that in the last example with neg * neg.\n\nAnd do the second and third minus signs mean subtraction or a negative number?",
        "A": "It doesnt matter if they are either subraction or negative, either way, it ends up with -6"
    },
    {
        "video_name": "pzQY-9Nmtws",
        "Q": "in 3:03 how does -2-2-2=-6",
        "A": "-2-2 is equal to -4. then subtract 2 or add -2 which equals to -6"
    },
    {
        "video_name": "pzQY-9Nmtws",
        "Q": "At 5:43 the answer is supposed to be a negative right ?",
        "A": "No... a negative * a negative = a positive. So, the answer is positive."
    },
    {
        "video_name": "gH4IsIEYof0",
        "Q": "I'm at 3:30 and wondering why you can't just multiply the whole fraction (1/(x^1/7)) by x^7. That would give you (1*x^7)/((x^1/7)*(x^7)) which again would give you (x^7)/x which is equal to x^6 making d=6 not -1/7. Am I wrong in believing that x^1/7 * x^7 = x^1/7 * x^7/1 which is equal to x^7/7=x?",
        "A": "If you multiply x^(1/7) * x^7, you need to add the exponents, just like x^2 * x^3 = x^5, not x^6. You are multiplying the exponents. Multiplication of exponents only occurs when one exponent is raised to another, like: (x^2)^3 = x^6. Now, you may be thinking, can you apply an exponent of 7 to the numerator and denominator? The answer would be no as that fraction would not be equivalent to the original fraction. Hope this helps."
    },
    {
        "video_name": "gH4IsIEYof0",
        "Q": "At about 3:40, Sal moves the positive 1 from the numerator to the denominator and changes it to a negative. So the unshown work would be: -1 times x to the seventh power. I understand up to that point, but what I didn't get was why he transferred the negative to the exponent. Shouldn't the negative be put on the base?",
        "A": "A negatively signed exponent indicates that the POSITION of the number has to move to the other part of the fraction (from denominator to numerator or vice versa). The rest of the number doesn t change. So, for instance, x^(-3) = 1 / x^3 or 17*x^(-2) / y^(-5) = 17 y^5 / x^2"
    },
    {
        "video_name": "MeVFZjT-ABM",
        "Q": "At 3:42 , how to get one over x? I don't know what he was doing here... Is it derivative of log x equals to one over x?",
        "A": "Yes, the derivative of log u = du / u (using natural logs, of course). Note that you have to use the chain rule, so you must multiply (not divide) by the derivative of whatever you are taking the log of. For example: Thus the derivative of log (7x\u00c2\u00b2) = d(7x\u00c2\u00b2) /(7x\u00c2\u00b2) = 14x/(7x\u00c2\u00b2) dx = 2/x dx"
    },
    {
        "video_name": "MeVFZjT-ABM",
        "Q": "At 0:55, can you say that 1/0 - 1/0 = 0/0, and therefore it's an indeterminate form and you can just apply L'Hopital's rule then? I know that they're undefined terms, and that you're not really supposed to work with them in math, but can you pretend? Like are there any situations where this wouldn't work? Basically, will 1/0 - 1/0 always end up being 0/0?",
        "A": "1/0-1/0 isn t 0/0 because the fractions aren t like fractions. While the denominators are both approaching 0, they are approaching 0 at different rates [speeds], so they have different denominators. I hope this helps!"
    },
    {
        "video_name": "MeVFZjT-ABM",
        "Q": "At 0:50 why doesn't Sal just subtract 1/0 minus 1/0 to get 0/0. I mean, the denominators are the same so if you subtract 1-1 to get 0 in the numerator you would get 0/0, which is an applicable form of L'Hopital's rule. Why can't you do it that way?",
        "A": "Because he want s to make sure that he dosen t get some students confused , & i am a student who contrast w / things at that point . do you >?"
    },
    {
        "video_name": "MeVFZjT-ABM",
        "Q": "At 2:38, how is the Natural log of x= 0? Can someone help me please? Thanks",
        "A": "Sal is looking at the limit here, seeing what happens to ln(x) as x => 1. He does this by substituting 1 in for x, which gives you ln(1). The natural log of 1 is basically saying e to what power gives you 1? And, of course, ANYTHING (including e) to the zero power = 1. So... ln(1) = 0."
    },
    {
        "video_name": "MeVFZjT-ABM",
        "Q": "At 0:43, Sal took the limit of an expression. It came out to 1/0. Fine. Then he simplified that expression and got 0/0. Weren't those two expressions equivalent to each other? If so, why wouldn't the limits be the same? By the same token, if the limits are different, doesn't that show that the expressions aren't really the same?",
        "A": "It has to do with the idea of taking the LIMIT. The functions had different exact values at the number itself. The first was not even defined at that value (0/0). However, the functions still have the same LIMIT even though some of the functions do not exist at the exact point itself."
    },
    {
        "video_name": "MeVFZjT-ABM",
        "Q": "at 3:27 he decides to use Lhopistal rule and to do that i understand that he should take the derivate of f(x) so why doesnt the apply the cuoficient rule",
        "A": "nvm i just got it thanks anyway"
    },
    {
        "video_name": "MeVFZjT-ABM",
        "Q": "At 3:33, he starts Differentiating (xlnx - (x-1))/(x-i)lnx. But isnt that in the form u/v and so while differentiating u/v we use the formula d/dx (u/v) = (du/dx*v - dv/dx*u) / v^2 ?",
        "A": "No! L Hopital s is not a standard differentiating technique. You treat a numerator and a denominator completely independently. Quotient rule does not apply."
    },
    {
        "video_name": "MeVFZjT-ABM",
        "Q": "So when the limit is not 0/0 or \u00e2\u0088\u009e/\u00e2\u0088\u009e, we cannot apply L'Hopital's rule as Sal said at 1:03 ? Does it mean that on any other form L'Hopital's rule cannot be applied and limit couldn't be determined? What do we do in such cases?",
        "A": "That s right. We can only use LH when it is in the form 0/0 or \u00c2\u00b1\u00e2\u0088\u009e/\u00c2\u00b1\u00e2\u0088\u009e. When we get other forms of indeterminate, we can t apply LH. It does not necessarily mean we can t determine the limit. What we do in this case is to algebraically manipulate it into the form 0/0 or \u00c2\u00b1\u00e2\u0088\u009e/\u00c2\u00b1\u00e2\u0088\u009e. For example, lim_x->0 x*ln(x) = 0 * -\u00e2\u0088\u009e, this is an indeterminate form but we can t use LH on it. So instead we can rewrite it to be lim_x->0 [1/(1/x)] * ln(x) => lim_x->0 ln(x)/(1/x) = -\u00e2\u0088\u009e/\u00e2\u0088\u009e now we can apply LH."
    },
    {
        "video_name": "MeVFZjT-ABM",
        "Q": "Isn`t it much more easier and fast @6:12 not to use the product rule for 1/x (x-1), but instead to re-form it into just 1/x by opening the brackets and then just to derive this, which results in -x to the second, which is also one and the answer is still the same... or this is wrong by some reason?",
        "A": "Yes, that would be easier and it s correct! You would get 1-1/x=1-x^(-1), the derivative of which is x^(-2), so that the denominator would be x^(-1)+x^(-2), which at x=1 is 2, giving you 1/2 as required."
    },
    {
        "video_name": "MeVFZjT-ABM",
        "Q": "Why in minute 5:00 he don't delete \"(x-1)/x \"? I think that because if x->1 then \"(x-1)/x\"=\"0\" and 1nx+0= 1nx... if somebody can explain me that, thank you!",
        "A": "well you cant just simplify because that would require plugging in numbers and then getting 0. what you could have done is take (x-1)/x and turn it into (x/x)-(1/x) which would just be 1-(1/x). then when you take the whole derivative of the function you end up with (1/x)/(1/x)+0-(-x^-2) which would be when you plug in the numbers 1/(1+0+1) which equals 1/2"
    },
    {
        "video_name": "Alhcv5d_XOs",
        "Q": "sorry for that dumb question: how can i learn to solve system of equations like Sal is doing here? 9:20? What is the factor to multiply? What equation is the one I need to choose for multiply it? What to subtract? Some confusion in my mind!",
        "A": "thank you! this help me a lot!"
    },
    {
        "video_name": "Alhcv5d_XOs",
        "Q": "2:03 doesnt \u00e2\u0080\u009dat least one ci is non zero imply linear independence? if not, how ?\nthx pl reply asap",
        "A": "At least one ci is non zero implies linear dependance. Sal explains this at 5:00. In short if you have non zero ci, you can represent vi as linear combination of other vectors, which is linear dependancy by definition. You can do this by dividing the whole equation by ci. And if all ci s are zero you can t do this. Hope this helps!"
    },
    {
        "video_name": "Alhcv5d_XOs",
        "Q": "Starting around 4:40, Sal tries to prove that some linear combination of all the vectors equaling zero (with at least one of the c's not equal to zero) implies that the set of vectors is linearly dependent. But Sal's proof depends on c_1 specifically being non-zero, even though the hypothesis only specifies that some c_k is non-zero, not that c_1 is non-zero. So it seems that generality is lost. Isn't this a flawed proof?",
        "A": "No, the c_1 could be any c_n."
    },
    {
        "video_name": "Alhcv5d_XOs",
        "Q": "in the beginning sal said that a set is linearly dependent if c1v1+c2v2... = 0 but at 8:13 in the example he said that if c1 and c2 are both zero than the set is linearly independent , how is that possibe because to satisfy dependence the above equation ie. c1v1 + c2v2 must be followed so c1 and c2 must be zero ?",
        "A": "a set is linearly dependent if c1v1+c2v2... = 0 if v1, v2 etc arent zero vectors, the only way c1v1+c2v2... can be zero is if c1, c2 etc are zero, if none of the vectors are linearly dependent."
    },
    {
        "video_name": "Alhcv5d_XOs",
        "Q": "In the last example (13:47), why is it valid to choose an arbitrary weight for c3 (or more generally, ci) in order to solve the two-equation system? I understand that we can't solve the system until we have only two unknowns b/c there are only two equations, but I don't understand why we can choose any number for c3 in order to \"constrain\" the system and make it solvable.",
        "A": "Because if we find at least one solution with one of the Ci-s not equal to zero, then we sufficiently prove linear dependence of set. So if we set C3 in this example to some non-zero number and solve the equation for c1 and c2, then we give sufficient prove to the linear dependence."
    },
    {
        "video_name": "Alhcv5d_XOs",
        "Q": "At 13:48 why did you assign a value to c3 instead of c2 or c1?",
        "A": "If you solve the system of equations, you will see that c3 = 1 simplifies the constants."
    },
    {
        "video_name": "Alhcv5d_XOs",
        "Q": "The calculations on 9:40... couldn't you check them with matrices as well?\nThe way I did that was to turn this matrix\n\n(2)---(3)---(0)\n(1)---(2)---(0)\n\ninto Reduced Echelon Form, and that ended up like this:\n\n(1)---(0)---(0)\n(0)---(1)---(0)\n\nThis should also show that it is an independent set, right?",
        "A": "yes, this answer is too short"
    },
    {
        "video_name": "Alhcv5d_XOs",
        "Q": "@ 14:22 why did you multiply the second equation by 2? Why 2 and how does he know to the number two? Why not 3? And why is he doing the multiplying? Why are we multiplying at all? Is it to make the numbers the same so we can subtract? Help please.",
        "A": "Yes, Sal was multiplying the second equation by 2 so that he could subtract the two equations and cancel out the c1 s, which is exactly what he did."
    },
    {
        "video_name": "Alhcv5d_XOs",
        "Q": "Isn't the vectors Sal talks about at 9:46, linearly dependent, because you can make c1= 1,5 and c2=-1, so their sum will be equal to 0.\nThe same way you can do c1=-1 and c2=1/2, and the result will still be the same.",
        "A": "yeah, that makes sense thanks!"
    },
    {
        "video_name": "D_smr0GBPvA",
        "Q": "At 0:57, how do you determine which cos double identity to use since there are three different ones? (i.e. cos(2x)=2cos^2x-1 versus the one Sal used)",
        "A": "You are free to use whichever one you find useful. If you want the result to be only in terms of the sine, then use the one that only involves the sine, etc. You can relate the three forms by using the appropriate Pythagorean identity."
    },
    {
        "video_name": "D_smr0GBPvA",
        "Q": "At 0:45, how does Sal know which angle of the triangle is angle cos(2 x angleABC)? He indicates it is the bottom right hand corner with the purple mark, but how does he know the angle is not the top left hand corner instead?",
        "A": "Because when given angle ABC, you are looking at the angle at vertex B."
    },
    {
        "video_name": "AFF8FXxt5os",
        "Q": "At 1:43, how is he graphing the vectors? at 3:37, how does he know whether it's clockwise or counterclockwise? thanks",
        "A": "f(x,y)=yi+xy ---> f(1,2)=2i+1j ---> means plot the vector with the components of 2i+1j at the point (1,2). Make sense now? Also he graphed ccw because when you plug in t values, the x and y values correspond to a ccw motion."
    },
    {
        "video_name": "AFF8FXxt5os",
        "Q": "What's up with the video? It ends after 4:54 for me, but it seems to be in the middle of the example... Or is it suppose to end at 4:54?",
        "A": "no problem here. it ends at 11:32. browser/flash problem i suppose! (or past site problem probably?)"
    },
    {
        "video_name": "AFF8FXxt5os",
        "Q": "why did the i and j (unit vectors) disappeared after the dot product at 9:25?",
        "A": "yes the result is a scalar, and the way you calculate the dot product two vectors is to multiply each element from each vector and add them like this V1 = a1i+b1j and V2= a2i+b2j then V1 dot V2 = a1*a2+b1*b2 which is a scalar without the i and J components."
    },
    {
        "video_name": "AFF8FXxt5os",
        "Q": "Why does Sal not write the integral sign with the circle at 3:58? It is a closed loop.",
        "A": "You could write the closed-loop integral symbol here, but it is not necessary. It is normally used for mathematic definitions, when you want to say this is only true for closed loops ."
    },
    {
        "video_name": "ZACf9EecFrY",
        "Q": "At 0:09 \"a polyhedron is a 3 dimensional shape that has flat surfaces and straight edges.\" This is confusing - how could a 3D shape have flat surfaces but not have straight edges? It seems impossible that a shape would have flat surfaces and curved edges.",
        "A": "think about a square most have even sides"
    },
    {
        "video_name": "ZACf9EecFrY",
        "Q": "1:10 Its not a rectangle is it or is it a square ...",
        "A": "That is not logical. If all squares are rectangles, then at least some rectangles must be squares. A rectangle with all equal sides is a square, so how can you say that a rectangle can never be a square? In other terms, squares are a subset of rectangles."
    },
    {
        "video_name": "ZACf9EecFrY",
        "Q": "At 6:05, can you also attach the tip of the green triangle with the tip of the orange triangle?",
        "A": "ya i think so"
    },
    {
        "video_name": "ZACf9EecFrY",
        "Q": "AT 1:09 Sal says that the base is rectangle but it looks like a square. So is a it a square or rectangle?",
        "A": "Sal called it a rectangle because you don t know for sure that it s a square. however, it s possible that it s a square because all squares are rectangles, but we just don t have enough info to tell for sure"
    },
    {
        "video_name": "XwEGKNPLQfQ",
        "Q": "At 4:07, it says \"if you take the hypotenuse times the cosine of theta, you would get x.\" I'm not sure how that was worked out. How does cosine relate to x?",
        "A": "cos \u00ce\u00b8(t) = x(t) / 20 because cos\u00ce\u00b8 = adjacent/hypotonuse in order set that equal to x(t), you multiple both sides by 20, it cancels out on the right, leaving 20 * cos \u00ce\u00b8(t) = x(t)"
    },
    {
        "video_name": "8ToCxaB7EOE",
        "Q": "at 3:57 why would you bother changing colors?",
        "A": "because he wanted another color to signify a different coordinate."
    },
    {
        "video_name": "8ToCxaB7EOE",
        "Q": "At 3:56 he says that point B has to be \"out here someplace\", indicating the space further along the x axis. But technically it could be further along the y axis, right?",
        "A": "No it cannot be on the y axis because the point HAS to be along the x axis if you still dont understand, watch that part a few times"
    },
    {
        "video_name": "8ToCxaB7EOE",
        "Q": "I dont get 1:25. can someone help?",
        "A": "This is just instructions from the math problem saying that Sal has to make both lines the same length. I hope that s what you were asking for, and if it isn t, please be more specific."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "At 13:12, why did Sal wrote x0^4? Shouldn't it be 1/16?\nAnd at 10:35, does that mean that the equation of the normal is a quadratic equation?",
        "A": "On your question about the 1/16: He does write 1/16 around 13:40 He just moved it to the end of the expression."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "Amazing answer but I still have a same problem as ppl above at 13:13 I still didn't get why 1/4xo^2 = (4/xo^2) X (xo^4) As I see, (4/xo^2) x (xo^4) = 4xo^2 hence, 1/4xo^2 = 4xo^2 ???\nThen , a proper answer (in my mind) will be 1/16 instead of xo^4. Sorry, but I'm really bad as specific terms of Maths (Eng is my 2nd language) so some body can please explain this more clearly for me (again)!",
        "A": "I hope this doesn t come too late, but let me try answering your question. I presume you mean the step going from the stuff under the orange-green radical sign to the green one? (I ll write x instead of x0 for clarity.) Let s go from the red thing (under the green radical) to the orange-green thing, you get 4/(x^2) * (x^4 + (1/2)x^2 + 1/16) = [4/(x^2)] * (x^4) + [4/(x^2)] * (x^2/2) + [4/(x^2)] * 1/16 = 4x^2 + 2 + 1/(4x^2) Which equals the orange-green thing. Hope this is what you meant."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "it would really help me if anyone could redirect me to a factorization video where Sal explains the type of factorization he did at 13:20 4/x^2.",
        "A": "But how can he first figure out a number that perfectly fit the form he need?"
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "At 21:06, Sal says x0 squared is equal to 1/2.\nSurely this impies that x0 is equal to plus OR minus 1/sqrt2?\nObviously that would put it in the wrong quadrant, but is there a way to mathematically justify discarding a solution like this?",
        "A": "You just answered your own question. x0 is defined as the interception point in the first quadrant, thus is has to be positive.If you are on your exams and feel like clarifying just in case, you could add in an extra line, ie. Because x0 is defined to be in the first quadrant, x0 > 0. Therefore, x0 = 1/sqrt2 Hope that helps"
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "At 18:45, why is the derivative 0 at maximum point? Has Sal done videos on it? If not, can someone please explain to me why is the derivative = 0 , at minimum/maximum point? Thanks in advance.",
        "A": "You can view it this way: when a function reaches a maximum point, the tangent line goes horizontally, i.e. it s slope equals 0. So, the derivative of a function at a maximum/minimum point = 0."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "At 11:00 why is x^2 =1?",
        "A": "The quadratic formula says that if y=0, then x= -b plus or minus sqrt (b^2-4ac) all over two a. The quadratic equation is ax^2+bx+c=y. Sal means that the coefficient of x^2 (a) is one and not x^2 itself."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "at 18:45, when Sal rewrites the second quadrant root as a function of x(0) ((x naught)), why doesn't he write -1/2x instead of -1/2x(0)???",
        "A": "Sal mistakenly left off the subscript of the x naught in the first term. There should be no singular x because the function was in terms of Xo (x naught)."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "Can anyone explain to me what Sal did from 12:38 to 13:47?",
        "A": "He is factoring the sum under the square root, and express it as a product of perfect squares (which is not always possible) so that he can get rid of the square root."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "Maybe I'm just forgetting my algebra, but at 7:00, Sal says that the line perpendicular to y=2x0 is y=-1/(2x0). I thought that the slope of the line perpendicular is the slope's negative reciprocal? Shouldn't that mean the line perpendicular is y=(-1/2)x0, instead of y=-1/(2x0)? This is really confusing me. Please tell me if I'm just completely messed up. Thanks!",
        "A": "2Xo * (-1)/(2Xo) = -1 (Sal s) 2Xo * (-Xo)/2 is not equal to -1 (your s)"
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "at 7:02, wouldn't the equation of the perpendicular line be -Xo/2 instead of 1/2Xo? I thought that when finding the perpendicular line you don't inverse the x",
        "A": "He took the inverse of the gradient, not the x. 2^-1 = 1/2 and the gradient has the opposite sign thus becomes -1/Xo."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "Just to confirm, when he takes the perpendicular line of the slope at 7:14, why does he include the Xo in the denominator?",
        "A": "Recall that the slope of a a line perpendicular to another line with slope m is -(1/m). Since the slope of the line was 2xo, then the slope of the perpendicular line must be -(1/(2xo)). Hope this helps."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "At 7:55, why is it xo^2 ? Forgive me, i'm very new to this, but every formula i have seen so far has been Y-Yo = m(X-Xo). Why is it Xo^2 instead of Yo^2 ?",
        "A": "the equation of the parabola is Y= X^2. So, let us consider the point of intersection of the normal line and the parabola as (Xo , Yo ). Now using the equation of the parabola ( Y= X^2) the value of Yo will be equal to Xo^2. The formula that you have taken is correct. Now , simply substitute the value of Yo in that formula."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "18:45 I don't understand why. Isn't our derivative going to equal zero at the point (0, 0)? What the difference between our extreme line and the other lines?",
        "A": "Think about what you know of derivatives. They represent a turning point - the value goes through a maximum or a minimum. Here , the equation for X-II ( value of X in the 2nd quadrant ) is X-II = -X0 - ( 1/2X0 ) ( what Sal wrote ). What is the value of X-II at the turning point of this equation ? Exactly what you know of a derivative - find the first derivative WRT X0 and set to 0 !"
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "from 9:50 : after we've got that equation of the line ; how do we - I mean how can we - equate y = x^2 , to the equation ; as one is the equation of parabola and the other of the line ?\n\nThankyou",
        "A": "When you set two equations equal to each other, you are solving to find out where they intersect each other."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "At 19:16, it is a little unclear for me why Sal takes the derivative of the equation in the yellow box. And also, how does setting it equal to zero mean that's the maximum or minimum? Because couldn't the function reach higher or lower values before or after the slope is 0?",
        "A": "It is a local max or minimum. This happens because at the top of a curve and at the bottom of the curve, the slope is 0. But, you are correct it might not be a real max or min, this test just gives you candidates for the max or min -- you have to make sure it is a real max or min."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "Minor point I would like confirmed: At 20:35 Sal takes the square root of both sides but does NOT indicate plus or minus in front of the square root. This is because Sal knows that x is positive, and only the positive square root will result in a positive x, so he can disregard the negative square root?",
        "A": "Correct. The math term for this is the principal root ."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "Question: At 11:23, why does x^2 equal 1? He mentions it very quickly, so it seems like it was mentioned before, but I can't seem to find why x^2 would equal 1.",
        "A": "At that part he s just using the quadratic formula. So for the equation ax^2 + bx + c = 0 at that point the coefficient on the x^2 term is 1. He s not saying that x^2 = 1 he s saying that a = 1 Make sense?"
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "Hi everyone:-D\nWhen Sal finds the X2 at 17:59, the answer is -X0-(1/2X0, right? But at 18:50, he replaces -X0 with -X. Why does he do that?",
        "A": "That was just a mistake. He should have said and typed X0. Note that he still treats it as if it were an X0 as he performs the derivative with respect to X0. Good catch!"
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "Can someone please explain the concept of how Sal used the derivative with respect to Xo at 18:39?",
        "A": "X2 is a function of normal line intersection in the second quadrant. Taking a derivative of that and solving for zero will show the extremum of the line, in this case the minimum. Just think back to what a derivative actually is and it should make sense once you understand what X2 is a function of. Hope that helps some."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "At 18:30, i dont quite understand why should we set the second quadrant intersection as a functoin and find its derivative then set it equals 0",
        "A": "It s a min/max problem (at extremas like mins or maxs the first derivative = 0)."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "Why is the slope of the perpendicular line the negative inverse of the slope of the tangent line ? At 6:35",
        "A": "It s one of the rules from Geometry... two lines are perpendicular if the product of their slopes is -1 (m1 * m2 = -1) so the slope of a perpendicular line can be determined by taking the negative inverse of the slope of a known line. Notice that vertical and horizontal lines are always perpendicular to each other (special case because vertical lines have undefined slope)"
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "At 13:22, why did he factor out a 4/x naught squared? I feel like i'm missing something. Someone please explain!",
        "A": "Ok, thanks for the help!"
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "At 20:00, Sal says that the equation will reach a maximum or minimum when it reaches zero. why is that so?",
        "A": "Not when the initial equation or function equals zero, rather when its derivative equals zero."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "There's a mistake at 14:00. He didn't factor out the square root properly. If you multiply it through, you don't get the original. 4/x^2(x^4 + 1/2*x^2 + 1/16) = 4x^4/x^2 + (4/x^2)(x^2/2) + (4/16x^2) = 4x^2 + 2 + 1/4x^2. You should get 1/4x^2 + (2 + 4x^2). Or am I mistaken?",
        "A": "You are mistaken. Remember that addition is commutative, therefore 1/(4x^2) + 2 + 4x^2 = 4x^2 + 2 + 1/(4x^2). Sal switched the order of terms when he rewrote the expression at 13:00. He did that in anticipation of rewriting x^4 + 1/2(x^2) + 1/16 as a square: (x^2 + 1/4)^2."
    },
    {
        "video_name": "viaPc8zDcRI",
        "Q": "At 20:04 I don't understand why it reaches a maximum or a minimum when x is equal to zero. If it is equal to zero, then won't that mean that the slope is 0?",
        "A": "You are correct. Any point on a function at which the slope is 0 is considered a minimum or maximum, because the tangent line is completely horizontal."
    },
    {
        "video_name": "4xFIi0JF2AM",
        "Q": "If you refactor the expression shown at 2:00 in a different way, you can see an algebraic demonstration of why the previous video (Determinants along other rows/cols) works.",
        "A": "Cool. There are 6 different ways to calculate the determinant (3 rows and 3 columns) by minors and cofactors. They all give the same 6 terms @ 2:00 when you multiply them out."
    },
    {
        "video_name": "3LgfZ4bQ-yc",
        "Q": "At 6:20 why can we say that 1/x is not affected as n approaches zero?",
        "A": "Because there is no n in the expression for 1/x. There s just an x."
    },
    {
        "video_name": "3LgfZ4bQ-yc",
        "Q": "I do not know what number e is, therefore I assume At 8:00 he shows that n = e^x - 1, so i am assuming number e is also defined the same way?",
        "A": "lim (1+1/x)^x = e x->infinity"
    },
    {
        "video_name": "z8vj8tUCkxY",
        "Q": "at 4:25 i lose you when you try to find how many radians are in 3 degressand then you start crossing stuff out..... totally confused why did you cancel out pi on both sides and why did you put 180/3?",
        "A": "He cancels the pi because he is multiplying the fraction, and when you have pi on top and pi on bottom then they cancel to become 1. But we don t need to write 1 cuz it does not change the other numbers. True with radians that he crossed out."
    },
    {
        "video_name": "z8vj8tUCkxY",
        "Q": "At 0:35 Sal said what the symbol for degrees is. Is there a symbol for radians?",
        "A": "There technically is a symbol for radians, but it is easily mistaken for the degree symbol. They look similar. Therefore you re more likely to just see 25 rad, 1 rad or whatever."
    },
    {
        "video_name": "z8vj8tUCkxY",
        "Q": "At 1:19, is there a way to divide 180 by 3.14 or should we just use the symbol \u00cf\u0080 for all the radian calculations?",
        "A": "Use the pi symbol in your calculations. Any questions that don t ask for it in terms of pi will have the calculator near the bottom of the popup you answer questions in, and you can divide numbers by pi there."
    },
    {
        "video_name": "z8vj8tUCkxY",
        "Q": "At 5:23 why are we allowed to cancel out units?",
        "A": "Simplification/ dimensional analysis. because you are multiplying fractions by one another you can simplify like terms. Think of the fraction line as a division sign. well if we did not cancel them out we would end up with degrees/degrees. which is one. so before we begin to multiply we can factor out these terms, values, units to make our job less difficult. We don t only use this skill in math we use it in lots of sciences. this is the dimensional analysis I mention. its a very good tool to learn."
    },
    {
        "video_name": "z8vj8tUCkxY",
        "Q": "1:30 why is it plural now?",
        "A": "check the video carefully"
    },
    {
        "video_name": "z8vj8tUCkxY",
        "Q": "7:00\nis -90 degree = +270 degree.",
        "A": "No bcoz -90 is -\u00cf\u0080/2 and 270 is 3*\u00cf\u0080/2"
    },
    {
        "video_name": "z8vj8tUCkxY",
        "Q": "At 6:53, Sal says that this is -90 degrees. How can an angle ever be negative? It's like decelerating a stopped car!\nAny help to this answer shall be appreciated.",
        "A": "There is a convention in mathematics that an angle or rotation is regarded as positive if the rotation is counter-clockwise. And negative if clockwise. You could regard -90\u00c2\u00b0 (a quarter turn clockwise) as the same thing as 270\u00c2\u00b0 (three-quarters turn counter-clockwise). BTW if by deceleration you mean negative acceleration, then you can decelerate a stopped car - throw it into reverse."
    },
    {
        "video_name": "z8vj8tUCkxY",
        "Q": "At 0:29, you ask how many degrees is that? I find it hard to remember the formulas can you help?",
        "A": "You don t really need to remember a formula. pi rads is 180 degrees. Everything else is just a ratio of that. If I tell you a pie has 5000 calories, do you need a formula to tell you how much half a pie has? Of course not, you just know that you multiply 5000 by 0.5. So how many degrees are in pi/2 radians? Easy, right?"
    },
    {
        "video_name": "z8vj8tUCkxY",
        "Q": "2:40 how is it possible that the units cancel out exactly similar to how 2/1 * 1*2 cancels out? also, the first degree unit * the denominator degree unit is not the first order in the order of operations. I have yet to make the transition from how maths to why maths.\n\nyours virtually,\nconfused",
        "A": "Lets do the problem 30 degrees in radians 30 degress * (pi radians / 180 degrees) = (30pi degrees * radians) / 180 degrees Cancel out the similar units = 30pi radians / 180 Divide =pi radians / 6 Move units to outside (pi/6) radians If you do the same thing with other units, say, meters and seconds. Like, dividing a velocity over a distance to find how long it will take (Meters cancel out, seconds give time) it will work out similarly"
    },
    {
        "video_name": "z8vj8tUCkxY",
        "Q": "At 6:55, what does neagtive 90 degrees mean? It doesn't seem very intuitive.",
        "A": "It only makes sense with some reference point. The unit circle provides the reference point for negative angles, so if 0 degrees is along the positive x axis, as I rotate an angle from that reference ray in the clockwise direction, my angles are negative with respect to the 0 angle. If I rotate it in the counterclockwise direction, the angle is positive. You are correct in that it is not intuitive to have a negative angle."
    },
    {
        "video_name": "z8vj8tUCkxY",
        "Q": "i do not understand 0:26",
        "A": "Degrees and radians are just two ways to measure arcs in circles. Given that there are 2pi radians in a circle but also 360 degrees in a circle, we get that 2 pi = 360 degrees."
    },
    {
        "video_name": "j511hg7Hlbg",
        "Q": "At 5:16, when he is getting the partial with respect to x of that expression, why didn't he consider y'? I mean, why it just suddenly disappear and is not consider? What is the explanation for that?",
        "A": "When a partial derivative of a function is calculated, all other variables are treated as constants. y doesn t have to be considered because the derivative of a constant is just 0. If you were referring to something else, I can t help you because when I tried to play the video, it wouldn t play."
    },
    {
        "video_name": "j511hg7Hlbg",
        "Q": "At 2:30 you show me \"mew\"... where can I learn about \"mew\"?\nThank you!",
        "A": "Mu is the name of some arbitrary function that we will use to solve this equation. Mu could be anything, but we want it to be something that ll help us solve this equation. Mu, in this sense, is known as the integrating factor, as Outhwest said."
    },
    {
        "video_name": "j511hg7Hlbg",
        "Q": "At 3:58, how come the partial derivative of u(x) respect to y stays as u(x) and the partial derivative of u(x) respect to x becomes u'(x). My point is that why isn't it u'(x) and 1, respectively?",
        "A": "partial derivative with respect to y means that you take y as the variable and all others are just constants, u(x) being a function of composed only of x is therefore part of the constant that s why it remains as u(x) when u(x)y is differentiated."
    },
    {
        "video_name": "j511hg7Hlbg",
        "Q": "At 8:45ish when he divided mu(x)/x = dmu/dx by mu(x) why did he end up with 1/x = 1/mu dmu/dx instead of 1/x = 1/mu(x) dmu/dx?",
        "A": "It s assumed that we know mu is a function of x, so he chose to write mu instead of mu(x) to make it look cleaner."
    },
    {
        "video_name": "j511hg7Hlbg",
        "Q": "At the end of the video,about \"10:00\" shouldnt there be an integrating constant C on the last equation .I mean \u00ce\u00bc = x+C .",
        "A": "It could. As he says at the beginning of the next video, we are just trying to find AN integrating factor. There are infintely many if you include constants, but the most convenient one is x+0."
    },
    {
        "video_name": "j511hg7Hlbg",
        "Q": "Hey I'm having a hard time seeing a pattern when Kahn takes the derivative or integral with respect to y or x at like 1:30 and later in the video too. Could some one help me know what to do when this comes up?\n-Thanks",
        "A": "If you are having problems with this, like at 1:30 . Possibly watch some videos on Partial Derivatives; where you take the derivative with respect to one variable and treat the other variables as constants."
    },
    {
        "video_name": "j511hg7Hlbg",
        "Q": "he used the product rule to take the derivative of \u00ce\u00bc(x) (x\u00c2\u00b2+xy)y' at 5:30. the term y' disappeared on the next step. WHY?",
        "A": "Sal was trying to test if the differential equation was exact, the method to do that is to compare the partial derivatives of the independent term with what is multiplying the y term, that is why he left behind the y , since it didn t enter into the method to test if the equation was exact."
    },
    {
        "video_name": "Y6wNiYcuCoE",
        "Q": "At 2:49 Sal gets the answer for 9^1/2 as 3. How does that work? How do you raise a number to a fractional exponent?",
        "A": "9^(1/2) is equivalent to the square root of 9. For a fractional exponent, the denominator is the nth root, and the numerator is the power. For example 9^(3/2) is sqrt(9^3)."
    },
    {
        "video_name": "Y6wNiYcuCoE",
        "Q": "At 4:32, why can the 810 not be multiplied by 375^t? I understand that 375 is raised to t, and 810 is not, but I guess I am looking for a more detailed answer about the properties of exponents. Why is not 303750^t?",
        "A": "It would violate PEMDAS rules. Exponents must be done before multiplication. You would be multiplying 1st."
    },
    {
        "video_name": "Y6wNiYcuCoE",
        "Q": "At 3:45, why can't you multiply 3 to the t power and 125 to the t power and get a product of 375 to the 2t power, instead of 375 to the t power? Aren't you supposed to add the two t exponents and get a sum of 2t?",
        "A": "You can only add exponent when you have a common base. 3 and 125 are not the same base, thus, you can t add their exponents. If you had 3^t * 3^t, that = 3^(2t)"
    },
    {
        "video_name": "hg2HR9zJFq4",
        "Q": "At 3:15 is that function? Because for any given x we are getting the y?",
        "A": "yes g is indeed a function!"
    },
    {
        "video_name": "hg2HR9zJFq4",
        "Q": "At 1:24, what is the symbol that looks like an 'E' representing?",
        "A": "This symbol: \u00e2\u0088\u0088, means is an element of or is a member of a set. For example A \u00e2\u0088\u0088 { A , B , C }."
    },
    {
        "video_name": "hg2HR9zJFq4",
        "Q": "At 1:45, which case would we use if we wanted to find out h(0) ?",
        "A": "You would use the 1st option. The domains for each piece are defined using interval notation. Since the 1st piece is defined for x = (-infinity, 0], the square bracket tells us that this is <=0. On the next piece, you will see x = (0, 8]. The parentheses on the 0 tell us that zero is not in this domain, while the square bracket on the 8 tells us the 8 is in the domain for this piece. Hope this helps."
    },
    {
        "video_name": "hg2HR9zJFq4",
        "Q": "Someone please explain what happened at 3:27. I would greatly appreciate it. Thank You!",
        "A": "It looks like Sal is trying to determine which line segments to use for determine the functions output if 4.0001 is the input. The middle line segment ends at X=4. Since 4.0001 is larger than 4, the last line segment is the one to use. That line segments creates the output value = -3. Hope this helps."
    },
    {
        "video_name": "hg2HR9zJFq4",
        "Q": "In the third example at around 2:40, why isn't the upper part of the function chosen (where g(x)=7) as that has an empty circle, and is of a value close to -3, but not exactly -3 ?",
        "A": "The upper part of the functions is for values greater than -3, but less than or equal to 4. g(-3.0001) is not greater than -3. It is less than -3. So you have to use 1st graph section on the left."
    },
    {
        "video_name": "lng4ZgConCM",
        "Q": "In this video at time frame of 1:05 it talks about y = M X +B. My question is where does the M and the B come from and what does it mean?",
        "A": "m and b are the slope and y-intercept of the best fit line. Go back 7 videos."
    },
    {
        "video_name": "8Wxw9bpKEGQ",
        "Q": "O...K... At 1:29 Sal is multiplying by -1 and I don't understand why. What's the deal?",
        "A": "You have to subtract the entire equation, which is basically multiplying each by -1."
    },
    {
        "video_name": "8Wxw9bpKEGQ",
        "Q": "at 1:48 it says that -x/x=1 and Sal says that it equals negative 1. What is correct?",
        "A": "x/x equals 1, we then multiply both sides by negative one, giving: -x/x = -1."
    },
    {
        "video_name": "8Wxw9bpKEGQ",
        "Q": "At 1:50, shouldn't -x/x=-1? It doesn't really seem to matter much because the absolute value remains unchanged, but I was a bit confused.",
        "A": "Only true if x does not equal 0"
    },
    {
        "video_name": "8Wxw9bpKEGQ",
        "Q": "At 1:50 did you mean to write '-1'?",
        "A": "of course he did... I don t understand why noobs always say those kinds of stuff"
    },
    {
        "video_name": "8Wxw9bpKEGQ",
        "Q": "at aroubd 2:02 you say -x divided by x is one but it actually is -1",
        "A": "And, at 1:51 in the video a box pops up and tells you that Sal meant -1."
    },
    {
        "video_name": "8Wxw9bpKEGQ",
        "Q": "at 0:53 What if the number was something like 6x^2 and the other was 3x would that be 6x^2/3x=2 or would that be 2x?",
        "A": "6x^2 / 3x = 2x You must be careful to evaluate the x s properly too :)"
    },
    {
        "video_name": "8Wxw9bpKEGQ",
        "Q": "1:16 I dont understand WHY we have to multiply it by negative one?",
        "A": "You are subtracting (x^2-2) from the dividend, which is X^2 -3x +2. So, it s X^2 -3x +2 - (x^2-2). And -(x^2-2) is the same as -1(x^2-2). You distribute the negative sign through the expression. Subtracting an expression is the same as adding the negative expression. Here s another example (the expressions below are all the same): 4 - 2 4 + -2 4 + -1(2) 4 + -2 4 - 2"
    },
    {
        "video_name": "gBxeju8dMho",
        "Q": "at 0:08, why does she write logarithmic",
        "A": "To describe a real snail s shell."
    },
    {
        "video_name": "gBxeju8dMho",
        "Q": "1:18 she says that you can't have the same number but 1 and 1 are next to each other in the Fibonacci series. Does that mean there is an exception to what she said?",
        "A": "No, there isn t an exception. The smallest Fibonacci number that counts for spirals is 3."
    },
    {
        "video_name": "gBxeju8dMho",
        "Q": "At 1:08, what are adjacent Fibonacci numbers?",
        "A": "Well, take the Fibonacci series, (1,1,2,3,5,8...). Adjacent mean next to each other, so adjacent Fibonacci numbers are two numbers in the Fibonacci series that are consecutive. Like 1-1, 1-2, 2-3, 3-5, ect."
    },
    {
        "video_name": "2XQH-J5KeHI",
        "Q": "At 2:16-2:30, why aren't the limits separated by equal signs? Does the squeeze theorem state that the limits of all three functions as x approaches to a certain point would be equal?",
        "A": "No, the squeeze theorem states that f(x) is less than or equal to g(x) is less than or equal to h(x). Technically all three could be equal, but that is not necessary the case, hence the less than or equal to sign."
    },
    {
        "video_name": "2XQH-J5KeHI",
        "Q": "3:33 also you could turn the second function into a form where you'd be able to plug in the 2 without making it indeterminate by factoring the top right?",
        "A": "Yes, once you cancel out x-2 from both the top and the bottom, you are left with x-1 which when evaluated at 2 gives 1."
    },
    {
        "video_name": "2XQH-J5KeHI",
        "Q": "At 2:12 , when sal takes the limits , he uses the (greater or equal to inequality) sign. But shouldnt it be actually only equal sign , as the limits are equal as x approaches 2.",
        "A": "The limits are in fact equal, and it s easy enough to see that without resort to the squeeze theorem. The point of this exercise, though, is to show how the squeeze theorem could be used to establish this limit, so we use the inequality until the final step."
    },
    {
        "video_name": "0eWm-LY23W0",
        "Q": "Im stuck i don't understand at 0:15 already.. its confusing",
        "A": "What he was trying to show was that, using the table above, you can plot at least two points on the graph. Since it s about getting points on each correct answer, you should know that there can t be any negative answers. There are x and y coordinates, which I think you know by now. x= 1 (Correct Answer) and y=5 (Points) So the graph there shows that you can plot two points. One I already gave you, which is (1,5), and the next one is (2,10). Hope this helped."
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "5:44 what happens if both the fraction and exponent is negative",
        "A": "First do the calculation of the exponent. Here it would boil down to 64/25 Then apply sign of the original fraction. So it would be -64/25"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "At 5:30, he creates a problem of -2 to the power of -3. But what about doing a problem where its a negative fraction to a positive power? I am very confused about when the answer is supposed to be positive or negative. Please help!",
        "A": "I know what you re trying to say. For example, -2^2 is just -2 x -2, which would be 4. Negative times negative: Positive Positive times negative: Negative Positive times Positive: Positive"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "At 6:48 in the video, why does 1/25/64 equal 64/25?",
        "A": "1/25/64 can be written as 1\u00c3\u00b725/64 1\u00c3\u00b725/64 can be written as 1 * 64/25 (bcz when u change \u00c3\u00b7 to * u take the reciprocal of the fraction) which gives us 64/25"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "In 4:49, how come the -3 exponent became a positive?",
        "A": "Well, the rule of negative exponents is that is the exponent is negative, then it becomes positive if 1 is divided by the whole no... A good way to visualise this is - 2\u00c2\u00b3 = 8 2\u00c2\u00b2 = 4 ( 8 / 2 ) 2^1 = 2 ( 4 / 2 ) 2^0 = 1 ( 2 / 2 ) 2^(-1)= ? ( 1 / 2 ) We already know it is 1/2 because 2^(-1) = 1/(2^1) = 1/2. Thus, 2^(-4) = 1/(2^4) = 1/16. Hope this helps :)"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "at 1:18 why is dividing by 2 the same as multiplying 1/2?",
        "A": "Because when you divide a number by 2, you are basically trying to find what number is half of the first number. For example, 9/2= 4.5. 9 multiplied by 1/2 is also equal to 4.5. I hope that helps!"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "Well, on about 2:38 he says that 2^-4 = 1/2^4.\nHow is that? Isn't a negative number less than zero?\nHow can it be equal to a decimal?\n\nsorry for bombarding with questions I'm just really curious.",
        "A": "All you have to understand is that if a number m is raised to the power minus n, it is equal to the 1 divided by the same power n. A good way to visualize this is - 2^3 = 8 2^2 = 4 ( 8 / 2 ) 2^1 = 2 ( 4 / 2 ) 2^0 = 1 ( 2 / 2 ) 2^(-1)= ? ( 1 / 2 ) We already know it is 1/2 because 2^(-1) = 1/(2^1) = 1/2. Thus, 2^(-4) = 1/(2^4) = 1/16. Hope this helps :)"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "At 6:23, how does he get 64/25 from 1/25/64 ?",
        "A": "1/25/64 means 1 divided by 25/64 Do the division of fractions -- Change divide to multiply by using the reciprocal of 25/64: 1 * 64/25 -- Multiply and you get: 64/25 Hope this helps"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "At 0:06 why did he put a one?",
        "A": "It helps some people to understand. Anything times 1 is just itself so the 1 doesn t change anything."
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "How do you write a fraction with a negative exponent? 4:16",
        "A": "Here s an example: (3/4)^(-2) Note: you need the parentheses around the fraction to show that the exponent applies to the entire fraction."
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "at 0:48 sal says negative one but meant one",
        "A": "Write that under Report a Problem."
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "At 0:10, Sal starts listing the powers of 2 and says we have to start with 1. Why? Why not just right 2 x 2 x 2 x 2?",
        "A": "That will be too much work and a waste of lead and trees."
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "Why at 0:06 (and at other places throughout the video) does Sal put a 1 in the beginning? Why is it necessary?",
        "A": "Sal puts the ones at the beginning because that is when it is the power of zero of a number. Or it is to represent the original number."
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "At 5:54, why isn't it (8/5) to the second power? Why is it 1/(5/8) to the second power? If you take the reciprocal of 8/5, isn't it 5/8?",
        "A": "You are right, it can also be (8/5)^2 but, Sal decided to use (1/(5/8))^2."
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "at 0:10 in the video, why is it 1 times 2 times 2 times 2 times 2. I thought it was just 2 times 2 times 2 times 2 ?",
        "A": "because 1x2=2 so you do not have to start with 1"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "I don't really understand when he uses terms like \"just flipping\" that confused me a little bit at 6:20?",
        "A": "The negative exponent means to swap the denominator and numerator or another way to put it is to flip the two values."
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "How come at 6:20 25/64 gets changed to 64/25?",
        "A": "You have 1 / (25/64) , the same thing as 1 \u00c3\u00b7 25/64 , which will become 1 x 64/25 . Simple example when you have 4 \u00c3\u00b7 2 , you change it to 4/2 which is the same thing as 4 x 1/2 . ( your taking the reciprocal of the second number and multiplying it to the first number.) So moving back to our problem, 1 x 64/25 = 64/25 and there you have it."
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "6:30 What happens when the negative power for a fraction is something other than two?",
        "A": "In the example you noted, let s say that the exponent is -3...(5/8)^-3...that would become 1/(5/8)^3 or 1/(125/512)...which becomes 512/125. Does that help/make sense?"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "At 2:58 why does he add the one? Ive read previous answers and I still don't understand. Please help!",
        "A": "He s multiplying 1. Multiplying 1 or multiplying by 1 makes no difference to the result, but he does it to make it easier to understand the zero exponent and negative exponents. Start with 5^3 = 1 x 5 x 5 x 5 Keep dividing by 5 (5^3) /5 = 1 x 5 x 5 x 5 / 5 = 1 x 5 x 5 = 5^2 (5^2) /5 = 1 x 5 x 5 / 5 = 1 x 5 = 5^1 (5^1) /5 = 1 x 5 / 5 = 1 = 5^0 (5^0) /5 = 1 / 5 = 5^-1 (5^-1)/5 = (1 / 5) / 5 = 1 / (5 x 5) = 5^-2 And so on"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "At 1:23, I'm confused. Can we transfer the negative from the exponent to the whole number so the number is -2 to the fourth power? Then, the equation would be 1x(-2)x(-2)x(-2)x(-2), which would be 16 since the one does not interfere with the equation and an even number of negatives multiplied together would be a positive.",
        "A": "the negative in the exponent does not mean any of the numbers are negative. A negative exponent means dividing. Think of this pattern: 2^3=8 2^2=4 2^1=2 2^0=1 2^-1=1/2 2^-2=1/4 See how we have a pattern of dividing by two every time? So going down in exponents equates to dividing so a negative exponent means divide and does not relate to the 2 at all!"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "Why does Sal start with a one at 0:07. Can't you just do 2*2*2*2 without the one.",
        "A": "I think he puts a 1 in because otherwise, some people might assume it should be a zero. The thinking goes 5^2 is two fives multiplied together, so 5^0 should be no fives, which must equal zero . Logical, but incorrect. Making sure the 1 is also included helps to avoid this error."
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "at 5:47, why does he do (5/8) squared instead of 5/8 squared without the parentheses?",
        "A": "This is because he wants to show that the whole expression (5/8) is being raised to the -2, not just the 5 in the numerator or something like that."
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "2:49 it says 2 to the negative 4 equals 1/2 to the 4. It helps so much. I should write it down somwhere as a reminder.",
        "A": "o.k.,that s cool"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "At 4:50, why is (-2)^-3 turned into 1/(-2)^3? I don't get it.",
        "A": "When you have a negative exponent, the entire term gets moved to the denominator. So, in order to simplify (-2)^-3, you can t put a number to a negative exponent and say the answer will turn out negative, that s not how it works. You need to move the whole term to the denominator, 1/(-2\u00c2\u00b3). And then simplify it: 1/(-8) = -1/8"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "At 5:25 why is the answer -1/8 and not 1/-8?",
        "A": "Since a fraction is basically a number divided by another number -1 divided by 8 would essentially be the same thing as 1 divided by -8."
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "5:20, could you write it as 1/-8 ??",
        "A": "the denominator can t be a negative number"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "at 5:25 I don't get how Sal got a minus n front of the answer. how?? I am super confused",
        "A": "Well, lets look at the question. -2 ^ -3. As Sal explains, that is the same thing as 1 / -2 ^ 3. Lets do -2 ^ 3: -2 * -2 * -2, which is -8. So the answer is 1 / -8, which is equivalent to -1 / 8. Hope that helped!"
    },
    {
        "video_name": "JnpqlXN9Whw",
        "Q": "Hello! I've got a question. (: I can't seem to understand this logically at the moment...\nAt 5:18, does the \"-\" only apply for the numerator, or the whole fraction?",
        "A": "I believe the proper way to write a negative fraction is to write it for the whole fraction because remember that a fraction is a number(numerator) divided by another number(denominator), so if the negative sign is put on the numerator, it will still be the same."
    },
    {
        "video_name": "br7tS1t2SFE",
        "Q": "What does vector and magnitude mean on 0:07?",
        "A": "A vector is an ordered set of numbers, commonly represented by an arrow. [x y z] is the general form of an R\u00c2\u00b3 vector, which often represents a point in 3D space with the x, y, and z coordinate. The magnitude of a vector is the length of the arrow or the distance from the point of origin of the vector. Vectors can be n-dimensional, so you can represent R\u00c2\u00b2 or 2D points with a vector [x y] or you can represent R\u00e2\u0081\u00b4 or 4D points with a vector [x y z w], and so on."
    },
    {
        "video_name": "br7tS1t2SFE",
        "Q": "im confused at 4:00",
        "A": "Sal uses components of vectors at 4:00. Look for the set of videos called Vector basics to learn about components of vectors."
    },
    {
        "video_name": "br7tS1t2SFE",
        "Q": "What is a column vector (4:06)",
        "A": "A column vector is a vector whose coordinates are written in a vertical column, like: 5 0 It s just the name for this style of writing down a vector."
    },
    {
        "video_name": "br7tS1t2SFE",
        "Q": "At 1:13, do direction and magnitude define what velocity technically is?\nBecause a vector is a type of velocity (or is it the other way around?), a vector and velocity both have a direction and magnitude?",
        "A": "A vector is a quantity having both a magnitude and direction. Velocity is a vector quantity as it has both magnitude and direction for example 20 mph towards North is the velocity of an object."
    },
    {
        "video_name": "br7tS1t2SFE",
        "Q": "At 0:05, Khan says a vector is something that has magnitude and direction. Can the vector have something else with it, besides magnitude and direction?",
        "A": "No. A vector only has a magnitude and a direction. That is all a vector is."
    },
    {
        "video_name": "br7tS1t2SFE",
        "Q": "3:14, does the arrow on top of the vector also have to be due east?",
        "A": "To clarify, the arrow pointing east has nothing to do with the example Sal used of 5mph east. It s simply notation to have the arrow on top of the variable be pointing to the right. If the vector was 10mph north the variable would still be written with an arrow pointing right (or bolded, in some textbooks as Sal mentioned)."
    },
    {
        "video_name": "br7tS1t2SFE",
        "Q": "at 0:50 you talk about direction. How can direction be specified in numbers and what is the direction angle?",
        "A": "From knowing the specified numbers, you could calculate directional angle through trig"
    },
    {
        "video_name": "br7tS1t2SFE",
        "Q": "This might be nitpicking, but at 3:40 aren't vectors in general supposed to be put behind this pair of (pointy?) brackets <5, 0> instead of (5, 0) which are generally more for representing a point?",
        "A": "No, there is no absolutely correct notation for representing vectors. If your reader knows what you are talking about, then there is no problem."
    },
    {
        "video_name": "br7tS1t2SFE",
        "Q": "At 1:33 you said \"we could have more than 3 dimensions\". How is that possible?",
        "A": "Simple, you use more numbers. Things in 2D use 2-tuple vectors, things in 3D use 3-tuple vectors, things in 4D use 4-tuple vectors, and so on. It s hard to visualize, but it s possible to represent with numbers."
    },
    {
        "video_name": "br7tS1t2SFE",
        "Q": "At 4:01 isn't that a matrix?",
        "A": "Yes, that is a matrix. Vectors can be represented using a single column or row matrix."
    },
    {
        "video_name": "br7tS1t2SFE",
        "Q": "so basically what he's saying at 2:38 is that you can refer to an infinite number of vectors with the same notation? Is the origin of the vector not important?",
        "A": "Exactly. The origin of the vector is only important for visualizing other aspects of a problem, the vector itself is represented by the arrow."
    },
    {
        "video_name": "br7tS1t2SFE",
        "Q": "At 4:01, why is it more common to write a vector as a \"column vector\", which is vertical stacked like (continued):\n| 3 |\n| 4 |\nbut not horizontally as [3 4]?",
        "A": "It s largely because of the way matrices work. When multiplying a matrix by a vector, it s the same algorithm as multiplying a matrix by an n-by-1 matrix, which means that the matrix has n rows and 1 column. If we were to write vectors horizontally, then the algorithm for matrix-matrix multiplication would differ from the matrix-vector algorithm. By writing vectors as columns, they can be thought of either as a vector or as an nx1 matrix."
    },
    {
        "video_name": "29P6bar7nHc",
        "Q": "At 2:21, what does absolute value mean?",
        "A": "Absolute value means \u00e2\u0080\u009cthe positive value of\u00e2\u0080\u009d. The positive value of -5 is 5: |-5| = 5 The positive value of 20 is 20: |20| = 20"
    },
    {
        "video_name": "vA-55wZtLeE",
        "Q": "at 9:17 i don't understand how you do do the subtraction",
        "A": "Well, if you subtract then the positive 3x would cancel with the other 3x. You want the x to be canceled so you can get y because that s what you want/can figure out first. Then you ll later get the value of x. Hope this helps! :)"
    },
    {
        "video_name": "vA-55wZtLeE",
        "Q": "at 3:53 could Sal have not made x= a decimal? In that case, 3.5 instead of 7/2?",
        "A": "Yes, he could have done so, but I guess he wanted to think in fractions, so he wrote 7/2. You can write it whichever way you like."
    },
    {
        "video_name": "vA-55wZtLeE",
        "Q": "@4:24 how is that 21/2 isn't it suppose to be 21/6?",
        "A": "3(7/2)=21/2. Only the numerator is multiplied by 3. 21/6 is just a not simplified version of 7/2."
    },
    {
        "video_name": "vA-55wZtLeE",
        "Q": "at 4:45\nwhat if 4y = something like 17/2? you cant simplified them. should you convert it to decimal? or just make it like that? so that could be 4y=17/2. divide them by 4 so\n\ny = 17/2/4? hmmmm that doesnt seem right.\n\nThis fraction is making me insane.",
        "A": "If you prefer, think of it as multiplying times 1/4 rather than dividing by 4. Then you end up with y = 17/2 * 1/4 = 17/(2*4) = 17/8"
    },
    {
        "video_name": "vA-55wZtLeE",
        "Q": "At 1:44 , what does the equation ax+by=c represent ? Is it kind of like the equation y=mx+b in finding the slope ? Also, at 2:46, what does the word explicitly mean ?",
        "A": "Ax + By = C is called the standard form of a linear equation. y = mx + b is called the slope-intercept form of a linear equation. You can convert the equation between these 2 formats. Explicitly means exactly."
    },
    {
        "video_name": "vA-55wZtLeE",
        "Q": "@4:24 how is that 21/2 isn't it suppose to be 21/6?",
        "A": "Hi! When you multiply a number by 3 it is the same as multiplying it by (3/1). When you multiply fractions you multiply the numerators and the denominators. So it is 21/2."
    },
    {
        "video_name": "vA-55wZtLeE",
        "Q": "At 8:30, could you also add the bottom equation to the top equation, not just subtract like he said in the video? Because if you could only do one or the other, how would you know which (+ or -) one to do? Also, at 9:10 to 9:40 I'm confused on why you would make the second equation negative. Why not keep it positive and add it to the top equation? Then you would get 6x+5y=4.63 I think, and couldn't you figure everything out from there?",
        "A": "You add or subtract to eliminate one of the variables and adding the equations would only have increased the variables. By subtracting the equations he eliminated the y variable. Also, you could do the problem if you simply added but it would be much more difficult."
    },
    {
        "video_name": "vA-55wZtLeE",
        "Q": "At 5:26, could you have expressed the answer as being (3.5, -2)?",
        "A": "Yes. If we were looking to graph the equation, then we would use 7/2 simply because it s easier to measure."
    },
    {
        "video_name": "vA-55wZtLeE",
        "Q": "@4:24 how is that 21/2 isn't it suppose to be 21/6?",
        "A": "The reason that it is 21/2 instead of 21/6 is because, when you multiply a fraction by something, you only multiply the numerator of the fraction by that something. If you multiplied the numerator and the denominator of the fraction by that something, then you would end up with the same fraction, only not simplified. 26/6 can be simplified into 7/2, which is the original fraction."
    },
    {
        "video_name": "vA-55wZtLeE",
        "Q": "I was just wondering, but at 5:41 in the video, were you supposed to write two minus signs or was it just a typo",
        "A": "There were supposed to be two minus signs, because the equation was subtracting (minus #1), and negative number number was being subtracted (minus #2)."
    },
    {
        "video_name": "tn53EdOr6Rw",
        "Q": "what does he means by \"just take a deep breathe\" on the time 0:37 at the video",
        "A": "He means just slow down and think about it."
    },
    {
        "video_name": "tn53EdOr6Rw",
        "Q": "At 0:54 the instructor says 9 to the 1/2 is 3, how did he get 3 exactly?",
        "A": "The 1/2 power is equivalent to the square root (\u00e2\u0088\u009a), so the primary root of 9 or \u00e2\u0088\u009a9 = 3."
    },
    {
        "video_name": "tn53EdOr6Rw",
        "Q": "So this has been my question for a long time, and in the video 2:57, the original answer we got was 1/-3 but the answer seem to be -(1/3) I know they are both the same but is it necessary to change the form?",
        "A": "It s more acceptable in terms of standard convention."
    },
    {
        "video_name": "_kbd6troMgA",
        "Q": "At 4:50 why is the derivative of (tan theta) sec^2 theta and not sec^2 divided by theta?",
        "A": "The derivative of tan(theta) is equal = sec^2(theta). At 4:50, the horizontal bar is not showing division per se, but the with respect to operator which he is using to explain the chain rule."
    },
    {
        "video_name": "_kbd6troMgA",
        "Q": "at 1:13: Why is this angle changing?",
        "A": "the angle is changing because the balloon itself is rising and you want to calculate the rate of change"
    },
    {
        "video_name": "_kbd6troMgA",
        "Q": "at 0:57 Sal says, \"When you take derivatives of trig functions you assume that you're dealing with radians.\" Does this mean that it is impossible to use degrees, or that it is just more convenient to use radians?",
        "A": "We don t use degrees in calculus. If you try to, you have to include the conversion factor for radians, so you are really just using radians. So, from here onward in math, you use radians."
    },
    {
        "video_name": "_kbd6troMgA",
        "Q": "at 4:05 why doesn't sall just make 500tan a function of x ,,,,,,, { (500 tan\u00ce\u00b8 ))=x] and then take dx/d\u00ce\u00b8\n.It seems to me in this type of question this method is alot easier. Please tell me if i'm wrong",
        "A": "The question is to find the rate of change in height with respect to time. You have to include the time element. Calculate it out your way and try a few numbers.. Pay close attention to the units and see if you get 200 meters per minute. Or anything in meters per minute."
    },
    {
        "video_name": "_kbd6troMgA",
        "Q": "At 5:09, Sal says \"... it's just going to be 1 over 500 dh/dt\". Is the long-form mental process (d(h^1)/dh = 1h^0 =1) over 500 (a constant) using the exponent rule? Is that the right way to think about it?",
        "A": "Derivative of a constant times a variable is equal to the constant times the derivative of a variable. For example, if you take the derivative of 3x\u00c2\u00b2, your result is 3 times the derivative of x\u00c2\u00b2. The derivative of x\u00c2\u00b2 is 2x, so 3 times 2x is 6x. It is just an associative thing to regroup as 3\u00e2\u0088\u0099dx\u00c2\u00b2 /dx rather than to do the differentiation all at once. Often in calculus you can make a lot of progress toward simplifying by moving out those constant factors."
    },
    {
        "video_name": "_kbd6troMgA",
        "Q": "0:57 why on doing Calculus...or on taking derivatives of trig functions.....we assume that we're dealing with radians ?\nwhy in calculus....when an angle is to be introduced for calculations....why it is ONLY done in RADIANS ?",
        "A": "Any mathematics above elementary trig will be almost exclusively expressed in radians."
    },
    {
        "video_name": "_kbd6troMgA",
        "Q": "At 5:02 where did 1/500 come from and why?\nThank you",
        "A": "He was taking the derivative of h/500. Since the denominator is just a constant, it can be factored out of the derivative, Thus, d/dt [h/500] = (1/500) d/dt (h) = (1/500)(dh/dt)"
    },
    {
        "video_name": "_kbd6troMgA",
        "Q": "I didn't understand this in the previous video either: At around 5:45, Sal says that we use sec^2(pi/4). But it's actually sec^2(theta), where theta = pi/4 + 0.2t. Why do we assume t = 0 here?",
        "A": "no, for this question, we use theta as a value which is an angle. It is already given that theta is pi/4. 0.2t is the rate at which the angle changes not the angle at the moment"
    },
    {
        "video_name": "_kbd6troMgA",
        "Q": "At 5:09 Why does h/500 become 1/500 (dh/dt)? Can someone walk me through the steps?",
        "A": "Sal is taking the derivative of the entire equation with respect to t, and on the right that means finding the derivative of h/500 with respect to t. His handwriting is hard to read here, but that s d/dt in white next to the expression [h/500]. The expression h/500 is the same as (1/500)*h. In this expression, 1/500 is a constant, so the derivative of (1/500)*h with respect to t is the same as (1/500) times the derivative of h with respect to t, and that s how we get (1/500)*(dh/dt)."
    },
    {
        "video_name": "qpbaglogObM",
        "Q": "at 5:31 it was pure trash?",
        "A": "The video is only 5:02 long. How can you comment about something at 5:31 when the video isn t that long?"
    },
    {
        "video_name": "qpbaglogObM",
        "Q": "At 4:08, what is the missing number?PLS HELP",
        "A": "The missing number is going to be Equal to 4 times 6 in other words its 24 :D I hope this helped :)"
    },
    {
        "video_name": "CYNUwiUzlPk",
        "Q": "what does he mean that angle CAB=ABC 2:07",
        "A": "Since it is an equilateral triangle, all 3 angles are congruent. angle CAB is the one at the top, and angle ABC is on the bottom left."
    },
    {
        "video_name": "CYNUwiUzlPk",
        "Q": "At 3:20 he seemed to have skipped a step. By what postulate do we know that the interior angles of a triangle are always 180?",
        "A": "It s by the Triangle Sum Theorem, which states that the sum of the interior angles of any triangle is equal to 180 degrees."
    },
    {
        "video_name": "CYNUwiUzlPk",
        "Q": "2:15 shouldn't it be angle CBA?",
        "A": "CBA and ABC mean the same thing - they refer to the same angle. You can say it either way, it won t make a difference. Hope this helps!"
    },
    {
        "video_name": "CYNUwiUzlPk",
        "Q": "At roughly 1:10, he says that angle ABC is equal to angle ACD. Where does segment D appear?",
        "A": "It is the line that is in the middle of the triangle that cuts it in half (the diagonal)"
    },
    {
        "video_name": "CYNUwiUzlPk",
        "Q": "Why did Sal draw a squiggle on top of the equal sign at 2:07?",
        "A": "It means congruent"
    },
    {
        "video_name": "SqcxYnNlI3Y",
        "Q": "At 6:25 why does he multiply it by K?",
        "A": "I know it might seem strange at first, but that is explained during 5:46 to 5:58. You can also review the video entitled Expected Value E(X) for more on this topic. Basically, he s trying to get what is called the probability weighted sum . You need to multiply each K (that s why it s there) times the probability that the random variable is equal to K, and sum all those values (for all possible values of K)."
    },
    {
        "video_name": "SqcxYnNlI3Y",
        "Q": "At 10:55 Sal simplifies p^k as p x p^k-1... I don't understand how this simplification is made. Can someone answer that for me?",
        "A": "Thanks for the explanation. I guess I m used to thinking of simplifying meaning a somewhat shorter or more basic expression (i.e.- simplifying fractions). This is all pretty tough for me, but your example helps. When I plug in real numbers it takes the abstraction and difficulty down a notch."
    },
    {
        "video_name": "SqcxYnNlI3Y",
        "Q": "At 12:30, I understand that a+1=k, and b+1=n. However, I cannot understand why n-k=a+1-(b+1).",
        "A": "Sal made a temporary mistake, it should be (b+1)-(a+1). For example: If 10 = 9+1 and 20 = 19+1, then 20-10 = (19+1)-(9+1) = 19-9."
    },
    {
        "video_name": "SqcxYnNlI3Y",
        "Q": "at minute 10:56 how does Sal go from P^k to P*P^k-1?",
        "A": "because it is the same... P^k means P*P*P... P^(k-1) means P*P*P... but because the exponent is one less than P^k , one P is missing so after that we can correct it like this: P*P^(k-1) By multiplying the P back in to the Term as you can see it is the same as before also the same would be: P*P*P^(k-2) because now k is two less than it should be but we multiplying the two P s back in I hope my English is not too bad and you got the point ;)"
    },
    {
        "video_name": "SqcxYnNlI3Y",
        "Q": "At 12:30ish, n-k is equal to b-a, is it not?",
        "A": "Yes I have the same..."
    },
    {
        "video_name": "SqcxYnNlI3Y",
        "Q": "Hi Sal, I have a question about taking out 'np' out of the sigma @ 11:07 in the video! I thought that if we take out 'np' out of the sigma, it would be n^2p (=n x np).. Could you explain how it becomes 'np,' instead of 'n^2p'?",
        "A": "Why do you think it s (n^2)*p ?"
    },
    {
        "video_name": "SqcxYnNlI3Y",
        "Q": "There is a mistake at 12:36. If a = k -1 and b = n -1, n -k = b -a",
        "A": "he corrected himself at 13:49"
    },
    {
        "video_name": "SqcxYnNlI3Y",
        "Q": "@11:25 Khan means to write (n-k)! on the bottom of the summation formula not (n-k!).\nCorrect?",
        "A": "That is correct."
    },
    {
        "video_name": "SqcxYnNlI3Y",
        "Q": "In the video at time 11:26 when rearranging the terms inside the summation, the denominator term ( n - k )! is written as ( n - k! ). I guess it should be ( n - k )! right?\nfactorial(thanks for these awesome videos) also read as Thanks for these awesome videos! hahaha",
        "A": "It s (n-k)! Otherwise you ll (more than likely) get a negative denomenator, and for probabilities, that doesn t work."
    },
    {
        "video_name": "K_OI9LA54AA",
        "Q": "@1:17, why is the change in y (y-b)?",
        "A": "because he defined the two points as (a,b) and (x,y) so the difference in the y would be y-b and difference in x would be x-a"
    },
    {
        "video_name": "K_OI9LA54AA",
        "Q": "At 1:13, is the slope represented as a fraction or a ratio?",
        "A": "Both. Ratios are an application of fractions. And, you can think of the slope as the ratio of change in Y to change in X. When you use the slope in math operations, the operations will be done as a fraction."
    },
    {
        "video_name": "K_OI9LA54AA",
        "Q": "At about 00:58 seconds or so. He says that those triangles are the deltas. I understand that but for full formula for slope does it matter which y or x goes first? Would you still get the same answer?",
        "A": "Slope is always rise over run. It doesn t matter which one you find first, but make sure they re in the proper place. Consider a line with rise 5 and run 4. The rise/run way is 5/4. But the run/rise way is 4/5. That is a different value, and would give us with a completely different line."
    },
    {
        "video_name": "K_OI9LA54AA",
        "Q": "I understand that the slope is defined as y2-y2/x2-x1. However, at 2:16, Sal says \"...this point right over here between the point ab is going to equal to 'm'.\" How the point a,b is equal to the slope?",
        "A": "actually, Tian, I think Sal means a and b as y-b/x-a. oh, and btw, (maybe its a typo) the slope is defined as y2-y1/x2-x1, not y2-y2/x2-x1"
    },
    {
        "video_name": "K_OI9LA54AA",
        "Q": "At 1:29 why did you use a different variable.",
        "A": "The b? When Sal says the change of y, he means the second y value minus the first y value. Y is the second point s y value, and b is the first point s y value."
    },
    {
        "video_name": "K_OI9LA54AA",
        "Q": "In 3:21, can you use the distributive property and say y-b = mx-ma?",
        "A": "Yes, It s the same thing. Sal does this in 5:41."
    },
    {
        "video_name": "K_OI9LA54AA",
        "Q": "At 4:20, Sal gives us a slope of 2 & a point of (-7,5). He then plugs these values into the point-slope equation y-b=m(x-a). Why does he substitute the variables (a,b) instead of (x,y)? Does it matter which point you choose? Can the equation be rewritten as 5-b=2(-7-a) instead of\ny-5=2(x-(-7)?",
        "A": "Yes it does not matter whether you put (a,b) or (x,y). Because (x,y) or (a,b) are mere names, and they can take any coordinates value as long as coordinates are present on line. Check below if you evaluate equation by putting (a,b) instead of (x,y) you will get same equation. 5-b=-14-2a 5+14=-2a+b 19=-2a+b 2a+19=b or b = 2a+19."
    },
    {
        "video_name": "4mdEsouIXGM",
        "Q": "At 6:23, what does Vi use to cut Wind's world apart?",
        "A": "xacto knife is what she used"
    },
    {
        "video_name": "4mdEsouIXGM",
        "Q": "at around 0:34: What does she mean by pine log?",
        "A": "Victoria (Vi) means a log from a pine tree"
    },
    {
        "video_name": "4mdEsouIXGM",
        "Q": "Did anyone realize at 6:12, god was just dog, but just upside down?",
        "A": "Yes, god becomes dog when flipped and rotated."
    },
    {
        "video_name": "4mdEsouIXGM",
        "Q": "At 2:10, what does \"nom n' life nom\" actually mean?",
        "A": "It doesn t really mean anything."
    },
    {
        "video_name": "4mdEsouIXGM",
        "Q": "how did Wind get the materials to build the dog house (6:01) and get a pet dog(5:59)?",
        "A": "from the pine log and the blue rod."
    },
    {
        "video_name": "4mdEsouIXGM",
        "Q": "At 5:29 Mr. Ug is building a fence to. How does vi draw both sides? Mr. Ug's side and winds side?",
        "A": "A mobius strip only has one side, and since the strip Vi was using was transparent, you could see the fence Wind built. By the way, Wind is Mr.Ug."
    },
    {
        "video_name": "4mdEsouIXGM",
        "Q": "Did she forget to erase Wind in certain areas like 5:15, or is that just my imagination?",
        "A": "She did erase the Wind drawings a bit out of order."
    },
    {
        "video_name": "4mdEsouIXGM",
        "Q": "At 3:03 Vi is hiding the 'n' that Wind wrote when she wrote 'BACK SOON'",
        "A": "I also saw that she hid it."
    },
    {
        "video_name": "4mdEsouIXGM",
        "Q": "at 3:08 does anyone see that it says buck zoon",
        "A": "Yes I did. It was meant to be done."
    },
    {
        "video_name": "09Cx7xuIXig",
        "Q": "At 5:04 he started saying Could a lot and thats the data he has but why not explain you can do it diffrent ways? it seems unsure in a way i guess",
        "A": "I think those couple of ways that Sal is referring to are just different orders by which you can pick those data points that allow you to construct the box plot. He starts plotting the middle half of the data, but he can also start plotting the lowest and highest value and then go with the other data points. As long as you end up with with a box-and-whisker plot with the correct information, the order that you choose for construcing it doesn t matter."
    },
    {
        "video_name": "09Cx7xuIXig",
        "Q": "At 1:41, isn't the 1 in the data set the lowest number to order, not 2?",
        "A": "He just accidently missed it when he was putting it together. Hope this helped:)"
    },
    {
        "video_name": "09Cx7xuIXig",
        "Q": "At 1:15 why didn't he do the one first? That number is lower than 2.",
        "A": "Check 1:51. He forgot xD"
    },
    {
        "video_name": "09Cx7xuIXig",
        "Q": "At 1:41 why does Sal not include one at the start and starts by using 2??",
        "A": "He missed the 1 s, realizes at 1:52"
    },
    {
        "video_name": "09Cx7xuIXig",
        "Q": "At 1:43, why did Sal miss the #1s in the data set?",
        "A": "He didnt but he said he did :)"
    },
    {
        "video_name": "09Cx7xuIXig",
        "Q": "At 7:59, 6 (the median) isn't exactly in the middle, is it? It's like so much more on the left than the right. Why isn't it in the middle?",
        "A": "Well it s in the middle of the set of numbers (not in value but in location when the set is arranged from least to greatest) as it has 8 numbers to its left and 8 numbers to its right. That s what they mean by median."
    },
    {
        "video_name": "09Cx7xuIXig",
        "Q": "@6:35 There is no fourth quartile right? I am learning in my math textbook that there is Q1 (quartile one), Q2 and Q3.",
        "A": "I,m pretty sure there isn.t but i could be wrong"
    },
    {
        "video_name": "09Cx7xuIXig",
        "Q": "Why are the numbers called data points? 2:40",
        "A": "Because when you re dealing with graphs or charts, you re graphing/charting data. Each individual number, word, etc that you re graphing/charting is called a data point, or, if the data point is a number, it can be called a value. This is similar to a collection of variables in processing.js if you ve learned a bit of that! :)"
    },
    {
        "video_name": "5ZdxnFspyP8",
        "Q": "At 0:32, I'm not getting where that -1 is coming from.",
        "A": "if x is one, then the opposite would be negative x."
    },
    {
        "video_name": "J1twbrHel3o",
        "Q": "@12:32 why sal say 9.3/4=2.32 where actual answer is 9.325 which is equal to 9.33?",
        "A": "Rounding from a 5 can go either higher or lower, since it s exactly in the middle of 0 and 10. Some people always round up, but many also round down. It is also very common to round so that we keep the last digit even (i.e. 2,4,6,8,0). The last method makes doing later calculations in your head easier."
    },
    {
        "video_name": "c-wtvEdEoVs",
        "Q": "At 4:26 why can't 3bc\u00e2\u0088\u009b(a\u00c2\u00b2b\u00c2\u00b2) be simplified further to become 3bc\u00e2\u0088\u009a(ab)?\nOr would it be 3a^(2/3)b^(5/3)c?",
        "A": "In radical form, 3bc\u00e2\u0088\u009b(a\u00c2\u00b2b\u00c2\u00b2) is completely simplified as the radical contains no perfect cubes. If you were to rewrite this in exponent form, you would get: 3a^(2/3)b^(5/3)c Which one to use depend on the instructions you are given."
    },
    {
        "video_name": "NcADzGz3bSI",
        "Q": "Is Division really a fraction? Because Sal said in 1:00 that it can also be written in fraction form.",
        "A": "Yes like if you said 4/2 that would be saying the same thing"
    },
    {
        "video_name": "Ws03IbNrjfM",
        "Q": "At 5:32, Sir Sal says that we have to just throw in \"another Z\" meaning one, but the z factor of 7 has z^2, so shouldn't the final answer be 21z^3 because we are multiplying it.",
        "A": "The LCM contains the highest number of each of the different factors in the given polynomials. The first polynomial has z to the first power; the second polynomial has z to the second power. So only a z^2 is required."
    },
    {
        "video_name": "Ws03IbNrjfM",
        "Q": "At 3:21 when he reduced everything down, what happened to 2z's z variable? Cause on the third phase of reduction you basically did get 3z(z+1)(z-3) which gives back every variable but not 2z, you only get 2 back.",
        "A": "Remember, when you factor the polynomial, you find factors of -3 that add to -2. The 2 factors are: -3 and +1. So, in factored form, you will not see the -2z. But, if you multiply (z -3)(z+1), you will get the -2z in the middle term. Hope this helps."
    },
    {
        "video_name": "dj3tcEbWBDQ",
        "Q": "So how can we check our answer while rounding up or down both numbers (according to what was said in 2:14)? Do we have to round the answer, or what?",
        "A": "That would work"
    },
    {
        "video_name": "dj3tcEbWBDQ",
        "Q": "how do we estimate without any paper 1:38)?",
        "A": "this means: try to calculate the required answer in your head, do not write down your calculation on paper, just give the answer either on a paper answerform or, in the Khan-course in the display."
    },
    {
        "video_name": "2dbasvm3iG0",
        "Q": "Why did Sal write a dot instead of a multiplication sign at 5:03?",
        "A": "because the dot at 5:03 means the same as the multiplication sign (x)"
    },
    {
        "video_name": "2dbasvm3iG0",
        "Q": "From 0:00 to 2:00 it says to simplify, Do we have to? Is there some other way to do it that is easier or is that the best way?",
        "A": "thanks Noble, so it is the best way? or is it One of the best ways? Smaller numbers are easier to work with"
    },
    {
        "video_name": "2dbasvm3iG0",
        "Q": "At 2:19, you can try to make the denominator the same and multiply the numerator same amount of times as the denominator. example, 2/3 and 3/4 is going to be 3\u00c3\u00974=12 and 4\u00c3\u00973=12, and 2\u00c3\u00974=8 and 3\u00c3\u00973=9 so 8/12<9/12 or 8/12 is smaller than 9/12.",
        "A": "thanks ok i will look at 2:19"
    },
    {
        "video_name": "2dbasvm3iG0",
        "Q": "what does sal mean on 2:49 - 3:01",
        "A": "It s hard to tell exactly which thing you re referring to, but maybe you mean when he is talking about prime factorization and finding the least common multiple (LCM)? Doing prime factorization (by making a factor tree) and using prime factors is another way to find an LCM of two or more numbers."
    },
    {
        "video_name": "2dbasvm3iG0",
        "Q": "I don't understand Algebra do you guys know something I could do? Also in this video at 1:48 I don't understand what it means....\nPlease advise and reply as soon as possible!\nSincerely,\nJessica Martin",
        "A": "Alright, so what Sal has done is he has simplified the 2 fractions, that is, 21/28 and 6/9.. The result is 3/4 and 2/3 respectively.. then he proceeds to compare these 2 fractions.. If you d like to know anything else, don t hesitate to ask.."
    },
    {
        "video_name": "2dbasvm3iG0",
        "Q": "At 2:35 could someone explain what a prime number is please?",
        "A": "A prime number is an integer (whole number) greater than one that has exactly two prime factors. That being said, the simpler definition would be any integer that only has itself and one as integer factors. Example: 2 = 2 x 1 3 = 3 x 1 5 = 5 x 1 Nine is NOT prime because 9 = 9 x 1 and 3 x 3 23 = 23 * 1"
    },
    {
        "video_name": "2dbasvm3iG0",
        "Q": "At 2:48 you said that its 2 times 2 times 3 is 12. Is it because you prime factorize it?",
        "A": "Yes, you are correct. :)"
    },
    {
        "video_name": "EGr3KC55sfU",
        "Q": "On 1:06 what happens when you can simplify the numbers ?",
        "A": "You just... simplify them! Of course, only if you CAN simplify them..."
    },
    {
        "video_name": "Bt60JVZRVCI",
        "Q": "At 5:20 sal divided the the denominator of 6/15 and the numerator of 6/15 by three. But why divide the numerator and the denominator by three? Is there a reason sal chose to divide it by three?",
        "A": "Sal chose three because you can t divide it another way. If it divides good with the numerator, it has to do with the denominator, too."
    },
    {
        "video_name": "1015d63VKh4",
        "Q": "At 2:21 Sal says that the derivative of lnx is 1/x. How does that work? And would it be similar for finding the derivative of a log?",
        "A": "ln(x) is the natural logarithmic function. The proof of (ln(x)) =1/x is at the end of the content list."
    },
    {
        "video_name": "1015d63VKh4",
        "Q": "In that last step at 4:09, can we take the exponent of x in ln(x^2), bring it back, and multiply it with the entire resulting expression as we normally do with logarithms? Basically what I'm asking is if the answer could be 4/x(cos(ln(x))) instead of 2/x(cos(ln(x^2))).",
        "A": "No, because you re taking the cosine of that log and cos(2x) does not equal 2 cos(x)."
    },
    {
        "video_name": "1015d63VKh4",
        "Q": "At 3:56, how does the (1/x^2)*2x simplify to 2/x?",
        "A": "(1/x\u00c2\u00b2)2x = 2x/x\u00c2\u00b2 = 2/x because the x s cancel out."
    },
    {
        "video_name": "1015d63VKh4",
        "Q": "At 2:24, Sal said the derivative of ln (x^2) is 1/(x^2). Shouldn't it be 2/x because the rule for natural log is (d/du)(ln u) = (d/du (u))/u? So (d/dx)(ln x^2) = [(d/dx)(x^2)]/x^2, right?",
        "A": "That s what he addresses in the next few seconds of the video. h(x) = x^2."
    },
    {
        "video_name": "Touv2APcBRw",
        "Q": "At 3:25, why is the factoring suddenly preceded by a negative? I've looked my notes over and watched the video multiple times closely, but I'm still just totally lost. Can anyone shed some light on what I missed here? Thanks!",
        "A": "It wasn t a negative sign, it s an = sign. The 2 horizontal lines are cramped together vertically"
    },
    {
        "video_name": "Touv2APcBRw",
        "Q": "Why is there a negative sign at 3:26? There shouldn't be a negative sign there.",
        "A": "That s an equals- he just squished it a little"
    },
    {
        "video_name": "Touv2APcBRw",
        "Q": "Wait at 1:49 when sal was rewriting a to the 3rd power and b to the 3rd power, how did he get the ab in the equation. Plz halp.",
        "A": "Sal is basically using the formula that a^3+b^3 can be factorised as (a+b)(a^2-ab+b^2). This is just one of the formulas you should try to remember as it helps a lot throughout school."
    },
    {
        "video_name": "Touv2APcBRw",
        "Q": "At about 3:24, why would you put a negative in front of the whole thing? There's already the negative on the last trinomial.",
        "A": "Thats an =sign, if you watch closely you see that he draw two lines on top of each other."
    },
    {
        "video_name": "hmj3_zbz2eg",
        "Q": "At 2:32 how did sal know that was a supllimentary angle?",
        "A": "In order to understand how to find supplementary angles, one must first know what a supplementary angle is. When two adjacent angles add up to form 180 degrees those angles are supplementary to each other. A supplementary angles always form straight lines when together, which is how Sal Khan knew that the angle was supplementary with the other angle."
    },
    {
        "video_name": "hmj3_zbz2eg",
        "Q": "at 1:25, how was he able to turn the angles into a equation to solve the degree?",
        "A": "He is solving for the unknown angle by assigning a variable x to it. He finds this x angle by taking the known angles (50 degrees and 64 degrees) and adding those to the x angle to equal 180 degrees, because all angles in a triangle always sum to 180. So the equation is x +50 + 64=180, or x + 114 = 180. Subtracting 114 from both sides, you end up with x=66."
    },
    {
        "video_name": "hmj3_zbz2eg",
        "Q": "On 2:09 the triangle is isosceles so shouldn't the two bottom angles be congruent just sayin",
        "A": "The triangle is not isosceles. There are no little equivalency lines showing you that two of the sides are the same length. It might look roughly isosceles, but it is not exactly; which is why we need methods to find out the angles."
    },
    {
        "video_name": "hmj3_zbz2eg",
        "Q": "i paused the video at 0:31, and i figured out the answer in 1 minute. He then went 11 times slower than i did, and did the exact same thing as me. so my question is, why does he have to be so slow?",
        "A": "He wants to show the answer to the question step by step. So people new to it will understand the topic or answer"
    },
    {
        "video_name": "hmj3_zbz2eg",
        "Q": "another way to think about this problem when asked at 0:40 is to see the large triangle, and set the three angles equal to 180. 64+31+(50+?)=180, and solve for ?=35. the rest of the video shows good methods necessary for other triangle problems.",
        "A": "They way I though about that was: 180-64-31-50 = 35 . I started to solve for a third angle of a large triangle: 180-64-31, then I realised, that 50+? would be a third angle of a large triangle, so without finding third angle of a large triangle, I subtracted 50 and got 35. Intuition tip: if you know 2 angle degrees of a triangle, and you want to find a third angle, you will always have to subtract 2 known angle degrees from 180 degrees: 180 - 1st angle - 2nd angle = 3rd angle"
    },
    {
        "video_name": "hmj3_zbz2eg",
        "Q": "At 3:28 ,does it always right",
        "A": "why odes a triangle absolutely have to have 180 degrees altogether (*__*)"
    },
    {
        "video_name": "hmj3_zbz2eg",
        "Q": "At around 4:43, what does Sal mean by a \"get-go?\"",
        "A": "From the get-go is a phrase that means right from the start . So he s saying, if you knew that property of angles right from the beginning."
    },
    {
        "video_name": "hmj3_zbz2eg",
        "Q": "At 5:38, instead of z + 145 = 180, couldn't you just do 180 - 145? I know that's the same thing, but it's faster.",
        "A": "yes it is basically the same thing. you still get the same answer but it is good to know both ways."
    },
    {
        "video_name": "hmj3_zbz2eg",
        "Q": "At 3:52, can all this math could be simplified?",
        "A": "You mean the equation with 7 and the rest equaling 180. Alright, there are two definitions at work. The sum of the angles of a triangle is 180 and the sums of angles dividing one side of a straight line sum to 180. So, since you said 3:52 you understand why one of those angles is 180-a-b. Well if I wanted to make it simpler I could just understand that and jump straight to having y on one side of the equation, as y = 180 minus the other angle y = 180 - (180-a-b) = 180 - 180 + a + b = a +b"
    },
    {
        "video_name": "hmj3_zbz2eg",
        "Q": "2:31, What does the word supplementary mean?",
        "A": "that 2 angles add up to 180 degrees"
    },
    {
        "video_name": "GEId0GonOZM",
        "Q": "At 3:30, what is an counterexample? How do you write an counter-example using inductive reasoning?",
        "A": "A counterexample is an example to prove a statement wrong. Using inductive reasoning, you can argue your case with a counterexample, if the counterexample is a fact. For example: If I became stronger, then I must have eaten eggs. Counterexample: Eggs aren t the only things that can make you stronger (milk, weight lifting etc)"
    },
    {
        "video_name": "T4JKO0OGjpQ",
        "Q": "OMG watch at 0:58 he messed up he said to the right! XD",
        "A": "yeah ur right, cause I put it at the time 0:53 and I were like what Sal would not ever make mistakes And I know that sometimes people make mistakes so it was kind of awkward for me."
    },
    {
        "video_name": "T4JKO0OGjpQ",
        "Q": "I understand know that at 0:51 you move it to the left if dividing by 10.",
        "A": "Also instead of left/right, you can think of it as smaller size (magnitude) vs. bigger size. Dividing by a number > 1 means making the result smaller. If you started with 123, then /10 results in 12.3 rather than 1230, because 12.3 is smaller."
    },
    {
        "video_name": "T4JKO0OGjpQ",
        "Q": "Where are the other videos that explain this part more in depth 0:50 ?",
        "A": "My question is so different can you do one with $3.55 and 160% question?"
    },
    {
        "video_name": "UCCNoXqCGZQ",
        "Q": "At 5:13, why does a tree appear?",
        "A": "They are using a tree as a visual example of exponential growth. That is all."
    },
    {
        "video_name": "UCCNoXqCGZQ",
        "Q": "at 2:05 ,brit says that the last square is the 63rd . but arent there 64 squares on a chess board?",
        "A": "he said 63 steps. It does take 63 steps to move from start to 64"
    },
    {
        "video_name": "UCCNoXqCGZQ",
        "Q": "At 4:30, Sal says \"this is 4 to the 0 power there are no branches.\" Does any number to the 0 power equal 1?",
        "A": "Every number, other than 0, when raised to the 0 power, is exactly 1. This means that Sal should have said that there is 1 branch."
    },
    {
        "video_name": "UCCNoXqCGZQ",
        "Q": "at 2:05 ,brit says that the last square is the 63rd . but arent there 64 squares on a chess board?",
        "A": "Brit says that it is the 63rd step, since the first square requires 0 steps."
    },
    {
        "video_name": "UCCNoXqCGZQ",
        "Q": "At 1:09 how come on the first one its 2 times 1 and for the rest its 2 times 2 times 2 ect.",
        "A": "I think it is because each time you have to double."
    },
    {
        "video_name": "aq_XL6FrmGs",
        "Q": "Around 1:09 Sal put arrows on the parallel lines , is it necessary to indicate that the lines are parallel ?",
        "A": "This video is about identifying parallel and perpendicular lines, so for the purpose of the video it is very important to mark parallel lines. In general, there are lots of traits of parallel lines, so if we know this is a fact, it is always best to mark them, thus we will be able to draw other conclusions from this fact."
    },
    {
        "video_name": "aq_XL6FrmGs",
        "Q": "I forget why does sal make that mark like in 2:35?",
        "A": "The arrows on top of the WX mean that the line of WX extends beyond the diagram, rather than being a line segment."
    },
    {
        "video_name": "aq_XL6FrmGs",
        "Q": "At 2:40, Sal says that there are 3 perpendicular line pairs. Aren't there 6 sets of perpendicular lines? The three that Sal mentioned and three more opposite them. Is that what \"a pair of perpendicular lines\" are?",
        "A": "Yes. A pair means 2, and there were 6 lines. So, 6 divided by 2 would equal 3, for 3 pairs"
    },
    {
        "video_name": "aq_XL6FrmGs",
        "Q": "@1:39 how come AB and Ct are not parallel?",
        "A": "At 1:39 the lines AB and CD aren t parallel because CD is leaning towards AB and will intersect later down the page."
    },
    {
        "video_name": "aq_XL6FrmGs",
        "Q": "At 2:40, Sal says that lines AB and WX are perpendicular. If you look at it, it doesn't really seem as if the lines are perpendicular. Can someone explain?",
        "A": "Since there is a right angle symbol there, you know that the two lines are perpindicular"
    },
    {
        "video_name": "aq_XL6FrmGs",
        "Q": "at 2:47 it says that ab and wx are right angles, but they don't look at all like right angles at all, any comments?",
        "A": "You can tell that ab and wx are perpendicular because of the right angle mark on the angle. If an angle is exactly 90 degrees, the angle mark is a box-like thing."
    },
    {
        "video_name": "aq_XL6FrmGs",
        "Q": "3:01 Why is it a right angle?",
        "A": "What he is saying here is that because the picture is not necessarily drawn to any scale or accurate angles, then if they also add a symbol, such as the right angle symbol, you need to pretend that the rest of the drawing matches this right angle symbol. It isn t saying that the drawing is geometrically accurate, but it is saying that if the symbol is there then you need to suspend your judgement of the accuracy of the drawing and just work based on the symbols shown."
    },
    {
        "video_name": "aq_XL6FrmGs",
        "Q": "At 0:22 it mentions lines in the same plane. What is a plane?",
        "A": "A plane is a flat surface. At 0:09, it mentions the image, which is on the screen. In this case the screen is the plane or area that has two demensions and what x and y make."
    },
    {
        "video_name": "aq_XL6FrmGs",
        "Q": "at 1:41 How can you tell if they are parallel or perpendicular? Do you just look if they are straight or they intersect?",
        "A": "Perpendicular means they have 90 degree angles where they meet. Parallel means they will never meet."
    },
    {
        "video_name": "fqQ6sslzyhY",
        "Q": "At 2:09, Sal says: \"We can imagine how our function looks like\". How do we know the function is north of the slope? Why it is not south of the slope? Or it doesn't make any difference where and in what shape the function is?",
        "A": "The function might be either north or south of the slope, or north on one side and south on the other. We are not told enough to know what the function looks like elsewhere, just the bit at x = -1. Sal made up a function that would meet the described conditions at x = -1, and the function he made up happened to be north."
    },
    {
        "video_name": "fqQ6sslzyhY",
        "Q": "7:21 What's the difference between point-slope form and standard form?",
        "A": "Point-slope form is (y-y1)=m*(x-x1) where the point (x1,y1) and the slope m are known. Standard form is y=(m*x)+b, where the slope m and the y-intercept b are known."
    },
    {
        "video_name": "fqQ6sslzyhY",
        "Q": "What I find tricky with this section is at 3:38 he goes to find the slope, but the question doesn't specifically ask for the slope.\nIf this question was given to me without Sal's help, I wouldn't know what to answer.\nTherefore, is \"tangent line\" just a fancy way of saying \"slope\"?",
        "A": "A tangent line is a line that touches the function at just one point. The tangent line, like any line, has a slope, which you learned about in algebra. The derivative of a function at a point c is defined as the slope of the line tangent to the point c."
    },
    {
        "video_name": "gWqYXa7r_V0",
        "Q": "At 2:37 Sal said \"0.1\" but meant \"0.01",
        "A": "Yup? it says 0.01 right there."
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "At 7:23 Sal mentions y = x. Maybe it's just because I'm really tired as I write this, but I can't seem to grasp the concept of y = x. What does this mean?",
        "A": "Imagine if y is 1, and y=x, then x is also 1. If y=2, then x=2. If y=3 then x=3. When you graph that you get the line that he shows there."
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "Doesn't the function at 0:11, Sal said you can put any real number. i is imaginary, right? But when I plug in i in the equation, I get the complex number 4+2i. Can't you input any number?",
        "A": "You can input any complex number into the function, however, the graphing part is restricted to real numbers only. The domain and range refer to the graph of the function, and you can only graph real numbers on the Cartesian plane. To graph imaginary numbers alongside real numbers, you need the Argand coordinate system."
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "At 4:31, How does or why does y/2 - 2 = x turn into or become x = 1/2y - 2? Where is the 1/2y coming from?",
        "A": "a/b = (1/b)\u00e2\u0080\u00a2a"
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "ok, what keeps confusing me is when sal says (e.g. at 5:22 ) that you can \"replace the y with an a or a b or an x...\". but don't you have to replace the y with the x to make it an inverse of x because the equation x = 1/2 y - 2 is the same line as y = 2x + 4? substituting in the x gives that symmetrical reflective line... if it sounds like i'm confused that's b/c i AM lol... all help appreciated! :D",
        "A": "You have to think by taking the invers that y becomes x and x becomes y. the point (2,3) becomes (3,2) and the point (-1,5) becomes (5,-1). Notice that point (1,1) or (-1,-1) doesnot change. All these point are on the reflective line y=x. so for the line f(x)=2x+1 you can write y=2x+1. Take the invers by swapping the x and y. then the invers is x=2y+1. put Y left and x right. then -2y=-x+1. divide by -2 on both sides. y=(1/2)x-1/2. write for y=f-1(x) then f-1(x)=(1/2)x-1/2. (f-1(x) is invers function) that all"
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "At 4:46, if f^-1 is the inverse, could you do f^2 or 3, or so on? If so, what would those mean?",
        "A": "f^2, f^3 have not been defined in Mathematics yet."
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "at 5:35 is there any special reason to rename f^-1(y) to f^-1(x). I suppose that both of these is the inverse function of f(x)?",
        "A": "Yes, since your original function is a function of x, the inverse must also be a function of x therefore: \u00c6\u0092(x) \u00e2\u0086\u0092 \u00c6\u0092\u00e2\u0081\u00bb\u00c2\u00b9(x)"
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "7:20 I don't understand the concept of \"y=x\". Can you please help me ?",
        "A": "The y=x is an equation for the line that goes through the points (-1, -1), (0,0), (1,1), etc. so that all values of x are the same values for y. This line goes through the origin and is 45 degrees to each of the axes in the first and third quadrants of the Cartesian coordinate system. Sal uses this concept to show that the two functions he is discussing are reflected in this line, just like reflecting through the y=0 or the x=0 (y and x axes)."
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "At 2:43, Sal says the inverse of f(x) is f(x)^-1. Is there a reason he uses that notation?",
        "A": "He uses that notation because saying something (in this case f(x)) raised to the negative first power is the same thing as the inverse of that something"
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "from 5:20 of the video where you rewrote the equation as an f(x), shouldn't you have written it as inverse f(x)=1/2y-2?",
        "A": "wait nvm. i made a mistake"
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "What does he mean from 6:05 to 8:45",
        "A": "Sal shows the relation between the graph from a (linear) function and its inverse. Specifically, he shows how the graph of the inverse of a function is a reflection over the line y = x."
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "At 4:55 does Sal mean to write = instead of > as it looks kinda confusing where he got > from? PLEASE LET me know if I am wrong.",
        "A": "its an equals"
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "Starting at 1:19 Sal draws another set as the range. Why is Sal drawing a second set next to the first one if the domain and the range are the same set, the set of real numbers? Shouldn't he draw one \"blob\" such that the function maps from elements in that set to other elements in that set?",
        "A": "No, because the domain is the numbers on x that can go in and the range is the numbers on y that can be a result of this function. So they could both be all real numbers but they could be different. In a square root function for example, the domain would be all positive numbers only."
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "@8:13 where the two lines cross, is that the numbers that would satisfy both equations? or is there any significance?",
        "A": "Benaiah, you are, by my knowledge, correct. The coordinates of the exact point where the lines meet are the x and y that solve in both equations. For some more info on the matter of solving for this point with any two functions, check out Systems of Equations."
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "At 4:50, why does Sal write F with a negative exponent of 1?",
        "A": "The original function is named F. To denote the inverse, we write the function name with a superscript of -1. It looks like an exponent, but it isn t. It is just the way to tell us that we have the inverse function to function F. Hope this helps."
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "i don't quite get the thing Sal does at 7:19. He is talking about y being equal to x, and then draws a dotted line in the middle. is there maybe a video that clarifies this relation?",
        "A": "Heh, yeah, that may have been a bit fast. The equation of that dotted line is y=x, and his point is that the function and the inverse were reflections of each other across that dotted line. In other words, if you drew the graph of a function and its inverse on a piece of paper and then folded the paper along the line y=x, the two graphs would line up."
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "At around 5:15, Sal labels the function 1/2y-2 as an inverse function, deemed f-1(y).\n\nI am not sure, but I think this is incorrect? F-1(x) should equal 1/2y-2, not F-1(y), right? F(y) is just a function, not an inverse. However, the inverse of x is equal to the y function.\n\nI hope I wasn't too confusing, but it would be appreciated if this got cleared up! :) thanks",
        "A": "f(x)= 2x +4 .... f^-1(x)=(x/2)-2 .... f^-1[f(x)]=(2x+4)/2 -2 = x .(identity) f(y)=(y/2)-2.... f^-1(y)= 2y+4.... f^-1[f(y)]=2(y/2 -2)+4 = y .(identity) Check plot at 7:45 to see that these are symmetric across X=Y. ..Venn rings can mislead."
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "When Sal was plugging in values of x (1:20) and relating the domain to the range, I noticed that the ratio from domain to range was 1:2. Is there a ratio like this for all functions and there domain and range?",
        "A": "Ender, Very observant. For a linear equation, the coefficeint on the x will give you the ratio you are taking about. If it had a 3x, you would see a 3:1 ratio. Notice that the inverse equation ended up with a 1/2x in the inverse function. So the ratio is reversed in the inverse to 2:1 instead of 1:2. With observation intuition like you have, math should be easy for you."
    },
    {
        "video_name": "W84lObmOp8M",
        "Q": "Time of the vedio. 4:30 Why does y go above 2 \"Y/2\" onece 2 is subtracted from either side? I don't uderstand the math here.",
        "A": "2 is actually divided from either side."
    },
    {
        "video_name": "I6TBBzIvgB8",
        "Q": "at 0:06 there are more factors of twelve like 2*6=12 or 3*4=12 so where are 3*4?",
        "A": "I think Sal included enough factors of 12 for people to get the idea."
    },
    {
        "video_name": "I6TBBzIvgB8",
        "Q": "@ 0:05 Sal says that 12 is arbitrary. What does that mean??",
        "A": "Arbitrary is a word that means based on random choice or personal whim , which basically means that it was a random number Sal picked for no reason."
    },
    {
        "video_name": "I6TBBzIvgB8",
        "Q": "I have a similar problem to the one at 3:06 , but mine is a subtraction problem. I am still confused on how to factor expressions with subtraction. My expression that I am trying to factor is 4x-8",
        "A": "2 is factor both 8 and 4 ,so you can distribute the equation 2(2x-4)."
    },
    {
        "video_name": "I6TBBzIvgB8",
        "Q": "At 3:45 What happened to the other 6? Wouldn't it be 12(x+5)?",
        "A": "6x+30 is the same as 6(x+5) 6*x + 6*5 is 6x+30 Same thing in reverse, 6*x+6*5. You take out the 6. Apparently, you take out the multiplication signs as well. But not the addition sign."
    },
    {
        "video_name": "I6TBBzIvgB8",
        "Q": "That is a HUGE leap to factoring out a fraction--not much explanation. How did he get the 1/2 out of 3/2x at 4:51?",
        "A": "3/2 times x is the same as 3/2x. Notice that 1/2 and 3/2 have the same denominator, which is two. Since 3 times 1/2 = 3/2, 3x times 1/2 = 3/2x. So 3/2x equals 1/2(3x)."
    },
    {
        "video_name": "I6TBBzIvgB8",
        "Q": "1:50 HOW does he get the 2?",
        "A": "So the number that you can take out of all of the terms is 2 2+4x So if you divide the whole thing by 2 you get 2(1+2x) Make sense?"
    },
    {
        "video_name": "I6TBBzIvgB8",
        "Q": "At 4:34, Sal asks that we try and solve 1/2 - 3/2x, but I am having a lot of trouble, and I don't understand it that well. Could anyone give an explanation on how to solve it?",
        "A": "I could have said. 1/2- 3/2X = 0, 2X (1/2) = -3, 2X/2 = -3, X = -3 1/2 - 3/2(-3), 1/2 - 3/-6, 1/2 +1/2 = 1 I made a mistake with the - sign in the other answer, but the proceedure was correct."
    },
    {
        "video_name": "I6TBBzIvgB8",
        "Q": "I don't really understand what he did at 1:56, will someone explain? how does 2+4x = 2(1+2x)",
        "A": "You re basically un-distributing, or factoring, the 2. Since 2 and 4x share the factor 2, you can factor it out to get 2(1+2x). To prove that they re the same, distribute the 2 by multiplying it by 1 and 2x, and you ll get 2 + 4x again."
    },
    {
        "video_name": "I6TBBzIvgB8",
        "Q": "It is amazing that at 3:36 he states that it is the reverse of the distributive property.",
        "A": "Which grade are you guys in?"
    },
    {
        "video_name": "V-BAdpvejTw",
        "Q": "At 6:06, why does Sal divide by two?",
        "A": "he is finding the average between two numbers. to find an average you add the numbers and divide that number by the number of terms"
    },
    {
        "video_name": "V-BAdpvejTw",
        "Q": "At ~6:10 how does sal get the -1 for the y value of the vertex?",
        "A": "Good question, he didn t really explain what he did there. Sal found that the x-value of the vertex was -3, so he substituted -3 back into the equation for x and solved it to find the y-value of the vertex. y = (x + 2)(x + 4) y = (-3 + 2)(-3 + 4) y = ( -1 ) \u00e2\u0080\u00a2 ( 1 ) y = - 1"
    },
    {
        "video_name": "7FTNWE7RTfQ",
        "Q": "Near 2:00, Sal lost me when he said BC and DC were congruent. Isn't DC way bigger? Help please?",
        "A": "The length of his triangles are not perfect. He is free handing this all, that is why it does not look congruent."
    },
    {
        "video_name": "7FTNWE7RTfQ",
        "Q": "At 0:30, how is angle ABC isosceles?",
        "A": "Sandy Knight didn t fully answer your question- the definition of an isosceles triangle is a triangle having two sides the same length- since AB and BC are marked congruent, that means triangle ABC is isosceles."
    },
    {
        "video_name": "7FTNWE7RTfQ",
        "Q": "At 4:11 isn't that triangle a right triangle?? So doesn't that mean that the triangle is a 45-45-90?",
        "A": "The triangle that results from drawing the diagonal in the square, is indeed a 45-45-90 triangle."
    },
    {
        "video_name": "TqGCNNlx6pU",
        "Q": "Hi, at 0:42, Sal says symbolizingly, \" we know, \u00e2\u0088\u00ab f(t) dt, a to b, = F(b)-F(a)\", and something about the second fundamental theorem of Calculus, abbreviated TSFTC. I have no clear idea about where F(b)-F(a) stated from, and what is TSFTC. I beg your help.",
        "A": "You may need to go back 2 sections to here (or even before) and work your way back to this video. I think you are missing some material. It may be best to start the integration section right from the start and get some experience and intuition before you tackle the first and second fundamental theorems of calculus in this section."
    },
    {
        "video_name": "F0LLR7bs7Qo",
        "Q": "Can you have a reciprocal for words or units? At 1:47 Sal says \"...because the units are the reciprocal of this\" and I wasn't sure...",
        "A": "sure reciprocal of m/s is s/m. reciprocal of banana/pie is pie/banana."
    },
    {
        "video_name": "F0LLR7bs7Qo",
        "Q": "At 2:23 how did you get 0.75?? At 3:41 how did you get 0.50??",
        "A": "These are computed by the fractions. 9/12=3/4=0.75 50/100 = 1/2 = 0.5"
    },
    {
        "video_name": "F0LLR7bs7Qo",
        "Q": "why does sal always change the numbers order? For example, he changes the statement; 12 feet per second to second per feet in the equation at 2:08. why does he do that?",
        "A": "So that he could eliminate the feet from the 9 feet and be left with just seconds. The rate of the squirrel is 12 feet per second but that s the same thing as saying in 1 second the squirrel can move 12 feet."
    },
    {
        "video_name": "F0LLR7bs7Qo",
        "Q": "at 2:57 Sal uses 50/100 and does seconds/feet beside it but shouldn't be\n50 seconds/100 feet ?",
        "A": "You could use either way..Sal usually use that way to cross multiply them"
    },
    {
        "video_name": "0_DYEFtlCiM",
        "Q": "At 2:56, Sal is going to multiply the inverse of A by column vector B. If they could not be multiplied, would column vector x be undefined?",
        "A": "yeah. if A doesnt have an inverse, then your system has infinitely many or no solutions, so the column vector x has to be undefined"
    },
    {
        "video_name": "q9XCrpLPfGk",
        "Q": "i do not get it how he got 1000 over 1 at 0:52",
        "A": "you should watch the videos in the previous chapter about unit conversions :) after that every conversion will be way clearer :)"
    },
    {
        "video_name": "0ZKtsUkrgFQ",
        "Q": "At 3:44 why did it show math",
        "A": "Because this IS math"
    },
    {
        "video_name": "0ZKtsUkrgFQ",
        "Q": "At 2:10, can you swap the score and name axis?",
        "A": "Nope then it wouldn t be the same."
    },
    {
        "video_name": "bLTfBvkrfsM",
        "Q": "At 6:22, why does Sal multiply 3 and 5, isn't the answer just 5?",
        "A": "No because 5 is not the greatest common factor, 15 is"
    },
    {
        "video_name": "bLTfBvkrfsM",
        "Q": "why did sal say GCD at 6:15?",
        "A": "Doesn t GCD stand for greatest common denominator?"
    },
    {
        "video_name": "bLTfBvkrfsM",
        "Q": "At 4:30 , what does Sal mean when he says\" So what are the most factors that we can take from both 21 and 30 to make the largest possible numbers\" ?",
        "A": "He wants to find as many factors in common to both numbers. 21 = 3 x 7 30 = 3 x 2 x 5 The only factor they have in common is 3. So, that is the largest possible common factor."
    },
    {
        "video_name": "bLTfBvkrfsM",
        "Q": "So, at 6:05, if they have two factors in common, then you multiply those numbers to get the GCD?",
        "A": "yes unless you can come up with a different way"
    },
    {
        "video_name": "bLTfBvkrfsM",
        "Q": "At 4:26, why is 30 split into 3 and 10? I thought for the prime factorization you wanted to use the smallest divisible numbers, which would have been 2 and 15. 15 would then split into 3 and 5, which would still give us the common divisor of three. But I guess I'm just confused about the proper usage of the factor tree.",
        "A": "The prime factorization will come out the same however you do it. For example: 30 = 3x10, 10=5x2 therefore 30=3x5x2. (5,3,2) 30 = 2x15, 15=5x3 therefore 30=2x5x3. (5,3,2) Either way you always will get the same answer. Hope this helps!"
    },
    {
        "video_name": "bLTfBvkrfsM",
        "Q": "At 0:4 what is GCD?",
        "A": "greatest common divsor"
    },
    {
        "video_name": "bLTfBvkrfsM",
        "Q": "at 4:51, why did he prime factorize 3 and 10 instead of 2 and 15",
        "A": "When you create prime factor trees, it doesn t matter which 2 pair of factors you start with. The only thing that matters is that the 2 numbers you pick must multiply and equal the original number (in this case, the 30). So, you can start with 2 * 15 as your 1st step. You would then factor the 15 into 3 * 5. You will have the same prime factors that Sal created. Hope this helps."
    },
    {
        "video_name": "bLTfBvkrfsM",
        "Q": "At 6:13 why did he multiply 3 and 5. Wouldn't the GCD be 5 because it is the greatest factor?",
        "A": "Good question.It won t be 5 even though it is the biggest factor out of 2,3 and 5 because the greatest common divisor(or the biggest number divisible by both numbers ) is the product of the factors that are common to both numbers.So in the problem Sal had the prime factorization of 105 (7*3*5) and 30 (2*3*5) and 3 and 5 were the common factors and so he multiplied them together and got 15 (If you check it 105 and 30 are both divisible by 15) and so that is why 15, not 5, is the GCD. Hope that helps! :)"
    },
    {
        "video_name": "cNlwi6lUCEM",
        "Q": "at 6:50 how does 14/4 become 7/2?",
        "A": "Yes, always simplify fractions on all tests. This is because the test moderators usually want the same answer from all the students."
    },
    {
        "video_name": "cNlwi6lUCEM",
        "Q": "Can someone please explain 7:13 to 8:02 for me? Specifically, I want the answer to the initial question, in the word problem, completely answered. Thank you so much. :)",
        "A": "Ok, thanks for that answer! :)"
    },
    {
        "video_name": "cNlwi6lUCEM",
        "Q": "at 8:41, how do you set up equations using percentages?",
        "A": "Search on Percent or percentage . There are several videos available on percentages, including some with examples of word problems or applications involving percentages."
    },
    {
        "video_name": "cNlwi6lUCEM",
        "Q": "At 6:18 you subtracted 2 from both sides but it should of been -2 because 4m - 2=12 is converted to 4m + (-2) =12 so then you would add 2 not subtract.",
        "A": "You can add 2 or subtract -2, they are both the same thing, he just chose to add 2."
    },
    {
        "video_name": "sPPjk4aXzmQ",
        "Q": "I'm still not understanding at 2:55 in the video where you say that ln x and ln |x| are mirror images when graphed? I still don't know how to tell off the top of my head. How do you go about figuring something like that?",
        "A": "Consider the function y = ln |x|. This function takes in the negative values of x which ln(x) cannot. Try to input negative values of x in the function y = ln |x| and we get the same value of y which we get for y = ln(x). Use a graphing calculator to see them visually to get a better understanding. Sorry for my bad English"
    },
    {
        "video_name": "sPPjk4aXzmQ",
        "Q": "AT 6:55 Sal says the derivative of ln|x| is 1/x for all except o but we can clearly see that the derivative are varied, for x> 0 its 1/x and for x<0 its -(1/x). But how does Sal say its the same derivative ?",
        "A": "When x < 0, we also have 1/x < 0, so we do indeed have d/dx log |x| = 1/x for every real number x \u00e2\u0089\u00a0 0."
    },
    {
        "video_name": "sPPjk4aXzmQ",
        "Q": "At 1:48: Why do we search for ln |x|? at that moment?",
        "A": "ln|x| includes both positive and negative values as the domain, while lnx does not."
    },
    {
        "video_name": "sPPjk4aXzmQ",
        "Q": "Why at 1:16 , in the (ln x + c) , why x>o ?\nPlease explain .",
        "A": "This is because the natural logarithm (ln) of 0 is undefined. It would be the same thing as saying e^x=0. There is no value of x that would give you 0 note: the derivative of lnx is x^-1 which also is undefined for 0."
    },
    {
        "video_name": "AGFO-ROxH_I",
        "Q": "At 1:26. What is the purpose of doing things like ft/mile when you can just make it easier by dividing the 5,280 feet by 3? (because there are 3 ft in a yard)",
        "A": "I honestly agree with you(some people find this method easier). You totally could find the number of feet and divide it by three, instead of going through the process of multiplying it by 1/3(with all the extra ft, mi, and yd stuff). I think Sal did it this way so that people understand what units they re working with in each step. It s really easy to get lost with the units in these types of problems."
    },
    {
        "video_name": "AGFO-ROxH_I",
        "Q": "At about 5:40,Sal just said 3520 yards = 300 laps without saying how he figured it out or what operation(s) he used .How'd he figure it out?",
        "A": "Well he probably left it to you to figure out but if you can t figure it out you can click report a problom at the bottom of the video. I think they would love to help you and not to confuse anybody else. Another way you can look at the viedo for some time. He probably explanes this method. Good luck!"
    },
    {
        "video_name": "AGFO-ROxH_I",
        "Q": "At 2:49, Sal checks his work by multiplying. Before this point, he did it in his head. Isn't that more work to do it in your head because afterwards you have to check it? Why didn't he just multiply on the screen? Wouldn't it have been easier for him?",
        "A": "You should check your work anyways, you can make a mistake even when you do it all out in writing."
    },
    {
        "video_name": "AGFO-ROxH_I",
        "Q": "Is what Sal is doing at around 2:12 an example of the factor-label system? Please don't answer, \"I don't know.\" I'd like a real answer.",
        "A": "Not quite sure what you mean by factor-label system. Converting units isn t about factoring, if I was to do something like convert miles to kilometers, I d get 1.60934 km/mile, so that I could cancel out the miles and get left with the number of kilometers, that is hardly a factor of a whole number."
    },
    {
        "video_name": "cPOjnyjii-4",
        "Q": "At 2:45: Why is there shadow on that point Sal describes?",
        "A": "At any time, you can draw a straight line from the light and through where the owl is. Wherever that line winds up hitting the ground, that s where the shadow is."
    },
    {
        "video_name": "M3PTPN7NH8M",
        "Q": "At 3:20, he said goal. Shouldn't it be basket?",
        "A": "Sal said: The goal is at 26 feet away at 3:14. In this context, the word goal means a target."
    },
    {
        "video_name": "wx2gI8iwMCA",
        "Q": "4:54\nhow will you add another mark to the numbers try these\nI = 1 II = 2 III = 3 IV = 4 V = 5 VI = 6 VII = 7 VIII = 8 IX = 9 X = 10 XI = 11\nXX = 20 XXXX = 40\nthen you can add a mark",
        "A": "Actually, it would NOT be XXXXI, because according to the rules of Roman Numerals , you can only have 3 of the same letters in a set . For, 41 would NOT be XXXI, it would actually be XLI. It would be XLI because another rule is that if a smaller numeral is in front of a bigger numeral, you subtract it. So 50 (L)-10 (X)=40 (XL). Here are some basic Roman Numerals : I=1 V=5 X=10 L=50 C=100 M=1,000 I hope this helped! -TheAmericanBerserker"
    },
    {
        "video_name": "wx2gI8iwMCA",
        "Q": "around 4:00, you said to group by tens, but wouldn't you, if you have hundreds, group ten groups of ten, in the thousands, 10 groups of those group of groups, so on?",
        "A": "no, in the thousands you count by hundreds"
    },
    {
        "video_name": "E9Q_Lc0g1xE",
        "Q": "at 12:03, he says \"horizontal\", doesn't he mean \"vertical\"?",
        "A": "Yep, he misspoke. He does name it correctly elsewhere in the video, though."
    },
    {
        "video_name": "DA2w8lRelIc",
        "Q": "Mike's plane takes off at 10:25am , his flights last for two hours and 35 minutes. What time does his plane land .",
        "A": "that is because time works on 60 numbers in a whole were as adding up works on 100."
    },
    {
        "video_name": "DA2w8lRelIc",
        "Q": "So if an old arcade game is scheduled to be unplugged at 9 AM tomorrow, and it is 8:50 PM, how long (in hours and minutes) will it be until the game is unplugged?",
        "A": "The game will be unplugged in 12 hours and 10 minutes."
    },
    {
        "video_name": "rHNY01R2VSQ",
        "Q": "1:40 - 1:48 why did the last 2 was changed to a 1",
        "A": "Because it was factored. If you do the distributive property, the 2 on the outside would multiply with the 1 inside the parenthesis equaling 2."
    },
    {
        "video_name": "JX5Zvh6swmo",
        "Q": "At 1:16 why can you not just leave it as (x+3)(x-3), rather than make it (x+3)^2?",
        "A": "I think you mean can you leave it as: (x+3)(x+3) -- where both are positive. Yes, you could. Sal is emphasizing that the factor is squared by writing it with the exponent."
    },
    {
        "video_name": "JX5Zvh6swmo",
        "Q": "At 0:35, Sal says that 1+9 doesn't equal 6, and negative 1 + 9 doesn't = 61x, but doesn't -3+9=6? Why won't that work?",
        "A": "because they need to multiply to equal 9 and add to equal 6, yours breaks the rules because (-3)(9) = -27. So you could have x^2 + 6x - 27 and your two would work great then."
    },
    {
        "video_name": "JVlfQEhzLMM",
        "Q": "This isn't the first time but at 1:24 instead of using an x, he uses a dot, what's the difference?",
        "A": "There is no difference. Since you are beginning to use variables, and the most common variable is x, that means confusion between the variable x and the operator x can result, so the multiplication dot is used instead of the multiplication x. You will get used to it before you know it!"
    },
    {
        "video_name": "JVlfQEhzLMM",
        "Q": "Why is it that (at 1:59) all of a sudden we add Drink A and B together? When did the problem say that we added the two drinks together and end up with C?",
        "A": "Reread the problem statement. In the 2nd sentence it says (somewhat condensed): he takes 2 liters of Drink A and ADDS 1.2 liters of Drink B . Then, in the 3rd sentence, the problem tells you that this new drink is Drink C. Hope this helps."
    },
    {
        "video_name": "JVlfQEhzLMM",
        "Q": "How do you solve \"dry mixture\" problems?\n\nFor example, here is one I made up:\n\nAmy has 7 pounds of a 3:4 pistachios to cashews mixture. How many pounds of pistachios is she supposed to add to convert it into a 8:4 mixture?\n\nI don't really understand how to solve these types of questions.",
        "A": "Let s say that there are 3x pounds of pistachios originally. That means that there were 4x pounds of cashews. Added together, there were 7x pounds of nuts that weighed 7 pounds. So, x=1. There were 3 pounds of pistachios and 4 pounds of cashews. You want to add pistachios to make it eight to four. Since this is already 3 to 4, then you just need to add the 5 pounds of pistachios to get from 3 to 8."
    },
    {
        "video_name": "xKH1Evwu150",
        "Q": "i dont understand what he did at 4:20",
        "A": "The equation reads t - 5 = 199. To solve this equation you need to get t to stand alone on one side, meaning you need to get rid of the - 5 on the left side. How do you get rid of -5? You +5 (add 5) Now the left side reads: t - 5 + 5. But you can not only +5 one side you also have to +5 the other side --> 199 + 5 So the whole equation reads: t - 5 + 5 = 199 + 5 (- 5 + 5 = 0 --> only t is left) t = 199 +5 t = 204"
    },
    {
        "video_name": "xKH1Evwu150",
        "Q": "At 1:14, how come it's not (t-5) + 210",
        "A": "Well the equation is showing 210*(t-5) or 210 times (t-5) not (t-5)\u00c2\u00b1 210."
    },
    {
        "video_name": "DAikW24_O0A",
        "Q": "at 0:29 why do u multiply 15% by x?",
        "A": "The way I would solve that problem, is by writing: 78/x = 15%/100% ; x = (78*100)/15 = 520"
    },
    {
        "video_name": "DAikW24_O0A",
        "Q": "He got it wrong\nat 2:45",
        "A": "No he did it right, at that video time, he is in the process of doing long division. (If you still think it seems wrong, it will help if you say what looks wrong, if you want more explanation.)"
    },
    {
        "video_name": "DAikW24_O0A",
        "Q": "Why do you put 0.15 on the bottom of both 0.15 and 78 as in 1:37?",
        "A": "To solve an equation, you must do the same operation to both sides of the equation. Your goal is to get to X = a number. So, you need to change 0.15X into just X. This is done by dividing by 0.15. Since left side is being divided by 0.15, you must do the same operation to the right side to get 78/0.15. Hope this helps."
    },
    {
        "video_name": "DAikW24_O0A",
        "Q": "at 3:02 this is hard",
        "A": "Stick with it. It s hard now, but it will become fun when you start to understand it. Then you ll be glad you were persistent."
    },
    {
        "video_name": "DAikW24_O0A",
        "Q": "at 5:17 Sal says gels , What in the world does he mean by gel.",
        "A": "He just means agrees with . I found this on google about the history of the word gel gel (v.) Look up gel at Dictionary.com 1902, to become a gel, from gel (n.). Figurative sense come together and agree well is from 1958. Related: Gelled; gelling. The figurative sense says come together and agree well"
    },
    {
        "video_name": "DAikW24_O0A",
        "Q": "At 2:05 how do you know how many zeros to put?",
        "A": "Think of it this way: Every number has a decimal point. 1 is actually 1.0 understand? Is there any difference between 1 and 1.0? No difference at all. Is there any difference between 1 and 1.00 ? They are exactly the same value. Is there any difference between 1 and 1.0000000 ? No difference. So, Sal is just guessing that he will need 2 or maybe 3 zeros and adds them to the end of the number. He could have added more, he could have added less, it wouldn t matter. Zero s after the decimal place don t matter."
    },
    {
        "video_name": "DAikW24_O0A",
        "Q": "@ 5:30 KHAN said that something gels with the answer , what the heck dos gel mean",
        "A": "Probably means it goes with or use common sense or makes sense"
    },
    {
        "video_name": "DAikW24_O0A",
        "Q": "at 3:10 i don't understand why you can move the decimal points in a division problem. Doesn't that change the value of the number?",
        "A": "Inverse operations. 78/.15 is the same as 78 * 100/15 = 7800/15 which is the same thing as moving the decimal in the long division algorithm, which doesn t change anything."
    },
    {
        "video_name": "DAikW24_O0A",
        "Q": "At 0:20 isn't the equation Mr. Khan has expressed a proportion in its own way?",
        "A": "Yep, it could be a proportion."
    },
    {
        "video_name": "BR5yFOt0zao",
        "Q": "At 1:00, why is he multiplying by 9/5 instead of 5/9? Wouldn't you use the regular fraction instead of the inverse?",
        "A": "Divide 5/9 = Multiply by 9/5."
    },
    {
        "video_name": "BR5yFOt0zao",
        "Q": "With the video's example solved at 2:26, if I wanted to do this problem in reverse and take 9/5C+32 = F and solve for C, how would I know that parenthesis are required? That is, how would I figure the answer would be C = 5/9(F-32) and not C = 5/9F-32?",
        "A": "Subtract 32 first. => 9/5C=F-32. Now, to divide by 9/5, you have to divide everything by 9/5 => C=5/9(f-32)."
    },
    {
        "video_name": "BR5yFOt0zao",
        "Q": "at 1:32 was is it 9/5times c cant it be 9c/5",
        "A": "Yep, it s the same thing. It s just a matter of personal preference. Hope this helps! \u00e2\u0080\u0094CT-2/002-24"
    },
    {
        "video_name": "BR5yFOt0zao",
        "Q": "at 1:08 why do you multiply 9/5 to the equation instead of 5/9. I don't get it.",
        "A": "Because to remove 5/9 from the equation, you divide both sides by 5/9. 5/9 divided by 5/9= 5/9*9/5=1"
    },
    {
        "video_name": "BR5yFOt0zao",
        "Q": "when the left side of the equation is multiplied by 9/5 in 1:22, shouldn't (F-32) be also multiplied by 9/5?",
        "A": "when he multiplies the left hand-side by 9/5, he also multiplies the other hand-side by 9/5 (watch the purple writings). Then he goes on to rewrite the equation in yellow (1:22). At 1:22, he doesnt multiply again, only simplifying the equation. So, he didn t make a mistake."
    },
    {
        "video_name": "BR5yFOt0zao",
        "Q": "Why does sal at 0:31 spell fahrenheit wrong?",
        "A": "Because Sal, just like many others, are human. I know you are from Mars, and I m a robot, but humans are different from us. They make mistakes. Don t worry, it was corrected though."
    },
    {
        "video_name": "dZnc3PtNaN4",
        "Q": "At 2:24, what does n choose(?) n mean?\nWhere he wrote () with one n on top of another inside.",
        "A": "(n) (r) =nCr =[n(n-1)(n-2)...(n-r+1])/[r(r-1)(r-2)...(r-r+1)] I hope this helps!"
    },
    {
        "video_name": "yV4Xa8Xtmrc",
        "Q": "at 1:10, why is it a negative sign and not positive when you break the original parenthesis apart?",
        "A": "What original parenthesis are you referring to?"
    },
    {
        "video_name": "yV4Xa8Xtmrc",
        "Q": "At 3:28, why does he add the negative sign?",
        "A": "Did you mean at 3:43? Sal just multiplied both sides of the equation by -1. C = 1 - 2sin^2 (\u00ce\u00b8) -C = -1 + 2sin^2 (\u00ce\u00b8) Or switching the order on the right side: -C = 2sin^2 (\u00ce\u00b8) - 1"
    },
    {
        "video_name": "fSu1LKnhM5Q",
        "Q": "um at 4:30, left column number 5, what's the difference between measure angle DCE (m<DCE) and just angle DCE (<DCE)?",
        "A": "Writing the letter m means measure... and it is a more formal way of just writing <DCE (m<DCE)"
    },
    {
        "video_name": "fSu1LKnhM5Q",
        "Q": "At 1:01 when he is writing down all of the postulates, why did he not also write the SSA postulate? It is the one where two sides and one of the non-included sides are all congruent.",
        "A": "I assume you meant one of the non-included angles. But SSA is actually not a congruency postulate. It can be shown why rigorously by using trigonometry, specifically the Law of Sines. There is however a special case of SSA for right triangles that is a valid congruency postulate, but we call this HL (hypotenuse-leg)."
    },
    {
        "video_name": "fSu1LKnhM5Q",
        "Q": "At 7:36 for statement 9, can someone give a more convincing argument to why they are supplementary (form a straight line)?",
        "A": "Segment DB is a straight line because it is a diagonal. Segment AC is also a straight line because it is a diagonal."
    },
    {
        "video_name": "fSu1LKnhM5Q",
        "Q": "At around 00:40, does it matter what SSS, SAS, ASA, or AAS postulate you use? Can you choose which one you want based on your opinion, or is there a specific order in which you use them to make a proof?",
        "A": "You choose the postulate based on what information you have. In this case, you know that all three sides of one are equal to all three sides of the other. That s where the SSS comes from. You cannot use any of the other postulates, because you do not know anything about the angle measures. All of the other postulates require that you know at least one pair of congruent angles."
    },
    {
        "video_name": "p46QWyHQE6M",
        "Q": "I understand that the Jacobian determinant returns a factor by which a small area gets changed when undergoing a certain transformation. How then do we approximate an area over a certain region that is much larger than a small neighbourhood? 8:43",
        "A": "Numerically adding all such individual approximations over the region of interest."
    },
    {
        "video_name": "Oo2vGhVkvDo",
        "Q": "What are single bonds and covalent bonds? (2:33)",
        "A": "A single bond is where one pair of electrons is shared by two atoms. A covalent bond is where atoms share multiple pairs of electrons."
    },
    {
        "video_name": "Oo2vGhVkvDo",
        "Q": "At 2:38 sal said absolute value whats that",
        "A": "The absolute value is the distance away from zero that the number is. For example the absolute value of -24 is 24 or the absolute value 13 is 13"
    },
    {
        "video_name": "Oo2vGhVkvDo",
        "Q": "At 2:38 Sal mentioned the absolute value. What is the absolute value?",
        "A": "The absolute value is the distance a certain number is from 0. For example, the absolute value of 5 is 5, because it is 5 units away from zero. The | | symbols represent absolute value, so if you have something like |-9|, you find the absolute value of -9, which is nine."
    },
    {
        "video_name": "Oo2vGhVkvDo",
        "Q": "At 2:45, why did Sal add 134 and 128?",
        "A": "The 2 numbers are on the opposites sides of 0 on a number line. So to find the difference you need the distance between the two numbers. This is why you have to add the 2 numbers, and this is why subtracting a negative is the same as adding a positive. 134 - 128 = 6 The difference between the 2 numbers. But 134 - -128 = 262"
    },
    {
        "video_name": "Oo2vGhVkvDo",
        "Q": "At 2:37 what is absolute value?",
        "A": "It is the distance away from zero. If you were to have a -3. The distance away from zero is 3. -3------------------------0------------------------- 1------------2-----------3"
    },
    {
        "video_name": "Oo2vGhVkvDo",
        "Q": "At 2:38 Sal mentioned the absolute value. What is the absolute value?",
        "A": "The absolute value is the distance from zero. Like when you have -28. The absolute value of -28 is 28 because the distance from zero to -28 is 28 to the left."
    },
    {
        "video_name": "Oo2vGhVkvDo",
        "Q": "We don't always HAVE TO use a number line, like how Sal made one at 1:50?Right?",
        "A": "No, you do not always have to make a number line, it just helps you visualize how the numbers are being subtracted or added, especially with negative numbers."
    },
    {
        "video_name": "eDv7dk9uNmM",
        "Q": "0:36 has a bug... \\(\u00c2\u00b0_\u00ca\u0096\u00c2\u00b0)/",
        "A": "What do you mean there is a bug? I watched the video and found no errors."
    },
    {
        "video_name": "7bO0TmJ6Ba4",
        "Q": "At 8:40, why does line XZ equal line AC?",
        "A": "Line XZ is not congruent to line AC. The 2 lines mentioned have the same ratio as the other corresponding sides of the triangle, because the triangles are similar."
    },
    {
        "video_name": "7bO0TmJ6Ba4",
        "Q": "at 3:31, I've always leaned ratios as 'something:something else', rather than something/something else (in the video he was doing ab/xy,) but in a test, could you show your ratios like ab:xy instead?",
        "A": "That would not be incorrect. However, expressing ratios as fractions allows you to import your knowledge of fractional arithmetic to the context of ratios, as they are entirely analogous."
    },
    {
        "video_name": "7bO0TmJ6Ba4",
        "Q": "At 0:40 Sal put 90 degrees as a normal angle not a right angle",
        "A": "Any angle with measure 90 degrees is assumed to be a right angle this is by definition."
    },
    {
        "video_name": "7bO0TmJ6Ba4",
        "Q": "at 5:01 was does sal make that side of the triangle 30 square root 3?",
        "A": "It s a right angled triangle. He wanted Pythagoras theorem to hold good for the triangle"
    },
    {
        "video_name": "7bO0TmJ6Ba4",
        "Q": "How do you know that the last pair of angles are congruent? (@ 1:58)",
        "A": "By the Third Angles Theorem and CPCTC, or corresponding parts of congruent triangles are congruent."
    },
    {
        "video_name": "7bO0TmJ6Ba4",
        "Q": "At \"3:55\" do all the corresponding sides have to have the same ratio to be congruent?",
        "A": "thx"
    },
    {
        "video_name": "7bO0TmJ6Ba4",
        "Q": "Are you always supposed to show 90 degrees with a box since at 1:29 Sal didn't?",
        "A": "Usually, a 90 degree angle is represented with a square (box) shape, for better understanding. Even when it says something like 90 degrees next to a 90 degree angle, it is still usually represented with a box. Sometimes, 90 degrees ISN T represented with a box. That exception just happened to occur in this video; in most of Sal s videos that include something about 90 degrees, the 90 degrees is almost always marked with a box"
    },
    {
        "video_name": "7bO0TmJ6Ba4",
        "Q": "At 5:05, you started saying that side AB is thirty square roots of three. I haven't learned that much about square roots yet, but I'm assuming this means 90. Am I correct? I just want to clarify.",
        "A": "No 30 times 3 is 90. 30 square root of 3 means 30 times the square root of 3. The square root of 3 is an irrational number that can be multiplied to equal 3. Since an irrational number is a decimal that goes on forever, we can t write it so we write the square root of 3."
    },
    {
        "video_name": "7bO0TmJ6Ba4",
        "Q": "Around 0:12 Sal mentions postulates. What exactly are postulates?",
        "A": "A postulate is a presupposition used for analyses."
    },
    {
        "video_name": "7bO0TmJ6Ba4",
        "Q": "How did Sal calculate the hypotenuse of the triangle at 5:03 so quickly ? Is there any method involved ?",
        "A": "You can divide that 60 by square root of 3. It s fixed ratio of 90-60-30 triangle."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "he lost me at 9:50. where did he get (5/2) and (3/4) from??",
        "A": "log base 2 of 32 is 5 times 1/2 is 5/2 and log 2 of 8 is 3 times by 1/4 is 3/4"
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "At 8:29, why didn't he write 2 as the base?",
        "A": "He must have spaced it. But he should have put it in."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "3:32 log_2^512 =9? Where did that come from?",
        "A": "2^9 = 512, so therefore log2(512) = 9."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "At 4:39, what does C stand for?\n\nThose are awesome videos. Keep making more.",
        "A": "It is simply a variable for the possible number that can go there."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "I Dont Get 6:38\nHow Can Log 17 357 = log 10 357 / log 10 17?\nCan Someone Explain?",
        "A": "Look at the formula and watch the proof..."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "At 6:40, can C = any number? Or does it have to remain as 10?",
        "A": "Yes, you can use any C. 10 or e are usually the most helpful in working many math problems, but the formula is true for all real numbers."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "Around 6:44 he says to press log on the calculator and add 357 then get bla bla bla, and to do the same with 17 and get bla bla bla. Then he mumbles something about dividing something, please somebody help me clear up what the mumbling means? Somebody that understands what is going on?",
        "A": "He basically tells you to divide the 2 numbers to get the answer."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "hello, at 9:53, how did you get 10/4 from 5/2?",
        "A": "You can create an equivalent fraction by multiplying both the numerator and denominator by the same value. 5/2 (2/2) = (5*2)/(2*2) = 10/4"
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "Around 2:17, Sal writes log base 2 of 8 is 3. The three in front makes this multiply to 9. Why then is the entire number of nine cubed on the other side of the equation? In my mind it would be log base 2 of 8 (=3) cubed.",
        "A": "the right side was written as: log base 2 of 8^3 this is showing that it is the log base 2 of (8^3) To make it be like you said: log base 2 of 8 (=3) cubed the notation would be: (log base 2 of 8)^3 Without the parenthesis, the cubed only applies to the 8, not the whole thing."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "At around 09:24 where he has 1/2 outside the parentheses, i followed the order of operations where you do the parentheses before multiplying which gave me 7/2. anybody got any idea?",
        "A": "You made some arithmetic errors. Probably just forgot to multiply the 1/2 to the log 2 of 8 (which is 3) before subtracting. It ends up being 1/2(5- 1 1/2)= 1/2(3 1/2)= 1/2(7/2)= 7/4"
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "around 5:37 sal mentions some number called 'e'. What is this number and how is it used in logarithms?",
        "A": "A transcendental number, the base of Napierian or natural logarithms, approximately equal to 2.71828."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "I don't get it, how does he go from log2(32/sqrt(8)) to log2(32-log2(sqrt(8)) at 8:47-9:13??",
        "A": "At 6:00 in the first Introduction to logarithm properties, he mentiones a property whereby logb A - logb C = logb A/C. the 32/sqrt8 is the same as the A/C in that rule, so he just reverses it to 32 - sqrt8."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "Where does power 1/2 come from at 8:14?",
        "A": "i just keep learning. :)"
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "at \"5:18\" sal talks about ln \"natural log\" what exactly is it in terms of log?\n\nAlso what is the difference between ln and log?",
        "A": "ln is a log with base e. In mathematics, e is a special number, kind of like pi is special. e is 2.71828183..."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "Could someone explain what happened at 6:08-7:18?",
        "A": "Our calculators can only compute log 10 or log e, so when we re asked for the value of a logarithm that doesn t have either of those bases, we need to apply the rule to change bases . log_17 (357) becomes log_10 (357) / log_10 (17). Or, it can also become log_e (357) / log_e (17) --- which can also be written more simply as ln 357 / ln 17. If you use your calculator to evaluate both of these fractions, you ll see that they do indeed come out to the same answer."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "hmmm... this might be a wee obvious but i cannae say im sure, at \"9:38\" (after Sal wrote 1/2 log2 32 -[minus]) where does he gets the 1/4 from???? one step above he had 1/2 log2 8... anyone? takk.",
        "A": "He was distributing the 1/2 that was outside the parentheses to the two logs inside the parentheses, kind of like if you have 1/2(A - 1/2B); then if you distribute the 1/2 inside the parentheses, you take 1/2 times A and you get 1/2A, and then you take 1/2 times 1/2B, which equals 1/4B (1/2 X 1/2 = 1/4). Using the distribution property gets you 1/2A - 1/4B in the case of my problem. Hope that isn t too confusing."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "whats e in the video at 5:30?",
        "A": "e is a constant that equals: 2.71828... Like pi, it never ends. e stands for Euler s number ."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "On 8:17, how did Sal manage to get 1/2 out of the square root of 32?",
        "A": "The square root of 32 is equal to 32 to the 1/2 power. This is explained in the Level 3 exponents video."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "why would you have to raise log2(32/square root of 8) to the 1/2 power??\nat 8:17. then also why would you put it in front of the log in 8:30",
        "A": "For example: sqrt(2^6) is 2^3. that s half of 6. In general, sqrt(x^n) = (x^n)^(1/2). And since (2^6)^1/2 = 2^3, and 3 is half of 6, (2^6)^(1/2) = 2^(6*(1/2)). In log form: log_2((2^6)^(1/2)) = log_2(2^(6*(1/2)) = (1/2)*6 = (1/2)*log_2(2^6)."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "What he do to get the 1/4 at 9:40? The part where he made a mistake...\nCan someone explain to me?",
        "A": "He multiplied 1/2 by 1/2. And this equals 1/4."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "Where did 5/2 come from? At 10:00, right after Sal distributes the 1/2 on both sides of the equation, he says \"this is five halves\". ??? Why is it 5/2?",
        "A": "okay, got it. Man, I hope there s another module for this stuff. Geez."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "At about 7:48, what is the cube root symbol?",
        "A": "This is the cube root symbol: \u00e2\u0088\u009b"
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "7:21This whole part (and I'm asking for most of us) makes no sense. WHAT IS IT THAT IS BEING SAID HERE? (thank you Lisandro Jimenes Leon)",
        "A": "You re probably confused on the whole square roots part. Sal rewrites the square roots as a number to the 1/2 power. So for example, \u00e2\u0088\u009a(8) can be rewritten as 8^(1/2). Hope this helps :)"
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "at the problem in 8:18, why does log2(sqrt(32/sqrt8) become log2(32/sqrt8)^1/2? and afterwards, why does 1/2 become the coefficient?",
        "A": "fixed it. but why is there the 1/2?"
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "At 6:15, how do Mr. Sal decide that 17 is B? How would those two expressions be the same?",
        "A": "since log_B(A) became log_17(357) and A=357, then B=17. Sal assigned log_B(A) to equal log_17(357)"
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "At 9:40 Sal changes the 1/2 to 1/4 and I don't understand why. Can anyone explain?",
        "A": "one 1/2 is outside the bracket which gets multiplied to 1/2 which is already there with log 8 base 2.thus it gets changed to 1/4 when we open the bracket."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "I don't understand 8:10.\n\nWhy is that the same as the other one? And how come both of the square roots were not removed?",
        "A": "It isn t a square root over a square root; it has a square root inside of a square root, if that helps."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "at 6:24, are you always going to have to use the base of 10 and/or e?",
        "A": "Not necessarily, just that your calculator automatically uses base 10 or e depending on whether you use log or ln. It doesn t matter, though; you understood that right?"
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "At 2:41, why did you do 2^9?",
        "A": "2^9 was done because we re trying to prove that log(8^3) to the base 2 is equal to 9. This converts to 2^9=8^3."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "I don't quite understand this part though, but anyway where did you get the 1/2 from at 8:15???",
        "A": "Hey, Rizky. He got the 1/2 because that s one of the rules. If you have a square root, you can get rid of the radical sign and make it a power of 1/2. So, that s all he did was raise the quantity under the sign to 1/2 power and got rid of the square root sign. This works for all roots. If it was a cubed root, we could get rid of the sign and raise the quantity to 1/3 power. If it was a fourth root, we could get rid of the sign and raise the quantity to 1/4 power and so on."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "Hi guys, the problem is I don't get the intuition of logarithms. When do we use logarithms in real life??Also from 8:57 I am kind of lost.Help needed. Thanks:)",
        "A": "Videos 47, 48, and 49 of the algebra playlist will hopefully explain the intuition behind logarithms."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "what is \"e\" (5:29)? help!",
        "A": "he explains it when he explains the ln button"
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "At 8:17, how did the square root of [(32)/(the square root of 8)] turn into (32/square root of 8) to the 1/2 power? Why wouln't it be (32/square root of 8) to the second power? Thanks? -- Pam",
        "A": "The sqrt(x) is the same as (x)^(1/2). The 1/2 power is defined as the square root. Similarly, the 1/3 power is the cube root, etc. This is just the way that people have defined it. Trust us, this definition will be more helpful later on, for example in the exponent rule and in the product rule in derivative calculus. If you watch the calculus videos, Sal generally ignores the quotient rule of derivative calculus (turning it into the product rule) by using this identity."
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "At 8:25-8:38 why did he remove the exponent 1/2 and put it on the left so that it turns into 1/2_log(32/sqrt(8))? He mentioned a property but which one is it?",
        "A": "log(x^y) = y*log(x) ( y=(1/2) ). In exponent form: (if u = log_a(x)): (a^u)^y = a^(y*u)"
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "At 5:20 what is the difference between log_10 and ln_e?",
        "A": "The value of e is 2.71828182846. Now,there s a lot of difference between (log_10) and log_(2.71828182846), isn t it?"
    },
    {
        "video_name": "TMmxKZaCqe0",
        "Q": "From 8:26- 8:53 how does the exponent 1/2 go in front of the rest of the expression?",
        "A": "It is a basic property of logarithms that log (x\u00e1\u00b5\u0087) = (b)(log x)"
    },
    {
        "video_name": "W9B3VYdC5T8",
        "Q": "So at 2:17 angles a, b, and c are each 60 degrees which makes the total 180 degrees, right?",
        "A": "Yes it is correct."
    },
    {
        "video_name": "tvj42WdKlH4",
        "Q": "1:54 2 is not a factor of 5 so why is a^2 a factor of a^5?",
        "A": "a^2 is the same as a * a a^5 is the same as a * a * a * a * a 2, of course, isn t a factor of 5. But a^2 is indeed a factor of a^5. a^2 is just three a s less than a^5"
    },
    {
        "video_name": "tvj42WdKlH4",
        "Q": "0:00-5:11- how would you do something like: 5^6/5^2?",
        "A": "Since there are no variables in this expression you could just multiply the whole thing out from the beginning to get 15625/25, and then divide to get 625. However, using the properties of exponents you could subtract the power of 5^2 in the denominator from the 5^6 in the numerator to get 5^4 (basically treating 5 like a variable). Either way it comes out to 625. Does that answer your question?"
    },
    {
        "video_name": "tvj42WdKlH4",
        "Q": "Can anyone explain what Sal said at 2:00",
        "A": "What Sal is trying to say is that in the expression a*a*a*a*a/a*a.....the common terms in the numerator and the denominator are the a^2 (or a*a) terms. So, you can cancel them out and you re just left with a^3 (or a*a*a) in the numerator. This is easier to do using the rule which Sal says at 1:14 ....Hope it helped!!"
    },
    {
        "video_name": "Himr2l8Rd18",
        "Q": "At 5:14, Sal says that it's a right Riemann sum. How do you know based on the formula that it is a right Riemann sum?",
        "A": "Because the sum has i = 1 and not i = 0 you are starting on the right side of the first rectangle as the first side would be located at i = 0 and the sum Sal uses starts at i = 1."
    },
    {
        "video_name": "Kjli0Gunkds",
        "Q": "At 2:26, how does Sal know if the sequence is finite or infinite?",
        "A": "If it s a finite sequence the limits will be given; like i goes from 1 to 10 ."
    },
    {
        "video_name": "Kjli0Gunkds",
        "Q": "At 5:18. Should it not be 28/h(n-1). I dont undersand why sal says 28/h(1). Can someone answer please, Thank you.",
        "A": "He is correct, he is trying to find h(2), so n = 2. When he substitutes this value into the equation, he as 28/h(2-1) thus 28/h(1)."
    },
    {
        "video_name": "Kjli0Gunkds",
        "Q": "At 0:00, the video starts with Sal saying 'function'.\nWhy are sequences not a type of function?",
        "A": "They are functions and that s what Sal said."
    },
    {
        "video_name": "Kjli0Gunkds",
        "Q": "@9:01 what exactly is the \"whole point of this video\"? Are you saying that in order to calculate or solve a recursively defined function/sequence for, say, the 70th term in that sequence, you first have to solve for every prior term in that sequence and if so, why would anyone use a recursive definition?",
        "A": "The reason Sal has included this video is to demonstrate recursive sequences. The last example shows the importance of base cases. Indeed, finding the 70th term in a sequence this way would take awhile. This is often used by computer programmers writing code. Code written recursively is often shorter and a bit more elegant."
    },
    {
        "video_name": "Kjli0Gunkds",
        "Q": "I've been trying with no success to find an explicit function for the recursive function h(n) = 14 if n=1 and 28/h(n-1) if N > 1 + whole number. See 4:50 in the video. Could somebody please help me understand how to write this formula?",
        "A": "h(1) = 14 = 8 + 6 = 8 + 6 * (-1)^(1 + 1) h(2) = 2 = 8 - 6 = 8 + 6 * (-1)^(2 + 1) h(3) = 14 = 8 + 6 = 8 + 6 * (-1)^(3 + 1) h(4) = 2 = 8 - 6 = 8 + 6 * (-1)^(4 + 1) \u00e2\u008b\u00ae h(n) = 8 + 6 * (-1)^(n + 1)"
    },
    {
        "video_name": "XhCSHu6dHb8",
        "Q": "Just a bit curious if this is a special case. At 10:12 we have seen that V is a null space of [1 1 1]. What would be the situation if V is not a null space or the equation is something like x1+x2+x3=5.\n\nDoes it mean that cannot use the Proj v(perp) x for simplicity. Does it mean we would need to persist with the method being described in the previous video?",
        "A": "That example is actually not valid as the set of solutions to x1+x2+x3=5 does not form a vector space."
    },
    {
        "video_name": "XhCSHu6dHb8",
        "Q": "At 5:20 Sal says that the vector v is by definition the projection of x on to v. How is that so? What definition is he referring to?",
        "A": "To the definition that any vector can be represented as the sum of it s projection on a subspace plus it s projection on the orthogonal complement of that subspace ."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "At 2:54, what's with the triangles for the change in y over the change in x? I don't remember my math teacher teaching me that.",
        "A": "The triangles stand for the Greek letter delta, which is used to represent change in . It s a fairly common notation in math and science, at least in the US."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "3:01 Shouldn't it be 0-6 over 5-2? I used y2-y1 over x2-x1.\n\nAlso, 1:31 How do you know if the y goes to b or y itself. It confuses me because b is the y-int, and then there is the actual y.",
        "A": "For Your first question, m=y2-y1 over x2-x1 is the same thing as what he says in the video. Your second question, you just got to remember"
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "At 13:54\nI understood everything but, why is it that y = -2x + b is [2= - 2 ( - 1 ) + b?",
        "A": "Just before this part of the video, Sal has figured out the slope to be -2, so he knows the equation is y = -2x + b. Now he wants to figure out b. To do this he plugs in one of the known points. He chooses (-1,2) which means x = -1 and y = 2 at this point. Solving for b using the point (x = -1,y = 2) ================== y = -2x + b therefore 2 = -2(-1) + b therefore b = 0 and the final equation for the line is simply y = -2x."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "AT 2:23 I don't get how he got -x",
        "A": "i agree watch the video again"
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "At around 4:47, why does Sal put 5 and 0 into the equation?",
        "A": "If you don t understand this part, you should review: how to find the slope of a line with 2 points. He was writing the formula to find the slope of 2 points, and just input x s and y s of those 2 points. That s it."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "After 11:00 why didn't you tell us that the function that maps x to y is the slope? We wouldn't have had to go through the delta stuff. It would have REALLY demonstrated what a function is in this case.",
        "A": "The function is y = mx + b , not the slope . Show us how to find m when 2 points are given without finding delta y and delta x"
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "At video point 14:57: why does Sal attach the x to the -2 in his equation f (x) = -2x ? Would the function of x be just -2?",
        "A": "No, if the function were f(x) = 2, then it would always be 2 regardless of the any(!) given x value. (Notice that f(x) = 2 would be a horizontal line that intersects the y-axis at (0, 2), which certainly would not contain one of the points we were given, f(1.5) = -3 . I hope this helps."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "At 6:39, i understand that Sal started with the second set of coordinates (-3,0) first but, i thought it didn't matter. I later tried doing the equation with the first set of coordinates first (3,5) and it doesn't work because you end up dividing 5 by 0. This doesn't work. Can someone explain how to know which coordinate set to use first? I am a little confused. Thank you, Elena",
        "A": "You were initially right: the order doesn t matter when finding slope. (5 -0)/(3--3) = 5/6 (0-5)/(-3-3) = -5/-6 = 5/6 Both give the same answer."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "2:57 to 3:51. In order to find the slope, Sal does (6-0)/(2-5) when in the last video, he said to subtract the first point from the second point. Here, he is doing the opposite. Is this OK? My algebra teacher says that's incorrect and we have to subtract the first point from the second point.",
        "A": "It does not matter which one is the first and which one is the second as long as you stay consistent in labeling and applying. So with two points (2,6) and (5,0) you could do (0 - 6)/(5 - 2) and get -6/3 = -2 The other way is (6 - 0)/(2 - 5) and get 6/-3 which is also equal to -2."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "At 3:26 when he is explaining how change in y over change in x equals m, why does he do 6-0 and 2-5. I thought that it should be y1-y2 over x1-x2, so shouldn't it be 0-6 and 5-2?",
        "A": "Which point is used for (x1, y1) vs, (x2, y2) does not matter. If you do your way: (0-6) / (5-2) = -6/3 = -2 You get the exact same slope that Sal calculated. Hope this helps."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "At 7:20 why did he choose the second point, what if the first piont??",
        "A": "It doesn t matter which point you select. Sal probably picked the 2nd point because the zero usually makes some of the math easier to do. If you were to use the 1st point to find b , you would get the same value for b . Here it is: 5 = 5/6(3) + b 5 = 15/6 + b 5 = 5/2 + b 10/2 - 5/2 = b 5/2 = b"
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "@ 4:11 isn't the formula supposed to be y2-y1/x2-x1?",
        "A": "Yes you have the correct formula for slope. And, Sal did use this formula at 4:11 in the video. The concept that you are missing is that it does not matter which point you label as (x1, y1) vs (x2, y2). Sal used the point (2, 6) as (x2, y2) and the point (5, 0) as (x1, y1). He calculated the slope as -2. If you reversed the point, you would still get the same slope. Watch... (0-6) / (5-2) = -6/3 = -2. Same Slope! Hope this helps."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "On 9:09, how did you get 4 as the change in y? I dont see anything on the graph that would imply that 4 is the change in y.",
        "A": "Hi, slope is the change in y over the change in x, Sal showed that for each 1 of x need to go up 4 in y. Therefore with the slope being the change in y over the change in x this is 4/1 which equals 4."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "at 2:55 how does m= change in y over change in x",
        "A": "That s the definition of slope. Graphically, it would be rise over run . You may need to review previous videos, specifically the one in this section entitled Slope of a Line ."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "Why exactly at 7:05 do the negatives in -5/-6 cancel out to make 5/6?",
        "A": "negative and negative equals a positive. So the same goes for -5/-6. Say if it was -5/6 then it could not work, because the negative doesn t cancel out. Hope this helps."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "at 2:05, why do add 4/5 to each?",
        "A": "Because in an equation both sides have to be equal. For example: 5d=20, we know that d=4 because 5/5 leaves d alone and 20/5=4. Hope I helped"
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "@ 1:07} Do you have to use m to represent the slope or do you choose any variable?",
        "A": "Well technically, you have to use m to represent the slope or else it wouldn t be right."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "At 3:28 why does he put the first coordinate first? the equation is y>2- y>1\nx>2- x>1",
        "A": "You can pick either one to be honest. you can start with the first coordinates or the second coordinates it doesn t matter the order."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "At 8:37, if b in the equation y = mx+b is 5/2 or 5 halves, how would it be plotted on the coordinate plane in fraction form as the y intercept? Isn't the y intercept supposed to be a whole number, so that it can be plotted on the coordinate plane?",
        "A": "The y-intercept can be any number, it need not be a whole number (usually is not). You just plot 2.5 as best you can, half-way between 2 and 3."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "at 3:53 , why does Sal put a negative when 6 - 0 is not negative?",
        "A": "because he went the other way around just to prove you would get the same value for the slope."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "8:56, how do you know that the change in X is 1?",
        "A": "Look at the graph. He went from x=2 to x=3 (a change of 1) then went up to find the next point where the line crossed two grid lines."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "at 1:48 how did he find the y-intercept of the equation if it was not written in standard form ?",
        "A": "He knew that line goes through the point (4/5, 0), right? 4/5 is an x value and 0 is an y value of this point. Also he knew that slope (m) is equal to -1. Therefore by substituting the values to the formula of y = mx + b we get 0 = -1 * 4/5 + b which is an equation with one variable and just like Sal did - 0 = -4/5 + b therefore b = 4/5."
    },
    {
        "video_name": "5fkh01mClLU",
        "Q": "At 3:54, isnt it Y2-Y1 and X2-X1?",
        "A": "Yes, Sal writes delta (the triangle) y over delta x. In mathematics we can write the greek letter Delta to mean the change in . So delta y / delta x means the change in y over the change in x which is the same thing as y_2 - y_1 / x_2 - x_1."
    },
    {
        "video_name": "xvvI_QRYxBY",
        "Q": "@ 3:10 Sal writes 154 instead of 150...is it correct?",
        "A": "Yes, it looks like L(7) is actually 150, so that would make the average rate of change (150-126)/(7-4) which simplifies to 24/3 which is 8 People/Hour."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "at 10:44, why did the 3X go under the 7X and not the 2Xsquared?",
        "A": "Because only LIKE terms can be added or subtracted. LIKE terms have the same base(s) AND the same exponent(s). For example, 3x and 7x are like because they both have x to the first power. 2x^2*y^3 and -5x^2*y^3 would be like terms, but 2x^2y^3 and -5x^3y^3 would not be like terms."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "At 3:15, why is it x^2 + x and not just x^2?",
        "A": "He has to multiply all of x+1 by x, so it becomes x^2+x."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "At 8:15, don't we need to mention that x+4 is not equal to zero ?",
        "A": "Yes, that would be proper."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "At 11:31, isn't -4 take +2, -6.",
        "A": "it was -4 -(-2) which became -4 + 2 which is -2"
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "At 5:19, why is the remainder equal to 4/(x+1) ? I'm really confused as I thought that the remainder should be just 4. Thanks.",
        "A": "The polynomial is (x\u00c2\u00b2+3x+6)/(x+1) (x\u00c2\u00b2+3x+2+4)/(x+1) ((x+1)(x+2)+4)/(x+1) (x+1)(x+2)/(x+1) + 4/(x+1) x+2 +[4/(x+1)] Everything in the polynomial, including the remainder, is being divided by x+1."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "At 3:12, why does Sal multiply all of (x-1) by x? He says x fits x times into x^2 but how does he know (x-1) fits x times into x^2? Thanks in advance!",
        "A": "x+1 doesn t fit x times into x^2, x^2 divided by x is x because x times x equals x^2. What you are supposed to do after you get the answer to x^2 divided by x is multiply that answer, which is x, by (x+1), so it equals x^2 + 1x then you subtract that to the problem, which is x^2+3x, which equals 2x. Hopes this helps."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "Why should we only consider the largest power when dividing? For example at 3:03, he divides x^2 by x. Surely x^2/(x+1) is different from x^2/(x)",
        "A": "This solution will become crystal clear when you start dividing by higher polynomials. Consider long division using the following notation: 17568 = 1*10^4 + 7*10*^3 + 5*10^2 + 6*10^1 + 8 & 10^0 Right? Divide this by 202 which is 2*10^2 + 0*10^1 + 2 Take out the null coefficient and divide 17568 by 202 using powers of ten. You start by diving the largest power of ten into the largest power of ten and then multiplying everything by that number and subtracting. Try it, you ll like it."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "At 8:14, if you're using the factoring method you can state \"x does not equal -4\" but if you use polynomial division how do you know which values the function will be undefined for?",
        "A": "It is the values of x for which the denominator would equal 0. Ordinarily we only explicitly state the x\u00e2\u0089\u00a0 whatever if that value is no longer obvious, such as having been canceled out (remember that you cannot cancel out 0/0). However, with most teachers, there is no harm in stating all of the things that x cannot equal."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "At 1:31, how does 2 go into 2x x times?",
        "A": "He just means 2 times x equals 2x, just like 2 times 4 is 8, so just think of eight as 2x and 4 as x"
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "At 3:04, \"x times x plus 1 is what\"? Why does he ask this? What is this about?",
        "A": "After you divide into the first term you have to multiply that result by what you re dividing by and put it under what you re dividing into (think of how you d do regular long division). x(x+1)=x^2+x"
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "From about 5:30, to check your answers, he put the first part over (x+1) and multiplied it by that too. Why did this work out?",
        "A": "(x+1) / (x+1) = 1 Whenever you need a common denominator to add two terms, you can multiply by 1 to achieve it. For normal fractions: 1/2 + 1/3 = 1/2 * 3/3 + 1/3 * 2/2 = 3/6 + 2/6 = 5/6 It works the same way with polynomials: (x+2) + 4/(x+1) = ( (x+2) * (x+1)/(x+1) ) + 4/(x+1) That provides a common denominator of x+1, allowing you to add the fractions together."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "So to check your work, if it doesn't have a remainder, you can just foil the divisor and the quotient to make sure you get the same polynomial. Are there cases when you'd have to do something different or that wouldn't work? (Like with a remainder, or if the divisor isn't a binomial, if such circumstances are even likely to arise?) So for example, how would you check the problem given at 9:30?",
        "A": "It should always work. You can also foil and add the remainder if there is one."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "At 8:15, i don't get how Sal subtracted the polynomials. Then he figured out how it simplifies. ? :<",
        "A": "He didn t subtract the polynomials; he divided them. x^2+5x+4 ---------------- x+4 x^2+5x+4 factored is (x+4)(x+1). (x+4)(x+1) ---------------- x+4 The x+4 in the numerator and denominator cancel each other out. x+1 Thus, x^2+5x+4 -------------- x+4 simplified is x+1. I hope this helps!"
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "When you do algebraic long division, don't you do it usually when your numerator can't be factored? Just wanted to say that because at 8:00 Sal factors the numerators. Then there wasn't really a need for long division. . .",
        "A": "Yes, if you can factor it, then it s better to factor it, but Sal was showing that long division works in all cases."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "at 10:56 in this video why did you move the 3x all the way above the 7x was it because 3x the varible X had 3 in front of it as its leading coefficient so you moved 3 spaces and ended up on 7X?",
        "A": "Sal moved the 3x above the 7x because they are both terms that have a coefficient and x^1. In polynomial long division, you want to make sure that you group the terms that have the same power of x with each other (ex. 3x and 7x, 2x^2 and 3x^2, etc.) Hope that helps :D"
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "At 2:24, where did the X+1 come from?",
        "A": "I m pretty sure Sal is just starting a new example with a simple polynomial, it has nothing to do with the previous problem."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "Wouldn't we also have to specify the Domain? For example, when Sal simplified by factoring at around 8:00, he canceled the (x+4) terms, wouldn't we have to state that x can't be equal to -4? Now that I think about it that is probably only important when evaluating functions. I post it anyway, should someone else wonder the same.",
        "A": "Yes, you are right. You don t even have to think about it, just draw the starting function and x+1 and it will be obvious they are not the same."
    },
    {
        "video_name": "FXgV9ySNusc",
        "Q": "At 1:09 did Sal mean x+2 divided by 2x + 4 or just 2 divided by 2x + 4? Please help!",
        "A": "Look at 0:19, he says that he s dividing 2x+4 by 2."
    },
    {
        "video_name": "OiNh2DswFt4",
        "Q": "At 4:30, Sal says that the integral of e^(-st) is (-1/s)e(^-st), and his proof of that is that when you take the derivative of (-1/s)e(^-st), you get e^(-st). However, doesn't that only apply when s is a constant? If s is a variable, and as far as I can tell, it is, then when you take the derivative of (-1/s)e(^-st) you would have to use chain rule, which wouldn't give you e^(-st). Am I misunderstanding something?",
        "A": "Reading this made me think about what Sal has done. So how you should visualize this transformation is by looking at this equation f(s) = \u00e2\u0088\u00ab f(t) * e^(-st) dt. So on the RHS we are integrating wrt t and s is to be considered as a constant. Hope this helped."
    },
    {
        "video_name": "OiNh2DswFt4",
        "Q": "At 3:18, what's an improper integral?",
        "A": "An improper integral is when one or both of the integration limits go to \u00e2\u0088\u009e. So in reality what you are doing is taking the limit when a variable goes to \u00e2\u0088\u009e, and integrating to that variable: \u00e2\u0088\u009e b \u00e2\u0088\u00abf(x)dx = lim \u00e2\u0088\u00abf(x)dx a b\u00e2\u0086\u0092\u00e2\u0088\u009e a"
    },
    {
        "video_name": "OiNh2DswFt4",
        "Q": "At 4:29 of the video Sal begins integration. He starts with -1/s times e to the -st but it gets hairy for me because what happened to adding 1 to the exponent??",
        "A": "It involves integration by substitution, wherein: Let -st=u => du = -s.dt Thus int e^-st = int (-1/s) e^u du = -1/s e^u Substituting back int e^-st = -1/s.e^-st"
    },
    {
        "video_name": "OiNh2DswFt4",
        "Q": "At 6:52-6:59, what are the differences between a \"googol\" and a \"googolplex\"?",
        "A": "Actually, a googolplex is even bigger: Exponentiation is right-associative, so googolplex = 10^10^100 = 10^(10^100) = 10^googol. That s A LOT of zeros. To give you some sense of how big these numbers are, there are about 10^80 atoms in the observable universe. A googol is 10^20 times this many - about the same ratio as the mass of a semi trailer to the mass of the entire Earth. Then a googolplex is a one with this many zeros - pretty much incomprehensibly large."
    },
    {
        "video_name": "4eLJGG2Ad30",
        "Q": "Is it necesarry to order the numbers like Sal did at 0:47?",
        "A": "No, but ordering them by magnitude makes the data easier to interpret."
    },
    {
        "video_name": "4eLJGG2Ad30",
        "Q": "At 0:25 what does he mean by most used, I barely see them?",
        "A": "It tell s you how much time but when it s like that it is a short video"
    },
    {
        "video_name": "4eLJGG2Ad30",
        "Q": "Which plot is the Histogram? Is it the one he`s making at 1:20 ?",
        "A": "The histogram is the one with the bars of color. It is the one that he shows at 4:40 in the show. The one when he makes a graph."
    },
    {
        "video_name": "v1ucHH06AxM",
        "Q": "At 1:52 he said that he don't know what b(1) is but we know that it is -7?",
        "A": "I wish he could make his videos more interesting to watch. They re very helpful, but they re putting me to sleep at the same time."
    },
    {
        "video_name": "c7ByaI3T7Dc",
        "Q": "At 5:14, he says that mathematicians will cringe when he plays with differentials in this way. What is a more rigorous way to talk about differentials. In other words, how could he do this in a way so that mathematicians WON'T cringe?\n\nMany thanks to anyone who could explain this!",
        "A": "mathematicians do not cringe at this approach,it goes back to leibnitz,before newton e.g.and is the common aproach in non standard analysis.sal is on a fine and productive foundation,in fact.great work on his part as usual. xrnusy."
    },
    {
        "video_name": "c7ByaI3T7Dc",
        "Q": "A little before 6:36, where Sal represents the limits in terms of partial differentials, wouldn't there be a partial r w.r.t partial x time i (r/y for j and r/z for k) or is that redundant? Because in my understanding, r is a function of x,y and z which are then functions of s and t.\n\nThank you.",
        "A": "It is the same. You are right in saying that r is a function of x, y and z, and they are functions of s and t. Therefore, you can replace x, y and z with s and t and get r as a function of s and t. Hope this helps."
    },
    {
        "video_name": "2nZsIeaiJUo",
        "Q": "AT 1:20 how is 2/3 equivalent to 4/6",
        "A": "It s equivalent because 2/3 takes the same space as 4/6, but they aren t at the same place"
    },
    {
        "video_name": "eoJGBr-tJXo",
        "Q": "At 5:42, sal was trying to say \"how many\", but he didn't and I thought he will say \"how many\" in the beginning. I want to know, is F-(vector field) dot n-hat represent an acceleration in somehow? and how would be possible to know how many? Should I know the total, for example, gases bounded in that contour first, then, minus the escaped-time multiplied by \"how fast\", to get how many? What does how fast mean if I am talking about bees exiting from a hole in bee house? Is it the divergence?",
        "A": "No, it would not be how many. It is, as Sal said, how fast. If you want to know how many have left in a given period of time, you would need to take the integral of the divergence over the period of time."
    },
    {
        "video_name": "dXB3cUGnaxQ",
        "Q": "At 1:50 Sal says that \"we are essentially sampling from this standard normal distribution and then squaring whatever number you got\". This is where I get confused. If we are sampling from this normal distribution, does that sample size has a sample size of n associated, and what is the relationship between that sample size n and the chi-square degrees of freedom referred to in the remainder of the video?",
        "A": "If X is Normally distributed with mean=0 and sd=1 , then X^2 is Chi-square distributed with df=1. Furthermore, if several Chi-square variables are independent, then if we add them together, we just need to add up their degrees of freedom. So if X1, X2, and X3 are all Normally distributed with mean 0 and sd 1, and all three are independent, then X1^2 + X2^2 + X3^2 is Chi-square distributed with df=3."
    },
    {
        "video_name": "R0EQg9vgbQw",
        "Q": "At 0:48, why does Sal start with a given hypotenuse of 1 for triangle ADF? Wouldn't this mean this proof only works under these circumstances?",
        "A": "according to me, he took it for easier calculation. you can try it out for other values of the hypotenuse also."
    },
    {
        "video_name": "R0EQg9vgbQw",
        "Q": "at 6:43 why is that angle labeled 90 - y?",
        "A": "We don t know what y is, but we know that those two angles add up to 90. That means the other one has to be y less than 90. Think about it: y + (that angle) = 90 (that angle) = 90 - y"
    },
    {
        "video_name": "R0EQg9vgbQw",
        "Q": "At 1:55(ish) Khan said that the sine was equal to the opposite over hypotenuse and than right after that he said that the sine was equal to the length of line DF. So is the sine the length of the opposite side or the length of the opposite side divided by the length of the hypotenuse.",
        "A": "If you will rewatch carefully you will notice, that the given hypotenuse is equal to 1. So sin(x + y) = DF / 1 . DF/ 1 = DF. I hope it will help."
    },
    {
        "video_name": "AxHAQ5l45FE",
        "Q": "At 2:38 Sal wrote 12 after describing the volume of a box instead of 12 to the third.",
        "A": "In this case he s counting the number of crates that can fit in the boxcar. If he had wanted to get the volume of the crates, he would have said 12 cubic meters (meters to the third), not 12 to the third."
    },
    {
        "video_name": "AxHAQ5l45FE",
        "Q": "At 2:21, how did you get 12 crates?",
        "A": "He got 12 crates by solving 3*2*2. 3 crates along the depth, 2 for the height, and two for the width. 3*2=6, and 6*2= 12 crates."
    },
    {
        "video_name": "2DPivVFCdqA",
        "Q": "I just got curious with the one statement at 3:08. It says it does not have to be that least common multiple, so I'm guessing, any multiple will do. So example, we have 3/6 + 5/8. What happens if we use 48 instead of 24, the least common multiple? How does that make sense? Does it make sense?",
        "A": "you will have to reduce after you add. 3/6 + 5/8 = 24/48 +30/48 = 54/48 = 1 6/48 =1 7/8 opposed to 3/6 + 5//8 = 12/24 + 15/24 =27/24 =1 1/8 which would you rather do?"
    },
    {
        "video_name": "a-e8fzqv3CE",
        "Q": "@ 2:36 when Vi id talking about that \"dumb\" problem, don't you do the multiplication first because you need to finish the parenthesis?",
        "A": "I think what she means is that after you do what s in the parenthesis, then there is a notation issue because there is both multiplication and division."
    },
    {
        "video_name": "a-e8fzqv3CE",
        "Q": "At around 1:54 how do you know which order to put the number in and which order to add them?",
        "A": "Use... PEMDAS! Solve the stuff in parentheses (the P ) first, solve the exponents second (the E ), than do all the multiplication and division from left to right, (the M and D ), and finish up with addition ans subtraction, solving the problem(s) from left to right. Does this help?"
    },
    {
        "video_name": "GB9Rqsah2gs",
        "Q": "At 3:05 Sal says that when you scale the circle's radius by a factor, you cause the area of the circle to increase by the square power of the scaling factor. Does that same rule apply to other shapes, or just circles?",
        "A": "This rule applies to all shapes; however, since the only shape with a consistent radius is a circle, it is logical to show this rule with a circle."
    },
    {
        "video_name": "GB9Rqsah2gs",
        "Q": "At 2:00 on this video Sal states that (2x)^2 is equal to 4x^2. Isn\u00e2\u0080\u0099t (2x)^2 actually equal to 4x? You lose the squared exponent when you multiply 2x by 2x to get 4x.",
        "A": "Jinx, Sal is correct. (2x)^2 = (2x)*(2x) = 2*x*2*x = 4x^2"
    },
    {
        "video_name": "GB9Rqsah2gs",
        "Q": "At 1:22 Where did he get 4*pi*x? Shouldn't it have been pi*4x? Or is it the same thing?",
        "A": "This is indeed the same thing. Regardless of the steps you take in solving it, the commutative property of multiplication says that the outcome is always the same when switching the order of multiplying factors. So an example would be: 4*6*8*5= 5*8*4*6. Hope you found this to be helpful!"
    },
    {
        "video_name": "GB9Rqsah2gs",
        "Q": "I'm still perplexed by this operation at 2:08, as Sal goes from (2x)^2 to 4x^2, without any other explanation than \"2x squared is 2x times 2x, which is the same thing as 4x squared\" and that does not at all seem correct. 2x times 2x uses up the square function, and he just added it back in after performing it. Am I still wrong here?",
        "A": "(2x)^2 =(2x)(2x)=(2)(x)(2)(x)=(2)(2)(x)(x)=(4)(x^2)=4x^2. (2)(2)=4 and (x)(x)=x^2"
    },
    {
        "video_name": "74iuGIaBgRc",
        "Q": "At 5:50, when the answer is revealed as 8/3 times the square root of 2/3, can you simplify that even more (at 5:33 when it is mentioned that 8 root 2 divided by 3 root 3 is an answer, can u simplify that to 8 root 6 over 9)?",
        "A": "Yes you could. And your version would be considered more simplified than Sal s version. Truly simplified radicals generally do not have radicals in the denominator. Sal s videos focus on one skill at a time, so he often will leave a radical in the denominator because he has not yet covered how to rationalize the denominator."
    },
    {
        "video_name": "74iuGIaBgRc",
        "Q": "At 4:19, Sale concludes that the answer is 3 times the square root of 10. This answer confuses me. Could not the square root of 180 be factored as the square root of 36 x 5? Then, we would have 6 times the square root of 5 divided by the square root of 2.\n6*(5)^1/2 numerator and (2)1/2 in the denominator. Comments?",
        "A": "Your work is incomplete. Answers with simplified radicals would have no radicals in the denominator. Your s still has a radical in the denominator. The easiest way to deal with the issue is as Sal did and do the division up front. If you take your approach, you can still get to Sal s answer, but you need to rationalize the denominator. Hope this helps."
    },
    {
        "video_name": "74iuGIaBgRc",
        "Q": "At 3:55 Sal says that sqrt of 180 is the same thing as 180 to the 1/2 power. Can somebody explain it, please? thank alex",
        "A": "Find out more on exponents, especially fractional exponents. For example 5 to the 2 power would equal 25 as you know. From the view of the number 25, 25 to the 2 power it would be 625, 25 to the 1 power it would still be 25 and 25 to the 1/2 power it would be 5."
    },
    {
        "video_name": "4oeoIOan_h4",
        "Q": "At 1:58, I didn't understand how you get .75x Could you please tell me?\nThank you",
        "A": "He was subtracting .25x from 1x not multiplying."
    },
    {
        "video_name": "4oeoIOan_h4",
        "Q": "Answered Thanks to Kim Seidel For Answering This Question First\n\nAt about 4:26, Sal said that if he were to round up, he would get $66.67. However, .75*66.67 gives 50.0025, which is slightly higher than $50, the max he can pay. So isn't it incorrect to say that you can go to $66.67? Shouldn't it be $66.66?\n\nThank you for your time, in advance.",
        "A": "His answer is actually ok. The reason is that our monetary system only carries two decimal places. He can t physically pay: $50.0025. The last 2 decimal places have no meaning in money. Instead, we always round these type of calculations to the hundredths place (the 2nd zero). Thus, $50.0025 rounds on hundredths to $50,00. Hope this help."
    },
    {
        "video_name": "xO_1bYgoQvA",
        "Q": "how whould it be better to not memorize multiplacation like sal says at 0:40?",
        "A": "Honestly, I dunno if that was a mistake or not."
    },
    {
        "video_name": "xO_1bYgoQvA",
        "Q": "15:03 nine times 1 is not eighty one! its 9!",
        "A": "He probably just made a mistake. Everybody makes mistakes even me. :D"
    },
    {
        "video_name": "xO_1bYgoQvA",
        "Q": "i dont get anything from 0:01 to 15:27",
        "A": "Then watch it again or ask a parent."
    },
    {
        "video_name": "gkyuLPzfDV0",
        "Q": "Question about Syntax - at 3:30, should the e_1, e_2, and e_3 have a vector arrow above them?",
        "A": "Yes, if Sal wanted to be completely unambiguous. However, sometimes you can get away without explicitly notating things as vectors if the situations implies that they are vectors (although this isn t the best practice for general use)."
    },
    {
        "video_name": "IiBC4ngwH6E",
        "Q": "what kind of rap is this 0:25 ..\ni am having a very hard time in understanding chain rule please help me!",
        "A": "If you find it hard to understand, try PatrickJMT. I sometimes find him better than Sal."
    },
    {
        "video_name": "IiBC4ngwH6E",
        "Q": "At 4:17, he said that f prime of g of x is equal to 1/2 * (3x^2 -x)^-1/2. But isn't that with respect to x. According to the chain rule, aren't we supposed to find the derivative of the outer function with respect to the inner function? So it should be d[f(g(x))] / d g(x). I can't understand what's happening here",
        "A": "F prime of g of x refers to the derivative of f(x), and then substituting in g(x) for x whenever there is an x in the f (x). This is definition of the derivative of the outer function with respect to the inner function. For example, if f(x) = x^2, and g(x) = 5x, then f (g(x))= 2(5x). Of course, when using chain rule, always remember to multiply by g (x)."
    },
    {
        "video_name": "IiBC4ngwH6E",
        "Q": "At 1:39 when you define f(x), why doesn't the negative sign carry over into the f(x)=sqrt(g[x]) function?",
        "A": "He defined f(x) = sqrt(g(x)), what negative sign are you referring to?"
    },
    {
        "video_name": "IiBC4ngwH6E",
        "Q": "I'm having a hard time understanding where f(x) and g(x) come from. From 1:37-1:52, how did you find that f(x) is the sqrt. of x and g(x) is 3x^2-x?",
        "A": "He takes them directly from the expression he is planning to differentiate. The square root is the outer function, 3x^2 - x is the inner function. The x in the definition of f(x) is not the same as the x in the definition of g(x). They are independent functions that he combines into f(g(x)). I guess you could say that f(g(x)) = \u00e2\u0088\u009ag(x) = \u00e2\u0088\u009a(3x^2 - x) and also f(g(x)) = f(3x^2 - x) = \u00e2\u0088\u009a(3x^2 - x)"
    },
    {
        "video_name": "IiBC4ngwH6E",
        "Q": "I was trying to understand the difference(s) between d/dx and f'(x) through some vids but 1:05 totally made me out of space. Can I rewrite d/dx [f(g(x))] = f'(g(x)) * g'(x) as d/dx [f(g(x))] = d/d[g(x)] [f(g(x))] * d/dx [g(x)]? Why do you have to use d/dx and f prime in a same math expression?",
        "A": "What you wrote is also valid. I would guess that Sal didn t use that formulation here because it is a bit harder to follow ..."
    },
    {
        "video_name": "o31cLUkS23E",
        "Q": "0:07 Khan tells us to pause the video and try the sum. Why does he ask us to do so?",
        "A": "so we can better understand it"
    },
    {
        "video_name": "o31cLUkS23E",
        "Q": "At 0:17 what does intuitive mean? My vocabulary is tiny :(",
        "A": "Intuitive means you might figure out how to solve the problem without having learned about it. When the numbers line up and the decimals are above each other it is intuitive to figure out how to place one number on top of the other."
    },
    {
        "video_name": "78tnNYNISOc",
        "Q": "At 5:34, I noticed that each smaller number starting with 3.01 was the previous number minus ln(3.1). Why is this?",
        "A": "3.1-3=0.1 3.01-3=0.01 3.001-3=0.001 Since each no is of the form 10^-n where n is decreasing. ln(a^b)=b lna So every number was a multiple of ln (3.1-3)"
    },
    {
        "video_name": "78tnNYNISOc",
        "Q": "At 5:35, is it just a coincidence that all of the y-values are multiples of -2.3 in increasing order? Is there any math behind this?",
        "A": "The Math Reason beyond the y values increase is that for being an Vertical Asymptote of some number the value of y should increase in order for it to get closer and closer to the value of x like You can see in the Graph of the Log function present in Sal\u00c2\u00b4s Video. Hope this helps, if you have anymore doubt please do Answer this Question ."
    },
    {
        "video_name": "BIpsQIJUCC8",
        "Q": "6:04 i think it's easier just to multiply 0.16 by 500 to get the decimel",
        "A": "I also think it is easier to multiply 0.16 by 500 to get the decimal"
    },
    {
        "video_name": "BIpsQIJUCC8",
        "Q": "I don't understand what you mean by all of those probabilities add up to 1 At 4:33",
        "A": "48%, 36%, and 16% add up to 100%. in order to change a percent into a decimal you move the decimal over two places to the left. (currently it is right after the second zero in the number 100) so that would be 1.00"
    },
    {
        "video_name": "b7JTVLc_aMk",
        "Q": "at 3:07 as a determinant sal deduct j vector but at same situation at 10:08 for combine second cross product he didn't! what is thereason?",
        "A": "Ok thats so cool and that is the answer."
    },
    {
        "video_name": "b7JTVLc_aMk",
        "Q": "14:22 the (-j) which is being rubbed off. Where did that minus sign come from.",
        "A": "The minus sign means -j^*(a1b3 - a3b1) = j^*(a3b1 - a1b3) . It s origin is this: a1*i^ x b3*k^ = a1b3*(-j^) ; and a3*k^ x b1*i^ = a3b1*(j^); therefore the y component of a x b is (a3b1 - a1b3)*j^ ."
    },
    {
        "video_name": "b7JTVLc_aMk",
        "Q": "At 2:20, what are i, j, and k? Are they meant to be axes (like x, y, and z axes) or are they unit vectors? Also, if they are axes, are the subscripts under a, b, and c meant to solely distinguish between components or are they pertaining to the axes as well?",
        "A": "1) i, j, and k are (1, 0, 0), (0, 1, 0), and (0, 0, 1). 2) Both"
    },
    {
        "video_name": "b7JTVLc_aMk",
        "Q": "At 2:24, why is the format of the cross product of vector bxc completely different to that of the intro video on cross products (video 7)? What am I missing?",
        "A": "The way he explained it in the introductory video is just an introduction to this way of doing it. It s exactly the same process without the determinant-like structure he used here. Neither method explains anything, they re just rote methods to arrive at the correct answer."
    },
    {
        "video_name": "TWv6f7rwjF4",
        "Q": "doesn't 2^0 is the same thing as 1 times 0 = 0. Why did Sal say its one? help please 1:11",
        "A": "The reason 2 to the zeroeth power is one is because you are multiplying 1 times zero twos not one times zero. 1 times 0 = 0. 2^1 = 1 times 2 = 2. 2^0 = 1 times zero 2 s or just 1."
    },
    {
        "video_name": "hIgnece9ins",
        "Q": "@7:23 he finds the third inflexion point. How come it didn't come up with 0 and 1. What's with using the second derivtive to find the critical point?",
        "A": "At the third inflexion point, the gradient of the line (and so the derivative of the function at that point) isn t 0. The reason it s an inflexion point is that the line changes from being downward concave to upward concave. In other words, the gradient of the gradient of the line changes from positive to negative, which is why you need to look at the second derivative."
    },
    {
        "video_name": "hIgnece9ins",
        "Q": "6:40 (I guess). Sal never tests 12 as an inflection point. He justifies it being a minimal point by saying its positive and thus concave upwards, however by the same logic 2/3 could be assumed a minimal point. I take it he doesn't test 12 as 12.0...1 and 11.0...1 plugged into 12x(3x-2) is always positive, thus a test was technically done and always should be?",
        "A": "Are you thinking that 12 might be an inflection point because 12 was the coefficient on one of the factors? The only points that are critical points are those values, when substituted for x, cause the derivative to go to zero. 12 is a coefficient on one of the terms in the derivative, but setting x to 12 will NOT cause the second derivative to be zero."
    },
    {
        "video_name": "hIgnece9ins",
        "Q": "At 15:16, how do we already know that the function is concave upward?",
        "A": "when the second derivative is positive, then the graph is concave upwards. when you evaluate x=1 for the second derivative, you get a positive value, thus it is concave up"
    },
    {
        "video_name": "hIgnece9ins",
        "Q": "Ok, i have see the video, but when Sal is graphing 19:04 he draw that the point (2/3,2/3) is upward than (1,1) it is imposible. Thanks a lot Sal!",
        "A": "It s actually at (2/3, f (2/3). So if you plugged 2/3 into the original function it would give you the Y value. But, since it s kind of a weird number, he just said f(2/3) to save time. I believe the actual point would be (2/3,38/27). I hope that makes sense"
    },
    {
        "video_name": "hIgnece9ins",
        "Q": "How do you tell if you're suppose to look at greater than or less than values for \u00c6\u0092''(x)? 11:30",
        "A": "You will always look at both. What makes x=2/3 and inflection point is that the second derivative switches from negative to positive values. In order to prove this you need to look at values that are less than and greater than the value of x"
    },
    {
        "video_name": "hIgnece9ins",
        "Q": "So @12:55 im looking for a test point for f\"(x) =12x(3x-2) when x>0, If i use x= .1 its negative & concave down, but if i use x=1 does that mean its positive & concave up? How do you know which test points to use?",
        "A": "You re looking for a test point where x > 0 AND x < 2/3 (since x = 2/3 is another point of inflection)."
    },
    {
        "video_name": "hIgnece9ins",
        "Q": "At 11:05 Sal says when x<2/3 12x is still positive, but what if x is negative?",
        "A": "sorry didn t watch the video just read your question"
    },
    {
        "video_name": "hIgnece9ins",
        "Q": "If you listen somewhere between 4:15 to 4:45, why does the instructor say that the critical points can be max or min or inflection points by taking first derivative and setting it to zero? I understand max and min part but how can a critical point be an inflection point if he took the first derivative and solved it by setting it to zero?",
        "A": "Setting the first derivative equal to 0 tells you where the instantaneous slope is 0, which may or may not be a max or min. Sometimes a curve flattens out to 0 slope and then resumes the same sign it had before, thus not a max or min. Think of driving up a mountain road. You will being going up, but there may be spots that flatten to 0 slope from time to time. Same thing here."
    },
    {
        "video_name": "hIgnece9ins",
        "Q": "Instead of the lengthy discussion starting at 9:00, could the third derivative have been used at 2/3 to determine that the second derivative changes sign in the way discussed?",
        "A": "I know this is a year late but, for future viewers, yes you could have. If the third derivative at 2/3 is not zero, you would know that it s an inflection point. Furthermore, if f (2/3) was positive (which it was) you would know that the slope is concave downwards below that point and concave upwards above that point. (You would know that the slope of the second derivative at that point is positive)"
    },
    {
        "video_name": "3_otNr9kRuY",
        "Q": "At 0:15, Sal begins to list all the possible outcomes for flipping 3 coins. Is there any way to figure out the number of outcomes without listing them or using a chart?",
        "A": "2 outcomes for one event; 3 events. Therefore 2^3 = 8 outcomes."
    },
    {
        "video_name": "3_otNr9kRuY",
        "Q": "at 0:25 you said we could get tails, heads, heads and at 0:35 you said we could get heads, heads, tails. So I was wondering aren't those two statements the same? If they are that would mean you counted too many outcomes.",
        "A": "No they are not the same. Let s look at an example where I flip a coin only twice. The possible results are 3 different ones: two heads, one of each, or two tails. But the amount of possible resulkts are 4 different ones: HH, HT, TH, and TT. You can clearly see that the chance of getting exactly 1 heads is 50% because it happens 2 out of every 4 sessions . Good question, it is very confusing to think about what CAN happen, and what ALL POSSIBILITIES are. One is intuitive, the other is calculated."
    },
    {
        "video_name": "0NyeDUhKwBE",
        "Q": "@6:37 As i know, only common terms not written twice (last eq'n) why did you wrote (x^3*y)+(x^2*y^2)/2 instead of (x^3*y)+(x^2*y^2)/2+x^2*y ??\nAnd i still didnt get that why that the eq.n equals to other with h prime @4:34",
        "A": "the answer of your questions is the same that after he founds the function psi he tested the this function and he takes partial derivatives of psi respect to the y and looked wheter it is equal to the function N of the problem"
    },
    {
        "video_name": "03yq7XsErqo",
        "Q": "0:31 I thought when you multiplied a negative with a positive the product will be negative.",
        "A": "It s -2 squared. So (-2)(-2), which is a negative times a negative. That equals a positive 4."
    },
    {
        "video_name": "03yq7XsErqo",
        "Q": "i don't get how a negative times a negative would equal a positive at 0:34 if there are no two negatives there?",
        "A": "He was evaluating (-2)\u00c2\u00b2, which is the same as (-2)\u00e2\u0080\u00a2(-2), which is the two negatives multiplied together that he was referring to. (-2)\u00c2\u00b2 = (-2)\u00e2\u0080\u00a2(-2) = 4"
    },
    {
        "video_name": "4I9iibPLdBw",
        "Q": "in 0:01 what does \"expression\" mean?",
        "A": "Oh! Thank you!"
    },
    {
        "video_name": "uPksX_O9ARo",
        "Q": "So at 3:30, the answer is 1, as x approaches negative infinity. If it approached positive infinity, would that answer still be the same?",
        "A": "In this case, yes. The highest-degree terms were raised to an even power, so the sign didn t matter."
    },
    {
        "video_name": "uPksX_O9ARo",
        "Q": "Hello everyone. I've got a question. In this video we multiply function by 1/x^2 at 0:50, although x aproaches (-infinity) so should not we multiply this function by -1/x^2 instead. Because in a practice section i met a lot of similar tasks and sometimes i do not understand by what i should multiply, by \"-\" or by \"+\".",
        "A": "The function that Sal uses, namely 1/x^2, is independent of the value that x is approaching. If you did use -1/x^2 instead, you d have to switch the signs of all the numbers in the expression, which is just more work, and another way to potentially get the wrong answer. Multiplying by a positive or negative expression depends on the case. Sometimes it makes things better and sometimes it makes it worse. With enough practice, you ll be able to tell more easily when to use the positive or the negative."
    },
    {
        "video_name": "5LMzbgfZ8cA",
        "Q": "At 0:36 why doesn't Sal use -1 ^n+1 instead of -1 ^n-1?",
        "A": "Both will work. Whenever you want to alternate the sign of terms, you can use (-1)^n in the formula, but this produces -1,+1,-1,+1... If you wanna do it the other way, ie +1,-1,+1,-1... You re basically shifting the order, so you can do (-1)^(n+k) where k is any odd number."
    },
    {
        "video_name": "pIO34mrW_Ro",
        "Q": "At 3:59, Sal already +4 to the sum. But then at 4:19 he added 4 again. Therefore his final answer is incorrect. The correct answer should have been 2,321,475.",
        "A": "Look again at the formula for S-sub-n: it contains two terms of the sequence, the first and the last. There is a four in each of these terms, and that means two separate fours :-)"
    },
    {
        "video_name": "145zOBT0LD4",
        "Q": "At 1:05, Sal says you can pair these first two, last two, etc.\nMy question is can you pair them in any order as long as you use all 3 equations?",
        "A": "yes, and it is important to use all three equations no matter what order you pair them in because you need to use all the constraints. Sal probably did it in that order because it was the most convenient."
    },
    {
        "video_name": "145zOBT0LD4",
        "Q": "After he establishes that Planes 2 and 3 are parallel, at 4:47 that \"it doesn't matter if this first equation interesects one or both of these...\". However, for future reference, if a plane (or line) intersects another plane (or line), would the first plane (or line) always have to intersect all planes (or lines) parallel to the second plane (or line)?",
        "A": "You are correct, as long as your definition of intersection does not allow for concurrence. If a line lies in a plane, then that line would not intersect planes parallel to it."
    },
    {
        "video_name": "145zOBT0LD4",
        "Q": "at 4:09 what if your final answer was -21 = 0 ?",
        "A": "-21 = 0 is exactly the same as 0 = -21"
    },
    {
        "video_name": "rtNuo7R3scY",
        "Q": "At 3:32 he begins to distribute the 2 to the fractions, multiplying them...but why did he only multiply the numerators and not the denominator? I thought he would have multiplied both top and bottom? I'm really confused.",
        "A": "Write 2 as a fraction. It becomes 2/1. I think you are trying to make it into 2/2 (this = 1, not 2). So, when Sal multiplies 2 with 1/5(m), he is doing 2/1 * 1/5 (m) = 2/5 (m). And, 2 * (-2/5) = 2/1 * (-2/5) = -4/5 Hope this helps."
    },
    {
        "video_name": "rtNuo7R3scY",
        "Q": "0:33 why do we add those first?",
        "A": "You can only add/subtract like terms. The 1st term (the -5.55 ) is unlike the other 2 terms because the other 2 terms contain the variable c . The first term has no variable c . Thus, it can not be added with the last 2. The last 2 terms are like terms and can be added. That is why Sal started there."
    },
    {
        "video_name": "rtNuo7R3scY",
        "Q": "At 3:59 can you divide anything there?",
        "A": "There is no value to dividing. You would just change the fractions into decimals and that is not needed."
    },
    {
        "video_name": "rtNuo7R3scY",
        "Q": "at 1:25 how did they get 4.20 out of the equation",
        "A": "-8.55c + 4.35c factored into (-8.55+4.35)c. (-8.55+4.35) simplify to -4.20 or -4.2"
    },
    {
        "video_name": "rtNuo7R3scY",
        "Q": "At 2:20 Sal reorganises the problem so he can combine the like terms - however since this is subtraction, doesn't this change the final result? As in, 30 - 10 = 20 but 10 - 30 = -20. Is this just the rule when combining like terms?",
        "A": "As long as the sign goes with the correct number, you can reorganize the terms. For example: 30 - 10 The minus sign belong to the 10. You can change this into -10 + 30 Both will still equal 20."
    },
    {
        "video_name": "rtNuo7R3scY",
        "Q": "At 1:30 did you intend for 420 to be the answer to the problem",
        "A": "You lost the decimal point and the sign. Sal has: -4.20 and then rewrote is as: -4.2 Note: -4.20 = -4.2"
    },
    {
        "video_name": "rtNuo7R3scY",
        "Q": "0:52 why did he put a plus there",
        "A": "Sal introduced the parentheses around the 2 terms he is added. He needs the plus sign in front of the parentheses to show the results is being added to the -5.55. If he had no sign there, it turns into multiplication. There is not multiplication in the original problem. Hope this helps."
    },
    {
        "video_name": "rtNuo7R3scY",
        "Q": "I don't understand putting a \"+\" after 5.55 (in video at 0:51 sec)",
        "A": "1) Subtraction is the same as adding a negative. Thus, putting in the + doesn t change the expression. 2) If he added the parentheses and didn t put in a + , then the problem would have been changed. The subtraction would have turned into multiplication. Hope this helps."
    },
    {
        "video_name": "rtNuo7R3scY",
        "Q": "At 1:50, is there a specific order to -5.55 - 4.2c? Can it go the other way around?",
        "A": "Yes it can be -4.2c - 5.55, this is more of the traditional way of doing it, variable and coefficient first."
    },
    {
        "video_name": "rtNuo7R3scY",
        "Q": "When he distributes the two at 3:23 why dose he only distribute it to the 1/5m and 2/5 and not the 3/5?",
        "A": "2(1/5 m - 2/5) + 3/5 The 2 is up against a parentheses, right? That means multiply . Everything inside the parentheses is multiplied by 2. But the 3/5 is not inside the parentheses so it doesn t get multiplied."
    },
    {
        "video_name": "rtNuo7R3scY",
        "Q": "At 2:47, how did he get -1/5 from subtracting two positive numbers, 2/5 and 3/5?\nIt would be greatly appreciated. Thank you.",
        "A": "To subtract fractions, we subtract their numerators. What is 2 - 3? Look at a number line. Start at positive 2. Take away 3 means move 3 units to the left. You end up on -1. That is how 2/5 - 3/5 becomes -1/5. Maybe you need to review adding and subtracting signed numbers. Hope this helps."
    },
    {
        "video_name": "m4eiYHL3PP8",
        "Q": "Can we also complete the square instead of factoring or doing the quadratic formula at 3:07?",
        "A": "Yes, you could use factoring or complete the square and get the same results."
    },
    {
        "video_name": "TTYDbGXgcCk",
        "Q": "At 2:20, when Sal Answers the question regarding the comments per video, doesn't the answer contradict the statement in the question that \"He gets the same number of comments for each cat video he uploads and he gets the same number of comments for each Dog video he uploads\" ?",
        "A": "No. Each cat video has exactly 750 comments, and each dog video has exactly 450 comments. Only the number of videos is variable."
    },
    {
        "video_name": "mr9Tow8hpCg",
        "Q": "9:45 : \"best way to get rid of the T is to multiply by T^{-1} on both side of the equality\"\n\nNo ! Use the fact that T is invertible and therefore bijective : if T(x) = T(y) then x = y",
        "A": "Sal hasn t used the word bijective yet in this series of videos, so he can t rely on that property to solve his problem. Therefore, based on the bed of knowledge that we ve built so far taking the inverse transformation of both sides might actually, be the best way to get rid of T. There may be simpler ways that Sal hasn t yet covered, such as you mention, but since they haven t yet been covered they aren t in our toolkit yet."
    },
    {
        "video_name": "mr9Tow8hpCg",
        "Q": "At about 3:30 Sal says that we know T is a linear transformation (and it has to be linear to represent it as a matrix), and the whole video is based on finding out whether T inverse is also linear.\n\nMy question is, are there any non-linear transformations that are invertible? I have the sense that the transformation from x to y such that y=x^3 should be invertible, but does anyone know for sure? Or does invertibility imply linearity?",
        "A": "No, a function that is invertible does not have to be linear. f(x)=x^3 is invertible and f^-1(x)=x^1/3. This makes sense because x^3 is definitely one-to-one and onto, and (x^3)^1/3=x and (x^1/3)^3=x"
    },
    {
        "video_name": "mr9Tow8hpCg",
        "Q": "At 10:27,\" I just change the associativity of this\" . I think we cannot use associativity before we proof that T-inverse is linear transformation . Look forward to answers. Many thanks",
        "A": "That really wasn t associativity per se, more like the definition of the composition operation. But it is true that any mappings follow the associative rule whether they have any special properties or not. It s really beyond the scope of linear algebra, but if you know any abstract algebra, it follows from the fact that the set of all mappings from a set to itself is a monoid under the operation of commutation. Hope this helps!"
    },
    {
        "video_name": "mr9Tow8hpCg",
        "Q": "at 10:43 does associativity only apply to linear transformations?",
        "A": "Yes, because only linear transformations can be represented as matrices."
    },
    {
        "video_name": "mr9Tow8hpCg",
        "Q": "Regarding syntax, at 8:33, T^-1(b), b, should have a vector arrow above it, and at 18:54, Ax, should also have a vector arrow above the x?",
        "A": "Yes. Yes. They are both vectors."
    },
    {
        "video_name": "HWFtcIRhr5c",
        "Q": "At 02:11 how does Sal know that 16^(1/4) is two. I thought it's 4 in the very first moment ( i know 4 is wrong^^). Should one really be able to do such things with the brain or should one use a trick for calculating it..",
        "A": "It s quite simple to be honest. Since its one over some 2^n in which n \u00e2\u0088\u0088 natural numbers. This means that we re dealing with square roots. If we rearrange it to 16^(1/(2^2)) (which looks complex but if you write it down its not) then we can see that we re just taking the square root two times. So square root of 16 is 4 then square root of 4 is 2"
    },
    {
        "video_name": "KcqO1fX9b_I",
        "Q": "Hey guys,\n\nAt 3:57 Sal explains that the third problem's limit, as x\u00e2\u0086\u0092\u00e2\u0088\u009e, is equal to \u00e2\u0088\u009e.\n\nHowever, while I was doing the Practice: Limits at infinity of rational functions, under one of the hints, it said, \"If the degree of the numerator is greater than the degree of the denominator, then the limit doesn't exist.\"\n\nCan someone explain to me why over here the limit is \u00e2\u0088\u009e and over there it doesn't exist?\n\nThanks,",
        "A": "Some texts define infinite limits as simply DNE. It just depends on the convention."
    },
    {
        "video_name": "KcqO1fX9b_I",
        "Q": "at 2:23 in the video sal says that 1/2x is equal to 0 how does he know that or how will i know that everytime i get a 1/2x will that be zero?",
        "A": "If you put 10 in for x, you have 1/20. If you put 100 in for x you have 1/200. and so on. The higher number you put in for x, the smaller the number becomes. Once you have 1 divided by an infinitely large number, then it s 0. Same for every equation that is 1/(any number)x."
    },
    {
        "video_name": "KcqO1fX9b_I",
        "Q": "Hi, simple question ; @2:03 you state that 3x^2/6x^4 = 1/2x. Now, I get the 1 and 2. But it is the x, that really bugs me - doesn't the answer become 1/2x^2?? or have I missed something??\nThank you so much!! :)",
        "A": "You misread the numerator. It was 3x\u00c2\u00b3."
    },
    {
        "video_name": "KcqO1fX9b_I",
        "Q": "At 2:00 how can he simplify that to 1/2x when there are two different powers?",
        "A": "He cancelled out common factors 3x^3/6x^4 can be factored to (3*x*x*x)/(2*3*x*x*x*x) You can cancel out the three and three of the x terms ( 3*x*x*x )/(2* 3*x*x*x *x) This leaves 1/(2*x)"
    },
    {
        "video_name": "Xe6YlrCgkIo",
        "Q": "At 3:53 I don't understand why Sal uses \"h\" as height when he has real value for that height which is 2.",
        "A": "We also need to know the height of the water at times near now so that we can answer the question of how quickly the height of the water is rising. Therefore, we do need to define h as a function of t so that we can calculate its derivative."
    },
    {
        "video_name": "Xe6YlrCgkIo",
        "Q": "At 3:41, I think it should be \"these are perpendicular lines\" instead of \"these are lines\". Is that correct?",
        "A": "Yes, because he is describing a cone. It is assumed that the lines are perpendicular, because if they weren t, we would be dealing with conic sections, something completely different."
    },
    {
        "video_name": "Xe6YlrCgkIo",
        "Q": "Can you explain why at 5:45 you isolated pi/12 and not take the derivative as a whole.",
        "A": "This is simply a matter of convenience. The derivative of a constant times a function is the same as that constant times the derivative of the function. For example, the derivative of 4x is the same as 4 times the derivative of x. Sal didn t change anything by setting the constant pi/12 aside but made it more convenient to write."
    },
    {
        "video_name": "Xe6YlrCgkIo",
        "Q": "At 10:59, does dh/dt going to be the same thing until the cup is full?",
        "A": "Nope. Because the cross-sectional area of the surface changes depending on the height, the rate at which the height changes will not be constant. In other words, the water level will rise quickly at first, but once we reach near the top of the container where it is much bigger, the water level will rise more slowly."
    },
    {
        "video_name": "Xe6YlrCgkIo",
        "Q": "Just before 4:45, where does he get the h/2 part for the area of water surface and why?",
        "A": "Since h is the diameter h/2 is the radius (as said at about 4:04)"
    },
    {
        "video_name": "Xe6YlrCgkIo",
        "Q": "@ 3:45 how would you find the surface area of the water if the ratio between the height and diameter of the cone was not 1:1?",
        "A": "My best guess is to put it like so r = h*(water height)/Diameter of cone max) r= 2h/4"
    },
    {
        "video_name": "Xe6YlrCgkIo",
        "Q": "At 8:26, Sal wrote pi/2 instead of pi/12.",
        "A": "He fixed it later on (:"
    },
    {
        "video_name": "Xe6YlrCgkIo",
        "Q": "At 3:58 he determined that the radius is h/2, which I understand, but how do you figure out what the radius is if the height and the diameter aren't equal, like they are in this problem? For example, in a problem I'm trying to do on my homework, the height of the cone is 12 and the diameter of the top of the cone is 10. How would you then figure out the radius of the surface of the water?",
        "A": "You re given the diameter at the top, and from that you can get the radius at the top and a ratio between that radius and the height of the cone. Use this information to find the ratio between water height and radius at that water height. A side view diagram will show that ratio remains unchanged because of similar triangles."
    },
    {
        "video_name": "Xe6YlrCgkIo",
        "Q": "Starting at 6:17, why does Sal write h(t)^3 over and over again just to take the derivative of it later on? Why couldn't he do that first and save space and time?",
        "A": "Yes he could and in a timed test situation that is what you and I would do. This might be a time when too much explanation ends up confusing some people."
    },
    {
        "video_name": "Xe6YlrCgkIo",
        "Q": "At 06:18: Why can we write that h is dependent on t?",
        "A": "The height changes as a function of time so naturally the height can be written as a function of time, h(t). We just don t know what that function is."
    },
    {
        "video_name": "Xe6YlrCgkIo",
        "Q": "At 08:47: Why can you rewrite dV/dt as pi/2*3h^2*dh/dt?",
        "A": "Did you accedently make you pi/12 into pi/2 by mistake at around 10 mins into the video, because you said pi/12 but wrote pi/2 and then use the pi/2"
    },
    {
        "video_name": "egjDLFX9VHg",
        "Q": "Why does he say 7 minus 2, rather than saying 2 minus 7 in the video at 5:09?",
        "A": "If you subtract the bigger number from the smaller one, you will get a negative number, which you will study later. You can t literally reverse your number and do the calculations like Addition, in a subtraction =)"
    },
    {
        "video_name": "OkFdDqW9xxM",
        "Q": "At 0:45, Is (log_x(b))/(log_x(a)), the same as log_x(b/a)",
        "A": "No. They are not the same at all. log (b/a) = log (b) - log (a) However, log(a) / log (b) cannot be combined by any simple rule, unless you just so happen to have a situation that the logs can be simplified and cancel each other out OR the logs happen to separately simplify to some convenient number. But, in general, you cannot combine a log divided by a log."
    },
    {
        "video_name": "OkFdDqW9xxM",
        "Q": "Hmm. At 7:00 he divides both sides by log_x(a). But what happens if log_x(a) is zero? Don't you have to qualify that your log function is only defined when that is non-zero? And if so, what restriction does that put on your log function?",
        "A": "log a = 0 only for a = 1, regardless of the base. And because the original logarithm had base a, a can t be 1, so we re not dividing by zero."
    },
    {
        "video_name": "OkFdDqW9xxM",
        "Q": "At 2:12 why does he use base 10? Please help!",
        "A": "Base 10 is a common logarithm base to use for the change of base formula because it is found on most calculators."
    },
    {
        "video_name": "OkFdDqW9xxM",
        "Q": "How did you get the log base 10? Will it always be base 10 when you use that formula in 2:07 ?",
        "A": "Assuming you mean in the change of base rule, you can replace base 10 for any other base you want. It could be base 7 or base 91 or base e. Usually, it s used for base 10 or base e, because they re easy to calculate using a calculator."
    },
    {
        "video_name": "OkFdDqW9xxM",
        "Q": "At 2:05: Why is x = 10?",
        "A": "He is trying to change the base 5 to base 10 so that it can be solved with the calculators log function(which is base 10)."
    },
    {
        "video_name": "OkFdDqW9xxM",
        "Q": "At 6:31, in the course of showing why the change-of-base formula makes sense, Sal divides by the log of a, base x. How do we know this does not cause division by 0?",
        "A": "If log_x(a) = 0, then x^0 = a. This is only possible if a = 1. So yes, it might cause a problem if a = 1, but if that is the case, then you can probably handle the problem without this formula."
    },
    {
        "video_name": "OkFdDqW9xxM",
        "Q": "at 2:03, why did he make x 10?",
        "A": "When you use the change of base, the base x can be anything but generally we want it to be 10 so we can use it on the calculator. As he explained it, our calculator is set up with a base of 10 and we can t enter any base we want when using log. Otherwise we wouldn t need to use this change of base rule in the first place."
    },
    {
        "video_name": "OkFdDqW9xxM",
        "Q": "Why did you make the base 10 at 2:03?",
        "A": "Because it was on his calculator. He could have just as easily used e as his base, it still would have been the same answer, but then he couldn t have done the numerator in his head."
    },
    {
        "video_name": "OkFdDqW9xxM",
        "Q": "At 1:36 doesn't Sal mean ln x =lne of x or is it log?",
        "A": "at 1:36 it is log."
    },
    {
        "video_name": "OkFdDqW9xxM",
        "Q": "At 2:11 why did you choose log base 10? Is that arbitrary? Is there a reason?",
        "A": "From about 1:17 to 1:44 Sal mentions that either base 10 or base e are usually chosen when applying the change of base rule. This is because our calculators are all set up to be able to compute log 10 numbers and natural logarithms (base e numbers)."
    },
    {
        "video_name": "OkFdDqW9xxM",
        "Q": "At 2:04, why do you use the base 10?",
        "A": "Logs are not easy to calculate. Simple scientific calculators have base e and base 10 logs programmed in."
    },
    {
        "video_name": "AtiOjlyOQf4",
        "Q": "At 0:36 why do you call them buckets? Why not column",
        "A": "Buckets are the little vertical lines/dashes he uses on the line plot; In the video, he gave a bucket for each age; one for 3, one for 4, one for 5, one for 6 and one for 7. The reason he called them buckets is because that s the name mathematicians gave for them and columns are usually used for tables. I hope this helped you understand why they are called buckets. :-)"
    },
    {
        "video_name": "AtiOjlyOQf4",
        "Q": "At 1:46, do the dots/x'es have to be color coded to represent each person?",
        "A": "No, that s just a technique used to make it easier to see which person goes where."
    },
    {
        "video_name": "RdehfQJ8i_0",
        "Q": "at 5:54, he says \"I want you to take this with a grain of salt.\" What does that mean again?",
        "A": "It means to not take it very seriously. Don t take it as a hard core fact."
    },
    {
        "video_name": "RdehfQJ8i_0",
        "Q": "At 17:38, he asks what is 9+10, what is the answer?",
        "A": "9+10 is the same as 10+9 which is equal to 19"
    },
    {
        "video_name": "RdehfQJ8i_0",
        "Q": "At around 5:52, Sal says, \"I really want you to take this with a grain of salt.\" What does that mean?",
        "A": "Taking something with a grain of salt is an idiom meaning that you need to understand the context and that it may not be absolutely true or correct. There may be other possibilities that could be too numerous to mention."
    },
    {
        "video_name": "RdehfQJ8i_0",
        "Q": "At 3:35, you he starts the math problem with a two. How come it doesn't start with a one?",
        "A": "If you look at the graph, the 0-9 has two samples which is where he got the 2"
    },
    {
        "video_name": "bcCLKACsYJ0",
        "Q": "5:19 - 5:25 how would you simplify? I mean how would you get 1 5/12? I don't know if you get my question but how would you get the 5/12?",
        "A": "You simplify a fraction by dividing the numerator and denominator with the same number. For example I use 8/12. 8 and 12 can be divided by 4. To simplify a fraction you also have to choose the biggest divisor you can get. 8/4=2. 12/4=3. You end with a simplified fraction 2/3."
    },
    {
        "video_name": "bcCLKACsYJ0",
        "Q": "At 3:01 how and why did you convert 32/30 into 16/15?",
        "A": "32 is 2*16 and 30 is 2*15, so we have (2*16)/(2*15). We can rewrite this as (2/2)*(16/15). 2/2 is 1 and 1*(16/15) is just 16/15. That is why people often refer to this as canceling out. Hopefully this helps."
    },
    {
        "video_name": "bcCLKACsYJ0",
        "Q": "At 4:22 He said the least common multiple of 2 and 12 is 12 and it is 3.",
        "A": "The least common multiple is the smallest number where all the other numbers are factors of it. It would be 12 since 2 and 12 are factors, and it is the smallest to satisfy that condition. 2 and 12 are not factors of 3."
    },
    {
        "video_name": "Gn2pdkvdbGQ",
        "Q": "at 2:40 how can u put line on top of the 3n for your answer thats what confused me",
        "A": "under because its a repeating decimal. A repeating decimal is when, no matter how many times you divide, you will end up with the same answer; in this case, 8."
    },
    {
        "video_name": "Gn2pdkvdbGQ",
        "Q": "At 6:25 isn't the answer 0.1827956?",
        "A": "While your answer is more accurate than the rounded answer in the video, you could keep dividing and find even more digits. for 17 divided by 93, the digits start repeating after the fifteenth digit, but they go on forever."
    },
    {
        "video_name": "Gn2pdkvdbGQ",
        "Q": "At 5:08 to 5:11 why did they place the number 93 between the 7 and 0 , shouldn't they have put it under the 17 ? Help please .",
        "A": "Thank you so much :)"
    },
    {
        "video_name": "Gn2pdkvdbGQ",
        "Q": "At 3:06 Sal talks about trailing zeros, what does trailing mean?",
        "A": "A simple explanation of trailing is: something which is being dragged or something which is being followed by . In mathematics, trailing zeros are a sequence of zeros in the decimal representation after which no other digits follows. These zeros can make calculation more visual but not changing the value of the number. That is: 2.00 is the same as 2 or 2.000 or 2.000000000000."
    },
    {
        "video_name": "Gn2pdkvdbGQ",
        "Q": "What does the line over the \"1.88\" at 3:53 mean?",
        "A": "it means the eights after the decimal point specifically go on forever"
    },
    {
        "video_name": "Gn2pdkvdbGQ",
        "Q": "What is the => that got wrote at 7:58 in this video?",
        "A": "It s just an arrow. He could have used an equal sign. He s just saying that .030 and .03 are two ways of writing the same number."
    },
    {
        "video_name": "Gn2pdkvdbGQ",
        "Q": "at 2:14 dont you only have to put the line over ONE of the threes? not both?",
        "A": "Either way is fine. It is important, though that you only put the line over the number pattern. For example: 0.143 with a bar over the 3 means 0.1433333... 0.143 with a bar over the 143 means 0.143143143...."
    },
    {
        "video_name": "Gn2pdkvdbGQ",
        "Q": "how come by 8:41 he simplifies 35 over 1000 divided by five why do u divide by five",
        "A": "because five divides evenly into both the numerator (top number) and the denominator (bottom number). for example, if i wanted to simplify the fraction 2/4, i know that the number two will divide evenly into both the top number and the bottom number. when i do that i will get 1/2. i know that half is equal to two quarters. in the same way 35/100 is equal to 7/200."
    },
    {
        "video_name": "sw2p2tUIFpc",
        "Q": "At 7:08 when you say srqt(cos^2(theta))=cos(theta) shouldn't it actually be the absolute value of cos(theta)?",
        "A": "Well when we mean square root we actually mean positive principal square root , when you put sqrt(25) in your calculator it gives you +5 and not -5. But that s not what I m talking about. The part where you have to be careful is whether the expression is (sqrt(x))^2 or sqrt(x^2). The former is just x because if it were negative it d be undefined if we re not talking about complex numbers. But the latter has to be IxI because x could be either positive or negative. That s why you need the absolute value."
    },
    {
        "video_name": "sw2p2tUIFpc",
        "Q": "How come at 14:58 Sal could just multiply that portion of the eqn by 2/2? Wouldn't that alter the values of the eqn and make it different from the original eqn?",
        "A": "Recall that x/x=1. So we are just multiplying by 1, which changes nothing."
    },
    {
        "video_name": "sw2p2tUIFpc",
        "Q": "At 04:31 Sal said (x-3)/2 is equal to sin theta squared but a couple of seconds later he said that (x-3)/2 is equal to sin theta",
        "A": "The exponent 2 is hidden by the little flourish at the end of the square root symbol :D"
    },
    {
        "video_name": "sw2p2tUIFpc",
        "Q": "At 9:45 isn't it suppose to be 2\u00ce\u00b8 + 2Sin 2\u00ce\u00b8 + C. Because it was 2 in front of Cos 2\u00ce\u00b8? Thank you for your time.",
        "A": "Remember the chain rule. When you derive sin(2x), you get (2x) * cos(2x) = 2 cos(2x)."
    },
    {
        "video_name": "sw2p2tUIFpc",
        "Q": "Helo, I am a bit confused at 15:00 when sal multiplies (x-3) by 2/2 and gets (x-3)/2 times sqrt(4) times sqrt(1- (x-3)^2/4 can some one describe this algebraically ? Thanks!!!",
        "A": "2/2 = sqrt4/2, right? This lets him move the sqrt4 under the radical as 4, and then it helps get rid of the fraction in there."
    },
    {
        "video_name": "sw2p2tUIFpc",
        "Q": "At 11:48, should it be written sin\u00ce\u00b8cos\u00ce\u00b8 + sin\u00ce\u00b8cos\u00ce\u00b8 (as opposed to sin\u00ce\u00b8cos\u00ce\u00b8 + sin\u00ce\u00b8 + cos\u00ce\u00b8)? I'm pretty sure it is..",
        "A": "yes youre right! I seen it and had to replay just to make sure but he said it write just wrote it wrong .. x"
    },
    {
        "video_name": "sw2p2tUIFpc",
        "Q": "At 8:30 Sal says he has proven this. Can anyone reference the video for me?",
        "A": "From the double angle formula and the Pythagorean identity, you have cos 2\u00ce\u00b8 = cos\u00c2\u00b2\u00ce\u00b8 - sin\u00c2\u00b2\u00ce\u00b8 = cos\u00c2\u00b2\u00ce\u00b8 - (1 - cos\u00c2\u00b2\u00ce\u00b8) = 2cos\u00c2\u00b2\u00ce\u00b8 - 1 That is a pretty common expression for cos 2\u00ce\u00b8, and Sal just rearranged the terms to get his identity."
    },
    {
        "video_name": "sw2p2tUIFpc",
        "Q": "at 9:16, it is asked what the derivative of cos(2theta) is, and a substitution is made. But I think a it was supposed to be 2*cos(2theta), and the 2* got dropped. i think it carries all the way to the end to simplify the end equation a bit, but im not sure if i am correct or not.",
        "A": "okay, i got you now. integration just feels wierd, and so i guess i got confused as to which way we were going. This problem highlights all of the integration wierdness, as subtitution looks like magic and is definately not intuitive. im glad its shown, but im not sure i like it. :p"
    },
    {
        "video_name": "sw2p2tUIFpc",
        "Q": "i get really confused at 9:30ish can someone explain to me how u get the anti-derivative of (2+2cos2theta)?",
        "A": "okay thanks fuzzy once again!!"
    },
    {
        "video_name": "S6BHQMk8C_A",
        "Q": "At 1:33, why is the Pythagorean Theorem used?",
        "A": "Try visualizing two horizontal and vertical lines at the x and y values of each of the two points. Now create a line between the points. This creates a right triangle, and you re trying the find the length of the hypotenuse to find the distance between the points."
    },
    {
        "video_name": "x6xtezhuCZ4",
        "Q": "At 3:04 Sal says that when you take 4/5 of 10 you get 8. How do you know you get 8?",
        "A": "Because 4/5 of ten can also be written as 8/10, which in turn, Gives us 8"
    },
    {
        "video_name": "x6xtezhuCZ4",
        "Q": "@3:06, Sal seems to make a jump from 4/5ths of 10 is 8 -- what was his thought process to jump to the answer 8?",
        "A": "4/5th of 10 is simply - 4/5 times 10. 5 times 2 is 10, so you can cancel 5 and 10. which gives you 4 times 2 = 8."
    },
    {
        "video_name": "x6xtezhuCZ4",
        "Q": "Can someone explain to me at 4:33, how to do it in the other way? I still don't get it.",
        "A": "theyshade in eiight"
    },
    {
        "video_name": "x6xtezhuCZ4",
        "Q": "at 3:04 can someone explain more detail why if you have something 10 then thats going to 8 of them? how he turn 10 into 8?",
        "A": "10 multiplied by 4/5 is 8. How do we get this? We divide 10 parts into 5 sections, making each section have 2 parts. Then we colour in 4 of the sections and this leads us to have 8 parts coloured in the end. Hope this helps!"
    },
    {
        "video_name": "x6xtezhuCZ4",
        "Q": "At 3:03 how does he get the number 8?",
        "A": "Sara Johnston 2x4=8. Look into the math problem carefully and you ll understand why he got the number 8 in his fraction. Hope this helps"
    },
    {
        "video_name": "0YErxSShF0A",
        "Q": "At 1:30 in the video i did not get it can anyone help me?",
        "A": "When you multiply or divide an inequality with a negative number, you flip the inequality symbol. Was this the thing that you didn t get?"
    },
    {
        "video_name": "0YErxSShF0A",
        "Q": "At 3:45, why is the portion of the number line with -6 to 4 shaded in? z<4 OR z>=-6, but if both is true, it shouldn't work for the inequality, unless it was z<4 and/or z>=-6. Right?",
        "A": "Because the value has to be <4 and > -6, only the part in the middle work. Fulfilling both requirements does not mean it is wrong because 2 rights don t make a wrong."
    },
    {
        "video_name": "0YErxSShF0A",
        "Q": "Why does he swap the inequality at 1:42?",
        "A": "The best way to describe is by example, - 2x < 4 if divide by -2 and flip inequality, answer will be x > -2. From beginning, I can add 2x and subtract 4 to get - 4 < 2x, then I can divide by 2 to get - 2 < x, and I can switch the inequality around to get x > -2. Thus, flip when dividing by negative."
    },
    {
        "video_name": "0YErxSShF0A",
        "Q": "I just want to make sure I'm understanding this. So at 3:39, Sal is saying that the numbers that are on the section of the number line where the inequalities overlap can be used to satisfy both inequalities?",
        "A": "Yes, that is what he meant. In that section will satisfy both inequalities."
    },
    {
        "video_name": "0YErxSShF0A",
        "Q": "I don't understand the video from 2:25. What is he doing after Sal solved both inequalities",
        "A": "He plotted the solutions on a number line (sometimes you are required to do to show you understand and it looks like you need some practices). By plotting it we can graphically visualize it easier."
    },
    {
        "video_name": "0YErxSShF0A",
        "Q": "I am not understanding the part starting from 2:36 the graph, I need help just understanding this. If you can explain this better please help me out. Thanks! :)",
        "A": "Harleen-- Apologies for the delayed answer, hope i can still be of help. At 2:36, Sal simply draws a number line to help us better visualize our answer. Is this what you where referring to?"
    },
    {
        "video_name": "0YErxSShF0A",
        "Q": "At 2:25 I got confused and you lost me. I do not understand the number line part. What does the number line have to do with the problem? How do you use the number line?",
        "A": "The number line is just another way of looking at the solution set, but you can deal very happily with inequalities without ever thinking about the number line - unless your examiner wants you to!"
    },
    {
        "video_name": "CZdziIlYIfI",
        "Q": "At 11:23, why is the integral of cos pi x equal to (1/pi) sin pi x? How would you get to that answer?",
        "A": "Well, you can see that it has to be true, since d(sin(pi x))/dx=pi cos(pi x) by the chain rule. To solve it, substitute u=pi*x. Then du=pi dx, dx=du/pi, so integral[sin(pi x)dx]=Integral[sin(u)du/pi]=cos(u)/pi=cos(pi x)/pi This is something called substitution, or a change of variables. If you haven t learned it yet, it might be what this lesson was foreshadowing xD"
    },
    {
        "video_name": "CZdziIlYIfI",
        "Q": "At 4:24, we have [x]=2 which is even, and the f(x) will be f(x)=3-x. Yes, we have negative slope going down, but the y-intercept should be 3 at x=0. But why not so here?",
        "A": "The y intercept si 3, but we are looking at the interval [2, 3), not at x = 0..."
    },
    {
        "video_name": "CZdziIlYIfI",
        "Q": "11:13 how is du=dx and what does it tell us?",
        "A": "Sal made the substitution u=x. I can take the derivative of both sides with respect to x: du/dx=dx/dx=1. I can then multiply both sides by dx: du=1*dx=dx. In some cases, differentials can be treated like numbers and can be manipulated like this."
    },
    {
        "video_name": "CZdziIlYIfI",
        "Q": "At 9:00 isn't it (x-1) not (1-x)? That would completely change the answer wouldn't it?",
        "A": "It is 1-x. The slope of that line is negative."
    },
    {
        "video_name": "CZdziIlYIfI",
        "Q": "At 5:20, when evaluating f(x)cos(pi x) at x=1, f(x) is zero, so wouldn't f(x)cos(pi x) also be zero?",
        "A": "At this point in the video, Sal is not evaluating f(x)cos(pi x), he is just helping us get a handle on what s going on here by evaluating cos(pi x). At x = 1, this is cos(pi), which as you know is -1."
    },
    {
        "video_name": "CZdziIlYIfI",
        "Q": "At 2:18 sec, why is the greates integer, defined in the question as less than or equal to x, the whole number lower than x? Why couldn't [x] at x=1 be equal to 1 as well?",
        "A": "At 3:39 Sal dots in the point where x=1 and [x]=1 making the point at (1,0)."
    },
    {
        "video_name": "CZdziIlYIfI",
        "Q": "At about 6:38, when he realizes that the entire thing will be symetric about the x-axis why can he not just say that the integral from -10 to 10 will be 0 if the integral from -10 to 0 is negative the integral from 0 to 10?",
        "A": "yes he can idk why he didn t"
    },
    {
        "video_name": "04N79tItPEA",
        "Q": "During 0:50ish, did Sal imply that there are an infinite amount of radii that can be measured from the center?",
        "A": "This is true. You can measure a radius from the center of the circle to anywhere you want on the edge of the circle because it should always be the same distance and if it isn t then the shape cannot be considered a circle."
    },
    {
        "video_name": "04N79tItPEA",
        "Q": "At 9:55, what is the formular to find circumference",
        "A": "The formula for circumference is 2*pi*radius."
    },
    {
        "video_name": "04N79tItPEA",
        "Q": "0:00 all the way down to 2:01 hasn't helped me at all.",
        "A": "Try listening to the video again till you understand the video on Parts of a Circle. You could also try listening to the video before the one on Parts of a circle."
    },
    {
        "video_name": "04N79tItPEA",
        "Q": "at 2:08 ,Is the radius half of the diameter cause it looks like that",
        "A": "yes it goes from the side of the circle to the center called the center"
    },
    {
        "video_name": "sI789G6FBb4",
        "Q": "At 4:44 Sal said let me right this expression as a rational expression. Is it rational? Because I see sqrt(3) at the numerator.",
        "A": "In a sense you re right, sqrt(3) is an irrational number. However, the term rational expression is generally used to describe anything written as a fraction (even though we have an irrational number in this fraction)."
    },
    {
        "video_name": "ND-Bbp_q46s",
        "Q": "0:24 he says that this is used in tests and stuff. So, we can make our own functions?",
        "A": "Shomik, Yes, you can make your own functions. It is sometimes very helpful to design a function that when you input two numbers, some particular formula is used to provide the output that you want. It is done all the time in computer programing."
    },
    {
        "video_name": "ND-Bbp_q46s",
        "Q": "At 1:14 where did you get 7 diamond 11 ?",
        "A": "that s an example. more ex. 4 diamond 5 , 100 diamond 56"
    },
    {
        "video_name": "ND-Bbp_q46s",
        "Q": "Did anyone notice Sal says 'sta-diamond\" at 3:12?",
        "A": "he was about to say star and fixed it."
    },
    {
        "video_name": "ND-Bbp_q46s",
        "Q": "at \"0:40\" i don't understand what does the diamond for.",
        "A": "It s an operator but it can change"
    },
    {
        "video_name": "LNSB0N6esPU",
        "Q": "At 5:05, how did the y just go under the x section and how did the y become 0?",
        "A": "Very similar to your other question. This time Sal is finding the X-intercept. To do this, we always set y=0 in the equation. 5x + 6(0) = 30 Multiply: 5x + 0 = 30 Add: 5x = 30 Divide: x = 6 Hope this helps."
    },
    {
        "video_name": "LNSB0N6esPU",
        "Q": "at 1:12 how in goshdarn world do you find the x and y values!?",
        "A": "All you need to do is decide on an x-value, plug it into the equation, and use the resulting y-value to create the ordered pair. The x-value doesn t need to be found; you can just pick one that makes sense to use in order to create the graph."
    },
    {
        "video_name": "LNSB0N6esPU",
        "Q": "At 4:48, how did Sal just take out the 5x without doing anything to the other side? How is that possible?",
        "A": "Sal is finding the y-intercept. To do this, you set x=0. When you substitute x=0 into the equation, you get: 5(0) + 6y = 30 Multiply 5(0) = 0. This changes the equation to: 0 + 6y = 30 Add 0 + 6y, and you get: 6y = 30 Sal just didn t write out these steps. I guess he thought it was obvious. Hope this helps."
    },
    {
        "video_name": "EHzKIR3RuxE",
        "Q": "At 4:23 doesn't continuity imply differentiability and vice versa?",
        "A": "No. Differentiability is a stronger condition than continuity. In particular, differentiability implies continuity but not the other way around. The classic example is the origin when considering the absolute value function \u00f0\u009d\u0091\u0093(\u00f0\u009d\u0091\u00a5) = |\u00f0\u009d\u0091\u00a5|. Clearly \u00f0\u009d\u0091\u0093 is continuous is at the origin but it is not differentiable at the origin. Comment if you have questions!"
    },
    {
        "video_name": "EHzKIR3RuxE",
        "Q": "At 2:10 Sal explains the extreme value theorem. He says that there has to be a minimum and maximum value between \"a\" and \"b\", but what if f(a) = f(b)? Couldn't we draw a graph that is just a horizontal line and there is no minimum or maximum?",
        "A": "In this case you can consider the minimum to be equivalent to the maximum which implies that the function is constant over the interval, instead of considering both the minimum and maximum to be nonexistent."
    },
    {
        "video_name": "EHzKIR3RuxE",
        "Q": "At 2:30, how about a horizontal straight line, it is a function that does not take on a min and max value!",
        "A": "Sure it does, they just happen to be the same value."
    },
    {
        "video_name": "9tmtDBpqq9s",
        "Q": "At 0:27-0:36, why would you add -3/4 to -7-3/6? I don't get that.",
        "A": "taking away a positive number from a negative is the same as adding its positive. if you had a problem like -3-4, that would be the same thing as -3 plus -4. just like in the fractions, -3/4-(-7)-3/6 is the same thing as -3/4 plus -7-3/6."
    },
    {
        "video_name": "tu1GPtfsQ7M",
        "Q": "At 8:55\nThere is y2 = v2 - Projv1v2 = [-1,0,1]-([-1,0,1]\u00c2\u00b7\u00e2\u0088\u009a\u00c2\u00bd[-1,1,0]) as I understood it that should have been all because y2 = [-1,0,1] and Projv1v2 = [-1,0,1]\u00c2\u00b7\u00e2\u0088\u009a\u00c2\u00bd[-1,1,0] but then Sal adds a multiplication of \u00e2\u0088\u009a\u00c2\u00bd[-1,1,0] can someone please explain why?",
        "A": "To get the projection you need to multiply the magnitude of the projection by a unit vector in the direction of the vector being projected upon; otherwise all you have is a scalar."
    },
    {
        "video_name": "tu1GPtfsQ7M",
        "Q": "@1:31, why is x1=c1, x2=c2. Isn't V defined in R2--it is a plane?",
        "A": "He actually wrote x2=c1 and x3=c2. He set x2 and x3 as free variables (c1 and c2 can be any real number). Given c1 and c2 (which define x2 and x3), x1 can t be any real number, it has to be equal to -c1-c2 (so that the coordinates correspond to a dot on the plane)."
    },
    {
        "video_name": "tu1GPtfsQ7M",
        "Q": "12:10, Why is \u00e2\u0088\u009a(3/2) the same as \u00e2\u0088\u009a(2/3)?",
        "A": "1/sqrt(3/2) = 1/(sqrt(3)/sqrt(2)) = sqrt(2)/sqrt(3) = sqrt(2/3)"
    },
    {
        "video_name": "CDf_aE5yg3A",
        "Q": "At 7:15, Sal gets the value = -infinity/-infinity. How does ln(sinx) and ln(x) each equal -infinity?",
        "A": "ln(0)= -\u00e2\u0088\u009e, and sin(0)=0. Look at a graph of ln(x)."
    },
    {
        "video_name": "CDf_aE5yg3A",
        "Q": "At, 5:20, how is ln (sin x) ^ 1/ln (x) = {1/ln(x)} * ln (sin x) ?",
        "A": "One of the rules for logarithms is that if a logarithm is raised to an exponent, it is the same as multiplying it by the value of the exponent by the rest of the equation. Since ln(sinx) is a natural logarithm, this applies, and so ln(sinx)^1/ln(x)=(1/ln(x))*ln(sinx). Feel free to comment back if you need clarification on this answer."
    },
    {
        "video_name": "CDf_aE5yg3A",
        "Q": "At 7:13 , why would you use infinity to see the approaching limit when the limit is approaching 0+?\n(this might be a concept i missed from calc though :P)\nWhy is it ok to use infinity to see the limit in that function?",
        "A": "The limit of ln (sin x) / ln (x) as x approaches 0. The numerator: ln (sin 0) = ln (0) = -infinity (made clearer if you graph ln(x)) The denominator: ln (0) = -infinity (same as top) Hope this helps"
    },
    {
        "video_name": "CDf_aE5yg3A",
        "Q": "At 7:16 why is this negative infinity and not positive infinity?",
        "A": "Because as X approaches ln(x) from the right, it equals negative infinity. Look at the graph"
    },
    {
        "video_name": "CDf_aE5yg3A",
        "Q": "At 10:30, he takes the derivative of x/sinx to get 1/cos x, after he applies 0 to cos x to get 1. How can he take the derivative of only x/sinx without also having to take the derivative of cosx (-sinx, which would result in another 0 last result)?",
        "A": "He used the limit product law to separate lim x -> 0+ of xcos(x)/sin(x) into lim x -> 0+ of x/sin(x) \u00e2\u0080\u00a2 lim x -> 0+ of cos(x). Simplifying this give us lim x -> 0+ of x/sin(x) \u00e2\u0080\u00a2 1 which equal to lim x -> 0+ of x/sin(x). lim x -> 0+ of x/sin(x) = 0/0, thus he took the derivative of x and sin(x) again by L Hopital s law."
    },
    {
        "video_name": "CDf_aE5yg3A",
        "Q": "When I got to the step that Sal does at 9:03 I applied L'H\u00c3\u00b4pital's rule to the whole thing again. I got the same answer, but Sal pointed out a slightly easier way by using the rule that \"the limit of the product of two functions is equal to the product of their limits.\"\n\nI had forgotten that rule - is there a name for it, and a proof of it?",
        "A": "It s part of your basic limit laws and can be proven using the epsilon-delta definition of a limit."
    },
    {
        "video_name": "CDf_aE5yg3A",
        "Q": "I know the derivative of a sum is the sum of the derivatives. But, I don't believe the derivative of a product is the product of the derivatives at 9:03. You should need to use the power rule f'g+g'f. It turns out that using the power rule for this example also returns lny = 1, but I think that's coincidental.",
        "A": "At the point referenced, he isn t using the product rule because there is no derivative being taken, only a limit. And lim AB = lim A * lim B (as x -> 0+)"
    },
    {
        "video_name": "CDf_aE5yg3A",
        "Q": "At 10:48, why do we apply the rule for x/sinx even though we already got the derivative? & how come we didn't get the derivative of the cosx as well?",
        "A": "The limit as x --> 0 of cosx = 1, thus the rule does not need to be applied because we found the limit. In contrast, the limit as x --> 0 of (x/sinx) results in 0/0 aka indeterminate form. L Hopital s Rule can be applied multiple times, same thing with derivatives (e.g., f (x) aka velocity, f (x) aka acceleration, f (x) aka jerk, etc...). Hope this helps!"
    },
    {
        "video_name": "KyYC8XzKsHU",
        "Q": "At 4:20 when Sal divided 3(x-y)^2, why did he not simplify [-2x+3(x-y)^2]/3(x-y)^2 to -2x/3(x-y)^2 +1?",
        "A": "Because all that was left to do was substitution the values of x and y, and since he was going to obtain a number any way, there really was no reason to further mess with the result."
    },
    {
        "video_name": "KyYC8XzKsHU",
        "Q": "at 5:11 what does he mean \"And that actually what it looks like on this graph\" after we get result 25/27 of the derivate function?",
        "A": "It refers to the slope of the given graph. He says It (dy/dx of the equation) is almost 1 but not quite , agian refering to the slope. He s basically saying that the slope on the graph looks like it would be a little less than 1."
    },
    {
        "video_name": "KyYC8XzKsHU",
        "Q": "At about 4:20, why would you not simplify by crossing out the 3(y-x)^2's and having -2x?",
        "A": "You don t want to cancel when sums/differences, only products."
    },
    {
        "video_name": "KyYC8XzKsHU",
        "Q": "at 2:25 you decide to distribute, Why? could you just put all non dy/dx terms on the opposite side of the equation and then add 1 to both sides making the final equation dy/dx= (1-2x)/3(y-x)^2",
        "A": "Yes, and that would be considerably easier than what Sal did, I think, but we need to be careful on the algebra. The final equation would actually be (-2x / (3(y-x)^2)) + 1. Looking at your equation, I think you may have added 1 to both sides a bit too soon."
    },
    {
        "video_name": "KyYC8XzKsHU",
        "Q": "I think that at 2:25 instead of distributing for dy/dx, substituting (1,4) for (x,y) will be easier",
        "A": "It probably would be for only finding the slope at that particular point, but I think the purpose of Sal s method here is probably to demonstrate how to find the general expression for the slope in an implicit differentiation problem. Hope that makes sense."
    },
    {
        "video_name": "KyYC8XzKsHU",
        "Q": "At 3:08 why didnt he just plug in the x and y which we already knew and then the algebra would become a lot easier?",
        "A": "I agree Raphael, and when I solved the problem myself before finishing the video, this is exactly what I did. I feel like it was easier than Sal s solution too, since my next step was just: 2 + 27(dy/dx) - 27 = 0 then: 27(dy/dx) = 25 and finally: dy/dx = 25/27"
    },
    {
        "video_name": "KyYC8XzKsHU",
        "Q": "2:05 I don't really understand why one has to add the bit after the 3(y-x)^2, isn't that everything done?",
        "A": "Because of chain rule. In this case it is f(g(x)) where f(x) = x^3 and g(x) = y-3. So d/dx(f(g(x))) = f (g(x)) + g (x)"
    },
    {
        "video_name": "KyYC8XzKsHU",
        "Q": "at 2:41 couldn't you solve for dy/dx by subtracting 2x from both sides then divide by 3(y-x0^2, and then add 1 to both sides, coming up with dy/dx=-2x/(3(y-x)^2)+1 instead of distributing?",
        "A": "Yes. If you take Sal s answer: (3(y - x)^2 - 2x)/(2(y - x)^2 and break it apart into two fractions: (3(y - x)^2)/(2(y - x)^2 - (2x)/((3(y - x)^2) and then simplify, you get your result."
    },
    {
        "video_name": "SKe3ItfdzTY",
        "Q": "At 1:06, Sal says that H is the height. Why? If h is the height, what is the 5 for? What does the 5 represent? I'm confused.",
        "A": "You\u00e2\u0080\u0099re trying to figure out the unknown height of a parallelogram with an area of 24 square units and side lengths of 6 and 5. Area = base * height, so plug in the information you already know: 24 = 6h 24/6 = 6/6 * h 4 = h Either side length works as a base, but dividing 24 by 5 gives you a decimal number, so it\u00e2\u0080\u0099s easier to work with 6."
    },
    {
        "video_name": "HiY5UElqfhs",
        "Q": "I don't quite understand what Sal is saying at 2:00 -\ncould someone please clarify?",
        "A": "He s basically just adding the two answers (6/5 and 71/10)"
    },
    {
        "video_name": "TugWqiUjOU4",
        "Q": "4:48: I was wondering why triangles ADC and CAB are not similar. They have all the same angles, and since they both share AC, which is therefore the same length for both of them, they cannot be larger than the other. To put it in another way, since in order to get a larger triangle with the exact same angles, you have to increase each side length by the same ratio, and one of these side lengths has not been increased, aren't both triangles similar?",
        "A": "Yes, they are both similar. I don t think he said that they aren t similar. If he did, it is a mistake. Also, triangle ADC is congruent to triangle CBA because they have three congruent angles and they share a side (CA). By the angle-angle-side congruency postulate, they are congruent. I hope that helps."
    },
    {
        "video_name": "TugWqiUjOU4",
        "Q": "At about 4:39, Sal says \"right over here DC is somehow equivalent to AC. But then he says the statement cannot be true. Why can't the statement e true?",
        "A": "Sal is looking at what would happen if the statement were true. He shows, by SOH CAH TOA, that if the statement in the box were true that DC must be equivalent to AC. But that would make triangle CAD isosceles, which would make it a 45-45-90 triangle, which it clearly isn t. So since the statement in the box implies something absurd, it must be false."
    },
    {
        "video_name": "TugWqiUjOU4",
        "Q": "4:43 he says we can't make the statement that sin (CBA) = DC/BC. However, the exercise specifies that we are talking about a paralelogram, so DC should be equal to AB, this is the only way the angles ABC and ADC can be equal. Is this too much background knowledge to include it in the exercise? Do I get a wrong answer if I mark the second statement true?",
        "A": "That is a trick question, of sorts. While it is true that DC \u00e2\u0089\u0085 AB, the issue is that side is what you need for cos (\u00e2\u0088\u00a0CBA), not the sine. Sin (\u00e2\u0088\u00a0CBA) = AC/AD = AC/BC"
    },
    {
        "video_name": "TugWqiUjOU4",
        "Q": "2:54 he said they are not compare able but in the video like 2 prior he said that since the 2 of the angles are the same and its a 90% angle the last angle will correspond. isn't that sort of a contradiction, which one is it?",
        "A": "It s not a contradiction. It s true that if you have 2 equal angles that the third is also equal, but 3 angles being the same doesn t mean the triangle s sides have the same length. Thus, the triangles are similar, but not congruent."
    },
    {
        "video_name": "TugWqiUjOU4",
        "Q": "I've heard Sal use the term \"arbitrary\" quite a few times. What does this exactly mean? Example: 2:29 of this video.",
        "A": "It just means based on random choice . In other words, when Sal chooses an arbitrary function or an arbitrary number , it just means that he is choosing those things randomly, for no particular reason."
    },
    {
        "video_name": "TugWqiUjOU4",
        "Q": "@4:24 Sal concludes that there's not enough information to say AB not = DC. Why is AB not equal to DC? The triangles ABC and CDA are congruent by AAS rule. So AB IS EQUAL to CD. So the second option would be true. Am I right?",
        "A": "You are definitely right that AB \u00e2\u0089\u00a1 CD. However, Sal is explaining that AC is not equal to CD. As a result, the second option is also incorrect."
    },
    {
        "video_name": "TugWqiUjOU4",
        "Q": "At about 3:30, why are not EF = DC? The other angle in all the other triangles is 59, so they are similar triangles... I thought the ratios of the sides of similar triangles were consistent... Am I right?",
        "A": "The ratios of the sides of the different triangles to the other sides of the SAME triangle are equal. For example, if the ratio of AC to CD was 3/5 then the ratio of EG to EF would also be 3/5. However we can t compare two sides of two differently sized triangles and expect the same result. If for example EF was bigger/smaller than CD, the result would not be 3/5 but something completely different. What you said is true ONLY if the sides of two triangles are congruent, meaning exactly the same size."
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "At 3:56 how do you know if its negative 2 or minus 2?",
        "A": "They mean the same thing. 0 + (-2) = 0 - 2."
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "1:57 . if there is a positive 4xy and a negative 4xy wouldn't it cancel out that xy's all together ?",
        "A": "Yes, you are correct... the positive 4xy and the negative 4xy cancel each other. As you see in Sal s final answer, there is no xy term."
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "Around 1:16, If you had parentheses around things, it would probably make it a lot harder. How would it work if you had parentheses?",
        "A": "You would multiply the stuff in the brackets by the number in front of the brackets. Example : 4( 1x + 2) simplifies to 4x + 8"
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "What happens if y equals a negative then what happens? 2:51",
        "A": "If y is a negative, then taking its opposite (-y) would yield a positive number."
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "at 3:56 how do you know if it is a negative number or a subtraction problem because there are not any partnthases",
        "A": "A negative number and a subtraction problem are basically the same thing. In a subtraction problem like 5-3, you are basically adding -3 to 5. 5-3=5+(-3). Hope this helps!"
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "at 4:26, wouldn't the equation be -y - x^2 + 2x + y^2 because in the video, it said -2x^2 + 3x^2?",
        "A": "-2x^2 + 3x^2 = x^2. It s positive not negative so the video s right."
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "At 4:14 why does he add instead of subtract",
        "A": "Because you are left with positive one x-squared terms. Negative 2x^2 plus 3x^2 is 1x^2."
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "At 2:51, how can you answer a problem like, 3m-5m, would it equal to -2m or could you not do it?",
        "A": "You can do that indeed, 3m-5m=-2m."
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "At 2:18 of the video, couldn't you be able to combine the 2x with the -2x squared and the 3x squared if u simplified them?",
        "A": "No you cannot, you can only combine terms with the same degree. You can combine the -2x squared with the 3x squared but you cannot combine those with the 2x. Hope this helped :) Happy holidays!!"
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "2:39-2:50 why would you subtract eventhough there is a + sign?",
        "A": "because same signs equal positive (+), and opposite signs equal negative (-)"
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "At 0:50 could you say \"y\" squared is the same as y*y or 2y. So the end result is 1y+4x. :)",
        "A": "2y is y + y not multiplied together, y*y is always y^2"
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "At 4:28 Sal writes -2x^2 + 3x^2 he say's it equal's 1x^2 but not 5x^2. Why?",
        "A": "The key here is the negative sign in the front of the -2x^2 term. -2x^2+3x^2 = x^2, not 5x^2. Just like -2 + 3 = 1 and not 5, and -2a + 3a = a (which is the same as 1a, but when the coefficient is 1 we usually don t write it... it s implied)."
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "at 1:19 - 1:30 why must we look for y's",
        "A": "the equation also has xs and we are trying to get rid of the ys"
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "At 0:00, what does Sal mean by a \" very, very, very hairy expression.\" ?",
        "A": "He means a difficult or hard to solve expression or problem."
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "At 0:08, dang, we aren't all Sonic the Hedgehog.! So i cheated at paused the video like a bad boy... ;D",
        "A": "3 years ago wow long time ago"
    },
    {
        "video_name": "FNnmseBlvaY",
        "Q": "At 2:47\n\nsubtraction is neither associative or commutative, so why can you reorder all these terms that were being subtracted from other terms. 3-2 \u00e2\u0089\u00a0 2-3",
        "A": "Let s treat it like addition. -3xy - 2x - 2xy is just the same as -3xy + (-2x) + (-2xy) The sign/operation (e.g. plus or minus sign) always goes with the term after it. With those addition signs, we could apply both the associative and commutative property to subtraction. Basically, you are correct that 3 - 2 =/= 2 - 3 but it is true that 3 - 2 = (-2) + 3 Hope this helps and have a happy new year! :D"
    },
    {
        "video_name": "oQhp3ndj28Y",
        "Q": "Sal references another video or videos at 4:20. What are those videos?",
        "A": "Probably the videos following this one that are explaining what e is."
    },
    {
        "video_name": "64bH_27Ehoc",
        "Q": "At 9:48. Sal can only do that because the numerator is 1, right?",
        "A": "Well our exponent properties tell us that the 1 in the numerator would also be squared but since 1 squared is just one it will stay the same."
    },
    {
        "video_name": "64bH_27Ehoc",
        "Q": "Around 11:50 I don't understand how he is combining the terms.",
        "A": "For example he made 3*4^2*(1/9)^2 = 3*(4/9)^2. I think it s because 4^2 * (1/9) will still equal 16/81 which is what (4/9)^2 will equal as well."
    },
    {
        "video_name": "64bH_27Ehoc",
        "Q": "At 11:52, Sal simplified 3 * 4^2 * (1/9)^2 to be 3 * (4/9)^2. Why isn't it 3 * (4/9)^2^2, or 3 * (4/9)^4, since there were 2 squares?",
        "A": "when multiplying numbers with exponents you add the exponents which makes sense, but this only works if the bases are the same, but in the video they arent Ex: 2^2*4^2=8^2...that is equal to 4*16=64 not sure if that makes sense, or answers your question"
    },
    {
        "video_name": "64bH_27Ehoc",
        "Q": "at 8:47, what does he mean multiply everything by four, and divide everything by four?\n\nSay we did n x 4 = 4n\n\nThen we divide 4n by 4, and that's n! So why not just keep it the same?",
        "A": "Sal did that so the 4 s on the inside of the brackets are raised to the same exponent as the 1/3 s."
    },
    {
        "video_name": "DR2DYe7PI74",
        "Q": "why in 0:30-0:42 the writing so small",
        "A": "Leave her alone.... she is NOT spamming. she just has an honest to goodness question & I m going to answer it."
    },
    {
        "video_name": "DR2DYe7PI74",
        "Q": "At 0:22 he writes 1/100 and says \"You can view that as 5 times 1/100\" why couldn't he just write 5/100?\nOr is this a mistake?",
        "A": "You version is not wrong. Sal is just trying to relate how the 5 and the place value (1/100) come together thru multiplication."
    },
    {
        "video_name": "8QihetGj3pg",
        "Q": "At 6:43 and before Sal show different vectors and how you can draw them. Is this really necessary to know how to draw all these different vectors? on the \"graph\"or can I just draw the vector i am most comfortable with.",
        "A": "He is showing how you can draw the same vector on different coordinates (better to say the same vector scaled for a given scalar). It is not necessary to draw all those different vectors, but it is useful to know you can draw them that way. Also, when you solve problems which include the addition, substitution etc. of vectors you are always going to draw the vectors which have certain coordinates or given dots (A, B, C...). You can always put them on the position where you need them to be."
    },
    {
        "video_name": "8QihetGj3pg",
        "Q": "at 3:05, isn't the arrow of vector b supposed point toward the east? because i remember on the last video, it says that the arrow on the top of the letter shows the direction of that vector. Please help.",
        "A": "The arrow on the top of the letter only indicates that the letter represents a vector quantity. It has nothing to do with direction. To see the direction of the vector, you must look at its components."
    },
    {
        "video_name": "8QihetGj3pg",
        "Q": "So essentially, addition of vectors can be commutative as Sal mentioned at 6:00, right?",
        "A": "Vector addition is commutative, yes."
    },
    {
        "video_name": "8QihetGj3pg",
        "Q": "At 0:53 and 0:56, what does Sal mean by saying these are both members of 'R2' if you think about it in terms of 'real coordinate spaces'?",
        "A": "It means they are 2D vectors which exist within a 2D space. R2 is the set of all 2-tuples, so all R2 vectors are represented by 2 numbers (like coordinates on a plane)."
    },
    {
        "video_name": "8QihetGj3pg",
        "Q": "At 1:06, what is meant by the sentence that vector a and vector b belong to R2. What is meant by R2? Please i want an immediate response!!",
        "A": "\u00e2\u0084\u009d means all real numbers, so \u00e2\u0084\u009d\u00c2\u00b2 means a set of 2 real numbers. All that it means is that you re working with a vector that holds 2 normal numbers and takes the form [x y]. An \u00e2\u0084\u009d\u00c2\u00b3 vector takes the form [x y z], an \u00e2\u0084\u009d\u00e2\u0081\u00b4 vector takes the form [x y z w], and in general an \u00e2\u0084\u009d\u00e2\u0081\u00bf vector takes the form [x\u00e2\u0082\u0081 x\u00e2\u0082\u0082 x\u00e2\u0082\u0083 ... x\u00e2\u0082\u0099]."
    },
    {
        "video_name": "8QihetGj3pg",
        "Q": "At 2:42 Sal shows us that vectors do not start at the origin.\nHe also says that all of those vectors are the same as vector A; I was just wondering if, if you had a negative copy of vector A would the magnitude still be the same as vector A even though it's a negative?",
        "A": "Yes. The negative sign changes the direction of the vector (-A points in the exact opposite direction as A), but the length (or magnitude) is unchanged."
    },
    {
        "video_name": "8QihetGj3pg",
        "Q": "4:31 the three vectors create a triangle, why?",
        "A": "They create a triangle because the third vector is a- b. Adding and subtracting vectors can be visualized by putting the end of one vector at the end of another (for addition, the arrow of the first points to the origin of the second, for subtraction the arrow of the second and first vector point to the same spot). This creates a third vector that s drawn from the origin of the first vector to the end of the second vector, and thus you get a triangle."
    },
    {
        "video_name": "8QihetGj3pg",
        "Q": "At 2:29, Sal mentions that for vectors, we only care about magnitude and direction. Now, what happens when we care about location? Is there a specific name for something that takes location into account? How would we notate it?",
        "A": "A vector whose tail is always at the origin is called a bound vector. A vector we can translate around is called a free vector."
    },
    {
        "video_name": "8QihetGj3pg",
        "Q": "Is there a way (or reason) we could (should) find the position of the vector at its beginning?\nSal says that all the vectors at 2:43 are equivalent, but what if we needed to know its origin. What could tell us this?",
        "A": "The origin is a point, and points can be represented by vectors as well. So if it s origin is [5,2], the the vector [5,2] is a vector we could use to represent the origin. The reason we draw vectors as arrows instead of simply plotting points is because it lets us visualize better how the vectors are interacting in various equations. Like here you see how adding two vectors can be visualized by two vectors connected at the end."
    },
    {
        "video_name": "8QihetGj3pg",
        "Q": "At 1:18, what are all those symbols above the graph?",
        "A": "a, b \u00e2\u0088\u0088 \u00f0\u009d\u0095\u00bd\u00c2\u00b2 means vector a and vector b belong to the 2D space of real numbers , meaning that both vectors are composed of 2 real numbers."
    },
    {
        "video_name": "8QihetGj3pg",
        "Q": "At 1:15, why does Sal write down both a and b are members of \"IR^2\"?",
        "A": "I think it s just a way of saying they are both two-dimensional vectors."
    },
    {
        "video_name": "4xfOq00BzJA",
        "Q": "Can 1000 be place anywhere at 1:41?",
        "A": "No, Sal is placing the 1000 linearly, as if a certain value was being added between each step in the scale..."
    },
    {
        "video_name": "QUem_2dkB9I",
        "Q": "Is it wrong if the first example at 1:12 I found 30 prime factorization first 5 and 6 then for 6 is 2 and 3 instead of what Sal did in the video which was for 30 is 2 and 15 and for 15 is 3 and 5? The least common multiple it was again 150 same as Sal.",
        "A": "When you do the prime factorization, it doesn t matter which 2 factors you start with. The only requirement is that the 2 numbers you select multiply to create the original number. You have the same prime factors as Sal. So, you have a correct factorization. Hope this helps."
    },
    {
        "video_name": "IpXL9HIkqUQ",
        "Q": "At 0:55 where did 2^5 come from?",
        "A": "Since 2^10 is the same as 2^5 * 2^5 he was just breaking it down into something easier to calculate to verify that he had the right answer for 2^10 before working the rest of the equation."
    },
    {
        "video_name": "j8cKFzj5V6g",
        "Q": "At 1:50, why didn't Sal write (z+8)(9z-5)(z+6)(9z-5)? I understand that the 9z-5 is already accounted for, but how so? Or why can't we add it in there twice?",
        "A": "It is not necesary. The objetive here is to have the same denominator in the both sides to make posible substract. You can multiply the same factor two times, but it is easier to just multiply in once, because you already have the same denominator on both sides."
    },
    {
        "video_name": "sHbX58y5D4U",
        "Q": "it shows the class at 0:15 but where were the classes Incorporated into the scatter plot? I only see the period and average score on the graph.",
        "A": "That s all u see because that s all she wanted to know. She wanted to know if the period has something to do with the average score! The classes taken weren t important. I hope this helps!"
    },
    {
        "video_name": "ENFNyNPYfZU",
        "Q": "How do you get a horizontal stretch (1:38, 2:27) that is curved once, while on the second graph there is two lines that curve twice?",
        "A": "At 1:38, you can get those curves with equations such as y = -x^3 - 2x^2 + x + 3 (I put a random one, not the one on the video). Generally, the higher the degree of the polynomial, the more times it can curve. The ones in 2:27 are just two parabolas with the skeleton equation y = ax^2 + bx + c. When I say skeleton , I mean the basic structure of the equation. Hope this helps!"
    },
    {
        "video_name": "ENFNyNPYfZU",
        "Q": "At 4:10, can I write f(x) = g(x+2) ?",
        "A": "Yes, that is technically correct. You have to consider what you are asked to do, though."
    },
    {
        "video_name": "ENFNyNPYfZU",
        "Q": "At 4:59, Sal draws the mirror image of f(x). Why does he do that? And, is there a way to this without drawing a mirror image?",
        "A": "I believe he s showing an example of -f(x) as a reflection of f(x) over the x-axis. Basically reversing y values since f(x) is the same as the y value."
    },
    {
        "video_name": "ENFNyNPYfZU",
        "Q": "How does the final problem work? (5:45 6:40)? Could someone explain it in more detail?\nThanks\nCelestaialHalcyon",
        "A": "Basically, he s trying to show us how to vertically stretch the function. By multiplying the function by three, he is simply tripling the y-values, and stretching the height of the function. Example: f(1) = 3 --> 3f(1) = 3(3) = 9 The input (aka the x value) remains the same, but the output (aka the y value) is being exaggerated."
    },
    {
        "video_name": "ENFNyNPYfZU",
        "Q": "At 4:19, is f(x-2) equal to f(x)-2? It seems like it is, but I get a bit confused sometimes.",
        "A": "No. If you have the -2 outside the parentheses it causes a vertical shift like in the first example . Only when you are adding or subtracting from x inside the parentheses do you get a horizontal shift (adding moves to the left, subtracting to the right). The confusing part is that horizontal shifts go the opposite direction of the operation, i.e. left for addition, but vertical shifts go the normal way, i.e. upward for addition."
    },
    {
        "video_name": "ENFNyNPYfZU",
        "Q": "At 2:38, Sal says that the function in red is equal to (-3, f(3))\nHow come it's not (-3,f(-3))?",
        "A": "I believe Sal made a mistake at that point to say and write (-3, f(3)). But then he corrected himself at 2:58-3:00, and wrote & said (-3, f(-3)). I saw that kind of mistake happens in some videos. I will take note and report them to Khan. So the videos will be corrected for errors"
    },
    {
        "video_name": "ENFNyNPYfZU",
        "Q": "At 6:16, how did Sal get to 3(-g(x))=-3g(x), especially -3g(x)?",
        "A": "Let s break this down... -g(x) basically tells you to multiply g(x) by the minus or a -1. So, we can say: -g(x) = -1 * g(x) Now, bring in the 3: 3(-g(x)) tell you to multiply 3 with -g(x), or 3 * -g(x). Put the pieces together: 3(-g(x)) = 3 * -g(x) = 3 * (-1) * g(x) Multiply the 3 and the -1 and you get -3. Thus: 3(-g(x)) = -3 g(x) Hope this helps."
    },
    {
        "video_name": "ENFNyNPYfZU",
        "Q": "from 5:45 to 6:40 how do u solve that final problem",
        "A": "Look at f and g at x = -8. f is 6 and g is -2. At x = -1, f is 3 and g is -1. at x=1 f is 4 and g is just about -4/3. At x=3 f is 7.5 and g is -2.5. In all of these f(x) is -3*g(x), or (the same thing) g(x) is -(1/3)*f(x)"
    },
    {
        "video_name": "E2VxbufzuWQ",
        "Q": "At 6:08, why is Sal evaluating the improper integral of e^-st*f(t) instead of just e^-st*f(t)? Because of integration by parts, shouldn't it just be e^-st*f(t) - the improper integral that comes afterwards?",
        "A": "I m not sure if I m understanding your question. He writes [ e^-st * f(t) ] in brackets so it can be evaluated from 0 to infiniti. You can evaluate that first or last (it doesn t matter) but you need to evaluate that otherwise you would have t in your answer and we know the result should only be a function of s ."
    },
    {
        "video_name": "E2VxbufzuWQ",
        "Q": "hello, can someone explain to me where is the constant C when we have done the antiderivative of f'(t) at 5:37!\nthank you for your answer",
        "A": "The integral is evaluated at the limits 0 to infinity 6:10, so the constant C is not necessary."
    },
    {
        "video_name": "E2VxbufzuWQ",
        "Q": "what did he do at 5:14 in the video",
        "A": "Are you referring to the point where he takes the derivative of both sides of the equation u=e^(-st)? He does this by using chain rule. He takes the derivative of e^(-st) with respect to -st, which results in e^(-st). He then takes the derivative of -st with respect to t, which results in -s. He lastly multiplied the two answers,which resulted in -s[e^(-st)]. This was the new right side of the equation. The derivative of u was simply u . The resulting equation from all of this was u =-s[e^(-st)]."
    },
    {
        "video_name": "E2VxbufzuWQ",
        "Q": "At 5:25 he states that the indefinite integral of v'=f'(t) with respect to t is just v = f(t). That sounds reasonable at first glance. But shouldn't there be a loss of information? If v had a constant term (with respect to t) the v' would toss that out. Don't you then need to put a c back in when you do the integration: v = f(t) + c?",
        "A": "yes it would. However, due to the fact that these are definite integrals, the c is going to be subtracted out in the evaluation. ie the integral from a to b of sin(x) dx = -cos(x) evaluated from a to b = -cos(a) + c - (cos(b) + c) = cos(a) - cos(b) with the cs canceling out as they are the same value for this particular function."
    },
    {
        "video_name": "E2VxbufzuWQ",
        "Q": "around \"4:40\", shouldn't there be a dt?",
        "A": "v can get confusing, that s why i stick to dv/dt, but its really not a big deal as long as you can keep up with when he s using it."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "at 3:07 it explains how i to the 4th power equals 1. At 1:47 it says i to the 0th power equals 1. How can they both equal 1",
        "A": "Well anything to the 0th power is 1. i represents root(-1) so i to the 2nd power would be -1. i to the fourth power is like taking i to the 2nd power times i to the 2nd power or -1 times -1 which equals 1."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 5:20 Sal mentioned principal square root and i just want a quick explanation of what that is",
        "A": "wat number can be multiplied by itself to get that number is the square root"
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "What's \"e\" in 0:22?",
        "A": "e is a irrational number like pi. It does have very specific applications in mathematics beyond scope of an introductory algebra course. A google search can bring up mountains of information about e if you re curious enough, but be prepared to feel overwhelmed!"
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 2:10, Sal points out that i^1 is basically i. But why didn't he say it as 'sqr rt (-1)?'",
        "A": "Well, that doesn t make any difference... Does it?"
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 4:09, when Sal was explaining how to get i^4, couldn't he just have done: i^2*i^2 and get 1 or was it necessary to make it i*i^3?",
        "A": "You are correct. i^4 = i*i^3 = i^2*i^2."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 0:46, i^2=-1, equal to i=sqrt of -1 because if you take the sqrt of both sides of i^2=-1, it turns into i=sqrt of -1? Thanks in advance!",
        "A": "Just by definition, i^2= -1 You could also write this as i= sqrt(-1) Squaring both sides: i^2= (sqrt(-1))^2 Squaring a radical removes the radical sign: i^2 = -1 I hope I could help! :)"
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 0:09, why imaginary numbers are called imaginary numbers? According to the video, imaginary numbers are necessary in the natural world of numbers (i to the fourth power is 1, a very natural number), so why call them imaginary?",
        "A": "Well, i is equal to sqrt -1, and you can t put that on a number line. When asked to put this on a graph, the i part is the y-value."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 0:48, he says that i is equal to the \"principle square root of -1.\" What's the principle square root? =D",
        "A": "Any nonzero number has exactly two square roots, and they are opposites of each other. The principle square root is the one that we decide by convention to use as the default. For positive numbers, the principle square root is positive. For negative numbers, the imaginary part of the principle square root is positive."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "I thought i was the square root of -1, not -1. At 3:30 to the rest, Sal says that i is -1.",
        "A": "Yeah, you re probably right. Maybe we should ask sal to inform people."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 4:04, I still do not understand now i^4 = 1.\nIf i^4 can be split into 2 products: i^3 * i, and i^3 is the same as -i, when you multiply -i * i, that should give you -i right?\n\nI'm so confused at this step. Could someone explain? Thanks.",
        "A": "No, i * i = -1, and -i = (-1) * i So -i * i = (-1) * i * i = (-1) * (-1) = 1 Rather than splitting i\u00e2\u0081\u00b4 into i\u00c2\u00b3 * i, it might be easier to think of it as i\u00c2\u00b2 * i\u00c2\u00b2 = (-1) * (-1) = 1"
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "On 3:35 did Sal just said that -i is 1 ? How come? I didn't get it :(",
        "A": "At that point in the video, Sal is not saying -i = 1 Sal is multiplying: i * (-i) i * i = -1 Then apply the minus sign from the original expression: - (-1) = 1 Hope this helps."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "i dont understand 2:35",
        "A": "Since i^3 = i*i*i which is equal to i^2*1, we now that the last i must be multiplied by i^2(-1). We also know that any number multiplied by -1 has its signs switched. Thus i^3 = -i."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At about 2:02, Sal says that any number raised to the zeroth power is 1. So what is the zeroth root of 1?",
        "A": "The zeroth root of x would be the same as x^(1/0), which is undefined because you re dividing by zero."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 1:50, he said that anything to the zero power is 1. Is this true with 0^0 also?",
        "A": "That is a point of disagreement between mathematicians. Some say that it is 1 others that it is undefined. Most likely, your instructor will tell you that 0^0 is undefined, but you would do well to ask. For reference sake, there is no formal proof yet presented as to what 0^0 equals."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "Sal says \"principal square root\" at 1:00. Is a principal square root different from a \"square root\"?",
        "A": "As you know, a number can have a positive and a negative square root (for instance, \u00e2\u0088\u009a144 can be + 12 or - 12) When we say principal square root , that makes it clear that we re talking about the positive square root."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 3:00 when i^3 = -i, what does that mean, exactly? Would it be -sqrt(-1)? Why don't the negatives cancel out to make it the square root of positive one, which is one?",
        "A": "No, the negative outside of the \u00e2\u0088\u009a does not come inside. But you are right, i^3 does equal -\u00e2\u0088\u009a(-1)."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 3:59, Sal said \"i\" to the fourth is the same thing is \"i\" to the zeroth power. This means \"i\" to the fourth is 1. Isn't 1 times \"i\" just \"i\"? Why is it 1?",
        "A": "If you raise anything to the zeroth power it is 1. For example 10^0 = 1 or 4 raised to the zeroth power is 1. If the imaginary number i has the same properties as all other numbers, then taking it to the zeroth power must mean it is equal to one. Hope this makes it a bit clearer"
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At about 0:46 , you say that i^2 = -1. then should i be + OR - sqr root of -1, or just the + sqr root of -1?",
        "A": "No, it is defined as the principal square root of -1."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 3:22, can I*4 be shown as I*2 times I*2?",
        "A": "Well I^4 is 1, I^2 is -1 -1 * -1 = 1 so yes"
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "at 0:50 what is a principal square root?",
        "A": "The principal square root is the unique nonnegative square root of a nonnegative real number. For example, the principal square root of 9 is 3, although both -3 and 3 are square roots of 9."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "I was just playing around with some numbers (specifically those at 4:25) and I got a slightly different answer. I wrote:\ni^6=(i^3)^2\ni^6=(i^3)(i^3)\ni^6=(-i)(-i)\ni^6=i?\nWhat was my mistake? I'm still new to this. Thanks.",
        "A": "You almost had it. i^6 = (-i)(-i) i^6 = -1"
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 4:00, Sal says that i^4 = i^3 * i.\nWouldn't i^4 = i^2 * i^2 be much easier to solve?",
        "A": "Yep, it would. Either Sal didn t realize that, or he just wanted to show how to solve i^3 multiplied by i instead of just taking the easiest i^2 and multiplying it by itself."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 3:43 you said that i*-i was the same as -1*i*i. How is this possible when to get negative one you need to have i squared?",
        "A": "-i is the same as -1*i so i*-i is the same i*-1*i You can rearrange the order of what you are multiplying and get -1*i*i You don t need i squared to get -1. If there is already a - sign, then you are multiplying by -1."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 3:10, wouldn't it be easier to find i^4 by mulitplying i^2 twice? Isn't that simpler than finding i * i^3?",
        "A": "That s the way I usually do it too. I guess Sal finds the other way easier."
    },
    {
        "video_name": "ysVcAYo7UPI",
        "Q": "At 3:22, can I*4 be shown as I*2 times I*2?",
        "A": "Yes it can. Good observation."
    },
    {
        "video_name": "Pms4cBWwPSU",
        "Q": "why is that when sal in 1:17 shows that the s is negative, but isn't a negative times a negative number is positive?",
        "A": "Yes, a negative times a negative will make a positive. However, in this lesson, Sal is multiplying 67 (s*s*s...*s*s, or s ^ 67) negatives! While an even number of negatives multiplied together result in a positive (-1 * -2 = +2), an odd number of negatives multiplied together will result in a negative (-1 * -2 * -3 = -6, try more). Since 67 is an odd number and s is negative, it means s ^ 67 will be negative."
    },
    {
        "video_name": "Pms4cBWwPSU",
        "Q": "1:38, why is T to the power of nine positive? assume T>0",
        "A": "No matter how many POSITIVE integers you multiply together, the answer is still positive."
    },
    {
        "video_name": "Pms4cBWwPSU",
        "Q": "Sal says at 0:51 that if you have a negative number being multiplied an odd number of times, it stays negative. Does this apply to everything? Are there exceptions?",
        "A": "there are no exceptions"
    },
    {
        "video_name": "IelS2vg7JO8",
        "Q": "At 1:38, Sal says that pi can be represented by 22/7. Is this approximate or is 22/7 also an irrational number equivalent to pi?",
        "A": "It is true; pi, or 3.1415926.....(3.14), can be interpreted as 22/7, which is more exact than any other method of solving for a measurement. 22/7 is an irrational number, but it is exact when it comes to solving a problem. Of course, the most exact answer is when you keep your answer in terms of pi, but obviously that is not always the case. Since 22/7 is a fraction, you don t need to worry about your answer being exact with extra decimal places."
    },
    {
        "video_name": "IelS2vg7JO8",
        "Q": "At 0:37 , he says that there is an advanced proof that can be explained in calculus. Does anyone have a rather simple proof on how this works?",
        "A": "There is a way to prove it without calculus by using cones and cylinders. But, it is NOT an easy proof. Personally, I think the proof using calculus is much easier to understand. But, at this level of math, you just have to accept mystery equations and formulas being tossed at you without proof. When you get to integral calculus, you ll be able to see where all these equations come from."
    },
    {
        "video_name": "IelS2vg7JO8",
        "Q": "At 2:26, Sal mentioned about the cubed radius. I am confused whether the whole equation is cubed or just the radius",
        "A": "Just the radius is cubed."
    },
    {
        "video_name": "IelS2vg7JO8",
        "Q": "I paused Mr.Sal at the time 1:42 and did the math since we have done it in class and I got 1436.75504",
        "A": "Sal just rounded to the nearest tenth."
    },
    {
        "video_name": "IelS2vg7JO8",
        "Q": "At 0:57 how is the radius half the diameter? also would it work if you used 14?",
        "A": "The radius is the length of a line drawn from the edge of a circle to the middle of the circle. The diameter is the length of a line drawn from the edge of a circle to the middle of the circle and back to the opposite edge. So the definition of the diameter is that it s twice the radius. Which means the radius is half the diameter."
    },
    {
        "video_name": "IelS2vg7JO8",
        "Q": "When Mr. Khan said the volume formula at 0:40, is the 4/3 rounded a bit or is it exact?",
        "A": "Exact, or he would say so."
    },
    {
        "video_name": "KCehC_3CBBY",
        "Q": "At around 1:13 Sal said, \"She used 1/3 of a cup laundry detergent, and she has a third left.\" But that's wrong because the word problem says, \"Gina had 2/3 cup of laundry detergent.\" Right?",
        "A": "I don t think so. 2/3-1/3=1/3 so Sal is correct"
    },
    {
        "video_name": "yEAxG_D1HDw",
        "Q": "on 4:07\nshouldn't it have been: (log_xA)^c? because sal included the ( ) on \"c(log_xA)\".\nThe \"C\" would have been raised to the whole log_xA, not just A. can someone clarify?",
        "A": "The parenthesis only mean multiplication in this case. C(log_x(A)) = C * log_x(A)."
    },
    {
        "video_name": "yEAxG_D1HDw",
        "Q": "At 1:48, Khan raised x^B=A to the C power to make (x)^BC=A^C. I don't understand how it is legal to raise both sides of an equation by the same power. I don't know the reasoning behind this. For example, if I raise x=9/4 by a power of 2, I get x^2=(9/4)^2. Then, to find x, I would square both sides to get x=+,-9/4 which is not the same as 9/4. For an algebraic operation to be legal, doesn't using an operator on both sides have to allow the operators to cancel out, leaving the same expression?",
        "A": "It requires setting domain limits to invalidate the negative answer from the new equation, but only for raising the sides by an even integer. If c were odd no domain restriction would be needed as only even would produce two answers."
    },
    {
        "video_name": "yEAxG_D1HDw",
        "Q": "what did sal do at 2:30?",
        "A": "Do you know the law of exponents (x^b)^c = x^(b*c) ? He s showing us that this is also one of the laws of logarithms."
    },
    {
        "video_name": "yEAxG_D1HDw",
        "Q": "Hello everyone. In 1:33. We know that log_x\u00e2\u0081\u00a1A=B \u00e2\u0087\u0092 x^B=A and that if we multiply the first expression times C and raise the second to C they are still equal. That means that multiplying times C and raise to C are equivalent operations. My question is: If this is correct what is it based on?",
        "A": "This is related to the idea that the logarithm is an exponent with respect to the original equation. Raising an exponential expression to a power is equivalent to multiplying its exponent by that number."
    },
    {
        "video_name": "mX91_3GQqLY",
        "Q": "Sal mention coprime numbers at 2:20. Could someone tell me what coprime numbers are?",
        "A": "It means they have no common factors (e. g. 27/32 = 3^3/2^5 = (3*3*3)/(2*2*2*2*2))."
    },
    {
        "video_name": "mX91_3GQqLY",
        "Q": "At 7:18, Sal says a/b is a reducible fraction. Well, if that's the case, then a/b can be reduced to a point where it becomes an irreducible fraction. All reducible fractions can be reduced to an irreducible one. Why can't we do the same with a/b if it's reducible, using the Euclidean Algorithm? By doing that, we will proove that \u00e2\u0088\u009a2 can be the ratio of two numbers in an irreducible fraction, i.e. that it's a rational number.",
        "A": "The video is based on proof by contradiction. By getting to the point at 7:18, he found the contradiction between his original assumption that is it irreducible to finding that a/b is reducible. This allows him to come to the conclusion that the original assumption was incorrect, and thus it must be irrational."
    },
    {
        "video_name": "mX91_3GQqLY",
        "Q": "At 1:05 of the proof, Sal assumes that a/b have no factors in common other than one. I am curious as to why this is assumed. Thanks.",
        "A": "Without that assumption it could be possible to reduce a/b to some integer c, and thus the final contradiction wouldn t hold. So we must be absolutely certain that a/b is already in lowest terms = GCD(a,b) = 1 by assuming this from the outset, otherwise statements regarding a/b are essentially meaningless."
    },
    {
        "video_name": "mX91_3GQqLY",
        "Q": "I get lost at 5:34 when Sal says 2b^2 = 4k^2. It seems to come out of nowhere. Why does the \"2k^2\" change to \"4k^2?\"",
        "A": "He shows that a is even. So therefore a = 2k, and a^2 = (2k)^2. So: (2k)^2 = (2k)*(2k) = 2*k*2*k = 2*2*k*k = 4(k^2) = 4k^2"
    },
    {
        "video_name": "mX91_3GQqLY",
        "Q": "At 5:45 How did Sal get from 2b^2=(2k)^2 to 2b^2=4k^2",
        "A": "2k * 2k = 4k^2"
    },
    {
        "video_name": "mX91_3GQqLY",
        "Q": "at 5:42, Sal says that 2b^2 = 4k^2 whereas he said 2b^2-2k^2 right before. am i missing something?",
        "A": "(2k)^2 = (2k)*(2k) = 2*k*2*k = 2*2*k*k = 4(k^2) = 4k^2"
    },
    {
        "video_name": "xTcwnhyPN5E",
        "Q": "FROM 00:00 - 2:08 I still do not get it!",
        "A": "Did you watch and practice the first module of this section: Areas Of Parallelograms? Watch those and this should make sense. This is just an extension of those vids. The only difference here is that you find the base and height by counting squares on the grid."
    },
    {
        "video_name": "w70af5Ou70M",
        "Q": "At 0:03, what does Sal mean by the rigorous definition of a limit?",
        "A": "Strictly mathematical, not just relying on intuition, you have to prove all the gaps that you generally assume to be true."
    },
    {
        "video_name": "w70af5Ou70M",
        "Q": "At 6:00, if 0 < |x-c| < delta, shouldn't | f(x) - L | < epsilon be 0 < | f(x) - L | < epsilon ?",
        "A": "No, the definition of limit allows f(x) - L to be equal to 0 at some points in the relevant range. This can happen with trig functions that oscillate above and below L, but with diminishing oscillations so that the function approaches L."
    },
    {
        "video_name": "w70af5Ou70M",
        "Q": "at 5:13 you mentioned that x is within delta of c, what does this statement means like it would be helpful if you can give an example. Thanks.",
        "A": "Example: Say Delta = 1, and C = 2. The x-values will be between 1 and 3 (C-Delta, 2-1, and C+Delta, 2+1 )."
    },
    {
        "video_name": "w70af5Ou70M",
        "Q": "At 3:32 Sal says \"But I can go even narrower than that boundary\" What does he mean by that?",
        "A": "Exactly as it means, because you need to find the values of c so that f(c) is within epsilon. So he can only go smaller from the boundary c+- delta, but once he goes more than that f(c) is outside epsilon then he lost the game."
    },
    {
        "video_name": "w70af5Ou70M",
        "Q": "Between 5:24 - 5:55 does he take the absolute value of x - c and f(x)-L\nIf Yes, why?\nThank you",
        "A": "Because |x-c| and |c-x| are the same, so it doesn t matter whether c or x is taller, i.e the answer is positive, regardless of order. For example, 5-3 and 3-5 are not same. But |5-3| and |3-5| are same, so you don t worry about order, which is a big problem in formulas."
    },
    {
        "video_name": "w70af5Ou70M",
        "Q": "at 1:22, did sal mean to say \"that says you want f of x to be within 0.01 of L\"?",
        "A": "You re right. If you turn on youtube-annotations it also says so. but hasn t been added to KA own s annotations system. Maybe post it under clarifications, so someone will add it eventually (I was also confused about this before accidentally turning on annotations). (If you don t post it I might post in the next few days if I don t forget about it.)"
    },
    {
        "video_name": "CJrVOf_3dN0",
        "Q": "7:28 Does SSS apply to non-euclidean geometry?",
        "A": "For non-Euclidean geometries, SAS, SSS, ASA, AAS all apply to triangle congruency. However, many of the other Euclidean postulates don t apply (especially those involving parallel lines and the sums of angles)."
    },
    {
        "video_name": "CJrVOf_3dN0",
        "Q": "At 3:34, what is the difference between axioms and postulates?",
        "A": "An axiom is really something that appears to be so obvious that it is used as a starting point in a theorem. It s often something that s given. On the other hand, a postulate is something that s suggested as true. It s an assumption made on the basis of reasoning from a particular argument rather than being totally self evident like an axiom. It can also be something that s merely based on a convention or definition. The thing that may be confusing is that we use terms like axiom and postulate interchangeably."
    },
    {
        "video_name": "CJrVOf_3dN0",
        "Q": "Is the axiom and postulate (6:35) similar to proofs?",
        "A": "Similar. An axiom is a core belief that we all agree to start with. These are often VERY fundamental, like a point takes up no space . A postulate is something that is more complicated ( such as A line has exactly one parallel through a non-colinear point ), but cannot be proven."
    },
    {
        "video_name": "CJrVOf_3dN0",
        "Q": "On 4:25 why did he say Angle BAC instead of angle ABC?",
        "A": "Angle BAC refers to the angle at vertex A. Angle ABC refers to the angle at vertex B. The middle letter is the letter that tells you at which vertex an angle is located."
    },
    {
        "video_name": "CJrVOf_3dN0",
        "Q": "At 08:08 were the side's actually congruent? cause i saw that one side had a smaller\nlength than the other",
        "A": "oh, thank you"
    },
    {
        "video_name": "CJrVOf_3dN0",
        "Q": "Around 6:30, Sal said that there was a slight difference between an axiom and a postulate, but I didn't really understand the difference. Can someone explain it to me?",
        "A": "There really isn t one beyond that fact that by little more than historical accident some corners of mathematics prefer one term over the other."
    },
    {
        "video_name": "CJrVOf_3dN0",
        "Q": "Um on 2:08 when u wrote AB = XY, weren't u suppose to have that curly thing on top of the equal sign?",
        "A": "thx so u only use the equal sign with curly thing on top to show that two triangles are congruent, not angles and sides?"
    },
    {
        "video_name": "CJrVOf_3dN0",
        "Q": "at 4:51 could you also call the angle just A because its not divided into to other angles or just to be sure of what is being referred to you'd call it angle BAC",
        "A": "When Mr. Sal was originally talking about the angle A (before creating the statement) he said that< A was congruent to < X. You could keep them like this ( what you were stating above) but we are dealing with triangles. As far as I have learned, there is no reason other than its like proper Geometry english ."
    },
    {
        "video_name": "CJrVOf_3dN0",
        "Q": "Like Sal said at 4:54. \"If line segments are congruent that just means they are the same size\". Going by what he said, does that mean that all lines that keep getting bigger infinitively are congruent too?",
        "A": "I have never seen the topic arise, but according to a strict definition of congruence, you are correct. What congruence actually implies is that you could apply a sequence of rigid transformations to one of the shapes, and it would coincide with the other. This certainly applies to lines. Good insight."
    },
    {
        "video_name": "0njioQqIxKY",
        "Q": "At 0:38, what is 3/4 of an hour, I kind of forgot.....",
        "A": "Well a hour has 60 minutes right? Thus 3/4 of 60 = 60*3 / 4 = 180/4 = 45 minutes!"
    },
    {
        "video_name": "17st-s5gg10",
        "Q": "At 1:14 if the question asked is the sum of factors then why are we multiplying instead of adding 2 cube, 3 cube & 5 cube?",
        "A": "Because Sal is first of all figuring out the prime factorization of 27000, which will help to work out which are the positive divisors."
    },
    {
        "video_name": "rSadG6EtJmY",
        "Q": "At 1:43 How can you identify k if x is not 1?",
        "A": "For example, if x is 2, you divide both side by 2. Then the 2 is gone from the right side and when you divide 2 from the left side (2.25) you get 1.125. That is the value of k. I hope it helps."
    },
    {
        "video_name": "rSadG6EtJmY",
        "Q": "i dont get the part on 3:33",
        "A": "The part where he shows the answer is 8? 18/2.25 = 8."
    },
    {
        "video_name": "99v51U3HSCU",
        "Q": "at 7:16, what are those transformations called? Vertical compression, vertical stretch, horizontal compression, horizontal stretch?",
        "A": "They are called vertical/horizantal stretches/shrinks. In the case at 7:16 with A(x-h)^2 =y-k it is a vertical streatch/shrink."
    },
    {
        "video_name": "99v51U3HSCU",
        "Q": "hi! this may be a very basic question, but at 0:58, how do you determine how high up on the graph the K will be?",
        "A": "He just chose an arbitrary position. The actual value of k is not important."
    },
    {
        "video_name": "99v51U3HSCU",
        "Q": "I SERIOUSLY DON\"T GET THIS!! at 1:15, how is x squared directly just ABOVE IT? and why is the y value positive k when it is y MINUS k = x squared! i seriously don't get this.\ncan someone please explain it to me? thanks",
        "A": "Well - for the yellow curve 1.) y=x^2 so the y intercept is when x = 0......so if x=0 then y=0 and you start at the origin. For the blue curve 2.) y-k=x^2 and again you are looking for the y intercept when x = 0 plug in 0 for x in equation 2.) y-k=x^2 and you get y-k=0. To solve, add k to both sides to cancel it from the left side and you end up with y=k when x=0.....voila!"
    },
    {
        "video_name": "99v51U3HSCU",
        "Q": "At 2:48, Is it possible to find the slope of a parabola?",
        "A": "No, quadratic forms do not have slopes they dont increase at a constant rate like linear function."
    },
    {
        "video_name": "w56Vuf9tHfA",
        "Q": "At 5:20, How did you get 5 3/4? Does it has to do something with 23/4?",
        "A": "Yes: 5 \u00e2\u0094\u008c\u00e2\u0094\u0080\u00e2\u0094\u0080 4\u00e2\u0094\u008223 -20 \u00e2\u0094\u0080\u00e2\u0094\u0080 R:3 23/4 = 5 + 3/4 or 5 3/4"
    },
    {
        "video_name": "w56Vuf9tHfA",
        "Q": "At 8:54 why isn't the reciprical of - 1/3 -3?",
        "A": "He corrects himself at 9:10."
    },
    {
        "video_name": "w56Vuf9tHfA",
        "Q": "at 8:37 Sal says \"we can take the reciprocal of both sides\"\n\nI don't remember seeing any material about that.\n\nIs there a video on it? if so could someone please link me?\n\nMakes sense anyway, I'd just like to cover it if there's a Khan section on it somewhere.\n\n\nSo if 1/2x = 1/6 then 2x = 6\n\nand if 2/19y = 5/23 then 19y/2 = 23/5",
        "A": "This is based on the concepts of proportions. A proportion written as: a/b = c/d can be changed into an equivalent proportion of b/a = d/c. You can search for the lessons on proportions. Or, I think you will find them in the PreAlgebra subject area."
    },
    {
        "video_name": "w56Vuf9tHfA",
        "Q": "At 9:00, shouldn't he have flipped the -1/3 to -3, not 3?",
        "A": "Yes, he should have. Ultimately, in this calculation, it s irrelevant, as in the next step he takes the absolute value of |b-k| (which would get rid of the negative sign), but it s careless. I am copying this and the above comment to the Clarification tab."
    },
    {
        "video_name": "w56Vuf9tHfA",
        "Q": "At 5:33, I don't understand how it is a downward opening parabola?\nHow are you supposed to know whether the parabola is upward or downward?",
        "A": "If the leading coefficient is negative, then the parabola opens downwards. If the coefficient is positive, then it opens upwards."
    },
    {
        "video_name": "w56Vuf9tHfA",
        "Q": "At 4:28, how did Sal determine that it was a downward opening parabola. After all, only the vertex was known.",
        "A": "The coefficient was negative. In this case, -1/3."
    },
    {
        "video_name": "E444KhRcWSk",
        "Q": "At 0:52 Sal says that knowing the equation is separable is a big clue. Are there any easy ways to tell beforehand if a differential equation is separable?",
        "A": "Not really, you have to algebraically manipulate the equation to try and put anything that has to do with x on one side, and everything that has to do with y on the other; if you manage to do that then it s separable, otherwise it s not."
    },
    {
        "video_name": "E444KhRcWSk",
        "Q": "At 1:38, why can't you divide by y^2 instead of 2^y^2, this would leave me with only dy/y^2=2^dx?",
        "A": "You can solve this problem either way. You can divide by y^2 or 2^y^2, it doesn t matter.... You ll still arrive at C = 1 and get the same answer. Hope this helps."
    },
    {
        "video_name": "wQ34EIfd-5A",
        "Q": "@1:13 Sal says you can just 'average' the co-ordinates to get the midpoint.... I'm not entirely following why this is true in 3 directions for a triangle? Does it just follow because it works in 2 dimensions?",
        "A": "The thing is that triangles have many centres (ortocentre, circumcentre, baricentre, etc.). One of those centres can be calculated by averaging the coordinates of the points (the baricentre if I remember correctly). The interesting thing is that all centres are the same point for equilateral triangles, which is what we are dealing with in this problem. So calculating the baricentre through averaging the points gives us the coordinates of the centre of the triangle."
    },
    {
        "video_name": "wQ34EIfd-5A",
        "Q": "Near 2:30 Sal mentions that he needs to find the distance of the two centers of the small tetrahedron for a sidelength. How, however, would it be equal to squaring the difference of both x, y and z coordinates? I don't think that forms a right triangle, and that's when you need squaring.\nSo what I mean is: To find the distance, why square the difference of sqrt 3 over nine and 3 sqrt 3 over nine? That's only done in the Pythagorean Theorem.",
        "A": "Pythagorean Theorem works in 3D too!"
    },
    {
        "video_name": "SeQeYVJZ2gE",
        "Q": "At 18:20 Sal says that the interval would get smaller if we took more samples because the n of 250 would increase. Wouldn't that be the effect of taking a larger single sample size rather than more samples?",
        "A": "He meant get a larger sample ."
    },
    {
        "video_name": "SeQeYVJZ2gE",
        "Q": "Around time 9:30. Why don't you use 3 standard deviations to find confidence interval? Why do you use the z-table?",
        "A": "It s overkill. 3\u00cf\u0083 gives a 99.7% confidence level, but the question only required a 99% confidence interval (CI). And other situations might only require a 90% CI, for example. Knowing the 68.3-95.4-99.7 rule is useful, but it s not sufficient for all circumstances, and that s when z-score tables come in handy. Also, it only applies to large sample numbers. Smaller samples use different multipliers for calculating confidence intervals, but that s another (the next?) video."
    },
    {
        "video_name": "SeQeYVJZ2gE",
        "Q": "at 15:08 why do we multiply 2.58 with 0.031?",
        "A": "where did we get that the SD is 2.2 above the mean ? where did u get this from ?"
    },
    {
        "video_name": "SeQeYVJZ2gE",
        "Q": "At 10:35 how is 0.495 calculated? I mean, it is obvious that 0.500 - 0.005 = 0.495 (since the normal distribution curve is symmetrical), but where is 0.005 coming from? Is it related with the emprical rule? Thank u all:)",
        "A": "What Khan failed to mention is that the Formula for Z (Table) is: Z(1-Alfa/2) Alfa = 1 - Confidence Interval = 1 - 0.99 = 0.01 Which gives us: Z(1 - 0.01/2) = Z(1 - 0.005) = Z(0.995) Hope I could help!"
    },
    {
        "video_name": "SeQeYVJZ2gE",
        "Q": "0:54 the number of teachers that thought it was a good idea was given a value of 1 and those that didn't a 0. Would it matter if those values were swapped?",
        "A": "The mean is .432 rather than .568. So 108 differences are 1-.432 and 142 of them are .432, same as before. So SD is the same. Therefore everything stays the same except the sample mean."
    },
    {
        "video_name": "SeQeYVJZ2gE",
        "Q": "At 18:10 he says you should take more samples. Does he mean take a sample from a greater number of teachers? For example 260 instead of 250? because if you take more samples of 250 teachers that will not make the standard deviation of the sample means smaller, right?",
        "A": "Yep, he is talking exactly about increasing the sample size, n."
    },
    {
        "video_name": "SeQeYVJZ2gE",
        "Q": "18:09\nre part 2 of the question, to narrow the confidence interval do you take more samples or do you increase the size of the sample? The formulas look like you increase the size of the sample (from 250) but the voiceover says take more samples. Is someone able to see where I'm getting it wrong?",
        "A": "Sometimes people use the word sample to mean one observation, other times to mean a group (in this problem, 250) of observations. I think Sal sometimes does this. So take more samples can therefore sometimes mean increase the size of the sample . I think Sal meant increase the size of the sample . Increasing the sample size will decrease SEM."
    },
    {
        "video_name": "XJBwJjP2_hM",
        "Q": "In 2:17 he changes 10^8 into 10^-8. How does this work. Wont that change the outcome? Please explain.",
        "A": "So, look carefully between 2:05 and 2:20 - in first case you have a fraction (10^13)/(10^8), where 10^8 is under the fraction; afterwards, Sal transformed the division into multiplication, where the number under the fraction is with a negative exponent, tha s how 10^13 divided by 10^8 becomes 10^13 multiplied by 10^-8;"
    },
    {
        "video_name": "XJBwJjP2_hM",
        "Q": "At 3:20, the result of the first four digits of the problem on the calculator was 0.3978, but he kept on saying it was 0.3979 through out the video. So is this just a typo?",
        "A": "But looking at the next number, 0.39786, since 6 is bigger than 5, it is appropriate to round up to 0.3979"
    },
    {
        "video_name": "XJBwJjP2_hM",
        "Q": "At around 2:18, what would you do if you had to divide 10 to the power of 6 by 10 to the negative first?",
        "A": "Just do the opposite: Instead of multiplying them and turning the positive exponent negative, make it positive. 10^6 times 10^+1 (or just 10) Another way of thinking about it is do the same thing as shown in the video, and since negatives cancel out, the 10^-1 is now just 10 when multiplying to 10^6 Hope that helped :)"
    },
    {
        "video_name": "XJBwJjP2_hM",
        "Q": "At around 3:21, why did he round it to .3979? Why didnt he just leave it at .3978?",
        "A": "Since the directions said round your answer to four decimal places. When there is something like that in the directions you round the that place up or down. For example if the number was .2548757 and directions said round the answer to the third decimal place the answer would be .255 since in .2548 the 8 is equal to or greater then 5. Hope that helps."
    },
    {
        "video_name": "XJBwJjP2_hM",
        "Q": "I often wonder about when is the correct time to round a number during a calculation. At 3:50 you rounded to the fourth decimal place. Should you wait to round until you have moved the decimal for the last time? Making it 3.9786*10 to the 4th?\nI have gotten the answer wrong by not rounding during the problem, but it is less accurate too.\nAm I correct in rounding after, or should it occur during the problem calculation?",
        "A": "You should estimate (round) an answer only at the very end, not before that. Although, for most of the calculations, rounding up to the 6th or 7th place is enough to still get a precise answer at the end. I d just like to make it clear that this is NOT a fact, this is only my personal opinion based on thousands of exercises I ve done."
    },
    {
        "video_name": "XJBwJjP2_hM",
        "Q": "on 2:32 i don't understand why he made 10^-8",
        "A": "Negative exponents are fractions so 1/10^8 = 10^-8. There is always a one over a number but people don t write it since it makes no difference."
    },
    {
        "video_name": "XJBwJjP2_hM",
        "Q": "When Sal says on 2:15 he says multiply but, at the end he subtracted how is that?",
        "A": "See, he multiplied 10^-8 there, and product of two numbers of same base but different power can be calculated by adding the powers, and he added 13 and -8 which is 5. Hope you got it."
    },
    {
        "video_name": "XJBwJjP2_hM",
        "Q": "Around 2:36 Sal said 10^13/10^8 = 10^5 , What if it was 10^-13/10^8 or 10^13/10^-8.. Do I subtract regularly like 13-(-8) or -13-8?",
        "A": "10^-13/10^8 = 10^-13 * 10^-8 = 10^-21 10^13/10^-8 = 10^13 * 10^8 = 10^21"
    },
    {
        "video_name": "XJBwJjP2_hM",
        "Q": "in 3:55 he says that 0.3979 x 10 to the fifth power is scientific notation but i thought that scientific notation was sopposed to be a number between 1 and 10",
        "A": "Actually, he says at about 3:50 that it looks like scientific notation at first glance and goes on to say at about 3:55 that it isn t actually. You are correct that it wasn t scientific notation. Sal then converts it to scientific notation, however."
    },
    {
        "video_name": "XJBwJjP2_hM",
        "Q": "At about 4:16, why can't you just leave the 0.3979 X 10^5 for scientific notation??Why do you have to change it to 0.3979 X 10 X 10^4?",
        "A": "standard form (you can call it scientific notation) converts numbers into: (number from 1 to 10, 10 not inclusive) * 10^x if you leave it as 0.3979 * 10^5, it s not in standard form.. you have to change it to be able to convert it to standard form.. =3.979*10^4 hope that helped"
    },
    {
        "video_name": "XJBwJjP2_hM",
        "Q": "At 5:10, does everyone really owe that much or do some people owe more than others?",
        "A": "Our national debt is not directly allocated to individual people. You will sometimes seem problems like the one in the video that show the average amount every person would owe to pay of the debt. This gives you a sense of how large the debt is. In reality, our federal income tax is used to pay for federal expenses. Individuals and companies pay income tax. And, the rates are based on income levels. So, the average cost per person does not directly align to how taxes are paid."
    },
    {
        "video_name": "R2JsjJyr0ck",
        "Q": "At 4:30 why is -1 in the numerator?",
        "A": "It follows from the power rule. Recall that if we define a function \u00c6\u0092: R \\ {0} -> R by \u00c6\u0092(x) = x^(-1), then the derivative of \u00c6\u0092 at x \u00e2\u0089\u00a0 0 is given by (-1) \u00c2\u00b7 x^(-2). Now apply this to the function given by \u00c6\u0092(x - 1)."
    },
    {
        "video_name": "tf4C8x8e7HQ",
        "Q": "At 3:36 and at other points Sal says that 1-x is the radius. Wouldn't it be only half the radius?",
        "A": "The triangle is the base of semi circle. That s why thee y value become the diameter not r."
    },
    {
        "video_name": "ViFLPsLTO1k",
        "Q": "in 1:40 he says times .25 but it is actually is 4.25",
        "A": "No, he meant 0.25. 4.25 would be the volume of the whole container, not the ring."
    },
    {
        "video_name": "ViFLPsLTO1k",
        "Q": "at 0:20 he said the water level changed how?",
        "A": "Ring has some volume and water also has volume water volume + ring volume would lead to increase in volume. volume is l*b*h since here in glass the base measurements can not be changed. so level of water increases because volume has increased."
    },
    {
        "video_name": "ViFLPsLTO1k",
        "Q": "Why did he say that at 1:44",
        "A": "Since the difference of the heights of both containers of water is 0.25, really he is finding the area of the gold ring indirectly by using l*w*h and then you have the VOLUME! Hope this clarifies it!"
    },
    {
        "video_name": "oAQnDsmu41E",
        "Q": "At 1:54, why is the inequality -1/(x\u00c2\u00b2-1) <= cosx/(x\u00c2\u00b2-1) <= 1/(x\u00c2\u00b2-1) true for all x?\nFor example, if x = 0, then -1/((0)\u00c2\u00b2-1) <= cos(0)/((0)\u00c2\u00b2-1) <= 1/((0)\u00c2\u00b2-1) becomes 1 <= -1 <= -1, which is false, right?",
        "A": "You re absolutely right. It seems like a mistake on Sal s part. I think it should have been something like 0 \u00e2\u0089\u00a4 | cos(x)/(x^2 -1) | \u00e2\u0089\u00a4 | 1/(x^2 -1) |. Good catch."
    },
    {
        "video_name": "8EmhGOQ-DNQ",
        "Q": "At 2:26 what did she mean by prime factorization? I know what it means, but how is the prime factorization related to the folding?",
        "A": "The prime factorization lets you fold in parts, instead of all at once. For example, to make 12ths (2*2*3) you can fold in half (2) then thirds (3) then half again (2) 2*2*3 = 12 layers of paper."
    },
    {
        "video_name": "8EmhGOQ-DNQ",
        "Q": "What did the last part at 3:58 mean? Why was she making that strip?",
        "A": "that was a mobius strip. it is a ring with a half twist. it has the unique property of having only one side."
    },
    {
        "video_name": "6o7b9yyhH7k",
        "Q": "At 6:37 why did Sal take y2=e^x?",
        "A": "He is telling us that this y2 = e^x is a solution to the differential equation: y + 2y = 3y. If the choice of e^x felt arbitrary, just know in the future, there are methods to calculating these solutions. I encourage you to verify that e^x is a solution to y + 2y = 3y."
    },
    {
        "video_name": "6o7b9yyhH7k",
        "Q": "At around 0:47 Sal writes down f\"(x) + 2f'(x). However in the equation before it says y\"+2y'. I don't really understand how this can be a function that represents the previous equation. Thanks!",
        "A": "If you recall, we can define a simple polynomial as y=x^2. Another way of writing this polynomial is f(x)=x^2, meaning that x^2 is a function with respect to x. Therefore, y=f(x), and higher derivatives such as f (x) = y and f (x) = y . Hope this helps!"
    },
    {
        "video_name": "6o7b9yyhH7k",
        "Q": "At 4:10 why did Sal took y(x)=e^-3x\nand after that how he took y'(x)=-3e^-3x at 4:50 an so on?",
        "A": "He is checking to see if y(x)=e^-3x is a potential solution for the equation y +2y =3y. To do that he needs to take the derivative of y(x), which is -3e^(-3x)=y and then y(x)=y."
    },
    {
        "video_name": "6o7b9yyhH7k",
        "Q": "At 1:05 we see d^2 y / d x^2. Where is that x^2 coming from?",
        "A": "This is exactly the same question as the one rappy3 asked 3 years ago. It was well answered Yamanqui."
    },
    {
        "video_name": "6o7b9yyhH7k",
        "Q": "so how do you come up with e^-3x (it just appeared from no wear) at 4:20",
        "A": "i think he just had that function to begin with knowing that it worked in that differential equation to demonstrate how you can plug in a function back into the differential equation to prove it would work"
    },
    {
        "video_name": "6o7b9yyhH7k",
        "Q": "At 1:05 we see d^2 y / d x^2. Where is that x^2 coming from?",
        "A": "It s part of the symbol for the second derivative, it means The second derivative of y with respect to x, since you are differentiating twice over x, then you put a x\u00c2\u00b2 in the denominator."
    },
    {
        "video_name": "_G9JwTwjulA",
        "Q": "At 3:36, Sal uses the arc length formula with 1 + dy^2 whereas in the previous video for #1d, he used dy^2 + dx^2 at the end. What determines which to use?",
        "A": "Nevermind. If anyone wants the answer, when it is parametric or x and y each receive their own equation, use this equation and for f(x)=y, use the 1 +..."
    },
    {
        "video_name": "FICRd7Lp67s",
        "Q": "Hi! In 0:17, The question is verbally asked the probability of the dice adding to ten. If I wanted to write the question in the P(n) form, could I use what Sal mentioned in a previous video (about sample and mean population formulas), the Sigma sign? Could I say P(sigma 10)=1/24?",
        "A": "Yes. Or you could do something like let D1 be dice one, D2 be dice two. P(D1 + D2 = 10) = 1/24. They both get the point across."
    },
    {
        "video_name": "FICRd7Lp67s",
        "Q": "why do you just not count the number at the top instead of counting everything \"2:56\"",
        "A": "Because probability is a ratio. It is always a certain number of chances out of a bigger pool. Like 1 in 4. It s the same as 1/4."
    },
    {
        "video_name": "FICRd7Lp67s",
        "Q": "At around 0:13 (I have captions, not sound) is it supposed to be a pyramid with a square base?",
        "A": "No, a pyramid with a square base would have 5 sides not 4. It would have to be a pyramid with a triangular base."
    },
    {
        "video_name": "FICRd7Lp67s",
        "Q": "at 1:39 why did you count the number of possible outcomes when at the top it tells you how many possibilities there are ?",
        "A": "That is probably how he did it but he wanted us to know how to in case it does not show it at some point in life."
    },
    {
        "video_name": "FICRd7Lp67s",
        "Q": "isn't it double counting 5:1 and 1:5 ?\nI mean if this problem was phrased in terms of rng's wouldn't it be wrong to count both results as a individual solution?",
        "A": "well, it depends on whether or not you are rolling different dice and whether it matters. If you are rolling a black and a white die, then if you get first 5:1 then 1:5, it would be different, but if you are rolling the same color die and it doesn t matter, then 5:1 would equal 1:5"
    },
    {
        "video_name": "UKYjhM_c_7s",
        "Q": "Is it basically 2+2? I got confused after 1:15.",
        "A": "Yes, it s 2+2. We know that Diya is 2 years old and Imran is 2 years older than Diya. Therefore, we just need to add 2 years to Diya s age (2 years old) to find Imran s."
    },
    {
        "video_name": "4ve7OaCnDLs",
        "Q": "at 1:20 you said lets do 20 x 7.5 but then in the working out you did 7.5 x 20. Why did it change? Why would you do 7.5 x 20? Should the answer be the same despite the order of the numbers?",
        "A": "When you multiply numbers, it does not matter what order you multiply them in. 20 x 7.5 will be the same as 7.5 x 20."
    },
    {
        "video_name": "4ve7OaCnDLs",
        "Q": "At 1:04, does a negative and a positive equal a positive and does a positive and a negative equal a negative?",
        "A": "Nice question, but if you were planning to add them, a postive plus a negative is either a negative or a postive, let s say -3+5; you would see if you had more negatives or postives. I use the team method, since the number five is greater in value, the answer HAS to be positive, just subtract 5 and 3 and you get 2. Subtracting will get you like this: -3-5= -8, or 3--5= 8 ( If you subtract a negative number you will always get a positive.)"
    },
    {
        "video_name": "4ve7OaCnDLs",
        "Q": "I can only understand the lattice version of multiplication. I can't tell what Sal was doing at 1:28.",
        "A": "It s a way of multiplying two digit or multiple digit numbers. I m pretty sure you can find a video on how to use that method in another topic, most likely Multiplication."
    },
    {
        "video_name": "4ve7OaCnDLs",
        "Q": "at 2:20 why cant a positive and another positive cancel out?",
        "A": "a positive and another positive does not cancel out because with negatives their well, negative. on the number line negative is to the left of the number line, and positive is the right of the number line. with negatives -5x-3=15, and 5x3=15 as well, thats because this: so you have -5 -3 times, which cancels out because of the second negative, two positives can t cancel out because it s on the right side of the number line i hope this helps ;)"
    },
    {
        "video_name": "YS_3XRj74oo",
        "Q": "What would the \"one positive, one negative\" rule from 2:30 be like with an equation that goes up to x^3? Or x^4 or above?",
        "A": "For x^3, it would be two positives, one negative or three negatives to keep the whole expression negative. For x^4, it s three positives, one negative or three negatives, one positive . Hope I m not missing anything else here."
    },
    {
        "video_name": "bZYTDst1MOo",
        "Q": "At 6:24, on the graph what if the discontinuity occurs in the middle of the function - meaning that the function is discontinuous just at a point between the interval a and b?\nWe can still find the absolute maximum and minimum values, right ?",
        "A": "yes, assuming the discontinuity is removable(hole) and not a jump discontinuity."
    },
    {
        "video_name": "bZYTDst1MOo",
        "Q": "If the absolute minimum is at \"c\", how is the minimum point at \"a\" as stated at 4:10? Ditto with \"b\"",
        "A": "a is actually a maximum and b is a relative minimum not an absolute minimum. :)"
    },
    {
        "video_name": "bZYTDst1MOo",
        "Q": "At 5:30 , the graph that sal drew, if it was continuous but the interval was not closed [thus (a,b)] would the function had any maxima or minima? I dont think then we had any problem finding the extreme points... because the interval is not closed but who cares, it's not like that the maxima and minima are at those interval points (like sal drew in the last graph explaining why the interval needed to be closed).",
        "A": "The whole point of this video is to show why the Extreme Value theorem requires closure at both endpoints [a,b] and continuity. The video is very clear and a good explanation why those two elements are required. Watch the video again, closely. At 5:30, Sal is proving the continuity requirement, not the endpoint requirement. He proves the endpoint requirement in the next graph."
    },
    {
        "video_name": "bZYTDst1MOo",
        "Q": "At 0:43, Sal says that there will be an absolute maximum and minimum for f(x) given this closed interval. Now what happens if the given function f(x) is a horizontal line? Does that mean that all of the points are absolute extrema?",
        "A": "Yes. Every point on the interval is both a maximum and a minimum."
    },
    {
        "video_name": "bZYTDst1MOo",
        "Q": "What is the meaning of the symbol that Sal writes at 3:03 right after d?",
        "A": "If you are referring to the symbol that looks a bit like this: \u00cf\u00b5, then it means is a member of a set. For example A \u00cf\u00b5 { A , B , C }. Sal explains: 2:52 could say there exists a c and d that are in the interval. 3:02 So they re members of the set that 3:04 are in the interval such that-- and I m just 3:09 using the logical notation here."
    },
    {
        "video_name": "bZYTDst1MOo",
        "Q": "When he says \"there are an\" at 1:11 he implies that there are only one maximum value. That isn't true because you can have multiple values of maximum and minimum. right?",
        "A": "Whereas there can be multiple relative extrema, there can be at most one absolute maximum and at most one absolute minimum."
    },
    {
        "video_name": "bZYTDst1MOo",
        "Q": "if there was more than one x point that results in an absolute maximum, minimum point, how would i represent that in the \"mathy\" notation provided in 2:55?",
        "A": "There exist(s) does not rule out the possibility that there exist more than one such point. Therefore, the notation presented there is valid in the case you specify as well. Note that one may use the symbol \u00e2\u0088\u0080 instead of for all / for every / for each , allowing us to write the even more compact: \u00e2\u0088\u0083 c, d \u00e2\u0088\u0088 [a, b]: \u00e2\u0088\u0080x \u00e2\u0088\u0088 [a, b] \u00c6\u0092(c) \u00e2\u0089\u00a4 \u00c6\u0092(x) \u00e2\u0089\u00a4 \u00c6\u0092(d). Edit Another way to put it: \u00e2\u0088\u0083 c, d \u00e2\u0088\u0088 [a, b]: \u00c6\u0092(c) = inf \u00c6\u0092([a, b]) \u00e2\u0088\u00a7 \u00c6\u0092(d) = sup \u00c6\u0092([a, b])."
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "at 10:41 where did the -1 go ?",
        "A": "the -1 gets squared: (-1)^2 = (-1)(-1) = +1 Hope this helps."
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "At 10:34 when he says to multiply the exponents you don't use the squared that is in with the -1?",
        "A": "Yes, you do...and -1^2 = 1"
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "At 11:04 Sal spoke so fast, he is so good at algebra. But could anyone rephrase what he said!",
        "A": "HE basically says that he can remake the problem using 3 times x squared times Y squared and rearranges it to where its easier to simplify."
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "What is the name of the exponent property Sal mentioned at 3:08?",
        "A": "It is called the distributive property of exponents."
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "At 5:15 Khan says you can add the exponents if the base is the same, does this work with negative exponents as well?",
        "A": "Most definitely. Remember that because a\u00e2\u0081\u00bb\u00c2\u00b2 is equivalent to 1 divided by a\u00e2\u0081\u00bb\u00c2\u00b2, a\u00c2\u00b3 \u00c3\u0097 a\u00e2\u0081\u00bb\u00c2\u00b2 = a \u00c3\u0097 a \u00c3\u0097 a \u00c3\u00b7 a \u00c3\u00b7 a = a\u00c2\u00b9 Two pairs of a \u00c3\u00b7 a cancel out, which can be translated to adding -2 to the exponent 3 to get the exponent of 1."
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "What did Sal mean at 3:10 when he said that we stumbled upon an exponent property/?? So confusing!",
        "A": "Sal is referring to the fact that ab^x = a^x +b^x. Look at this example: (2*3)^3=2^3 + 3^3 because: 2^3 + 3^3 = 2*2*2*3*3*3 =216 and (2*3)^3 = 6^3 = 6*6*6 =216. Thus, ab^x = a^x +b^x."
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "13:10 So if any number to the zero power is 1, is the 1 that comes from that equation to the power of 1 now? (Example: 8^0=1=1^1?)",
        "A": "That is true. 1^0 = 7^0"
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "at 8:09 can somebody rework that please",
        "A": "I ll try, but I don t know exactly which problem you need re-worked. If I have, (Thinking of number).. if I have (7^3) \u00e2\u009c\u0096 (7^5) that equals 7^3+5= 7^8. Let me know if that clears things up for you, and if not, I ll help you work through another. Hope this helps! - Tempest"
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "At 5:10, Sal references \"the same base\". Is there a video about exponential bases? I can't seem to find one.",
        "A": "Bases (base 10 and base e ) are briefly touched on in passing in some of the videos in the Logarithm section, but I have a hunch that s not what you re asking about. Basically what he is saying is that there are certain rules you can follow when the two base numbers you are multiplying are the same, regardless of their exponent. The computation 5^3 x 5^2 has a common base of 5. However, 5^3 x 3^2 does not. Because it does not, you cannot follow the rule and add the two exponents together and get your answer."
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "At 10:39 why is the one taken out of the equation.\nHe said -1 squared is one and then the one disappeared.",
        "A": "The 1 disappears because any number multiplied by 1 is equal to itself. So Sal could have written it as 1*(x^4*y^2) but that s the same as just (x^4*y^2)"
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "At 1:24, why is Sal using a . instead of a multiplication sign?",
        "A": "A dot is the more commonly used symbol for multiplication as you get to complex algebra. Sometime X is mistaked as a variable, which is why."
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "AT 10:00 where did the -1 come from I read somebody's comment on it but it made no sense to me. They said, \"The negative one is there because Sal made the original terms -x^2. Negative x^2 is the same as -1 * x^2. How is it the same they didn't explain how it is the same. It's \"seems\" -1 popped up out of no where. Thank you!",
        "A": "OK, to make it simple forget about the variables. Lets say you had -8. And someone just rewrote it as -1 times 8. it equals -8. so -1 times 8 is the same thing as -8. Does that help?"
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "After Sal finishes simplifying the second bracket at around 10:25 how come he doesn't put a bracket around y^2 but one around (-1)^2?",
        "A": "The parentheses are needed around the -1 to show that the exponent is applied to the minus sign as well as the 1. (-1)^2 = (-1)(-1) = +1 If it was written as: -1^2, you will get a different result/meaning. -1^2 = -(1*1) = -1 Since the y is a single symbol, parentheses are not needed. Hope this helps."
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "At 10:26, why do you put an -1",
        "A": "he was distributing the power of 2 to all three parts of the expression inside the parentheses (-1)(x^2) and (y)"
    },
    {
        "video_name": "zM_p7tfWvLU",
        "Q": "At 3:03, do you have to use parentheses or could you just write that 3x to the third power?",
        "A": "If you write: 3x^3, then the exponent applies only to the x If you want the exponent to apply to both the 3 and the x , you need to use the parentheses: (3x)^3 Hope this helps."
    },
    {
        "video_name": "jCwRV1QL_Xs",
        "Q": "At 2:34 you say if you take all of the x's that satisfy do I have a valid subspace? But subspace of what? of Rn?",
        "A": "Yes, of Rn, because x is a member of Rn."
    },
    {
        "video_name": "jCwRV1QL_Xs",
        "Q": "1:07\nI believe that the first condition, that the subspace contains the zero vector, is unnecessary given the third condition --that the subspace is closed under multiplication by a scalar quantity.",
        "A": "yes. technically, closed under vector addition and scalar multiplication is enough to define a subspace. it s usually mentioned that a subspace must also contain the zero vector because it s usually easy to check this condition and it is a necessary condition."
    },
    {
        "video_name": "jCwRV1QL_Xs",
        "Q": "At 05:28, Let 'a' be any real number, so we know that 0a = 0, for all real number, then the 0 vector will be a member of all sets of vectors and we do not need to check, right?",
        "A": "You re right. There wasn t a need to go through the multiplication to show the zero vector is a member of the null space."
    },
    {
        "video_name": "jCwRV1QL_Xs",
        "Q": "Hi, got a question might sound a bit stupid...\nTo call some set a subspace, mustn't there be a reference? Here, doesn't mean that we are trying to prove if the vector X set is a subspace to the R^n range? Or to the matrix A?\n\nBTW,\nAt 7:15, sal writes: AV1+AV2=0, but it does not mean V1+V2 is closed in vector set X, right? Same how to prove any c times v1 would also be closed in vector X set?\nIf answers are no, how can we say Vector X set is sub space?\nThanks a lot in advance!",
        "A": "a subspace is a subset of a vector space which itself is also a vector space vector spaces are closed by vector addition and scalar multiplication this is by definition"
    },
    {
        "video_name": "jCwRV1QL_Xs",
        "Q": "6:50 should be v_2 not v^2",
        "A": "Yes, you are right."
    },
    {
        "video_name": "uuDQkhx6TAI",
        "Q": "At 4:20, Could you just use the rotation rules to answer these type of questions?\nex. R 270 (x,y) -----> (y, -x)",
        "A": "yes by following the rules you are essentially doing the short cut method"
    },
    {
        "video_name": "uuDQkhx6TAI",
        "Q": "At 0:05, 0:06, or 0:07, what does a 270\u00c2\u00b0 turn mean, and at 1:05, 1:06, 1:07, 1:08, 1:09, or 1:10, why does he say a 270\u00c2\u00b0 turn is equal to a -90\u00c2\u00b0 turn and it's easier to think of it as a -90\u00c2\u00b0 turn when it's a 270\u00c2\u00b0 turn?",
        "A": "The two angles are both pointing in the same direction. There are 360 degrees in a circle so let s think about it as a circle. A +270 degree angle goes counterclockwise and a -90 degree angle goes clockwise. If you draw it out, the two angles actually meet. BUT thing to remember: a +270 degree angle is NOT the same as a -90 degree angle, the +270 degree angle is larger. Except when you re talking about rotating something, it doesn t matter as long as you get to the same end position. Hope this helped :) Happy holidays!!"
    },
    {
        "video_name": "Qst1UVtq8pE",
        "Q": "These examples all start with x=0 or x=1 at 3:18 which are simple enough to determine the value of a from what I can see, but the exercises only have x=2 or x=3. How would one solve g(2)=a.r^2=3 and determine the value of a?",
        "A": "Well you could move r^2 over and say a = g(2)/r^2. Then you determine the value of r and plug it in."
    },
    {
        "video_name": "Qst1UVtq8pE",
        "Q": "At 4:33 Sal raises 2/3 to the x power without parentheses. Wouldn't parentheses be required to indicate that you're raising the whole fraction to the x power?",
        "A": "Yes. You are absolutely correct. He does it right when writing the formula on the answer screen. But at 4:33 he didn t and that made it 2 to the xth power over 3. Not 2 over 3 to the xth power."
    },
    {
        "video_name": "Qst1UVtq8pE",
        "Q": "How would you do the process at 3:05 with an x value that isn't 0?",
        "A": "You can look in the hints of the exercises. Sal has a bad example here because with an x=0 you can take a lot of shortcuts."
    },
    {
        "video_name": "Qst1UVtq8pE",
        "Q": "At 1:25, why change in y is equal to change in f? Could someone explain to me how this could work?",
        "A": "Because the y value for an equation could be replaced by a function call such as f(x). This makes it so that there could be multiple functions at once without the functions ever bothering each other with a single y variable. If there was a y = x^2 + 2x + 2, a y = x - 6, and a y = x^3 - 8; that wouldn t work out so it is better to separate the y. f(x) = x^2 + 2x + 2 g(x) = x - 6 h(x) = x^3 - 8 Neat, don t you think?"
    },
    {
        "video_name": "Qst1UVtq8pE",
        "Q": "at 4:28 sal doesn't put parentheses on 2/3, but when he types the answer in at 4:55 he puts parentheses on 2/3. which one is the correct one? with or without parentheses?",
        "A": "Ultimately they are the same thing. The first one at 4:28, he puts the parentheses around the 2/3^x to imply that we are not doing 3*2/3 then to the power of x. So that we are doing 2/3 to the power of x, then multiplying by 3. The second version, by putting the bracket around 2/3 he is implying that we are doing 2/3 to the power of x, then multiplying by 3. The outcome is the same."
    },
    {
        "video_name": "Qst1UVtq8pE",
        "Q": "why does he do triangles when he says change for example at 1:30",
        "A": "Those are not triangles, they are the Greek capital letter delta, which is the standard algebraic symbol for change in ."
    },
    {
        "video_name": "Yq9W6B7XnmU",
        "Q": "At 3:15, does Sal mean one over three instead of three over one, or is that a improper fraction.",
        "A": "It s an improper fraction, he just wants to multiply like that."
    },
    {
        "video_name": "Yq9W6B7XnmU",
        "Q": "at around 1:00 why doesn't Sal just add the ones and then the fourths",
        "A": "Because it is a mix fraction, don t forget it is not 1/4 but 3/4."
    },
    {
        "video_name": "oIV1zM8qlpk",
        "Q": "At 0:38, he says \"transversals intersecting parallel lines. What does that mean?",
        "A": "Imagine an equal sign being parallel lines, then imagine a line going through both of them, that is what he means"
    },
    {
        "video_name": "oIV1zM8qlpk",
        "Q": "At 0:22, what does transversals mean?",
        "A": "a transversal is a line that goes through parallel lines."
    },
    {
        "video_name": "oIV1zM8qlpk",
        "Q": "I think at 2:17 you made an error. Isn't it suppose to be <EBD is congruent to <BAC, not <CAB? I think you switch it around. If you put <EBC is congruent to <CAB that would be an incorrect statement... Wouldn't it? That's what my teacher told me. 0.o",
        "A": "I think you misunderstood your teacher :). <EBD and <CAB are definitely congruent, they have the same measure. <BAC and <CAB are the same angle, just with a reversed notation."
    },
    {
        "video_name": "oIV1zM8qlpk",
        "Q": "at 0:36 , what does \"Transversal\" mean?\n\nI have heard sai say it, but never knew what it meant.",
        "A": "A transversal, in respect to parallel lines, is a line which intersects the two parallel lines. When Sal refers to the transversals at 0:30, he is referring to on the lines which intersect the pair of parallel lines"
    },
    {
        "video_name": "oIV1zM8qlpk",
        "Q": "During my Geometry class, at around 1:45, I was watching my teacher do a proof similar to this. Anyway, I asked if you could do this without using exterior angles, but he didn't know.\nSo...could you prove opposite angles congruent without using exterior angles?",
        "A": "Yes, you can draw a line through A and D or C and B ( through any 2 points there is exactly one line ) and then prove triangles ABC and CBD congruent by the SSS postulate. After that, you can prove opposite angles are congruent by CPCTC ( corresponding parts of congruent triangles are congruent )."
    },
    {
        "video_name": "oIV1zM8qlpk",
        "Q": "can someone help me he lost me at 1:45",
        "A": "of course i can help (i will just watch it then see if i can help"
    },
    {
        "video_name": "Xp6V_lO1ZKA",
        "Q": "At 4:12 Sal says that the probability of winning is 1/3. Wouldn't it be 1/2 because after they show you the other door you can rule that one out (therefore having a fifty-fifty chance)?",
        "A": "No. If you don t switch, then you don t benefit from him showing you the door. If you picked the right door originally (1/3 chance) then you still have the right door. If you didn t pick the right door originally (2/3 chance) then you still don t have the right door."
    },
    {
        "video_name": "Xp6V_lO1ZKA",
        "Q": "0:11 when was he the host?",
        "A": "Let s Make a Deal was presented by Monty Hall from 1963 to 1986."
    },
    {
        "video_name": "Xp6V_lO1ZKA",
        "Q": "About 5:10 Sal said that \"if you pick wrong, you should always switch.\" But how do you know whether you're wrong or not?",
        "A": "You don t. However, in the initial situation, you have a 2/3 chance of picking wrong. You have a 1/3 chance of picking right. In the case of the 2/3, you should switch, but in the case of the 1/3, you should not switch."
    },
    {
        "video_name": "Xp6V_lO1ZKA",
        "Q": "at 3:07 when they say we are going to expose this goat what do they mean?",
        "A": "The whole problem is about having 3 doors that are closed. When a door is opened, you can see whatever is behind it (in this case, a goat). So, in other words, a goat is exposed."
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "2:05 why did he multiply 2 \u00c3\u00972\u00c3\u00973\u00c3\u00975",
        "A": "Because those are the prime factors of the numbers. When you get down to the prime factors you are able to take out multiple expressions and find out what can be taken out or added to in the equation. This is usually done in square roots or during Greatest Common Factor or Least Common Multiples."
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "At 6:55 , he has a third three from the 24. Why doesn't he use that? I did just a while ago, and 9x24 is 216, and 216 divided by 27 is eight. 0.o",
        "A": "You ve found one example where it is true but it is not true for all numbers divisible by 9 and 24. For example 144 is divisible by 9 and 24. 144/9 = 16 144/24 = 6 But 144 is not divisible by 27. So you include only 2 of the 3 s because that is the maximum found in any of the factorizations."
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "0:01. All numbers divisible by 12 and 20 are also divisible by: ?",
        "A": "2 and 4 (2 squared)"
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "At 7:33 how does 4 work? I thought that for the number to work it had to have the same number of numbers in it as the example Sal gave\ne.g. 9 = 3*3 or 8 = 2*2*2\nPlease explain someone....",
        "A": "4 works because its prime factorization is 2*2. 2*2 is part of 2*2*2*3*3, so all numbers divible by 9 and 24 are also divisible by 4. In other words, we re not looking for an exact match. There is no 3 in 4, but we don t really care, because we just want to know if 9 and 24 combined have at least two 2 s. Maybe they also have lots of numbers in them, but that doesn t matter, we just want two 2 s for 4 to work. Does it help?"
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "I'm confused by those extra numbers at 2:14. If you can add extra numbers onto the prime numbers, why isn't 8 at 4:23 one of the answers? Couldn't you just add another two onto it where the AxBxC is, so long as you have at least some of the prime numbers?",
        "A": "I believe it s assuming that any extra numbers are NOT another 2, or that they are variables (current unknowns). If Sal DID include another 2, or if at least one of the variables turned out to be a 2, then yes, 8 would become an answer."
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "Can someone explain what he did at about 1:40 on?",
        "A": "Commonly in Algebra if it s the same number on the top, and the same number under it, you can cancel out both of them. It s like subtracting/dividing both the top & the bottom by whatever the number is."
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "In 0:17 what do you mean by divisibale?",
        "A": "Able to be divided. 4 is able to be divided by 2. 4 is divisible by 2."
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "At 4:06, why does Sal say that 12 is equal to 2 *2 * 3? Why can't he just say that 12 equals 2 * 6?",
        "A": "He s using price factorizations, and 6 isn t prime. Prime factorizations are often used to avoid any hidden common factors to guarantee finding the LOWEST common multiple of two numbers."
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "So when Sal said 2x2x2x3x5xaxbxc at 2:42 what does he mean exactly?",
        "A": "a, b, and c are random numbers. Sal means that a number divisible by both 12 and 20 is also divisible by any combination of 2, 2, 2, 3 and 5, but not necessarily limited to those. 540 for instance is divisible by 12 and 20, and thus also divisible by any combination of 2, 2, 2, 3 and 5. It is not limited to combinations of those numbers though: 540 is f.i. also divisible by 9, which is not in there."
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "Every time I get to 5:42 in the clip there is a green flash. Were you having technical issues?",
        "A": "That happens to me at 5:47, 5:50, and 5:56."
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "Starting at 2:54, the video indicates that any combination of the prime factors will create a correct answer. One of my questions on the exercises did not work with this idea. The numbers were 18 and 20.\n18 = 2 x 3 x 3\n20 = 2 x 2 x 5\nanswer choices (24,30,124,86,134)\nBoth 24 and 30 can be created by using the prime factors.....\nnevermind...i see now i added a 3rd 2 into the mix",
        "A": "Peter, you say you see you added a third 2 into the mix. Do you mean you took ALL the prime factorizations for BOTH numbers 18 and 20? Instead of what Sal seems to do which is, ignore the 2 from the second number 20 because we already have a 2 from the first number 18? That s the part I don t understand. Why don t we use ALL the prime factorizations? Why does a 2 in 20 get cancelled out because there s a 2 in 18?"
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "Why does he say there are only two 3's at 6:54? There should be three 2's and three 3's.",
        "A": "If a number is repeated in the prime factorization of both numbers, it is not repeated in the set of numbers for the field. Watch the video again - you ll see that he does it for the first set as well."
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "At 0:50 Sal says the prime factorisation of 12 is 2*2*3, but could this be written in any order for example: 2*3*2 or 3*2*2 it would still equal the same number?",
        "A": "yes since it is equal"
    },
    {
        "video_name": "zWcfVC-oCNw",
        "Q": "At 5:47 why don't we include the third 3. Other wise the prime factorization would have been 2*2*2*3*3*3.",
        "A": "He is finding the Least Common Denominator, not the multiplication of two numbers."
    },
    {
        "video_name": "gfVY7cW7VLQ",
        "Q": "At 12:30, why is it that v1 = 1*v1 + 0*v2 + 0*v3 as a vector in the new basis B={v1,v2,v3}?",
        "A": "All vectors in that R^3 can be represented as linear combinations of v1, v2, and v3. That is, all vectors in R^3 can be expressed as x = c1v1 + c2v2 + c3v3, where c1, c2, and c3 are scaling factors. If x = v1, what are the values for c1, c2, and c3?"
    },
    {
        "video_name": "gfVY7cW7VLQ",
        "Q": "Around 8:00 he is doing the transformations of the vectors. What exactly are the transformations of? The mirror image perpendicular to the plane?",
        "A": "Yes, it is the mirror image of the vector through the plane."
    },
    {
        "video_name": "gfVY7cW7VLQ",
        "Q": "at 7:52, i dont understand why T(v3) is -v3. Since it is perpendicular to the plane spanned by v1 and v2..shouldnt the projection of v3 on the plane be 0?",
        "A": "He mixed projection with reflection. If it were the case for reflection, then T(v3) has to be equal to -v3 which he mentions at 18:00."
    },
    {
        "video_name": "15s6B7K9paA",
        "Q": "what does he mean at 0:08?",
        "A": "Are you puzzled by the term absolute value ? If so, this means the numerical size of that expression regardless of whether it is positive or negative . It s also known as the MAGNITUDE. For example in IX+8I =0 the absolute value of X is 8 (not -8)"
    },
    {
        "video_name": "15s6B7K9paA",
        "Q": "When Sal was doing the first problem at 1:36 , Don't you divide 6 from the right side, instead of subtracting? I think this because on the right side, 6 is being multiplied to the absolute value?",
        "A": "6 is being multiplied to the absolute value. However, Sal is correct. To solve this problem, you have to put like terms together. Subtracting the 6 abs(x+10) groups the two absolute values together."
    },
    {
        "video_name": "15s6B7K9paA",
        "Q": "At about 2:00 he states that you take 4|x+10| -6|x+10| and get the answer -2|x+10|. Why does the |x+10| not cancel each other out? You are subtracting the same number from a positive same number. Why does it stay the same?",
        "A": "y = |x+10| 4|x+10| - 6|x+10| = 4y - 6y = -2y = -2|x+10| From the comments below, another way to view it: 4|x+10| - 6|x+10| = (4 - 6) * |x+10| = (-2) * |x+10| = -2|x+10| Or, 4|x+10| - 6|x+10| = |x+10| + |x+10| + |x+10| + |x+10| - |x+10| - |x+10| - |x+10| - |x+10| - |x+10| - |x+10| = - |x+10| - |x+10| = -2|x+10|"
    },
    {
        "video_name": "15s6B7K9paA",
        "Q": "At 3:01 the video begins to explain why an absolute value cannot be negative. This makes sense, but for some reason my math teacher didn't know that there was such a thing as no solution to absolute value equations. Is this a relatively new math standard?",
        "A": "Hi, The absolute value concept has been around for a while, I have no Idea why your teacher did not know about this. Absolute value equations do have no solutions... For ex |x|=-9 you cannot have a absolute value that has a - solution, so there is no solution."
    },
    {
        "video_name": "etl9KKf6se0",
        "Q": "At 1:06, isn't it supposed to be -5/9?",
        "A": "Yes, it is. He said -5 but meant -5/9."
    },
    {
        "video_name": "etl9KKf6se0",
        "Q": "surely 1:06 is -5/9",
        "A": "This is a known error. A little pop-up box in the lower right corner of the video explains that Sal really meant -5/9."
    },
    {
        "video_name": "92U67CUy9Gc",
        "Q": "At 3:30 why is it that -3 becomes -6? But when you multiply them it's supposed to be a positive?",
        "A": "Sal was pointing out that -6 is just two times -3 . And this is true, because a positive times a negative is a negative. So no, 2 * -3 is not a positive, it s -6, as Sal said. I hope that helps."
    },
    {
        "video_name": "92U67CUy9Gc",
        "Q": "at 4:44, I do not understand 1/x= -3 x 1/y. I cannot solve it back to y=3x. I got various answers...",
        "A": "1 / x = -3 * 1 / y Multiply both sides by x, getting 1 = -3x * 1 / y Multiply both sides by y, getting y = -3x"
    },
    {
        "video_name": "92U67CUy9Gc",
        "Q": "At 2:53, why doesn't y = -2x count as inverse variation? The values move in opposite directions; as x increases y decreases. Thanks for any help.",
        "A": "It s still a constant being multiplied by x. Inverse variation is a constant being divided by x."
    },
    {
        "video_name": "92U67CUy9Gc",
        "Q": "At 4:48, how did Sal get 1/x = -3 * 1/y from y/x = -3? I don't understand the steps. Thanks for any help.",
        "A": "Both sides have been multiplied by 1/y . Whenever you multiply both sides of an equation by the same term, the left hand side and the right hand side equal each other (i.e.: L.H.S=R.H.S )"
    },
    {
        "video_name": "92U67CUy9Gc",
        "Q": "At 4:48,How do you do that?",
        "A": "We start with y/x = - 3 and divide it by y. y can be written y/1 y/x / y/1 = y/x * 1/y = y/xy Cancelling out the y in the numerator and denominator gives you 1/x"
    },
    {
        "video_name": "92U67CUy9Gc",
        "Q": "At 3:50 How do you know what x means without it telling you? I'm lost, so very lost.\ny = -3x\nx | y\n1 | -3\n2 | -6\nWhy is it x2 then 1/3? Please help me! Thank you.",
        "A": "Sal is just taking different values for x to show the corresponding changes in y. For example, if we scale up x by 2, then y is also scaled up by 2 or if we scale down x by 3, then y is also scaled down by 3 and so on. Hope this helps! :)"
    },
    {
        "video_name": "92U67CUy9Gc",
        "Q": "Does K always have to be the variable? At 0:32",
        "A": "You don t have to use k as the variable, but that it is the most common variable for direct and inverse variation."
    },
    {
        "video_name": "92U67CUy9Gc",
        "Q": "At 8:50, Sal draws a symbol like this: <=>, what does it mean? I know that => means imply, but I am not sure what the symbol means when it has double-sided arrows. Thanks in advance.",
        "A": "If you write A <=> B, it simply means that A implies B (which we write as A => B) and also B implies A (which we write as B => A). In other words, the logical or mathematical argument being used could also be used in the other direction if necessary."
    },
    {
        "video_name": "92U67CUy9Gc",
        "Q": "At 4:38, Sal shows how to do it with a negative number, but how would you do it with something like y=9^87. Do we have to solve it first or can we just leave it in that form?",
        "A": "the quotient of a number and 18 is 5. find the number"
    },
    {
        "video_name": "3AxyrK2jYc8",
        "Q": "At 5:54, how did Sal end up with 2sqrt(5)? I keep getting 2sqrt(10).",
        "A": "When you have 20, you have to find a perfect square that can come out, 20 = 4 * 5, and 4 is the perfect square, so \u00e2\u0088\u009a20 = \u00e2\u0088\u009a4 \u00e2\u0080\u00a2 \u00e2\u0088\u009a5 which is 2 \u00e2\u0088\u009a5."
    },
    {
        "video_name": "_jiI0AV8Vr4",
        "Q": "at 1:26, why does sal do 81/9 first instead of 5*81/9?",
        "A": "you had to follow BEDMAS/PEMDAS sal did 81 divided by 9 because division goes first before going to multiplication"
    },
    {
        "video_name": "_jiI0AV8Vr4",
        "Q": "I have a question. At 3:33, Sal says we have multiplication and division. He proceeds to do the division first. 81 / 9 = 9. Shouldn't you do the multiplication first? 5 * 81/9? Working left to right? Because there are no Parentheses and there are no exponents. In the first video \"Intro to order of operations\" Sal states that you should work left to right when mult./div. and add/sub.?",
        "A": "With PEMDAS multiplication and division are of equal importance in order of operations. If you multiply by the fraction 81/9 you would get the same answer. The reason Sal divided first was because it made the problem easier to solve, because 5 times 9 is easier than 5 times 81/9. If you don t believe me try it on your calculator and you will get the same answer either way."
    },
    {
        "video_name": "OhRr26AfFBU",
        "Q": "At 3:00 Sal said \"If we go one more standard deviation then, you know where this is headed, 99.7% of a deviation in that range.\" But why isn't it 100%?",
        "A": "The Normal curve doesn t ever hit 0, so technically any place that we chop it off, we ll be chopping off a little bit of the probability. It so happens that at +/- 3 standard deviations we ve captures 99.7% of the area, and for many folks that is close enough to being basically everything."
    },
    {
        "video_name": "OhRr26AfFBU",
        "Q": "at 4:03 you said that all the weights were kg, however at the beginning of the video you said the standard deviation was in grams. why?",
        "A": "It s not grams, it s Kilograms. He just made a mistake."
    },
    {
        "video_name": "Z5EnuVJawmY",
        "Q": "Just a question at 3:16, the side h is adjacent to 90 degrees. But suppose we want to know the opposite one, which will that be? 108 is hypotenuse, h is adjacent and even the third side is adjacent. How is it possible that it doesn't have an opposite side?",
        "A": "All three sides are not adjacent to the angle theta. The hypotenuse is always the longest side so even though it is adjacent to theta, it is still the hypotenuse. The opposite side would be the water line."
    },
    {
        "video_name": "Z5EnuVJawmY",
        "Q": "At 3:35, how does Sal know which function to use?",
        "A": "Listen again to the video from 2:45 to 3:10 and then to 3:30. Sal indicates that h is the adjacent side and that the 108 is the hypotenuse . As per SOH CAH TOA, that combination of sides indicates using cosine."
    },
    {
        "video_name": "Z5EnuVJawmY",
        "Q": "At 5:18, Sal pulls a scientific calculator simulator out. How can I download/install this and from what source?",
        "A": "Here s a TI_83 that I use - wabbitemu TI calculator emulator just download it. Then download the TI_83 rom file off the internet and set it up and you ll be good to go."
    },
    {
        "video_name": "Z5EnuVJawmY",
        "Q": "1:34 couldnt we just say that angle x and angle theta are alternate interior? cuz the two lines (side of triangle and the line h) are parellel (both are 90 degrees)",
        "A": "yes, you could. good job!"
    },
    {
        "video_name": "Z5EnuVJawmY",
        "Q": "At 5:10 why can Sal just divide 139 times 108 though 180 ? How does that work ? I just don`t understand how he can turn h/108 is equal to 139/180 into h is equal to 139*108/180 . Can someone explain that to me ? Please =)",
        "A": "He multiplies both sides by 108."
    },
    {
        "video_name": "Z5EnuVJawmY",
        "Q": "At 4:39 how can we assume the pyramid to be an isosceles triangle",
        "A": "I would make that assumption because they only give one measurement for the edge of the triangle and don t specify which edge it is. Also, they give the measurement of the edge that is out of the water and in the diagram put that measurement on the other side of the triangle. If it were not an isosceles triangle, those measurements would have to be on the same side of the triangle."
    },
    {
        "video_name": "GOA9XWEo7QI",
        "Q": "At 1:28 when he says that the arc is 174, shouldn't it be 360-174 because the arc AC doesn't include b? Like why did he do the arc with b in it?",
        "A": "No, a minor arc is defined only by the two end points, it does not matter which points are in between. If you wanted a major arc between A and C, you need a point opposite B and you name a major arc by three letters to distinguish it from a minor arc."
    },
    {
        "video_name": "GOA9XWEo7QI",
        "Q": "AT 1:28, you said that arc AC is 174. i thought that would be arc ABC? And arc AC would be 186. because you are not passing through point b.",
        "A": "I think that arc AC in this case is 174, because it is the minor arc, and arc ABC would be 186 (cause it s the major arc), even though it doesn t really correspond to the picture? Hmmm I m not sure.... I m sorry I couldn t help more!"
    },
    {
        "video_name": "GOA9XWEo7QI",
        "Q": "At 1:06, Sal is highlighting arc ABC. Wasn't he supposed to highlight the arc AC, since that's what the question is asking?",
        "A": "I also think so. As we already know that arc ABC is 174. We can find arc AC by doing 360-174 = 186 deg"
    },
    {
        "video_name": "GOA9XWEo7QI",
        "Q": "At 0:25, isn't the major arc also Arc AC, so wouldn't that be really confusing that both the minor arc and the major arc are called the same thing? Shouldn't the minor arc be called Arc ABC in this case?",
        "A": "Hi Hisham Malik, The major arc has to be denoted with three letters so it will be called arc ACB. The minor arc is always two letters so in this case it is AC. Always remember that a major arc is always 180\u00cb\u0099 or more. Hope that helps! - Sam"
    },
    {
        "video_name": "wJ37GJyViU8",
        "Q": "At about 3:30 in the video, the line facing out of the protractor seems to be colinear to 120 degrees. If that is true, could a student simply find the difference between 180 degrees and wherever that line outside the protractor seems to be colinear (in this case 120 degrees) and determine that the measure of the angle is 60 degrees?",
        "A": "That s a good observation. On the face of your question you d be correct in doing it that way. The only problem that might come is if you measure that angle incorrectly. The best thing to do is to rotate to protractor tool so that the angle is included within the tool and easily measurable."
    },
    {
        "video_name": "wJ37GJyViU8",
        "Q": "At 3:30 is that called a reflex angle?",
        "A": "yes it is called a reflex angle because it is the outer part of the angle"
    },
    {
        "video_name": "wJ37GJyViU8",
        "Q": "3:00 is there a protractor that is a full circle?",
        "A": "No, then it would not be a protractor."
    },
    {
        "video_name": "wJ37GJyViU8",
        "Q": "At 0:33, how do you know which way you're suppose to put the angle?",
        "A": "When measuring an angle, you want to make sure that one side is up against the 0 mark, and the other side is actually inside the protractor. As long as you end up doing that, you should be ok!"
    },
    {
        "video_name": "NRS2ZJVVzOI",
        "Q": "How come at time 0:53, that animal-nosed triangle is considered obtuse, even thought visually, it looks like an acute triangle?",
        "A": "The two side angles are indeed acute, but the bottom center angle (BAC) is obtuse, which means the whole triangle is obtuse."
    },
    {
        "video_name": "dsvLLKQCxeA",
        "Q": "At 6:07, was it not known to Vi that you can use a tiny little dot in between instead of an x?",
        "A": "Well, yes, you can use a dot as a multiplication symbol, but that way of doing things is only introduced when you start doing algebra, and the x is the first multiplication sign you learn. Also, you can confuse the dot with a decimal point or something."
    },
    {
        "video_name": "dsvLLKQCxeA",
        "Q": "Wait....did Vi just mention Minecraft at 7:25 ?",
        "A": "Triangle party included the symbol of the three brothers from HP and snakes and graphs she drew the dark mark on her arm. Vi is great and sneaking that kind of stuff in. JO IS MY QUEEN!"
    },
    {
        "video_name": "dsvLLKQCxeA",
        "Q": "@1:10,hungarian horntail!harry potter!",
        "A": "Its actually 1:06"
    },
    {
        "video_name": "dsvLLKQCxeA",
        "Q": "At 1:29 in the video, it says you have to paint 1 dimension, how can you paint one dimensions? Even if you just paint a line of paint, it will still be 2 dimensional. Also, if you had 2 dimensions, you wouldn't paint a 1 dimension line, but 1, 2 dimension surface. Wouldn't you?",
        "A": "What Vi is saying is that you would have to paint the edge of the 2 dimensional figure. Because edges are by nature one-dimensional, the paint only needs to be applied in one dimension. Note that this is mathematical paint which has no thickness to it."
    },
    {
        "video_name": "dsvLLKQCxeA",
        "Q": "at 4:45 what is the difference between the two types of triangles? Is there more than one type, besides isosceles and the other two?",
        "A": "Here all the triangles are equilateral, which means that all the sides are equal and all the angles are equal. Most of the patterns she uses contain equilateral shapes. The three kinds of triangles are equilateral (all angles and sides are congruent), isosceles (two sides are congruent and their two opposite angles are congruent), and scalene (no sides or angles are congruent)."
    },
    {
        "video_name": "7fYDCUIvZnM",
        "Q": "What does Sal mean at 1:06 when he says, 'the actual coordinate in R2 on the Cartesian coordinates'? What does R2 mean?",
        "A": "R^2 just refers to the coordinate plane"
    },
    {
        "video_name": "7fYDCUIvZnM",
        "Q": "Up to ~ 4:26, I think it's going to be easier to understand if you include the t-axis, even if it's just a quick sketch off to the side. As it stands, your original diagram for (x(t), y(t)) fails the VLT , and is going to confuse the viewer. In R3, of course, (x(t), y(t)) is a perfectly valid function -- which is the point. Parameterizing takes you from R2 to R3.",
        "A": "We choose not to, because parameterization will have been well-understood by anyone at this point. Additionally, convention dictates that we don t show an axis for the parameter."
    },
    {
        "video_name": "gl_-E6iVAg4",
        "Q": "Pay strict attention to the video starting at 3:24, guys. Sal gives out valuable info that could help you with your EOCs (End-of-Course exams)!",
        "A": "very true. He does give a good strategy with solving this"
    },
    {
        "video_name": "gl_-E6iVAg4",
        "Q": "why do we have to use the distributive property anyway? why cant we just do the numbers in the parentheses like sal showed at 0:23 through to 1:08",
        "A": "Because you might run into something like 4(x+8)=48 and x+4 is impossible as it is an unknown variable."
    },
    {
        "video_name": "QYoNHOjr7Rg",
        "Q": "At 2:05, there is a mistake. Graph B slope is 4, not 5.",
        "A": "No, there is no error. Sal went from the point (0, 1) to the point (1, 6). This makes the slope = 5/1 = 5 You may be thinking the 2nd point is (1, 5), but it isn t, The point is 1 above the 5 on the y-axis. Hope this helps."
    },
    {
        "video_name": "KlKYvbigBqs",
        "Q": "at 1:43,why sal Is saying degenerate triangle? does it mean the shape of the triangle has changed",
        "A": "Degenerate triangle means it is a triangle that doesn t have 2 dimensions. It becomes a straight line with no area."
    },
    {
        "video_name": "KlKYvbigBqs",
        "Q": "*2:22* why does x=4?",
        "A": "Because 10 - 6 is 4 (i.e. the 10 side minus the 6 side). Sal s talking about the degenerate case where the triangle is a line so the remaining side must be 4."
    },
    {
        "video_name": "N-7tcTIrers",
        "Q": "Is it just me, or do all of Vi Hart's video's have an air of awesomeness starting from 0:01, wait, no, -10.00",
        "A": "True, and by the way I am awesome."
    },
    {
        "video_name": "N-7tcTIrers",
        "Q": "At 6:03, Vi explains the concept of fractional powers instead of square roots. Could there be a fractional power to replace square root 2.(See \"What's up with Pythagoras?\")",
        "A": "Yes. \u00e2\u0088\u009ax = x^1/2"
    },
    {
        "video_name": "N-7tcTIrers",
        "Q": "At 4:47, why is there a blue squiggle in the corner of the page? Is that from one of her other videos?",
        "A": "I believe it is part of her doodling in math Squiggle Inception vids"
    },
    {
        "video_name": "N-7tcTIrers",
        "Q": "Well...@~3:51 she's saying 5 * 7 is counting in a +5 sort of way...So how do you express 3.6 * 4 as +1 +1 +1 +1 +1 ... or +4 +4 +4 +...?! How do you get .6 of a 4 and what does .6 mean?...okay, wow. I sound like Vi.",
        "A": "if we re counting in a 4 sort of way, 0.6 would be 2.4 if you were counting in a 1 sort of way, or a base 10 sort of way i guess"
    },
    {
        "video_name": "N-7tcTIrers",
        "Q": "Around 7:42 or so, I find that things get really confusing. Would someone please break it down and explain?",
        "A": "a fraction of 125= 1/3 of 1, or *5. *25 is what she means to get to 25, or 125 to 2/3."
    },
    {
        "video_name": "N-7tcTIrers",
        "Q": "At 5:31, how is the moon big enough to cover the sun? Isn't the sun much larger?",
        "A": "The moon doesn t completely cover the sun, you are still able to see a thin layer of the sun around the moon but I wouldn t advise in looking at the sun in an eclipse, you ll permanently damage you eyes."
    },
    {
        "video_name": "N-7tcTIrers",
        "Q": "at 4:30 isn't 27 just 9 in a base 3 sequence\n?",
        "A": "It would be 9 when counting in a +3 sort of way, because +3, 9 times is 3x9. But when counting in a x3 sort of way, it s 3 steps, or 3 to the power of 3. Since 3^4=81, 3 is 1/4 of 81 and 27 is 3/4 of 81 in a timsy sort of way. If you still don t get it and you (like me) don t like to walk away from something without fully understanding it, rewatch the video until you can explain it to someone."
    },
    {
        "video_name": "mLE-SlOZToc",
        "Q": "at 2:48, Why is the answer 6/7? I just don't understand",
        "A": "There are fourteen marbles in the bag: 9 red, 2 blue, 3 green. The non-blue marbles are the 9 red and 3 green, which add up to 12. So the probability of getting a non-blue marble is 12/14, which reduces to 6/7 if you divide both the denominator and the numerator by 2."
    },
    {
        "video_name": "mLE-SlOZToc",
        "Q": "At 2:30, Sal says \"If this isn't in Lois---\" and breaks off. This makes me curious, what is \"Lois\" form?",
        "A": "I think he was going to say lowest terms - about the same thing as simplified form. Hope this helps"
    },
    {
        "video_name": "mLE-SlOZToc",
        "Q": "At around 2:51, couldn't you just do 1-2/14 to figure out the probability of picking a non-blue marble?",
        "A": "Yes that is a third way of doing this. It is not really much different than what Sal does at 2:05 when he subtracts 14 - 2 which is the same as 14/14 - 12/14 (getting common denominator)."
    },
    {
        "video_name": "mLE-SlOZToc",
        "Q": "around 9:20 wouldn't it be four eighty-first",
        "A": "Yes. You are completely correct. But the problem with your question is that Sal actually did put four eighty-first"
    },
    {
        "video_name": "mLE-SlOZToc",
        "Q": "At 7:10 why did he divide by two pi?",
        "A": "circumference of a circle of radius r is 2pi*r, in the problem we require area of bigger circle for which its radius is to be known, so to find the radius, circumference (which is 2pi*r) has to be divided by 2pi"
    },
    {
        "video_name": "mLE-SlOZToc",
        "Q": "At 4:56, when it says that \"The circumference of a circle is 36pi.\" Does that mean that the circumference is 36 multiplied by pi?",
        "A": "That s right. pi is like an algebraic variable in that we usually just gather those like terms together and leave it be, but in the end we can always multiply it out to find a plain real number that is a good estimate."
    },
    {
        "video_name": "mLE-SlOZToc",
        "Q": "2:31 What are our constraints?",
        "A": "Our constraints are selecting a non-blue marble."
    },
    {
        "video_name": "mLE-SlOZToc",
        "Q": "At 6:04 and 6:06, Sal says that there is an infinite number of points in a circle. But wouldn't you eventually run out of space? How is there an infinite number of points!",
        "A": "Because each point takes no space. For any two points, there is a point halfway between them."
    },
    {
        "video_name": "mLE-SlOZToc",
        "Q": "Around 6:19, Sal describes the points that are in the small and large circle. Since this isn\u00e2\u0080\u0099t on a graph, are the points scattered? I am a bit confused on the what the question means by \u00e2\u0080\u009cpoint also in the smaller circle.\u00e2\u0080\u009d Thanks :)",
        "A": "Can someone help me understand circumfrenece and ratio? Please I m a bit confused and I don t want to fall behind in class."
    },
    {
        "video_name": "mLE-SlOZToc",
        "Q": "at 4:55 how can pi be 36?isn't pie 3.1415927?",
        "A": "Yes, pi is 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148, but 36\u00cf\u0080 means 36 multiplied by \u00cf\u0080."
    },
    {
        "video_name": "MkpbtCRwcCE",
        "Q": "At 0:28 couldn't you put 592 over a thousand taking the decimal point out.like 592/1000?",
        "A": "Yes, ... you could change: 59.2/100 into 592/1000. They are equivalent fractions. Then, convert to a decimal and get 0.592 Hope this help.s"
    },
    {
        "video_name": "MkpbtCRwcCE",
        "Q": "what did he say at 1:20",
        "A": "he just said the answer in a decimal. (Ans: 0.592)"
    },
    {
        "video_name": "B_vAlneziHo",
        "Q": "At 2:00 Sal shows how to find the number of combinations in-between flowers and pots. What about Multiple combinations, for example, to throw in 5 different designs you can add to the pots? How do you do that?\nOr what about six-digit combinations to safes? How can you compute the number of possible combinations?\nPlease respond, because I have exams and really need these answers!",
        "A": "Thanks to the Counting Principle, you can find the total combinations by multiplying the number of choices for each slot. For a six-digit combination safe, where each digit is chosen from the digits 0 to 9, you ll find \u00e2\u0082\u0081\u00e2\u0082\u0080C\u00e2\u0082\u0081 = 10 choices (ways to pick from ten things one at a time) for each of six slots. 10\u00e2\u0088\u009910\u00e2\u0088\u009910\u00e2\u0088\u009910\u00e2\u0088\u009910\u00e2\u0088\u009910 = 10\u00e2\u0081\u00b6 = 1,000,000 = 1 million total combinations for a 6-digit safe"
    },
    {
        "video_name": "yVWxxQ7SMOw",
        "Q": "At 0:58, how does Sal know that the denominator is continuous at x=-1",
        "A": "Because you get a real number by plugging -1 into the denominator: sqrt(x+5) - 2 = +/-2 - 2 = 0, -4. The limit of the denominator exists if you consider this by itself. The problem occurs when the expression is used as the denominator, and you have the instance where it evaluates to 0. Now there is a divide by zero issue."
    },
    {
        "video_name": "yVWxxQ7SMOw",
        "Q": "at 5:00 Sal simplifies (x+1)(\u00e2\u0088\u009a(x+5)+2)/(x+1) into \u00e2\u0088\u009a(x+5)+2 but I was taught you couldn't cancel out to numbers on the opposite side of a division, if the numbers cancelling out are multiplying other numbers",
        "A": "Algebra allows cancellation of like terms in the numerator and denominator. So if you have x*y/x, you can cancel out the x terms to leave y."
    },
    {
        "video_name": "yVWxxQ7SMOw",
        "Q": "At 8:56 , We know that x is approaching -1 from values greater than -1, and y =4, so why is the function line downward sloping rather than a straight line? What is the basis of drawing the function line? Is there a video about it?",
        "A": "Because that s what that function looks like. It s a standard square root curve, shifted five units left and two units up. It should be neither level nor straight. Try graphing it on paper, or on a graphing calculator, such as the free one at Desmos.com online. Graphing functions is primarily covered earlier in math on Khan, much of it in Algebra."
    },
    {
        "video_name": "yVWxxQ7SMOw",
        "Q": "At 6:46, I don't get the part where Sal says \"f(x) is equal to g(x) for all x not equal to 1\". Does that mean f(x) is equal to g(x) when x is not equal to 1? If so, why is that relevant to the reason he uses f(x) instead of just plugging -1 into simplified version of g(x)?",
        "A": "At 2:20 Sal assigns g(x) as the function he s trying to take the limit of. g(x) has to be constrained at x=-1 because it leads to a 0 in the denominator. Later on, Sal assigns f(x) to the conjugate, which does not need to be constrained. He can t just plug -1 into the conjugate of g(x), because he s already constrained it at -1."
    },
    {
        "video_name": "G_mUcQeXak8",
        "Q": "At, 1:12, if it doesn't matter, then why do they even put zeroes after a number?",
        "A": "The zero in the original number is needed: 72,830 is not the same as 7283. What Sal is trying to show you is that when you write the number in expanded form, the zero can be put in or dropped (your choice). The reason is that doing + 0 will not change the value of the other numbers when you added them together. For example: 2 + 5 + 0 = 7 but so does 2 + 5 = 7. Hope this helps."
    },
    {
        "video_name": "n34jx1FIN8M",
        "Q": "At 1:00, \"double-angle-identity\"? where is that explained? I don't find it anywhere in Khan Academy. Is there actually a video about it?",
        "A": "If you use the search bar at the top of Khan Academy s page, you can find it easily. It will be in the trigonometry playlist."
    },
    {
        "video_name": "n34jx1FIN8M",
        "Q": "At 4:00, why does Sal change (1/2 cos 4x) into (1/8 4 cos4x)? Is that necessary?",
        "A": "I believe he simply prepared ahead of time for the u-substitution. It s not 100% necessary, but it can be helpful."
    },
    {
        "video_name": "n34jx1FIN8M",
        "Q": "sal I understand how you generated the first double angle formula but the second double angle formula.. i get cos^2(2x)=1/2(1+cos2x).... how did you get cos4x...is it a mistake or most likely a hole in my algebra? 2:48",
        "A": "If you plug in 2x as your x value, the equation becomes: Cos^2(2x) = 1/2 (1 + cos(2(2x))) so therefore, 1/2 (1 + cos(4x)) Hope that helped!"
    },
    {
        "video_name": "n34jx1FIN8M",
        "Q": "At 2:05, why does Sal move the 1/4 from inside the integral to the outside? Thanks!",
        "A": "He did it because it s possible to move constant factors in or out of the integral, and because it makes it easier to integrate."
    },
    {
        "video_name": "t5ry2rasWvI",
        "Q": "And @5:20 I thought you could never have a negative under the radical. Or is that only for the arithmetic?",
        "A": "The key here is you don t know the value of X. As long as X <= -3, the value inside the radical will be positive."
    },
    {
        "video_name": "t5ry2rasWvI",
        "Q": "At 5:26, what is y when sqrt(x+3) is negative? I don't understand!",
        "A": "sqrt(x) means positive (principal) sqrt always. If someone wants the negative sqrt they must write -sqrt(x). This is a rule. When -(x+3) is negative (that is, to the right of -3), y = sqrt(-(x+3)) is undefined. At -3, -(x+3) is 0. As x moves to the left of -3, -(x+3) increases from 0, and so its sqrt, y, increases."
    },
    {
        "video_name": "BIGX05Mp5nw",
        "Q": "At 3:15 you indicated that the 4 times 8 is 32, which it is but why didn't you multiply the quotient as a whole number all together so that it would read 18 times 4? I always thought that division was supposed to be like that. Is my interpretation of long division incorrect?",
        "A": "At 3:20 you don t have to have the quotient equivalent to the whole number all the time. Remainders are still needed in division."
    },
    {
        "video_name": "BIGX05Mp5nw",
        "Q": "At 1:03 Sal told us to write the remainder like 2r1. But can I write ... instead of r?",
        "A": "Write it in any way that will help you remember that the solved problem has a remainder as part of the answer."
    },
    {
        "video_name": "BIGX05Mp5nw",
        "Q": "what does \" 4 is going into 70 ten times \" means at 2:45 ?",
        "A": "280 = 4 x 70 Because, we multiplied it. With 0 s, Just multiply numbers without zeroes then, insert (the number of 0 s) to your product. Example: 50 x 50 = 5 x 5 x 100 = 2500 (2 zeroes inserted in the product)"
    },
    {
        "video_name": "BIGX05Mp5nw",
        "Q": "0:10 is 3 divided by 7 the same as 7 divided by 3?",
        "A": "No it isn t. 7 divided by three is 2.3. But 3 divided by 7 is 0.428571."
    },
    {
        "video_name": "BIGX05Mp5nw",
        "Q": "in a the video at 2:02 why are the 3 separate from the four?",
        "A": "The 3 is the answer to the question, while the 4 is part of the equation. They have to be separate."
    },
    {
        "video_name": "BIGX05Mp5nw",
        "Q": "At 0:30, my student is asking \"Why 2 groups of 3?\" I keep trying to explain but she doesn't understand. How can I explain it to her?",
        "A": "You are dividing by three, so you have to make groups of three. There are only 2 sets of 3 in 7, with one left over. Does that help?"
    },
    {
        "video_name": "zVLjWIftX_o",
        "Q": "how do you answer the question presented at 5:31 ?",
        "A": "2 balls + 3 balls = 5 balls"
    },
    {
        "video_name": "c6-FJRda_Vc",
        "Q": "In 0:48, can't you just divide 30 by 6, or is it not the right thing to do?",
        "A": "Yes we can also do like that in that case k=5 Now since y=kx therefore 45=5*x If divide both sides by 5 we get x=9"
    },
    {
        "video_name": "_MIn3zFkEcc",
        "Q": "At 1:41 what happens to the four?",
        "A": "Ummm.....there is no 1:41. The video is 1:26 minutes long. But, the four is rounded off and is no longer needed, if that helps :)"
    },
    {
        "video_name": "_MIn3zFkEcc",
        "Q": "From 1:14 to 1:21 Sal says that 9.564 rounds to 9.6 wouldn't you say 9.600? Would 9.6 be the same as 9.60 and 9.600?",
        "A": "You say 9.6 + as many 0 s as you want. You can even say 9.6 with no 0 s."
    },
    {
        "video_name": "TbaltFbJ3wE",
        "Q": "2:00\nWhy did he say \"middle row top\" column instead of 'top row middle column'?",
        "A": "Prob ly just to be diferent :)"
    },
    {
        "video_name": "TbaltFbJ3wE",
        "Q": "At 0:44, Sal refers to the object as both a \"matrix\" and an \"array\". Are those terms entirely synonymous, or do they have some subtle difference? I think I've seen \"array\" more in computer science, and \"matrix\" in pure mathematics. Is this observation correct?\nThanks! :)\n-Yehonathan",
        "A": "They are pretty much the same. Both are refered to as a group of numbers, but as you pointed out, people usually use array in computer science and matrix in mathematics."
    },
    {
        "video_name": "TbaltFbJ3wE",
        "Q": "at 2:15 it was middle column and top row.. isnt it?",
        "A": "Yes. He means what you know."
    },
    {
        "video_name": "TbaltFbJ3wE",
        "Q": "At 2:09, you can tell that the scalar scales the entire matrix by 1:3, just like the scale on the map. Is the term scalar after scale?",
        "A": "Absolutely! I love how etymological insights can lead to an intuitive understanding of these concepts."
    },
    {
        "video_name": "TbaltFbJ3wE",
        "Q": "At 0:40 can Sal do another video on scalar multiplication but this time with multiplying by a negative scalar?",
        "A": "Negative scalars are very similar, you just multiply them by negative numbers."
    },
    {
        "video_name": "D5fmcpNygQk",
        "Q": "0:49 why you circling numbers?",
        "A": "To show the numeral that he was multiplying."
    },
    {
        "video_name": "D5fmcpNygQk",
        "Q": "what do the arrows mean from 1:45 - 4:24 i did not understand even by watching the vid",
        "A": "I think its to direct you to where he is going to talk about next."
    },
    {
        "video_name": "D5fmcpNygQk",
        "Q": "1:13 Sal forgot to write the + symbol while adding 363 and 4840. That's bad right\n?/",
        "A": "Yes, but if he remembered to add it up then it doesn t make a difference."
    },
    {
        "video_name": "D5fmcpNygQk",
        "Q": "at 3:16 he says that you could count to the right of a decimal. How is that possible",
        "A": "it is possible because say place value you need to look to the right to see if that number is bigger or lower than 5"
    },
    {
        "video_name": "D5fmcpNygQk",
        "Q": "Wait in 2:23 You have to divide. Can't you just move the decimal over by 5? I mean that is easier. (for me at least)",
        "A": "He is giving you the intuition on why you move the decimal.........it s because you divide....... Rock on"
    },
    {
        "video_name": "PdSzruR5OeE",
        "Q": "The way I understand this that in order to apply L'H\u00c3\u00b4pital's rule the limit of f(x) & g(x) as x -> c MUST be 0. This results in the indeterminate form. I don't understand what third part means.\nThe limit of f'(x) / g'(x) at x -> c must be L. Technically he says \"exists and equal L\" (2:30). What is L? Does this just mean that L is not indeterminate? Thanks.",
        "A": "L is the limit of the original expression. L should be a finite value, and if L can be found then this is the limit; where lim as x app c of f (x)/g (x) = L I should add that 0/0 is one form of indeterminate form, another is infinity/infinity."
    },
    {
        "video_name": "PdSzruR5OeE",
        "Q": "at 0:50, is that 6/0? that's indeterminate? I thought that was just undefined.",
        "A": "Indeterminate form is 0/0. 6/0 is just undefined, but it is not in indeterminate form."
    },
    {
        "video_name": "5xitzTutKqM",
        "Q": "i got lost at 3:21 and from there i don't really how you get absolute value of the tangent even though it is the sector of the circle",
        "A": "I m not completely certain I understand what your question is, so please feel free to followup with a comment on this answer ... The area of the sector is theta/2. |tan(theta)|/2 is the area of largest triangle (the one bounded by the blue line). We are using absolute values because we want the math to work in the fourth quadrant. This is important because the limit is as theta approaches zero, which could be from above (i.e. in the first quadrant) or below (i.e. in the fourth quadrant)."
    },
    {
        "video_name": "5xitzTutKqM",
        "Q": "At 8:52 why is the limit of 1 is 1 when theta approaches zero?",
        "A": "The lim(1) when \u00ce\u0098\u00e2\u0086\u00920 means: on the graph y=1, what does the y-coordinate approach when the x-coordinate (or in this case \u00ce\u0098) approach 0. But on the graph y=1, the y-coordinate is always 1 no matter what the x-coordinate is. In other words, lim(k) as \u00ce\u0098\u00e2\u0086\u0092n = k, where k,n are any real numbers."
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "at 3:16, sal said that the principle square root of neg one is \"i\", can ANY neg number's square root be symbolized by \"i\"?",
        "A": "It can be expressed as the square root of the absolute value of that number multiplied by i. For example : sqrt(-6) = sqrt(6)*i"
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "At 8:20, I'm not sure what is meant by (-i)^2=-1? Thanks in advance :)",
        "A": "He is saying that the principal root of -1 is i, and that the negative root (-i) squared is also one. i and -i are imaginary numbers."
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "At 4:59,the green hand writing, I do not understand what he means. Can someone explain that section?",
        "A": "He is substituting the solution (3+i)/2 for x in the original equation 2x\u00c2\u00b2+5=6x. So where ever there is an x, remove it and put (3+i)/2 there. This is a way to check if the solution is correct."
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "I'm not sure what Sal means @8:21 when he says (-i)^2=-1. I was under the impression that i^2=-1 and based upon what Sal writes (-i)^2 would be equal to 1 instead?",
        "A": "(-i)\u00c2\u00b2 is in fact equal to -1 However, -i\u00c2\u00b2 = 1 Here is the distinction: (-i)\u00c2\u00b2 = (-1)\u00c2\u00b2 * (i)\u00c2\u00b2 = (1) * (-1) = -1 -i\u00c2\u00b2 = (-1)* i\u00c2\u00b2 = (-1)*(-1) = 1 So, the answer depends on whether the coefficient of -1 is included in the squaring."
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "Wait, at 2:29, in the radical, Sal has +36, if b^2 is -6, it should be -36, if I'm right.",
        "A": "b=(-6) so therefore, (-6)^2 is 36. Hope that helps"
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "At 3:16 - 3:17, does Sal mean that the principle square root of -1 is i. If he does, then can't at 5:08 - 10:15, he just re-write the i as the sqrt of -1?",
        "A": "That would be un-simplifying the expression. i is commonly used as the \u00e2\u0088\u009a-1. From this point on, you won t really see i written as \u00e2\u0088\u009a-1. If you re not used to that, then watching these videos will definitely help you do so. I hope this explains why Sal didn t substitute \u00e2\u0088\u009a-1 for i at the end!"
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "Oh, and would somebody please take the time out of their day to explain from 5:09 onward?",
        "A": "He just wants to prove that the solutions for x are indeed correct, so to do that you have to replace x in the original equation for the numbers you get out of the quadratic formula, you get 2 of those, hence the 2 demonstrations, one for (3+i)/2 and (3-i)/2"
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "I didn't catch why when raising the (3-i) to the power of two [at 7:25] how it becomes 6i. He mentioned that he added 3 to -1i and multiplied 2 into 6i. Basically i don't understand why he ends up with 9-6i-1 when in my head I would get 3+1",
        "A": "He has two binomials: (3 - i)(3 - i) To multiply them, you use FOIL. Multiply: 3(3) + 3(-i) -i(3) -i(-i) = 9 - 3i - 3i -1 Combine like terms: 8 - 6i Hope this helps."
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "At 3:30, he made -4 become -1(4).\nMy question is:\n\nI thought\n Sqrt(ab) = Sqrt(a) x Sqrt(b) when a and b are above zero, but he used -1 which is not above 0\n\nThen the property should be Sqrt(ab) = Sqrt(a) x Sqrt(b) when a and b are above -1. Right?",
        "A": "The property is this: \u00e2\u0088\u009a(ab)=\u00e2\u0088\u009aa\u00e2\u0080\u00a2\u00e2\u0088\u009ab when a and b are both positive or 0, or when exactly one of them is negative."
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "At about 3:37 in the video Mr. Khan divided the numerator and denominator by 2. Why did he do this when the bottom number is 4? Isn't that the number you would divide by. Where did he pull a 2 from?",
        "A": "Does this answer your question? : (6 +/- 2i)/4 means either (6 + 2i)/4, or (6 - 2i)/4 (6 + 2i)/4 = (1/4)*(6+2i) = 6/4 + 2i/4 (distributive law) = 3/2 + i/2 (6 - 2i)/4 = (1/4)*(6-2i) = 6/4 - 2i/4 (distributive law) = 3/2 - i/2"
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "At 8:20, Sal says that (-i)^2 is -1. Why is this the case?\n\nI understand the definition of i^2 = - 1, but not quite this part.",
        "A": "It is the same as squaring any other negative. (-a)\u00c2\u00b2 = (-1)\u00c2\u00b2(a\u00c2\u00b2) = a\u00c2\u00b2 Thus (-\u00f0\u009d\u0091\u0096)\u00c2\u00b2 = (-1)\u00c2\u00b2(\u00f0\u009d\u0091\u0096\u00c2\u00b2) = (1)(\u00f0\u009d\u0091\u0096\u00c2\u00b2) = \u00f0\u009d\u0091\u0096\u00c2\u00b2 = -1"
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "At 3:44 why does the 2 after the plus or minus sign get eliminated? I understand simplifying 6/4 to 3/2, but wouldn't eliminating the 2 on numerator eliminate the 2 denominator since 2/2=1? Please explain. This step has always confused me!",
        "A": "(6 +/- 2i)/4 means either (6 + 2i)/4, or (6 - 2i)/4 (6 + 2i)/4 = (1/4)*(6+2i) = 6/4 + 2i/4 (distributive law) = 3/2 + i/2 (6 - 2i)/4 = (1/4)*(6-2i) = 6/4 - 2i/4 (distributive law) = 3/2 - i/2"
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "At 0:41, I noticed that Sal re-ordered the equation. Does it really matter that he does that because I use it with the order not like Sal has it, butI know the equation good enough to use it and get the correct answer.",
        "A": "It should be fine to do it as long as you abide by the rules of math."
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "3:25, how is sqrt of -4, 2i? wouldn't it be 4i?",
        "A": "You need to take the square root of the 4 itself. This = 2. 2^2 = 4, thus sqrt(4) = 2. To get 4i, you would need to have sqrt(-16)."
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "5:50 why is i^2 = -1",
        "A": "the complex number i is defined as the square root of -1 Therefore i^2 is the square root of -1 x square root of negative 1 You re left with i^2 = -1"
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "i get most of what Sal is saying here, the one thing that confused me was at 08:23 sal said that (-i)squared = -1, i thought i squared was -1? can anyone help me",
        "A": "-i is equal to -1*i, so with exponent laws, that is equal to (-1)^2*i^2, (-1)^2=1, so that is equal to i^2=-1."
    },
    {
        "video_name": "dnjK4DPqh0k",
        "Q": "Im am extremely confused on how you get \"i\" on this, 3:19",
        "A": "There is no such thing as the square root of -1, so people use the variable i to mean the square root of -1. So when Sal separates the equation, he is doing the square root of -1 * the square root of that number. EX. sqrt(-9)= sqrt(-1*-9)=sqrt(-1)*(sqrt(9)= 3*i= 3i"
    },
    {
        "video_name": "Tqpcku0hrPU",
        "Q": "Why is divided by 1/8 when the exponent decrease? 1:23",
        "A": "I think you misheard him. He said 1/a. When you decrease the exponent of a by 1, you re dividing by a (or multiplying by 1/a, same thing) because there s one less multiplied a when you do this. Example: a^4--->a^3 a*a*a*a--->a*a*a There s one less a. The change was a multiplication of 1/a because the exponent decreased by 1. I hope this makes sense now!"
    },
    {
        "video_name": "Tqpcku0hrPU",
        "Q": "At 1:52, Sal talked about a thing called Pie(?)\nwhat is that?",
        "A": "Pi is 3.14159265 which goes on forever like infinity but just a number."
    },
    {
        "video_name": "Tqpcku0hrPU",
        "Q": "at 2:13, why would a to the 0 not just equal a? just because your going backwards with an exponent does not mean that you need to start dividing. a multiplied by itself 0 times would just equal a, right? Why do you suddenly divide when you hit 0? I'm so confused...",
        "A": "a^1 = a The exponent of 1 means you have 1 occurrence of a Remember, exponents are repetitive multiplication. To decrease an exponent, you would need to use the opposite of multiplication, which is division. To get from a^3 to a^2, you divide by a . To get from a^1 to a^0, you divide: a/a = 1 Hope this helps."
    },
    {
        "video_name": "Tqpcku0hrPU",
        "Q": "i dont understand what he's doing at 0:48",
        "A": "Sal is repeating patterns to show why a x a = a^2 and why a x a x a = a^3 and vice versa e.g. why a^-2 = 1/a^2"
    },
    {
        "video_name": "Tqpcku0hrPU",
        "Q": "So at 0:12 is 1/(a^b) the same as (1/a)^b?\nThanks",
        "A": "Yes, because if you take the whole fraction 1/a to the power of b, one to any power is always one, so the top will stay 1, and the bottom will be a^b."
    },
    {
        "video_name": "Tqpcku0hrPU",
        "Q": "At 3:04 why did he divide a^0 by a instead of by 1/a?",
        "A": "Well, a^1 = 1 x a = a a^0 = 1 x nothing = 1. In other words, 1 times zero a s a^-1 = 1 x 1/a = 1/a I can t remember where it is, but somewhere Sal explains that a dividing by a fraction is equal to multiplying by it s reciprocal. In other words: 4 x 2 = 8 4 / 2 = 2 4 x (1/2) = 2 4 / (1/2) = 4 x 2/1 = 8 Ain t math the best? ;) EDIT: if he wanted to use the fraction form, it d be a^0 x 1/a, OR a^0 / (a/1)"
    },
    {
        "video_name": "Tqpcku0hrPU",
        "Q": "at 2:33 why a^-0 is equal to 1?",
        "A": "Because to go from a^1 = a to a^0 you must divide by a (or multiply by 1/a which is the same.) a/a = (a x 1/a) = 1."
    },
    {
        "video_name": "Tqpcku0hrPU",
        "Q": "At 1:52, Sal mentions pi. I have never heard of this! Could someone please explain??",
        "A": "Pi is just the name of a number. It s roughly 3.142. You will see it a lot because it crops up in almost anything to do with circles and geometry. It s a measure of how many times the circumference of a circle (all the way around the circle) is larger than the diameter (a line straight across the middle). The number works out the same for any circle at all, so it s been given its own name: pi, which is the Greek letter equivalent to the English letter p ."
    },
    {
        "video_name": "ESSMQH6Y5OA",
        "Q": "at 2:57 why (KxN)(NxK)=KxK",
        "A": "If you have 2 matrices and you want to multiply them, you ll have one that s an nxm matrix and another that s an mxk matrix. When you multiply them, you dot the rows of the first matrix with the columns of the second. A is nxm B is mxk A = \u00e2\u008e\u00a1a\u00e2\u0083\u0091\u00e2\u0082\u0081\u00e2\u008e\u00a4 \u00e2\u008e\u00a2a\u00e2\u0083\u0091\u00e2\u0082\u0082\u00e2\u008e\u00a5 \u00e2\u008e\u00a2..\u00e2\u008e\u00a5 \u00e2\u008e\u00a3a\u00e2\u0083\u0091\u00e2\u0082\u0099\u00e2\u008e\u00a6(row vectors of A) B = [b\u00e2\u0083\u0091\u00e2\u0082\u0081 b\u00e2\u0083\u0091\u00e2\u0082\u0082 ... b\u00e2\u0083\u0091k] (column vectors of B) AB = \u00e2\u008e\u00a1a\u00e2\u0083\u0091\u00e2\u0082\u0081\u00e2\u0080\u00a2b\u00e2\u0083\u0091\u00e2\u0082\u0081 a\u00e2\u0083\u0091\u00e2\u0082\u0081\u00e2\u0080\u00a2b\u00e2\u0083\u0091\u00e2\u0082\u0082 ... a\u00e2\u0083\u0091\u00e2\u0082\u0081\u00e2\u0080\u00a2b\u00e2\u0083\u0091k\u00e2\u008e\u00a4 \u00e2\u008e\u00a2a\u00e2\u0083\u0091\u00e2\u0082\u0082\u00e2\u0080\u00a2b\u00e2\u0083\u0091\u00e2\u0082\u0081 a\u00e2\u0083\u0091\u00e2\u0082\u0082\u00e2\u0080\u00a2b\u00e2\u0083\u0091\u00e2\u0082\u0082 ... a\u00e2\u0083\u0091\u00e2\u0082\u0082\u00e2\u0080\u00a2b\u00e2\u0083\u0091k\u00e2\u008e\u00a5 \u00e2\u008e\u00a2... ... ... ...\u00e2\u008e\u00a5 \u00e2\u008e\u00a3a\u00e2\u0083\u0091\u00e2\u0082\u0099\u00e2\u0080\u00a2b\u00e2\u0083\u0091\u00e2\u0082\u0081 a\u00e2\u0083\u0091\u00e2\u0082\u0099\u00e2\u0080\u00a2b\u00e2\u0083\u0091\u00e2\u0082\u0082 ... a\u00e2\u0083\u0091\u00e2\u0082\u0099\u00e2\u0080\u00a2b\u00e2\u0083\u0091k\u00e2\u008e\u00a6"
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "At 2:44, what would you do if the percent had a decimal? Like, if you had 17.78%, then would you write it as 17.78/100?",
        "A": "no you would not."
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "At 1:55, how about if we need to find a percentage 9/50, well, how do you do that.",
        "A": "That works, but with the state of this question, the denominator being 50, a factor of 100, I would just multiply both the numerator and denominator by 2."
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "In 2:24, what does hunoreohus mean?",
        "A": "Hundredths. n/100 is equivalent to saying n hundredths."
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "at 1:06 the guy puts 18 hunoreoths. Is that 18 hunderths or what?? If he said it,sorry i dont hear since this computer has no volume.",
        "A": "Inaya, Sal was trying to write hundredths . His d did look like an o . Sorry it caused you confusion. Sometimes it is hard to write legibly on a computer screen. The vidoes that Sal has made for the world to use are amazing, but not always perfect."
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "At 1:20....Why would it be 18/hundreth and not 18/hundred Sal never said why it wouldn't be HUNDRETH and not HUNDRED because its physically over a HUNDRED!",
        "A": "its hundredth and not hundred because its after the decimal. Its hundred if its before the decimal"
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "at 1:23 why mr.sal said 18 hundreoths , it looks confusing ! can someone help me? thnx",
        "A": "Dear William, What Sal means is the decimal divides the wholes numbers like 1, 10, 100 etc. and the decimals which start at the tenths, I take the decimal point as a oneth except that sounds weird, right? So eight hundredths is just like the hundreds except it s behind the decimal point, so to sperate the two they put ths at the end Hope this helps, Ethan"
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "At 2:40, do you have to make it in decimal form?",
        "A": "Well, the question asked for the decimal for of 18%. It is from every per cent or 0.18 because it is 18 out of 100."
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "At 2:09, it says 9/50 is in the simplest form, but if you want to get the decimal as the final answer, can't you just divide 9 by 50 to get 0.18/1 as a fraction, or does a fraction have to only be a whole number?",
        "A": "A fraction always has an integer in both the numerator and denominator positions."
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "isn't 18/100 .18, not 18.0 at 2:36?",
        "A": "In the video, it is written 18%=18/100=18.0%=0.18, which is true."
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "At 0:32 how does cent mean 100. Isn't cent usually used for money??",
        "A": "Cent is also a root word for measurement and such. For example, CENTimeters. Also, cent is 100 in french"
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "At4:20,how do I turn 1/6 into a percent?",
        "A": "There is not a 4:20 in this video,this video only goes intill 3:32 but I can tell you how to turn 1/6 into a percent. 1/6 as a decimal is about .167. You move the decimal to the right two spaces and we get 16.7..... put a percent sign next to it and we get 16.7% as the answer"
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "0:08: How would you do it if the 100 had a decimal? As such, 18/100.02.",
        "A": "I do not see how you tripped me up. I fully understand that."
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "at 1:38 what is a numerator and a denomonator",
        "A": "Numerator is the top number in a fraction Denominator is the bottom number in a fraction. In the fraction 2/3, the numerator is 2 and the denominator is 3."
    },
    {
        "video_name": "-gB1y-PMWfs",
        "Q": "In the video at 0:47 to 0:55 it what a little funny that he can't make mind",
        "A": "Shiela went shopping and spend $120 on a coat. She then used 2/5 of the remaining money to buy a dress. She is left with 1/5 of her original amount of money. How much money did shiela have at first ? how to resolve this with algebra and without algebra. Kindly assist"
    },
    {
        "video_name": "s8_14yxp1lQ",
        "Q": "At 2:47, Sal says that 66/21 times 21/66 m equals one what do we do with the one do we just leave it?",
        "A": "Yes, you can just leave the 1 or drop it because 1m and m are the same thing. Anytime you have variable and there is no number in front of it, the number is understood to be a 1. Hope this helps."
    },
    {
        "video_name": "FksgVpM_iXs",
        "Q": "It's been a while since I've watched these videos... Does someone mind explaining why there are no solutions? (@4:19 Sal says 'there is no solution to the system')\n\nI was doing this in my notebook, shouldn't there be two solutions?\nx = 3 +/- (2i * sqrt(30))/6\n\nThanks!",
        "A": "Algebraically, since you have an imaginary number there is no real solutions. There are two solutions...the issue is that they are imaginary. When he demonstrated graphing the two parabolas he demonstrates that they do not (and will not) intersect. The only solution(s) to a system is when any part equation intersects with the other equation and since they don t there is no real solution (only imaginary). Hope that helps."
    },
    {
        "video_name": "FksgVpM_iXs",
        "Q": "At 1:15 how do you get that second equation?",
        "A": "Sal just expanded the (x-4)^2 so that it became (x-4)(x-4), which then became x^2-8x+16 and fitted that into the equation."
    },
    {
        "video_name": "FksgVpM_iXs",
        "Q": "5:42 How did that +1 and -1 come from?",
        "A": "This is part of the technique called completing the square , which there are videos on if you d like to review it. Simply put, the idea is to add some term which allows you to factor the quadratic into a perfect square, but you can t just simply add a term, you also have to subtract it as well in order to keep the equation the same."
    },
    {
        "video_name": "FksgVpM_iXs",
        "Q": "At 5:42 what rule did he use to add the -1 +1 ,I could never had thought about that which is very neat but I don't know how he thought of that what rule of factorisation is that.",
        "A": "Most of the operations he uses on the second equation are part of a process called completing the square. I m sure there are videos here on how it works. If you find this interesting you re going to start having lost of fun in the following sections."
    },
    {
        "video_name": "FksgVpM_iXs",
        "Q": "5:27 why is he changing the signs on the original equation? It was Y= -x^2 +2x -2 so shouldnt be Y= -1 (-x^2 +2x -2) ? How can he just swap the signs like this? Somebody please help!",
        "A": "you can multiply the whole equation by negative 1 on both sides if you want to change the signs. Some people prefer not to work with a negative x^2 term so they change it. Remember you re gonna set the quadratic equal to zero to find a solution so multiplying by negative 1 doesn t affect that side."
    },
    {
        "video_name": "FksgVpM_iXs",
        "Q": "Ok I keep having issues when I decide to divide out a -1 instead of factor it out.\nSal factored out the -1 for the 2nd equation and got y = - ( x-1 ) ^2 -1 ( 6:17 )\nBut I got y = (x - 1)^2 + 1.\n\nIf I make x = 1, I get y = 1, (1, 1) which is wrong according to Sal's parabola. Why is this?",
        "A": "You re changing the y value so for whatever x value you put in, you get the opposite integer. It s a reflection across x = -1."
    },
    {
        "video_name": "kjNgXVZ1aZ4",
        "Q": "At 4:40 Grant writes cos^2 + sin^2 while it should be cos^2(t) + sin^2(t).",
        "A": "He does so every time this identity appears. I don t think it s wrong it s just laziness"
    },
    {
        "video_name": "K_fgnCJOI8I",
        "Q": "Smashing lesson, as usual! I am using the summer to refresh a little my basic math, and your videos are truly excellent.\n\nBy the way, I am seriously considering burning a dvd with all your lessons as a present for my little cousin. But if you, directly or indirectly, sold pre-made dvds with your lessons then I (and, I suppose, many others) would be happy to support your work in this way. Just a suggestion, of course...\n\nA small issue I have at about 05:47:\n\n\"The gradient of F(x,y) is going to",
        "A": "Gah, short question boxes. Here is the real question. A small issue I have at about 05:47: The gradient of F(x,y) is going to be a vector field that tells you the direction of steepest descent . Maybe I am mistaken (it s been far too long since I studied basic calculus), but shouldn t the gradient tell us the direction of steepest ascent instead? Thanks a lot for your work! Pietro P.S. Do you plan to add some lessons about Fourier and Complex analysis later on?"
    },
    {
        "video_name": "K_fgnCJOI8I",
        "Q": "couple questions\n\n1 ) at min 6:45 shouldn't the yellow surface he painted be f(x,y) in lower case? since the gradient field, F(x,y) is the field pointing in the direction of fastest increase of surface f(x,y) ?",
        "A": "He defined F(x, y) as the function, and f(x, y) as the gradient of F. F is the scalar field, and f is the vector field."
    },
    {
        "video_name": "K_fgnCJOI8I",
        "Q": "At about 10:10 . Where did the unit vectors go? Thank you.",
        "A": "Both f and dr have the unit vectors in them. Thus, when you dot the two together: i.i = |1|^2 = 1 j.j = |1|^2 = 1 Hope that helps."
    },
    {
        "video_name": "XaJQse2u5TQ",
        "Q": "at 3:21, Why did you write\n(1/5 + 1/5) + (1/5 + 1/5) + (1/5 + 1/5)\ninstead of just telling us?",
        "A": "Looks like he did it to explain what multiplying by 3 means. He has explained multiplication in the earlier math lessons so I also suppose he is doing it for repetition s sake in case someone didn t remember."
    },
    {
        "video_name": "_3BnyEr5fG4",
        "Q": "at 4:40, is it four times negative four or four ex minus four?",
        "A": "4*(x-1) = 4*x + 4*-1 = 4x-4"
    },
    {
        "video_name": "_3BnyEr5fG4",
        "Q": "At 8:12, why wouldn't the line be defined with zero?",
        "A": "The equation of the line works for any real value of x, but the growth pattern starts with x=1 block, then x=2 blocks, ..., so only positive integers are being used."
    },
    {
        "video_name": "_3BnyEr5fG4",
        "Q": "At 6:10 how did going from one to two get expressed as 2-1?",
        "A": "In the slope formula, (y2-y1)/(x2-x1), The 2 is the y2 and the 1 is the y1. And, naturally, the 5 is the x2 and the 1 is the x1. By the way, I m answering this for anyone else who may have this question as you probably already figured it out!"
    },
    {
        "video_name": "_3BnyEr5fG4",
        "Q": "In 7:42 he says that the 4 and -4 cancel each other out, I didn't know that you could do that. Not that I am sad about it, I just don't get it.",
        "A": "The 4 and -4 cancel because they were being added together. Another way to write what he did would be 1 = 4 + b (1) - 4 = (4 + b) - 4 1 - 4 = 4 - 4 + b -3 = (4 - 4) + b -3 = (0) + b (see how the 4 and -4 combine for 0?) -3 = b"
    },
    {
        "video_name": "_3BnyEr5fG4",
        "Q": "at 5:32 what does linear mean",
        "A": "It simply means that if you graph the equation it s going to be a straight line. It s a simple as that, LINEar = line. :)"
    },
    {
        "video_name": "FKJjqEdfB9Y",
        "Q": "I'm confused. At 0:35, why did you put that orange line there? What does it represent? How did you get it?",
        "A": "That s the width of the rectangle"
    },
    {
        "video_name": "-lWVpoPaPBc",
        "Q": "At 2:44, would the zero product property still work if you divide the equation by (x-4) and solve from there?",
        "A": "It would not be the use of the zero product property, but it would work the same because it applies the rule that 0 divided by any number is zero. You would have to do it twice anyway because you would also have to divide by (2x+1)."
    },
    {
        "video_name": "PsNidCBr5II",
        "Q": "At 16:30 the c > 0 should be c >= 0. Clearly ||x + Z|| = ||x|| + ||Z|| = ||x|| where Z is the zero vector. Was the c >= 0 a typo, or was Sal intending some other subtle point that he apparently missed?",
        "A": "No. Per the Cauchy-Schwarz inequality, if x = c*y, then c must be positive, and it is already implied to be non-zero. I think that s what you re getting at."
    },
    {
        "video_name": "PsNidCBr5II",
        "Q": "Around 14:50; why do you necessarily know that the y-vector is also positive? Obviously, c needs to be positive; but, again; why is it a given that the y-vector is positive?",
        "A": "Vectors aren t positive or negative. They have a direction, but y could have all positive components, some positive and some negative, or all negative components. We aren t assuming anything about y."
    },
    {
        "video_name": "PsNidCBr5II",
        "Q": "at around 7:24 you talk about dot products being negative but how could a scalar quantity be negative? isnt negative just used to specify direction of a vector?",
        "A": "A scalar is just a number, that includes zero, all positives and all negatives. The direction of the vector can be specified in many ways, when representing the vector in cartesian coordinates, the directions is given by the relation of each of the components of the vector, so a single sign does not completely specify the direction. When you multiply a vector by -1, the effect is to reverse the vector."
    },
    {
        "video_name": "PsNidCBr5II",
        "Q": "When Sal was doing the dot product of the two binomials (x+y)^2 (x and y being vectors) I wasn't much of a fan how he did it. I was always taught to use the FOIL method when multiplying two binomials and I just think the way he explained it and the way he used the distributive property was lengthy and confusing. In fact, in most of the videos I've watched, I haven't seen him use the FOIL method. Is there a particular reason why he avoids it or is it just a preference? (around 3:40 in the video)",
        "A": "I ll tell you why I use Sal s method: because I find it more intuitive. When I find something intuitive, an mnemonic like FOIL is just one more thing to memorize. But if FOIL helps to keep track of what you re doing, by all means use it. For me, it s a matter of preference. I suspect that s what s going on here as well."
    },
    {
        "video_name": "PsNidCBr5II",
        "Q": "At 8:42, why is x dot y replaced with the product of length x and length y?\nThis seems to be the main substitution that give us the inequality... but I don't understand when he says \"this whole term has to be less than this whole term\"",
        "A": "x \u00c2\u00b7 y is always less than or equal to ||x|| ||y||."
    },
    {
        "video_name": "PsNidCBr5II",
        "Q": "At 08:40 mr Kahn said \"this (i.e., 2(x*y)) is definitely less than that (2* absolute(X)* absolute(Y))\"\nmustn't it be less than or equal too?",
        "A": "Yes, this is true. I just reported an error in the video for this exact thing."
    },
    {
        "video_name": "PsNidCBr5II",
        "Q": "At 5:38, Sal says \"I should stop using the term magnitude.\" Why?",
        "A": "In 1 or 2 dimensions the vector is graph-able, and length communicates the idea in a more commonly understood way. The terms are interchangeable (personally, I was taught to use magnitude)."
    },
    {
        "video_name": "l6T4RhlgkG0",
        "Q": "I am confused. What about a function with a negative slope. Like for example, y = -x +10.\nClearly, as the change in x approaches zero, the function gets bigger.\n\n5:50",
        "A": "If the function has a negative slope, as dx approaches zero, u(x) does get bigger. However, in this case u(c) > u(x), and our expression lim x\u00e2\u0086\u0092c {u(c) - u(x)} still goes to zero."
    },
    {
        "video_name": "udG9KhNMKJw",
        "Q": "At 2:53, could you put HA in any of the bottom spaces? If you can't, why?",
        "A": "Yes, HA could be put in any of the bottom spaces. He is merely pointing out one possibility of where HA, HB and HC could turn up."
    },
    {
        "video_name": "udG9KhNMKJw",
        "Q": "I don't quite get 6:47 how he got 6/15",
        "A": "Wait. I thought it was 6/15. Whats with 5/16"
    },
    {
        "video_name": "udG9KhNMKJw",
        "Q": "At 00:52 Sal calculated total 32 equally likely possibilities. Are there no renundancies/ duplicates (different orders such as HHHTT and TTHHH which we consider as one combination) in these outcomes which we need to eliminate. It is only in the numerator that we eliminate such duplicates.",
        "A": "This was confusing for me first. But remember: What is the chance of getting any kind of specific combination?`For instance what are the chances of getting 4 heads and a tail? Well its (1/2) raised to 5 it is one possibility out of total 2 raised to the 5."
    },
    {
        "video_name": "a3acutLstF8",
        "Q": "How do you know to divide it instead of multiply? 1:39",
        "A": "Look at the relationship between the variable and the number. We always do the opposite. For example: 1.2x is 1.2 times x . You use division (opposite of multiplication). y/12 is y divided by 12 . You use multiplication (opposite of division). Hope this helps."
    },
    {
        "video_name": "a3acutLstF8",
        "Q": "At 1:49, why did he multiply each number by ten?",
        "A": "So that it would be easier to divide 0.6 into 1.2 like 6 into 12 which is easily verifiable as 2"
    },
    {
        "video_name": "a3acutLstF8",
        "Q": "Around 5:20, instead of giving the big answer for why we are adding 10/10 to only one side of the equation, you could have just said that the fraction 4.5/0.5 would be equivalent to 45/5 since were multiplying both the numerator and the denominator by 10 so if we still divide 45/5, it would be equal to dividing 4.5/0.5",
        "A": "Sal didn t say anything about adding. He said to multiply the left side with 10/10 (which is equal to 1). And also I think he meant to say what you said. I hope I answered your question!"
    },
    {
        "video_name": "V0xounKGEXs",
        "Q": "At 1:41: Can a parallel line ever have an obtuse, acute, or right angle?",
        "A": "no because angles intersect and are not parallel"
    },
    {
        "video_name": "V0xounKGEXs",
        "Q": "At 2:00, wasn't Sal talking about intersecting lines?",
        "A": "Yes, he was. The lines were not parallel nor perpendicular (intersect at a right angle) so they are intersecting lines."
    },
    {
        "video_name": "V0xounKGEXs",
        "Q": "Around 1:31 the person starts talking about parallel lines. Does a certain amount of space have to be between the lines to be parallel?",
        "A": "Nope. As long as they have the same slope (and as long as they re not the same exact line), they are parallel."
    },
    {
        "video_name": "V0xounKGEXs",
        "Q": "It is 1:32 and why does not parallel ines never intesect each other?",
        "A": "Because that s what parallel means: they NEVER intersect. Think of railroad: the lines go in the same direction, but they don t intersect. Of course, they will intersect over time, but you get what I mean."
    },
    {
        "video_name": "V0xounKGEXs",
        "Q": "Why is it necessary that the angle where they intersect should be a right angle like Sal told at 0:56 ?",
        "A": "He s just making sure that you know what the word perpendicular means. He s defining it. Two lines that are perpendicular have an angle of 90 degrees. 90 degrees is another way to say it is a right angle ."
    },
    {
        "video_name": "V0xounKGEXs",
        "Q": "At 0:49, how do we know the right triangle?",
        "A": "if a triangle has a 90 degree angle in any corner it is a right triangle"
    },
    {
        "video_name": "V0xounKGEXs",
        "Q": "At 1:55 Sal draws a line, He says that it is nether parallel nor perpendicular, What is it?",
        "A": "A line. You see there is nothing next to it or joining it... Hope that helps..."
    },
    {
        "video_name": "Muba9-W2FOQ",
        "Q": "At 2:36, when Sal makes the graph, does it matter on which axis that the patterns go on? Or does it have to go in a specific place?",
        "A": "Pattern B does have to go on the X axis (horizontal line) and Pattern A must go onto the Y axis (Vertical line). Hope this answers your question!"
    },
    {
        "video_name": "Muba9-W2FOQ",
        "Q": "can someone help me i got confused at 3:20 to 3:50",
        "A": "Sal is just explaining how the relationships can be plotted on the Coordinate graph."
    },
    {
        "video_name": "iimpwYBiKNg",
        "Q": "At 1:46 why do you divide each side by 6?",
        "A": "to take away 6w"
    },
    {
        "video_name": "iimpwYBiKNg",
        "Q": "At 0:50 - 1:04 why do you add it ?",
        "A": "Because he needed to know what is w. what he knew was that length is twice the width. So he marked width as w and lenght as 2*w(lenght is twice that widht. He did not know what w is both he knows what width and lenght is together(60) . Solution to this is simple equation in this case w(left side) + 2w(bottom) + w (right side) + 2w(top) equals to 60. in other words the perimeter is sum of all of sides together. I hope this helps."
    },
    {
        "video_name": "iimpwYBiKNg",
        "Q": "At 1:30, why do you divide by 6?",
        "A": "To isolate the w (representing the width), which allows us to find out what w is equal to (in this case 60/6, which equals 10). For following explanation, P = perimeter, w = Width, and L = length: The reason we have 6w is because P = w + w + L+ L, and we know that the length is twice the width, so L = w + w . So, P = w + w + w + w + w + w, or P = 6w"
    },
    {
        "video_name": "F2mfEldxsPI",
        "Q": "At 4:12, when he is subtracting by the population mean, the denominator shouldn't be N instead of n since we are talking in this part of the formula about the population and not just the sample?",
        "A": "He says he is actually subtracting from the sample variance a pseudo-sample variance, using the true mean but changing nothing else. Therefore, the denominator is n for both expressions for the red graph (and would be n-1 for the blue s and n-2 for the green s). If he was finding the difference between the sample and population variances, you would be correct. But for this simulation, he uses the pseudo-sample variance to best demonstrate the unbiased estimate."
    },
    {
        "video_name": "F2mfEldxsPI",
        "Q": "at 4:23 it should have been \u00ce\u00a3(xi - x\u00cc\u0085)^2/N - \u00ce\u00a3(xi - x\u00cc\u0085)^2/n, isn't it?",
        "A": "\u00ce\u00a3(xi - x\u00cc\u0085)^2/n - \u00ce\u00a3(xi - \u00ce\u00bc)^2/N, yep. Sal makes a lot of typos."
    },
    {
        "video_name": "lpXCvX6ZenM",
        "Q": "At 6:12 he mentions that thinking of (dv/dk)(dk/dt)=(dv/dt) as fractions cancelling out isn't 'rigorous' (a notion he always repeats when doing this calculation. Is there a video in which he explains a more rigorous way to think of this?",
        "A": "I dont think so. In general, it seems that he prefers to use intuitive, geometric arguments to justify mathematical results. If you want a more rigorous discussion, try out Chris Tisdell on Youtube."
    },
    {
        "video_name": "A7zPgMV4eTM",
        "Q": "So in 1:41, do you get the answer by duplicating the squares, like Sal did? Or do we do it by visualizing how many units can be packed inside the figure? This is all very confusing.",
        "A": "You visualize how many are packed in the figure."
    },
    {
        "video_name": "QWLcNxQ3KvQ",
        "Q": "@2:37 Sal inputs 4 and gets mapped 1 and -1, which he can mark on the graph. But what if x (for x=y^2+3) is equal to a number less than 3? The graph doesn't go that far back.",
        "A": "Ah, that s because for values of x that are < 3, you will be in the position of trying to take the square root of a negative number, which we cannot do...if y = sgrt(x -3) and we try, say, x = 2, you get y = sqrt(2 - 3) or y = sqrt(-1) which yields no solution."
    },
    {
        "video_name": "QWLcNxQ3KvQ",
        "Q": "At 2:35 Sal says that the shape cannot be a function. In that case, how do you explain parabolas?",
        "A": "First off, that shape is a parabola, albeit a sideways parabola, but it can not be a function of x because it fails the vertical line test. It can however be a function of y since it passes the horizontal line test. A traditional parabola that you are thinking of is a function of x because it does pass the vertical line test."
    },
    {
        "video_name": "QWLcNxQ3KvQ",
        "Q": "If we graph a the function y = principle square root of (x - 3) then we only have the top part of the graph as shown @ 1:57. Now is that a function of x because there is no value of y when x = 0?",
        "A": "It is a function. Its domain (= what x-values it takes) is x \u00e2\u0089\u00a5 3."
    },
    {
        "video_name": "QWLcNxQ3KvQ",
        "Q": "As an example to show it is not a function an x-value is chosen to show that it corresponds to two y-values (2:32). However, could you not argue that if you were to look at the x-value of 3 there is only one y-value associated with it (0)?",
        "A": "Yes, but for all other x-values there are multiple y-values for other, so it is still not a function."
    },
    {
        "video_name": "TPov_v9mLWc",
        "Q": "At 5:24, why is the unit normal vector equal to k-hat. Is it because k-hat is normal to the surface, since k-hat is in the z-direction? To find the normal vector in the case of a region on the xy-plane, where z = 0, we didn't need to calculate the cross product to find out that k-hat was normal to the region right?",
        "A": "Yes, any upward looking surface in the xy plane will have a normal vector straight up with magnitude 1, with is the exact definition of the unit vector k\u00cc\u0082. And not only is this the normal vector of any surface on the xy plane, but also of any surface parallel to the xy plane, so any (upwards looking) surface with a constant z-coordinate will also have k\u00cc\u0082 as it s normal vector."
    },
    {
        "video_name": "DROZVHObeko",
        "Q": "At 5:03 Sal divides 5! with 2!. Can't he just divide with 2?",
        "A": "Yes, Sal could have just wrote 2, but in a different problem n! will not always equal n. Therefore, Sal wrote it as 2! in order to possibly avoid confusion in different problems. But yes, he could have wrote 2, but I would remember it is as 2! for understanding."
    },
    {
        "video_name": "9McJ3GobPaY",
        "Q": "Why did he put quotation marks around ones at 1:20?",
        "A": "well, ones isn t really a thing. It is a unit, but it does not name an object. The quotations show that he isn t counting ones (he is counting plums), but he is counting that many single plums. Hope this helps!"
    },
    {
        "video_name": "n34dqyVCXs4",
        "Q": "at 1:59 I don't get how you get 4xsquared. Is it because xsquared is negative?",
        "A": "Combining like terms. -x\u00c2\u00b2+5x\u00c2\u00b2=4x\u00c2\u00b2 note: -x\u00c2\u00b2 is the same as -1x\u00c2\u00b2 Is it because xsquared is negative? Yes"
    },
    {
        "video_name": "n34dqyVCXs4",
        "Q": "At 2:18, why don't you add 4x^2 and 2x?",
        "A": "Because x^2 and x are two different things. Its kind of like apples and oranges as far as the two. Same with the reason he doesn t add 2x and 10."
    },
    {
        "video_name": "n34dqyVCXs4",
        "Q": "At 1:56, wouldn't the -x^2 and the 5x^2 cancel out to just 4?",
        "A": "When we add/subtract like terms, we only add/subtract their coefficients (the numbers in front). The variables do not change. -1x^2 + 5x^2 = (-1+5)x^2 = 4x^2 The X s do not cancel out. The only way the variables would disappear is if the coefficient becomes 0: 0x^2 = 0*x^2 = 0."
    },
    {
        "video_name": "n34dqyVCXs4",
        "Q": "towards the end at 2:10 .. Sal says \"There are two constants i.e. 0 degree terms\"\nIs he referring to the power of 9 and -1 ?\nIf so .. 9 to the power of 0 i.e. [9^0] would be equalt to 1 right??\nas anything to the power 0 is one...",
        "A": "0 degree terms means the variable is raised to 0."
    },
    {
        "video_name": "bJ_09eoCmag",
        "Q": "for video 16:32 there should be negative sign on X parameter.r(90,90)= -bi+ak",
        "A": "It looks like r = +bi + ak is correct as it is, since he defined a left hand coordinate system for this surface about halfway through the video (tricky)"
    },
    {
        "video_name": "gSEYtAjuZ-Y",
        "Q": "what's the thing at 1:03?",
        "A": "A bucket its what you organize all of your data in."
    },
    {
        "video_name": "gSEYtAjuZ-Y",
        "Q": "From 0:51-1:41, he's making the ranges for the histogram. My teacher taught me that you have to find the mean change of all the numbers to make the ranges or \"buckets,\" but Sal didn't do that. Is my teacher wrong, or is Sal wrong?\n\nBasically, do you have to calculate the average change between the numbers in your set to make the buckets or not?",
        "A": "Neither of them are wrong. I don t calculate the range for buckets. However, it is not wrong to do so."
    },
    {
        "video_name": "gSEYtAjuZ-Y",
        "Q": "At 3:14, Sal writes histograph has instead of histogram. Is histograph really a thing?",
        "A": "Its the same thing. You could call it any way you want."
    },
    {
        "video_name": "gSEYtAjuZ-Y",
        "Q": "In what situations would a histogram be 'so much more useful' than a bar graph? Or would this just be a different way to represent data?\n\n**plus at 4:38 he says bar to describe the representation of the data :|",
        "A": "Histograms represent a single thing (e.g. heights of the people in your class) and bar graphs represent discrete categories (e.g. how many people like pizza versus hamburgers versus salad)."
    },
    {
        "video_name": "W-Nio466Ek4",
        "Q": "Couldn't we say at 4:25 that for a^2 to come about we need to have that p again, which must be located in b^2, which means that a and b are not coprime?",
        "A": "We have a^2 from the hypothesis a/b = sqrt(P) , and so (a^2)/(b^2) = P, and a^2 = P(b^2). That s how a^2 comes about. It continues to have the same value. You don t have to get it again ."
    },
    {
        "video_name": "W-Nio466Ek4",
        "Q": "at 4:26, Sal said a^2 is a multiple of p, so a is a multiple of p.\nbut how about this case?:\n6^2 = 36, and 36 is a multiple of 4\n6 is not a multiple of 4.\nhow does this work?",
        "A": "He showed explicitly from prime factorization that it applies when p is prime. For the example you gave, you used 4, which is not prime."
    },
    {
        "video_name": "W-Nio466Ek4",
        "Q": "at 5:32 he says (kp)^2 is equal to k^2*p^2. Shouldn't that, instead, end up being k^2+2kp+p^2? (kp)^2 is equal to (kp)*(kp). And you would just FOIL, right?",
        "A": "What s (2*3)^2 = (2*3)*(2*3)? Is it: 2^2+2*2*3+3^2 = 25 ? 2*3*2*3 = 2*2*3*3 = 4*9 = 36 = 6*6 ?"
    },
    {
        "video_name": "W-Nio466Ek4",
        "Q": "I believe Sal meant at 6:24 B instead of P. Am I right or wrong ?",
        "A": "B and P sound similar, but he didn t really make a mistake."
    },
    {
        "video_name": "9kW6zFK5E5c",
        "Q": "Even though I have watched the videos leading up to this, I don't understand at 14:15 why all a, b and c have to be equal to zero. Any help would be appreciated. Thank you.",
        "A": "Sal is just setting them all to be equal to zero. He can set them to be whatever he want, so he set them to be 0 to make things simple."
    },
    {
        "video_name": "9kW6zFK5E5c",
        "Q": "At 9:20, shouldn't c3 = (1/11)(3c - 5a + b)?\nWhere did the b go?",
        "A": "Im sure that he forgot to write it :) and he wrote it in 10:54"
    },
    {
        "video_name": "9kW6zFK5E5c",
        "Q": "at 16:20 he says that the span of 2 vectors could never span R^3, how is that correct? Isn't it so that only a 3-tuple is required for you to have an R^3 plane? Why 3 of them are needed? 3-tuple meaning a vector with 3 dimensions?",
        "A": "A 3-tuple is a vector with 3 elements. It has dimension of 1. The span of it is a line in 3D (it s also a subspace). If you had two of these vectors and they were independent (e.g. <1,1,0> and <0,1,0>) then you have a subspace with 2 dimensions. This is a plane in R^3. This cannot span all of R^3 either. You can never get anywhere in the z direction using the vectors in my example. It s only when you have a collection of 3 independent vectors that you can get to anywhere you want to be in space."
    },
    {
        "video_name": "9kW6zFK5E5c",
        "Q": "At 15:54 Sal says \"if you have exactly 3 vectors and they do span R3, they have to be linearly independent\"\nSo if you have 3 vectors in R3, and one of them is (0,0,0), are they still linearly independent?\n(this independence thing is messing with my mind!!)",
        "A": "If one of the 3 vectors is the zero vector, they are not linearly independent. However, at the same time, they also do not span R3."
    },
    {
        "video_name": "9kW6zFK5E5c",
        "Q": "For the equations Sal got at 3:54, couldn't you just convert that to an augmented matrix and put the matrix in reduced echelon form and solve a b and c that way?",
        "A": "You could and these concepts are introduced in later videos in the Linear Algebra series."
    },
    {
        "video_name": "9kW6zFK5E5c",
        "Q": "@9:14 Sal you forgot the b from the 2nd equation.\n\n11c3= -5a+b+3c",
        "A": "he corrected it at 10:46"
    },
    {
        "video_name": "9kW6zFK5E5c",
        "Q": "instead of setting the sum of the vectors equal to [a,b,c] (at around 01:53)could you not just set the sum of the vectors equal to zero, prove the set's linearly independent and say that implies the span is R3 as none of the vectors are redundant? finding the equations for c1, c2 and c3 seems pointless, unless i missed something",
        "A": "Sean, I think Sal is trying in this video to relate the concepts of linear independence and span. The earlier videos have covered linear independence and linear dependence.... and they ve also covered span. But none of the earlier videos have proved (to my knowledge, anyways) that to span R^n requires at minimum n linearly independent vectors. Also, none of the videos have covered the concept that n linearly independent vectors always spans R^n."
    },
    {
        "video_name": "9kW6zFK5E5c",
        "Q": "does just putting zeros in at the end really prove linear independence? at 14:40, he just stuck (0,0,0) in, does that really prove that 0 0 0 is the ONLY solution? Sal is a bae",
        "A": "A set of vectors: {\u00e2\u0083\u0091v\u00e2\u0082\u0081, \u00e2\u0083\u0091v\u00e2\u0082\u0082, ..., \u00e2\u0083\u0091v\u00e2\u0082\u0099} is linearly independent if: c\u00e2\u0082\u0081\u00e2\u0083\u0091v\u00e2\u0082\u0081 + c\u00e2\u0082\u0082\u00e2\u0083\u0091v\u00e2\u0082\u0082 + ... + c\u00e2\u0082\u0099\u00e2\u0083\u0091v\u00e2\u0082\u0099 = \u00e2\u0083\u00910(where each c is a scalar) only has a trivial solution. (meaning all of the c s have to be 0). Sal didn t just choose the zero vector at the end, it s part of the definition of linear independence. He said he chose it, but he was really just pretending to. After choosing the zero vector, he shouldn t have been able to solve for every value of c if the set was linearly dependent."
    },
    {
        "video_name": "9kW6zFK5E5c",
        "Q": "at 1:50 can (a.b,c) also be (0,0,0) ?",
        "A": "Yes because technically (a,b,c) don t have a value therefore you are correct"
    },
    {
        "video_name": "9kW6zFK5E5c",
        "Q": "Sal, could you please please please explain why you're just randomly adding one row to another at 6:00? It's very very confusing and hard to follow, especially when you could have you're the time not saying trivial stuff but actually explain the not the obvious stuff..",
        "A": "This is just algebra, ie., if a = b and c = d, then a + c = b + d. The rows are successively added to get them into echelon form to solve the system of equations by back substitution."
    },
    {
        "video_name": "jFazrvLodrA",
        "Q": "I'm confused. At 1:39, you state the function mentioned in the last video (f(x)=x^3+1) is neither even nor odd, but there is y-axis symmetry. Doesn't that make it even?",
        "A": "It is not reflected across the y axis - on the left it s going down on the right it s going up. Also, it s not reflected about the origin, so it s not odd."
    },
    {
        "video_name": "jFazrvLodrA",
        "Q": "At 00:32 what is \"x\"?",
        "A": "Our function says f(x) will be x^2. So whatever term we have for x, we will square it."
    },
    {
        "video_name": "zKotuhQWIRg",
        "Q": "so why is there no solution in 1:50",
        "A": "80 does not equal -14."
    },
    {
        "video_name": "zKotuhQWIRg",
        "Q": "A 1:45, Sal said that there is no solution to the equation. What does this mean if the equation has no solution? It doesn't seem right for this to happen.",
        "A": "It can happen. It just means there is no value that can be assigned to the variable that will make the equation be true. Solutions to equations will always make the equation be true (the 2 sides become the same number when you substitute the value for the variable)."
    },
    {
        "video_name": "2B4EBvVvf9w",
        "Q": "At 31:20 sal says when we subtract sets then we remove the elements, what about the 6 in set B?",
        "A": "There is no sixes in set A so you just ignore the six. By the way, I think you mean 3:20 instead of 31:20."
    },
    {
        "video_name": "2B4EBvVvf9w",
        "Q": "Hi Sal, at 3:55, you mention a notion written down as A\\B. Could you elaborate on on this differs from B\\A?",
        "A": "A\\B: the elements of A that are not in B. B\\A: the elements of B that are not in A. For example if A consists of the elements 1,2,3,4 and B consists of the elements 2,4,6,8 then A\\B consists of the elements 1,2,3,4 except for 2,4 (that are also in B), so that leaves 1,3. And B\\A consists of the elements 2,4,6,8 except for 2,4 (that are also in A),so that leaves 6,8."
    },
    {
        "video_name": "Hoyv3-BMAGc",
        "Q": "3:50 Why the interval is called open interval? Can't we simply use interval? What is the difference?\nFrom this I guess that there should be something called as \"closed interval\". How it is different from open interval?",
        "A": "An open interval uses the parentheses symbols. These tell you that the end points of the interval are not part of the set. For example: 1 < x 5 can be described using the open interval: (1, 5) A closed interval is denoted by square brackets. They tell you that the set includes the end points as part of the set. For example: 1 <= x <= 5 is a closed interval and can be represented as: [1, 5] Hope this helps."
    },
    {
        "video_name": "Hoyv3-BMAGc",
        "Q": "Starting at 0:14, what IS an absolute maximum/minimum? Thanks. (Although Sal said it was obvious, but I want a clearer image of what these terms mean).",
        "A": "Absolute max is the left endpoint and the abs min is the right endpoint. The valley is the local/relative min. The hill is the local/relative max."
    },
    {
        "video_name": "Hoyv3-BMAGc",
        "Q": "At 0:35, he says that point 'a' is a maximum. However, if used the definition of a maximum, it would be:\nf(a) >= f(x) for all x within an open interval of (f(a - h), f(a + h)) for h > 0\nx is less then 'a' within f(a + h), however, x is not less then 'a' within f(a - h).\nf(a - h) is off our graph, so we don't know if f(a - h) is greater or less then f(a).\nWith that knowledge, wouldn't the point 'a' not be a maximum?",
        "A": "At the start, he already defined the domain to be (a,b) and a=0. f(a) is the absolute maximum of the graph for the interval (a,b) because there are no other points higher than it. Even if we extended the graph into x<0 and that f(a-h)>f(a), f(a) would still be the absolute maximum for interval (a,b). This is because we are only talking about the interval (a,b). Anything outside our defined domain (or interval) should be disregarded."
    },
    {
        "video_name": "Hoyv3-BMAGc",
        "Q": "When Sal defines the qualifying factors for a relative maximum value f(c), he doesn't seem to take into account that the variable \"h\" could bring the interval past f(d) or f(b), which would render f(c) not a relative maximum. If you don't understand this question, just go to about 5:00 in the video. Please respond! I need clarification!",
        "A": "I think Sal was being a bit sloppy when he wrote for h > 0. That could mean for all h > 0, which would lead to the kind of problems Andrew identified. A more careful wording would be that there exists an h > 0 for which the condition holds. This allows us to stay very close to c."
    },
    {
        "video_name": "Hoyv3-BMAGc",
        "Q": "At 3:39, why does Sal use () instead of []? What is the difference?",
        "A": "Also, simple parentheses are used to show an OPEN INTERVAL, which is what Sal said he wanted. Using square brackets would have indicated a CLOSED INTERVAL (in other words, the end points would have been included)."
    },
    {
        "video_name": "Hoyv3-BMAGc",
        "Q": "Where is \"h\" coming from? That's where I got confused? (3:39)",
        "A": "the h is a value small enough so that the interval focuses just on the small part of the graph around x=c and ignores the other curves in the rest of the graph. basically you want to make sure f(x) is larger than the points immediately left or right of it (rather than all the points on the graph) before calling it a relative maximum. hope this makes sense"
    },
    {
        "video_name": "xks4cETlN58",
        "Q": "Hi just wondering, at 1:39 why do we use principal square root and not both possible values? Also, what is a principal square root?",
        "A": "1) Sal took the principal root because that s how the function was defined. 2) The principal root is the positive, real square root, represented by the \u00e2\u0088\u009a sign."
    },
    {
        "video_name": "xks4cETlN58",
        "Q": "So in looking at the graph beginning at 03:40, would the sin and cos of x also fit in this category of being limited within two horizontal asymptotes?",
        "A": "That s a good question. I don t know the answer, but I think the answer is no. sin and cos don t approach one, they reach one - again and again. My understanding of asymptotes is that they are what a function approaches as x heads towards infinity."
    },
    {
        "video_name": "xks4cETlN58",
        "Q": "at 1:39, why do we take the principal square root of x and not just the square root of x?",
        "A": "The definition of a function is that an input has one output. So, if f(x)=sqrt(x), unless we used the principal square root, f(4)= 2 and -2 If this is a function, the input 4 cannot have two outputs! That is why when using the square root in a function, we use the principal square root."
    },
    {
        "video_name": "xks4cETlN58",
        "Q": "At 2:57, aren't we supposed to first cancel out x and then put x --> infinity?\nIf no then why do we get 1 in this funtion- f(x) as x --> 0 in x/x = 1?",
        "A": "Sal did cancel them out. For positive values of \u00f0\u009d\u0091\u00a5 we have |\u00f0\u009d\u0091\u00a5| = \u00f0\u009d\u0091\u00a5 \u00e2\u0087\u0092 \u00f0\u009d\u0091\u00a5 \u00e2\u0088\u0095 |\u00f0\u009d\u0091\u00a5| = \u00f0\u009d\u0091\u00a5 \u00e2\u0088\u0095 \u00f0\u009d\u0091\u00a5 = 1 For negative values of \u00f0\u009d\u0091\u00a5 we have |\u00f0\u009d\u0091\u00a5| = \u00e2\u0088\u0092\u00f0\u009d\u0091\u00a5 \u00e2\u0087\u0092 \u00f0\u009d\u0091\u00a5 \u00e2\u0088\u0095 |\u00f0\u009d\u0091\u00a5| = \u00e2\u0088\u0092\u00f0\u009d\u0091\u00a5 \u00e2\u0088\u0095 \u00f0\u009d\u0091\u00a5 = \u00e2\u0088\u00921"
    },
    {
        "video_name": "xks4cETlN58",
        "Q": "at 2:51, how is the limit -1. Why can't it be positive 1? Does the negative infinity make it negative 1?",
        "A": "if you plug a small enough number, like -10000, which is almost the same as infinity for this problem, then x/abs(x)=-10000/abs(-10000)=-10000/10000=-1. the absolute value thing makes it do that"
    },
    {
        "video_name": "xks4cETlN58",
        "Q": "Why do we use the absolute value of x around 2:30?",
        "A": "Sal s claim is that the given function f(x) behaves like x/|x| for very large x (referenced around 1:50). If you want less hand-waving, you could multiply both the numerator and denominator of f(x) by 1/x, for the limit as x-->infinity, and simplify to show you do get a limit of 1. Similarly, you d multiply top and bottom of f(x) by -1/x when examining the limit as x-->negative infinity."
    },
    {
        "video_name": "xks4cETlN58",
        "Q": "In the video at @4:51, Sal automatically assumes that the asymptotes are horizontal asymptotes. How do you determine whether the asymptotes are horizontal or vertical?",
        "A": "Vertical asymptotes appear when we approach a point of dividing by zero. That s not possible for the function considered in this video because x^2 + 1 will always be a positive number greater than or equal to 1."
    },
    {
        "video_name": "MNbat1lrJW4",
        "Q": "At 9:49 how it's evident that 95% chance that men are more likely to vote for the candidate than women?",
        "A": "@mnada101 I agree that it was somewhat an abrupt ending. This is how I interpret this result. Sal calculated 95% confidence interval for p1-p2. Since it is positive on both ends, one can conclude that men are more likely to vote for the given candidate. In other words, proportion of men voting for this candidate in 95% cases will be higher than proportion of women voting for the same candidate. Hope this makes sense )"
    },
    {
        "video_name": "MNbat1lrJW4",
        "Q": "At approximately 0:39, He put Sampling clist located above the green figure. I don't comprehend what he meant Can you please clarify that??",
        "A": "Sampling Distribution (that s a d, not a cl)"
    },
    {
        "video_name": "MNbat1lrJW4",
        "Q": "Maybe I was distracted during a video, but at @5:08, why do we multiply the 95% z score by the combined sample standard deviation (.022)? Is this because we found the estimated z score of what the sampling distribution should be at 95% confidence, and then have to normalize it based on our random sample findings?",
        "A": "The z score tells us how far away from the mean, in standard deviation units, a point will need to be to contain a certain percent of a normal curve. Because we want to know in real numbers (or at least in real percentages) instead of in standard deviations, we use the factor of 1 standard deviation = .022, and multiply that against the 1.96 we found on the z=table (for a value of 97.5%), for a result of .043. So, .043 below, and .043 above the mean is 95%."
    },
    {
        "video_name": "BNHLzEv6Mjg",
        "Q": "At 0:03 how is C. =0.1x + 1?",
        "A": "It does not say: C. =0.1x + 1 The picture has one of 3 lines labeled as line C and the equation: y = 0.1x + 1."
    },
    {
        "video_name": "BNHLzEv6Mjg",
        "Q": "at 1:09 what does he mean we he says single solution",
        "A": "Dustin is correct. Also worth noting that, for a line, there can only be one solution, infinite solutions, or no solutions. That means a pair of lines can cross once, be completely overlapping, or parallel. They wouldn t cross twice or three times, because then they would not be lines. :)"
    },
    {
        "video_name": "5RzDVNob0-0",
        "Q": "at 0:27 why can you just multiply it?",
        "A": "Because you have to do the parenthesses first (:"
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "What did you do, at 2:33. What does +3 and -6 have to do w/ anything? Why did you re-write the 3 and 18 after the answer? plz explain\nHelp...",
        "A": "Those are the factor s of -18. He rewrote the answer so that you can factor by grouping"
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "at 3:24, I still don't understand- do you always have to keep factoring out, or can you just stop? My math teacher taught me that you could stop, but I needed to make sure that I got it.",
        "A": "Thank you! I figured, but I didn t know which was correct, because I m actually in a video conferencing math class, so I have 2 math teachers with not necessarily the same ideas."
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "At 4:15, my teacher taught me that if -6 and 18 are both divisible by 3 it would be -3(2k - 6)?",
        "A": "Yes, because if you distribute the 3 into those, you would get -6k + 18."
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "0:24 all the numbers were divisible by 8 what if they were not divisible like 6x^2 - 13x -6?",
        "A": "you would factor by grouping regularly if it is not factorable, you would use the quadratic equation"
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "Why at 1:20 dose sal factor out the 8 why can't he just divide it out",
        "A": "There isn t an equation. You can t just divide an expression by a number without changing its meaning. If it were an equation you could divide both sides by 8 and be okay."
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "Around 4:17 what I dont understand when hes factoring out the k+3 how come the k+3 on the left stays in the new answer but the other one doesnt and instead turns into k-6.",
        "A": "He is using factor by grouping. The idea is similar to combining like terms. When you add 2x and 3x it becomes 5x. k(k+3) -6(k+3) follows the same logic."
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "So I really dont understand. At 0:13 he says to simplify but what if you couldnt? like say 3d^2 + 23d +14? What would you do?",
        "A": "Well, 3d^2 + 23d +14 can be factored to (3d + 2)(d + 7). If we distribute/FOIL the equation, we get 3d^2 + 21d + 2d + 14 or 3d^2 + 23d +14 ."
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "Why did he group instead of just factoring immediatly at 8(k^2-3k-18)? Around 1:03",
        "A": "Because Sal wanted to explain it step-by-step"
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "how would u solve:\n1: 2a^2+3a-2\n2: 3a^2+7a+6\n3:7a^2+69a-10\nI JUST DONT GET IT !",
        "A": "There is nothing really to solve in any of these, but you can factor them 1: (2a - 1)(a + 1) 2: can t factor as written, would need to use the quadratic equation to find the roots 3: (7a - 1)(a +10)"
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "Around 1:25 Sal said that you can write it in the form x^2+bx+c,but what about the 8 outside the parenthesis?",
        "A": "He is just looking at the part inside the parentheses. Once he factors that, he uses that to write the next line - which includes the 8. (See 3:16)"
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "At 3:55 the instructor says to insert a plus sign into the grouping. Is this always the case? When I'm factoring other trinomials I'm finding that inserting a plus sign there does not always yield correct factorizations",
        "A": "You can always place a plus sign, followed by a negative number. This results in a subtraction anyway."
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "why did you put -18 at 3:09",
        "A": "It was a part of the original factoring, so you must transfer it down with the rest...It gets lonely sometimes :)"
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "At 0:15, he mentions that the terms are divisible by eight. What if there is no common factor? Also, at 1:21, he mentions that you need to find two numbers, whose sum is equal to the coefficient on x. What if, in a problem unlike the one described, there are no two factors, whose sum is equal to the coefficient of x? How do you solve a problem then? Thanks!",
        "A": "1) If there was no common factor, then you would be starting at a problem like: k^2-3k-18; or you would use factoring quadratics by grouping if the leading term has a coefficient that is not 1. 2) There are polynomials that are not factorable. These are just like prime numbers. They are considered to be prime polynomials. If this occurs when you are trying to solve an equation, you would use the quadratic formula to solve the equation."
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "2:40\nhow did you come up with +3 and -6?",
        "A": "Sal was looking for numbers that make -18 when multiplied together, such as 3 and -6."
    },
    {
        "video_name": "GMoqg_s4Dl4",
        "Q": "is it necessary that quadratic expression have a leading 1 coefficient (1:18) ?\n1x^2+bx+c",
        "A": "The answer is yes. But in practice, the leading 1 coefficient is not wrote to minimize the expression. Every time you see a variable, you can always infer a coefficient of 1 before it. In this case, coef 1 is the A which we use in Ax^2+Bx+C to help resolve the factorization. if it was any other number, it would needs a supplementary step to factorize the equation. In this situation, you just need to find the factor that gives a+b = B & ab =AC (where A = 1 here, so we ignore it)"
    },
    {
        "video_name": "2QeDRsxSF9M",
        "Q": "At 0:45, shouldn't one say \"not reject H0\" instead of \"accepting H0\"? That is because, if, for example, the expected percentage for the first category were 9,999999% instead of 10%, one would probably still \"not reject H0\" - so to avoid a type 2 error (accept H0 when H0 is false) one always \"not rejects\" and never accepts H0. Or why would it be possible to \"accept\" in this example? Thanks.",
        "A": "Yes, more properly we should say Fail to reject Ho, rather than Accept Ho."
    },
    {
        "video_name": "2QeDRsxSF9M",
        "Q": "At 8:50\nWhat does rejuct mean?",
        "A": "It means reject :)."
    },
    {
        "video_name": "24WMbh1BBKc",
        "Q": "from 4:00 to 4:30 in the video i don't get the corresponding angles?",
        "A": "In order for two triangles (or other polygons) to be congruent or similar, the two triangles must have 3 sets of congruent angles, If they are the same, then each angle on one triangle must have an angle on the other that corresponds (or matches) it."
    },
    {
        "video_name": "24WMbh1BBKc",
        "Q": "In 2:32, on the top, does it say conslant? It has red under it, but I can't read what it says.",
        "A": "It says constant . I think Mr. Khan might have forgotten to cross his t , causing it to look like an l . :)"
    },
    {
        "video_name": "24WMbh1BBKc",
        "Q": "at 7:35 didnt understand the \"slope\" part?",
        "A": "Slope line. It means the triangles are the same because the side they were formed from the line had the same slope."
    },
    {
        "video_name": "24WMbh1BBKc",
        "Q": "What is 'arbitrary'? Sal mentions it at 4:40, and 4:37.\nAt 4:40: \"Then this would be true, really, for many arbitrary sets of points.\nBut please explain it mathematically, not just a link to dictionary.com.\nThanks a bunch!\n- Temp",
        "A": "Arbitrary means that you choose it for no particular reason. So it would be true for many sets of points you that you may choose."
    },
    {
        "video_name": "24WMbh1BBKc",
        "Q": "I don't like this video because it seems to be using circular reasoning. At about 5:30, the orange line is called a transversal, but doesn't that assume that it is indeed a straight line with a constant slope? Is not a constant slope a necessary property of a straight line? For the slope to be varied, the line would have to be slanted, or warped in some fashion. It seems to me that the constant slope is part of the basic assumptions in this argument. Please correct my mistake!",
        "A": "Yes, a transversal is a line and a line has a constant slope. I don t believe that this information affects the logic of Sal s explanation."
    },
    {
        "video_name": "mkyZ45KQYi4",
        "Q": "at 7:30 to find the denominator of the fraction o the probability of getting at least one heads, can you simple raise 2 to the power of the amount of flips? Then the numerator would just be one less, right?",
        "A": "Yes, because P(at least 1 head out of n flips) = 1 - P(all tails out of n flips) = 1 - (1/2^n) = (2^n/2^n) - (1/2^n) = (2^n - 1) / (2^n)."
    },
    {
        "video_name": "mkyZ45KQYi4",
        "Q": "at 8:25 ish, sal uses a calculator. does he usually do that?",
        "A": "i agree with gsankar51..."
    },
    {
        "video_name": "mkyZ45KQYi4",
        "Q": "@4:05 a pop-up appears saying Sal wrote \"P(Not all tails) OR P(all tails) = 1\" but meant \"P(Not all tails OR all tails) = 1.\" What's the difference? I thought they were the same. This correction affects the rest of the video where he essentially says P(A) = 1 - P(notA)",
        "A": "I m not exactly for sure why that pop-up appears. Your assessment of the situation is correct... P(A) + P(not A) = 1 (i.e. the probability of anything happening is 1). So, if A = all tails and not A = not all tails, then P(at least one head) = P(not A) = 1 - P(A). The term for this is a probability complement, since if you break a group into two parts, A and not A , then A + not A = the original group. A is the complement of not A and vice versa. I agree that the pop-up adds a bit of confusion."
    },
    {
        "video_name": "mkyZ45KQYi4",
        "Q": "At 2:45, Why Sal is doing P(Not getting all tails) = 1-P(TTT) in 3 flips.",
        "A": "It s a shortcut - that way he only needs to calculate that one probability instead of the probabilities of all the not all tails permutations."
    },
    {
        "video_name": "mkyZ45KQYi4",
        "Q": "For the correction that pops up around 4:02 - aren't the 2 the same thing?",
        "A": "No. The probabilities are numbers. So saying P(Not all tails) OR P(all tails) is similar to saying .5 or .375. The correction puts the OR inside the probability statement, meaning we compute the probability that either event occurs."
    },
    {
        "video_name": "mkyZ45KQYi4",
        "Q": "At 4:02 there is a text box that pops up indicating that Sal wrote \"P(Not all Tails) OR P(all tails) = 1\" but mean \"P(Not all Tails OR all tails) = 1\". What is the distinction here between putting the mutually exclusive events in one P() as opposed to each set of events having their own P()? What are the implications of both versions?",
        "A": "Well, really, the problem is that P(Not all Tails) is a number, so it s not right to put OR next to it. It should be + , not OR . What he wrote is like writing 15/16 OR 1/16 = 1 , but of course he should write 15/16 + 1/16 = 1. That s all."
    },
    {
        "video_name": "mkyZ45KQYi4",
        "Q": "At 2:55 where is the 1 - coming from?",
        "A": "The 1 is representing a set of all possible events that could be happening. Sometimes there are events that are mutually exclusive which means that there is one event or a second event and those are all the possible events. In that case you can subtract the probability of one event[P(x_event)] from all events (1) and get the probability of the other event. (P(x_event2)) or vice versa."
    },
    {
        "video_name": "mkyZ45KQYi4",
        "Q": "At around 3:48, when Sal writes P(Not all tails) OR P(All tails)..., if we have to do that in class, would we write:\n\nP(Not all tails OR all tails)\n\nor would we write:\n\nP(Not all tails) OR P(All tails)?\n\nThank you.",
        "A": "P(Not all tails OR All tails) = P(Not all tails) + P(All tails). This is the correct style for set notation, which Sal isn t using and hasn t talked about at this point in the probability module. (It obviously doesn t use set notation, since I don t know how to make those symbols on a computer)"
    },
    {
        "video_name": "oORnGaJp1pk",
        "Q": "2:54, How did he get -2?",
        "A": "First we try to solve for f(-5). We know that y = f(x). If y = f(x), then by asking what is the value of f(-5), we mean what will be the value of y if we take x as -5. from the blue color graph we know that when x = -5, y = -2, Therefore we can say that if f(x) = y then f(-5) = -2. Hope that helps!"
    },
    {
        "video_name": "CpDfay5NeCg",
        "Q": "The part of the graph Sal is indicating at 2:40 looks to me like it's below 0 on the y-axis....so why is he saying the function is positive?",
        "A": "I think you misheard him. He states the function isn t positive there."
    },
    {
        "video_name": "CpDfay5NeCg",
        "Q": "At 0:07 Sal says that f prime (x) is the first derivative of f(x)? What is meant by the term \"first derivative\"? How is it different from the normal derivative?",
        "A": "What you call the normal derivative is the first derivative. For example: f(x) = x^3 first derivative is: 3x^2 Then we have something called the second derivative, which is simply when you derive something twice. You do this by deriving the first derivitive, like this: second derivative: 6x And to get the third derivative, you derive the second. third derivative: 6"
    },
    {
        "video_name": "CpDfay5NeCg",
        "Q": "How come 'x should be in sitting in first or second quadrant\" 1:15 when in second quadrant x is negative?",
        "A": "What he said was f(x) should be sitting in the first or second quadrant . f(x) is the y-value, which is positive in those quadrants."
    },
    {
        "video_name": "MhLfun2Vask",
        "Q": "At 3:33 he says that it can't be top right and bottom left circumference of the circle, but I don't understand why is that? And why is top left and bottom right can be the part we want?",
        "A": "It can t be top right or bottom left because the tangent needs to have a positive slope, that is to look like a forward slash /, not a backslash \\. The bottom right part is impossible to have a common tangent with the hyperbola, so the only part of the circle possible to draw the tangent on is the top left."
    },
    {
        "video_name": "T8U7hCmUdpo",
        "Q": "In 1:55 of the video 2+3=5 not 6. Am I right?",
        "A": "Where are you getting the 6 from? At this point in the video, Lindsay is work with the fraction 1/4. She counts the pieces and comes up with 5. She never references a 6 ."
    },
    {
        "video_name": "T8U7hCmUdpo",
        "Q": "In 1:35, you say that 2 is the numerator, and that makes no sense... Do you mean Denominator or is it numerator ?? That still makes no sense, just explain it...",
        "A": "She said, Maybe I wanna change this and I wanna say, I want 2, I want 2 as my numerator? In order to get change the numerator to 2, you must multiply both the numerator and denominator by 2. 2/8 simplifies to 1/4 because you are dividing both the numerator and denominator by the GCF (Greatest Common Factor), which, in this case, is 2."
    },
    {
        "video_name": "8aHUrjgF6s0",
        "Q": "at 5:31 grant wrote ||dT/ds||=||dT/dt||/||ds/ds|| instead he should worte ||dT/ds||=||dT/dt||/||ds/dt||",
        "A": "Yes, you are right!"
    },
    {
        "video_name": "8aHUrjgF6s0",
        "Q": "At 5:13, isn't there a mistake, when the narrator writes \"d s (arrow)/d s\" in the denominator, when it should be \"d s (arrow)/ dt\"?",
        "A": "maybe a clerical error"
    },
    {
        "video_name": "8aHUrjgF6s0",
        "Q": "At 1:39, why do we need a measure of how perpendicular the first and second derivative are to each other? If we are measuring how much the first derivative is turning, shouldn't the largest value be when the second and first derivative (put head-to-tail as in the video) make a very acute angle, and the smallest when they make a very obtuse angle? With the perpendicular case just being the middle way, indicating \"moderate\" rotation?",
        "A": "No, if you break the second derivative vector down into components that are parallel and perpendicular to the first derivative, the component that is parallel corresponds to a growing or shrinking of the first derivative and only the component perpendicular to the first derivative will correspond to turning. The second derivative has to be scaled according to the length of the first derivative though. Hope this helps"
    },
    {
        "video_name": "v4IdaXvyE7U",
        "Q": "At 3:01 he tells that you'll asymptote toward the x-axis. What does he mean by that? What's an asymptote?",
        "A": "An asymptote is a line (in this case a horizontal asymptote at y = 0) toward which a function approaches, but never reaches, it gets closer and closer, but can never be that number. The old riddle is that if you step toward a wall and cut each of your steps in half, will you ever reach the wall? While you could not do this in real life, in theory you could never reach the wall. that is f(x) = (1/2)^x."
    },
    {
        "video_name": "he4kcTujy30",
        "Q": "in 2:00 what is he taking about",
        "A": "7 hundredths is the same thing as 7 times 1 hundredth. 7x1 is 7."
    },
    {
        "video_name": "he4kcTujy30",
        "Q": "in 2:00 what is he taking about",
        "A": "Decimals in extended form are just like regular numbers in expanded form, except you use different place value. For example, 365 in expanded form is 300+60+5 or (3 x 100)+ (6 x 10)+ (5x1). 3.65 in expanded form would be 3+0.6+0.05 or (3 x 1 ) + (6 x 0.1) +(5 x 0.01) or (3 x 1)+(6 x 1/10)+(5 x 1/100)"
    },
    {
        "video_name": "he4kcTujy30",
        "Q": "At about 0:20, Sal said \"How do you EXPAND....\". What does he mean when he says \"expand\"?",
        "A": "Which takes up more room on a piece of paper: 905.074 or (9 x 100) + 5 + (7 x 1/100) + (4 x 1/1000)? The value of the number hasn t changed, it hasn t expanded . But it sure takes up a lot more room. That s what he means by expanded form."
    },
    {
        "video_name": "dvoHB9djouc",
        "Q": "What is that weird \"E\" looking thing at 1:36?",
        "A": "That is the greek letter sigma. In math, it means summation."
    },
    {
        "video_name": "dvoHB9djouc",
        "Q": "At 3:30, why square the distance to the mean to get a positive number? Why not just take the absolute value?",
        "A": "We square the distance so that outliers will get more weight. If we just took the absolute value, the outliers would have the same weight as others. The idea is that data points that are way off the mean are telling you something important. Data points that are just a little off aren t telling you too much."
    },
    {
        "video_name": "dvoHB9djouc",
        "Q": "What does the i mean in the summation formula? Just wanted to know. 2:19",
        "A": "i is the element number. It goes from 1 (the first) to n (the last)."
    },
    {
        "video_name": "lXShNH1G6Pk",
        "Q": "At 6:06, \"everything we're doing is in the open interval\", why is it open? Does it have any particular reason?",
        "A": "It s on an open interval because the original equation is not defined at -\u00cf\u0080/4. So after manipulating the expression into something that could indeed be evaluated at -\u00cf\u0080/4, it s important to remember that the actual equation we were given still can t be. So we call it an open interval at -\u00cf\u0080/4, meaning it s still undefined at that point. We re still working with limits that approach the point, not with the point itself. It s a subtle point, but it s important in the pure mathematics sense."
    },
    {
        "video_name": "lXShNH1G6Pk",
        "Q": "At 4:55, could I have just directly substituted in 2cos^2theta - 1 for the denominator?",
        "A": "Sure, but how would that help?"
    },
    {
        "video_name": "lXShNH1G6Pk",
        "Q": "At 1:48, how did Sal know off the top of his head what sin((-pi/4)) was? Is there a series of videos I could watch to learn these radian values?",
        "A": "-\u00cf\u0080/4 is an angle on the unit circle. Most of us have it memorized. You may want to memorize them as well for future benefits if you continue to do math. You can google unit circle and get a lot of pic/charts showing those special angles and their cosine s and sine s values."
    },
    {
        "video_name": "lXShNH1G6Pk",
        "Q": "at 2:33 you have conver 2 cos pi/4 as the same as cos pi/2...\nmy question is why do must you convert in this way as the cos pi/4 is sq root 2 over 2... x 2 is the sq rt of 2.. and the cos of the sq rt of 2 is not 0",
        "A": "It is cos(2\u00ce\u00b8) which is not the same as 2cos(\u00ce\u00b8). Sal should had use parenthesis like I did to avoid confusion for students. cos(2\u00e2\u0080\u00a2\u00cf\u0080/4) , you simplify what inside the parenthesis first then compute cosine function."
    },
    {
        "video_name": "lXShNH1G6Pk",
        "Q": "At 6:43 how limit is not defined?? sinx is negative in the interval (0,-pi/2). So sin(-pi/4)= -1/root 2. so overall limit is 1/2. (since the simpified xpression approaches -pi/4 but not defined when equals to -pi/4).",
        "A": "At that time he wasn t saying -\u00cf\u0080/4 but saying \u00cf\u0080/4, which does indeed make the denominator 0. I m not sure why he threw in \u00cf\u0080/4, but the limit is defined at x = -\u00cf\u0080/4 which he did from7:10 on"
    },
    {
        "video_name": "kBVDSu7v8os",
        "Q": "Why does he use the chain rule at 2:43? When using the power rule he didn't have to use the chain rule before? I'm just pretty confused",
        "A": "See answer to the same question by brookwarrior20."
    },
    {
        "video_name": "kBVDSu7v8os",
        "Q": "I'm really confused as to why at 1:51 he established a relationship x^2+h^2=100, but didn't plug in x = 8? My only idea is that because variable x and variable h are being subject to change (h is falling and x is sliding out). Therefore, we do not plug x = 8 into the equation until after we have established a derivative equation.\n\nSimilar to the cone problem from the previous video. We did not plug h=2 until we had established a derivative equation.\n\nIs this a good way of thinking/approaching this issue?",
        "A": "Yes, I think you have the right idea. x = 8 is just an initial value at t = 0 If you plug in 8 initially you d be making it a constant (the same way the length of the ladder is a constant 10). Then when you take the derivative dx/dt = 0 which is clearly wrong. So you need to set up the equation like Sal has it because both x and h are going to be changing with time."
    },
    {
        "video_name": "8eSUbi_aYL4",
        "Q": "At 1:49, how do we know that a-sub-5 is the same thing as a-sub-i?",
        "A": "That is what the question asks, what is a sub 5, so any value can be used for I, but in this case we as for what happens when I = 5."
    },
    {
        "video_name": "8eSUbi_aYL4",
        "Q": "At the time of 2:51, I can't understand what the video is trying to say. Can you help me better understand?",
        "A": "They mean that a sub 1 is -7. This just means that the first term in the sequence is -7. Whenever you see a letter-sub-a number, it mean that term in the sequence."
    },
    {
        "video_name": "32rdijPB-rA",
        "Q": "At 5:31 Why is the cofactor determinant of f equal to (dh-eg) and not (ah-bg)? I might have misunderstood the pattern.",
        "A": "You are correct; Sal made a mistake there."
    },
    {
        "video_name": "a1Ye5RcWOqg",
        "Q": "At about 7:15 Sal put an n in the sampling distribution for the women, but we already know that it is 1000, correct? We put that number in on the men's side. Help, I'm trying not to get confused!!!!! lol...",
        "A": "Yes, n=1,000 for the women as well; it s corrected in the next video when it comes time to calculate it."
    },
    {
        "video_name": "BItpeFXC4vA",
        "Q": "At 0:10 Whats does place values mean?",
        "A": "Place value is the value of each digit within a number. For example: In the number 32, the 3 is in tens place. Any digit in that place is worth ten. So the 3 is worth 3*10 = 30 Hope this helps."
    },
    {
        "video_name": "wo7DSaPP8hQ",
        "Q": "At 2:20, I don't get get the multiplying by 2/3's part. Can anyone explain me that part?",
        "A": "The problem said both farms were down by a third. That means each farm had a 1/3 of their production or whatever taken away from it, leaving 2/3 s of the production. To get that value (two thirds of M or R) you multiply 2/3*M and 2/3*R."
    },
    {
        "video_name": "wo7DSaPP8hQ",
        "Q": "At 2:46, why do you only have to switch the iniquality when it is negative, and not when its positive?",
        "A": "because if you multiply(or divide) by a negative number you basically mirror the problem on the number line: for instance if you multiply by -1, 3 will be -3 and -2 will be 2. But while on the positive side 3 is larger than 2, so an higher absolute value (3) is larger than a smaller absolut value (2). This is reversed for negative numbers: -3 is smaller than -2 So 3>2 will become -3<-2 when multiplied by -1"
    },
    {
        "video_name": "wo7DSaPP8hQ",
        "Q": "3:23-3:56 Why is Sal using 2/3? The problem says about a third which is about 1/3.",
        "A": "Because the farms were down by a third, meaning x - 1/3x which equals 2/3x"
    },
    {
        "video_name": "wo7DSaPP8hQ",
        "Q": "At 3:54 is it possible to figure out how many more apples M had than R?",
        "A": "No, because you don t know how many apples each farm grew."
    },
    {
        "video_name": "wo7DSaPP8hQ",
        "Q": "2:47 why are you multiplying each side by 2/3?",
        "A": "Because they have 1/3 less apples each year. They only produced 2/3 of what they would normally do since the cold weather made them produce 1/3 less apples."
    },
    {
        "video_name": "wo7DSaPP8hQ",
        "Q": "At 2:29s, Sal says that \"going down by a 1/3 is the same as multiplying by 2/3\". If I want to verify this, how can I do that? I seem to have a problem understanding ratios, it's really frustrating (especially with fractions).",
        "A": "I don t see how that s possible either, except for the number 1... He must not be talking about regular numbers. Maybe he meant X minus one-third X = two thirds X... I checked it out and yes that does work. Maybe he made a mistake in his video. But I m pretty sure he mean X minus one-third X = two thirds X. If you have anymore questions, ask me. :D"
    },
    {
        "video_name": "wo7DSaPP8hQ",
        "Q": "At 4:03 isn't the \"2/3M\" supposed to be closer to \"M\"?. Right now I think its around \"1/3M\"",
        "A": "It is 1/3M if you are talking about the distance from R to M, but we are not. We are talking about the distance from 0 to M, in which case, it is pretty close to 2/3M."
    },
    {
        "video_name": "5fK8HEYNRuQ",
        "Q": "at 1:31 how is 12/4 + 1/4 equal 3 1/4 how did you get those numbers?",
        "A": "Because Sal added an improper fraction and he made the improper fraction answer he got into a fraction with a whole number"
    },
    {
        "video_name": "5fK8HEYNRuQ",
        "Q": "2:48 in this video would the answer be 3/8?",
        "A": "another way that isn t quite as nasty as what he did is to do 3 1/4 - 2 7/8. First subtract out the whole numbers that is 1 + (1/4 - 7/8) = 1 + (2/8 - 7/8). since it s going to be negative unless I regroup/borrow from that one, I m going to make the one a part of the fraction now. 10/8 - 7/8 = 3/8"
    },
    {
        "video_name": "-Y-M9aD_ccQ",
        "Q": "2:20 What if the points form perfect straight lines with slope 0 or infinity? what can we comment on the correlation between X and Y - 1 or -1?",
        "A": "this was a good question to ask in order to get the right answer"
    },
    {
        "video_name": "YjWCDdNlXxc",
        "Q": "At 0:10 can't the equation just be (Area of the semi-circle= \u00cf\u0080*r2*1/2) ? Can't we just use this and get to the answer without having to find the area of the whole circle like Sal did?",
        "A": "The point of the exercise is to make you go thru the process to make sure you understand how all the elements work together. I think in general you are correct, but be careful with your notation: Area of a circle = \u00cf\u0080r^2."
    },
    {
        "video_name": "497oIjqRPco",
        "Q": "At 2:31, Sal says that \"it's a digit between 1 and 9, including 1 and nine.\"\nHe should have said, \"it's a digit greater or equal to 1 and less than 10.\"",
        "A": "but what about 9.9?"
    },
    {
        "video_name": "497oIjqRPco",
        "Q": "At 1:40, he simplified 7/5 to 1 and 2/5. But how did he simplify it to 1.4 so quickly?",
        "A": "He likely has memorized that 2/5 = 0.4 It pays to memorize frequently occurring fractions & their decimal values. Example to memorize 1/2 = 0.5 3/4 = 0.75 1/3 = 0.333... with the 3 repeating 2/3 = 0.666... with the 6 repeating 1/5 = 0.2 2/5 = 0/4 3/5 = 0.6 4/5 = 0.8"
    },
    {
        "video_name": "497oIjqRPco",
        "Q": "At 1:58 does Sal mean \"ten to the 5th PLUS ten to the negative seven?\"\n\nOr did he really mean \"ten to the 5th TIMES ten to the negative seven?\"\n\nThanks",
        "A": "He did mean ten to the 5th TIMES ten to the negative seven because when two numbers with exponents are multiplied, you add the exponents together. For example: 10^2 TIMES 10^3 equals 10^2+3 or 10^5. Therefore, ten to the 5th TIMES ten to the negative seven is a multiplication problem, though to solve it you add (or subtract) the exponents. Hope this helps."
    },
    {
        "video_name": "497oIjqRPco",
        "Q": "Is 9.3*10^3 considered scientific notation? I am not sure because at 2:40, Sal said for a number to be written in scientific notation, the number has to be between 1 and 9 including 1 and 9.",
        "A": "9.3*10^3 is correct scientific notation. I mean, your only other options are 0.93*10^4 or 93.0*10^2 and both of those are definitely not correct."
    },
    {
        "video_name": "497oIjqRPco",
        "Q": "At 1:40, can you reduce the 7 and the 10^7? And the 5 and the 10^5?",
        "A": "No, if the two 10 s (base numbers) were 5 and 7, they could be factored out (if diagonal from the existing 5 and 7) or further increase the exponent factor if they were across from each other, but the fact that they match the exponent values does not allow for any reduction."
    },
    {
        "video_name": "497oIjqRPco",
        "Q": "at 2:45 Sal said greater than or equal to 9. Can't you have 9.54*10^8",
        "A": "If you see the answer to Ishan s question, 2 questions above this one, then I think you will have your answer. Maybe Sal was rushing a bit and meant to say (again) any number less than 10? Which would be all numbers up to and including 9.99999999... So, my guess is, YES, you can have 9.54*10^8!"
    },
    {
        "video_name": "497oIjqRPco",
        "Q": "at 1:04, isn't it positive 9 instead of -9?",
        "A": "That was a mistake."
    },
    {
        "video_name": "Cf0-6ngH2gs",
        "Q": "At 4:25, Sal multiplies ar^(n-1) by -r and gets -ar^n. I do not quite get how that works and would like some help on it.\n\nThanks in advance!",
        "A": "OK, this is a really REALLY great question. When you multiply ar^(n-1) and -r together the first thing you can do is distribute the negative sign, which gives you -ar^(n-1) * r. The variable r can also be expressed as r^1. So you get -ar^(n-1) * r^1. Next you can pull out the -a which gives you (-a)(r^(n-1)) * r^1. Then you can simplify and get (-a)(r^(n-1+1)). Once again that can be simplified very easily to -ar^n. I hope that was helpful. #GoGators"
    },
    {
        "video_name": "Cf0-6ngH2gs",
        "Q": "At around 6:30 seconds, our general formula for a FINITE number n is proved. My question is, why does this differ from the infinite geometric series formula (S=a/1-r, where absolute value of r is less than 1). I understand the whole bit about the infinite series and that it must converge (hence the condition for r), but how and why are these two generalizations off by a factor of (1-r^n).",
        "A": "When we consider the sum of an infinite geometric series, we look at what happens to the partial sum formula (derived at 6:30) as n gets arbitrarily large. Remember that |r|<1. So when n gets really big, r^n gets very small, since we re multiplying by a number less than 1 a bunch of times. So when n goes to \u00e2\u0088\u009e, r^n goes to 0, and a(1-r^n)/(1-r) becomes a(1-0)/(1-r)=a/(1-r)."
    },
    {
        "video_name": "YKQLZOq4HTA",
        "Q": "I don't agree with this answer. At 3:46, the answer is graphed as 2 < p < 22, but because p-12+4<14 OR -(p-12)+4<14, the correct answer is p<2 OR p<22?",
        "A": "It s the absolute value of p-12. Sal already proved that 0 didn t work so p<2 is false."
    },
    {
        "video_name": "YKQLZOq4HTA",
        "Q": "At 2:28, how did l p-12 l turn into p-12?",
        "A": "Because he solved simultaneously for both cases when | p-12 | is negative distance and when | p-12 | is positive distance from 10. Sal could have also solved each case separately, which might have been clearer: if | p-12 | is positive then (p-12) < 10 and p < 22 if | p-12 | is negative then (p-12) > -10 and p > 2"
    },
    {
        "video_name": "YKQLZOq4HTA",
        "Q": "So when we have a negative absolute value like the one at 4:23 |-12|, do we always make it positive and why? Because if we didn't make it go positive, the inequality with a value of zero in the place of the p would be correct as it would give us -8>-14 right??",
        "A": "That s what the absolute value thing means. Absolute value means the size or magnitude of the number from zero. So when you have the absolute value of a negative number ,say -5 then you have to find the space u moved from 0 to -5 regardless of direction, in this case 5. So you can t ignore the absolute value, l-12l must equal to 12."
    },
    {
        "video_name": "YKQLZOq4HTA",
        "Q": "[Around 2:33]\nOkay, so I understand using -10<p-12<10 would help in this situation, but my teacher taught us to use that as a \"answer holder\" for AND inequalities, so why use them to find the algebraic answer when you can just do something like\np-12<10\nand\np-12>-10 ?\n\n{Example base equations:\nA < X < B\nA < X\nB > X\nX + A > B }",
        "A": "It s your choice"
    },
    {
        "video_name": "YKQLZOq4HTA",
        "Q": "At 2:23, why did Sal add the -10 to the inequality?",
        "A": "it is an inequality so he has to solve for p-12<10 and p-12<10."
    },
    {
        "video_name": "yuqB-d5MjZA",
        "Q": "Is there a name given to the vector of first order partial derivatives shown at 5:25? Why do we use such a vector? Thanks.",
        "A": "Those can be called by a couple of different names, depending on the uses, including gradient of f (used in this video and denoted as \u00e2\u0088\u0087f) and the Jacobian matrix, denoted as J. The extension of this for second partial derivatives is the Hessian matrix, denoted as H. Hope this helps."
    },
    {
        "video_name": "4yzV8m8X3Kk",
        "Q": "At 2:00 the z component is added. Why isn't it z instead of 0? I thought with z = 0 all the time you just get a vectorfield in the xy-plane.",
        "A": "Think of the 3rd component as of z*0. If you input z=1 or 2 or 3 you will have still have to plot y^3-9y and x^3-9x in a flat plane, but since z=1,2 or 3 each plotted vector that lied in xy plane will have to start higher. By adding z*0 as 3rd component you don t change the direction or magnitude of the plotted vector, but the starting position (the beginning of a vector communicates the input, the direction and magnitude is the output- which doesn t change )"
    },
    {
        "video_name": "W0VWO4asgmk",
        "Q": "At 2:02, I run into my first problem of trying to make sense of this. \"As you move closer and closer to the expressions value, what does the expression equal? It essentially equals 4.\" Well, That doesn't make sense to me because as you move closer and closer to the expressions value on the x axis, you have an infinite number of numbers which increase exponentially at the rate at which you are moving towards 2, not just one number 4. 4 is the number on the expression that you are apr. 2 on x axis",
        "A": "I pretty much understand, I would just word it differently. I would say, as you approach 2 on the x axis, you approach 4 on the expression."
    },
    {
        "video_name": "W0VWO4asgmk",
        "Q": "At. 5:45, is the limit notation similar to saying that the limitation on the function as it approaches 2 is 4?",
        "A": "Yes- it is saying that as you get closer and closer to f(2), you get 4. The limit as x approaches 2 for f(x) does not equal f(2) because at x=2, the function does not lead up to 3. (It leads up to 4)."
    },
    {
        "video_name": "W0VWO4asgmk",
        "Q": "6:34 can someone rephrase,\n\n''limit as x approaches 2 of f of x does not equal f of 2.''",
        "A": "The value of y as x begins to approach 2 does not equal the value of y at 2."
    },
    {
        "video_name": "XOIhNVeLfWs",
        "Q": "Sal distributed the numbers at 2:32 to get rid of the fractions. Do you have to do that, or is that optional?",
        "A": "If you are asked to write the equation in standard form, then you need to get rid of the fractions. Standard form is: Ax + By = C where A, B and C are integers (so no fractions)."
    },
    {
        "video_name": "XOIhNVeLfWs",
        "Q": "At 5:06, it show the answer as -14x+21y=12, but I thought \"A\" always has to be positive.",
        "A": "If A is positive, then the slope has a sign opposite to B. If A is negative, then the slope has same sign as B. (because in standard form Ax and By are on the same side)"
    },
    {
        "video_name": "XOIhNVeLfWs",
        "Q": "At 3:55 isn't Ax in this case -14 supposed to be non-negative?",
        "A": "It is not required. Some sources tell you that the A must be positive. Other sources, tell you A should be an integer. I would recommend that your follow the instructions provided by your teacher."
    },
    {
        "video_name": "XOIhNVeLfWs",
        "Q": "At 1:08, what if you don't know b?",
        "A": "If there is nothing in front of the Y then it is assumed to be 1."
    },
    {
        "video_name": "XOIhNVeLfWs",
        "Q": "Ok, so I completely get how to do this, but can anyone explain why we have to simplify the integers in the equations? At 1:47, Sal just says \"Because there's obviously a bunch of forms in this way\" which is, like...um, what...",
        "A": "Simplifying just makes the equation simpler and easier to convert to other forms. It also makes it quite nice to find the x-intercept, y-intercept, and slope. :)"
    },
    {
        "video_name": "EXLVMGSDQbI",
        "Q": "At 1:38, Sal says that \"the fastest growing function would be a line.\" Why couldn't some type of curve also be the fastest growing function? Would it be possible for such a curve to have 7 as the greatest instantaneous speed?",
        "A": "The problem said that the f (x) \u00e2\u0089\u00a4 7. Thus, the maximum slope is 7. Thus, the function that meets that requirement and increases at the maximum allowed rate would have to have a constant slope of 7 (the maximum slope allowed here). Only straight lines have a slope that is constant."
    },
    {
        "video_name": "EXLVMGSDQbI",
        "Q": "(at about 2:45) Why is it f(10) - 3 for y? I'm wondering what the logic is behind putting the f(10) there.",
        "A": "We re trying to find the highest possible value of f(x) when x=10, as stated in the initial problem. Since we have a known point for the function and a bound on the derivative (slope), we can use the highest possible derivative to extrapolate from the known point f(-2)=3 to find the greatest possible value for f(10)."
    },
    {
        "video_name": "EXLVMGSDQbI",
        "Q": "at 1:45 why do we assume that point (-2,3) has a slope of 7? We know we want the max value of f(10) and so would need to use the maximum tangent line slope of 7 to get this, but no one said anything about (-2,3) having a tangent line whose slope is 7. So how is this decided? Essentially why does the tangent line with slope 7 go through point (-2,3)",
        "A": "The tangent line, as part of its definition, must touch the function f at one -- and only one -- point. In addition, it must also have a slope equal to the value of the derivative of the function f. In order for this to work, you must have a slope of 7 at the point (-2,3) for the tangent line."
    },
    {
        "video_name": "O4Qnsubo2tg",
        "Q": "At 4:14 is there a mathematical reason why the favourable outcomes line up on the diagonal?",
        "A": "That is a result of how he decided to visualize this. Imagine we flip the table around a little and put it into a coordinate system. Along the x-axis you put marks on the numbers 1, 2, 3, 4, 5, 6, and you do the same on the y-axis. We are interested in rolling doubles, i.e. getting the same on both dice. If we let x denote the number of eyes on the first die, and y do the same for the second die, we are interested in the case y = x. But this is the equation of the diagonal line you refer to."
    },
    {
        "video_name": "O4Qnsubo2tg",
        "Q": "At 3:26, couldn't he just count the number of boxes to find the number of possible outcomes?",
        "A": "Absolutely, with a grid the size of this, but it is possible to find and roll very large dice (in terms of # of sides - 100 sided dice exist!) and very large numbers of dice (some games involve simultaneously rolling 10 or more). So it s useful to learn how the math works here, in the simple case, so you can use it to sort out probabilities elsewhere, where it might be very hard to just count."
    },
    {
        "video_name": "O4Qnsubo2tg",
        "Q": "At 0:03, he says \"Find the probability of rolling doubles.\" What are doubles? Are they two dice that have the same number or something?",
        "A": "Yes - when two dice are rolled, and each die lands on the same number, it is known as a double..."
    },
    {
        "video_name": "h6mZLTpTHcU",
        "Q": "Is there a specific reason why Sal chose G(4) and G(8) at 0:38 ?",
        "A": "So it seems. It s a clever choice because it shows us that the chunk of net area between x=4 and x=8 is actually zero, not affecting the net area gathered in the interval [-3,4]."
    },
    {
        "video_name": "ogEddosP1G8",
        "Q": "At 2:07, even if the 0 had an x, like 0x = -20, it still wouldn't work, right? Because everything times 0 = 0, not -20.",
        "A": "Yes, in fact, x is undefined in that equation because to solve for it, you would have to divide by 0 which is undefined."
    },
    {
        "video_name": "ogEddosP1G8",
        "Q": "At about 1:02 we see Sal declare that he is going to add the two equations. Watching the video I can see it produces clear results. What I don't understand is how we would come to the conclusion that is what needs to be done.",
        "A": "To add or subtract the equations you must have 2 numbers or variables that if added or subtracted would get 0. Then you would do the same operation to each other variable. In his case 5x and -5x if added would be 0."
    },
    {
        "video_name": "ogEddosP1G8",
        "Q": "At 1:20, Sal is multiplying one equation by -1. If you do this to one \"side\" - as he refers to \"left\" and \"right\".... isn't he supposed to do the same to the other side? How can he just pull this out of the air?",
        "A": "He did do that to both sides. 5x -9y = 36 -1 (5x -9y) = -1 (36) -5x +9y = -36"
    },
    {
        "video_name": "ogEddosP1G8",
        "Q": "How would you know if a system has exactly one solution, and i didn't get how in 5:37 (4/5, 1) represents exactly one solution",
        "A": "I m not sure what you mean. That is the answer. You only have one place on the graph where the two lines intersect. Therefore, you only have one answer that satisfies both equations."
    },
    {
        "video_name": "ogEddosP1G8",
        "Q": "At 5:27, how do you get 5x=4",
        "A": "Sal did a step in his head. 5x+3y=1 when y = -1 5x+3(-1)=1 5x-3=1 5x-3(+3)=1(+3) Sal skipped over this step; he did it in his head. 5x = 4"
    },
    {
        "video_name": "ZN7YaSbY3-w",
        "Q": "At 0:10 sal represented the vector like this [2]\n[1]\nI thought that vectors are represented by an ordered pair like this (2,1) .",
        "A": "All representations of a vector are equally valid, as long as your reader knows what you are talking about."
    },
    {
        "video_name": "ZN7YaSbY3-w",
        "Q": "At 5:31, how can a negative number (which is a scalar) affect the direction of a vector? By theory it should only affect the magnitude of a vector.",
        "A": "Negative scalars and 0 affect the direction of a vector. Negatives flip them so they point in the opposite direction, zero makes them the zero vector (which is the only vector with no direction). A scaled vector is still always co-linear with the original though."
    },
    {
        "video_name": "ZN7YaSbY3-w",
        "Q": "At 1:03, so saying a scalar is just a number is the same thing as saying a scalar quantity has just a magnitude but no direction, right? Thanks!",
        "A": "Yes, you are right, a scalar is simply a number, which represents a quantity but no direction."
    },
    {
        "video_name": "ZN7YaSbY3-w",
        "Q": "At 1:21 can you use the distributive property?",
        "A": "Great insight :D It s the same procedure: 3 * (2\u00c3\u00ae + 1\u00c3\u00a2) = 6\u00c3\u00ae + 3\u00c3\u00a2 As you dealing with a vector, you will generate another vector. Hope it helps! P.s.: the \u00c3\u00a2 supposed to be a j with ^, but I don t know how to put the ^ over the j"
    },
    {
        "video_name": "ZN7YaSbY3-w",
        "Q": "At 5:31, how can a negative number (which is a scalar) affect the direction of a vector? By theory it should only affect the magnitude of a vector.",
        "A": "By taking sign convention into account when you multiply a vector with a negative number it means you move on to the opposite direction. Like if you were moving to the left now you move to the right of origin..... hope it helps"
    },
    {
        "video_name": "qrVvpYt3Vl0",
        "Q": "At 0:38 i dont get it, so for A+, seven people have only one blood drop?",
        "A": "Each blood drop = 8 people. So 7*8=56 people."
    },
    {
        "video_name": "qrVvpYt3Vl0",
        "Q": "At 0:29, could the scale also be called a key?",
        "A": "yes they are synonyms"
    },
    {
        "video_name": "qrVvpYt3Vl0",
        "Q": "At 0:43, what does he explain ?",
        "A": "He is saying that there are 7 blood drops in the A+ row. 7 drops, with 8 people per drop, 7 * 8 = 56 people."
    },
    {
        "video_name": "IFU7Go6Qg6E",
        "Q": "Would it not have been much easier (and faster) to recognize a 30-60-90 triangle at 3:10 and use simple geometry to realize the sides are at ratios of 1, 2, sqrt(3) -> as S/2, S, (S/2)*Sqrt(3)?",
        "A": "Absolutely. But you can learn to trust a new method by first ensuring that it is consistent with other methods in cases where you can apply the other method. Then when you are outside of a familiar context, you can apply the more general method."
    },
    {
        "video_name": "IFU7Go6Qg6E",
        "Q": "At 0:44 when Sal says,\" We are going to minimize the combined area of the triangle and the square\" Does he mean he wants to find the smallest possible area of the square and triangle combined?",
        "A": "Yep, he wants the total area to be as small as possible."
    },
    {
        "video_name": "IFU7Go6Qg6E",
        "Q": "i am so lost @ 5:23 for the combined area of the triangle and square we added the two area's together. why is he multiplying by (x/3)^2 when he already found the area to be ( squareroot of 3/ 4 ) ?",
        "A": "The area is not \u00e2\u0088\u009a3/4 It s \u00e2\u0088\u009a3/4 s\u00c2\u00b2 And he determined earlier that s = x/3 In this problem there are some unknowns. You aren t trying to calculate the area as a number because you may not have enough information to do that initially. So Sal is creating a function for finding the area. It may help to watch the next video in which he calculates the actual area."
    },
    {
        "video_name": "E0TNh9uWesw",
        "Q": "at 1:50, why did Khan add the 8 to both sides? Wasn't he supposed to subtract them?",
        "A": "To solve for x in the equation 8x - 8 = 7x - 6, we must isolate x on one side. You can do this by applying an opposite operation. Example: 8x - 8 = 7x - 6 +8 +8 <--- Since it says - 8, we add 8 to both sides. This is an opposite operation used to desert x. Hope this helps."
    },
    {
        "video_name": "E0TNh9uWesw",
        "Q": "what is the process called in 2:17 to find x ?",
        "A": "oh okay. thank you"
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "At 2:15 she says some people think Fibonacci numbers should start with zero instead of one. any opinions on this?",
        "A": "The argument comes from the definition of a sequence. The commonly accepted definition of a sequence is a function that maps from the set of natural numbers to some other set. Some believe that a sequence should be a mapping from the set of natural numbers and zero to some other set. If that is the case, there would be one more slot at the beginning of the sequence to be taken by 0."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "At 2:57 if she said it's the season for decorative pinecones, does that mean this was made in December?",
        "A": "Yes, it was made in December."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "at 2:30 is she making a a spiral out of fibonacci numbers on purpose?",
        "A": "I think that it is on purpose."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "at 4:14 what is the thing that she found along the side of the road",
        "A": "I believe that in one of her more recent videos she states that it is from a Mongolia tree. If you google Mongolia cone you will be able to see the did varieties."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "at 2:00 she brings up the fibonacci numbers. Did Fibonacci invent this? what can you use the sequence for?",
        "A": "He didn t invent it; it already existed, but he did discover it. It can be used in architecture, design, engineering, and many other things."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "At 4:14 you said that you found something on the side of the road. Isn't it a Mongolia pod? I know very little about plants, so I want to hear from professionals.",
        "A": "By the answers to other questions I can tell that it is a magnolia seed pod. The country Mongolia did not grow from seed!"
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "i don't get what she does at 4:34",
        "A": "If you look carefully at 4:28, you ll see that there are spirals in each of the clusters withing the cauliflower. I think that she looked at the spirals withing the clusters and counted them."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "At 1:10, Vi makes a spiral that is always a rectangle according to her, but what would happen if it went to infinity?",
        "A": "It would be a rectangle with infinite length and breadth, of course."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "At 1:36, aren't all the side lengths on the squares Fibonacci numbers?",
        "A": "Yes, you are correct. That was actually how the squares were designed."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "At 1:42 I can't see the spiral on the pine cone too well. I do see the spirals on the flowers, though. Anyone want to help me understand?",
        "A": "I don t know if you watched the whole video and found this out already, but she draws along the spirals with marker and, later, glitter glue. I found this helpful, since I couldn t see the spirals on my own either."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "Are there two 1's in the fibanacci series? At 2:19 she's showing how to get to the next number, Is she just showing us two ones to help us understand where the 2 comes from, or would you really put 1 twice if you were listing fibanacci numbers?",
        "A": "Short answer, yes. The third number is actually 1 and not 2. Long answer, the fib series is defined by the first two numbers 0 and 1. After that each number is the sum of the previous two. So the third number is the sum of 0 and 1 because 0+1 = 1 Here is the first few numbers: 0,1,1,2,3,5,8,13 You can see that 2 is 1+1, 3 is 2+1, 5 is 3+2, 8 is 5+3 and so on."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "At 1:42 isn't it the Golden ratio ?",
        "A": "The spiral which Vi drew is called the Fibonacci spiral. And while the Fibonacci spiral approaches the golden spiral, they are two separate things. The golden ratio is defined to be the ratio between two quantities when it is equal to the ratio between the sum of the two quantities and the larger of the two quantities. Algebraically, this is written as a/b = (a+b)/a = golden ratio. This number is approximately 1.6180339887...."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "Does anyone know what that thing from 4:15 is?",
        "A": "It s like a pinecone, and there s a specific kind of tree that makes it. I can t remember what it s name was..."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "At 4:11,can the weird pine thing have a fourth spiral?",
        "A": "You can always find a third and fourth and fifth spiral, assuming that the piney things go out far enough. The further out the piney things go, the more spirals you can have. This one does indeed have a third spiral. However, it is very difficult to see whether it has a fourth spiral, which is found in the intersections between the second and the third spirals. This clearly is too small to have a fifth spiral."
    },
    {
        "video_name": "ahXIMUkSXX0",
        "Q": "Seeing the number of the Fibonacci series at 02:07 makes me worder: Is there a formula that generates this series (just as there is a formula that generates triangle numbers)?",
        "A": "i dont think there is a formula other than adding the last 2 numbers to get the next one"
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "Quick silly question, at 7:42 Salman changed the inequality sign because he divided both sides of -20x =< -2. by -20. My question is why did he do it in the last step even though we have done multiplication on the previous steps? does it matter which step you flip it on?",
        "A": "NO. Just kidding. Yes, you have a great point. Also, you are correct. It doesn t matter of which order of steps you take, the thing that matters is if the steps are correct. -Hope this helped."
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "Af 3:21, how and why does Sal put down -negative infinity as an interger, when no such number exists?!",
        "A": "Negative infinity does exist. It is the lowest possible number."
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "At 3:18 i dont under stand how you go x<-9",
        "A": "27/3 equals 9 and its a negative 3."
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "at 2:46 why does sal switch the grater sign to a less than sign",
        "A": "The signs are switched because it is a rule in inequalities that when a negative number is multiplied or divided on both sides of the equation, the signs will be switched. So if it is < at first it will be > after a negative number is divided or multiplied on both sides of the equation. This rule only applies when the number is negative. Hope this helps a little bit. :)"
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "At 2:33, Sal says that you swap the inequality sign when you divide by a negative number. But I'm pretty sure my teacher taught me that when you divide by a negative, you would change > to a less than OR EQUAL TO symbol, not just to a <. So confused...who is right....",
        "A": "Laura, If you had an inequality like -x < 3 You would divide both sides by -1 and reverse the sign so you get x > -3 If you had -x \u00e2\u0089\u00a5 3 You would divide both sides by -1 and reverse the sign so you would get x \u00e2\u0089\u00a4 -3 The equal sign stays if it was there before dividing by a negative. If no equal sign was there before dividing by a negative, then no equal sign is there after dividing by a negative. I hope that helps make it click for you."
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "Since you use ] for \"including\" do you use ) for not including? Time in video 7:14",
        "A": "yes square brackets mean inclusive, so the interval includes those values. round brackets mean you exclude them, so the interval goes as close to the value as possible without actually getting to it."
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "7:24\nHow would you do it if you had to go backwards (You were given the solution and asked to find the inequality that has that solution)?",
        "A": "Just like in simple math. If I said add two numbers together that equal six 2+4=6, and we are done. as Sal likes to say. So ... .. Pic a number -1 make an expression where X = -1 4x+3<-1. You see it worked and just like the addition there are only a couple of possibilities compared to all the possibilities that could work. Just check your work!"
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "At about 4:40 in the video, Sal talks about the distributive property AND inequalities. How can you solve 5(-3x/5) greater than or equal to 3 (2x/3+7/3)?",
        "A": "Pirate, 5(-3x/5) \u00e2\u0089\u00a5 3 (2x/3+7/3) The 5/5 on the left becomes 1 and disappears -3x \u00e2\u0089\u00a5 3 (2x/3+7/3) Now distribute the 3 on the right -3x \u00e2\u0089\u00a5 3*2x/3+ 3*7/3) The 3/3 in both terms on the right becomes 1 and disappears -3x \u00e2\u0089\u00a5 2x+ 7 Subtract 2x from both sides -5x \u00e2\u0089\u00a5 7 Now divide both sides by -5 (and remember to reverse the sign when multiplying or dividing by a negative number x \u00e2\u0089\u00a4 -7/5 I hope that helps make it click for you."
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "At 7:27, I could understand the box bracket on 1/10, I can't understand the bracket on the negative infinity. Shouldn't it be box bracket also?",
        "A": "No, because infinity is not a number (so it can t be contained in the bracket). The bracket indicates that the solution ends at that point (up to and including that point on the number line), but infinity, by definition, goes on forever."
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "At 3:44, you said beacause were not using an equal sign we dont colur in the circle, is < with _ underneath the same as an equal sign?",
        "A": "Yes that s what he meant"
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "at around 1:20 , what is the -infinity, -1 mean? What is its purpose? How would I use it in future equations? Along with the other one around 7:12 ? When would infinity be positive? When would there be brackets on which side? examples? when would it be (\u00e2\u0088\u009e, x)? What about\n\n[\u00e2\u0088\u009e, x]?\n[\u00e2\u0088\u009e,x)?\n[-\u00e2\u0088\u009e,x)?\n[-\u00e2\u0088\u009e,x]?\n\neach and every possiblity, why would each one be?",
        "A": "Okay, so -\u00e2\u0088\u009e represents the smallest number possible, for inequalities saying the number is less than? And then there is a bracket on only the right side when it is less than or equal to? And for a greaterthan/greater than equal to, it would be just \u00e2\u0088\u009e not -\u00e2\u0088\u009e?"
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "0:00 remember this restart.\nhello I need help with this please answer this now!\nMulti-step_inequalities",
        "A": "You haven t actually asked a question. Can you be more specific? What part of the video don t you understand? If you can solve multi-step equations, then you can solve multi-step inequalities. You use the same techniques. The one difference is that if you multiply or divide by a negative value, then you need to reverse the inequality."
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "In question 3 (5:39) do you have to subtract 8x (from the right side) from both sides, or can you add 12x (from the left side) to both sides.\nJust wondering",
        "A": "You can do it either way. Usually you want to make the variable posititive. Just so it is easier to divide in the end. For me it is harder to divide a positive by a negative than a negative by a positive."
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "in 7:05 how does the negatives cancel out?",
        "A": "One way to look at it is to write the right hand side as (-1)2/(-1)20 Now you can easily see that the (-1)/(-1) cancel out, and you re left with 2/20"
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "at 1:22 why is the infinity negative",
        "A": "negative infinity just means anything less than the number -1"
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "In the beginning, 1:24, how do you know the infinity is negative? The 'x' is positive, so shouldn't the infinity sign also be positive?",
        "A": "They are saying that x < -1 which means x can be all values from -\u00e2\u0088\u009e to -1 not including -1. In fact, x cannot be positive at all in this problem."
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "At 0:52, how do you know to divide by four? Is it always 4 or did you do something to get it?",
        "A": "Your goal when solving an equation or inequality is to get the variable by itself. Sal has 4x < -4 Look at the 4x . He needs to change this into just x . So, the 4 must be moved. 4x is 4 times x . We always use the opposite operation, the opposite of multiplication is division. So, Sal needs to divide both sides by 4. Hope this helps."
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "At 3:25, why does Sal put negative infinity, as that would be an impossible number to reach, even on a number line?",
        "A": "negative infinity is not a number that you can reach, but it is a valid mathematical concept"
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "At 1:20, you write (- infinity, -1) (and then again at later points). Why are we writing a coordinate pair for something that is tangibly impossible? (we can say it's x anywhere y 1, but we can't graph it)",
        "A": "Infinity is not x anywhere . It s x way, way out there"
    },
    {
        "video_name": "xOxvyeSl0uA",
        "Q": "Ok, maybe this is a dumb question, but at 1:39, how can 5x possibly be greater than 8x plus 27? Is this an actual mistake in the video, or am I the one wrong here?",
        "A": "What if x was negative?"
    },
    {
        "video_name": "MFoXK57sw9k",
        "Q": "At 0:46 his number line tick marks don't match. How is it just two tick marks to get from 18 to 27?",
        "A": "Look again... it takes 3 tick marks to get from 18 to 27."
    },
    {
        "video_name": "MFoXK57sw9k",
        "Q": "1:00 So basiclly ur turning neg. and pos. into their opposites?",
        "A": "yes, that is what he is doing"
    },
    {
        "video_name": "MFoXK57sw9k",
        "Q": "At 0:23, you see brackets around a negative number. What do they notate?",
        "A": "The brackets mean the absolute value of the number. In other words, how far is it from zero? The absolute value of a number is always positive."
    },
    {
        "video_name": "bJb4n19sFrs",
        "Q": "why did he spell bananas bauauas on 1:14",
        "A": "His handwriting is just a bit off... if you look closely it is still an n . He is also just writing fast."
    },
    {
        "video_name": "bJb4n19sFrs",
        "Q": "At 1:08, Khan said that \"in order to have a proportional relationship, the ratio between these two have to be constant.\" What does that mean?",
        "A": "It means that the number of 1 thing has to equal another. For example, a simple proportion relationship is that there are 20 students per 1 teacher in a classroom. In order for it to continue to be proportional, you would need to have 40 students and 2 teachers. When you do 40/2 = 20 which is the same thing as 20/1. This shows that is proportional."
    },
    {
        "video_name": "6vlBOHckmzU",
        "Q": "I could barely understand what Sal said at 4:31 about P(small). However, it would be better if someone fully explained it by both ways (the shorter one as Sal did and the longer one). If I calculated the probability using the normal method, how would it be?",
        "A": "P(grand prize) = 1/10 x 1/10 x 1/26 = 1/2600 P(only letter correct) = 9/10 x 9/10 x 1/26 = 81/2600 P(1 number and the letter right) = (1/10 x 9/10 x 2P2) = 18/100 x 1/26 = 18/2600 NOTE: 2P2 because you can get the first number right or the second number right. Order matters. Expected value of grand prize = 1/2600 x $10,405 = $4 Expected value of smaller prize = (81/2600 + 18/2600) x 100 = $3.81 Total expected value of prizes= $7.81 cost = $5 Expected profit = $7.81 - $5 = $2.81"
    },
    {
        "video_name": "ViLt2WI0XSg",
        "Q": "When sal says f(n) at 1:06 is this a function?",
        "A": "f(n) is part of function notation. In this case, n is the input (or the independent variable) and f(n) is the output (or the dependent variable). We say f(n) to indicate the value of the function when the input is n. So yes, it is a function. If it s easier to think about, you can mentally replace n with x, and f(n) with y. A lot of people are used to seeing x s and y s instead!"
    },
    {
        "video_name": "ViLt2WI0XSg",
        "Q": "At 2:20 could you use the equation y=18-7x or would it be wrong because it is not explicit?",
        "A": "The way to look at it is that an arithmetic sequence is just a bunch of dots that will go up along a line starting with the first term at 1. Your equation would be based on drawing a line through all the dots. They are related, but not exactly the same thing. The domain of the sequence would be { 1, 2, 3, 4, ...}, but the domain of the equation would be all real numbers."
    },
    {
        "video_name": "OFNGpKGg9IQ",
        "Q": "At 4:05, when Sal is finding the outer radius, he writes 4 - x^2 + 2x. Shouldn't it be 4 - x^2 - 2x as in the original function?",
        "A": "The original function is x\u00c2\u00b2 - 2x. The outer radius is 4 minus this function: that is, 4 - (x\u00c2\u00b2 - 2x), which is, after removing the parentheses, 4 - x\u00c2\u00b2 + 2x."
    },
    {
        "video_name": "OFNGpKGg9IQ",
        "Q": "So at around 3:20, or sometime before that, Sal says Area is pi times stuff, minus pi times other stuff. But is that not volume already? Just that he has not written the integral sign yet?",
        "A": "The formula he s using there is just the area of a circle, minus the area of the inner circle (the cutout of the washer). It doesn t become volume until we provide a third dimension (that is, some thickness to the washer). We do that by multiplying by dx and then integrating. You might not be thinking of the dx as having independent significance, but without it we re looking at something that has only two dimensions: area, not volume."
    },
    {
        "video_name": "jxf7XqvZWWg",
        "Q": "What does Sal mean when he refers to the \"depth\" of the disc as being \"dy\" at 0:31?",
        "A": "Imagine added height in the y direction. The circle is defined only as 2D, so you need depth to make it 3D to make volume."
    },
    {
        "video_name": "AOsWph2FNLw",
        "Q": "On factorials at 6:48, well, cant it be like 2!=2*1*0*-1*-2........and so on?",
        "A": "Factorials stop at 1, for instance 2! = 2 * 1. 5! = 5*4*3*2*1. 1! = 1, now a special property is when 0! which we state is just 1. And you will never have a negative factorial"
    },
    {
        "video_name": "AOsWph2FNLw",
        "Q": "At 8:45 he says that you can use this formula for finding the number of ways to stick 5 things in 2 chairs without differentiating between those two things. What does he mean when he says differentiate?",
        "A": "He means essentially that we don t care if Object #1 and Object #2 are different, we won t be thinking of them as different. So say you have 5 people that you need to put into two chairs. One chair is on the left, the other is on the right. Jack and Jill are two different people, but we don t care if Jack is on the left and Jill on the right, or if Jill is on the left and Jack on the right. All we care is that we have Jack and Jill, instead of Jack and Sally, or Bob and Charles, etc."
    },
    {
        "video_name": "n-S63_goDFg",
        "Q": "At 0:16, Sal uses base 2.What is base 2 on vectors?",
        "A": "@ :32 he says R^2 , and these column vectors are R^2 vectors."
    },
    {
        "video_name": "2XB4K6T-yKo",
        "Q": "At 4:25 why don't we use the product rule for g(x) = -3 * log pi (x) or why is'nt -3 = 0 when grabbing the derivative of a constant",
        "A": "You also could use the product rule, but because the derivative of a constant is 0, the first term u (x)*v(x) in your result would disappear and you would be left with u(x)*v (x) which is what we are doing anyway based on our differentiation rules."
    },
    {
        "video_name": "2XB4K6T-yKo",
        "Q": "At 4:08, why have we written f prime of x equal to the derivative of ln x?",
        "A": "Be careful, it is not equal to the derivative of ln(x), f (x) is equal to the derivative of log base 7 of x. As Sal wrote that is 1/ln(7) times the derivative of ln(x), which ends up being 1 / [ln(7)*x]."
    },
    {
        "video_name": "ivg3dLTarbs",
        "Q": "In 2:07 why do you use two integral signs to integrate over the surface",
        "A": "Because you need a double integral to integrate over the surface. A surface is 2 dimensional. In some books the write just one integral sign, but they also think of it as a double integral. So it s just a shortcut."
    },
    {
        "video_name": "C2PC9185gIw",
        "Q": "From 0:39 on Sal says: \"This isn't always going to be the case... Let's assume that A has n linearly independent eigenvectors\" From the previous videos I was under the impression that a set of eigenvectors for A would always be L.I. eigenvectors. Why isn't this always the case? Could you provide me with an example? Thanks!",
        "A": "You are right - (nonzero) eigenvectors with distinct eigenvalues will always be linearly independent. So if the characteristic polynomial has n distinct roots, we are fine. If roots are repeated (they have an algebraic multiplicity greater than 1 ), we may be in trouble. Example for a non-digonalizable (defective) matrix: 1 1 0 1 You will later learn about generalized eigenvectors and the Jordan normal form , which will allow you to diagonalize A regardless."
    },
    {
        "video_name": "C2PC9185gIw",
        "Q": "At 6:50 you write d as a bunch of column vectors, if I'm performing a transformation of a 3x3 second-order tensor and have 3 unique unit eigenvectors, are these components of the eigenvectors the column of the transformation matrix?",
        "A": "D is a diagonal transformation matrix (with the eigenvalues on the diagonal) which transforms vectors whose coordinates are in terms of a basis B of corresponding eigenvectors. D isn t a matrix of eigenvectors. C is a matrix whose columns are those eigenvectors (the basis B ), and which transforms vectors from the B coordinate system into standard basis vectors. C^-1 transforms them back from standard basis to basis B ."
    },
    {
        "video_name": "OPpmp-kAuE4",
        "Q": "At 2:50 how can Sal just pull up the screen and have another problem?",
        "A": "It s the program he is using. That problem was off the screen but the program lets him drag the picture up so the rest of the picture comes into view."
    },
    {
        "video_name": "lZfXc4nHooo",
        "Q": "At 4:32 there is 32 1/5. But what do you do when the integer (not the exponent) is a fraction?",
        "A": "In the case of the video, you re talking about the 5th root of 32. If you were to have something like (1/32)^1/5, you d take the fifth root of the numerator and the denominator individually. That is: (1)^(1/5) / (32)^(1/5)"
    },
    {
        "video_name": "lZfXc4nHooo",
        "Q": "At 4:19, is a fifth root possible?",
        "A": "Yes. At the time stamp you noted, Sal is actually doing the 5th root of 32. The index on a radical can go as high as needed. You could fine the 100th root of 1, it = 1."
    },
    {
        "video_name": "lZfXc4nHooo",
        "Q": "At 0:45 Sal talks about 4^1/2. What would 4^-1/2 power be?",
        "A": "a negative power tells you to reciprocate the base so 4^-1/2 = 1/4^1/2"
    },
    {
        "video_name": "lZfXc4nHooo",
        "Q": "At 1:20, Sal asks, \"But what is the square root of 4, especially the principle root mean?\" and then he asks, \"what is a number that if I were to multiply it by itself...I'm going to get 4.\" I get that +2 is supposed to be \"the principle root\", but what about -2, how does that work if the exponent is a improper negative fraction? And how does he use that scratchboard so well? Vote if u agree",
        "A": "(-2)^2 = (-2)(-2) and a negative times a negative is a positive 4."
    },
    {
        "video_name": "hgtMWR3TFnY",
        "Q": "@12:40.. the std deviation is the avg distance from the mean. If the normal distribution is infinite, how can one quantify an average distance?",
        "A": "The probability that a random variable takes some value b is lower and lower as the further b is away from the mean. For example: the probability of obtaining a value further than 4 standard deviations away is 0.006%. You can expect to get barely 6 outcomes that large/small in 100,000 trials. I think that is the intuition behind it..."
    },
    {
        "video_name": "EJTPGyWqhqc",
        "Q": "at 2:10 in the veido aren't you supposed to multiply",
        "A": "No because 1 and 1/6 is a fraction and 1 equals 6/6 so 6 plus 1 ( from the 1/6) is 7 so 1 and 1/6 is equal to 7/6 but also you could multiply 1 x 6 and add 1. The denominator stays the same, but the numerator does NOT stay the same so 1 and 1/6 equals 7/6."
    },
    {
        "video_name": "v6Ct2OEmGNU",
        "Q": "At 3:22, how did you know how to draw the actual slope? I understood that the slope would =2 but since I will be drawing the slope on a graph paper, how will I be able to actually draw it?",
        "A": "If the slope is 2 that means you go up 2 units for every 1 unit you go to the right."
    },
    {
        "video_name": "sLw902V40L0",
        "Q": "at 5:45 why is the integral of 1 = t+c?",
        "A": "Since d/dt (t) = 1, we have \u00e2\u0088\u00ab 1 dt = t + C, where C is any real number."
    },
    {
        "video_name": "sLw902V40L0",
        "Q": "how did the antiderivative of -6 became -6t at 6:54 ?",
        "A": "he was just taking the integral/antiderivative of v(t). if you follow why t became t^2/2, then take the antiderivative of -6 with the same idea in mind."
    },
    {
        "video_name": "9hM32lsQ4aI",
        "Q": "i do not get what he means at 2:51",
        "A": "at 2:51 it is obvious 100+40 simply equals 140"
    },
    {
        "video_name": "9hM32lsQ4aI",
        "Q": "in 0:58 when he says \" what this really means,\" what does HE mean",
        "A": "Watch the video. He says it in the video!"
    },
    {
        "video_name": "9hM32lsQ4aI",
        "Q": "I don't get why at 1:11 he said one houndred and zero tens.",
        "A": "He said that because it is easier to picture carrying that way. This is because in adding you only add the numbers in the same place, so ones with ones, tens with tens and hundreds with hundreds. So when you add 160 and 140, you would first add the tens place and get 10 tens, which you would then hundreds column and put the zero in the tens column, thus one hundred and zero tens. Got it?"
    },
    {
        "video_name": "hXg-6YgAARk",
        "Q": "Just to clarify:\nAt 3:19, shouldn't the antiderivative of sin(x)+Cos(x) contain 2 constants( in other words, 2C) due to the fact that we are taking the anti-derivative of sin(x) and cos(x) individually?",
        "A": "f(x) = sin(x) + cos(x) F(x) = \u00e2\u0088\u00ab(sin(x) + cos(x))dx F(x) = \u00e2\u0088\u00ab(sin(x))dx + \u00e2\u0088\u00ab(cos(x))dx F(x) = -cos(x) + sin(x) + C F(x) = [sin(x) - cos(x) + C]"
    },
    {
        "video_name": "hXg-6YgAARk",
        "Q": "how is the antiderative of 1/a = ln abs a?? o.O 3:55",
        "A": "is that like a special rule like e^x?"
    },
    {
        "video_name": "hXg-6YgAARk",
        "Q": "At 3:41, I believe he is referring to the next video instead of the last video. Antiderivative of x^-1. Am I mistaken?",
        "A": "You seem right. Good job on noticing that!"
    },
    {
        "video_name": "hXg-6YgAARk",
        "Q": "Should the answer to the second problem at 3:57 specify that a \u00e2\u0089\u00a0 (is not equal to) zero, like we saw in the \"Antiderivative of x^-1\" video? Or is this different because there are other terms in the indefinite integral besides 1/a?",
        "A": "When taking the natural log, it should be assumed a\u00e2\u0089\u00a00"
    },
    {
        "video_name": "hXg-6YgAARk",
        "Q": "At 2:45, Sal means that the derivative of e^x is equal to e^x itself, right?",
        "A": "Yes, he was pointing out that it made an interesting correlation to the graph of the tangent line."
    },
    {
        "video_name": "Oc8sWN_jNF4",
        "Q": "Is the guy supposed to have three eyes at 2:56?",
        "A": "Yes. Vi was probably showing that 4D space would be really really REALLY weird."
    },
    {
        "video_name": "Oc8sWN_jNF4",
        "Q": "So at 7:17, is she suggesting, like John Green, that some infinities are in fact bigger than other infinities? Is that mathematically possible in any way??",
        "A": "It is. There are probably lots of complicated reasons why, but say there s a hypothetical universe with infinite stars. Each star has 3 planets, and each planet has 2 moons. Even though there are infinite stars, planets, and moons, there still have to be more planets than stars."
    },
    {
        "video_name": "Oc8sWN_jNF4",
        "Q": "in 7:10 , why did vi hart say that if the middle of the line was here any other dots were infinitly far away from it and the line would just go on when you just double the length of the start of the line and the dot??",
        "A": "Because it is true. There is an infinite amount of squiggled up line between any two points on the Koch Curve, and so if you were to unravel the Koch Curve, the two points would be infinitely distant from each other. This means if one point is on the paper, then the other point must be infinitely distant from your view."
    },
    {
        "video_name": "Oc8sWN_jNF4",
        "Q": "at 0:33 isn't it supposed to be \"Oh no(comma), time keeps happening\"?",
        "A": "Well, don t forget that this was done all in capital letters, and those don t bode for grammatical sense. Besides, the way Vi displays this shout is more like a monotone, because it s wavy and shifty (which is not how a common speech bubble is represented)."
    },
    {
        "video_name": "Oc8sWN_jNF4",
        "Q": "At 1:38, Vi says \"4 squared over 9 squared. But what is written is 16/27. Isn't that supposed to be 4/81?",
        "A": "hm... it s supposed to be 16/81..."
    },
    {
        "video_name": "Oc8sWN_jNF4",
        "Q": "at 4:44 why am i writing this just watch this",
        "A": "because it is so posed to be a video of entertaiment and math."
    },
    {
        "video_name": "Oc8sWN_jNF4",
        "Q": "At 6:04, square root 2 squared =2? I'm sorry I'm in 4th.",
        "A": "If you made it to that point in the video and understood everything up to that point ( I am assuming so since that is the only question you asked) then this should be easy. Root 2 = 2^(1/2) and squaring that = (2^(1/2))^2 Since a power raised to another power is the same as the powers being multiplied together you have (1/2)*(2/1) giving you 2/2 or just one. Using that exponent rule to simplify the equation you now get 2^1 or simply 2."
    },
    {
        "video_name": "Oc8sWN_jNF4",
        "Q": "at about 7:05, Vi says that if you try to match up corresponding points, you'll get infinity. But if you have an infinite number of points on the point you're trying to match up, you'll never be able to match it up because it keeps getting infinitly pointier, and there will never be a definite point you're trying to match up. And if infinity keeps getting keeps getting infinatly longer, does that mean that it's getting infinatly shorter at the same time? In that case is there such thing as infinity at all?",
        "A": "Just because it gets infinitely pointy does not mean that there is no definite point on the entire line. For instance, if I were to look at a single corner, perhaps of one of the earlier triangles, then I can be absolutely sure that the point will never move again, no matter how many iterations we go through."
    },
    {
        "video_name": "8CJ6Qdcoxsc",
        "Q": "At 3:10 Sal took x (the smallest number) and made it 4x. Did he do that because they're were 4 numbers (or integers)?? Why did he get 4x?",
        "A": "The 4 numbers are represented by: x = 1st number x+2 = 2nd number x+4 = 3rd number x+6 = 4th number Sal did not change x into 4x . That can t be done. The 4x comes from adding all 4 numbers together. x + x+2 + x+4 + x+6 = (x + x + x + x) + (2 + 4 + 6) = 4x + 12 Hope this helps."
    },
    {
        "video_name": "8CJ6Qdcoxsc",
        "Q": "At 0:31 why aren't we taking 7,9,11,13 or 9,11,13,15. Is it necessary to start from 11?\nOr can we take a group of 4 consecutive integers from any random order?",
        "A": "At this point in the video Sal is just showing an example of 4 consecutive odd integers. So, yes, other examples could be: 7,9,11,13 or 9,11,13,15. The important thing to realize is that these numbers have a defined pattern. Each number is 2 higher that the prior number. You can see the pattern if you look at an example."
    },
    {
        "video_name": "8CJ6Qdcoxsc",
        "Q": "At 4:01, dragging 12 to the side of 136 and subtracting 12 from 136 works as well, doesn't it?",
        "A": "yeah, that s what he s doing in the vid in the first place"
    },
    {
        "video_name": "8CJ6Qdcoxsc",
        "Q": "At 3:27, wouldn't order of operations tell you to divide by 4 first before subtracting 12? And if not, then why?",
        "A": "When you do an operation on the two sides of an equation, the usual order of operation doesn t count. What s important is that you do the same operation on both sides, to mantain the equality true."
    },
    {
        "video_name": "9Eg97Rtg-pE",
        "Q": "At 4:37. Sal simply plugs in h(x) to be h(3) when he was solving for h'(3). Are you allowed to do that? Why are you able to do that? I am not sure because while he was writing that he admitted that he was guessing.",
        "A": "The equation for h (x), when h and g are inverses, is 1/(g (h(x))). So to find h (3), he plugged in 3 for every x in the equation, giving h (3)=1/(g (h(3))). This would be the same if you were finding any other equation, you plug in the number you are looking for anywhere there is an x."
    },
    {
        "video_name": "LcEqOzNov4E",
        "Q": "At about 3:49 how does the slope decrease\nPlease explain anyone",
        "A": "It goes down since that is the sign representing the f value"
    },
    {
        "video_name": "KUhdMbx5ges",
        "Q": "4:04 the point is not called inradius, i think",
        "A": "Incenter."
    },
    {
        "video_name": "KUhdMbx5ges",
        "Q": "At 0:39, I was already lost. Can someone explain bi-secting?",
        "A": "Bisecting is simply to divide into two parts. If you were to bisect a 90\u00c2\u00ba angle, you would get two angles measuring 45\u00c2\u00ba each. An angle is always bisected by a line, ray, line segment, or in theoretical cases a vector."
    },
    {
        "video_name": "KUhdMbx5ges",
        "Q": "so the point at 4:02 is called the incenter?",
        "A": "Yes it is called the incenter"
    },
    {
        "video_name": "_b-2rZpX5z4",
        "Q": "At 1:23 Sal says (x/(x+1))^2 could be written as x^2/(x^2+1). Why is it that the denominator would not expand into a quadratic like (x+1)^2 would?",
        "A": "I am not able to hear the video just because I can t have my sound on right now, and he didn t write that out. You are correct, though... (x/(1+x))^2 would be x^2/(1 + 2x + x^2)."
    },
    {
        "video_name": "_b-2rZpX5z4",
        "Q": "At around 1:32, Sal didn't make a condition/constraint where x cannot equal to -1. Why? Wouldn't g(x) be undefined when x is equal to -1? If so, then would the f(g(x)) also be undefined when g(x) is equal to -1?",
        "A": "I would agree with your assessment. If X = -1, both g(x) and f(g(x)) would be undefined. F(x) should also be restricted to values where X not = 0 as square roots are usually restricted to where the contents of the radical are >=0 to stay within the set of real numbers."
    },
    {
        "video_name": "pbLiN8D9gAk",
        "Q": "At 1:10, why do you add 45,000+33,000 instead of subtracting 33,000 from the 120,000?\nShouldn't it be:\nL+ 45,000 + 33,000(-33,000) >= 120,000 (- 33,000)",
        "A": "Your way of subtracting 33,000 from both sides is one way to solve the problem, and all you have to do afterward is subtract 45,000 in the next step. Another way is to add them together first and subtract. Both methods are similar and will result in the same answer, so you should not be worried if you solved the problem in a different way, as long as you got the same answer as the video."
    },
    {
        "video_name": "VWO1m0S-a9Y",
        "Q": "Can there be more than three numbers in a ratio, like 4:3:2:1:5?",
        "A": "Yes, you may not be doing this yet but if you take Chemistry it is very important."
    },
    {
        "video_name": "tXlcE_K_C-Y",
        "Q": "What's with the patriotic colors? did this video come out near the 4th of July? 1:15",
        "A": "I think it is just a cool colors"
    },
    {
        "video_name": "tXlcE_K_C-Y",
        "Q": "At 6:07, if all the marbles were the same. Would this be possible?",
        "A": "The chance of that happening is extremely small, IF they were the same."
    },
    {
        "video_name": "36v2EXZRzUE",
        "Q": "At 0:58,he lost me with what he said It has to Change?",
        "A": "Hes saying that line graphes is useful if you want to show something that chances over time. In the video it is the stock that chaces from month to month. On the other hand: if you wanted to show how many people in a group like chokolate and how many like coffee or tea, another kind of graph or diagram would be more useful."
    },
    {
        "video_name": "rpI-X9Gn5a4",
        "Q": "At 3:34 how was he able to use -6 if x isn't equal to -6? It is only greater than -6, so he should have used -5 instead right?",
        "A": "The reason Sal used -6 is because he was working with the greater than symbol. If he were using greater than or equal to then you would be correct. Since it is greater than we need to know greater than what? the input has to be greater than -6 so the output then has to be greater than g(-6) (even though it is not in the domain of the function). I hope this helps :)"
    },
    {
        "video_name": "rpI-X9Gn5a4",
        "Q": "@ 4:47. So I get that x can't be larger than 4 and simultaneously less than -3. He's using 4 as the input first. So say I was to input -3 originally and see at the end that my compound inequality was wrong, could I just switch the -3 and 4? Would this rule apply to every problem like this (where the normal steps would produce an incorrect compound inequality)?",
        "A": "I am not sure exactly what you are asking, are you saying that 4 < x < -3? This is not proper annotation and is actually incorrect because this says x > 4 and x < -3 which has nothing in common. If you put in -3, you would have to recognize this as a maximum, so it would still have to go on the right."
    },
    {
        "video_name": "rpI-X9Gn5a4",
        "Q": "At 5:30, shouldn't the range of the second piece of the function be 4<g(x)<-3 instead of -3<g(x)<4 ?",
        "A": "No, because what you are suggesting with 4<g(x)<-3 is that x should be bigger than 4 but less than -3 which is not possible."
    },
    {
        "video_name": "rpI-X9Gn5a4",
        "Q": "at 3:15 he says: \" as it's approaching negative 6.\" Does this mean we use negative 6 itself or something as close as possible?",
        "A": "Your second statement is correct, and you are approaching it from the right, so approaching would mean something like -5, -5.9, -5.99, -5.999, ... which as you stated you are trying to get closer and closer without actually reaching -6. With rounding, it would actually end up at negative 6, and 6.9 with the 9 repeating is actually -6."
    },
    {
        "video_name": "rpI-X9Gn5a4",
        "Q": "at 4:28 he doing it backwards and completely lost me please explain?",
        "A": "same. I got all lose at this point"
    },
    {
        "video_name": "MZpULgKhaEU",
        "Q": "At 3:13 can the fraction 109/300 be simplified any more?",
        "A": "109 is a prime number, which means the only whole numbers you can divide it by are 1 and 109 itself. This means it has no factors in common with 300 with which it can be simplified. So no, it can t be simplified any more."
    },
    {
        "video_name": "MZpULgKhaEU",
        "Q": "At 0:55 is it 1/3 + (9%) or 1/3 + (9%) * 1/3 or (9%) * 1/3 ?",
        "A": "The second one you gave is the correct one: 1/3 + (9%) * 1/3 Expansion of 9% means that 100% of the original volume (v=1/3) has an additional 9% (of the volume v=1/3) added to it. v + (9%) * v or to take if a few steps further: (100%) v + (9%) * v refactored gives: (100% + 9%) * v Let me know if you don t think the additional steps are helpful; I can easily remove them."
    },
    {
        "video_name": "mVlCXkht6hg",
        "Q": "How come he puts 365 under 2pi at around 6:15?",
        "A": "With y=cos(kx), the period is determined by k (a coefficient in front of x), and the period itself is whatever value of x makes kx equal to 2pi (because one full rotation is 2pi radians). When k=1, as in y=cos(1x) or just y=cos(x), the period is 2pi because kx equals 2pi when x equals 2pi. To stretch the period out to 365, kx must be equal to 2pi when x equals 365. If k times 365 equals 2pi (that is, 365k=2pi), then k must equal 2pi/365."
    },
    {
        "video_name": "mVlCXkht6hg",
        "Q": "@ about 3:50 Sal starts to explain which trig function to use. He poses the question, \"Which trigonometric function starts at your maximum point?\"\nHe treats it like he's already taught us this, but I have no idea. Is there an introductory video describing the differences between the graphs of sine, cosine and all the other trigonometric functions?",
        "A": "The videos about the unit circle covered this."
    },
    {
        "video_name": "mVlCXkht6hg",
        "Q": "At 3:25, isn't it incorrect to say that the period is 365 days since there are leap years, which has 366 days?",
        "A": "it s just an approximation"
    },
    {
        "video_name": "mVlCXkht6hg",
        "Q": "At 4:05, someone please explain the relation between unit circle angle functions and the curve?",
        "A": "Check out an earlier video on the unit circle. Notice that when the sine is smallest (near theta = 0 and pi) the roc (rate of change) is the largest, and when sine is near + or - 1 the roc is near 0. roc for cosine is similarly max near pi/2 and 3pi/2, and approaches 0 close to theta = 0 and pi."
    },
    {
        "video_name": "mVlCXkht6hg",
        "Q": "At 4:08 in the video, Sal starts talking about that to use a trigonometric function that starts at your starting point. For example, you use sine if your point starts at 0, and cosine if your point starts at 1. But what if your starting point is at -0.2? What trigonometric function can you use if your starting point starts at a negative?",
        "A": "All of the trig functions are valid for negative numbers. You would have to be more detailed to establish which one would be appropriate. But you might wind up with something simple like f(x) = sin(x) \u00e2\u0088\u0092 0.2"
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "At 2:44, how did you get that equation?",
        "A": "The answer to this question is probably much, much too complicated to be adressed in a singlke post. All that can be said is: the equation is called the quadradic equation and Sal looks at it in a video entitled Proof of quadradic equation"
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "At around 6:00 Sal says that the square root of 2.25 is 1.5 OR it could be -1.5. He solves the equation (or rather doesn't solve it) because the equation comes out to 1.5 = -1.5, (which does not work.) Why can't you use the negative square root of 2.25? It seems very valid and would prove the equation.",
        "A": "The principal square root of 2.25 is 1.5, not -1.5, which is the negative root. We cannot use the negative root, because the original problem statement asks about the principal square root of x, not the negative square root of x. If it were -sqrt(x), then we would use -1.5, and if it were \u00c2\u00b1sqrt(x), then we would use both roots."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "At 02:18 how did the 24x become a 25x when you subtract the x from both sides?",
        "A": "This is because you are subtracting one x, from an initial value of negative 24x. And we know -24-1=-25."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "2:32, Sal lost me at that part, when he started talking about the quadratic formula. Couldn't he just have done some other method?",
        "A": "Yes he could have factored."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "At 6:13, What is an extraneos solution? Please give this a like if you have the same question. Thank you.",
        "A": "An extraneous solution is a solution that falls out of the algebra for solving a square root equation, but does not actually satisfy the equation. Extraneous solutions arise when we raise both sides of an equation to an even power. This happens because polynomials of even degree are not invertible (for most y-values they achieve, there are multiple x-values that correspond)."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "But wait! Didn't Sal say at 5:40 that -1.5 was another square root of 2.25? But then he ignored that, and claimed that 2.25 was an extraneous solution, because 1.5\u00e2\u0089\u00a0-1.5",
        "A": "Remember, the square root has two roots: the principle root and the negative root. The default is that we always use the principle root (the non-negative one). We only use the negative root if there is a minus sign in front of the radical. sqrt(2.25) is asking for the principle root. It equals 1.5 -sqrt(2.25) is asking for the negative root. It equals -1.5 Hope this helps."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "ON 3:01, when starting to do the quadratic formula, what if we would get a negative number on another example in the square root, would that still be possible to do?",
        "A": "If you mean if b^2-4ac under the radical is negative , then yes, you can do it with complex numbers, which have been covered in earlier algebra II videos."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "at 2:47, 25^2 = 625.. but isnt b= -25x^2?",
        "A": "the principle square root is not always a positive number, however, it means that it could plus or minus that number. For example: The principle square root of 25 is either positive 5 or negative five. it is expressed with the plus or minus symbol with the five directly after it. In that respect, there are two solutions with a principle square root."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "At 1:30 why did Sal multiply the 2x and the 6 then multiply the answer by two?",
        "A": "Because the square factor form is as following: (ax-b)^2 ax-b x ax-b _________ ax^2 -abx -bax +b^2 Focus in the part (-abx -bax), since multiplication scalar is commutative, they are twice themselves. Rewrite them in order, (-abx -abx), which is equal to 2(-abx). If a = 2 and b = 6, representing them in the expression 2(-2*6x), 2(-12x) that is equal to -24x."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "At 7:34, What is the principale square root? If you have the same question please give this one a like. Thank you.",
        "A": "The principal square root of x is the positive number that, when squared, equals x. Square roots are multivalued, so we use the principal root when we want only the positive root."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "At 5:38, Sal uses the \"principlal\" root of 2.25. Is the principal root always the positive root?",
        "A": "Yes, the principal root is always the positive root. If the problem has a minus in front of the square root, then it is asking for the negative root. For example - sqrt(2.25) = -1.5 (the minus in front is asking for the negative root). hope this helps."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "time 9:05 x=[-1(2x-6)}^2 not equal to 4x^2 -24x+36.\nit works when we take the sq but if we shift the sq root to the left it give us a different statement?",
        "A": "[-1(2x-6)}^2 absolutely does equal 4x^2 -24x+36 Here s the work: Apply the exponent to both factors: (-1)^2 (2x-6)^2 Simplify the exponents: 1 (4x^2 - 12x -12x + 36) = 1 (4x^2 - 24x + 36) = 4x^2 - 24x + 36"
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "At 1:34, how did he get -24x?",
        "A": "He omitted some steps, which you will also do once you get better. Here is what happened: (2x-6)\u00c2\u00b2 // we started with this (2x-6)\u00c2\u00b2 = (2x-6)(2x-6) // Now use FOIL to get 4x\u00c2\u00b2 - 12x - 12x +36 // Now gather like terms to get 4x\u00c2\u00b2 - 24x + 36"
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "At 8:27, where did that -1 come from?",
        "A": "It is to get the two possible solutions , although not necessary if you both positive/negative solution on both sides for the equation."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "Why is 25 listed as +/- at 2:42?",
        "A": "A quadratic has a plus/minus solution since essentially, a negative squared yields a positive and so does a positive squared. So for example, the square root of 16 has two solutions, 4 and -4 since: 4 * 4 = 16 and -4 * -4 = 16"
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "After 6:20, I was kind of lost, and confused. I guess Sal is trying to say that if the equation is plus or minus 2.25, it would of satisfied the negative equation but since it is not, 2.25 is an extraneous solution?",
        "A": "That s what he was saying @9:40 Just keep in mind that when you square a square root, there may be an extraneous solution. So test them both with the original equation and rule out the extra one."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "At 1:55, how did you get -24x ?",
        "A": "(2x-6)^2 is the same as (2x-6)(2x-6). Using the foil method, multiplying first, outside, inside, last, referring to the positions of the numbers, you get: 2x(2x)=4x^2 2x(-6)=-12x (-6)(2x)=-12x (-6)(-6)=36 Combine the two -12x and you get -24x. It is also helpful to memorize the format for (ax+b)^2, which is a^2x^2+ 2abx+b^2"
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "1:40 Wait how did they get 4x^2 - 24x + 36? I get the 4x^2 and the 36 but where did the 24x come from?",
        "A": "=(2x-6)^2 =(2x-6)(2x-6) =4x^2-12x-12x+36 =4x^2-24x+36 When we are using distributive properties we must times all the terms in the parenthesis. So 2x from the first parenthesis times the -6 from the second parenthesis gives us -12, then as we move forward we get a total of -24x."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "At 4:42, why do get the square root of the number, and do all that other stuff?",
        "A": "He got the answer x=4 and x=2.25 He just put x=4 and x=2.25 back into the original equation \u00e2\u0088\u009ax=2x-6 to test which one is true and which one is extraneous solution."
    },
    {
        "video_name": "711pdW8TbbY",
        "Q": "So at 1:40 he gets 4 x squared-2 4 x+36?? were did Sal get the 36 from i do not see 6 square and also he mention that the equation ( at 1:37) he say that its a polynomial expression or multiplying binomials. What do he mean by that?",
        "A": "(2x-6)^2 is the exponent form for multiplying the binomial (2x-6) times itself. (2x-6)*(2x-6) is multiplied using FOIL (or using special products) Using FOIL: (2x-6)*(2x-6) = (2x)*(2x) -6(2x) -6(2x) -6(-6) = 4x^2 -12x -12x +36 = 4x^2 -24x +36 Hope this helps."
    },
    {
        "video_name": "0uCslW40VHQ",
        "Q": "At 1:22, if exponents are just shorthand for repeated multiplication, shouldn't it be at the same level as multiplication and division?",
        "A": "No. Exponents are greater than multiplication. If you had 5*4, you d get 20. But if you had 5^4 (5 to the fourth power), it s 5*5*5*5, which is 625. And 625 is greater than 20."
    },
    {
        "video_name": "0uCslW40VHQ",
        "Q": "At 0:43, Why does he use PEMDAS? Everyone I know uses GEMS.\nGrouping\nExponents\nMultiply/Divide\nSubtract/Add\n\nIn order from left to right for the letters M and S.",
        "A": "PEMDAS stands for Parenthesis, Exponents, Multiplying, Division, Addition. and Subtraction. It really is the same, but it has more steps."
    },
    {
        "video_name": "0uCslW40VHQ",
        "Q": "I formally know this as B(rackets) E(xponents) D(ivision) M(ultiplication) A(ddition) S(ubtraction). At 0:54 it said multiplication and division were at the same level, likewise for addition and subtraction... does this mean when the two of this are in an equation, to work from left to right?",
        "A": "In most cases, you should work from left to right. Start with brackets, and then work on your exponents, then do multiplication/division from left to right, and finish with addition/subtraction."
    },
    {
        "video_name": "LJqNdG6Y2cM",
        "Q": "shouldn't the cos x be negative at 5:08 ?",
        "A": "I see ,it appears that he distributed the negative sign on the whole thing and that made it what it is, thanks for the reply"
    },
    {
        "video_name": "LJqNdG6Y2cM",
        "Q": "At 1:10, when Sal says that if g'(x)=cosx, then g(x)=sinx, why isn't g(x)=sinx+C? I know that when we apply the rule it works and when we take the derivative the result will be the original, but wouldn't operating with sinx+C be more correct? why when we do that we get a different result? Thanks !",
        "A": "It does, thanks (:"
    },
    {
        "video_name": "LJqNdG6Y2cM",
        "Q": "At 3:00, where did he get the negative sign? It can't be the same negative sign that was in front of the second integral he found the anti derivative of because he applied that negative sign at 4:42.",
        "A": "the negative at 3:00 comes from the fact that f(x)g(x) is e^x(- cos x). Which is the same thing as -e^x(cos x) at 4:42 when he subs it back in, he has - [-e^x(cos x) + antiderivative(e^x(cos x))] when you distribute that negative out front the first part becomes positive and the second part becomes negative, which is how he writes it"
    },
    {
        "video_name": "RzEFGpe0rv4",
        "Q": "I did this about 0:05 into this video. I think that the answer will/does come out to 33. I checked using addition, but I'm still not entirely sure I understand what I did.\n\nI seem to keep breaking numbers down to their place values and adding/subtracting from the ones place leftward. Is this a good habit to have developed?",
        "A": "Probably, i think it is a good habbit"
    },
    {
        "video_name": "xo4VpX2IIMk",
        "Q": "@ 1:11 20 ft x 20 ft would be a square, wouldn't it?",
        "A": "Yes, it would but a square is also a rectangle"
    },
    {
        "video_name": "xo4VpX2IIMk",
        "Q": "at 1:14 sal said a rectangle but meant a square",
        "A": "It was likely just a simple mistake, but even so, all squares are technically rectangles because they have four sides and four right angles."
    },
    {
        "video_name": "xo4VpX2IIMk",
        "Q": "how come on 1:43 the guy in this video says 15 but writes 16?",
        "A": "simple mistake.... i know your thinking: teachers are humans too? but yes, they are and humans make mistakes ;D"
    },
    {
        "video_name": "xo4VpX2IIMk",
        "Q": "I DONT KNOW HOW TO MULTIPLY 2 digit numbers! :( I'm really bad at this is here any video? 4:10 simply cant understand it so fast",
        "A": "look back to the third or fourth grade curriculum"
    },
    {
        "video_name": "xo4VpX2IIMk",
        "Q": "at 0:02 how is the mouse moving by its self what?",
        "A": "That is because Sal is using a drawing program"
    },
    {
        "video_name": "xo4VpX2IIMk",
        "Q": "At 3:07, why did we not include the 5? Is it because it did not have a rectangle to contribute to the problem in order to calculate the total area?",
        "A": "The video didn t include the 5 because it is showing the length of the distance between the teal rectangle and the purple one. The purple rectangle already show the length and width(20 and 20). The 5 is also not the whole length of the rectangle. It is trying to throw you off."
    },
    {
        "video_name": "xo4VpX2IIMk",
        "Q": "at 1:00,what did sal ment?",
        "A": "He means that you would divide the shape into rectangles which is easy to find the areas of then add them to get the area of the big shape that consists of many rectangles."
    },
    {
        "video_name": "21vbBiCVijE",
        "Q": "At 2:35, Sal says that he sits on AD, and at 2:50 he says that he also sits on BE. What does he mean by that? He also says at 5:44 that he will see in about five seconds. Was Sal blind?",
        "A": "The point I sits on, or is on, the line segments AD and BE. Basically, AD and BE pass through I. Sal said that we ll see what he s talking about in about 5 seconds. I hope this clarifies what Sal said!"
    },
    {
        "video_name": "21vbBiCVijE",
        "Q": "In 0:30 , it talks about an angle bisector. What is an angle bisector?",
        "A": "It is a line segment which divides an angle into two equal halves."
    },
    {
        "video_name": "21vbBiCVijE",
        "Q": "At 5:54 what did he mean by \"tangent\" to the three sides?",
        "A": "Tangent means that a line touches a circle at exactly one point on the circle. So when he said that the circle was tangent to the three sides, he meant that the circle touched the lines at exactly one point."
    },
    {
        "video_name": "CLrImGKeuEI",
        "Q": "Hey 5:31 what is happening? I am unsure what to do in this situation. Please elaborate for me.",
        "A": "When Sal initially reduces the fraction at about 4:40 in the video, he made an error. At 5:31, he shows you the correct way to reduce the fraction. Both terms in the numerator must be divided by the -2. (-8+2\u00e2\u0088\u009a15)/(-2) = -8/(-2) + 2\u00e2\u0088\u009a15/(-2) = 4 - \u00e2\u0088\u009a15 (-8-2\u00e2\u0088\u009a15)/(-2) = -8/(-2) - 2\u00e2\u0088\u009a15/(-2) = 4 + \u00e2\u0088\u009a15 Hope this helps."
    },
    {
        "video_name": "CLrImGKeuEI",
        "Q": "At 5:54, aren't you supposed to divide sq.root of 15 by -2",
        "A": "He did, the leading term on sqrt(15) was a 2. (2*sqrt(15))/2 same thing as (a*b)/a = b"
    },
    {
        "video_name": "CLrImGKeuEI",
        "Q": "At 3:15, how did he get the sqrt of 60 like that? Using what he said compared to a calculator gives me two different answers. Did I simply misunderstand him, or am I more confused that I thought?",
        "A": "Watch the video again and pay careful attention to the step that Sal takes to change 60 to 15 * 4 and take the square roots of each term. My calculator shows that the sqrt of 60 is identical to 2 * sqrt 15."
    },
    {
        "video_name": "CLrImGKeuEI",
        "Q": "At 3:14, why can't you just simplify the square root of 64-4 into 8-2. Why does that change your answer so much?",
        "A": "You cannot separate square roots into parts if they have addition/subtraction between terms inside."
    },
    {
        "video_name": "CLrImGKeuEI",
        "Q": "at 5:43 how did u divide to -2 two times?I think what u did first time was right",
        "A": "he has an expression (A+B*c)/B = (A/B) + (B*c)/B = (A/B)+ C It is just 2 fractions combined (added/subtracted) with a common denominator."
    },
    {
        "video_name": "CLrImGKeuEI",
        "Q": "At 3:19 do u have to use the factor tree, can't u just do the square root of factoriazation",
        "A": "64-4=60 and 60 is not a perfect square number. YOu cannont square root 64 and 4 seperately because they are two terms, not just one term"
    },
    {
        "video_name": "CLrImGKeuEI",
        "Q": "Could Sal have taken the square root of 64, then the square root of 4 (if it hadn't been negative) instead of taking the square root of 60 at 3:20 ?",
        "A": "You have to do everything under the square before you square."
    },
    {
        "video_name": "pvyqucC-0NU",
        "Q": "2:31 Couldn't Sal just minus 42 and 63 to get the answer",
        "A": "Subtract 42 from 63?. Yes, he could have. He just put in an extra step to help people understand what happened."
    },
    {
        "video_name": "u7dhn-hBHzQ",
        "Q": "Around 2:56, Sal calls this technique local linearization. Just to clarify, is local linearization equivalent to the linear approximation? And is process creating a tangent line to a curve and looking at a specific point? I'm unsure about the definition.",
        "A": "Yes, my AP Calculus teacher also introduced this topic as Tangent Line Approximation. They all mean the same thing: using the tangent line at a point on the graph to approximate the value of another point very close to the original point."
    },
    {
        "video_name": "u7dhn-hBHzQ",
        "Q": "at 9:36 you got an overestimate slightly, but if the graph you were doing was concave up then it would be a slight underestimate right?",
        "A": "No, because the graph is concave down f (x)= .5 * x^(-1/2) f (x)= -.25 * x^(-3/2) At the point x=4.36 the second derivative is negative, therefore the graph is concave down and is a slight overestimate"
    },
    {
        "video_name": "u7dhn-hBHzQ",
        "Q": "At 3:40 (ish) I'm not sure I see why L(x) takes the form f(a) + f'(a)(x-a). Is it just a manipulation of y = mx + c about different axes?",
        "A": "It s the point slope formula for the point (a, f(a)) where a is the center of the local approximation."
    },
    {
        "video_name": "cWn6g8Qqvs4",
        "Q": "On 2:20 forward we multiply we get the i^2=-1.\nCould we revert the problem and get the real number to be an imaginary one ?\nTaking that example x=15<=> x=-15*i^2\nI'm assuming yes, but I don't know until which point we can \"create\" Im imaginary numbers from real ones.",
        "A": "You could of course do that but there s really no point. All you did was: \u00f0\u009d\u0091\u00a5 > 0 \u00f0\u009d\u0091\u00a5 = -\u00f0\u009d\u0091\u00a5\u00f0\u009d\u0091\u0096\u00c2\u00b2 That s similar to saying that -15 is the same thing as -(-15) which is true but doesn t point out anything new."
    },
    {
        "video_name": "cWn6g8Qqvs4",
        "Q": "Why is i^2 is -1? 2:25",
        "A": "When you square a square root, you get whatever number is left inside that square root. So i = sqrt(-1) and i * i = sqrt(-1) * sqrt(-1) = -1. Note: The reason you would not get 1 when you multiply i*I is because the rules for real numbers do not apply here. Remember that we are working with imaginary numbers."
    },
    {
        "video_name": "cWn6g8Qqvs4",
        "Q": "At 0:52, Sal says that (1-3i) is a number. Then he breaks it up and distributes it over (2+5i). I thought you couldn't break up a number, but had to treat the whole thing? What's going on?",
        "A": "no we can break up a number eg; (5)2 = (2+3)2=10 see"
    },
    {
        "video_name": "cWn6g8Qqvs4",
        "Q": "at 2:03 why is it not -15i",
        "A": "i^2= -1 (This is a rule)"
    },
    {
        "video_name": "cWn6g8Qqvs4",
        "Q": "What does FOIL mean and how is it used? 3:54",
        "A": "It s a mnemonic for how to multiply binomials. It stands for First, Outside, Inside, Last. If you have (a+b)(c+d) and want to multiply it out, you first multiply the First terms: ac the Outside terms: ad the Inside terms: bc the Last terms: bd Then you add these up, and the result is ac+ad+bc+bd"
    },
    {
        "video_name": "XoEn1LfVoTo",
        "Q": "2:57 so really you just do the opiste of the sign. so +5 you would -5.",
        "A": "Yes. If you have plus 5, you must subtract 5 on both sides, and if you have -5, you must add 5 on both sides. This causes them to cancel out and go away."
    },
    {
        "video_name": "XoEn1LfVoTo",
        "Q": "Why would you divide both sides by three?\n\nTime-4:08-4:10",
        "A": "3x=12 in this equation you want to know the value of x...so in easy words to make x as one and to cancel the 3 it s the easiest way to divide by 3 in both sides.thus the equation also remain unchanged."
    },
    {
        "video_name": "XoEn1LfVoTo",
        "Q": "In 2:33 do you have to cancel out +5 first and subtract it from 17? Couldn't you just divide 3 from 3x then divide 3 into 17 first, and then cancel out +5 and subtract? Seems like it defies PEMDAS since you add/subtract before multiplying/dividing the coefficient from x.",
        "A": "PEMDAS is the order of operations in an expression. It s not the rules for solving an equation. To solve equations you do the same thing to both sides to get x by itself. You can perform these operations (on both sides) in whatever order is most helpful in that. PEMDAS is still always true at every step, but it doesn t tell you what to do to both sides."
    },
    {
        "video_name": "XoEn1LfVoTo",
        "Q": "At 4:05, what about two step equations with fractions. EX. 7m/8=21",
        "A": "7m/8=21 multiply 8 to both sides 7m=168 divide by 7 on both sides m= 24 (7*24)/8=21 168/8=21 21=21"
    },
    {
        "video_name": "XoEn1LfVoTo",
        "Q": "At 2:20 sal said\n\"what is a symbolic way of representing.....\"\n\nwhat does he mean by \"symbolic?\"",
        "A": "Symbolic means that we want to represent it with an equation/symbols instead of just talking about it."
    },
    {
        "video_name": "XoEn1LfVoTo",
        "Q": "so, at 0:35 what would be the way to solve for x?",
        "A": "3x + 5 = 17 3x/3 = 12/3 3x + 5 = 17 x = 4/1 = 4 - 5 - 5"
    },
    {
        "video_name": "XoEn1LfVoTo",
        "Q": "Could the fraction answer at 8:25 be reduced to -4/3?",
        "A": "No -8/7 = - 1 1/7. If you want to reduce a fraction you have to divide the top (numerator) and bottom (denominator) by the same number. So if you do (-8\u00c3\u00b72)/(7\u00c3\u00b72) you get -4/3.5 or -4/7/2 or -8/7"
    },
    {
        "video_name": "XoEn1LfVoTo",
        "Q": "At 1:56, Sal says, \"the form that we are used to, what is that form\"",
        "A": "Sal is talking about getting the Variable( in this case x) on one side and ALL the variables on the other. so he was saying to subtract the 5 form both sides and then we will get 3x= 12. then divide by 3 and we would get x=4.... this is the form he was talking about. Hope this helps!"
    },
    {
        "video_name": "XoEn1LfVoTo",
        "Q": "At 5:05 Sal says that 3s/8 is the same thing as 3/8*s, but I don't understand this. Let's say s=5. 3s/8 is then equal to 15/8 but 3/8*s is equal to .. ok so as I ask this question and do the math I see they do in fact equal the same thing, but I have no idea why this is and feel it is going to be confusing for me in the future. Is there a way to explain WHY these two things actually work out to the same value?",
        "A": "Write the S as a fraction. s = s/1 3/8 * s = 3/8 * s/1 = (3*s) / (8*1) = 3s/8 To multiply fractions, we multiply numerator * numerator and denominator * denominator. The same steps work with the 5. 5 a s a fraction = 5/1 3/8 * 5 = 3/8 * 5/1 = (3*5) / (8*1) = 15/8"
    },
    {
        "video_name": "XoEn1LfVoTo",
        "Q": "@2:30 how do you know whether to get rid of the 5 or 3x first as in divide 3x by 3?",
        "A": "You can do either one 1st. But, generally it is easier to get rid of the 5 1st and save dividing by the coefficient of X (the 3) for last. If you move the 5 1st, the steps are as shown in the video. If you divide by 3 1st, you must divide the entire equation by 3. 3x/3 + 5/3 = 17/3 x + 5/3 = 17/3 Then subtract 5/3 to get: x = 17/3 - 5/3 = 12/3 = 4. Notice, we got the same answer by dividing 1st, But, a lot of the work involves fractions. Thus, we generally save that division step until last to avoid the fractions."
    },
    {
        "video_name": "0uHhk7P9SNo",
        "Q": "0:02 where is that?",
        "A": "should be one of the practice problems"
    },
    {
        "video_name": "0uHhk7P9SNo",
        "Q": "At 5:10, it gets a little confusing. I was wondering why Sal was going over the outcomes of a tie for the last option, and why those were ruled out.",
        "A": "because we went through the tie outcome in the second line"
    },
    {
        "video_name": "0uHhk7P9SNo",
        "Q": "Not a question. Just a comment. At about 4:20 Sal says \"I don't see a pattern\". I find it amazing that he makes such a statement, showing either that he does things totally unprepared (which is NOT what I would expect a teacher to do, since it wastes the students' time) or he's acting naive. The first thing that jumped at me when shown the red highlighted items was that the black (non-highlighted) outcomes were all the ties.",
        "A": "Yh, I m pretty sure Sal just likes to come down to the level of the learner, to show you how to go about problem-solving a new type of problem. I love this approach of his when it s a subject i ve never done before."
    },
    {
        "video_name": "0uHhk7P9SNo",
        "Q": "At 4:30, these questions should be reworded as \"the subset consists of ONLY outcomes where...\". to represent Sal's logic, otherwise this is not grammatically correct. The sample subset DOES contain ALL the outcomes where your friend wins OR there is a tie. It's extremely confusing and it is used in the quiz after the video as well.",
        "A": "My thoughts are the same as yours. I believe that what Sal explains is not correct. The first 3 statements are actually to tell us to check what s not included in the subset to see whether or not they contain what meets the question. And if it does, then the statement is false - because the subset doesn t contain all stated outcomes. Based on this logic, the correct answer is No. 4 still."
    },
    {
        "video_name": "AntQBDFOjUw",
        "Q": "At 3:00, isn't the equation not differentiable at x = 1? Do you need to make a note of \"for x =/= 1\" when defining the derivative in this way?",
        "A": "Good spot! You are correct, Sal should have specified that the result is true for x \u00e2\u0089\u00a0 1 . I ve submitted a clarification ."
    },
    {
        "video_name": "0BgUKHTW37E",
        "Q": "At 5:13 why did you go down -6? Because the first time you did it, it was -3. I am confused was this a mistake thanks!",
        "A": "as you can see, he also went over two, so it amounts to the same thing. For rise over run, its the same slope."
    },
    {
        "video_name": "0BgUKHTW37E",
        "Q": "at 0:56 , wouldnt it be y=-x+5 instead of y=5-x im confused",
        "A": "y-x=5 he added x at both side y-x+x=5+x y=5+x he crossed out the -x+x since -x+x=0"
    },
    {
        "video_name": "0BgUKHTW37E",
        "Q": "If it says y =-3x +5 at 5:02, doesn't it mean that if you one in the y direction, you go -3 in the x direction?",
        "A": "A slope of -3 is the same as a slope of -3/1. Because anything divided by 1 is the same number. Remember, slope is rise over run . The top number is your rise ... your up and down number. The bottom number is the run ... the left and right number. So a slope of -3 means either: For every 3 steps down (-3) you go 1 step to the right (1) OR For every 3 steps up (3) you go 1 step to the left (-1)"
    },
    {
        "video_name": "0BgUKHTW37E",
        "Q": "at 0:55 should always add by x or can we use y?",
        "A": "will he was solving for y, but when he want to solve x he could do the problem using x. I hope you understand my answer!?"
    },
    {
        "video_name": "SPVqgkOZMAc",
        "Q": "at 6:22, why sal multiplied c's to the answers, would it not bean answer withoat c's ?",
        "A": "It would have been A solution but he s getting the general solution. I believe the previous video has the properties he used."
    },
    {
        "video_name": "SPVqgkOZMAc",
        "Q": "At 5:00, can we divide both sides by e^(rx) to get the characteristic equation?",
        "A": "Yes, since Sal correctly said that e^(rx) is always positive (for real r and x), so you wouldn t be dividing by 0 when you divide both sides."
    },
    {
        "video_name": "SPVqgkOZMAc",
        "Q": "at 4:27 sal says \"e^rx cannot be zero\", why is that so?",
        "A": "e raised to any power can t be 0"
    },
    {
        "video_name": "AJNDeVt9UOo",
        "Q": "At 0:24 i cant even hear what the guy is saying.",
        "A": "Listen to it again. Try to block out any other sounds."
    },
    {
        "video_name": "AJNDeVt9UOo",
        "Q": "0:24 What of the variable is a fraction?",
        "A": "That s ok. Variables can be any real number. If the variable is a fraction, the only difference is that you need to use the correct steps to add, subtract, multiply, divide fractions to do the work involved. For example: 2x + 1/2 when x=2/5 1) substitute: 2 (2/5) + 1/2 2) multiply: 4/5 + 1/2 3) add: LCD = 10. 4/5 + 1/2 = 4/5 (2/2) + 1/2 (5/5) = 8/10 + 5/10 = 13/10 Hope this helps."
    },
    {
        "video_name": "AJNDeVt9UOo",
        "Q": "At 0:46 after you evaluate the expression do you use order of operations?",
        "A": "The order of operations must be used any where in math."
    },
    {
        "video_name": "AJNDeVt9UOo",
        "Q": "At 0:41 do I have to use the dot and parenthesis or can I just use the dot as the multiplication. What I mean is do we have to do 5*(1) and just do 5*1 instead?",
        "A": "You do not need both the dot and the parentheses. 5*1 or 5 (1) are both ok."
    },
    {
        "video_name": "AJNDeVt9UOo",
        "Q": "the ratio of red counters to blue counters is 9:11. There are y blue counters. Express the number of red counters in terms of y.",
        "A": "For every 11 blue counters, there are 9 red counters, thus for every 1 blue counter, there are 9/11 red counters. With y blue counters, there would be 9/11 y red counters."
    },
    {
        "video_name": "cvB8b4AACyE",
        "Q": "For the question on 3:16 can I write the solution in interval notation as [-7,5] ?",
        "A": "yes, it goes either way :-)"
    },
    {
        "video_name": "KV_XLL4K2Fw",
        "Q": "i got confused at 1:21 how did 8=80 and also i needed help after that to how did 80 become 104 and how is it 19/13?",
        "A": "1) The 8 never becomes 80, so I don t know what you mean with that part of your question. 2) To subtract 8 - 85/13, you need a common denominator (LCD=13). 8 = 8/1 *13/13 = 104/13. 3) To subtract fractions with a common denominator, you subtract the numerators. 104/13 - 85/13 = (104-85)/13 = 19/13 Seems like you need to review how to work with fractions. You really need to understand fractions to do the lessons with linear equations and slopes. Hope this helps."
    },
    {
        "video_name": "KV_XLL4K2Fw",
        "Q": "Please help me to understand the calculation in the video (at 1:26 timestamp) where Sal says 80 + 24 is 104 which then becomes 104/13 subtract 85/13. How did he get the 104?\nThanks!",
        "A": "All he did was break it down (8*13) 8(10+3) Using distributive property 80 + 24=104"
    },
    {
        "video_name": "KV_XLL4K2Fw",
        "Q": "How did he get 104 at 1:20?",
        "A": "He made the number a fraction so =104/13 so t could be a fraction to make it easier to subtract from the other fraction that has the denominator as 13."
    },
    {
        "video_name": "QR2vxfwiHAU",
        "Q": "@ 11:04what if you had the foci points and the vertices points? HOW do you fine the standars equation?",
        "A": "Lets a represents the major radius, b represents the minor radius, and f represent the focus. f\u00c2\u00b2+b\u00c2\u00b2=a\u00c2\u00b2 The foci are always on the same axis as the major radius. So if given the foci and one of the two radius, you can use the formula above to figure out the missing radius."
    },
    {
        "video_name": "QR2vxfwiHAU",
        "Q": "this is just impossible.\ntake the diagram at 9:15 for example. if you place two focci to the right and left of the vertical axis and place the final 'edge' focus directly above it, d1 plus d2 in this case would be a lot smaller than 2a.\nCan someone please explain this to me?",
        "A": "The foci have a specific place they are located based on the equation of the ellipse so that property always hold true. They are not placed arbitrarily; instead, they are based on what a and b are. The foci are c units away from the vertex and are on the major axis. c is given as follows: c^2 = a^2 - b^2"
    },
    {
        "video_name": "QR2vxfwiHAU",
        "Q": "At 9:55, why couldn't the final focal length equation (F=sqrt(a^2-b^2)) be simplified to f=a-b?",
        "A": "sqrt(a\u00c2\u00b2) - sqrt(b\u00c2\u00b2) does equal a-b But the sqrt(a\u00c2\u00b2) - sqrt(b\u00c2\u00b2) \u00e2\u0089\u00a0 sqrt(a\u00c2\u00b2-b\u00c2\u00b2) And the sqrt(a\u00c2\u00b2-b\u00c2\u00b2) is not a-b Try it with some numbers sqrt(5\u00c2\u00b2-3\u00c2\u00b2) = sqrt(25-9) = sqrt(16) = 4 5-3 = 2 so sqrt(5\u00c2\u00b2-3\u00c2\u00b2) \u00e2\u0089\u00a0 5-3 I hope that helps"
    },
    {
        "video_name": "QR2vxfwiHAU",
        "Q": "At 12:13, how do you decide the focal points?",
        "A": "It is just the centre of circle so for (x+a)^2 (y+b)^2 it will be (-a,-b)"
    },
    {
        "video_name": "QR2vxfwiHAU",
        "Q": "at 2:50 Sal says that d1+d2=2a. is it like this as \"a\" is the semi major axis?\nalso do the foci always lie on the major axis of a ellipse?\nplease help me.",
        "A": "The answers for both of your questions is yes. The sum of the distances between any point of the ellipse and the foci is equal to 2a (what means, the sum is equal to twice the legth of the semi major axis). And yes, the foci always lie on the major axis."
    },
    {
        "video_name": "QR2vxfwiHAU",
        "Q": "did he just assume the foci at 12:05 ?",
        "A": "Actually, We know foci will always lie on semi major axis. So, Sal took any two arbitrary point on semi major axis. and because we know distance between a focus and center of ellipse is f = sqrt(a-b), where [a>b] we calculated our distance and hence we got the coordinates of our foci."
    },
    {
        "video_name": "QR2vxfwiHAU",
        "Q": "Why is it not +- Square root of 9-4, at 12:36",
        "A": "because it s distance no negative distance I guess?"
    },
    {
        "video_name": "QR2vxfwiHAU",
        "Q": "At 10:04 why did Sal didn't put a plus minus sign in front of the square root",
        "A": "Because you cannot have negative distances you are unable to add a plus minus sign in front of the square root. Or you can think of the pythagorean theorm. You have probably never added a negative sign if you had a radical so apply it to here"
    },
    {
        "video_name": "QR2vxfwiHAU",
        "Q": "You kept calling it the \"semi minor radius\", however, when you first mentioned them you said that it was called the semi minor axis, and could also be called the minor radius. So I don't really get it.at 11:26 you counted 4 steps instead of three being the number you wrote as a=3 so why did u go 4 steps instead?Why is it not +- Square root of 9-4, at 12:36. please help others and sal",
        "A": "At 11:26, the centre is (1, -2), so he goes right 3 from (1, -2) which gets him to (4, -2). He went left 3 from there as well so he got (-2, -2). 12:36: In geometry, you can t have negative length."
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "Is it correct to define the span as the span of the SET of vectors, not as the span of VECTORS (as Sal does at 3:35)?",
        "A": "There s no real difference between a set of vectors and vectors (plural)."
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "In 3:18 Sal says that the span of the two vectors he had is a line. What I don't still understand is, if you can place vectors wherever you want, then is it not possible to get a bunch of collinear vectors and draw them parallel to each other so that your span is R2?",
        "A": "I was wondering about the same thing. The span of 2 vectors is all the possible linear combinations of the 2 vectors (with every possible magnitude and direction (in this case they are all in the same direction)), not the same vector located at every single point of a space (which space?). Yes, any vector can be located anywhere, but that doesn t mean it s located everywhere, or even in more than one place, in any vector space. And how would we determine which space all these parallel vectors are in?"
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "at 5:13 when khan said linearly dependent, is it true that all Co-linear set of vectors are linearly dependent.",
        "A": "If a set of vectors is collinear, then any of the vectors is a linear combination of any one of the remaining vectors; so the definition of a linearly dependent set ( at least one of the vectors is a linear combination of the others ) is satisfied."
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "At around 8:15, you say that the third vector (v3) can be made up of a linear combination of the other two vectors (v1 and v2). However, any of those vectors could be made up of a combination of the other two: v1 = v3-v2, and v2 =v3-v1. Is this always the case, or are there some times when there really is only one \"redundant\" equation?",
        "A": "You are right. It is always the case that any one of them can be thrown out. The last one usually seems to get the raw deal though :)"
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "At 11:27, Sal says that (7,0) and (-1,0) are linearly independent because there is no way one can be a combination of the other. But the vectors (2,3) and (7,2) mentioned at 8:10, are also not a combination of the other. So why are (7,0) and (1,0) linearly independent but (2,3) and (7,2) not?",
        "A": "The vectors (2,3) and (7,2) are linearly independent. Sal was saying that the vectors (2,3), (7,2) and (9,5) together are not linearly independent. This is because you can make any one of the three vectors as a linear combination of the other two."
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "@12:32 vector 9 5 (or v3)is a linear combination of v1 and v2. So what conclusion we draw? Are v1, v2, v3 linearly dependent or it is just v3 which is linearly depented on both?",
        "A": "The three vectors are linearly dependent. If you did a little algebra, you could also express v1 in terms of v2 and v3 or v2 in terms of v1 and v3, so it really only makes sense to talk about the whole set of vectors as being linearly dependent."
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "9:37 says that span (v1,v2)=R^2, but V3 is linearly dependent, so i am assuming that the set{v1,v2,v3} is linearly dependent but the set{v1,v2} are linearly independent?...just clarifying??..so does this mean only 2 linearly independent vectors can span a vector space?",
        "A": "you need as many independent vectors as dimensions of the space - so 2 to span R2 but 3 to span R3, 4 to span R4 etc."
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "At 9:55, Sal asks if the sets (2,3) (7,2) (9,5) are linearly dependent or independent and then demonstrates the 3rd vector to be the sum of 1st and 2nd and consequently a linearly dependent space. If I change the 3rd set to (10,4) and the other 2 sets remain the same, the third set is still linearly dependent of the first two as I can express it as a sum of the first two applying scalars to the first 2 sets, correct?",
        "A": "Yes. In fact, any set of 3 vectors that are R^2 vectors will be a linearly dependent set, no matter what your vectors are. In general, say V is a vector space of dimension n, and S is a set of vectors from V, where the cardinality/size of S is greater than n. Then S is a linearly dependent set. As an example, if V=R^2, then the dimension of V is 2, so any set that contains more than 2 vectors from R^2 is guaranteed to be linearly dependent."
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "11:45-11:50, Sal mentions Linear Independent set ,Just out of curiosity, does the vector's orthogonality have anything to do with it's dependence/independence?",
        "A": "If two (non-zero) vectors are orthogonal, then they are linearly independent, but the converse is not necessarily true."
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "For the example at about 4:25, Sal refers to the span only ever as \"the line\" on the graph. However, I'm trying to find out what it would be algebraically or 'formally'. In that case, would the span ({[2,3],[4,6]}} = c[2,3], where c is an arbitrary scalar value?",
        "A": "I believe you would represent the span as the equation of the line. So you would solve for the line in with slope 3/2 that crosses through the point (2,3) and represent the span as an equation of that line."
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "In the final example you say the set of vectors are linearly independent @ 14:40 because they cannot be described as a combination of each other. But say if you added a fourth 3-D vector, [2, -1, -7], to the set, would this make the set linearly dependent since [0, 1, 0] + [0, 0, 7] + [2,-1, -7] = [2, 0, 0] ?",
        "A": "Yes. In fact once you have have 3 linearly independent vectors, then adding any other 3d vector will make the set no longer independent."
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "At around 06:40, are you introducing a third 2-tuple or a 3-tuple?",
        "A": "In order to ever represent a 3D space you need at least \u00e2\u0084\u009d\u00c2\u00b3 vectors, all of them. So in that diagram (even though he never wrote it) Sal must have been using at least \u00e2\u0084\u009d\u00c2\u00b3 vectors all along, both to represent only the plane, and to eventually represent all of \u00e2\u0084\u009d\u00c2\u00b3. I say at least because any higher dimension vectors can be used to represent lines, planes or volumes, and since it was only a diagram without written vectors or numbers, we could have been talking all along about \u00e2\u0084\u009d\u00c2\u00b2\u00e2\u0081\u00b0 vectors."
    },
    {
        "video_name": "CrV1xCWdY-g",
        "Q": "In the last example at 15:10, Sal says the 3 vectors are linearly independent, but then he says that the span of these three vectors is R3. Isn\u00c2\u00b4t this contradictory??",
        "A": "No, this is not contradictory. In fact, if three vectors are linearly independent, their span MUST be R3."
    },
    {
        "video_name": "eEUK_ThrHuQ",
        "Q": "At 1:44, there is something about row vectors. Who can explain that to me, using basic terms and where it would be applied?",
        "A": "In exactly the same way that you add and subtract multiples of rows in systems of equations with 2 or 3 variables, you solve augmented matrices, whether they re augmented with one fixed vector or with an identity matrix."
    },
    {
        "video_name": "eEUK_ThrHuQ",
        "Q": "At 8:26 we divided by a, so shouldn't we assume that in order for matrix to be invertible, a can't be zero as well?",
        "A": "True, but the 1st entry in the first row wouldn t be 0."
    },
    {
        "video_name": "qGmJ4F3b5W8",
        "Q": "Around 3:20 are the lines suppose to be straight when we solve?",
        "A": "yes"
    },
    {
        "video_name": "qGmJ4F3b5W8",
        "Q": "0:59 What is it called when x outputs an answer that can be y and z?",
        "A": "We simply refer to it as not a function . There isn t a specific name."
    },
    {
        "video_name": "1jDDfkKKgmc",
        "Q": "At 10:03, Sal says he wishes the bizzare formula thing (Sorry I have a short attention span) equals 1 instead of -1. What if you substitute -i for i? Am I getting into something completely different? And how can you create formulas with imaginary numbers?",
        "A": "-i can also mean the opposite of i. And what about i^3? Visit the powers of negative numbers video."
    },
    {
        "video_name": "1jDDfkKKgmc",
        "Q": "At 3:04, is the plural form of radius really radiuseseseseses?",
        "A": "The plural of radius is radii (also radiuses)"
    },
    {
        "video_name": "1jDDfkKKgmc",
        "Q": "Starting at 3:12, what is that Greek letter? It sounded like Sal said Pay Thayta. Is it a variable that stands for a number like pi? Thanks :)",
        "A": "Theta is a variable that is usually used to represent unknown angles"
    },
    {
        "video_name": "1jDDfkKKgmc",
        "Q": "At 10:10 -- Euler's id doesn't just have important \"numbers\" in math, it's composed of the fundamental operations and the fundamental relation too -- yes, it has e, i, pi, one, and zero; but also addition, multiplication (in the exponent), exponentiation, and equality. Not a hack, imho.",
        "A": "Yeah, the relation at 16:20 is just unsettling. Call me old fashioned."
    },
    {
        "video_name": "1jDDfkKKgmc",
        "Q": "Little grammar question...\nIs the plural form of radius that Sal is trying to say at around 3:30, supposed to be radians or radii?",
        "A": "radii is the correct answer"
    },
    {
        "video_name": "1jDDfkKKgmc",
        "Q": "What does he mean by radians at 3:19 in the video??",
        "A": "Radians are a way to measure angles, similar to degrees. 2pi radians are equal to 360 degrees. If you imagine drawing a radius in a circle, then picking up that line and bending it into a curve so that you can put it along the circumference, the distance that that curve goes is equal to one radian."
    },
    {
        "video_name": "1jDDfkKKgmc",
        "Q": "At around 5:30, I believe Sal had it the other way around: he said that the cosine is the distance on the x axis and the sine is the distance on the y axis, but isn't the sine the distance on the x axis and the cosine the distance on the y axis?",
        "A": "No, sal is correct. Sin function relates to the y-axis and the cosine function relates to the x-axis."
    },
    {
        "video_name": "WWv0RUxDfbs",
        "Q": "8:36 why is it 5 choose 3 instead of a permutation? The different orders of the flips seems to matter, would we still use combinations then?",
        "A": "It doesn t matter what combination we discover. In this case, the orders of the flips do not matter; all that matters is how many heads we get at the end."
    },
    {
        "video_name": "WWv0RUxDfbs",
        "Q": "a tiny bit before 6:12, Sal said that 5!/4! was 5. How did he figure that out? Just wanted to know, because there's prolly some kind of formula for figuring that out.\nThanks!",
        "A": "This can be seen by applying the definition of factorials and canceling like factors from top and bottom. 5!/4! = (1*2*3*4*5)/(1*2*3*4) = 5/1 = 5. Have a blessed, wonderful day!"
    },
    {
        "video_name": "Z7C69xP08d8",
        "Q": "minute 1:47 why do we subtract 2x and not anything else I don't understand that",
        "A": "We are trying to get all values with x in them onto one side of the equation. To remove it from the right hand side, we subtract it so that the 2x s cancel out. Then to keep the equation balanced, we also subtract 2x from the left."
    },
    {
        "video_name": "Z7C69xP08d8",
        "Q": "At 2:27 Sal says \"now we've simplified this to a linear equation\"\nI'm wondering, what is it that makes something \"linear\" or \"non-linear\"? Is it just the way it is formatted, or the number of variables, or what?",
        "A": "an equation is called linear, if all variables have power of 1 (they don t have second power etc). It s called linear because when you make a graph, it is represented by a straight line. If the variables have powers greater than 1, lets say x^2+y=5, then the graph is no longer a straight line. (x^2 = x squared )"
    },
    {
        "video_name": "Z7C69xP08d8",
        "Q": "Beginning about 0:40, Sal has us start to solve by multiplying both sides of the equation by x. That certainly works, but when he asked us, just before that, to think about how we would solve the problem ourselves, I started by adding 10/x to both sides. Then you could subtract 2 from both sides and you have 5=25/x. Then you can multiply both sides by x, yielding 5x=25, and then dividing both sides by 5 to get the x=5 answer. Somehow, to me, that seems cleaner. Thoughts?",
        "A": "I believe both ways would work, but however you want to do it, if it works better and all the time, do it. :)"
    },
    {
        "video_name": "Z7C69xP08d8",
        "Q": "In 1:45 I don't get why 7 becomes 7x and 2 becomes 2x can someone tell me?",
        "A": "First, he multiplies both sides by x to get the x out of the denominator. Then, he distributes the x into both (7 - 10/x) and (2+15/x). x(7 - 10/x) = 7x - 10 x(2 + 15/x) = 2x + 15"
    },
    {
        "video_name": "Z7C69xP08d8",
        "Q": "at 0:10 I said to myself \"x=5, NEXT VIDEO!\"",
        "A": "great! Khan makes learning so easy"
    },
    {
        "video_name": "Z7C69xP08d8",
        "Q": "At 1:07 how does x times negative 10 over x equals 10? And at 1:25 how does x time 15 over x equals 15? How did the x in both equations cancel each other out?",
        "A": "The process used is the same process used to reduce any fraction. For example: 20/2 = 10 This is done by cancelling out a common factor of 2 from the numerator and denominator. 20/ 2 = (2*10)/2 = 10 The only difference is that the common factor in your problem is a variable, rather than a specific number. Hope this helps."
    },
    {
        "video_name": "Z7C69xP08d8",
        "Q": "1:40\n\nWhat happens to the x when it goes into -10, why isn't that -10x? What rule decides that the x appears with the 7 (ie 7x in the equation) but not with the 10?\n\nSame thing with the other side, there is some reason that 15 is not 15x as well even though x goes into it",
        "A": "It actually does. Look. Let s try to open up the parentheses slowly: x(7 - 10/x) = 7x - 10/x * x The x s cancel each other, leaving, 7x-10. :)"
    },
    {
        "video_name": "Z7C69xP08d8",
        "Q": "Around 1:30, Sal says this simplifies to a linear equation. Doesn't a linear equation require that there be two variables?",
        "A": "A linear equation can have 1 variable. It then creates a horizontal or vertical line. For example, in the video, Sal gets x=5 . This creates a vertical line. It is interpreted as x=5 for all values of Y. So, the line contains points like: (5, 0); (5,-3); (5,11); etc. Hope this helps."
    },
    {
        "video_name": "pZ6mMVEE89g",
        "Q": "At 4:05 Sal derives the \"characteristic polynomial\". This seems to be a simple quadratic equation that can be solved (as long as b^2-4ac is >= 0).\nSo does that mean that most 2by2 matrices have an eigenvalue ?\n\nAlso, does the fact that the 2 eigenvalues exist mean that the columns are linearly dependent? It doesn't seem like that to me though.",
        "A": "You can actually find the eigenvalue even if b^2 - 4ac is < 0. That would just mean that the eigenvalue is imaginary. Matrices can have imaginary eigenvalues."
    },
    {
        "video_name": "VjLEoo3hIoM",
        "Q": "At 04:00, why does Sal come to a conclusion that both the events need to be multiplied ? What is the explanation to multiply both the events ?",
        "A": "To get the probability of both events being true. If you are asking why you multiply, it is because, for example, if there is a 1/2 probability of the 1st being green and a 1/3 probability of the 2nd being green, the probability of the 2nd being green and the 1st is green is 1/2 of the time the 2nd is green (1/3) since an of means multiplication, the probability of both being green is 1/2 x 1/3."
    },
    {
        "video_name": "VjLEoo3hIoM",
        "Q": "at 6:25 i don't get what the question sal asked meant. Can some one help me out",
        "A": "He meant that if you picked a green marble on the first try, you could replace it in the bag. Which mean the probability of having a green marble on the second try wouldn t depend on the first pick, so would still would have 3 out of five chance to pick a green marble. So 3/5 times 3/5 equals 9/25, equals 0.36 or 36%. 36% of 1$ is 36cents, which mean ON AVERAGE (after many many trials) , you d get 36 cents for 35 cents spent. So it s better, but I still don t like gambling!"
    },
    {
        "video_name": "VjLEoo3hIoM",
        "Q": "Why isn't 2/4 simplified in 5:02?",
        "A": "He simplified it at 5:14."
    },
    {
        "video_name": "VjLEoo3hIoM",
        "Q": "At 5:50, Sir multiplies the chances of winning the game with the prize of 1$ to find the average value of money the player would win after gambling many times over. But, 30% chances of winning means winning 30 times if gambled 100 times over,.....then expected value (if lets say gambled 100 times over) should have been = [(1$)x(30times)]-[(0.35$)x(100times)] as each time one plays he/she shells out 0.35$. So,this comes out to be -5$, ie: a loss of 5$.\nSo how did the player win 0.30$ as shown in the video?",
        "A": "You are getting to the same result, Sal is saying a win is only worth 30 cents so it is not worth paying 35 cents in order to play, it would produce an expected 5 cent loss. You are counting the initial payment as part of the payout, while Sal is comparing the cost to play vs the expected win. Both are valid ways to look at the problem."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "At 1:26, Sal mentions 'a, sub k'. What does that mean? Is it a mathematical expression?",
        "A": "It s the letter a, with k as a subscript. In a\u00e2\u0082\u0082, the 2 is a subscript. It s just a way to name variables. If you re typing in an environment where you can t make subscripts, it s also fine to use an underscore, like a_2. In the context of sequences, a_n refers to the nth term of the sequence whose name is a."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "3:01\nI thought it would be A(k)=3k-2:\n1*3 - 2 = 3-2 = 1.\n2*3 - 2 = 6-2 = 4.\netc.\nI thought you weren't supposed to have parenthesis in the function",
        "A": "Your version and Sal s are equal. If you simplify Sal s version, you get your version: 1 + 3(k - 1) = 1 + 3k - 3 = 3k - 2. Hope this helps."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "At 4:56, he says that you start with 3 so when writing the formula for the sequence you also start with 3. I'm confused; I thought you were supposed to write a formula that would allow you to find the first number in the sequence instead of using the number to find the other terms.",
        "A": "The explicit definition allows you to find any item in the series (including the first one) without having to know all the previous items. The formula is a(k) = 3 + 4(k-1), if you substitute k=1, the formula gives you the first item in the series a(1) = 3. The same way that if you want to find the 1000th element you just do a(1000) = 3 + 4(999) = 3999. In both cases you didn t need to know all the previous items in order to get the result."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "I don't understand what is happening after 2:54.\nWhy 1+3(k-1)? Where did that come from? .-.",
        "A": "The (k-1) is there because we don t apply the +3 until AFTER the first term. It basically says start at 1 and 3 for each term AFTER the first."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "At 1:23, what does Sal mean when he says, \"a sub k\"? what is \"a sub k\"? From that point and on, I started to get really confused about what he was talking about.",
        "A": "The kth term. a sub 1 (= a_1) is the 1st term. a_357 is the 357th term."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "So in Intro to sequences at 4:17, it says you can use another letter for the \"a\", but they keep using sub \"k\". Yet in this video they used sub \"n\". Does it matter what the sub-letter is or is it something specific for different sequences?",
        "A": "I think i through n are commonly used for integers. Sometimes i and j are the variable integers and m and n are their limits of variation. For the dependent variable I m not sure."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "0:12-- So a sequence is basically a rule (mathwise, I mean.)?",
        "A": "A sequence is a list of numbers. That list may or may not be defined by a simple rule."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "At 6:20 and 7:40, why is it necessary to define the first term of tge sequence?",
        "A": "Because if you don t define the first term, a-sub-3 or a-sub-9 could be absolutely any number. If a-sub-1 wasn t defined, it could be equal to 6 or 13 or even some really crazy number such as 13,243,546,576,879,809. So it s pretty important to make sure we have a-sub-1 defined before we try to define a-sub-k for any other value."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "At 1:30 when he writes number 4 does he mean by the number of terms in the sequence. I'm not sure.",
        "A": "Yes, he does. The expression on the left hand side there means the 1st through 4th terms of the sequence named a ."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "At 5:05, is Sal talking in general? Because otherwise what term are we 'at'?",
        "A": "Yes: this is a way of talking in general, since the formula is the same for every element in the sequence. In the formula Sal gives here, we are at the k-th term, where k can be any integer greater than zero."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "At the beginning, for the first two functions at 5:55, couldn't you have done a(k)=3k+1 and a(k)=4k+3.",
        "A": "If you did that, then for the first one, a(1) = 3(1)+1 =4, a(2) = 3(2)+1 =7 etc. which means you have a completely different sequence. What you would want to do is find a(0) which in this case is -2 and use that with a(k) = 3k - 2 to get the explicit formula. Same with the second one which would have a(0) = -1, so it would be a(k) = 4x - 1."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "So for the first finite sequence a_k Sal writes at 3:13, he really means to write a_k = 1 + 3(a_k-1) right? This would be the explicit format while a(k) = 1 + 3(k-1) would be recursive? I want to make sure that I am not mistaken.",
        "A": "With the explicit definition you can find the value of any term in the sequence, just by plugging in the kth term that you re interested in. With the recursive definition, you need more information. Knowing the value of the preceding term in the sequence is required, otherwise you will not be able to figure the kth that you want to know. The explicit definition of a sequence is a function for the whole sequence while the recursive one is a function of the preceding term."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "At about 3:00, when he defines the function for k, can that be simplified? Or in this notation would you just leave it?",
        "A": "You could simplify if you want to, but normally it s left there like that, so you can see the parts easier."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "at 4:43 should k=1 be k=3 instead? where the first term is 3?",
        "A": "K represents a number s position in the sequence. In this case, k = 1st number of the sequence which, as you said, is 3. Yes, it does get confusing. :)"
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "I still can't understand how he computed the sequence at 2:59. Can someone please explain?",
        "A": "Sal started with the first term of 1. He noticed that to get the second term he needed to add 3, that to get the third term he needed to add a second 3, that to get the fourth term he needed to add a third 3. In other words, to generate any member of the sequence, he needed to start with 1 and then add 3 times (the term number -1). That s how he got 1 + 3 ( k - 1 ). Hope this helps!"
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "at 3:10, why does he use k-1, why is that needed?",
        "A": "k-1 is the number of differences that have to the be added to the first term to get the kth term. For example, if you want to get the 2nd term, you add 1 difference to the first term, and if you want the 3rd term, you add 2 differences to the first term (or 1 difference to the 2nd term), and so on."
    },
    {
        "video_name": "KRFiAlo7t1E",
        "Q": "At about 3:00, when he defines the function for k, can that be simplified? Or in this notation would you just leave it?",
        "A": "Yes it can be simplified to==>> 3k-2;which gives you the k th value of the function.According to me both the notations are fine,but the one in which he has left it s easier to understand."
    },
    {
        "video_name": "43AS7bPUORc",
        "Q": "Why is he solving for x in this problem at 3:56? Is it because we are rotating around the y-axis? I am super confused.",
        "A": "He solved for x because he needed to find the horizontal distance, and x is the horizontal axis. In the other video, he left the function solved for y because he was measuring the vertical distance"
    },
    {
        "video_name": "43AS7bPUORc",
        "Q": "At 4:14, Sal say's that x=\u00e2\u0088\u009a y, but if y=4, this function would only have one x-value (i.e., x=2), whereas x=-2 is also a solution to the graph.",
        "A": "Yes, he specifies that he is only looking at the positive side of the functions. Since you are spinning the function around the axis to use the disc method, you only need to look at one side. The spin will sweep it through the negative side as well. If you were solving this instead, you would have to do a + or - when you take the square root."
    },
    {
        "video_name": "43AS7bPUORc",
        "Q": "At 5:55, why did we use the interval from 1 to 4 instead of from 0 to 4?",
        "A": "See 0:12 - it is 1 to 4 because that is how he decided to define the shape of the object whose volume is being measured."
    },
    {
        "video_name": "43AS7bPUORc",
        "Q": "Anyone that might explain why Sal takes the principle root of y at 3:50? For some reason it is completely lost on me..",
        "A": "He did that because he is rotating around the y-axis and he is trying to find the volume. As Sal showed, you need to find the radius of each disk so as to apply it into A = (pi)r^2 and then V = A(dy). Notice that it is in terms of dy, not dx. Therefore, the equation y=x^2 needed to be changed into terms of x, otherwise you would be finding a radius and thus an area and thus a volume of a solid that is irrelevant to this problem."
    },
    {
        "video_name": "zNtbi4OJs9I",
        "Q": "At 1:09, how can I use algebra to find the measure of more than two supplementary angles?",
        "A": "If there are three angles on a straight line, then you add all three of them together on one side of the equation and put 180 on the other side of the equals sign, then solve."
    },
    {
        "video_name": "1reblXFlM6I",
        "Q": "At 1:40, Sal uses the base 10 for the division. But can we use natural logarithm \"e\" instead of 10 and get the same answer?",
        "A": "The change of base formula applies for any arbitrary base\u00e2\u0080\u0094 base 10 and base e are often most convenient."
    },
    {
        "video_name": "1reblXFlM6I",
        "Q": "2:32 So you can introduce a logarithms into an equation and treating it like an algebraic term?",
        "A": "You bet! As long as you do the same thing to both sides you can do logs, or exponentials, differentiation and integration, anything you think will make it easier for you to solve the problem."
    },
    {
        "video_name": "1reblXFlM6I",
        "Q": "At 2:38, why did he take both sides of equation with log base b, what is the reasoning behind it?\nMore precisely how did someone discover that doing above would prove the formula?",
        "A": "If you dig deeper you will understand that your question is essentially asking: Is it naturally true that any number can be written in the form of other number to log (base other number) of (that any number )? The answer is actually yes and to me it seems axiomatic right now (after a quick search for a proof), though I don t feel completely satisfied."
    },
    {
        "video_name": "qPx7i1jwXX4",
        "Q": "I'm sure at least a few of us who are here have been taught to (when there's a need for it) to use the equation y = mx + c where m is the slope coefficient and c is at which point of y, x = 0 is crossed. I'm somewhat confused at the order of terms and constants at 1:21 - how can one write the c and -mx terms the opposite way? Does it even matter?",
        "A": "y=mx-b is the same thing as y=mx+(-b). When Sal said 12-2x, it was also saying 12+(-2)x. The to is just negative because you are losing 2 inches every day. If you gained 2 inches per day, the equation would be y=12+2x. The 12 can also be negative."
    },
    {
        "video_name": "qPx7i1jwXX4",
        "Q": "why does he use so many colors at 4:30",
        "A": "To correlate the points on the plane to the data table."
    },
    {
        "video_name": "qPx7i1jwXX4",
        "Q": "At 1:48 , is the 2x multiplication?",
        "A": "The x is not a multiplication sign if that s what you mean, but the expression 2x is using x as a variable to represent the number of days since Monday and multiplying it by 2 since 2 inches of snows melts for every day that passes."
    },
    {
        "video_name": "C7CY4l99S7M",
        "Q": "at about 6:30, how is it not equal even though \"y = x^2\". Wouldn't \"y = kx\"be the same if k is x? I understand its not constant, but it applies to the rules I believe.",
        "A": "For it to be a proportional relationship, k needs to be the same number every time. In the case of y=x^2, k is always x. But that means that for every x value, k changes! That s why it isn t proportional."
    },
    {
        "video_name": "C7CY4l99S7M",
        "Q": "when your doing negative around 4:20\ncan you go up instead of down",
        "A": "You don t want to go up, because then you would make a bigger negative number."
    },
    {
        "video_name": "C7CY4l99S7M",
        "Q": "At 2:19, how can we see that they are proportional?",
        "A": "We can see if the graph is proportional if it passes through (0,0) and is a straight line."
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "around 3:13(first example)\nI thought you would shade to the left if its less than and to the right if its greater than.Have I been doing it wrong or am I missing something? I understand what he says but, I think I may have learned it wrong...",
        "A": "That is what one might think and it seems to make sense as first because the left side is the smaller number, but that is not true. where to shade is depending on the slope of the line from left to right. Imagine the slope is a ladder against a wall. If Y is less than the line, you would shade under the ladder (slope). If Y is greater you would shade above it. Hope that helps."
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "At 1:25 why do we start the counting in y-axis above the y-intercept?",
        "A": "He is graphing y=4x+3. The y-intercept is 3. The slope is 4. Remember the slope is rise/run. We need 2 points to graph the line. The first point is (0.3), the y-intercept. The second point is a rise of 4 from the first point and a run of 1. So we go up 4 and over by 1 to graph the second point on the line. Does that help?"
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "At 5:47 he adds the x to -1/2 to make -1/2x. I thought the 1 in the 1/2 was x because he started with y is greater than -x/2-6",
        "A": "You can t say the 1 in the 1/2 is the x. You can t lose the variable. To split the x/2 apart you can do this: x/2 = x/1 * 1/2 = x * 1/2 = 1/2 * x"
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "At 5:44, how did u draw the slope on gragh?",
        "A": "The slope of a line is rise over run, and in this inequality is -1/2. For every -1 along the y axis, we move +2 to the right on the x axis."
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "Can someone please help me understand what he means between 1:18 and 1:26 where he says \"Which means if we go 1 in the x-direction, we're going to go up 4 in the y.\" Does he mean rise over run? :/",
        "A": "Yes, the 4 out of y=4x+3 refers to the slope. The four is the rise, and 1 is the run. Therefore, from 3, we go up 4 in the y, and to the right 1 in the x. BTW, the slope being only 4 is just like 4/1"
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "How can I plot the rise over run?\n(1:30)\nInside the y-intercept is 3. Then, one was plotted in (1,4), how come?",
        "A": "Given: y = 4\u00e2\u0080\u00a2x + 3 Compute y: x y -1 \u00e2\u0094\u0082 4\u00e2\u0080\u00a2-1 + 3 = -4 + 3 = [-1] \u00e2\u0094\u0082 0 \u00e2\u0094\u0082 4\u00e2\u0080\u00a20 + 3 = 0 + 3 = [3] \u00e2\u0094\u0082 1 \u00e2\u0094\u0082 4\u00e2\u0080\u00a21 + 3 = 4 + 3 = [7]"
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "Just to make sure,\nx/2 is the same as (1/2)x right? 5:45",
        "A": "Yes it is. Multiply out 1/2(x) and you get 1x/2 which is just x/2."
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "At 6:5 , after taking y intercept why we didn't take x intercept ?\nWe sal used slope property to daw the line ?",
        "A": "because u only need the y intercept to graph the inequality hope this helps"
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "4:22 How can the area \"under the equation graph\" be shaded in the first place? According to euclidean geometry Lines continue until infinity in the space that the line is drawn in. How large would the \"shaded area\" be in that case, if you wanted to shade an area, under an infinitely long line....?",
        "A": "You may be confusing equations (which are just the lines) vs. inequalities. When you have an inequality, there are points above / below the line that make the inequality true (so are solutions). This is just like if you solve: x + 1 = 10 which has one solution: x = 9 compared to ... x + 1 > 10 which has many solutions: x > 9. All values larger than 9 will be solutions. Hope this helps."
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "At 1:25, why doesn't he start counting from the bottom?",
        "A": "he is counting from the y int. of 3."
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "At 5:20 Sal says that y > -x/2 - 6 is the same as y = -1/2 - 6; how is this so?",
        "A": "What he is saying (and he makes a mistake which he corrects later at 5:40) is that -x/2 is the same as -1/2 x . To graph y > -x/2 - 6 he is starting by graphing the line y = - x/2 - 6 which is the same line as y = -1/2 x - 6 ."
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "at 1:25, he says the slop is four but he moves over 1 space on the x intercept? I thought you move over 4 spaces on the x intercept?",
        "A": "The rise-over-run concept is used to measure all slopes, positive and negative. When Sal says the slope is 4, he is saying that the slopes value is 4. Not passing over the x-intercept 4 spaces, as you have interpreted. The rise takes the dividend, and the run take the divisor. The output quotient is the value of the slope. To tell if a slope is negative or positive, if it slides upwards, it s positive. If it s negative, it slopes down. Let me know if this helps!"
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "3:00 isnt sal graphing y is less than or equal to 3 because he is marking 3?",
        "A": "Sal is placing a point on 3 because it is the y-intercept of a line, or the part of a line where it crosses the y-intercept. To graph a line you must first put a point on the y-intercept, then calculate the slope."
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "At 5:16, What if y that is known and _x - _y\nsuch as\n28 > 3x-3y?",
        "A": "A way to figure out where to shade in you can test Point (0,0). This happens when you plug in the zero to the y and x and solve to see if the equation is true"
    },
    {
        "video_name": "unSBFwK881s",
        "Q": "At 1:21, what do you mean by slope, and how would you do it if the second number was less than the first, I don't exactly get it",
        "A": "y=mx+b is the formula m=slope b=y-intercept"
    },
    {
        "video_name": "2oHUkHYtQoM",
        "Q": "At 1:33 why can't the instantaneous velocity just equal to  30.2/2  ?",
        "A": "30.2 is the distance traveled, not the velocity. So we have to find out how much that distance was changing at that point in time."
    },
    {
        "video_name": "2oHUkHYtQoM",
        "Q": "at 2:39 Sal says to take the Tagent we use the average of [1.5, 2] and [2, 2.5]. why don't we take the average of the [1.5, 2] and [1, 1.5]?",
        "A": "I believe it is because we want the line tangent to t=2, if you notice t=2 is between [1.5,2] and [2,2.5]. If you took the average of [1, 1.5] and [1.5, 2] you would get the line tangent to t= 1.5."
    },
    {
        "video_name": "2oHUkHYtQoM",
        "Q": "At 1:49 Sal says we can get the slope of the tangent line @ t=2 seconds\u00ef\u00bb\u00bf by averaging the \"slopes of the tangent lines between 1.5 and 2 and 2 and 2.5\" ... Aren't they secant lines, i.e. the\u00ef\u00bb\u00bf slope of a line between two points on a curve?",
        "A": "We don t actually have a curve in this case, just a list of data points, but conceptually they re points on a curve and of course there s no such thing as a tangent between two points on a curve, so Sal misspoke when he referred to them as tangents. He meant to say secant lines, or just lines."
    },
    {
        "video_name": "2oHUkHYtQoM",
        "Q": "Doesn't it say t is not the x value? Why did you put t in the x spot in the denominator around 3:40+?",
        "A": "If we were to graph this we would traditionally list the horizontal axis Time ( t ), and the vertical axis Speed ( s ). We just are used to using x but there is no x spot . As John stated, X is usual our independent variable , but for this problem we labeled it T for time."
    },
    {
        "video_name": "UXQGzgPf1LE",
        "Q": "At 5:35, doesn't (e^x)(e^cos(x)) simplify to e^(cos(x) + x) ?",
        "A": "I didn t watch the video, but yes, you can write (e^x)(e^cos(x)) as e^(x + cos(x))."
    },
    {
        "video_name": "qkfODKmZ-x4",
        "Q": "Hi, just making sure I'm not missing something\u00e2\u0080\u00a6\n\nAt 12:49, Sal says \"the set in R2 that connects these dots by the same transformation will be mapped to the set in R3 that connects these dots.\" Shouldn't \"R3\" be \"R2\", because the new triangle is still in R2?\n\nThanks :)",
        "A": "Yes it s a mistake."
    },
    {
        "video_name": "qkfODKmZ-x4",
        "Q": "at 12:46 Sal says the \"transformation will be mapped to the set in R^3\".\n\nI presume \"R^2\" is intended here. Or do I misunderstand?",
        "A": "You are correct, Sal made a mistake: a 2x2 matrix as your A for T( x )=A x should map x from R^2 into R^2."
    },
    {
        "video_name": "IL3UCuXrUzE",
        "Q": "At 0:30, where did the five come from?",
        "A": "He subtracted the three on -2x + y = 3 So: = -2x + y = 3 -3 -3 --------------- And you get the rest."
    },
    {
        "video_name": "IL3UCuXrUzE",
        "Q": "Hey guys! Does anybody of you know how in min 0:37 he went from the second to the third equation? Thanks!",
        "A": "All he did was subtract 5 from both sides and factor each side."
    },
    {
        "video_name": "IL3UCuXrUzE",
        "Q": "at 0:33 how did he get 5",
        "A": "he subtracted 5 from both sides of the equation: y-5=2(x-1) y-5=2x-2 y-5+5=2x-2+5 y=2x+3 hope that helped"
    },
    {
        "video_name": "IL3UCuXrUzE",
        "Q": "Guys how did Sal get from y=2x+3 to (y-5)=2(x-1) at 0:37? I'm really confused...",
        "A": "Even though he does not give a good reason for doing so, all he did was subtract 5 from both sides and factor, so y - 5 = 2x + 3 - 5 or y - 5 = 2x - 2 or y - 5 = 2(x - 1). This just puts the equation into a point slope form. With the 2x, he could have chosen to subtract any odd number (so when we add or subtract from 3, it will be an even number which is divisible by 2)."
    },
    {
        "video_name": "IL3UCuXrUzE",
        "Q": "Question: What if the mx in Y = mx + B is a fraction?\nHow would i get my X? I'm saying this because it wasn't included in the video.\nAt 1:35.",
        "A": "If m is a fraction then it makes things easier if you multiply both sides of the equation by the bottom of the fraction."
    },
    {
        "video_name": "IL3UCuXrUzE",
        "Q": "At 0:32 why does he write y-5 ?",
        "A": "Sal used one of the points from the line (1, 5) to write the equation in point-slope form. I think he is trying to show you the 3 different formats commonly used for linear equations. All of the equations create the same line."
    },
    {
        "video_name": "IL3UCuXrUzE",
        "Q": "Look on 1:04 somethings weird...",
        "A": "He just adjusted the margin. It s easier to see if you turn off the subtitles and pay close attention"
    },
    {
        "video_name": "IL3UCuXrUzE",
        "Q": "At 3:53, or somewhere around there, Sal said delta x is one\nIn between the x and one, there's a little symbol. Is it a greater than, or a equal to sign?",
        "A": "That symbol is an equal sign."
    },
    {
        "video_name": "IL3UCuXrUzE",
        "Q": "What does the video mean double as in 7:20 in the video",
        "A": "The y value there doesn t double, it ll just increase by two. So it means plus two. For example, in y = 2x + 3: if x = 2, y = 2 * 2 + 3 we get y = 7 if x = 3, y = 2 * 3 + 3 we get y = 9 y didn t double, it only increased by 2. Hope this helped."
    },
    {
        "video_name": "IL3UCuXrUzE",
        "Q": "What's the standard time to finish each section on average?(For example, this section:Linear Equations and Functions)\nMy mom wants me to finish the entire 8th grade by today (started today...at 3:00 too...gotta eat as well), and to me, it seems impossible at this rate.\nIt is currently 8:00 P.M.",
        "A": "Yes one can do it.but all can not do in one day."
    },
    {
        "video_name": "qtsMgdZ98Yg",
        "Q": "At 3:07 how is log base b over log base four equal to log base 4 to b",
        "A": "It s not log_b/log_4 . It s log_x(b)/log_x(4) . That = log_4(b) from the change of base rule."
    },
    {
        "video_name": "PWiWkqHmum0",
        "Q": "At 1:35 when I used my calculator I got a different answer it said 18.6 instead of 4.6 I am quite puzzled because I have repeated the calculation correctly and still I have the same wrong answer 18.6.",
        "A": "Are you sure you ve input the decimals properly? The answer is 4.6. The numbers are 4.0, 4.2, 5.0, 4.3 and 5.5."
    },
    {
        "video_name": "PWiWkqHmum0",
        "Q": "At 1:00, did Sal say \"Mu\" or \"U\" or what?",
        "A": "he said mu its a greek letter for population mean"
    },
    {
        "video_name": "PWiWkqHmum0",
        "Q": "At 5:11: Sal said that the number of data points was \"unitless\". Isn't the unit the number of data points because it's what the number measures?",
        "A": "This stems from an tool in applied maths called dimensional analysis. In dimensional analysis, you can only add quantities if they have the same units and their sum preserves their units and various other rules. Say we have a collection of lengths of a certain thing: 3m, 5m, 6m, and 8m. Then we want the mean to also be a length in meters. So \u00ce\u00bc = (3m+5m+6m+8m)/4 = 22m/4 = 5.5m. We want the units to be meters, therefore the quantity 4 has to be unit-less."
    },
    {
        "video_name": "PWiWkqHmum0",
        "Q": "at 2:34 what does the purple symbol mean?",
        "A": "it means sigma"
    },
    {
        "video_name": "C3RPDVGZdFw",
        "Q": "at 0:14, he said area is height X width. isn't it length X width?",
        "A": "it is length because of the height of the side. And the width of the bottom."
    },
    {
        "video_name": "C3RPDVGZdFw",
        "Q": "Through 3:05 and 3:25 he multiplies 35 WHOLES to 1/72 and he comes up with 35/72 what I don't forget is how he would multiply 35 wholes with 1/72 and gets 35/72. How is this possible?",
        "A": "Well, 35 wholes equals 35/1. Now 35/1 x 1/72 would be 35/72, because 35 x 1 is 35 and 1 x 72 is 72. Hope this helps!"
    },
    {
        "video_name": "C3RPDVGZdFw",
        "Q": "At 3:03 I don't understand how 35*1/72 is 35/72 can you explain please",
        "A": "35=35/1, so you would multiply 35/1 by 1/72. To do this, you would multiply the numerators by each other and the denominators by each other. 35 times 1= 35 (so this is your new numerator). 1 times 72= 72 (so this is your new denominator). This would lead to an answer of 35/72. A numerator is the number at the top of the fraction, and a denominator is the number on the bottom."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 7:10, wouldn't phi have to eventually repeat? I mean, wouldn't it run out of number sequences eventually?",
        "A": "no actually cause once every combination of three digits are used it would move on to four digits then five then six then seven..."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "2:40\nHow can Phi also be 1 1/2?\ninfinite one's like that in a loop would equal 2 correct?\ni'm confused ;/",
        "A": "Phi isn t 1 1/2, it s 1 1/phi. (Watch the video again starting at 0:40 to see Sal walk through how he gets to that answer) The first time I watched the video, I kept thinking it should all equal 2, too. But if it did, and we put a 2 in place of phi in the above equation, we get 2 = 1 1/2 and that s just not right. The Golden Ratio blows my mind all the time. There s no harm and lots of good in watching that video again and again."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 4:41 he creates a quadratic equation how does he do that?",
        "A": "The quadratic equation is a formula that has been determined ahead of time and applies to any problem like that."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 3:30, why did Sal ignore the negative square root of (1 + phi)?",
        "A": "As Sal mentioned at 5:10, we only care about the positive answer. This is because we started off with two lengths a and b. However, a length cannot be negative. Imagine if you had a rectangle with dimensions -2x-5. How would you draw this, or construct it? This is why we often disregard negative answers when we re dealing with length, area, volume, or so forth. Those answers just don t really make any logical sense. I hope this helps."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 8:10 of the video Sal mentions that the Golden Ratio shows up in music.....can anyone point out examples of this?",
        "A": "The harp and the string placements and sounds are what Sal is referring to."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "@11:30, How is the reciprocal of (b/a-b) = 1/(a-b/b). It seems like he took two reciprocals.",
        "A": "He did not take the reciprocal, but two reciprocals. It was a blunder on his part, but the statement b/(a-b) = 1/((a-b)/b) still holds true. x = (x^-1)^-1 (x^a)^b = x^ab (x^-1)^-1 = x^(-1)(-1) = x^1 = x"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "Wait,I have a question. At around 2:10 what happens, are we still looking for the perfect 'a' value to give a perfect ratio of 'a' and 'b' or have we started a new problem? As he keeps talking about the 'phi' thing, he kinda gets off topic. Sorry, I'm just confused. He starts talking about getting a quadratic equation to solve 'phi', but how did we get 'phi' in the first place. I kept replaying it, but it seems like 'phi' just popped out of nowhere. Please help. :)\n\n-Nature Goddess",
        "A": "Just as 3.1415blahblahblah is referred to by the greek letter pi , the golden ratio aka 1.61blahblahblah is referred to by the greek letter phi"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "hmmm i dont get it at 5:49 how did it became 1 + 4? /2 ? where does 4 come from?",
        "A": "Sal used the Quadratic Formula::: x= -b +_ \u00e2\u0088\u009ab^2-4ac and all of that over 2a. so he got the 4 from 4ac which in this problem 4(1)(-1) so we would get -4. so b^2 - - 4 is the same as b^2+4. that s how he got +4. Hope this helps!"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "Okay, please help! At 11:33, how can sal say that B/A-B is = to its reciprocal?? Earlier in the video, we established that A/B(phi) is equal to its reciprocal, B/A +1.\nSo, to wrap it up, at the 11:33 mark, why don't we need that 1 that we got from A+B/A",
        "A": "yes, thank you. I did not notice the double reciprocal ."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 1:24, Sal says that since b/a is the inverse of a/b that all the sudden it becomes 1/phi. Why is this? How do you know that b/a becomes 1/phi?",
        "A": "at 0:44, he says a/b is equal to phi. if you write 1/phi it is the same thing as 1/(a/b) which is equal to b/a."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 8:56, I don't get the golden ratio on the star.",
        "A": "that is actually sort of obvious because phi=1+sqrt5/2, and a pentagon has five sides."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 3:20,Sal says that phi squared is\n= to phi+1.If that's true,how come if you look carefully and see that phi=1?Because phixphi=phi+1.",
        "A": "Phi is not equal to 1. Let phi=x: x^2=x+1 x^2 - x=1 (subtraction) If you plug in 1: (1)^2 - (1)=1 1-1=1 0=1 This is not true By the way, phi x phi = phi^2, not 2phi. So you cannot subtract phi from phi^2 because they don t have the same exponent."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "What symbol does Sal draw @ 6:06.\n\n? = 1 + sqrt(5) / 2",
        "A": "Its Phi, one of the Greek letters. :)"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 3:30 Sal switches the order while doing the square root. Why is that? I have seen it being done before somewhere but I never wondered why? Help me please.",
        "A": "It was easier to do it that way. Writing the 1 in front left space to write the repetitive sequence."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "at 0:23 how does a/b = a+b/a?",
        "A": "The ratio between the larger part (\u00f0\u009d\u0091\u008e) to the smaller part (\u00f0\u009d\u0091\u008f) is the same as the ratio of the whole thing (\u00f0\u009d\u0091\u008e + \u00f0\u009d\u0091\u008f) to the larger part (\u00f0\u009d\u0091\u008e)."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 12:19, how did \"phi minus one\" turn into a positive \"one over phi\" wouldn't that be negative \"one over phi\"?",
        "A": "No, if you check out from about 1:30, you see that Sal finds (really early on) that \u00ce\u00a6 = 1 + 1/\u00ce\u00a6, so 1/\u00ce\u00a6 = \u00ce\u00a6-1. 1] \u00ce\u00a6 = a/b = (a+b)/a By definition 2] \u00ce\u00a6 = (a+b)/a = a/a + b/a Separate out the numerator 3] \u00ce\u00a6 = a/a + b/a = 1 + b/a Simplify a/a 4] \u00ce\u00a6 = a/b, so 1/\u00ce\u00a6 = b/a Going back to (1) 5] \u00ce\u00a6 = 1 + 1/\u00ce\u00a6 Substituting (4) into (3) 6] 1/\u00ce\u00a6 = \u00ce\u00a6 - 1 Subtract 1 from both sides and swap sides"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 5:40, how did that formula was derived from phi^2-phi-1?",
        "A": "That is the quadratic formula in action. If you don t know what that is, I suggest you revisit quadratics in Algebra 1."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At around 11:50 what exactly did he do to the ratio? How did it go from b/a-b to 1/a-b/b to 1/a/b-1",
        "A": "Short version: the following is an identity when neither p nor q are zero: p/q = 1/(q/p) Why? Multiply both sides by q/p and it s obvious. Since this is an identity, we can work as follows: b/(a-b) = 1/((a-b)/b) = 1/(a/b - b/b) = 1/(a/b - 1)"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 5:56, why phi= (1 +- square root of 1+4)/2 ?",
        "A": "Sal basically used the quadratic formula, which states that x (or in this case phi) = (-b\u00c2\u00b1 sqrt(b^2-4*a*c))/2a. In case you do not understand the a, the b, and the c, they correspond to the first, the second, and the third coefficients of the formula, respectively. Since the binomial is \u00cf\u0086^2-\u00cf\u0086-1, a is 1, b is -1, and c is -1 as well. Another way to write the binomial to make it a little easier to understand is (1*\u00cf\u0086^2)+(-1*\u00cf\u0086)+(-1). Hope this helps!"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At, 6:00, Why do we only care about the positive answer?",
        "A": "We only care about the positive answer because the lines aren t going to have a negative length (I m pretty sure that s impossible). In the question we have 1 \u00c2\u00b1 \u00e2\u0088\u009a1+4 which is 1\u00c2\u00b1 \u00e2\u0088\u009a5 If we do 1 - \u00e2\u0088\u009a5 we get a negative answer and that would mean we have to have a negative length on the lines."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 1:15 I understand that a/a is equal to 1. And how b/a is just the inverse of a/b but can anyone tell me how sal came up with 1 over phi? Im just a little confused.",
        "A": "The golden ratio is a hard concept to really grasp. Watching the video over several times may help. I will try to explain how Sal came up with the fact that phi = 1+1/phi First, by definition phi is the number for which a/b = (a+b)/a so phi = a/b and phi = (a+b)/a (a+b)/a is the same thing as 1/a * (a+b) Now using the distributive property that is a/a + b/a so phi = 1+b/a And we already know that phi also equals a/b and 1/(a/b) = b/a so 1/phi = b/a And if phi =1+b/a the phi = 1+ 1/phi I hope that helps!"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 4:25, how did you create a quadratic equation from phi = 1 + (1 / phi) to phi(squared) = phi plus 1?",
        "A": "If you multiply both sides of the original equation by phi, phi becomes phi(squared), 1 becomes phi, and (1/phi) becomes 1. Therefore, a new equation : phi(squared) = phi plus 1 , is formed"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 9:00, how do people figure that out? I mean, do they just randomly draw polygons and test them out? That must take so long...",
        "A": "Well, when you are dealing with shapes, many things end up being symmetrical especially including stars. And when mathematicians measured it out to find out how big it was then they found out this fascinating number."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 5:38, where does Sal get -4ac?",
        "A": "The formula Sal uses to solve quadratic equation is x=(-b+sqrt(b^2-4ac))/2a, called quadratic formula. There is a whole topic about it in the algebra chapter, or you can just look up its derivation on wikipedia"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 2:20 and 3:50, what makes it go on forever? It doesn't make sense.",
        "A": "It goes forever because phi is in its own definition. Look up recursion."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "was anyone else's mind blown when Sal explains the fractioncepition at 02:13?",
        "A": "LittleMonster, I am not sure where you got the term fractionsception, but Yes, it is hard to wrap my mind around the infinitely recursive defintion of phi."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 0:43 he just adds in 'phi', why does he do that? it is really confusing me",
        "A": "Phi is the Greek-letter variable we use to talk about the golden ratio just like we use pi for the ratio between a circle s circumference and diameter."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "at 5:18, if we are assuming that a and b are positive distances, does that mean it is possible to have a negative distance",
        "A": "most of the scalar quantities like distance and speed are never negative . The vector quantities , because they have direction involved can be negative like acceleration and displacement"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "at 4:00 i cant hear him can anyone tell me what he said?",
        "A": "...the same number that can be expressed this way, the same number where if I just subtract 1 from it you get its inverse. It can also be expressed... Make sense?"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 3:15, how did he get \"\u00ce\u00a6^2=\u00ce\u00a6+1\"?",
        "A": "\u00ce\u00a6 - 1 = 1 / \u00ce\u00a6, right? Multiply both sides by \u00ce\u00a6. You will get \u00ce\u00a6^2 - \u00ce\u00a6 = 1. Then just move the -\u00ce\u00a6 to the opposite side by adding \u00ce\u00a6 to it. And you end up with \u00ce\u00a6^2 = \u00ce\u00a6+1. There you go ;D"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "at minute 1:17, I cannot follow how b/a becomes 1/ b/a ... can anyone point me to the right video to understand how it works?",
        "A": "I m not sure if there is a video that explicitly explains this manipulation but it is quite simple: We are told \u00cf\u0086 = \u00f0\u009d\u0091\u008e/\u00f0\u009d\u0091\u008f. Take the reciprocal of both sides (or equivalently, raise each side to the -1 power) and this gives 1/\u00cf\u0086 = \u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008e. So whenever we see a \u00f0\u009d\u0091\u008f/\u00f0\u009d\u0091\u008e, we can simply substitute that with 1/\u00cf\u0086 because both are equivalent. Comment if you have questions."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 5:01 how do we know that a=1 and b=-1",
        "A": "When working with phi, we see phi as a variable. And when there is nothing in front of a variable, we say it has a coefficient of 1 So for example: 2x has a coefficient of 2 and x has a coefficient of 1. The formula goes, when: ax\u00c2\u00b2 + bx + c = 0 then the discriminant D = b\u00c2\u00b2-4ac So in this case phi is x and a is 1 because there is nothing in front of phi. b is -1 because there is nothing in front of phi and it is negative. Hope that helped."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 11:38, how do you change b/a-b to 1/a-b/b...I get how it's the reciprocal, but I don't get how he changed it...can someone show me?",
        "A": "b/(a-b) raise to the -1 power twice [[b/(a-b)]^-1]]^-1 use inner -1 power to get the reciprocal = [(a-b)/b]^-1 use outer -1 power to get 1 divided by (a-b)/b = 1/[(a-b)/b]"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "On 2:55, how does \"phi=1+1/phi\" be squared into ''phi(squared)=phi + 1\"?",
        "A": "It s not squared into , just multiply both sides by phi. So, phi = 1 + 1/phi phi * phi = phi * 1 + phi * 1/phi phi^2 = phi + phi/phi phi^2 = phi + 1"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "at 3:17 you say that phi^2 is equal to phi plus one. Shouldn't it be equal to 1 plus 1 considering you got rid of phi when you multiplied it to the other side thus creating phi^2?",
        "A": "No, \u00cf\u0086 = 1 + (1/\u00cf\u0086) Multiply both sides by \u00cf\u0086: \u00cf\u0086 \u00e2\u0080\u00a2 \u00cf\u0086 = (1 \u00e2\u0080\u00a2 \u00cf\u0086) + [\u00cf\u0086(1/\u00cf\u0086)] \u00cf\u0086\u00c2\u00b2 = \u00cf\u0086 + (\u00cf\u0086/\u00cf\u0086) \u00cf\u0086\u00c2\u00b2 = \u00cf\u0086 + 1 Comment if you have any questions."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "Why is this about 15:00 long?",
        "A": "Because reasons. ;)"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "how did Sal get from phi-1=1/phi to phi^2=phi+1 by multiplying...? 3:02",
        "A": "Multiply both sides by phi: phi(phi-1)=phi(1/phi), phi(phi-1)=1, Use the distributive property of multiplication: phi^2-phi=1, Add phi to both sides: phi^2-phi +phi=1+phi, And you re done: phi^2=phi+1"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "During the eclipse Sal talks about at 5:28, why is the moon the exact size of the sun, offering complete 1 for 1 coverage?",
        "A": "hahaha"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "around 4:45 Sal talks about videos with the quadratic formula does any one know where i can view these videos",
        "A": "If you do a search on quadratic formula, you can find videos on that topic."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "On 2:15 i don't understand why he keeps dividing?",
        "A": "Phi = 1 + 1/Phi, so you can take the Phi under the 1 and turn that into 1 + 1/Phi, making it look like Phi = 1+1/1+1/Phi. You can do this because Phi is defined as 1+ 1/Phi so it is a number defined by itself. You can also keep going. 1+1/1+1/1+1/Phi = Phi. And all of these are equivalent. 1+1/Phi is equal to 1+1/1+1/1+1/Phi. The reason he keeps dividing is to show the people watching this video that you can exchange Phi as many times as you want for it s definition, 1+1/Phi."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 3:10 how does is 1 + 1/p squared equal to p + 1?",
        "A": "p=1+1/p multiply both sides by p p^2 = p(1+1/p) = p + 1"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "I dont understand, my computer is a 16:9 ratio. if i do 16/9, it'll equal 1.77777777777777",
        "A": "so 16/9 is not 1.7777777 but 1.618 instead?"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 11:41, how does he go from 1 / (a-b / b) to 1 / (a/b - 1) ?",
        "A": "Let s focus only on the denominator: (a-b)/b. Because you have subtraction in the counter, you can simple divide a and b by b . So: (a-b)/b = a/b - b/b = a/b - 1. Hope it helps :)"
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "on 10:47 what does he mean by bxb square?",
        "A": "A bxb square is just a square with side length b. Just like if you had a 2x2 square it would have side length 2."
    },
    {
        "video_name": "5zosU6XTgSY",
        "Q": "At 1:06 how does b/a = 1/phi? Thanks!",
        "A": "It works that way because of the golden ratio. He just simplified his first equation just like in normal algebra"
    },
    {
        "video_name": "Xhc7Hens0f8",
        "Q": "At 3:00 in the video, how is the value of ln(x) = -(1/2). I could not understand the calculation here and after it till x = 1/sqrt(e). Hoping someone would be kind enough to explain the calculation going on there, thank you.",
        "A": "At 3:00 Sal divided 2x from both sides, -x/(2x) is the same as -1x/(2x). The x s cancelled out to have -1/2. From 3:15 he just uses properties of exponents to convert e^(-1/2) = 1/(e^(1/2)) = 1/\u00e2\u0088\u009a(e)"
    },
    {
        "video_name": "-rxUip6Ulnw",
        "Q": "at 0:15, why did Sal not distribute",
        "A": "You can distribute, but on some cases, like this, it s much easier to do (8-3) * 8 - 3 then doing (8 * 8) - (8*3) * 8 - 3. You can solve the problem using less work, too."
    },
    {
        "video_name": "x1z0hOyjapU",
        "Q": "At 5:13, why we need to transpose the matrix line to multiply it with the vector (do the dot product)?",
        "A": "Because we defined a dot product only for vectors, which by convention are represented as column vectors. So a matrix row is not technically a vector. But if we define our row as a transpose of a column vector, we can now use our definition and calculate the dot product of that column vector with any other vector."
    },
    {
        "video_name": "x1z0hOyjapU",
        "Q": "(At the end, around 17:20): It seems pretty clear that S(T(x)) = A*(Bx), because (e.g., if B is 3X4 and A is 2x3) Bx is an R3 vector and A*(Bx) transforms Bx; but have we proved this?",
        "A": "It doesn t need to be proven because it is essentially the definition of matrix products. Matrix products (with appropriate sizes) just represent composition of functions."
    },
    {
        "video_name": "BpPHve1cCCs",
        "Q": "At 3:38 where does that formula where he's finding the lim as x approaches 1(from left) of g(x)/(x-1) come from, specifically the x-1 on the bottom?",
        "A": "From the definition of the derivative. The definition is the limit as x approaches 1 of (g(x) - g(1)) / (x - 1). But g(1) is 0, so we are left with g(x) / (x - 1)."
    },
    {
        "video_name": "Ehd3cgRBvl0",
        "Q": "at 2:11 I did not understand what she was talking about",
        "A": "If you multiply a number by 10, for example take a number like 9, which is in the ones place (a number by itself, like 9 is in the ones place), all you have to do is move the number from whatever place it is in (ones, hundreds, thousands etc) to one place above. For example,take 9 which is in the ones place. Multiplying it by 10 will move it up to the tens place (90 = 9 x 10). Multiplying 90 again will move it to hundreds place (900 = 9 x 100)."
    },
    {
        "video_name": "Ehd3cgRBvl0",
        "Q": "1:43 and 4:33 can the same thing be done with decimals",
        "A": "Yes! 0.898 x 10 = 8.98 (move all digits left 1) 8.64 / 10 = 0.864 (move all digits right 1)"
    },
    {
        "video_name": "5ZMHdtGySvI",
        "Q": "at 2:00, why is it so confusing?",
        "A": "lol here let me help. It is very confusing because you are probably a 3rd grader or lower and Sal is a math expert, you wont figure this out instantly but dont stop trying! Good Luck!"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 0:27 sal mentions binomials but I haven't learned about binomials yet. Could someone please explain in a nutshell what a binomial is?",
        "A": "A binomial is an expression with two terms in it added together. A term is some combination of variables and constants multiplied together. Examples of binomials include: a+b, x+3, y\u00c2\u00b2+4y, 7xy-3p."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 2:46, Sal factored out and \"undistributed\" the two (s+5) and ended up with (s+5)(s-7)=0, which I don't get how he did that. Can someone please explain what he did and what is \"undistribution?\"",
        "A": "\u00c2\u00a7\u00c2\u00a7, Sal had s(s-7) + 5(s-7) If he had s(apples) + 5(apples) he would have (s+5)(apples) It works the same with (s-7) as it does with (apples) s(s-7) + 5(s-7) = (s+5)(s-7) I hope that helps make it click for you."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 1:49, why does s^2+5s turn into s(s+5)?",
        "A": "since s is squared, you need to set it up as if you are about to distribute the s to the other s and 5. when you distribute (to check) it should lead back to s^2 + 5s"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "Hold on, in this video, Sal says that you use a + b to for the sum and a * b for the other thingy ( around 0:55 ) but my textbook says a * c and b for the sum... what's the difference/ what's right?",
        "A": "I m not going to say your book is wrong because they may have defined things differently, but Sal s approach is definitely correct."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "why in 4:29 does he say + instead of minus",
        "A": "He says + 10 because (-2 x -5) = 10, as the two negatives cancel each other out."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 2:44, Sal \"undistributes\" the (s+5). How does that work?",
        "A": "Well when your multiplying (S+5)(S-7) and your distributing to get the answer, you first distribute the either one term or another: (S+5)(S-7)= S(S+5)+7(S+5) Now you just need to distribute again: S(S+5) - 7(S+5)= S^2+5S + -7S-35. So when your re-distributing, your just reversing what you distributed: S(S+5)-7(S+5)= (S+5)(S-7)"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 0:38, Sal says that there is a shortcut when 1 is the leading coefficient. Why does it only work then? Why can't it work when its a different number?",
        "A": "I believe he is talking about what he does at 5:20. Having a 1 as the leading coefficient makes it easier to make the binomial because you can just forget about it (as Sal does) and move on to the second and third pieces of the equation."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "So the equation is s^2-2s-35=0. I finally understand how to factor but I do not understand where the 5 and 7 derive from at 1:36.",
        "A": "Watch from 0:41 and it says that to factor you do a+b=-2(which is from the 2nd number the equation) and a\u00e2\u0080\u00a2b=-35(3rd number) and 5 and -7 are fit for the roles of a and b. Remember there is something called the power of rewind :)"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 4:30, Isn't S= -5 and S= 7? Sal had stated in his video that It is S= -5 OR S= 7.\nThanks guys!",
        "A": "OR means that either solution for S can be substituted into the equation. Only these two values for S can satisfy the equation i.e., make it equal to ZERO. Every other value for S will not work."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 1:36 Is it okay to switch the places of the 5 and the -7? I was trying to work it out and I have it written down as: s^2-7s+5s-35=0 instead of the way it's explained. This is very new to me, sorry if it's a silly question!",
        "A": "That s no problem. If you write it that way some of your steps will be different, but the final answer should be the same."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 2:32, where does the -35 go?",
        "A": "I don t see -35 go at 2:32, i see that at 2:12. So, -7s-35 is go for -7(s+5). because if you multiply -7(s+5) corretly, it will be equal to -7s-35."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 2:28, I don't clearly understand how Sal factors out the\n\n( s + 5 ) in the expression s( s + 5 ) - 7 ( s + 5 ) into ( s + 5 )( s - 7 ). How does he do it?",
        "A": "Sal used the distribution property, if you had A*B - C*B you can change this to B(A-C). A * B - C * B = B * (A-C) So s * ( s + 5 ) - 7 * ( s + 5 ) = ( s + 5 ) * ( s - 7 )."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 0:49 why does he do A + B = -2?",
        "A": "It s the formula for finding the solutions to the quadratic. What he is saying is you need 2 numbers that when added together equal -2, but when multiplied equals -35. Note: since the multiplied is negative, one of the two numbers will be negative and the other will be positive."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 2:36 with s(s+5)-7(s+5)=0 how is one of the (s+5) factored out? What makes it factored out into just (s+5)(s-7)=0. To me it looks like it magically disappeared for no reason.",
        "A": "Oh ok gotcha thanks."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 4:44, so -5 and 7 could both be answers?",
        "A": "Yes! All quadratics have two solutions for each variable!"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "What would you do if it is x^2-3=0 at 1:39 ?",
        "A": "(x^2 \u00e2\u0080\u0093 3) does not factor, so you would not use factoring to solve this, you would use square roots."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "from 3:39 why does this work?\nassume, for example, S+5 = 0 in (s+5)(s-7)=0.\nfactor out s from the equation so that we have s(5)(-7)=0.\ns(5)(-7)=-35s\n-35s=0\ndivide both sides by -35\ns=0/-35\ns=0?!",
        "A": "It is not supposed to equal zero you can t divide zero by anything"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 0:25, why do you have to factor and not solve the equation the normal way?",
        "A": "It s another way to easily find the solutions. Without using the quadratic equation, I mean. Good luck, Shai @forever.artist.infinity"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "Sal says at 4:44, that when 1 is the leading coefficient, the second method can be used. So if one isn't the leading coefficient, like if it's 3x^2 for the first term, is it better to use factoring by grouping instead? Or can we use the second method & factoring by grouping interchangeably? Because I prefer the second method, it seems easier. (sorry if my question is confusing, English isn't my native language!)",
        "A": "You can only plug in the factored numbers (ex. 3 and 4) into the (x+3)(x+4) if a is equal to 1. If it is not, pull out a GCF to try to set a equal to 1 or insert the factors to replace b, group, and then pull out common numbers. Then simply rearrange the equation."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "at 0:59, how do you get a*b= -35 ?",
        "A": "a = 7 , b = -5 a = 5, b= -7"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "wait, at 3:51, if s+5=0 and s-7=0, then wouldn't s+5=s-7? but that doesn't make any sense!",
        "A": "It s s+5=0 OR s-7=0 , because the equation (s+5)(s-7)=0 would be true either way."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 3:14 if either s+5 OR s-7 is equal to zero. Then how can the equation have 2 roots. I mean it's either s=-5 or s=7 right? IF you're gonna say both are equal to zero , how can we just assume?",
        "A": "That is an interesting question. The letter s represents a variable. That is, a quantity that is allowed to change. When s has a value of -5, the value of the polynomial is 0. When s has a value of 7, the polynomial has a value of 0. This is not an assumption. You can calculate it."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "Wait, it there a mistake or am I just confused? At 4:22 you plug in -5 for s, and first get 25, but shouldn't -5^2 equal -25?",
        "A": "Sal is calculating s^2 using s=-5 s^2 = (-5)^2, not -5^2 If you are doing -5^2, you are finding -s^2 for when s=+5 You need to square the entire value of s . You aren t squaring the minus. Hope this helps."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "at 6:18 he wrote down (s+5) (s-7), but before that he said that the factors were s= -5 and s= 7. Did he mean to switch those two numbers like that? and would it be the same thing?",
        "A": "Yes, when you have your roots you flip the sign. It s because it s written as (x-y), so when you have -5, it becomes (x+5)"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "I didn't understand the steps following 5:21 upto 5:50. Can someone please help?",
        "A": "We have a quadratic in standard form: ax^2+bx+c. Let us assume that the a value of the quadratic to the right is one. If we want to express this quadratic in factored form, we will have to find two numbers that when we take their sum, they sum to the b value/coefficient of the quadratic and multiply to the c value. This is because when we try to expand the factored form using these two numbers, we want to ensure that the result is equivocal to the initial equation (ax^2+bx+c). Hope this helps :)"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "What does he mean at 2:40 when he said he undistributed?",
        "A": "He means that he factored out (s+5)."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 2:15, how would you do it if the equation is x(x+3)=0?",
        "A": "use the distributive property and put the outside x into three and the other x then divide 0 into three parts it will either be a decimal or a fraction depending on how you did it your answer is x"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "at :53 to 1:22, you have to find the numbers that equal to the variables?",
        "A": "Yes. You have to find the set of numbers that makes both statements true. a + b = -2 and ab = -35 In this case 5 and -7 solves both."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "How do you know which numbers to find the a(b) or a+b for?\n#the confusion is real!\n0:45",
        "A": "I had the same question, I think you just guess and check which is why I prefer the quadratic equation, but I am not sure. I hope this helps, Good Luck!"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At around 1:35, how did Sal go from s^2-2s-35 to s^2+5s-7s-35? Please explain??",
        "A": "He rewrote -2s as 5s-7s. Since 5s-7s=-2s he can substitute one for the other. While it may look more complicated, changing the perspective in the problem can make it easier to solve."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "at 6:07 wouldn't it be (s-5)(s+7) and not (s+5)(s-7) since the solutions are -5 and 7?",
        "A": "No. If we want to find the solutions for the equation: (s + 5)(s - 7) = 0 We have to first note that either (s + 5) = 0 or (s - 7) = 0 or both. This is sometimes called the Zero Product Principle and it is perfectly intuitive. We must consider both possibilities so we make two equations: s + 5 = 0 s = -5 s - 7 = 0 s = 7 Therefore, if the solutions for s are -5 and 7, then the equation would be: (s + 5)(s - 7) = 0 Comment if you have questions."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "how come around 0:55 a+b=-2?",
        "A": "When you factor the expression s\u00c2\u00b2 - 2s - 35 Your answer will be in the form (s+a)(s+b) where a+b = -2 (because -2 is the middle coefficient) and a*b = -35 (because -35 is the last coefficient) Now 5 * (-7) = -35 and 5 + (-7) = -2 So a is 5 and b is -7 so you get (s+5)(s-7) I hope that is of help."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "Why explicitly when it is equal to zero? Around 0:15",
        "A": "Good question, later on, in more advanced concepts, you will deal with equations that are not necessarily equal to zero. These are called inequalities, for example the equation may be less than or, greater than or equal to zero hope that helps"
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "at 1:18 where it says a+b = -2 and a*b = -35 what equation could you use to find a and b because if a+b = 3.5355 and a*b = 4.34554 I doubt you can do that in your head and plugging in variables would take forever until you find a and b",
        "A": "Where did you get these numbers from: a+b = 3.5355 and a*b = 4.34554? Did you really start with the polynomial: x^2 + 3.5355x + 4.34554? I have never seen a textbook that would have you factor a polynomial with coefficients in that form."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "Starting at 1:55 can you explain how you got those number after you factor them",
        "A": "He s basically doing the opposite of the distributive property. Let s look at the first part as an example. s^2 + 5s factored out is s(s + 5). Use the distributive property on that and see what you get. You ll get what we started out with, s^ + 5s. It s the same thing with -7s -35. If you factor out the common factors, it s -7(s + 5). Now use the distribute property. You get the original -7s - 35."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "Around 4:12 of the video you had said to add 7 to both sides in order to make it equivalent. Why didn't you just move the 7 to the other side?",
        "A": "Moving the -7 to the other side would change the whole equation and make it incorrect. To get rid of the -7, you need to do the opposite of subtraction, and add 7. But remember that whatever you do to one side of an equation, you have to do to the other (so it stays equivalent)."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "at 1:56 how is s(s+5) the same thing as s^2 +5s",
        "A": "s(s+5) = s*(s+5) = s*s + s*5 = s^2 + 5s If you have trouble with that, I suggest you watch some videos in Algebra I."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "At 3:45, how can both of them be equal to 0?",
        "A": "Those are both answers. If you graph the parabolic function, there would be two zeros."
    },
    {
        "video_name": "2ZzuZvz33X0",
        "Q": "what did you do at 0:54? I really didn't understand how you got there. Could you explain it better?",
        "A": "You want to factor the expression into a form (s + a)(s - b) , [you know one is a minus since the last term is negative] (s + a)(s - b) = s\u00c2\u00b2 + (a-b)s - ab , so we know we need 2 numbers with a difference of 2 and whose product is 35. The numbers that work are a = 5, b = 7."
    },
    {
        "video_name": "tRHLEWSUjrQ",
        "Q": "I stopped at 1:16 above and I can already say its P= 0.9 squared.",
        "A": "Great! Just remember, some people catch on faster than others :)."
    },
    {
        "video_name": "tRHLEWSUjrQ",
        "Q": "At 0:05, Sal reads \"zero point eight one\", and at 1:56, he reads \"zero point eighty one.\"\nWhich one is right?",
        "A": "You can also think like this, he could have said it the multiple ways he did to attempt adapting to whatever way you think of it."
    },
    {
        "video_name": "tRHLEWSUjrQ",
        "Q": "Khan, On 1:47, I think you did it wrong because 0.9 or .9 has a value of .90 and not .09. .9 \u00c3\u0097 .9 = 8.1 and not .81. Was your way a mistake or is it supposed to be that way ?",
        "A": "0.9 X 0.9 is supposed to be 0.81. You are multiplying 0.9 by something less than 1. So your answer should be less than the original 0.9. 8.1 is greater than 0.9. 0.81 is less than 0.9. If you want to check, you can multiply 0.9 X 0.9 with brute force. 0.9 X 0.9 ____ 81 000 ____ 081(.0) Now you move the decimal two points to the left, to get 0.81"
    },
    {
        "video_name": "q87L9R9v274",
        "Q": "I don't get how at 1:48, the slope field can be used in the XY and dy over dx table.",
        "A": "It s the other way around, the table can be used to create a slope field; meaning you can choose values for x and y, and calculate the dy/dx (which is the slope), and from that construct a slope field. But that was only a comment in passing, since for Euler s method you need to have a initial condition, you don t need a slope field."
    },
    {
        "video_name": "q87L9R9v274",
        "Q": "Did Sal just say computers are smarter than humans at 0:50?",
        "A": "Nope. 0:49 do you just give up? 0:51 And the answer to that question is no. 0:53 You do not just give up 0:55 because we now have computers, 0:58 and computers are really good at numerical methods. Perhaps they re just smarter than some humans."
    },
    {
        "video_name": "q87L9R9v274",
        "Q": "at 5:39\nWhy when X increases by 1, Y increases by (dy/dx)?\n\nIs y value proportional increment of x value?",
        "A": "Yes, the change in y is proportional to the change in x. You can say that \u00ce\u0094y = dy/dx * \u00ce\u0094x."
    },
    {
        "video_name": "tIpKfDc295M",
        "Q": "What was the other name for gradient? At 1:48 ish",
        "A": "It is nabla or del. del f is how you would pronounce the gradient of f. (or grad f). nabla is an upside-down Greek delta."
    },
    {
        "video_name": "tIpKfDc295M",
        "Q": "At 1:05, when we take the derivative of f in respect to x, therefore take y = sin(y) as a constant, why doesn't it disappear in the derivative?",
        "A": "A constant disappears in the derivative when it is added. f(x) = c * g(x) => f (x) = c *g (x) In this case g(x) = x\u00c2\u00b2 and c = sin(y) But if we had : f(x) = g(x) +c Then we would have: f (x) = g (x) When the constant is added, it disappears :)"
    },
    {
        "video_name": "tIpKfDc295M",
        "Q": "at 2:35, how the result was square of x multply cosy, it should be siny, shouldnt it?",
        "A": "No. Taking the partial derivative with respect to a variable means you treat all other variables as though they are constant. So if you re taking the partial derivative with respect to y, x gets treated as a constant and you differentiate y. The original function was (x^2)sin(y), so x^2 gets left as a constant and sin(y) gets differentiated. The derivative of sin is cos, so the correct answer is (x^2)cos(y)"
    },
    {
        "video_name": "tYknkDjp-bQ",
        "Q": "At 2:29 Sal says, \"What are the non-prime factors of each of these numbers?\" Why non-prime factors specifically?",
        "A": "Sal is showing two different ways to find the greatest common factor. The first way, at 1:30, he does prime factorization. The second way, at 2:26, he uses non-prime factorization. So if he didn t make the specification in the second example of non-prime factors he would just be repeating what he had shown at 1:30."
    },
    {
        "video_name": "tYknkDjp-bQ",
        "Q": "at 5:00 i dont get it bro",
        "A": "He doesn t want to cut off the original problem from the screen, if that s what you mean."
    },
    {
        "video_name": "STcsaKuW-24",
        "Q": "At 6:55, or so, Sal starts using another way to factor the equation. I don't get that way, but I get the undoing of FOIL. . . need help!",
        "A": "Okay, the American, I ll try to explain. You might want to watch some of the videos in the previous play list in the algebra section. If you have (x+c)(x+d) You ll have: x^2+cx+dx+cd or x^2+(c+d)x+cd Now let s look at the problem Sal gave us: b^2-4b-60 This is the exact same format. -4 is c+d and 60 is c*d So the only two numbers that make sense are: 6 and -10 so c=6 and=-10 We can say: b^2+6b-10b-60 (b+6)(b-10) Done!"
    },
    {
        "video_name": "STcsaKuW-24",
        "Q": "At 1:43, shouldn't you distribute the b first and then multiply it with 1/2?",
        "A": "Communative property of multiplication- You can do it in any order."
    },
    {
        "video_name": "STcsaKuW-24",
        "Q": "At 1:55, where did the -2b come from? Also, I'm don't know how he gets b squared over 2?",
        "A": "From 1:47 to 2:05, Sal is using the distributive property on the right-hand side. The distributive property say that if you have a(b + c), it s the same as ab + ac; the factor outside the brackets gets multiplied by each of the terms inside the brackets. Sal had (b/2)(b - 4). The factor outside the brackets is (b/2). The terms inside the brackets are b and -4. So (b/2)(b - 4) is the same as (b/2)(b) - (b/2)(4). (b/2)(b) = b^2 / 2. -(b/2)(4) = -4b / 2 = -2b. So (b/2)(b-4) = b^2 / 2 - 2b."
    },
    {
        "video_name": "STcsaKuW-24",
        "Q": "At 1:35 when he says \"let me make it clear... lets do it this way\"\nHow did he know to multiply the 1/2 with the b before distributing?\nIs that part of the order of operations? Are you supposed to do any multiplications that come before the parentheses first before distributing always?",
        "A": "you can do it both ways, the order of multiplications doesn t matter, just like (2*3)4 equals 2(3*4). Distributing before multiplying in the video yields: 1/2*(b^2-4b)=1/2*b^2-2b which is the same result as in the video"
    },
    {
        "video_name": "STcsaKuW-24",
        "Q": "At about 2:35 in the video, at (b squared/2-2b) times 2, where did he get the 2 that he multiplied on both sides of the equation?",
        "A": "On the right side of the expression, we have 1/2*b*(b-4). Thus, we will have 1/2*b^2-2 after multiplication. Since we are looking for b and not a fractional part of it, it is quite convenient to multiply the expression by 2 to get rid of 1/2. However, we cannot multiply only one side of the equation and we need to multiply 30 by 2 as well. 2*30 = 2(b^2/2 - 2b)"
    },
    {
        "video_name": "STcsaKuW-24",
        "Q": "At 1:33 why do you multiply 1/2 b and (b-4) ? ( in simple words please)",
        "A": "the area of a triangle is the base times the height times one half. so b is the base the height is b-4 and 1/2 is one half"
    },
    {
        "video_name": "4qfdCwys2ew",
        "Q": "At 9:38, shouldn't it be lim t->0+ ? (Tau approaching 0 from the right?)",
        "A": "Yes, but that is implied since the function is not defined for a negative \u00cf\u0084."
    },
    {
        "video_name": "drhoIgAhlQM",
        "Q": "wait, @ 0:18 where did the exponent 1/3 come from? what does that mean?",
        "A": "like if we are told to find the square root of a number , we can write than number inside the radical sign or we can write number raised to the power 1/2 , so similarly if we are told to find the cube root , it can also be written as the number raised to the power 1/3 instead of placing that radical sign :) I hope this helped you !!"
    },
    {
        "video_name": "drhoIgAhlQM",
        "Q": "06:48 Does every equation that contains radical expressions have a solution",
        "A": "I d say they almost all (if not all) have one. The only problem is that some of them are actually not real numbers."
    },
    {
        "video_name": "drhoIgAhlQM",
        "Q": "at 1:46 y does he keep adding 7 onto the end ?",
        "A": "He is not adding 7. He is multiplying by 7. That is because the prime factorization is -1*7*7*7."
    },
    {
        "video_name": "drhoIgAhlQM",
        "Q": "Is there a video of 0:37 where he goes through the rules of how you know what something is divisible by?\n\nIt's another part of my swiss cheese math brain I need filled in.",
        "A": "actually it is basic common sense. all numbers have divisibility rules mostly. it is not hars to recognize them. if you are to lazy to search up the rules, then use this trick= if a numbers prime factors are not divisible by x, then the number is not divisible by it. example= 6 has prime factors of 2 and 3. 2 and 3 are not divisible by 23,so 6 is not divisible by 23."
    },
    {
        "video_name": "6agzj3A9IgA",
        "Q": "At 3:41, why do you have to move the 45 to the other side?",
        "A": "normally (in Algebra 1 problems) there will be a negative number there (usually a nice, perfect square). So to solve it you will need to move it to the other side and square root it to find the answer or answers. Example: x^2 + 6x - 27= 0 x^2 + 6x +9 -9 -27 = 0 (x + 3)^2 - 36 = 0 (x + 3)^2 = 36 x + 3 = +- 6 x = -3 +- 6 Note: +- means plus or minus"
    },
    {
        "video_name": "6agzj3A9IgA",
        "Q": "@ 2:00, wouldn't he usually subtract 70 from both sides, then add 25 to both sides? I don't understand what he's trying to tell us by doing it this way.",
        "A": "Doing what you suggest is perfectly valid. The important point is to understand why the number 25 was chosen (in other words, how to complete the square ). Once that number is chosen you can either use Sal s method of adding AND subtracting that number from the SAME side of the equation, or you can use your method of adding that number to EACH side of the equation."
    },
    {
        "video_name": "6agzj3A9IgA",
        "Q": "at 0:25 how many roots can there be?",
        "A": "Quadratics can have 2 roots or 1 (if it s a square)"
    },
    {
        "video_name": "6agzj3A9IgA",
        "Q": "3:50 Why do we have to take the square root of (x + 5)^2 and -45?",
        "A": "we have to take the square root to solve for x. Given that (x+5) is squared, normally, to get solve for x we should get rid of that ^2, to do so, we should take the square root on both sides of the equation, BUT : given that on the other side there is a negative number, we cannot solve the equation, for there is no REAL number that solves the square root of a negative number. Hope this helped :)"
    },
    {
        "video_name": "6agzj3A9IgA",
        "Q": "At 5:13, Sal says that there is no solution for the equation. Couldn't you multiply both sides by -1, and then get 45 = -(x + 5)^2? Then you could get the square root of +45, and the negative sq root of x + 5? I know this is wrong, so why can't I do this? Isn't it -1 x (x + 5), so I could separate them and get the sq root?",
        "A": "You are only adding a negative to the other side, making that side unsquarable. A negative number cannot be squared."
    },
    {
        "video_name": "6agzj3A9IgA",
        "Q": "@2:07 I understand the +25 but why -25",
        "A": "Nevermind i get it now."
    },
    {
        "video_name": "6agzj3A9IgA",
        "Q": "At 0:58 when you divided everything by 4, do you have to factor it out? Or are you just simplifying it?",
        "A": "You do not have to, it just simplifies the expression and makes the rest of the process easier."
    },
    {
        "video_name": "6agzj3A9IgA",
        "Q": "At 2:05 why does Sal subtract 25 from the same side? Wouldn't you just add 25 to the other side of the equation like he has been doing for completing the square?",
        "A": "it doesn t matter whether you + or -."
    },
    {
        "video_name": "6agzj3A9IgA",
        "Q": "At 2:10, Sal said we can add a billion and subtract a billion without changing the equation. I agree completely with that.\nBut can we add infinite and subtract infinite without changing the equation?",
        "A": "No, because you can t actually add or subtract infinity. It is not a number, it is an abstract concept. Infinity minus infinity is not zero. Similarly, infinity divided by infinity is not one. Therefore, you cannot apply the same rules of inverse operations to infinity as you would to an actual number."
    },
    {
        "video_name": "pjrI91J6jOw",
        "Q": "How many birds are there in all in the example in 2:11 to 2:58? Is it just 2^1+2^2+^3+2^4+2^5+2^6+2^7+2^8+2^9+2^10+2^11+2^12+2^13+2^14+2^15+2^16?",
        "A": "There are seventeen types of birds. But in total there are 131,071"
    },
    {
        "video_name": "pjrI91J6jOw",
        "Q": "at 1:05, how come she could say 5 sounds(about) in once?",
        "A": "She layered several sounds of her voice on top of each other, then inserted that into the video."
    },
    {
        "video_name": "pjrI91J6jOw",
        "Q": "At 1:05, why does it sound like multiple people are speaking",
        "A": "Because it was lots of words at once"
    },
    {
        "video_name": "pjrI91J6jOw",
        "Q": "eww... at about 1:32 (i'm not sure of the exact time because i had my eyes closed) it showed her dismantling a raw bird... not very appropriate. also, at 2:36 she says that \"the world has plenty of birds.\" a lot of birds are legally endangered, and killing 200 would probably be illegal. and at 3:00, i almost threw up.",
        "A": "well what do you think you do at thanksgiving ,but that is gross."
    },
    {
        "video_name": "pjrI91J6jOw",
        "Q": "0:10 aren't chickens bigger than ducks?",
        "A": "For the most part, no. Chickens have an average weight of 3-4 lbs, whereas ducks have an average weight of 5-7 lbs. Note of course that this is the weight of the chicken at the optimum time for butchering, which is what we care about for a Thanksgiving dinner."
    },
    {
        "video_name": "pjrI91J6jOw",
        "Q": "1:05 made me laugh! ;D",
        "A": "This should be in Tips and Thanks, not Questions."
    },
    {
        "video_name": "pjrI91J6jOw",
        "Q": "How did she make her voice different close to 1:05?",
        "A": "She layered several sounds of her voice onto each other."
    },
    {
        "video_name": "K4KDLWENXm0",
        "Q": "at 10:12, shouldn't it be 2*0.96 is the 95% confidence interval for the u?",
        "A": "you describe an interval by two values, like, for example [1; 9] or, as Sal writes it, would be 5 +/- 4 (which is the same information actually). 2*0.96 is the length of the interval measured in units of one stdev. it s not a precise usage of mathematical terms in this video, I know, but trust me, you don t want to see statistics rigurously taught by a mathematician!! Sal s way is much better :)"
    },
    {
        "video_name": "hzSef_HVqz4",
        "Q": "At 1:52, I attempted to rewrite it a different way, which ended up being wrong. Why is it wrong?\nI went::\n(x^3 + 3x^2 - 5) / x^2\nThen I converted the denominator to x^-2, and multiplied it in front of the numerator. Isn't it the same as dividing it?\nThis resulted: x^-2 (x^3 + 3x^2 - 5)\nI multiplied everything inside the brackets with the x to the negative to power and got:\nx^-1 + 3x^0 - 5\n\nSo integrating it became:\nx^0+ 3x - 5x + c\n\nYikes....",
        "A": "It s wrong because you didn t multiply everything inside the brackets correctly. x\u00c2\u00af\u00c2\u00b2 ( x\u00c2\u00b3 + 3x\u00c2\u00b2 - 5) = x + 3 - 5x\u00c2\u00af\u00c2\u00b2 Integrating would give x\u00c2\u00b2/2 + 3x + 5/x + C"
    },
    {
        "video_name": "hzSef_HVqz4",
        "Q": "At 3:56 can't we integrate the x^5 as( x^6/6) and then take the 3rd root of it?, like( x^6/6)^1/3 and the answer will be x^2/2??",
        "A": "Most definitely not. Generally \u00ca\u0083 u(v(x)) dx \u00e2\u0089\u00a0 u[ \u00ca\u0083 v(x) dx ] For this example you can demonstrate this yourself. If you differentiate x\u00c2\u00b2/2 you get x. Not the original function, which was \u00c2\u00b3\u00e2\u0088\u009a(x\u00e2\u0081\u00b5)"
    },
    {
        "video_name": "5ABpqVSx33I",
        "Q": "At 5:33, why don't the one-tailed p values on both sides add up to the two-tailed p-value in a t-distribution like they do in a normal distribution?",
        "A": "They do. What made you think otherwise?"
    },
    {
        "video_name": "5ABpqVSx33I",
        "Q": "At 1:50 Sal says that we don't know what sigma (xbar) is. But isn't that just the sample standard deviation (s)? Surely you always know your sample standard deviation because you have collected the data. I may have missed something earlier on, but this video has just made it occur to me.",
        "A": "You always know your sample mean and sample standard deviation (X\u00cc\u0085 and s), but it s much rarer to know the population mean and standard deviation (\u00ce\u00bc and \u00cf\u0083) for lots of statistical problems."
    },
    {
        "video_name": "-6Fu2T_RSGM",
        "Q": "Around 0:30 he starts talking about point-slope form, but when he writes it out, it as:\ny-y1=m(x-x1).\nis this the same as:\nm=y2-y1/x2-x1 ?\n(I couldn't make the sub become small)",
        "A": "Yes, those are much the same thing. Traditionally, it is called the point-slope form when it is written as: (y-y\u00e2\u0082\u0081)= m(x-x\u00e2\u0082\u0081). When written the way you wrote it is just called the slope form (though some texts still call it the point-slope form). Related to that is the two-point form which is: (y-y\u00e2\u0082\u0081)= [(y\u00e2\u0082\u0082-y\u00e2\u0082\u0081)/(x\u00e2\u0082\u0082-x\u00e2\u0082\u0081)] (x-x\u00e2\u0082\u0081) Which can be rearranged to the symmetric form: (x\u00e2\u0082\u0082-x\u00e2\u0082\u0081)(y-y\u00e2\u0082\u0081)= (y\u00e2\u0082\u0082-y\u00e2\u0082\u0081)(x-x\u00e2\u0082\u0081)"
    },
    {
        "video_name": "-6Fu2T_RSGM",
        "Q": "At 7:32 why did he multiply by 3!",
        "A": "He did it to remove the fraction 2/3x, just to make the equation easier to read."
    },
    {
        "video_name": "-6Fu2T_RSGM",
        "Q": "how did he know to multiply by 3 like he did at 7:44?",
        "A": "He multiplied by 3 because there was a 3 in the fraction s denominator. Multiplying by three caused the numerator and denominator both to have a 3 in them, which canceled out and got rid of the fraction. But, he had to multiply all the other terms in the equation by 3 as well so that he didn t change the value of the answer. Here are the steps he left out: (2/3)x + y = 4 (3)(2/3)x + (3)(y) = (3)(4) 2x + 3y = 12"
    },
    {
        "video_name": "-6Fu2T_RSGM",
        "Q": "at 5:51 when he multiplies the -2/3 by 3 how does he end up with negative two?",
        "A": "This is any fraction multiplied by its denominator ends up its numerator, so -2/3x3=-3"
    },
    {
        "video_name": "-6Fu2T_RSGM",
        "Q": "im confused on how he knows what to put on there and where 2:00",
        "A": "To find the equation of any line from two given points we need to find the slope. And if you have seen the previous videos in the playlist you must be knowing that slope(m) = (change in y) / (change in x) Now he had two points : (-3,6) and (6,0) Just for ease consider : x1 = -3; y1 = 6; x2 = 6; y2 = 0. Slope = (y2 - y1) / (x2 - x1) |or| (y1 - y2) / (x1 - x2) => Slope(m) = (0 - 6) / (6 - -3) = -6/9 = -2/3 Those are the values he substituted at 2:00"
    },
    {
        "video_name": "-6Fu2T_RSGM",
        "Q": "on 3:58 where do u get -2/3",
        "A": "he simplifyed the fraction"
    },
    {
        "video_name": "-6Fu2T_RSGM",
        "Q": "At 7:24 Sal says we're in standard form, but there's still a fraction. I learned (and teach) that it's not standard until there are no fractions, so once he simplifies and draws his box around the final answer is standard form. Some hard-line algebra teacher once asked me with a smirk: \"is standard form A/P X + B/Q Y + C/R = 0 ?\" His point which stuck like glue was to get rid of those denominators... :-). So is it actually still standard form if there's a fraction?",
        "A": "If values of variable A P B Q C R is given, then we will put those values and see if it can be simplified further. But in Standard Tests If you can simplify fraction easy fraction you should, otherwise you don t really have to."
    },
    {
        "video_name": "-6Fu2T_RSGM",
        "Q": "At 7:40 why did he also multiply Y by 3?",
        "A": "so you can change the fraction into a nuber without afecting the equation"
    },
    {
        "video_name": "bJF9R8_-0O0",
        "Q": "0:05, What does \" Congruent \" mean.",
        "A": "When two things are said to be congruent, it means that all of their measurements are identical. A square that has sides measuring 4 inches will be congruent to another square that also has 4-inch sides. Even if the other square is placed in a different place or flipped around, the two squares are still congruent to each other."
    },
    {
        "video_name": "eS-_ZFzHjYA",
        "Q": "At 3:09, Sal finalizes our answer as a number with a decimal in the numerator and fractions in the denominator.\n\nTo express our answer in simplest terms, do we need to combine the two fractions in the denominator?\n\nAlso, is it the custom in mathematics when simplifying fractions to give our final answer in either decimals or fractions, but not both, or is it \"permissible\" to leave our answer in its current form.",
        "A": "Since Sal s goal is to find a decimal approximation for x, it is fine to leave his work in the form it s in at 3:09. If this was to be his final answer, then yes it is conventional to rationalize denominators, not leave fractions with decimal parts, and simplify fractions using a common denominator. The advantage of leaving it the way he does is that the fraction he creates is still an exact value for x. Since he uses a calculator to compute the final answer, no clean-up was necessary."
    },
    {
        "video_name": "eS-_ZFzHjYA",
        "Q": "At 3:21, What do we do if we find that the discovered point is NOT a minimum?",
        "A": "Then it must be a maximum point. It does not make that much sense, because the problem would have had to say: With a perimeter of 100 m, what geometric figure gives you the greater area: a square or a triangle? To answer that question one doesn t need calculus. Just calculate the area of a triangle and that of a square, separately, and then compare them. You can cover a greater area with a square, so your x must be equal to zero. Take a look at Justin Keith s answer in the comment below."
    },
    {
        "video_name": "eS-_ZFzHjYA",
        "Q": "At 4:00, you determine that there is a minimum. Is there not a maximum value as well?",
        "A": "The expression is polynomial of degree one and can have at most one root. Hence, there exists only one critical point in the domain, which is minimum in this case."
    },
    {
        "video_name": "eS-_ZFzHjYA",
        "Q": "At 5:30 which side of the cut string would you use for which shape? Or would it not matter?",
        "A": "x is equal to the amount of string used in the triangle, and 100 minus x (the remainder of the string) is used to make the square! Yay math!"
    },
    {
        "video_name": "hpWdDmgsIRE",
        "Q": "At 05:24 he says that 21% did not get sick and then writes 21% in the row labeled \"sick\". Did he make a mistake, or am I missing something here?",
        "A": "He just said the wrong thing. Good catch."
    },
    {
        "video_name": "AreOMvPCxv8",
        "Q": "Is it a coincidence that \"x\" in this derived formula is equal to what seems to closely resemble the law of cosines? At 5:09. I thought it was interesting.",
        "A": "it is no coincidence. Heron s Formula is an alternative to the law of cosines, which could also be used to solve for the area of a triangle only knowing the sides."
    },
    {
        "video_name": "AreOMvPCxv8",
        "Q": "@3:24 how does\n(c - x)squared\nexpand to\nc squared - 2cx + x squared?",
        "A": "This is done by just using the formula of the square of two # s. That formula is (a-b)^2 is always equal to a^2-2ab+b^2. If you try it with any two # s it will always work. For example (9-7)^2 is equal to 9^2-2(9X7)+7^2= 81-2(63)+49= 130-126=4. You will also get 4 as your answer if you just subtract 9-7 (2) and square it."
    },
    {
        "video_name": "AreOMvPCxv8",
        "Q": "4:49\ni have been taught that the formula is \"S= root of S(S-a) (S-b) (S-c)\n?\nerr . . . . . i'm from India.",
        "A": "yeah, but what the hell is s(s-a) (s-b) (s-c)?"
    },
    {
        "video_name": "AreOMvPCxv8",
        "Q": "why does he use c-x on 1:40??",
        "A": "Oh I get it!! c-x will give the rest of the measurement!!"
    },
    {
        "video_name": "AreOMvPCxv8",
        "Q": "at 7:35 where h = square root of a2 -(( c2+a2-b2)/2c)2 can it be simplified to h = a i ((c2+a2-b2)/2c) cuz the square root and the square signs cancel out",
        "A": "A square root of two squares added together cannot cancel out. Example: sqrt(a^2 + b^2) is NOT a+b Think of an example: sqrt(3^2 + 4^2) is NOT 3+4 or 7 = sqrt(9 + 16) = sqrt(25) = 5"
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "at 5:27 can you combine the bases as 9^9 seeing as 3 squared = 9?",
        "A": "Yes, that is one option. Somehow it seems cleaner to have a base that is not a perfect square, though."
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "So... (2*7)^3 will further simplify to 14^6? @ 5:12",
        "A": "Yes jim.falcone3693, since we have 2 x 7 in parentheses, we can simplify that using the order of operations, but we can take the situation another way. if we use the distributive property, we can bypass the order of operations. For example: (2*7)^3 2^3 * 7^3 (Apply the exponent to both 2 and 7) 8 * 343 2744 Hope that helped! Samuel"
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "at 4:47 Sal does the problem (2*7)^3=2^3*7^3. Why is it not 14^3?",
        "A": "You forgot to review the part at 2:58 where he explains how to deal with (4 x 5)^3. See that part again and you should understand it."
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "0:38 when you are multiplying exponents with the same base and you add the exponents and you get your answer for eample 2 to the 15th power dont you still have to multipky 2 15 timees to get your final answer?",
        "A": "Usually, yeah, unless the question says otherwise. He s just not showing that step. 2^15 = 32768"
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "At 2:20 can you do 3 to the 7th =2,187 to the -9th?",
        "A": "You can, but it s not the point he was making. He was simply demonstrating the rule of the exponents."
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "What does he mean at 6:18?",
        "A": "When you multiply exponents with the same base, you can add the exponents and keep the base."
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "Wouldn't it be more correct to say that (4*5)^3 = 20^3 because parentheses come before exponents? (3:10)",
        "A": "(4*5)^3=4^3*5^3=20^3 There are multiple ways to evaluate that expression, and both ways are equally correct. I would make it 20^3 because that s easier to evaluate, but Sal s trying to show you exponent laws."
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "at 5:30 Sal said that 9 can turn into 3 so it would match the problem: 3^3 times 9^8 he said u could change the 9 since it can be ^ by 3 times 3 into (3^2 times 3^2)^8)^-2 but what if the question was this 4^3 times 17^5 and u can not change the 17 into something matching the 4 like Sal did. What would you do then.\n\nSorry i wrote the question confusing if u don't understand please say it so i could try to re-type it into a understandable way.",
        "A": "Well, that situation wouldn t work with this process because 4 and 17 have virtually nothing in common. You would have to solve 4^3 and 17^5 separately."
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "for the problem at 2:16, would you not include the negative if the exponent was an even number instead of an odd number?",
        "A": "Having a negative exponent is not the same as having a negative base taken to a power. (-3)^2 = 9 = 3^2 is what you re thinking of I believe but (3)^-4 is not = (3)^4 (3)^-4 = (1/3)^4"
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "At 3:05, Sal says that (2*9)^100 is 2^100*9^100. It could also work if you did (2*9=18)^100 which is of course 18^100. Correct????",
        "A": "If you get the same answer both ways then yes"
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "Around 6:41 if you have 18 mulitplied by -2, How is it -36 when the larger number is Positive. Shouldnt it be +36?",
        "A": "It doesn t matter how big the positive/negative number is. If there s one negative, the answer is negative. If there are two negatives, it s negative (For this, the same goes for positive.)"
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "The problem Sal starts at 3:00, shouldn't it have been (4*5)^3, then 9^3 because of PEDMAS?",
        "A": "(4*5).(4*5).(4*5)"
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "At 2:10, Sal gives an example of exponents with 3 to the 7th to the -9th. How can this be possibly be answered?",
        "A": "You would multiply the two exponents. 7*(-9)=-63. Keep the same base. Your answer should be 3^-63. 3^-63 is also equal to 1/(3^63).For more details, you should check out the exponent rules of multiplication video on Khan Academy."
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "At 4:40 , when he does (2*7)^3 , he distributes the exponent and makes the problem 2^3*7^3, do you have to do that? Would solving what's in the parenthesis first and then raising your answer to the exponent give you the wrong answer?",
        "A": "No, you don t have to do that. In fact, you could also do it as (2 x 7)(2 x 7)(2 x 7). Any of those ways will still get you the same answer."
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "At 3:08, couldn't you just do 20 to the 3rd power?",
        "A": "distributive property, my friend; when those numbers get big, and one day, they will, you ve just gotta distribute, my friend, and share the power."
    },
    {
        "video_name": "rEtuPhl6930",
        "Q": "@6:25 Would it have been easier just to to turn (3^2*9^8)^-2 into (9*9^8)^-2 which would then simplify to (9^9)^-2 ----> 9^-18 or is there a reason we want 3 as the base?",
        "A": "Yes, that does work and is correct, but he wanted to demonstrate the breaking down of one exponent in order to be able to combine them. Had it been 3^3 instead of 3^2, your trick would not work."
    },
    {
        "video_name": "aGwT2-RERXY",
        "Q": "at 00:33 you said to make a parallel line. do you have to do a construction for a parallel line, or just draw any parallel line?",
        "A": "The benefit of the parallel line going through the point, it allows you to see that the angles are going to work out."
    },
    {
        "video_name": "aGwT2-RERXY",
        "Q": "at 9:54 what is a perpendicular bisector?",
        "A": "A perpendicular bisector is any line that intercepts another line. In the videos of the whole section over perpendicular bisectors, he draws a perpendicular bisector at the midpoint of the lines of the triangle. I hope this answers your question, and if you or anyone out there needs any help, please do tell me."
    },
    {
        "video_name": "9Wfn-WWV1aY",
        "Q": "At 4:03, why can we assume that v is a function of x?",
        "A": "At that point, you don t actually know if v is a function of x or not. By writing it as dv/dx you are saying that we are going to differentiate v with respect to x . If we actually had an expression of v and it was a function of x we would simply have the derivative of it, otherwise it would go to zero. So we are not actually assuming v to be a function of x but rather taking the derivative of v with respect to x. Hope this helps!"
    },
    {
        "video_name": "9Wfn-WWV1aY",
        "Q": "At 5:00 Sal says that the derivative of v with respect to x is equivalent to 1/x. But isn't dv/dx = d(y/x)/dx which is (-y/x^2), not 1/x? I think Sal was thinking of dv/dy.\ndv/dy would be d(y/x)/dy equaling 1/x as the y would be derived to 1.",
        "A": "He was not thinking of dv/dy. He differentiated both sides of y=vx with respect to x and solve for dv/dx. You can also compute dv/dx directly, however dv/dx = d(y/x)dx is not (-y/x^2) like you suggested since y is a function of x. So dv/dx = d(y/x)/dx = [(x(dy/dx)-y]/x^2, by substituting dy/dx with the expression from the original problem, the right hand side simplifies to 1/x."
    },
    {
        "video_name": "9Wfn-WWV1aY",
        "Q": "at 5:23, don't you have to integrate according to a same d-variable?\nSal integrates the left hand side according to dv and the right hand side according to dx",
        "A": "It s the same as when you have, for example, dy/dx = 2x. You multiply both sides by dx to get dy = 2x dx. Then integrate to get the very familiar y = x^2 (the + c is not particularly relevant here). You always have different variables on each side, otherwise there would be no point in integrating. For example, if you had dv/dx = x^2 + 4x, it would become v = (x^3)/3 + 2x^2 by integration."
    },
    {
        "video_name": "9Wfn-WWV1aY",
        "Q": "Although the derivation of y in 6:55 is correct, apparently y=xln|x|+xC is not a solution to y'=1+y/x because if you derive this value of y with respect to x, you obtain y'=1+ln|x|+C . Analysing from this point the right solution would be y=xln|x|+C, because if you derive this expression with respect to x you obtain y'=1+ln|x| where ln|x|=y/x in both cases. So why this is happening? Am I making some mistake? I'm pretty confused...",
        "A": "Your mistake is that you are assuming y/x=ln|x|, but this is only true if C=0. I believe what confuses you is that y=xln|x|+C might indeed be a valid solution to the problem. However, it implies the following: y=xln|x|+C => y/x=ln|x|+C/x y =1+ln|x| => y/x=ln|x| As you can see, a contradiction arises if C is not 0. Let s try the same reasoning but with y=xln|x|+xC: y=xln|x|+xC => y/x=ln|x|+C y =1+ln|x|+C => y/x=ln|x|+C No contradictions this time."
    },
    {
        "video_name": "zqduFg0s5Q4",
        "Q": "At 1:33, was Sal saying that grams are actually a unit of mass instead of weight?!",
        "A": "Yes, it seems confusing. Mass is how much space in an item, Weight is how much gravity pushes on an item. You don t say Hey, this weighs 50 grams . You say This has a mass of 50 grams . And don t say This has a mass of 1 pound . Say This weighs 5 pounds ."
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "At 7:55, how do we know that psi 1 is congruent to psi 2?",
        "A": "I watched it a few times. It doesn t seem that Sal says they are congruent. What he does say is that psi 1 is 1/2 of theta 1 and psi 2 is 1/2 of theta 2, which is based on what he did earlier. He also says that psi 1 + psi 2 = psi, which is clear from the diagram."
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "About 7:00, he said that the notation won't apply to all inscribed angles. Is there a special case where the notation doesn't apply?",
        "A": "Thank you Sal. You are an excellent math instructor and this site is beyond helpful. However, you tend to use a lot of algebra. I find it would be more helpful to use simple arithmetic."
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "Why can't you add 1/2 beta1 and 1/2 beta2 together? Video time: \"8:55-9:01\"",
        "A": "it s important to know that 1/2x(sub1), and 1/2x(sub2) is almost the same thing as 1/2x+1/2y. If you distribute out, the 1/2 you get 1/2(x1+x2.) It is not the same as 1/2x+1/2x because that would be the same as 1x. that is why the subscript 1 and 2 makes the ultimate difference."
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "at 00:38 you mentioned si for the first time. What does si signify?",
        "A": "Psi (\u00cf\u0088) (not si) is a Greek letter. It s just a variable, like theta (\u00ce\u00b8). He s using it because there are two important angles so he needs a different letter for each."
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "At 0:11 and 0:45, Sal says \"vertex\", what does that mean?",
        "A": "The Vertex is the point where the two line segments meet. So at 00:11 when he points to the vertex he is staying, the place where the two line segments meet, which is on the circumference of the circle"
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "At 0:25, you mentioned si for the first time. What does it mean?",
        "A": "Psi is just a Greek letter that he uses as a variable."
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "At 8:22 how do you know psi 1 is = to 1/2 beta 1, and same for psi 2 and beta 2?",
        "A": "At 6:00 minutes of this video Sal proves psi = 1/2 beta when one ray is a diameter, in the example you re asking about he creates a diameter so he can use this result to know Psi 1 = 1/2 beta 1 and the same for psi 2 and beta 2."
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "At 1:28, what does \"arc 'subtended' by psi\" mean? What is theta?",
        "A": "The arc subtended by psi means the piece of the circle between the rays of the angle -- the part he colored pink. He starts explaining this at 1:00. Psi and theta are both Greek letters that are commonly used to label angles in geometry."
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "What does that \"pitchfork\" sign mean? At 0:30.",
        "A": "As stated above, it it is the Greek letter psi, and is used to represent the angle subtending the arc formed by the two chords sharing a common vertex on the circumference of the circle."
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "At 12:37, isn't the Theta 1 and Theta 2 a 90 degree angle? Cause Salman have drawn the angle as similar as 90 degree.",
        "A": "no,you can t say that thetas is 90 degrees just by observing and not measuring them"
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "at 0:35 can somebody please explain this to me in ways that are easier to understand just trying to comprehend what he had said is making my head hurt XD",
        "A": "Honestly he s proving how this works in the best way possible throughout the video. The short and sweet of what he s trying to prove: The angle measure of an inscribed angle (formed by a point on the circumference of the circle and 2 rays passing through the other side of the circle) is half the measure of a central angle (2 rays going through the outer edge of the circle which originate from the center)."
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "at about 02:00, Sal talks about theta, and at about 00:38, he talks about psi. What do these mean?",
        "A": "Theta and Psi are letters in the Greek alphabet. They are commonly used to represent angles, the same way letters like a, b, c, or x, y, or z are commonly used as variables to represent unknowns in algebraic expressions. He could have used any letter to represent the angles. Hope this helps. Good Luck."
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "What does Subtended mean at 1:32",
        "A": "It s the amount of arc that s cut out by the two lines extended from the angle we re talking about. - that should be clear from the vid @1:32"
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "I was confused at 9:25 and a few secounds earlier, he adds the two thetas, (1/2, and 1/2) wouldnt that equal 1? Not 1/2",
        "A": "the pitch fork symbol is called Psi (\u00cf\u0088) and it is just a greek letter that we use to represent unknown in geometry. Theta (\u00ce\u00b8) means the same thing, at least in geometry. If you speak Greek, Theta or Psi would be in your alphabet. to Cath1040"
    },
    {
        "video_name": "MyzGVbCHh5M",
        "Q": "How do you know that si 1 is equal to 1/2 theta 1 and si 2 is equal to 1/2 theta 2 at 8:17 to 8:30?",
        "A": "At 5:50, psi = 1/2*theta is proven for the special case where one of the rays of the inscribed angle is along the diameter of the circle. This is then used from 8:17 to 8:30 to prove the more general case by subdividing the general inscribed angle into two inscribed angles with a side along a diameter. This is done because it makes result of the previous proof applicable to the current proof."
    },
    {
        "video_name": "LhrGS4-Dd9I",
        "Q": "at 6:16, could you not use CPCTC or Corresponding Parts of Congruent Triangles are Congruent?",
        "A": "Yes you could. He just expressed it in a different way."
    },
    {
        "video_name": "LhrGS4-Dd9I",
        "Q": "wait at 7:45 shouldn't he say AC is parallel to BD by alternate interior angles. I learned to do it with the converse. since alternate interior angles are used to find if the lines are parallel, can i still use the converse?",
        "A": "He did say that they were congruent by alt. int. angles. Also, you most certainly can use the converse of the theorem to solve this problem. Hope this helps :)"
    },
    {
        "video_name": "LhrGS4-Dd9I",
        "Q": "Why does Sal use the congruent symbol (\"\u00e2\u0089\u0085\") at 1:10? We use the equal to (\"=\") symbol at school. Is there any difference between the two symbols if we use them to indicate that the measures are equal, like in this case?",
        "A": "if he used equal he would habe to change it to m<Abc instead of just <ABC. equal refers to a specific number."
    },
    {
        "video_name": "LhrGS4-Dd9I",
        "Q": "Why didn't he just put CPCTC for 4:05? What is the theorem for the two triangles at 6:40?",
        "A": "He didn t put CPCTC because everyone has their own way of saying that corresponding parts of congruent triangles are congruent. The theorem for the two triangles is called the Converse of Alternate Interior Angles Theorem. This basically states that it the alternate interior angles of two triangles are congruent, then the two triangles are congruent."
    },
    {
        "video_name": "kj3ZfOQGKdE",
        "Q": "At the point 1:47, it is stated that Y = -x - 2. What does that mean?\n\nI've looked at the coordinates along the line, and they don't really seem to match this. What does Y = -x - 2 mean? How did he come to this conclusion? And how do I apply that to the problem? What does it do?",
        "A": "at 1:47 , y is a variable, -x is the slope - 2 is the y intercept so the formula is y = mx - or + b y - variable m- slope b - y intercept"
    },
    {
        "video_name": "kj3ZfOQGKdE",
        "Q": "At around 3:00, I'm confused. Isn't the slope of that line a postive 1 and not negative? The line that connects point I to the line of reflection?",
        "A": "Yes, it is + 1. Sal drew things a bit sloppily so that his equals sign looks like a minus sign."
    },
    {
        "video_name": "hShqXKpI6KA",
        "Q": "Can somebody explain to me why the denominators always cancel? Sal does this at 2:51 of this video",
        "A": "Oh, like when he crosses off the units (words) after the numbers. Remember that a number divided by itself always equals 1. (9 / 9 = 1; 5 / 5 = 1). The same thing is true for units or words. (inches / inches = 1; puppies / puppies = 1) So, when you divide two things with the same units by each other, the units cancel out, or become just 1. Example: 6 meters / 2 meters = 3 The meters word is gone because meters/meters = 1, and 1*3 is just 3."
    },
    {
        "video_name": "fZO-JylMFqY",
        "Q": "At 0:43, how do we know the negative does not belong to the 3 in the fraction? Could it not be that that line moves up 7 and moves to the left 3 times?",
        "A": "The minus can be: (-7)/3 or 7/(-3). So, yes, the line going up 7 units and left 3 units is the same as moving down 7 and right 3 units."
    },
    {
        "video_name": "LrS5_l-gk94",
        "Q": "At \"7:47\" I don't understand why he adds a^2=cd to b^2=ce.",
        "A": "No, its easier than that. If you have, say, x=2 and y=5, you know that x+y=2+5. Sal is just adding known quantities together because the result helps us find the piece we are looking for."
    },
    {
        "video_name": "LrS5_l-gk94",
        "Q": "Nothing I remember or watched, so far, prepared me for the whole \u00e2\u0080\u009ctriangle A is similar to triangle B\u00e2\u0080\u009d postulate; or with specific reference to this video, the argument of \u00e2\u0080\u009ctriangle ADC is similar to triangle ACB\u00e2\u0080\u009d at 03:17. I understand that they share one side and one angle, but I still don't understand why that qualifies the \u00e2\u0080\u009csimilarity\u00e2\u0080\u009d relationship. Moreover, what does it even mean when we say \u00e2\u0080\u009ctriangle X is similar to triangle Y\u00e2\u0080\u009d? And when would two triangles qualify for that relationship of similarity?",
        "A": "Good question. If a triangle is similar to another triangle, it must have at least one same side, and at least two similar angles. In the video, when Sal says that two triangles are similar, he shows two equal angles, as well as one common side. I hope this helped u!"
    },
    {
        "video_name": "LrS5_l-gk94",
        "Q": "around 3:00 sal said that Triangle ADC is similar to triangle ACB how's that possible ?",
        "A": "Those two triangles each have a right angle, and they have angle A in common. Therefore, they are similar by the AA similarity condition."
    },
    {
        "video_name": "LrS5_l-gk94",
        "Q": "At 3:22, he uses the ~ . Does that mean anything?",
        "A": "~ means is similar to ."
    },
    {
        "video_name": "LrS5_l-gk94",
        "Q": "At 5:30, I have no idea what he did or how he explained it. Please help me and explain! But to be honest, I wish Sal was my math teacher! :)",
        "A": "He already defined \u00f0\u009d\u0091\u0090 = |\u00f0\u009d\u0090\u00b4\u00f0\u009d\u0090\u00b5|, and then (5:30) he defines \u00f0\u009d\u0091\u0091 = |\u00f0\u009d\u0090\u00b4\u00f0\u009d\u0090\u00b7| and \u00f0\u009d\u0091\u0092 = |\u00f0\u009d\u0090\u00b7\u00f0\u009d\u0090\u00b5|. \u00f0\u009d\u0090\u00b7 is on \u00f0\u009d\u0090\u00b4\u00f0\u009d\u0090\u00b5 \u00e2\u0087\u0092 \u00f0\u009d\u0091\u0090 = \u00f0\u009d\u0091\u0091 + \u00f0\u009d\u0091\u0092"
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "in 1:00, how did you know it's 10/8?",
        "A": "that is because 8x10/8=10 and 8 and 10 in the question Sal has to answer they are both numerators"
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "how does multiplying n by n cancel it out at 5:50 ? Doesnt that make it n^2?",
        "A": "it is (10/n) * n which is equal to 10"
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "At 0:46 can't you just say 8n=10*36?",
        "A": "Yes, that is another way to do the problem."
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "At 4:30 why would you put 360 over 8 instead of just dividing it",
        "A": "at 4:30 it is easy"
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "I can't figure out the answer to 9:10=10:k.",
        "A": "Think of this as division instead of ratio So 9/10=10/k Multiply by k 9/10*k= 10 So k=10*10/9 and 100/9 = 1.11 or roughly 11.10 = 11.1:10 or 111:100 ..."
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "At 3:49 when you are multiplying 36x10 what I usually do (which I didn't see as one of your ways of cross multiplying) is, I cross multiply and then in this specific problem I would divide by 8 and it would = N.\nI was just wondering if anyone else does it this way because for me it is easiest.",
        "A": "You are doing the same thing as the video, just not writing down all the steps - and that is OK!"
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "at 2:55 what did khan do to get 36 i really do not understand",
        "A": "If you are asking about when he multiplied 8 X 9/2, what he did was multiply 8/1 (which is the same as just 8) and 9/2 giving 72/2 which then simplifies to just 36. Hope this helps :)"
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "At around 4:30 how did sal get 4 x X = 360 where did he get threehundred sixty",
        "A": "36 x 10 = 360 It s in the step above, which starts at about 3:20"
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "Hey everyone, at 2:50 Sal converts 10x(9/2) to 10/2x9. I understand how both fractions are equivalent, but my hard head can't seem to grasp how it is that he divides 10 by 2 and also 2 by 2, and finally how he can transfer the divisor of 2 from the 9 to the 10. Does this have something to do with PEMDAS or an algebraic law? Can someone please clarify why it is that this can be done and what the rule is? Thanks a bunch in advance, Vanessa.",
        "A": "It works because in fractional form you can multiply all the numerators first and then divide by the denominator."
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "At 0:32, where did he get the 10/8 from. Did he cross multiply only the 8 with the n?",
        "A": "Yes... kind of... Sal s approach was to figure out what time 8 will create the 10. It would be easier to understand if he did the cross multiplication where you would get: 8n = 10(36) 8n = 360 Then divide both sides by 8 to get to n=45"
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "Wait, wait, wait, wait a minute. In 4:21, it shows you have to divide 360/8. But what if in a question the dividing thing is 21/5? Would the answer have to be decimals or rounded up? Or would the problem, not work and you have to use another technique?",
        "A": "You d probably use a fraction to represent the extra. 5 goes into 21 4 times. We have an extra of 1. The denominator is 5 so the answer would be 4 and 1/5"
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "At 1:08,what if you can't divide the numerator and the denominator by the same number? What I mean is that; what if there are three numbers that just don't go into each other?",
        "A": "Except for dividing by zero, any number can be expressed as a fraction of any other number. If I didn t understand you, give me an example of three numbers that just don t go into each other."
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "At 0:33 how did 8*10/8 work exactly to get to 10 ? I don't understand ,",
        "A": "Well, 8*10 = 80, so divide that by 8 and you ll get 10. Or you can think of it this way: (8*10)/8, because you have to multiply before you divide using PEMDAS. You can simplify this fraction by dividing the numerator and the denominator by 8, and you ll get 10/1, or just 10."
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "at 0:43 how did the you get 5/4?",
        "A": "because he simiplfily"
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "How did Sal, in 0:34, know that the number was 10/8?",
        "A": "Great Answer!! I got it now, thanks."
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "$200:8 hours = $50:but how many hours",
        "A": "the answer is x"
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "At 0:50 how did you get 5/4???",
        "A": "The easiest form of 10/8 is 5/4 thats were it is from"
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "the ratio of men to women is5:8 if there are 126 more women how many men are in that village",
        "A": "On one side of the equation we have: 5 men/8 women. On the other hand we have x men/(x+126) women. So we cross multiply. 5(x+126) = 8x 5x + 630 = 8x 630=3x 210=x. So we have 210 men and then (210+126)=336 women. Which is in perfect 5/8 ratio."
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "At 1:03 Sal said \"36 times 5 divided by 4\" when he should have said 5 over 4.",
        "A": "The bar in the middle of a fraction is a division symbol. So, Sal s words are a correct way of saying the problem."
    },
    {
        "video_name": "GO5ajwbFqVQ",
        "Q": "what odes \"n\" mean at 0:15?",
        "A": "N is a variable, if that is what you re looking for."
    },
    {
        "video_name": "xxAFh-qHPPA",
        "Q": "At 5:20 why don't we just multiply 29.12 x 10^1? Is there a reason we break 29.12 into 2.912 x 10 THEN multiple the 10 (aka 10^1)? Because my first instinct was: 29.12 x 10^1 = 291.2\n\nOr is breaking the 29.12 into 2.912 x 10 just some sort of accepted math rule? Something that's too complicated to explain at this level or something.",
        "A": "To have a number written in scientific notation, the 1st number is always written in the form where there is only one digit to the left of the decimal point and that one digit can only by from 1-9, no 0. This is just the definition of scientific notation. 29.12 x 10^1 does not meet that definition. It has 2 digits to the left of the decimal point. This is why it gets changed into 2.912 x 10^2"
    },
    {
        "video_name": "TgDk06Qayxw",
        "Q": "At 0:48 what are chorasumptions ( note: I don't know how to spell it)",
        "A": "Core assumptions - meaning the things you declare/know to be true that are then used as building blocks to prove other more complicated theorems. SSS would be considered a core assumption."
    },
    {
        "video_name": "TgDk06Qayxw",
        "Q": "At 2:37 he said he had a chord that wasn't a diameter? Well thats not what I was taught.",
        "A": "A diameter is a line that goes straight through the middle of a circle, however a chord is a line with two points on the edge of a circle, but it doesn t have to be the middle."
    },
    {
        "video_name": "TgDk06Qayxw",
        "Q": "At 4:40, is line segment AB congruent to line segments AC and AD?",
        "A": "Yes, I believe so, because all radii of a circle are equal. Someone please correct me if this is inaccurate."
    },
    {
        "video_name": "TgDk06Qayxw",
        "Q": "At 1:43, is the SSS more of a postulate or an axiom?",
        "A": "SSS is neither an axiom nor a postulate it is a criterion and all other s like asa,sas are also criteria"
    },
    {
        "video_name": "TgDk06Qayxw",
        "Q": "At 2:35 Sal said \"...a segment of a secant line..\"; what is a secant line?",
        "A": "A secant line is a line that cuts a curve into two or more parts. Here it cut the circle at points C and D."
    },
    {
        "video_name": "TgDk06Qayxw",
        "Q": "8:18: How are you going to find a chord with the property that it is bisected by the radius? Obviously you can construct infinitely many chords, even of either given length or given location. But I do not see how you can position a chord in such a way that it is bisected by a given radius, without resuming to the property you want to prove, that is SSS.",
        "A": "well, it s not a given radius, you can draw a radius in any position, so, for any chord you draw, there is a radius which bisects it. hope i helped : ) P.S : you can try it out by drawing a circle and any chord, you ll see that for any chord there is a particular position of the radius which bisects it, and that is going to be perpendicular to the chord."
    },
    {
        "video_name": "TgDk06Qayxw",
        "Q": "At 5:45 what does this symbol means? _|\nin \"Prove: AB _| CD\"",
        "A": "The symbol \u00e2\u008a\u00a5 means perpendicular to . AB \u00e2\u008a\u00a5 CD means AB is perpendicular to CD that is AB cuts CD at a 90 degree angle."
    },
    {
        "video_name": "lUA1LYCoIAg",
        "Q": "At 4:44 shouldn't he say what makes this four a zero instead of what makes this zero a four?",
        "A": "At 4:44 he is saying what makes this x minus four a zero , not what makes this four a zero, so no."
    },
    {
        "video_name": "uoN_Ax3dfQo",
        "Q": "At 04:12 min, I think population of bears increases by 2/3.",
        "A": "It actually shrinks because every time you times a whole number by a fraction the number is always lower then the whole number you times with"
    },
    {
        "video_name": "uoN_Ax3dfQo",
        "Q": "4:02 Sal says the bear population shrinks by a factor of 2/3, but this statement is just a double negative, if something shrinks by a factor <1, then it increases. So would it shrink by the reciprocal of 2/3? (So 3/2), or would it shrink by the difference of 1 and the original factor added to 1. Which would be (1 - 2/3 + 1 or 4/3).",
        "A": "No, shrinking by a factor of 2/3 means the original amount is MULTIPLIED by 2/3, which means you have 2/3 of the original left."
    },
    {
        "video_name": "uoN_Ax3dfQo",
        "Q": "At 4:12 why wouldn't we say that the bear population shrinks by a factor of 1/3 since each year the population is 1/3 less than it was the year before?",
        "A": "This is one of those cases where we could have gone either way with the language. But think of it this way; if you grow by a factor of 2, then you double. if you shrink by a factor of 3 then you are 1/3 the size. So if your size becomes 2/3 the original size, then you shrank by a factor of 3/2."
    },
    {
        "video_name": "8xbIS2UkQxI",
        "Q": "At 4:13, doesn't an oval contains 1 curved side and not 0 sides?",
        "A": "it has 0 sides"
    },
    {
        "video_name": "8xbIS2UkQxI",
        "Q": "At 1:44. Doesn't an oval technically have infinite sides? Every tangent the oval has, there is a corresponding slope for a side which is infinitesimally small. Since there are infinite tangent lines, aren't there infinite sides?",
        "A": "Geez bro this is low for ur level mate. Anyway, no because in a perfect world the slope would be one line, but we haven t advanced that much"
    },
    {
        "video_name": "8xbIS2UkQxI",
        "Q": "How does the oval (at 1:28) have no sides? I thought it has one side that goes around it...",
        "A": "A side is a line segment, but an oval has a curve. Thus a oval has no sides!"
    },
    {
        "video_name": "8xbIS2UkQxI",
        "Q": "At 3:32 sal says the sides arn't the same so does that mean no shapes can be exactly the same as eachother.",
        "A": "That is incorrect because in 4th grade math there are things called similar triangles that means that it s exact or really close to another triangle."
    },
    {
        "video_name": "8xbIS2UkQxI",
        "Q": "So part of the definition of a square is the definition of a rectangle (2:31 is the part I'm referencing)",
        "A": "BUT, not all rectangles are squares, because many rectangles have sides of different length."
    },
    {
        "video_name": "8xbIS2UkQxI",
        "Q": "Many people say the oval has one side only. Hopefully at 1:32,the oval is a \"one sided shape\"?",
        "A": "@Mekkhi. I flagged you because you 1. Did not answer the question. 2. Solicited Votes. @Hk.kh. An oval has zero sides. It goes on for infinity, like the circle."
    },
    {
        "video_name": "8xbIS2UkQxI",
        "Q": "At 3:20, can someone clarify for me what Cousin Fal is saying?",
        "A": "He is asking Sal how to classify shapes based on whether or not all of a shape sides are equal."
    },
    {
        "video_name": "duomXwMSPv8",
        "Q": "At 4:10 how did he get what he did, i don't understand",
        "A": "He simplified the expression."
    },
    {
        "video_name": "__nkbr6DeTg",
        "Q": "Can't you use the 'Scratchpad' instead of just pointing at it so you can mark it at 0:05 ?",
        "A": "It s allmost haiioween! but Haloween allready past.Thats sad."
    },
    {
        "video_name": "__nkbr6DeTg",
        "Q": "at 0:08 Sal starts talking really slow like he wants to make sure you understand what he's saying and really important. why did he do that?",
        "A": "Since this section is mainly for small children, (kindergarten), he wanted to go slow to make sure they understand. He wanted to make a point that it was very important, because in higher grades, you will dig deeper into that concept."
    },
    {
        "video_name": "6hri9k_2R8o",
        "Q": "How do you draw tangent lines at a point like shown at 3:25 of the video?",
        "A": "To get the SLOPE of tangent line at a point on the graph, you must, Take the derivative of the graph, and then plug in the x value of where the point is into the derivative."
    },
    {
        "video_name": "6hri9k_2R8o",
        "Q": "how did they get \"a\" and \"b\" at 0:50?",
        "A": "They didn t get any specific value for a and b, Sal was just illustrating the theorem."
    },
    {
        "video_name": "uXTRmzXlorI",
        "Q": "In 0:38, How does the Identity help us do our math.",
        "A": "In future when we try to do much complex math , if we can identify a chunk of equation is actually equals to 0, without doing other math step directly using identity , we can solve our problem . Similar useful identity of 1 has."
    },
    {
        "video_name": "P35LyN9g0oI",
        "Q": "At 1:18, is there a proper term for the 'star thing'?",
        "A": "Ampersand #Revive Yess! It does look like a shuriken! I am Japanese, so I should know. It is called a throwing star in English language, I think."
    },
    {
        "video_name": "P35LyN9g0oI",
        "Q": "At 2:27, isn't the figure different ? The arrows on the bottom are facing right and the arrows on the sides are going up in the first parallelogram, but in the second parallelogram, the arrows on the bottom are going left and the arrows on the sides are going down instead of up. Are the arrows not counted as part of the figure or what?",
        "A": "Arrows just are symbols to show that sides are parallel to each other, one arrow sides are parallel, and two arrow sides are parallel. The direction they are pointed does not really mean anything. They are not part of the figure per se, just to represent something about the figure."
    },
    {
        "video_name": "8iKsY8C5VW8",
        "Q": "Quick question, at 2:54 when Sal is talking about P(A | B), the calculation would be\nP (A and B) = P(A) * P(A | B) or\nP (1/6 * 1/4) = 1/6 * P(A | B) or\nP (1/24) = 1/6 * P(A | B) and then dividing both sides by 1/6\nYou get: 1/4 = P(A | B) or the answer is 1/4, but the real answer is 1/6... Am I doing the formula wrong? Why is this not working?",
        "A": "Double-check your first line (of equations) with the formula for conditional probability."
    },
    {
        "video_name": "lR_kUUPL8YY",
        "Q": "How come he says at 0:01 He says million instead of the number",
        "A": "The number is 7 million 346 thousand 521 and 32 thousandths. (not sure about about the spelling though :P)"
    },
    {
        "video_name": "59s_CQjTD6w",
        "Q": "iin 5:20.... The PENTAGON has five lines of symmetry, not four. Kind of irrelevant, but I thought I'd point it out anyways :)",
        "A": "In the video, it said in the bottom right corner, At 5:15 Lindsay counts 4 lines of symmetry, but they are actually 5. I hope this helps!"
    },
    {
        "video_name": "MJBjGnR6vlk",
        "Q": "Around 1:55, he talks about u=0 and t = 172 in terms of the days. I understand the reasoning, however, I don't understand how to recognize when we have to shift the graph? In this problem, why do we have to start on the first day of the year instead of having the 172nd day in the equation?",
        "A": "If you didn t shift the cosine argument, then the maximum temperature for a cosine function would be at theta = 0 radians = the beginning of Jan 1st. 172 actually is in the equation (as the amount subtracted from t to get the right starting point for the cosine (it shifts cosine 172 days to the right)."
    },
    {
        "video_name": "MJBjGnR6vlk",
        "Q": "At 2:20 , Sal says cosine hits its maximum point. What does it mean?",
        "A": "Look on the unit circle, where y = sint and x = cost, and t is the angle of the radius of the circle with the +x axis. What are the max and min values sint and cost can have? At what angles t do these values occur?"
    },
    {
        "video_name": "MJBjGnR6vlk",
        "Q": "At 2:20, it seems like Sal just decides to use cosine instead of sin, just because it is easier. Is it optional whether to use sine or cosine, or is there a good reason that he used cosine?",
        "A": "He uses cosine in the video because we are given the maximum point with the day, June 21. The reason why he doesn t use sine is because we are not given when the average day is, so we cannot use it. After finishing the function with cosine, we can then find out where the average day is, but then we have already made a function, so why make another one? I hope this helps."
    },
    {
        "video_name": "MJBjGnR6vlk",
        "Q": "At 1:22 of the video Sal states the equation \"u = 0\" for zero days after June 21st......\nI believe this is incorrect reasoning, since the problem asks for the Length of the day on the \"t\"-th day of the year......Since there would be no \"zero\"-th day of the year, I believe \"u\" would equal 1 ....in other words equating to the FIRST day of the year......and therefore the proper solution should be...\nL(t)=357cos(2pi(t-171)/365)+739.5\n\nsince the 172nd day minus 171 days equals the FIRST DAY......January 1st!",
        "A": "When you plug in 172, you want to get the maximum day length. The cosine function is maximized at 0, therefore the correct expression is (t-172)."
    },
    {
        "video_name": "MJBjGnR6vlk",
        "Q": "2:19 what does he mean when he says that cosine hits it maximum when the input is 0 and sine does not?",
        "A": "He is saying that cosine reaches its maximum when x= 0, and sine does not because sin0=0. Sine reaches its maximum when x=pi/2."
    },
    {
        "video_name": "S5LEdJfCsHM",
        "Q": "How did Bernard KNOW that the date wasn't the 14th at 8:08?",
        "A": "not alice, cheryl"
    },
    {
        "video_name": "S5LEdJfCsHM",
        "Q": "Brain teaser based on Korean drama show My Daughter Geum Sa Wol:\nSo Kook-Ja (Grandma Emiko) baked 2 batches of Bum Bum Emiko Cookies.\nThey were both for her granddaughters. There was a batch of rainbow sprinkled cookies and a batch of chocolate chip cookies. Which batch was for Dalleh? Which batch was for Jjilleh?\nHints:\nBoth granddaughters love the colors of the rainbow.\nThe girls both have sweet tooths.\nBe sure to watch My Daughter Geum Sa Wol TODAY from 6:30 PM to 9:00 PM!",
        "A": "I think that you could ve added more information. Personally, Jjileh had sprinkles."
    },
    {
        "video_name": "S5LEdJfCsHM",
        "Q": "What does khan mean by deduce 7:52",
        "A": "To deduce an answer is to figure out the answer using the information given. The given information must eliminate some choices in order for the correct answer to be deduced."
    },
    {
        "video_name": "721RrH6auoU",
        "Q": "at 2:30, where did you get 4 from for the oak tree?",
        "A": "The number of branches increased by 12 every 3 years. So, over 1 year, it increased by 4 branches."
    },
    {
        "video_name": "tJrSILRXOUc",
        "Q": "at 0:19 , why does he go to the right?",
        "A": "he is just trying to make a point about the numbers greater than 6"
    },
    {
        "video_name": "KiVGac1aBt8",
        "Q": "At 0:50, Sal says, \"Mary owns 4,000 to 200 shares\". I feel this may trip some students up when they hear that, because we are working with ratios. A common syntax for ratios comparing two numbers with \"to\", like 1 to 4 (1:4)",
        "A": "He actually said Mary owns Four-thousand-two-hundred shares."
    },
    {
        "video_name": "h-TPSylHrvE",
        "Q": "in the video (4:40) you stated that the angle formed between pi/2 + theta and pi/2 is theta, hence, the coordinates of theta and pi/2+theta should be some. Therefore sin(pi/2 + theta) = sin(theta) and cos(pi/2 +theta) = cos(theta). Is this true?",
        "A": "A radius of the unit circle that forms the angle \u00f0\u009d\u009c\u0083 with the \u00f0\u009d\u0091\u00a5-axis has an end point (cos \u00f0\u009d\u009c\u0083, sin \u00f0\u009d\u009c\u0083) on the unit circle. We can say that the angle \u00f0\u009d\u009c\u0083 has coordinates (cos \u00f0\u009d\u009c\u0083, sin \u00f0\u009d\u009c\u0083) . If we rotate the radius \u00f0\u009d\u009c\u008b/2 around the origin it now forms the angle \u00f0\u009d\u009c\u0083 with the \u00f0\u009d\u0091\u00a6-axis (instead of the \u00f0\u009d\u0091\u00a5-axis, as before), which means that the coordinates of \u00f0\u009d\u009c\u0083 + \u00f0\u009d\u009c\u008b/2 are (-sin \u00f0\u009d\u009c\u0083, cos \u00f0\u009d\u009c\u0083). But it also has the coordinates (cos (\u00f0\u009d\u009c\u0083 + \u00f0\u009d\u009c\u008b/2), sin (\u00f0\u009d\u009c\u0083 + \u00f0\u009d\u009c\u008b/2)) \u00e2\u0087\u0092 \u00e2\u0087\u0092 cos (\u00f0\u009d\u009c\u0083 + \u00f0\u009d\u009c\u008b/2) = -sin \u00f0\u009d\u009c\u0083, sin (\u00f0\u009d\u009c\u0083 + \u00f0\u009d\u009c\u008b/2) = cos \u00f0\u009d\u009c\u0083."
    },
    {
        "video_name": "h-TPSylHrvE",
        "Q": "considering the right hand triangle\nthe three angles are theta, pi()/2 and pi()/2-theta\nsurely cos theta =sin(pi()/2 - theta)\nand not that given at 6:12",
        "A": "Your problem is that you re not using the unit circle definitions of sin and cos. Forget about right triangles. Draw both angles t and t+pi in their proper quadrants, and compare cost (x/r at angle t) with sin(t+pi) (y/r at angle t+pi) (pick an actual angle t, like pi/6)."
    },
    {
        "video_name": "h-TPSylHrvE",
        "Q": "At 4:07 Sal says that the length of the magenta line is Sine theta plus Pi. How did he come to that?\n\nWouldn't the length of that line be equal to Cos theta since we are rotating the triangle?",
        "A": "Yes, you re saying the same thing Sal said - that they re equal. By using the unit circle definition of sin(a) and cos(a) ( = y/r and x/r at angle a); sin(t+pi/2) = cost"
    },
    {
        "video_name": "h-TPSylHrvE",
        "Q": "Can someone please explain that at 3:43 How did Sal figure out that, sin(Pi/2 - theta) is the Y coordinate? I mean, how could that Positive Y axis be the sin(Pi/2 - Theta)?",
        "A": "It s theta + pi/2. Not pi/2 - theta."
    },
    {
        "video_name": "h-TPSylHrvE",
        "Q": "At 3:55, how does Sal conclude that the line in magenta is sin (theta+pi/2)?",
        "A": "Take the angle t = pi/6. What s its x coordinate on the unit circle? What s the y coordinate of pi/6+pi/2? What are the sine and cosine of each angle?"
    },
    {
        "video_name": "h-TPSylHrvE",
        "Q": "At 4:03, how is sin (theta + pi/2) equal to the line in purple and how is it the length of the y-co-ordinate?",
        "A": "What s the y coordinate of pi/6 + pi/2? What s the sine of (pi/6+pi/2)?"
    },
    {
        "video_name": "ZElOxG7_m3c",
        "Q": "At 0:40 couldn't you just use the Pythagorean Theorem?",
        "A": "No, because it s not a right triangle (or, at the very least, we can t prove it to be a right triangle)."
    },
    {
        "video_name": "ZElOxG7_m3c",
        "Q": "At 1:42, do you always subtract 2?",
        "A": "Actually, it s 2 times b and c. But, yeah, the 2 is always part of the formula. However, sometimes the formula is written with different letter combinations. (b^2 = a^2 + c^2 - 2ac * cos B and c^2 = a^2 + b^2 - 2ab * cos C) For those combinations, just remember that whatever is to the left of the = shouldn t be repeated on the right until after the cosine."
    },
    {
        "video_name": "ZElOxG7_m3c",
        "Q": "In 0:55, Sal says the word \"theta\". What does the word \"theta\" mean?",
        "A": "Theta \u00ce\u00b8 is a Greek letter frequently used as a variable for an angle."
    },
    {
        "video_name": "ZElOxG7_m3c",
        "Q": "At 3:53, what is the difference between degree mode and radian mode?",
        "A": "1 radian = 180/pi degrees. Therefore cos(180 degrees) = cos(pi radians). You have to choose the mode that matches the types of degree measures you are using. In radian mode, you enter the radian and it will give you the degree. In degree mode, you put the degree in the trig function and it will give you the radian."
    },
    {
        "video_name": "ZElOxG7_m3c",
        "Q": "At 3:20, when he square roots both sides, shouldn't he take + or -?",
        "A": "In trig, when you take the square root, you can ignore the negative root. He is only taking the arithmetic square root, which only has a positive answer. Also, a negative answer doesn t make sense because a side length can t be negative. Hope this helps!"
    },
    {
        "video_name": "ZElOxG7_m3c",
        "Q": "How do we find the cosine of theta at 2:08? There is no right angle in that triangle.",
        "A": "To go off of what Sara said, the whole purpose of the Law of Cosines, is to be able to find theta and side lengths in non-right triangles."
    },
    {
        "video_name": "zlq_jQYD1mA",
        "Q": "I can see that the example given at 8:38, 7\u00c3\u00b7(7\u00e2\u0080\u00a210) = 7\u00c3\u00b77\u00c3\u00b710, is a general principle.\na\u00c3\u00b7(b\u00e2\u0080\u00a2c) = a\u00c3\u00b7b\u00c3\u00b7c, assuming that b and c are non-zero numbers. But why?",
        "A": "Good question. I m not sure if this is the correct answer, but this is what I came up with: Let s solve for a by using algebra. a\u00c3\u00b7(b\u00e2\u0080\u00a2c) = a\u00c3\u00b7b\u00c3\u00b7c Multiply both sides by 1 to free from the parentheses a\u00c3\u00b7b\u00e2\u0080\u00a2c = a\u00c3\u00b7b\u00c3\u00b7c Divide both sides by b and c a = a It works! Since a = a, the equation contains an infinite amount of solutions. So that means the formula should work for any combination of numbers that you choose. Did that help? Viewers, please comment below if my answer is incorrect OR if you have a better answer."
    },
    {
        "video_name": "zlq_jQYD1mA",
        "Q": "Why is 2/2 the same as 1/2 at 1:05 they are different fractions",
        "A": "It isn t the same it s the answer, 1/2 = .5 or 1 half, while 2/2 = 1 whole or 1.00."
    },
    {
        "video_name": "zlq_jQYD1mA",
        "Q": "why 3 is 30 tenths in the minute 2:45??",
        "A": "because 0.3x30=3 so 3 is the same thing as 30 tenths"
    },
    {
        "video_name": "qlWZJ21O63s",
        "Q": "What does this mean at 1:24??",
        "A": "Hi Steven Lee, Sal says, 3 more in the x direction and 3 less in the y direction but meant to say 3 less in the x direction and 3 more in the y direction . Also when Sal says right over there, he means that the point of the image corresponding to the pre-image is plotted in that position (3,-3) Hope that helps! - Sam"
    },
    {
        "video_name": "qlWZJ21O63s",
        "Q": "at 2:13 how did u get -3",
        "A": "Point F is at (6, -6). To map this with a scale factor of 1/2, we simply multiply each coordinate with the scale factor: (6 * 1/2, -6 * 1/2) = (6 / 2, -6 / 2) = (3, -3)."
    },
    {
        "video_name": "PEeUTQ0Gri8",
        "Q": "1:04 It would be correct if you started at 1, and counted by 2's. Right or Wrong?",
        "A": "No, it would not be correct."
    },
    {
        "video_name": "PEeUTQ0Gri8",
        "Q": "At 1:56 we does not have a special formula for checking wrong numbers.",
        "A": "You need to check if every symbol has a number beneah it, and count the symbols yourself. if you can count to 10 you should be able to do it. Good luck :)"
    },
    {
        "video_name": "PEeUTQ0Gri8",
        "Q": "At 0:22 why does sal not skip a couple of flowers",
        "A": "Well ,I think its 5"
    },
    {
        "video_name": "ViqrHGae7FA",
        "Q": "At 8:05, he says that x sub i-1 is for the left side, how should it be written if you want to use the midpoint or the right side?",
        "A": "For right endpoints, the notation is simply x sub i, left endpoints are, as Sal points out, x sub i-1, midpoints are x sub i* (star), which looks like x sub i with a star directly above i."
    },
    {
        "video_name": "ViqrHGae7FA",
        "Q": "At 1:58 and 2:00, he says \"x knot.\" What does that mean?",
        "A": "It is x naught. That means x\u00e2\u0082\u0080 . Naught is another word for zero."
    },
    {
        "video_name": "ViqrHGae7FA",
        "Q": "At 3:55, Sal says that for the purposes of the video, he will assume the width of each rectangle is constant, but then says that the sum can be written with unequal width but \"it get's a little bit fancier.\" I have only seen summation notation for Riemann sums when the divisions are equal. How would a Riemann sum with unequal divisions be written in summation notation?",
        "A": "I ve never seen it written, but it is interesting to think about ways to go about it. What if deltax_n = f(deltax_(n-1), f_(n-1), f_(n-2))? Maybe something like deltax_n = abs{1/[(f_(n-1) - f_(n-2)]} * deltax_(n-1)"
    },
    {
        "video_name": "Su4DKLbK8mA",
        "Q": "I don't understan through 1:22 of the video how you use a number line...",
        "A": "you use it by ploting lines to plot your numbers"
    },
    {
        "video_name": "Su4DKLbK8mA",
        "Q": "Where does the 10 come from 2:05",
        "A": "Because Sal decided to divide it into 10 pieces instead of 5."
    },
    {
        "video_name": "XFwQV-KCudw",
        "Q": "I'm not quite sure what Sal Khan is saying when at 2:13-2:27 he says that the function needs to be the same, along with the domain and inputs. Can someone please explain? Also, at 6:05, I'm not really getting what Sal is saying when he says putting z is not equal to 3/2 is redundant.",
        "A": "@ 2:13: You re trying to simplify, not change, the expression. You want the simplified expression to be exactly the same as the original one, but without any factors that have no net result. So therefore it must have the same domain (and range). @ 6:05, (2z-3) is still in the denominator, so everybody knows that it s undefined at z = 3/2."
    },
    {
        "video_name": "XFwQV-KCudw",
        "Q": "at 5:12 , how did Sal get Z not equal to zero?",
        "A": "Because division by zero is never allowed, Sal is checking each of the factors in the denominator to see what values would make them zero. z is a one of the factors in the denominator, so z is not allowed to equal zero. ( 2z - 3 ) is the other factor in the denominator, so z is not allowed to equal 3/2."
    },
    {
        "video_name": "XFwQV-KCudw",
        "Q": "At 5:11 Why Z cannot be equal to zero? What would happen if it is equal to zero? Another question is that why at 5:24 the denominator is set to equal to zero while numerator is ignored?",
        "A": "It s only 0 in the denominator that s a problem. Why? Because if you get a non-0 number divided by 0 there is no finite number which can be the result. The expression blows up . So that s why it s undefined - because it s impossible. But when you have 0 as a factor of the numerator that s fine, it just makes the whole numerator 0, and the whole expression 0."
    },
    {
        "video_name": "XFwQV-KCudw",
        "Q": "I dont understand what happened just before 3:11 i mean how did Sal come to that conclusion or answer. Please explain.",
        "A": "Sal had the fraction: (2x+1) / 2. He split the fraction into 2 terms and reduced them. (2x+1) / 2 = 2x/2 + 1/2 = x + 1/2 Hope this helps."
    },
    {
        "video_name": "XFwQV-KCudw",
        "Q": "I'm a little confused. At 0:38 when Sal says that 0 is undefined, what does that mean exactly? Is 0 an undefined answer? Also is there a reason to write that x does not equal zero for every problem?",
        "A": "What we are avoiding is division by zero. Since in the example the denominator is 14x, if x=0 then (14)(0)=0 and you have division by zero - not allowed! Now let s say the denominator was 14 - x. In this case, then, if x was equal to 14 you would have 14-14=0. In this case you would need to say that x does not equal 14 in order to avoid division by zero."
    },
    {
        "video_name": "_WOr9-_HbAM",
        "Q": "At about 2:32, Sal is discussing the statement at the top of the right column. He says that the value of f(x) is approaching 1 as x approaches 0 from the negative side. I don't see it. He only illustrates with points after the apex of the curve. What about the points on the left side of the curve, where the value of y went from 0 to some value greater than 1? At some point on the curve, it appears that y = 1, which would seem to eliminate it as a limit.",
        "A": "I think you are mixing limits with critical points (that will be taught in derivative applications). The limit is not where the graph reaches an apex. The limit is to go as close as you wish to a certain point. In the video, Sal is explaining how he gets as close as possible to x=0. So he only cares about the values that are very very close to 0."
    },
    {
        "video_name": "_WOr9-_HbAM",
        "Q": "at 3:59 how is the is it approaching 0",
        "A": "The graph goes down to 0 from the left side of x=1."
    },
    {
        "video_name": "_WOr9-_HbAM",
        "Q": "At 4:28 Sal talks about the limit as x reaches 1.5. But as it reaches 1.5 f(x) appears to be 1. However when x is 1.5 f(x) is 1. So my question is, is it a limit when from both directions f(x) appears to be reaching a number even if it is also the number?",
        "A": "In this case, f(1.5) is a defined point of the graph and equals to 1 - But I would say that it is still a limit (even if it probably is a little pointless in practice to calculate a limit to a properly defined point)"
    },
    {
        "video_name": "_WOr9-_HbAM",
        "Q": "I am not sure I understand 3:06 where Sal says the limit exists, they are both approaching the same limit but the negative side is not equal to the positive side so according to previous videos this limit does not exist?",
        "A": "Watch out! The video was dealing with one sided and two sided limits. When we ask if the limit of f exists at some x=c. with out qualification (eg positive side, negative side) then all that needs to be shown is that the value that f approaches as x approaches c from from both sides of c are equal."
    },
    {
        "video_name": "_WOr9-_HbAM",
        "Q": "At 2:37, how is the second statement true? I thought You had to look at the 2nd line on the graph when the statement said as x approaches 0 from the positive side, but instead Sal just looked at the positive side of the first line. Help please.",
        "A": "you see it says 0 from the negative side so the line ends up on 1 so when it approaches from the -ve side take the line from left to the point and when it says 0 from +ve side take the line from right . pretty much they end up on the same point which is 1"
    },
    {
        "video_name": "_WOr9-_HbAM",
        "Q": "At 4:10, he says that the lim f(x) does not exist when x ->1. Doesn't the solid circle mean that it is at 1, therefore making the statement lim f(x) does exist when x->1?",
        "A": "lim(x\u00e2\u0086\u00921^-) \u00e2\u0081\u00a1f(x) = 0 lim(x\u00e2\u0086\u00921^+) \u00e2\u0081\u00a1f(x) = 1 As there is a gap, lim(x\u00e2\u0086\u00921)\u00e2\u0081\u00a1 f(x) is not defined. For Sal s version, try this:-"
    },
    {
        "video_name": "_WOr9-_HbAM",
        "Q": "at 2:06 where yu say x is approaching one from the left but isnt it not bubbled in at the f(x)= 1\nwhat does the bubbled in and not bubbled circles stand for?",
        "A": "A bubbled circle means the function touches that point. A non-bubbled circle means the function comes infinitely close to that point but it undefined on that specific point. In this example, it is still defined, but it is just used to mean to go as close to it as possible without touching it to see what is approaches."
    },
    {
        "video_name": "_WOr9-_HbAM",
        "Q": "At 1:05, when the tutor was explaining why it is false, can someone give a bit more of a explanation on why it is false cause I'm looking at the graph and I keep coming up with zero.",
        "A": "I don t believe Sal actually made a mistake. He states that we are coming from the positive side, so we would approach from numbers greater than 1. At those numbers it appears that we approach 1, as he stated. If we had approached from the negative side then yes, we would be approaching 0. I believe you mistook the positive sign as a sign to go up the number line. That would explain why you are coming from the negative side."
    },
    {
        "video_name": "o4pbAQgJYjI",
        "Q": "how is it that @ 5:16 he says x increases by 1 but it shows -x... And why does y decrease by 1... I'm so lost \u00f0\u009f\u0098\u0094",
        "A": "As you are starting to learn this, -x is best thought of as -1/1 x, so you have two choices -1/1 means that when x changes 1, y changes -1 or using rise/run, you could say down one right one 1/-1 means that when x changes -1, y changes 1 once again in rise/run, you could say up one left one. A negative slope will always go down from left to right. Since the first one is the proper way of writing a negative fraction (negative is moved to the numerator) that is the one he choose to use. Hope this helps"
    },
    {
        "video_name": "o4pbAQgJYjI",
        "Q": "0:40 How do you know that it is = to zero?",
        "A": "Sal says that x=0 to isolate (figure out) the y value, and likewise, does the same for the x value. So if the equation is -x-3y=9, with x=0, you would do -1(0), so then you end up with -3y=9. Then you would do 9/-3 to solve for y. Hope this helps!"
    },
    {
        "video_name": "f8svAm237xM",
        "Q": "At 0:15, I dont' get the joke on the \"Angle Side Side\" name he gave it. I don't often get jokes, so can someone explain that to me",
        "A": "A(ngle)-S(ide)-S(ide)- A-S-S- spells a bad word. That s why you don t want to say it that way or you ll make people giggle! S-S-A doesn t spell anything bad, so you want to say it that way when you talk about this non-postulate."
    },
    {
        "video_name": "f8svAm237xM",
        "Q": "At 5:55, Sal talks about 'the RSH postulate'. I learnt pretty much the same thing before, in school, but then they called it 'the HL theorem' (Hypotenuse-Leg). So can these terms be used interchangeably? Has anyone heard of it before?",
        "A": "Yeah, they re just two different ways of saying the same thing. Maybe the biggest problem in math is that there is no agreement about what things are called, so having the flexibility to use the language that your different teachers like is an important skill to develop."
    },
    {
        "video_name": "f8svAm237xM",
        "Q": "The RSH postulate that Sal mentions at 6:22 seems to be an exact copy of something I learned on school, called the Hypotenuse Leg (HL) postulate.\nAre these terms the same, or is there a catch I'm missing?",
        "A": "Yep, same thing. Both use the hypotenuse, right angle, and one leg, just different names."
    },
    {
        "video_name": "f8svAm237xM",
        "Q": "Refering to 5:15, if the angle of SSA is obtuse, there's only one solution left.\nSo I can't figure out why SSO (side-side-obtuse) couldn't be a postulate... it might be helpful !\nAny trap ?",
        "A": "It could be if you like. Most people don t consider it one because it would be pretty unusual to know that two triangles are obtuse. Note that this is what the RHL postulate says -- that SSA is true if you know that the two triangles are both right triangles."
    },
    {
        "video_name": "f8svAm237xM",
        "Q": "at 3:42 you can't have 2 right angles in a triangle also right?",
        "A": "wow. thx for answering so fast"
    },
    {
        "video_name": "VvuuRpJbbHE",
        "Q": "at 1:26, -44/2= -22 but -22 * 2 = 44 doesn't that mean that A can't equal -22?",
        "A": "-22*2= -44 not 44 because only a negative number times a negative number makes a positive number, whereas in this problem there is a negative number (-22) times a positive number (2), which makes a negative number. That means that -44=2a and a= -22."
    },
    {
        "video_name": "VvuuRpJbbHE",
        "Q": "At 1:30, why is -44=2a? Is the middle number always going to be equal to 2a?",
        "A": "Yes. It is easier to think of it like this: a= -44/2 I would encourage you to watch the original video on completing the square for more detail."
    },
    {
        "video_name": "VvuuRpJbbHE",
        "Q": "What property does sal use during 0:45?",
        "A": "That s just the FOIL method for expanding the product of two binomials."
    },
    {
        "video_name": "VvuuRpJbbHE",
        "Q": "So, C=(-22)^2 or C=(x-22)^2. According to 2:11 and 3:15.\n\nThank you",
        "A": "No, C = (-22)^2 *c = constant (x-22)^2 is the factored form of the tri-nomial x^2-44x+484"
    },
    {
        "video_name": "VvuuRpJbbHE",
        "Q": "like at 3:11 , how do you 'factor' it into a binomial?",
        "A": "When a binomial is squared it becomes a perfect square trinomial. On the right-hand side a generic model of this is shown. Using this model, Sid works backwards to solve the problem. On the right, the generic perfect-square trinomial is shown, x^2+2ax+a^2 Using this, Sid finds that a = -22, therefore (because the model shows a binomial being (x+a)^2) the answer to the problem is (x-22). The first 2 steps in purple were unnecessary."
    },
    {
        "video_name": "1oFv56vSvvA",
        "Q": "At 2:17 , Is milligram 1000 gram or is it kilogram that has 1000 grams, if so then how much is milligram?\nWhat are the order of weight in metric and SI, please I have trouble remember these in order?",
        "A": "1000 milligrams is one gram, 1000 grams is one kilogram."
    },
    {
        "video_name": "bb4bSCjlFAw",
        "Q": "Is there ANY way to express the curve graphed graphed at 4:00 with a normal x and y relation (like x^2+y^2=x-y, but just a relation that would actually look like 4:00)? At first I thought that you might be able to do so by graphing the inverse of tcos(t) = inverse of tsin(t), but those functions don't have defined inverses",
        "A": "You can do this for some parametric functions - the ones where you can isolate a parameter. But I don t see how you can isolate t on this one so I assume this spiral curve can only be expressed parametric."
    },
    {
        "video_name": "KD46pC_KFWk",
        "Q": "At 3:26, when you say that a candidate cannot have 0 votes, how did you deduce that from the question? The statement underlined in purple only says that a member votes once for one candidate out of a total of 27. It does not say anything about the number of votes each candidate gets. For example, each member of the committee might have votes for the same candidate. In this case, that candidate would have all the votes and the rest would have 0.",
        "A": "Having someone with 0 votes contradicts the second condition(the one underlined in green) as Sal says."
    },
    {
        "video_name": "yUUPP70Fhpo",
        "Q": "Around 10:00 and 13:00 , even though I understand, that we want a C that is as small as possible, I don't see why we can just pick the lowest value from the evaluated inequality, when we really don't know what C is?",
        "A": "c is what can take on a variety of values in this case, the inequality holds true if c is greater that what we find it to be. we simply evaluate to find what the smallest value of c can be."
    },
    {
        "video_name": "yUUPP70Fhpo",
        "Q": "At 3:25, Sal assumes that the (n+1)st derivative of f is continuous. But isn't continuity already implied because the function is differentiable? In other words, is the assumption of continuity redundant?",
        "A": "Technically no. If f is differentiable then f is continuous, but the derivative of that function doesn t have to be continuous. An example would be f(x) = (x^2)*sin(1/x) if x != 0 0 if x = 0"
    },
    {
        "video_name": "yUUPP70Fhpo",
        "Q": "At 4:30 , how can we integrate both sides of inequality? I mean, indefinite integral is the set of all primitive functions of function \"under\" the integral - so we are comparing two sets of functions with relation \"larger than\"? I don't get it.",
        "A": "You can use a definite integral instead, which yields a valid proof."
    },
    {
        "video_name": "yUUPP70Fhpo",
        "Q": "At 8:10, why don't we have to write in the + C when integrating E^(n+1) (x), yet we have to on the right? It is not as if we can combine them to make the one C on the right, since the C on the left would be within absolute values, along with E^(n) (x). |a+b| is not equal to |a| + |b|, and we can't use the Triangle Inequality, either.",
        "A": "Let s say that we have f(x) = x^2. When you integrate f(x), you should write F(x) = (1/3) * x^3 + C, not F(x) + C = (1/3) * X^3 + C."
    },
    {
        "video_name": "yUUPP70Fhpo",
        "Q": "I dont get it at 11:50. If you would integrate S(M(x-a)) = S(Mx-Ma) = (Mx^2)/2-2Ma/2+(a^2)/2 + C = (Mx^2)/2-Ma + (a^2)/2 + C. What happens to the (a^2)/2 term? Is it a part of the C constant?",
        "A": "Actually, it the term is M*((a^2)/2), not just (a^2)/2. Yes, the M*((a^2)/2) term is part of C. You can actually prove it to yourself. Just set Sal\u00e2\u0080\u0099s result (using u-substitution) equal to your result. Remember to label the constants as C1 and C2 to avoid confusion. When you put both C1 and C2 on the same side, you\u00e2\u0080\u0099ll see that the constant from the u-substitution result minus the constant from your result is M*((a^2)/2). I hope that helped."
    },
    {
        "video_name": "yUUPP70Fhpo",
        "Q": "When taking the antiderivative of the remainder at 8:10 why does that not have a constant added to it?\n\nShouldn't anti[E^(n+1)(x)] be equal to E^(n)(x) + C ?",
        "A": "It should indeed be E^(n)(x) + C, which Sal says if you wait until 8:45, after which we goes further and finds an appropriate value for C. You just need to have patience."
    },
    {
        "video_name": "Lvr2YsxG10o",
        "Q": "at 1:55 does Sal say SHAVED?\nAnd he does a 2:20 aswell!",
        "A": "At 1:55 the transcript says that he says I ve shaded one of them , but it does sound like he says I shaved one of them . At 2:20, he says If we shave- if we- not shave, if we shade this entire row.. . If you re ever wondering what he s saying , just look at the transcript! :))) Hope that helped!"
    },
    {
        "video_name": "Lvr2YsxG10o",
        "Q": "At 0:31, why is Sal referencing Latin? It makes sense but why do it.",
        "A": "He is basically saying that 20 per(cent) means 20 out of 100! Because in latin cent means 100."
    },
    {
        "video_name": "Lvr2YsxG10o",
        "Q": "At 0:35 percent means every 100",
        "A": "cent=100 if you have a 100 grid and someone asks you to shade in 60% you would shade in 60 squares."
    },
    {
        "video_name": "Lvr2YsxG10o",
        "Q": "isnt .20 the same thing as 20%\n\n0:48",
        "A": "Yes, because to change a decimal to percent you are supposed to move the decimal two digits right."
    },
    {
        "video_name": "Lvr2YsxG10o",
        "Q": "At 0:35, percent means per hundred ,right?",
        "A": "i think so Bro"
    },
    {
        "video_name": "Lvr2YsxG10o",
        "Q": "I know the meaning of a percent, but he only tells about the hundred squares(in 1:03). What about if there are numbers like 23 or 545? Do I just divide by 100 or is there an easier way?",
        "A": "you can simplify the number or make it larger"
    },
    {
        "video_name": "z0-1gBy1ykE",
        "Q": "in 1:58 he starts doing the other part which is -pi/3. what if the pi part has a number next to it? what do you do when that happens?",
        "A": "The process is the same. Let s say you want to convert -2pi/3 Multiply by 180/pi -2pi/3 * 180/pi = -360pi/3pi Reduce to -120 degrees Hope this helps."
    },
    {
        "video_name": "z0-1gBy1ykE",
        "Q": "You said at 0:46 that the degree symbol is literally the unit of the number, is there a symbol for radians?",
        "A": "In my opinion radians are units just like degrees are. For some reason, units for radians often are not used"
    },
    {
        "video_name": "z0-1gBy1ykE",
        "Q": "At 0:21, Sal says that 2pi radians is 360 degrees. When you do basic geometry, 2pi radius (radii) is 360 degrees. As both statement are equal, are radius's and radians the same or different? Please give the definition of both terms.",
        "A": "They re different. Radius: the distance between the center of a circle and any point on its circumference Radian: One radian is 180/pi degrees. An arc of a circle (that has a radius of 1 unit) with a central angle of a radians has a length of a units."
    },
    {
        "video_name": "z0-1gBy1ykE",
        "Q": "At 2:52, what would you do if the two pi's so not cancel out",
        "A": "If you can t simplify it any further you d have to leave it as it is."
    },
    {
        "video_name": "z0-1gBy1ykE",
        "Q": "At 3:05, do we cross multiply (meaning (3x180)/-1) or do you multiply them straight ( [-1x180]/3 )",
        "A": "What s (1/3)(1/2)? (1/2)(1/2)? (1/3)(6/1)? (1/3)*6? (1/2)(pi)? (1/2)(pi/2)? (2/1)(1/pi)? (1/3)(180/1)? Are they cross or straight? Does it matter what the numbers are?"
    },
    {
        "video_name": "2BgWWsypzLA",
        "Q": "@4:00. Why not put the k back into 10^-4 = .4008(10^-4k). Or am I missing something entirely.",
        "A": "They are the same. (10^-4)^k = (10^-4k) = 1/(10^4k)"
    },
    {
        "video_name": "2BgWWsypzLA",
        "Q": "Why does Sal do anything with the denominator at 5:22? I dont understand why he would expand those like that. It certainly gives a different answer if you do that. I was going through the questions and didn't use fractions like that but kept the decimals and my answer was 1.96 instead of 196. Can someone explain this to me?",
        "A": "From his example: 1 - 0.0001 = 0.9999, which is also equal to (9999/10,000). He probably just does this to remove decimals, since 0.4008/(9,999/10,000) = (0.4008*10,000)/9,999 = 4,008/9,999. This is still equal to 0.4008/0.9999."
    },
    {
        "video_name": "Ez_-RwV9WVo",
        "Q": "At 4:04 i believe i dont get on how Sal gets his mid point",
        "A": "The points are (4,-5) and (8,2) so the x values are 4 and 8; the y values are -5 and 2. The midpoint is going to be the averages for each of x and y. To get an average, add the numbers and divide the total by how many numbers you added. x:4+8=12 -> 12/2=6 -> The x value of the midpoint will be 6. y:2+(-5)=-3 -> -3/2=-1.5 The y value of the midpoint will be -1.5. Put those together and you get a midpoint of (6,-1.5)."
    },
    {
        "video_name": "lvAYFUIEpFI",
        "Q": "At 6:30 on the video, I am trying to calculate an equation that i have. I need to find the farthest point of this equation. x^+(y-20)^=225. Can you help me with this?",
        "A": "I think that s a circle with the origin at (0,20) and a radius of 15. The furthest point (from the radius) would be 15 units away. All of the points would be 15 units away from the radius, by the way."
    },
    {
        "video_name": "lvAYFUIEpFI",
        "Q": "At 6:45 if the semi major and minor axis are both are r, then does that mean that they are equal? Then wouldn't the equation become a circle?",
        "A": "Devina, Yes when both axis are equal, that is a special case when the ellipse is a circle. That is very perceptive. Sal even mentions that two second later in the video at 6:47"
    },
    {
        "video_name": "lvAYFUIEpFI",
        "Q": "Why is x=-2,and y=1(-2,1)?? It should be (2,-1)!!! (At 12:06) Why does he want to make it equal 0? :D",
        "A": "10:20-11:00 talks about zero."
    },
    {
        "video_name": "lvAYFUIEpFI",
        "Q": "I am confused about what Sal says about it being \"the same part on the ellipse\" at 10:50",
        "A": "He is saying that because it s still the same ellipse, just shifted in right and down."
    },
    {
        "video_name": "lvAYFUIEpFI",
        "Q": "one doubt: at 8:48 why is it (x-5)? n not (x+5)",
        "A": "When x = 5 -> (x - 5) = 0 which means that the centre of the ellipse is at (5, y)."
    },
    {
        "video_name": "lvAYFUIEpFI",
        "Q": "i understand the example last done in the video 11:30.i understand the shifting the semi major axis(a) and the semi minor axis(b) but when i tried to graph it was not the same as khan had drawn?",
        "A": "did you check your work?"
    },
    {
        "video_name": "lvAYFUIEpFI",
        "Q": "At 5:40, Sal plots the ellipse at (0, 0), while later on at 12:48, he plots it off the origin. For the first ellipse, why didn't he finish off the equation and solve for x and y? That would place the center of the ellipse at (3, 5), not (0, 0).",
        "A": "Because the standard form of an ellipse is always equal to 1, and if you were to solve for x and y, then it wouldn t be in standard form any more, also, it would become a different shape, I think. Hopefully this helps!"
    },
    {
        "video_name": "lvAYFUIEpFI",
        "Q": "At 6:45 how can be the other side 1 when r2 is coming on the other side?I think it should be zero (0).",
        "A": "Any number divided by itself = 1, not zero 8/8 = 1 -10/(-10) = 1 r^2 / r^2 = 1 Hope this helps."
    },
    {
        "video_name": "lvAYFUIEpFI",
        "Q": "Why do you have to divide X^2+y^2 with r^2? Why do you have to write X^2&Y^2 there(3:06...)?",
        "A": "In order to compare it with the general ellipse formula. where (x^2/a^2 ) + (y^2/b^2) = 1 Hence, a and b are equal so it becomes r^2 (or any other constant). :)"
    },
    {
        "video_name": "lvAYFUIEpFI",
        "Q": "At 4:48,there is an example given of ellipse\nx^2/9 + y^2/25 =1\nif we multiply 225 on both sides we get,\nx^2*225/9 +y^2*225/25 =225\n25*x^2 + 9*y^2 = 225\n(5x)^2 + (3y^2) = (15)^2\nThis in the form of equation of a circle\nbut it is an ellipse also!\nHow?",
        "A": "For standard form of circles, the coefficient is not part of the base of a power and needs to be factored out in order to get a denominator. In addition, the denominators are a result of dividing both sides by the constant in order to get 1. The denominators of circles are equal, and circles are, in a sense, ellipses with eccentricity of zero. Even if you have a coefficient greater than 1, you can still write it as a fraction, e.g. 9x^2 = x^2/ (1/9)."
    },
    {
        "video_name": "lvAYFUIEpFI",
        "Q": "At about 13:00, shouldn't the y direction gone down 2 relative to the center of the ellipse? Sal went to y=-2, instead of y=-1 (2 down from the center of the ellipse) Was this just a mistake?",
        "A": "Yes, it was just a mistake. Like you said, the y direction should go down 2 relative to the center of the ellipse."
    },
    {
        "video_name": "I01O0r8b-po",
        "Q": "at 4:40 wouldn't that be alot harder to do than just doing what he did before?",
        "A": "Just do your best and follow the rules to have fun on Khan Acadamy."
    },
    {
        "video_name": "I01O0r8b-po",
        "Q": "at 4:40, 137 divided by 12 is not exactly 11, its 11.416666666666666666666667",
        "A": "Well I beleive thats whats Sal is writing there 11 5/12 = 11.4166666667 12 fits 11 times into 137 and youre left with a rest of 5/12 and 5/12 = 0.41666666667"
    },
    {
        "video_name": "yIQUhXa-n-M",
        "Q": "at about 2:57 it is 1 over cos y. why cannot we just say 1 of sec y instead of having to write the fundamental square root?",
        "A": "Using the secant doesn t get you to a final solution. The solution need to be in the form of dy = (some function of x) dx If we use the sec y, we still need to find a way to put the answer in the correct form. Using the reciprocal of the cosine makes that easier."
    },
    {
        "video_name": "yIQUhXa-n-M",
        "Q": "At 0:58 why y=sin^-1*x equal to sin*y=x?",
        "A": "Algebra. Starting with y = sin\u00e2\u0081\u00bb\u00c2\u00b9x, if you take the sin of both sides, you ll get sin(y) = x."
    },
    {
        "video_name": "yIQUhXa-n-M",
        "Q": "At 3:35, how did sin^(-2)y become (sin y)^2?",
        "A": "I assume you mean sin^(2)y become (sin y)^2. The two expressions are exactly the same. It s just a convention that most people will usually put the power right after the sin or cos but it doesn t change the expression at all."
    },
    {
        "video_name": "yIQUhXa-n-M",
        "Q": "At 3:10 , what does sal mean by principal root ?",
        "A": "All square roots have two \u00c2\u00b1 solutions: Principle solution is only the + solution."
    },
    {
        "video_name": "yIQUhXa-n-M",
        "Q": "At 1:39 I'm a bit confused why he doesn't just stop with cos y. I thought the derivative of sin x = cos x flat out. I don't know why he is using chain rule on the \"inner function\" of y to then need to multiply by dy/dx.\n\nI'm also not sure why he is getting dy/dx instead of just 1. He is treating it as a function and not a variable. I'm not sure I get that especially given that x is also defined as a function but does equal 1 when differentiated.",
        "A": "This is implicit differentiation; since the derivative is being taken with respect to x, the derivative of y, with respect to x, is dy/dx. If the derivative was with respect to y, then the derivative of sin(y) would be cos(y). But since it is with respect to x, the chain rule must be used: the derivative of the outside function sin(y), which is cos(y), multiplied by the derivative of the inside function y, which is dy/dx. Therefore, the derivative of sin(y) comes out to cos(y)(dy/dx). Hope this helps!"
    },
    {
        "video_name": "xZBbfLLfVV4",
        "Q": "At 6:02 the matrix minor determinant should be positive 3 how come it's negative 5?",
        "A": "(-1 x 1) - (2 x 2) = -1 -4 = -5"
    },
    {
        "video_name": "NWJinKmWzx8",
        "Q": "At 1:52, he appears to be doing the Distributive Property. Can there be other ways to do so?",
        "A": "I don t know can you please tell me? also, amen about the medicine, just the thing, I am a 5th grader in Horton Middle School."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "at 11:10, is 3x4 also a free variable or only the variable right next to the pivot variables?",
        "A": "When you get the matrix to row echelon form, all the leftmost non zero coefficients in each row are pivots. All the other coefficients that are not eliminated in the reduced row echelon form are for free variables."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "11:10 Is x4 in the top equation a free variable?",
        "A": "It is indeed a free variable."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 12:54 is that the span of the matrix?",
        "A": "It seems to me that, yes, it s the span of all the vectors that satisfy these two equations, as Sal goes on to explain. So it s the span of the solution set, which is the same as the span of the augmented matrix. I think the span of that same matrix unaugmented (without the constant vector) might be greater than the span of this (or maybe any) augmented matrix."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "I missed something. Where did he get the X2 and X4 variables at 14:26? Why are these included?",
        "A": "What he s doing implies that the free variables x2 and x4 are on their own x2 and x4 axes of R^4, which I have doubts about. 1) The original 3x4 transformation matrix is from R^4 to R^3. 2) The rref matrix has only 2 rows, which seems to mean there are only x1 and x3 coordinates in the solution."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "at 4:35 or there abouts Sal says 2 - 1 is -1~ seems like it should be 1 and than a few seconds later says 12 - 7 is negative 5, and it should be positive 5. ??",
        "A": "He is replacing the 2nd row with the 1st row - the 2nd row. It is actually 1-2 = -1 and 7-12=-5"
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "at 4:49 you replaced r2 with (r1 - r2). is that legal? shouldnt row 2 turn into (r2 - r1)",
        "A": "Both operations are equivalent, because if you multiply both sides of row 1 - row 2 by -1, you get row 2 - row 1 (try it). a-c = b-d <=> c-a = d-b"
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "@7:00 Sal subtracts the first row from the third, could he also subtract the second row from the first?",
        "A": "Did you mean 5:00? If he subtracted the second from the first, then the second row would have a leading digit before the first row did. However, in RREFs of matrices, the leading digit of a row always has to be before the leading digit of all the rows after it. If you subtracted the second row from the first, then you would get farther away from this rule. I hope this helps!"
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "Can someone explain with a little more details what would be a plane in R4 (17:00) ?",
        "A": "Thanks! Ok, but is the span of the two basis vectors a flat surface in R4, like a flat surface in R3?"
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "at 0:40 int the video sal says a matrices is a rays of numbers, what is a rays?",
        "A": "ar\u00c2\u00b7ray /\u00c9\u0099\u00cb\u0088r\u00c4\u0081/ Noun An impressive display or range of a particular type of thing: a bewildering array of choices ."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "From 14:34 onwards, how did Sal get to the values of X2 and X4 which he plugs into the vector? (I'm not seeing any of those values in the (new) reduced system of equations.) Thanks.",
        "A": "He just assumes that the x2 coordinate in R4 is x2, and the x4 coordinate in R^4 is x4. I am also doubtful about this."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "Hi! Time 14:21. How did you determine X2 and X4?",
        "A": "They can t be determined because there aren t enough equations to determine all the variables. In this video the system reduces to 2 equations in 4 variables, with x1 and x3 both as functions of x2 and x4, and not fixed. Since there are only two equations, two of the variables must depend on the other two, which are independent. Since the pivot variables are now functions of x2 and x4, it s convenient to make x2 and x4 the free (independent) variables (they can be any real numbers)."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 11:23 in the video, is 3X^4 also a free variable?",
        "A": "Yes, he already said x4 is a free variable in the second equation. It s obviously a free variable in the first equation too. (3x4 is not a variable, but I think you meant x4 in the first equation.)"
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "0:07 -0:09 how do you know you're not constraining it enough?\nwhat does \"constraing\" in this case mean?\nplz reply ASAP\nThx!!",
        "A": "If there are more variables than equations (in either the problem or solution set equations), you can t get a fixed (constrained) value for every variable. constrain: limit, restrain, control from Latin constringere to bind together, tie tightly, fetter, shackle, chain, from com with, together (see com-) + stringere to draw tight (see strain (v.)). Related: Constrained; constraining."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "@11:07 isn't 3x4 an independent variable?",
        "A": "Yes, the free variables (x2 and x4 in this example) can be thought of as independent variables and the pivot variables (x1 and x3 in this example) can be thought of as dependent variables that depend on the free variables. Good job on figuring that out!"
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 6:27 Sal multiplies the middle row by -1, but don't you have to multiply all of the rows?",
        "A": "You can replace any row with: 1) it s multiple. 2) Its difference with another row (or with any multiple of another row)."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 5:48 imagine that instead of having a row 3 that looks like [0 0 -2 4] it would look like [0 1 -2 4]. Should we exchange row 2 with row 3?",
        "A": "Yes, it seems that way to me. Then for 1 2 1 1 | 7 0 1-2 4 |-10 0 0-1 2 |-5 I reduced it to 1 0 0 3 | 2 0 1 0 0 | 0 0 0 1-2 | 5 which is x1 = 2 - 3(x4) x2 = 0 x3 = 5 + 2(x4) I think exchanging rows doesn t change the result, but might make the work harder or easier. Exchanging columns however can change which variables are free, and the coefficients in the reduced matrix."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "16:51 shouldn't the solution be a 3d space? And not only a plane? Because we're adding 3 vectors!",
        "A": "No, because we re only adding linear combinations of 2 vectors. One vector is fixed, so it s not treated the same way."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "(4:04) Why do replace the middle row with the 1st row minus the 2nd row? Why don't we just do the 2nd row minus the 1st row? so that the 2nd row is 0 0 1 -2 5?",
        "A": "It would have been faster if Sal did that, but it doesn t really matter."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 8:20, how does he know that he found the reduced row echelon form of the initial matrix? What is the definition of it?",
        "A": "He explained it like 10 seconds after your time stamp."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "@7:55 Sal gets rid of the first 1 don't you have to keep the leading 1?",
        "A": "You don t necessarily have to keep the leading 1. If there are 2 leading 1s in the same column, then you definitely want to get rid of the lower 1 because there can only be one non-zero digit in any column with a 1 that is leading in its row."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 08:50 Sal defined the pivot entry as the only non-zero entries in their columns. Using this definition, the second column would also contain the pivot element 2. After watching more I saw that it makes sense to only call the entries in the first and third column pivot elements, but what's the exact definition of pivot elements then?",
        "A": "The first non zero element in a row of the ref matrix."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 4:27, can someone explain to me why Sal does the first row - the second row",
        "A": "He wants to get rid of as many non zeroes in the matrix as this makes everything much easier to work with. By takin first-second row you get two zeroes in the second row. In other words he wants to create pivot elements to get the thing in reduced row echelon form, see 08:36"
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "@9:47 don't you have to get rid of the first two in the second column to get a one with 2 zeros?",
        "A": "Why would we want to do that? What reason?"
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "can you scale a row by 0? like at 7:00 ?",
        "A": "No. Never. Sal didn t do that. He zeroed out the row by adding a multiple of another row to it."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "@16:53 Does R4 mean 4 demensions?",
        "A": "Yes, R^4 means 4 components in each vector, which could be interpreted as 4 dimensions. Good job on figuring that out!"
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "at 9:02, 2 is also a non zero entry in the second column. Why 2 is not a pivot entry?",
        "A": "Because you can only have one pivot per row. And the pivot in each row (if it exists) is the first non zero number you see on the very far left. 2 in that row is the not the farthest non zero number in that row, 1 is, which is why 2 is not a pivot in that row. If a number is not a pivot in its respective row, then it also is not a pivot in it s respective column."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 11:00, What is pivot variable?\nAt 11:05, What is free variable?",
        "A": "Basically, in reduced row-echelon form, your pivot variables are any variables associated with the columns that contain any leading one s in your matrix. The free variables are the variables associated with the columns that DO NOT have any leading ones."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 11:35, why's 3X4 (the last term in the top equation) not considered a free variable?",
        "A": "X4 is the free variable, which he already circled. Note that he s not circling the coefficients."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "Will this be involved in my high school Maths textbooks? I suppose it's kind of difficult. Also, is it a formula, from 5:04 to 8:35, to reduce row echelon forms, or I can do whatever I want to achieve the goal mentioned around 3:55? Thank you, and your patience is highly appreciated ! :)",
        "A": "There are three row operations: (i) Swap any two rows (ii) Multiply a row by a scalar (iii) Add a row to a scalar multiple of another It is usually introduced at the end of pre-calculus. If you do not get to it in high school pre-calc, it will definitely pop up in your college pre-calc course."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 9:54, Sal says the leading entry of the second row, the value 1, is the only non-zero entry in that row. This is in a row with: 0 0 1 -2. So my question is, why's -2 not considered one of the non-zero entry (as it isn't a zero)?",
        "A": "I think he misspoke there."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At around 16:30. I thought a plane had to be in R^3. Or can a plane be created with two any non Zero non co-linear vectors in R^n?",
        "A": "A plane can be created as the span of any two LI vectors in R^n. They still exist in the 4th dimension, it s just harder to visualize how that works."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 11:10 he points out that free variables are those variables that are not associated with pivot variables, but only makes a circle around x2 in first equation and x4 in second equation. Isnt x4 in first equation a free variable?",
        "A": "Yes, at 11:07 he specifies which are the pivot variables (x1 and x3) and which are the free variables (x2 and x4); he just didn t circle the x4 in the first equation."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 16:20 since we are summing the vectors 'a' and 'b', shouldn't the graphical representation had the 'b' vector's tail be on 'a' vector's head? instead of both 'a' and 'b' tails being on the position vector [2 0 5 0]?",
        "A": "Yes, you re right. We have here vector addition, which is always head to tail (in any order). The drawing is confusing. You cannot add a to b as drawn."
    },
    {
        "video_name": "L0CmbneYETs",
        "Q": "At 11:19 , why isn't 3x4 also a free variable?",
        "A": "It is, x3 and x4 are free variables in both equations. 3x4 and -2x4 are both the variable x4 but with different values on the constant."
    },
    {
        "video_name": "REiDXCN0lGU",
        "Q": "First of all, I have two questions. One, whenever Sal was doing the example in 2:00 I was wondering why did Sal have to 7x^0 because I was confused by that. And second question. When he talking about the term of coeffiecient, I didn't quite understand the definition he explained. can someone please help me understand what it means?",
        "A": "Yeah it does. Thanks so much. :)"
    },
    {
        "video_name": "REiDXCN0lGU",
        "Q": "at 1:44, why over complicate things and say it's x^0 which is equal to 1. Therefore, 7 is a coefficient? thanks",
        "A": "It may seem like an over complication now but this definition explains: .... why the 7 is by itself , .... the adoption of the convention to not write out the x^0 term, .... why the lone term is still a coefficient. Once you get into the calculus you will have a better understanding and appreciation for the rigorous definitions that mathematics is built upon, for they give us the power to be sure of our conclusions."
    },
    {
        "video_name": "REiDXCN0lGU",
        "Q": "@ 3:51, Sal showed us what polynomials were but never what they weren't what are examples of non polynomials?\nThanks\nSummer :)",
        "A": "A polynomial is a group of multiple numbers, or (for example) three terms. So the only think I can think of that isn t a polynomial would be something like 1 or 5x."
    },
    {
        "video_name": "REiDXCN0lGU",
        "Q": "At 1:50, is 7 commonly accepted as a term?",
        "A": "Yes, as all other numbers are."
    },
    {
        "video_name": "REiDXCN0lGU",
        "Q": "On 2:25, how is 7x = 7x^0? Isn't any number to the 0th power 1? Shouldn't this be 7x^1?",
        "A": "It is not that 7x = 7x^0. Sal has written that 7 = 7x^0 Since, x^0 = 1 Thus, 7x^0 = 7*1 = 7"
    },
    {
        "video_name": "REiDXCN0lGU",
        "Q": "if a term isn't showing an exponent (1:20) do you still have to create one and then identify it?",
        "A": "If a term does not have an exponent, then the exponent is assumed to be 1."
    },
    {
        "video_name": "REiDXCN0lGU",
        "Q": "My math book states that a constant term in a polynomial does not have a coefficient, because there is no variable that is multiplied to it. But in the video around 1:35, Sal says that the constant can be viewed as 7*x^0. I see the reasoning behind both of the ideas, but i am not sure which is correct. Are both ways acceptable? If not, which one is correct?",
        "A": "Both ways really are agreeable notions which is why Sal said that you could view this as such."
    },
    {
        "video_name": "REiDXCN0lGU",
        "Q": "Around the 1:45 mark, Sal mentions that the 7 could be listed as a Coefficient. I've always understood that the number at the end without a variable was a constant., not a coefficient. Which is it?",
        "A": "Well you just confused yourself. A CONSTANT is a number without a variable. A COEFFICIENT is a number with the variable on the end but when writing the coefficient you only write the number not the variable."
    },
    {
        "video_name": "REiDXCN0lGU",
        "Q": "at 1:30isnt it x over1?",
        "A": "Remember: any number over 1 is that number. this same rule applies to variables. in this equation, x over 1 is the same as x. and anything to the 0 power equals 1. (zero is a controversial number that can t be determined for sure that it is equal to 1; many times it is determined to be undefined ). hope this helps! :-)"
    },
    {
        "video_name": "REiDXCN0lGU",
        "Q": "At 2:24 Sal said that the exponent on 7 was 0. Wouldn't 0 be the exponent that is being applied to x?",
        "A": "Yes, it would be. I m thinking he was saying that 0 was the exponent of the term that included 7, and not the exponent of 7."
    },
    {
        "video_name": "eu_GFuU7tLI",
        "Q": "At 6:00, the partial derivative of psi with respect to x of x^2/y is just 2xy, not the extra plus x^2*y'. Partial derivative means to take the derivative of one variable in a multivariable situation, holding the other variables constant. I'm not sure why there is the extra x^2*y' part. Thanks",
        "A": "He s not taking the partial derivative. He s just doing regular differentiation (using the chain rule) and treating y as a function of x. Hope that helps."
    },
    {
        "video_name": "eu_GFuU7tLI",
        "Q": "At 06:25 when we differentiate psi w.r.t X, why can't we take derivatives of y terms as zero directly ? Is it because when psi is function of both X and Y then we can't put derivatives of Y terms w.r.t. x as zero...Can u pls help me get conceptually cleared ? Thanks a ton Sal !",
        "A": "implicit differentiation."
    },
    {
        "video_name": "eu_GFuU7tLI",
        "Q": "At 5:29, shouldn't Sal have said \"product rule\" instead of \"chain rule\"?",
        "A": "you re not even sure of yourself."
    },
    {
        "video_name": "eu_GFuU7tLI",
        "Q": "why at 1:30 are you dividing by dx? in my textbook we have M(x,y)dx+N(x,y)dy=0... not sure why you're doing this... thank you!",
        "A": "He is dividing by dx because the standard form for solving exact equations is M(x,y)+N(x,y)y =0. y can also be written as dy/dx. So before solving this equation, it first must be in its standard form which means he must divide the entire equation by dx."
    },
    {
        "video_name": "eu_GFuU7tLI",
        "Q": "Can't you just separate the equation at 0:33?",
        "A": "Yes you can but here Khan is just trying to show how to use the Exact Equation approach."
    },
    {
        "video_name": "eu_GFuU7tLI",
        "Q": "at 5:38 why are we even using the product rule? when I put d/dx(x^2*y) into Wolfram Alpha it gives me 2xy as the answer. This is very confusing to me.",
        "A": "d/dx(x^2*y) treats y as a constant, so 2xy is the answer."
    },
    {
        "video_name": "eu_GFuU7tLI",
        "Q": "At 1:04, Sal divides the differential equation by dx. But, I thought that derivatives such as dy/dx are not supposed to represent fractions? Are we actually dividing dy by dx to get dy/dx or is there a more subtle meaning and Sal is using an abuse of notation?",
        "A": "dy/dx is a fraction. dx is just an infinitesimal change in x, while dy is an infinitesimal change in y. dy/dx is just the same thing as \u00ce\u0094y/\u00ce\u0094x, as \u00ce\u0094x approaches 0. There is no abuse of notation. This is actually what the derivative means."
    },
    {
        "video_name": "eu_GFuU7tLI",
        "Q": "At 5:29, isn't he using the product rule rather then the stated chain rule?",
        "A": "Well, he s using the product rule to differentiate x^2y as well as the chain rule to differentiate y which is a function of x."
    },
    {
        "video_name": "uC09taczvOo",
        "Q": "1:35 Does it matter if you say one fourth, or are you supposed to say one quarter?",
        "A": "One fourth and one quarter are technically the same thing when talking about fractions, so you can use either."
    },
    {
        "video_name": "uC09taczvOo",
        "Q": "0:47 something cant be more negative",
        "A": "0:47 -1.2 is more negative than -1, because it is farther away from zero on the number line."
    },
    {
        "video_name": "uC09taczvOo",
        "Q": "At 0:27 he says three thirds is one but it isn't quite one, is it?",
        "A": "Three thirds is exactly one. Think about it like a pizza: If you cut a pizza into three equal pieces then the three pieces (each are one third of the pizza) are equal to one whole pizza"
    },
    {
        "video_name": "R-2Uw10QgIo",
        "Q": "At 3:45, Sal said that the integral of 1/u was the absolute value of ln u. Why did Sal put the absolute value sign?",
        "A": "You cannot take the natual logarithm of a negative number."
    },
    {
        "video_name": "YclQE2XtaLw",
        "Q": "At 06:03: Why is p within 0.43+ or - 1 also approximately 95 %?",
        "A": "Because the +/- 1 comes from the 95% rule, that says that all members of the sample (or a population) is within the +/- 2 * standard deviation of the mean. So 0.43 is the mean and the 1 is the 2 * standard deviation."
    },
    {
        "video_name": "YclQE2XtaLw",
        "Q": "At 3:41 If you don't have the standard deviation, do you need to figure that out?",
        "A": "To calculate SEM we use sample s since we have it and we don t have population s (sigma). So this reduces our accuracy."
    },
    {
        "video_name": "bhrmEy0wvD0",
        "Q": "When evaluating the last integral at time 2:46, shouldn't it be positive (1/3)sin(theta) because the negatives cancel out? I know that this doesn't change the answer because either way that term is zero but just checking.",
        "A": "Nope! You re integrating cos and not differentiating cos. d(sin(\u00c3\u00b8))/d\u00c3\u00b8 = cos(\u00c3\u00b8). d(cos(\u00c3\u00b8))/d\u00c3\u00b8 = -sin(\u00c3\u00b8). So there is no negative from the integration of cosine :)"
    },
    {
        "video_name": "XyDMsotfJhE",
        "Q": "3:35 Why do we care about the x and y value that makes the expression 0? This is the only part I don't get in this video.",
        "A": "One way to think of it is that the equation is describing the route you would take to get from the center of the circle all the way back to the origin, or (0,0). So a circle centered at (5,-2) with a radius of 5 would have the equation (x-5)\u00c2\u00b2 + (y+2)\u00c2\u00b2 = 25. What it ultimately means is that, given any point (x,y) that falls on this circle, (x-5)\u00c2\u00b2 + (y+2)\u00c2\u00b2 = 25. This is the Pythagorean description of a circle."
    },
    {
        "video_name": "XyDMsotfJhE",
        "Q": "3:49 so we don't do anything with the exponents?",
        "A": "Yes because it represents the distance formula or pythagorean theorm: a^2+b^2=c^2"
    },
    {
        "video_name": "XyDMsotfJhE",
        "Q": "Can someone just explain what he did at 1:20 and on, because I really don't want to go watching an entire playlist just to understand what he did. (If there is any fast way of explaining it of course)",
        "A": "all he did was complete the square. take the number next to the second variable (the one that is not squared), and divide it by two, then square that number. Ex. x^2+8x+__=0 for this, the blank is 16. However when we put sixteen in the blank, we are forced to add another 16 to the other side of the equation to balance. The answer would be (x+4)^2=16"
    },
    {
        "video_name": "XyDMsotfJhE",
        "Q": "wait at 2:30, why is it (y-2)^2? shouldn't it be (y+2)^2",
        "A": "If you expand (y - 2)^2, it would be (y - 2)(y -2), which would equal y^2 - 2y - 2y + 4, or y^2 - 4y + 4, which is what Sal originally had. Hope that helped!"
    },
    {
        "video_name": "XyDMsotfJhE",
        "Q": "at 1:44, what would you do if you had the equation x^2 + y^2 + 8x + 2y - 8 = 0?",
        "A": "Hi!, You would do the same thing. Group the x s and you get (x^2+8x+16)+(y^2+2y+1)=8+16+1. That s basically it. Hope it helps :)"
    },
    {
        "video_name": "XyDMsotfJhE",
        "Q": "At 0:58, why isn't it minu y^2 when the minus sign is in front of the y^2?",
        "A": "Actually it s a + , Sal just circled it so that the circle goes right on the vertical bar of the + sign xD"
    },
    {
        "video_name": "loAA3TCNAvU",
        "Q": "At 3:26 how did Sal get the 3 ?",
        "A": "The length of that whole right side of the figure is given at the start as 6.5. The length of the left side is given as 3.5 at the start of the video (figures in white). If that whole side is 6.5 and we know part of it is 3.5, we know that the leftover part must be 3 to add up to 6.5."
    },
    {
        "video_name": "loAA3TCNAvU",
        "Q": "video time: 3:16, I don't understand.",
        "A": "If the entire right side is 6.5, and we know that a certain part of it is equal to 3.5, then we can use subtraction to figure out the remaining length (6.5-3.5=3). Once we know that remaining section, we can find the area of the bottom triangle using 9*3 and divide that by 2."
    },
    {
        "video_name": "loAA3TCNAvU",
        "Q": "In 1:43 I am confused where he got 1/2 x 7 is 3.5 how is that possible?",
        "A": "Multiplying something is the same as dividing it by 2, so 1/2 \u00c3\u0097 7 is the same as 7 \u00c3\u00b7 2. 2 goes into 7 three and a half times, so 1/2 \u00c3\u0097 7 = 3.5."
    },
    {
        "video_name": "loAA3TCNAvU",
        "Q": "Why can't you just find the area of the trapezoid at 3:05, instead of splitting it into two different shapes? Like you can use the formula A= [a+b]/2 x h to find the area of a trapezoid.",
        "A": "That also works, but that formula is based on the shapes of two triangles. See if you can figure out how? (also put some brackets around a+b)"
    },
    {
        "video_name": "loAA3TCNAvU",
        "Q": "At 3:10 - 3:25 Sal had to find the distance of the bottom portion of the weird shape, how did he figure that out?",
        "A": "Sal added together 2 and 7 to find the height of the bottom-most triangle."
    },
    {
        "video_name": "loAA3TCNAvU",
        "Q": "How did he get 3 at 3:24",
        "A": "The whole line is 6.5, and the top part is 3.5. So the bottom part must be 6.5 - 3.5 = 3."
    },
    {
        "video_name": "loAA3TCNAvU",
        "Q": "In 1:43 I am confused where he got 1/2 x 7 is 3.5 how is that possible?",
        "A": "Well, multiplying something by 1/2 is the exact same as dividing it by 2. 1/2 of 7 is 3.5, since seven is an odd number, it won t have a whole number when it is divided by 2, or should I say, multiplied by 1/2. For example, the standard formula for the area of a triangle is the base multiplied by the height and then multiplied by 1/2, but most people like to just say divide by 2."
    },
    {
        "video_name": "loAA3TCNAvU",
        "Q": "At 3:34, Sal mentioned that the 6.5 (long side) side had to be divided by two which then he got the top half as 3.5 and the lower half at 3. I did my own calculations and divided 6.5 by 2 and 3.25. So technically, are Sal's calculations incorrect?",
        "A": "That was confusing and reading your post made me think but I believe he s taking the numbers away rather then dividing? Hope it helps, Lili"
    },
    {
        "video_name": "loAA3TCNAvU",
        "Q": "I don't understand what it is talking about in 1:19",
        "A": "basically he is splitting the shape into separate parts. he was calculating the triangle. if you know how to calculate a triangle it is base * height / 2 but the way he does it is 1/2 * base& height"
    },
    {
        "video_name": "loAA3TCNAvU",
        "Q": "At 3:33, how can you find the measurement of that triangle with the given information. You don't know for sure that it is a right triangle, to use the formula. How can you assume that?",
        "A": "We know that it was a right triangle because of the square near the bottom vertex of the triangle is used to indicate that it is a right angle triangle."
    },
    {
        "video_name": "loAA3TCNAvU",
        "Q": "At 4:12, is it possible to just simply half the rectangle?\nbecause a triangle is half of a rectangle",
        "A": "yes, i think that is what he is doing. *1/2 is splitting it"
    },
    {
        "video_name": "loAA3TCNAvU",
        "Q": "At 1:14, he multiplies the area by 1/2 and then by two. But why two? Where does the two come from, what does it do? I thought to find the area of a triangle, you just multiplied by 1/2?",
        "A": "It appears that the triangle has a base of 2 and a height of 3.5, so the 2 is not to cancel the 1/2, it is just putting in b and h into the equation A = 1/2 bh."
    },
    {
        "video_name": "Ly86lwq_2gc",
        "Q": "At 7:03 why did he multiply square root of 3 over square root of 3 with the other fraction? Why does it make sense to do that? Thanks!",
        "A": "There is a preference of not having a radical in the denominator - such as \u00e2\u0088\u009a 3 in 1/(3\u00e2\u0088\u009a 3). To not have a square root in the denominator, one can multiple both the top and bottom by the radical value ( \u00e2\u0088\u009a 3 in this case). It is safe to do this because multiplying by ( (\u00e2\u0088\u009a 3) / (\u00e2\u0088\u009a 3) ) is multiplying by one (identity property of multiplication)."
    },
    {
        "video_name": "Ly86lwq_2gc",
        "Q": "At 3:42, what does \"SAS\" mean?",
        "A": "It means that you have one side that has a similar ratio to a corresponding side on another triangle, an included angle that is congruent to the corresponding angle on the other triangle, and another side that has the same similarity ratio as the fist side to its corresponding side."
    },
    {
        "video_name": "Ly86lwq_2gc",
        "Q": "At 6:05, Sal said that the longest angle of the larger triangle was 18 square roots of 3. But isn't 27 larger?",
        "A": "No, 18 root 3 = 31.1769..., which is greater than 27, because root 3 is slightly greater than 1.7 and 18 times 1.7 is 30.6"
    },
    {
        "video_name": "Ly86lwq_2gc",
        "Q": "at 5:02 wouldn't they both share the vertex D ratherb then C :'o?",
        "A": "Yes and no it depends on the school and teacher..."
    },
    {
        "video_name": "Ly86lwq_2gc",
        "Q": "At 5:50, why is 27 < 18 square roots of 3? How did Sal come to that conclusion?",
        "A": "If you know that the \u00e2\u0088\u009a3 is a little bigger than 1.5 (1.7_____), then number sense would say that since 18 \u00e2\u0080\u00a2 1.5 = 27, this relationship must be true."
    },
    {
        "video_name": "Ly86lwq_2gc",
        "Q": "At 6:11 would that be considered an \"improper\" fraction because there are square roots in the denominators?",
        "A": "You can rationalize the denominator by multiplying by rad 3/rad 3"
    },
    {
        "video_name": "Ly86lwq_2gc",
        "Q": "At 6:06 isn't 27 the longest side of the triangle?",
        "A": "No, 18 root 3 = 31.1769..., which is greater than 27, because root 3 is slightly greater than 1.7 and 18 times 1.7 is 30.6"
    },
    {
        "video_name": "Ly86lwq_2gc",
        "Q": "I'm having trouble digesting @ 6:43.\n\n1 / 3rad3 = rad3 / rad3, they don't seem equal to me. I understand that multiplying by one normalizes it but isn't this an equality problem?\n\n1 / 3rad3 = .6933\nrad3 / 9 = .1924 they are not equal\n\nI am missing something but I can't see it...",
        "A": "To figure out if the pair of triangles are similar, you need to determine if the ratios of the sides are equal. Sal multiplies 1/3sqrt3 by sqrt3/sqrt3 (he does not equate them) to get rid of the sqrt in the denominator, making it sqrt3/9."
    },
    {
        "video_name": "Ly86lwq_2gc",
        "Q": "At 8:38, he says that we have SAS. That means that the triangles must be congruent, so if two triangles are congruent aren't they also similar? If not, why?",
        "A": "SAS is the name for both a congruency postulate and a similarity postulate, but the postulates themselves are different, even though they share a name. He is referring to the similarity postulate of SAS."
    },
    {
        "video_name": "Ly86lwq_2gc",
        "Q": "I didn't really get it at 2:24 can anyone help me?",
        "A": "For the triangles to be similar, the ratios or proportions of the sides of one triangle have to be proportional to the ratios of the sides of the other triangle."
    },
    {
        "video_name": "kAm9x3uWF9I",
        "Q": "he finishes the problem at 1:49, and it is 7/100\n(the answer)",
        "A": "The video keeps going honey. Then the 7/10 represents how plus 4/10 also equivalent to 40/100 gives you the same answer as the right-hand side. The only difference is you are reading this math problem from left to right rather right to left."
    },
    {
        "video_name": "6m642-2D3V4",
        "Q": "I don't get how you get the answer (3,0) at 2:01",
        "A": "So, when Sal was working out the equation, he plugged 0 in for y. 0=3x-9. From here like he did, you add 9 to the other side and divide by three. This gives you the x coordinates because you plugged zero in for y. If you plugged zero in for x, then you would be solving for y. Hope this helps."
    },
    {
        "video_name": "6m642-2D3V4",
        "Q": "On 2:57 and a little before, how does Sal Khan get -9? I didn't really get it.",
        "A": "At that point in the video, Sal is finding the Y-intercept. This is the point on the Y-axis where the line crosses that axis. This point will always have an X-value = 0. So, he substitutes in X=0 and solves for Y. The equation is: y = 3x - 9 -- Substitute x=0: y = 3(0) - 9 -- Multiply: y = 0 - 9 -- Subtract: y = -9 So, the y-intercept is at: (0, -9) Hope this helps."
    },
    {
        "video_name": "6m642-2D3V4",
        "Q": "In 0:13 he says \"where it intercepts the X Axis\".\nSo it HAS to cross the x/y axis?\nOr can it be used for any point at all?",
        "A": "By it do you mean this method? If so then not exactly, for instance, if you have an equation like 3y=2, this equation would be a horizontal line and would not cross the x-axis only the y-axis. And all lines on a graph cross either the x or the y-axis eventually."
    },
    {
        "video_name": "YBYu5aZPLeg",
        "Q": "At 0:57 what is delta x?",
        "A": "Delta x is the change in x. The line changes from 0 to 2, so the difference between 2 and 0 would be 2. Thus, delta, which means change in, would equal 2."
    },
    {
        "video_name": "YBYu5aZPLeg",
        "Q": "So at 0:35 when he's labeling the y-intercept, is that what you're supposed to do first when graphing a two-variable inequality?",
        "A": "Sal is not graphing the inequality in this video. He is finding the linear inequality that matches the graph. To find the inequality, you need 2 pieces of info: 1) The slope: this can be calculated using any 2 points on the line. 2) A point on the line. The points do not have to be the y-intercept. Hope this helps."
    },
    {
        "video_name": "YBYu5aZPLeg",
        "Q": "At 1:47 what is delta?",
        "A": "Delta means change. change in y or x is the same as saying delta x or delta y."
    },
    {
        "video_name": "YBYu5aZPLeg",
        "Q": "1:46 what is delta",
        "A": "The greek letter delta (a triangle) is often used as an abbreviation for the words change in . Sal is using it for change in Y / change in X . Hope this helps."
    },
    {
        "video_name": "YBYu5aZPLeg",
        "Q": "At 0:26, Sal says \"intercept\", what exactly does that mean?",
        "A": "The point it crosses that axis. The y-intercept is the place where it crosses the y-axis, the x-intercept is where it crosses the x-axis, the z-intercept..."
    },
    {
        "video_name": "JvDpYlyKkNU",
        "Q": "at 1:25, how did he know which letters to minus the other letters? In other words, why couldn't it be h-x, and k-y?",
        "A": "Following the order x-h and y-k (rather than the other way around) was perhaps somewhat arbitrary at the beginning (when these formulas were first developed), but this is now the standard convention. Conventions like these are useful because they give us all a common language (so to speak), but then we are obliged simply to learn them and follow them (especially when communicating with others)."
    },
    {
        "video_name": "JvDpYlyKkNU",
        "Q": "at 3:25, why wouldn't r be -7 instead of 7, its the same either ways, right?",
        "A": "The radius represents the distance from the center to the edge of the circle. Distances are never negative: the distance from your bed to your bathroom is the same as the distance from your bathroom to your bed - because it involves the same amount of movement."
    },
    {
        "video_name": "JvDpYlyKkNU",
        "Q": "@2:49, Sal said and wrote \"(x--3)^2\" but didn't he mean \"(x-3)^2\"?",
        "A": "No, he explicitly says: Instead of x plus 3 squared, we can have x minus NEGATIVE 3 square . That s because (x + 3)\u00c2\u00b2 = (x - -3)\u00c2\u00b2, since by the laws of sign -(-3) = 3"
    },
    {
        "video_name": "JvDpYlyKkNU",
        "Q": "why did he make the 3 negative at 3:00? That totally confused me",
        "A": "The circle is being translated 3 units to the left which is where the negative -3 and value k=4 for the case of (y-4)^2). The reason being is (x+3)^2 needs value of x, 3 less to retain the same position compared to just x^2. Try taking an another look at the video"
    },
    {
        "video_name": "JvDpYlyKkNU",
        "Q": "I am confused at 3:33. Why isn't H and K 3 and -4, respectively?",
        "A": "Look at the circle. Where is the center?"
    },
    {
        "video_name": "wQXYtsyfbqg",
        "Q": "At 3:03 I don't understand the process of getting 2/3x^3/2 from the square root of x, could you please explain?",
        "A": "The square root of x is the same as x^(1/2), so use the reverse power rule to add one to the power, which turns it into x^(3/2), and divide it by the fraction, which is the same as multiplying by the reciprocal, which in this case is 2/3, and turns the whole thing into (2/3)x^(3/2)"
    },
    {
        "video_name": "wQXYtsyfbqg",
        "Q": "How were you able to get the equation you had at 2:05 and how did you substitute all the numbers in that equation.",
        "A": "The equation at 2:05 was derived from the second fundamental theorem of calculus. Instead of going from one expression to the difference of two expressions, he went from the difference of two expressions to one expression."
    },
    {
        "video_name": "xw6utjoyMi4",
        "Q": "Where are the conditional probability videos that he mentions at 2:00?",
        "A": "Conditional probability refers to the probability given that another event has happened. So all of the equations where you see the given line: | are calculations of conditional probabilities."
    },
    {
        "video_name": "xw6utjoyMi4",
        "Q": "While watching this with captions I seen it say [UNINTELLIGIBLE] pls tell me what that means and why its here thanx :D fyi the time is 4:52",
        "A": "It means the person talking is mumbling or making noises and it s impossible or very difficult to hear what they re saying."
    },
    {
        "video_name": "xw6utjoyMi4",
        "Q": "At 13:22 Sal divides ((15/64)*1/3) by 0.24165 which is P(4/6H|U)but at 16:10 he is talking about the whole P(4/6H)as P(4/6H|F)+P(4/6H|U). Should not be the denominator P(4/6H)=0,24165+0,23437=0.47602. Then the result of the whole example should be ((15/64)*1/3)/(0.47602)=0.1641 (16.42% not 32.3%)? Am I wrong? Thank you for answer.",
        "A": "Actually, at 10:56 Sal calculates the value for P(4/6 H | U) and finds it is 16.384%. At 11:34 Sal explains that 0.24165 is the probability -- not knowing which coin I picked out -- that s the probability of getting four out of six heads."
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "1:58 shouln't the derivative of 2sinx be (sinx-2cosx)?",
        "A": "No, thinking in terms of the chain rule, which states d/dx [f(x)g(x)]= [f (x)*g(x)]+[g (x)*f(x)], the application here can be seen in this way f(x)=2 g(x)=sinx; f (x)=0! g (x)=cosx so putting that together we get: (0*sinx)+(2)(cosx)=2cosx (as stated in the video). Hope that helps! :)"
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "I am just wondering but isn't the derivative of sinx: - cosx because if so at 7:43 the answer would be 0/2 but I don't know if that changes anything.",
        "A": "You are probably getting confused because the derivative of cos X = -sin X"
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "Is it certain that i will reach an end in this procedure? Sal tells us to use it AGAIN in 5:30, hehe. If i get stuck in an infinite loop, how do i know?",
        "A": "@Zachary: No, that s incorrect in this case. There are no x terms left that will decay; they are all inside the trig functions. We can keep taking the derivatives and it ll never run out. For instance, let s say we had sin(x)/cos(x), and took the derivatives of them separately (so no quotient rule will be used!). First derivative: cos(x)/-sin(x) Second: -sin(x)/-cos(x) Third: -cos(x)/sin(x) Fourth: sin(x)/cos(x) Fifth: cos(x)/-sin(x) ... etc, etc."
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "At 0:10,what does he mean by \"limit\"?",
        "A": "limit means turn this variable into : something"
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "At 1:58, when finding the derivative of 2sin(x), 2 is just a constant term. Why does it appear again in the derivative?",
        "A": "2 wasn t just transferred down, it was regenerated by the product rule . To prove it, I ll work out the derivative of 2sin(x) 2sin(x) :: Pull this equation into two parts, the 2 and the sin(x). ( 2 )( sin(x) ) :: Now find the derivative of both using the product rule . (0 * sin(x)) + (cos(x) * 2) :: The derivative of a constant is 0, so 2 is now 0. 2cos(x) :: We arrive at our answer. If you re still confused, review the product rule some more!"
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "At 5:25 why don't you just use double angle formula on sin2x and then divide both numerator and denominator by sinx?",
        "A": "If you do that you will get the same answer: 6 Of course, Sal is demonstrating L Hopital s rule in this video so he d probably rather show it that way. Just Keith: I thought you could do a divide by zero like this in a limit problem (only) because we are looking for the value approaching zero but NOT at zero. And I think Sal has done it in other videos. Am I mistaken?"
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "Sorry for this question but why does he say at 7:42 that \"this limit has to also be equal to 6\"?",
        "A": "how many times Im able to use LhopitalS rule?"
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "At 5:23, Sal still wants to use L'Hopital's rule, even if he's taken the derivative twice. Is there a limit as to how far L'Hopital's rule can go? What is the maximum amount you can take before either giving up or finding out what the limit is?",
        "A": "There is no limit. L H\u00c3\u00b4pital s Rule says if plugging in the limit of f(x)/g(x) gives the right indeterminate form, then lim f(x)/g(x) =lim f (x)/g (x) . So if plugging in the limit gives the right indeterminate form for f (x)/g (x), then l H\u00c3\u00b4pital s Rule applies. It doesn t matter that f (x)/g (x) is already a derivative. We can create a chain of this limit equals this limit equals this limit equals... as long as we want."
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "7:19 to 7:35: I didn't completely understand when you say the limit is 6 for all of it. My question may be wrong but I am confused by the fact that how the 3 order derivative of two functions f(x)'''/g(x)''' is equal to the original function f(x)/g(x).",
        "A": "I can relate to your confusion. He is evaluating the limit of the function, not the actual function or its derivatives. The L Hospital s derivative operation(s) is merely a technique to resolve undifined results, 0/0 or \u00c2\u00b1\u00e2\u0088\u009e/\u00e2\u0088\u009e. The proof of L Hospital s method is based on the mean value theorum."
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "at around 2:54 ish he shows the derivative of sine2x is 2 cos 2x and the derivitive of 2 sinx is 2 cosx. i haven't been on the academy in a while so please forgive me, but where did he pull that 2 in front of the cos 2x from?",
        "A": "the chain rule"
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "Is Sal taking the second derivative to use for the rule? Because right at 5:26 there is the original f(x) over g(x) (2sinx-sin2x/x-sinx) and then he takes the limit as x approaches 0 of the derivative (2cosx-2cos2x/1-cosx) and that equals 0/0, so he takes the limit of the second derivative. Is he using the second derivative because the first derivative provided a zero answer? Can you keep differentiating the functions untill you get a non-zero answer that you can use as L for the rule?",
        "A": "Yes."
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "Could someone explain what he did at 4:40? I get that he took the derivative, and that the derivative of cosx=-sinx, but how did he get the 4 in front?",
        "A": "you find the derivative of cos(2x) with the chain rule : it s the product of the derivative of the intern function by the derivative of the extern function : d/dx[cos(2x)] = d/dx[2x]d/dx[cos](2x) = 2 * -sin(2x) So, d/dx[-2cos(2x)] is -2 * d/dx[cos(2x) = -2*2-sin(2x) = 4sin(2x)"
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "why wasn't the quotient rule used to to find the derivative at 2:00.",
        "A": "because we don t want the derivative of the quotient but the quotient of the derivatives"
    },
    {
        "video_name": "BiVOC3WocXs",
        "Q": "I'm having a hard time with this. Am i correct when saying that you should use the chainrule when dealing with for example: sin2x (1:58) or 2cos2x (4:05)? That's how i got the answers eventually, but i'm not sure why I shouldn't use the product-rule, since it's 2.cos2x... Can anyone give me the intuition for this?",
        "A": "Yes, in both parts you are correct. You do take the chain rule of sin2x and 2cos2x, whixh is how he got 2cos2x (co2x times 2) and -4sin2x(-sin2x times 2 times 2). You re also correct about using the product rule for 2cos2x, for if you do it, it would yield -4sin2x. But what you can also do (and what sal did) was leave out constant terms that are in front of the function, like 2(cos2x); then take the derivative of the function, and put the constant back in front. It doesn;t work for variables though."
    },
    {
        "video_name": "FVSgVMVZZ-4",
        "Q": "At 6:22, Isn't using SSA not like \"right\" to use? I mean, It's not a postulate so I would think that C is not what they wanted.. right?",
        "A": "C is not correct. SSA can only be used if the A in SSA is obtuse. If the angle is right, then RSH is the proper postulate."
    },
    {
        "video_name": "0Dd-y_apbRw",
        "Q": "I'm confused. At 2:06 wouldn't it be 8.57 \u00c3\u0097 10^(-4)? Or is there a rule that you don't count the zero before the decimal?",
        "A": "Good question ! If you look very closely we did count the zero before the decimal ! So if you have a number like 0.002 we write 2*10^(-3), since there are three zeroes before the 2 and if we have 2000.0 we write 2*10^(3) since there are three zeroes behind the 2."
    },
    {
        "video_name": "0Dd-y_apbRw",
        "Q": "I don't understand 10:50\n\nHow can you write 10^11X(10^-6)^-1\n\nWhere does -1 come from ? PLease help ! Thanks",
        "A": "He is using the fact that 1/(x^1) = x^-1 Therefore (10^11)/(10^-6) = (10^11) * 1/(10^-6) = (10^11) * ((10^-6)^-1) = (10^11) * (10^6) = 10^(11+6) = 10^17"
    },
    {
        "video_name": "0Dd-y_apbRw",
        "Q": "At about 9:00, couldn't you turn the scientific notation of 20.48 x 10(to the fifth power) into 2048000.0 and then made that into scientific notation of 2.048 x 10(to the sixth power)?\n\nIf what I am talking about makes sense.",
        "A": "Yes, your method works as well."
    },
    {
        "video_name": "0Dd-y_apbRw",
        "Q": "At 11:37 do the minus signs cancel each other out?",
        "A": "Yes. Subtracting a negative is the same thing as adding a positive. The two negatives really just cancel each other out, and so you can write it as a positive instead, which is what Sal did in the video. Nice observation."
    },
    {
        "video_name": "0Dd-y_apbRw",
        "Q": "At about 1:27 he says arbotory or like something what does it mean",
        "A": "(of a constant or other quantity) of unspecified value."
    },
    {
        "video_name": "0Dd-y_apbRw",
        "Q": "At 11:00, Sal did a calculation (off to the upper right), and he added a -1 to the equation. Either he didn't explain why, or maybe I didn't catch the explanation, but I don't understand where that -1 came from. What was it for? Where did it come from?\n\nThanks in advance for any help with this.",
        "A": "When the 10^(-6) is raised up to the numerator, the sign of the exponent changes. This is just following the properties of negative exponents. 1/10^(-6) = 10^6/1 = 10^6 Sal achieved the same thing, but he showed the change in the exponent by applying the -1 exponent (-6 * -1 = 6) Hope this helps."
    },
    {
        "video_name": "AXqhWeUEtQU",
        "Q": "At 6:57, when you do the partial derivative with respect to y, why did you plug in 1 for the x first? I am confused ...",
        "A": "It s because hes doing the differentiation in the specific coordinate case of x=1 y=2. He has just substituted the value 1 in place of the x."
    },
    {
        "video_name": "vlSEpdWf2hQ",
        "Q": "At 1:19, what is that graphing option on the left? How did he know to click the option on the right? Would it have made any difference had he chosen the other one?",
        "A": "The one he has chosen is for exponential function, and the other one is for logarithmic function."
    },
    {
        "video_name": "cDpBtkU2cf8",
        "Q": "4:40 what is he talking about",
        "A": "he is just saying that u have another way or u can switch it around."
    },
    {
        "video_name": "cDpBtkU2cf8",
        "Q": "! = down\n\nin 2:29 he can also do\n>!\n>!\n>!\n>!\n\nand it will still be 12",
        "A": "Well, he is only trying to explain the different way using which you can multiply. All of them are twelve."
    },
    {
        "video_name": "cDpBtkU2cf8",
        "Q": "At 2:10, how did you get 2 groups of 6?",
        "A": "Hey! i have seen you before MAIJ! i saw u comment on Hannah B s program! I am friends with her. I am Tim BTW. Glad to see you helping. I am part of a team that helps out 1-5 in multiplication."
    },
    {
        "video_name": "OBVGQt1Eeug",
        "Q": "At 1:23 why did Sal multiply the amount of saline (50) by the percent (25)?",
        "A": "To find the amount of saline in pounds."
    },
    {
        "video_name": "k3qZCxL5TeE",
        "Q": "At 0:36, is the order of all digits the same as the way he wrote it?",
        "A": "To create the largest number using a series of numbers, you use the larger ones first in the number because, for example, 1,000,000 is larger than 1,000 because the one has been put in a higher place value."
    },
    {
        "video_name": "k3qZCxL5TeE",
        "Q": "At 2:00, why would he show the least possible number if that wasn't even the topic",
        "A": "Just to show what to do in reverse scenario"
    },
    {
        "video_name": "k3qZCxL5TeE",
        "Q": "did sal khan say disit at 0:15?",
        "A": "He meant to say digit."
    },
    {
        "video_name": "k3qZCxL5TeE",
        "Q": "the video starts @ 0:00 and ends @ 2:18. thank me later",
        "A": "okay sorry to be wrighting on your pad but DOES ANYBODY LOVE THE CORDINATE PLANES! THEY ARE SO MUCH FUN!"
    },
    {
        "video_name": "-QHff5pRdM8",
        "Q": "At 3:22 -ish Sal says an integer is not a whole number because it is negative, so does that mean whole numbers are integers?",
        "A": "The set of integers include all the natural numbers, their negatives, and 0: ... -3, -2, -1, 0, 1, 2, 3 ... The set of whole numbers includes all positive integers and 0. 0, 1, 2, 3 ..."
    },
    {
        "video_name": "-QHff5pRdM8",
        "Q": "At video Intro to rational & irrational numbers (4:15)Sal says PII dosn't end. So, how can we calculate PI/PI if we cannot read last digit from PI?. Is it just based on assuming that it is the same number. In this sense, is this math based on assumption? Just wondering, thanks :)",
        "A": "It is not based upon assumption. It is based upon the identify property of multiplication. It tells us that any number multiplied by 1 will equal itself: a * 1 = a By dividing both sides of the property by a , it also tells us that any number divided by itself = 1 (except for x=0): a/a = 1 So, if you divide Pi by itself, it must = 1."
    },
    {
        "video_name": "-QHff5pRdM8",
        "Q": "What does principal root mean? He says it at 8:00",
        "A": "Square roots have 2 answers: the principle root (which is the positive root) and the negative root. The default is to use the principle root. It a problem needs the negative root, there will be a minus sign in front of the square root. Examples: sqrt(9) = 3 This is asking for the principle root. - sqrt(9) = -3. The minus in front of the square root tells us that this is asking for the negative root. Hope this helps."
    },
    {
        "video_name": "-QHff5pRdM8",
        "Q": "At 7:05, the wierd symbol that Sal calls pi is irrational? What is pi and how is it irrational?",
        "A": "Pi is irrational because it goes on forever but its digits don t repeat"
    },
    {
        "video_name": "-QHff5pRdM8",
        "Q": "at 4:23 Sal puts 22/7 or pi in rational numbers. I thought pi was an irrational number",
        "A": "22/7 is not the same as pi. It is just a rough estimate we let you use, just like 3.14 3.14<pi<22/7"
    },
    {
        "video_name": "-QHff5pRdM8",
        "Q": "At 2:28, Sal has some square roots listed. Why are some square roots rational and others aren't? Numbers whose square root is a repeating decimal should be rational right? I am only saying this because repeating decimals are rational numbers.",
        "A": "NO. A repeating decimal is only rational if the digits keep repeating themselves. It s irrational if it s like pi, which goes on forever but the digits keep changing: 3.1415926..."
    },
    {
        "video_name": "-QHff5pRdM8",
        "Q": "At 4:24, if 22/7 is Rational, then how is Pi (3.141592653857...) considered Irrational?",
        "A": "A Pi is a non-terminating decimal. A non-terminating decimal is the proper name to a never ending decimal."
    },
    {
        "video_name": "-QHff5pRdM8",
        "Q": "At 4:22, why isn't 22/7 not a irrational number because it is the fraction form of pi, and pi is an irrational number.",
        "A": "An irrational number that never ends or terminates. 22/7=3.142857143 it does end, unlike pie that is infinite. If a rational number doesn t it has to repeat in a pattern, then it is considered irrational, therefore it can t be written as a fraction."
    },
    {
        "video_name": "-QHff5pRdM8",
        "Q": "At 4:18 you put 22/7 in rational number but at 6:58 you said pie is irrational .As pie and 22/7 is same why was 22/7 put in rational when pie irrational ?",
        "A": "Because \u00cf\u0080 and 22/7 are NOT the same. Who told you that? \u00cf\u0080 is an irrational number, which can be approximated by 22/7 (or 3.14) but it is not the same as either of those two numbers."
    },
    {
        "video_name": "-QHff5pRdM8",
        "Q": "At 3:44, Sal says that 0.25 can only be described as a \"fraction of two integers\". But it's being shown as a decimal on the screen!",
        "A": "Sal is saying that the only other way to represent 0.25 is as a fraction. You can not represent it solely as an integer or a whole number. Therefore it would be a rational number. Whereas another number, lets say 10, can be described as both a whole number (10), an integer (10), and a fraction (10/1). But with 0.25, it can ONLY be described as a fraction and a decimal."
    },
    {
        "video_name": "MtRXjXdXDow",
        "Q": "At 2:28, where did negative 1/2 come from",
        "A": "It comes after having taken the derivative of x\u00e2\u0081\u00bb\u00c2\u00b9. You can write 1/(2x) as x\u00e2\u0081\u00bb\u00c2\u00b9/2 and then take its derivative. You ll get the same answer."
    },
    {
        "video_name": "MtRXjXdXDow",
        "Q": "could someone explain why 2:57 is not 1 + x^4/4 + 1/4 x^-4? I know what he did to get his 1 + x^4/4 -1/2 + 1/4 x^-4 but I don't understand why, is that just one of those math tricks one should know or something?",
        "A": "What do you think he did to get the middle term to be -1/2 instead of the +1/4 you think it should be ? I recommend you try multiplying out (\u00c2\u00bdx\u00c2\u00b2 - \u00c2\u00bdx\u00c2\u00af\u00c2\u00b2)(\u00c2\u00bdx\u00c2\u00b2 - \u00c2\u00bdx\u00c2\u00af\u00c2\u00b2) again. You will get Sal s answer - all he did was use FOIL."
    },
    {
        "video_name": "u5dPUHjagSI",
        "Q": "at 1:40, he said to multiply the top equation by 1/6 and I don't understand how he got that number. How did he know to multiply the top equation by 1/6?",
        "A": "6x-6y= -24 He said to multiply the equation by 1/6 (or divide by 6) because 6 is a common factor of all of those numbers. I hope this helps!"
    },
    {
        "video_name": "u5dPUHjagSI",
        "Q": "at the time 1:40 you multiplied by 1/6. Couldn't I multiply the top equation by 5 and the bottom equation by 6 and get a correct answer instead of having to use a fraction? I'm not sure how you even thought of 1/6 to use. I realize it's simpler since the #'s would be smaller but it's easier for me the way I am showing you. Would this be ok?",
        "A": "Yes, you are right, but multiplying it by 1/6 kept the numbers small, making it easier and less susceptible to any mental errors."
    },
    {
        "video_name": "u5dPUHjagSI",
        "Q": "At 3:18 he explains you eliminate the opposite variables x and -x, but I had a problem that ended up with no exact opposite variables. What do I do? Look below.\n6x-2y=16 times all by 1/2 equals 3x-y=8\n5x-5y=-20 times all by 1/5 equals x -y=-4\nHow will I solve that?",
        "A": "When you end up with exact same variables instead of exact opposites, you can subtract one equation from the other instead of adding them. Or you can multiply one of the equations by -1 to get opposites, and then add (which is really the same as subtracting)."
    },
    {
        "video_name": "v_OfFmMRvOc",
        "Q": "At 1:02 I think Sal is applying the chain rule to the right hand side. He's taking the derivative of cos y with respect to y (that just means y is the independent variable right?) . Then (this is what I don't understand) he takes the derivative of y with respect to x, but writes it as dy/dx. Why does he write it that way? Is that like a variable for the actual derivative of y?",
        "A": "I think you have the right idea. He is doing this problem as implicit differentiation . This is just the previous section before this video. dy/dx is just the derivative of y with respect to x and that s the way it s usually written but you also could see y or f (x) as other notations."
    },
    {
        "video_name": "v_OfFmMRvOc",
        "Q": "At 2:00, Couldn't you have converted 1/sinY to cosY which is equal to X?",
        "A": "1/sin(y) = cosec(y), not cos(y)."
    },
    {
        "video_name": "v_OfFmMRvOc",
        "Q": "Why y=cos^-1 *x is equal x=cosy. I understand cos^-1 is not same as 1/cosx, but I dont get what transformations does Sal make at 0:49 in the equation to get that(x=cosy).",
        "A": "Remember that cos^-1 is the inverse function of cos. So, you can go from the original equation of y=cos^-1 (x) to x=cos(y) by taking the cos of both sides: cos(y)=cos(cos^-1(x)) (the cos and arccos cancel) cos(y) = x"
    },
    {
        "video_name": "Jhu1M53AF0E",
        "Q": "At around 4:00, can I replace \"4 cos \u00ce\u00b8 sin \u00ce\u00b8\" with \"2 sin 2\u00ce\u00b8\" so that later on my integral would become \"-cos 2\u00ce\u00b8\" instead of 2 sin^2 \u00ce\u00b8\"?",
        "A": "Yes, you can. 2sin\u00c2\u00b2(\u00ce\u00b8) is the same as 1 - cos(2\u00ce\u00b8), and that 1 is just a constant that can be ignored, because it is the same when evaluated at 0 and at 2\u00cf\u0080."
    },
    {
        "video_name": "Jhu1M53AF0E",
        "Q": "3:22 how does 1/2cos(2x) = d/dx (1/4sin(2x) *substituting x for theta",
        "A": "Because d/dx(sin(2x)) = 2*cos(2x) (by Chain rule)."
    },
    {
        "video_name": "NPXTkj75-AM",
        "Q": "How would x equal 10 at 3:57 . I worked it out it out and the answer that I got was nothing close to -18.\n-1(10)-4(7)=-18\n-10-28=-38\n-38 is not even close to -18.\n\n-1(10)--4(7)\n-10-(-28)=18\nThis problem could be worked out in two ways. I think. Either way, none of these ways could help you arrive at the answer of -18. Could I please receive some help.",
        "A": "If the solution is (10,7) a straight substitution in the first equation would be x - 4y = 10 - 4(7) = 10 - 28 = -18 which is what we expect second equation is -x + 3y = -10 + 3(7) = -10 + 21 = 11r Where did you get the -1 in front of the x for the first equation? Are you mixing up the two equations somehow?"
    },
    {
        "video_name": "NPXTkj75-AM",
        "Q": "At 5:00, how did Sal know how the lines would look and which direction they would be pointing at?",
        "A": "I think there is a good chance that he already had a sketch worked out ahead of time and just drew it on the screen."
    },
    {
        "video_name": "NPXTkj75-AM",
        "Q": "At 1:11 how do you know when to add or subtract the second equation. What is the first equation was -x-4y=-18 and the second equation was x+3y=11? Would you add or subtract? I am confused.",
        "A": "Since -x + x = 0 would eliminate the x terms, you have to add. If you had x - 4y = - 18 and x + 3y = 11, you would have to subtract because x + x = 2x, but x - x = 0."
    },
    {
        "video_name": "0WUXQNjdRvM",
        "Q": "At 0:03 why does * mean times? Why not just use X or Times instead? I'm confused. :(",
        "A": "* is just another symbol for multiplication. they all mean the same thing."
    },
    {
        "video_name": "0WUXQNjdRvM",
        "Q": "at 0:28 why dose it have thes)( huh?",
        "A": "it means do that part of the problem first then the rest of the problem"
    },
    {
        "video_name": "0WUXQNjdRvM",
        "Q": "I did not understand what he did on 1:32/ 3:17.",
        "A": "When you multiply the second digit of the second number, add a 0 to that answer, because you are multiplying the value from the tens column. If it s from the tens, add a zero. If it s from the hundreds, you would add two zeros."
    },
    {
        "video_name": "0WUXQNjdRvM",
        "Q": "2:08 But 40 times a 3 is really 120, not 20.\n\nHmmm... he could have written out the 12 in front of the 0 to form a 120, and then write two more 0 as placeholders under the red 0, and then say 4 times the 2 above it which is 8. He would ended up with...\n\n92\n120\n800\n------\n1012",
        "A": "the ansewer is correct"
    },
    {
        "video_name": "0WUXQNjdRvM",
        "Q": "at 0:11 Sal aid the hardest part a bout the multiplication was the little dot. how many different ways can you write a division problem",
        "A": "hey are you in Mrs. Ps class. if you are STOP PICKING ON MADDY. if you are not sorry."
    },
    {
        "video_name": "fuSOY9r1R6w",
        "Q": "At 13:30, why does x'(a+b-t) not become -x'(u)? I mean, I could write x' as dx(a+b-t)/dt, right? So, when I substitute u = a+b-t in the 'numerator', shouldn't I also have to substitute dt = -du in the 'denominator'? I would thus get dx(a+b-t)/dt = dx(u)/(-du) = -dx(u)/du = -x'(u) \u00e2\u0080\u0093 and this would eventually negate the whole result (same for the y terms).",
        "A": "When we made the substitution a+b-t = u, we changed the variable which we differentiate with, t disappeared from our problem."
    },
    {
        "video_name": "pGaDcOMdw48",
        "Q": "At 6:33, shouldn't it be b^2 cos^2 theta and not b^2 cos theta?",
        "A": "Sal corrects that later in the video."
    },
    {
        "video_name": "pGaDcOMdw48",
        "Q": "At 6:33, he was squaring b*cos(theta), but he only squared b.\nWhen you square two numbers that are being multiplied, don't you square one, and multiply that by the square of the other?",
        "A": "Yes (b*cos(theta))^2 = b^2 * (cos(theta))^2 ... He corrects himself at 6:57...:)"
    },
    {
        "video_name": "pGaDcOMdw48",
        "Q": "I don't get what they said at 5:15",
        "A": "Click under settings interactive transcript . He said So we figured out m, we figured out e, and now we want to figure out a, and this should jump out at you. He then said how you can use the pythagorean theorem; a^2=m^2 + e^2"
    },
    {
        "video_name": "pGaDcOMdw48",
        "Q": "At 6:36 how does expanding \"(c - bcos\u00ce\u00b8)^2\" equal \"(c^2) - 2cbcos\u00ce\u00b8 + b^2cos^2\u00ce\u00b8\"?\n\nWhere does \"2cbcos\u00ce\u00b8\" come from of there is already c^2 and b^2cos^2\u00ce\u00b8?",
        "A": "We are squaring a binomial of the form: (\u00f0\u009d\u0091\u008e \u00e2\u0080\u0093 \u00f0\u009d\u0091\u008f) So we evaluate: (\u00f0\u009d\u0091\u008e \u00e2\u0080\u0093 \u00f0\u009d\u0091\u008f)\u00c2\u00b2 = (\u00f0\u009d\u0091\u008e \u00e2\u0080\u0093 \u00f0\u009d\u0091\u008f)(\u00f0\u009d\u0091\u008e \u00e2\u0080\u0093 \u00f0\u009d\u0091\u008f) = \u00f0\u009d\u0091\u008e\u00c2\u00b2 \u00e2\u0080\u0093 \u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f \u00e2\u0080\u0093 \u00f0\u009d\u0091\u008f\u00f0\u009d\u0091\u008e + \u00f0\u009d\u0091\u008f\u00c2\u00b2 = \u00f0\u009d\u0091\u008e\u00c2\u00b2 \u00e2\u0080\u0093 2\u00f0\u009d\u0091\u008e\u00f0\u009d\u0091\u008f + \u00f0\u009d\u0091\u008f\u00c2\u00b2 You can review this in the algebra section. You should be fluent with this sort of algebra to continue on to trigonometry."
    },
    {
        "video_name": "pGaDcOMdw48",
        "Q": "At 6:16 in the video, how does the (c-b cos\u00e2\u0081\u00a1\u00e2\u0088\u0085)^2 become c^2-2cbcos\u00e2\u0088\u0085+b^2 cos\u00e2\u0088\u0085? I just don't see the expansion behind this and it would be great if I could get help.",
        "A": "(A-B)^2= (-A)(-A) + (A)(-B) + (-B)(A) + (-B)(-B) = A^2 - 2A.B + B^2 Substitute c for A and -b cos(theta) for -B."
    },
    {
        "video_name": "pGaDcOMdw48",
        "Q": "for the first problem in the video where did you get the 2bc cos 'theta' from ??? 6:30",
        "A": "It comes from the formula: [ (x-y)^2 = x^2 + y^2 -2xy ] So that 2bc cos (theta) comes from the above formula when x=c and y=bcos (theta)..!! Hope u undrstud..!!"
    },
    {
        "video_name": "pGaDcOMdw48",
        "Q": "At 7:35, why is sin^2+cos^2=1 ?",
        "A": "sin^2 + cos^2 = o^2/h^2 + a^2/h^2 = 1 (o= opposite side, a = adjacent side, h= hypotenuse) Multiply this by h^2: o^2 + a^2 which = h^2 This is the pythagorean theorem."
    },
    {
        "video_name": "pGaDcOMdw48",
        "Q": "At 7:32, he said that the sine of theta plus the cosine of theta is 1, but how do you prove it? Thank you.",
        "A": "Be careful. Sal said that the sine SQUARED + cosine SQUARED is 1. Think of the unit circle. Pick a point on that circle (to make it simpler, choose one in the 1st quadrant) and drop a perpendicular to the X-axis. Consider the triangle that is formed. The y value is sin theta; the x value is cos theta; the hypotenuse is 1 (because it s a unit circle). Then use the Pythagorean theorem : sin^2 + cos^2 = 1. And we re done! (Someone else may be able to provide a more rigorous proof.)"
    },
    {
        "video_name": "pGaDcOMdw48",
        "Q": "at around 3:45 how does he get from cos=d/b to d=bcos of theta?",
        "A": "cos=d/b so then if you multiply both sides of the equation by b then the b s cancel out on the right side and ur left with bcos=d"
    },
    {
        "video_name": "pGaDcOMdw48",
        "Q": "@ 7:03, where does he get the 2 from in 2cbcosine(theta)? Sorry for the stupid question. :)",
        "A": "He got the 2cbcos\u00ce\u00b8 term from expanding (c - bcos\u00ce\u00b8)^2. Refer to the expansion of a square. (a + b)^2 = a^2 + 2ab + b^2. If we substitute our term in, you would get c^2 + 2*c*(-bcos\u00ce\u00b8) + (-bcos\u00ce\u00b8)^2. You then simplify it to c^2 - 2cbcos\u00ce\u00b8 + b^2cos^2\u00ce\u00b8."
    },
    {
        "video_name": "pGaDcOMdw48",
        "Q": "Why is (at 6:35) it that (c-bcosQ)^2 = c^2 - 2cbcosQ + b^2cos^2Q ? I understand that he multiplied it out but I don't see how he got to that. Can someone please write out the logic?",
        "A": "This is the form: (a-b)^2 = a^2 -2ab+b^2 Right?"
    },
    {
        "video_name": "A52fEdPn9lg",
        "Q": "Why is it c over (x-2)^2? It should be c over (x-2). (at 01:47):)",
        "A": "it is explained at around 9:20 ish... he explains that with each term, you go up a power."
    },
    {
        "video_name": "A52fEdPn9lg",
        "Q": "At 2:08, why was Khan using c / (x - 2) ^ 2 instead of c? (x - 2 )",
        "A": "If you have c/(x-2), you can just add b/(x-2) to it, and view the whole thing as one term which doesn t work for partial fraction expansion. (see 0:52- 1:10)."
    },
    {
        "video_name": "A52fEdPn9lg",
        "Q": "At around 1:42 and after that, Sal breaks up the problem into 3 different fractions. Please check the video for them, as they are very long and I don't want to write them here. Why doesn't he just break it into 2 parts, with denominators (x+1) and (x-2)^2?",
        "A": "I m not sure exactly why this is so, but when you have a squared factor such as (x - 2)^2, you need a fraction with (x - 2) and one with (x - 2)^2."
    },
    {
        "video_name": "A52fEdPn9lg",
        "Q": "at 2:10 I don't understand how is that if you ad B/(x-2) + C/(x-2)^2 you would get an expression something X + something over (x-2)^2",
        "A": "B/(x-2) + C/(x-2)^2 Multiply B/(x-2) by (x-2)/(x-2) to get an equivalent fraction with the denominator (x-2)^2 B(x-2)/(x-2)^2 + C/(x-2)^2 Now we can add the numerators directly ( B(x-2) + C ) / (x-2)^2 Now expand the numerator ( Bx - 2B + C ) / (x-2)^2 There s your something X + something over (x-2)^2"
    },
    {
        "video_name": "A52fEdPn9lg",
        "Q": "Why does he write(x-1)(x-2)^2??? (at 3:27)",
        "A": "Because that s the original denominator."
    },
    {
        "video_name": "A52fEdPn9lg",
        "Q": "Why isn't it split into A/(x-1)+ B/(x-2)^2 ? At around 9:20 he explains that it isn't, but I want to know WHY it's not.",
        "A": "You wouldn t be decomposing the whole fraction, you d be left with something that could still be decomposed."
    },
    {
        "video_name": "kQkeayQke8A",
        "Q": "At 2:20 why is A greater than B based on what he said?",
        "A": "Well, A is the first letter in the alphabet so naturally people assume that A is greater than B"
    },
    {
        "video_name": "kQkeayQke8A",
        "Q": "Why in 4:36, negative A is to the right? I was taught that negative numbers are to the left and positive integers are to the right. Please explain!",
        "A": "Negative A means that it is the opposite of A, because a negative number is always opposites with the positive number that is equally as far away from 0 on the number line as the negative is. The reason that that is important is that if, for example, A were equal to -4 the opposite of A would be 4. With that in mind -A could equal both a negative or a positive number."
    },
    {
        "video_name": "_uTAdf_AsfQ",
        "Q": "\"They do span the column space of A but they're not the basis\" (15:30) what does that mean?",
        "A": "In order to be a basis, the vectors must all be linearly independent. As he proves later in the video, v_3 and v_4 are linear combinations of v_1 and v_2, meaning the vectors are not linearly independent. Therefore they cannot be a basis. However, linearly dependent vectors can span. In that case, one or more of the vectors tells you no new information and the vectors can still span without those dependent vectors."
    },
    {
        "video_name": "_uTAdf_AsfQ",
        "Q": "Around time 6:31 in the video, How does not having a pivot entry relate to the variable being free? It seems like when you multiply out the matrix by the vector, x3 and x4 would still get multiplied by entries in rows 1 and 2, so it would matter right?",
        "A": "Only for the 3rd and 4th column vectors. The scalar coefficients of the 3rd and 4th column vectors are (v3 and v4) are x3 and x4, the free variables."
    },
    {
        "video_name": "_uTAdf_AsfQ",
        "Q": "at around 2:47, when you are trying to reduce a matrix, are you allowed to go through and do 2row1-row2, when you are working on row2? I didn't learn the proper way to get a matrix into row reduced echelon form and the \"rules\" of doing it.",
        "A": "yes, it would yield the same resulting row we can also multiply a row by -1 if we need a better looking solution"
    },
    {
        "video_name": "mbb3msmX2xs",
        "Q": "At 2:29 wouldn't 2 be the GCF?",
        "A": "No... since both 12 and 32y can be divided evenly by 4, then 4 is the GCF Hope this helps."
    },
    {
        "video_name": "mbb3msmX2xs",
        "Q": "At 3:10, there are two \"4\"s but why would the equation turn out to be 4(3+8)? Where did the other \"4\" go? And why would the unknown variable \"y\" stick onto 8? Why not the 4?",
        "A": "He undid the distribution. If you recall, a(b+c) = ab +ac, what he did was reverting it. Or we call it factoring out the common factor. 8y is one term, we are factoring, not solving for y so it doesn t need to be isolated."
    },
    {
        "video_name": "mbb3msmX2xs",
        "Q": "Couldn't you just rewrite the expression shown at 0:05 as \"4(x+4.5)\"?",
        "A": "Yes, you could, but that isn t always the most advantageous of solutions. Often, when we factor, part of our goal is keeping the numbers easy to manage, so we try to keep all of our coefficients and constants as integers."
    },
    {
        "video_name": "mbb3msmX2xs",
        "Q": "at 2:47 how do you know what the sighn is going to be?",
        "A": "12 + 32y Basically, he is just breaking up those numbers into smaller pieces. He looks at the 12 and the 32y separately, and imagines what type of multiplication he could use for them that both include the same multiple. So he chooses 4, because both numbers (12 and 32) can be multiplied by 4 Step 1: look at 12 + 32y Step 2: Break up the 12+ 32 y (4 x 3) + (4x8) y ...then continue with the distribution like in the video..."
    },
    {
        "video_name": "Y9sP4V3bKh0",
        "Q": "What does Sal mean at 0:47 (when he says that a function can be defined while the limit isn't)?",
        "A": "It means f(a) exists, but lim_(x->a) f(x) doesn t exist. For example, f(x) = { 2 for x < 0 , 4 for x \u00e2\u0089\u00a5 0 We can see that for f(0) = 4, but lim_(x->0) f(x) doesn t exist because the left sided limit as x->0 is 2 but the right sided limit as x->0 is 4. Since the left sided limit and the right sided limit don t agree, the limit as x->0 doesn t exist."
    },
    {
        "video_name": "mFftY8Y_pyY",
        "Q": "At 2:38 Sal writes 6+5(3). Could you not write 6+5 to the power of 2?",
        "A": "No. 6+5(3) is also 6+5 x 3 so you can t square 6+5 and get the same answer. 6+5(3) = 33. 6+5 to the power of 2 or squared is 121."
    },
    {
        "video_name": "mFftY8Y_pyY",
        "Q": "At 5:01, Sal says the number of incremental houses is 50 - 1, which is 49. If you were doing this problem, couldn't you just say 49, instead of 50 - 1?",
        "A": "By saying 50 minus one he is going through step by step and explaining the entire process."
    },
    {
        "video_name": "1wsF9GpGd00",
        "Q": "At 1:19, is B also a strict subset of C?",
        "A": "Yes, because C contains all that is in B and more."
    },
    {
        "video_name": "1wsF9GpGd00",
        "Q": "@0:38 , shouldn't the numbers in set C be in numerical order?",
        "A": "No, not necessarily. The items in a set don t have to be in any particular order."
    },
    {
        "video_name": "1wsF9GpGd00",
        "Q": "At 2:08, Sal says that every set is a subset of itself. Likewise, can you say that every set is a superset of itself?",
        "A": "Sort of. If A is a subset of B, B is the superset of A. However, saying A is a subset of A leads to A is a superset of A being redundant an as a result largely ignored."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "At 5:01 you found the y value. How do you do it?\nAlso, for the x value you just factor the equation out and get your answers, right?",
        "A": "You use x=2 and substitute into the equation to find y . The 1st part of the equation turns into zero, leaving just 16. Sal kind of talks you thru this at about 4:45 in the video."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "@2:58 I don't understand how the -8 is part of the equation",
        "A": "Inside the bracket, the (x-2)^2 goes. -4-4 remains now, -4-4=-8 therefore, -8 is part of the equation"
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "A little after 7:10 in the video after he uses the formula to get 2 for x he says when x is equal to 2 y is equal to 16. How did he get the 16 for the y? Thanks.",
        "A": "Once you ve found the x-coordinate of the vertex (here it s 2), finish solving. y = -2(2-2)^2 + 16 will solve to... y = 16"
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "At 1:30, what form is Sal writing that equation in and why did he do that?",
        "A": "He s still writing in the quadratic form, just factoring out the two so it s easier to handle. Much easier to work with an equation like this: 128x^2+256x+512=0 when it s represented like this: 128(x^2+2x+4)=0 I ve done the same thing as sal here - divided out the 128 from the equation and substitute it back in. It s still the same number, with the same solutions and values, just easier to work with and factor."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "At around 2:48, why is it (x-2)^2?",
        "A": "(x-2)^2 = (x-2)(x-2) using FOIL, it expands to x^2 - 2x -2x + 4 or x^2 - 4x + 4. So he took that part of the equation and factored out a perfect square, the purpose of the video was to complete the square to find the vertex and axis of symmetry."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "At around 7:10 when he was explaining the quick way of finding the axis of symmetry, I understand that the answer from the formula will give you the value of 'x', but how can we know what the 'y' coordinate is? I thought it was going to be '8', but it seems to be '16', how did he find that out by only using the axis of symmetry formula if in the case he didn't complete the square yet? So nevertheless, do we still have to complete the square? Thank you!",
        "A": "Sal didn t solve for the y-value of the vertex at the end there, he brought the 16 down from when he completed the square. If you wanted to find the y-value of the vertex after you found the x-value using the formula, then put the x-value back into the original equation and solve the equation. In this case, -2x\u00c2\u00b2 +8x + 8 -2(2)\u00c2\u00b2 + 8(2) + 8 -2(4) + 16 + 8 -8 + 24 16 = y"
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "at 2:18, why does sal subtract 4 twice?",
        "A": "Follow the colors. They kind of show where each one comes from. The -4 in yellow comes from the 8 in the original problem after he factors out -2. 8/(-2) = -4. The -4 in pink comes along with the +4 also in pink. Sal is completing the square to make graphing the parabola a little easier to do. To complete the square, you have to add 4. But, we don t actually want to change the polynomials value, so he also subtracts 4. Thus, he really added zero yet changed the format of the equation."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "At around 4:00 to 4:10, he says something like \"more negative\" and \"more positive\". What does he mean?",
        "A": "He means the magnitude, a.k.a. absolute value. more negative - further away from zero in the negative direction. more positive - further away from zero in the positive direction."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "i really don't understand how he got 4x at 1:40",
        "A": "It s -4x and he got it from factoring out -2."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "At 7:10 he gets \"2\" for the x axis of the vertex but he doesn't explain how you would get the y axis. If I were to encounter this on a test how would I find the y axis?",
        "A": "When the equation is in the form: y = -2 (x-2)^2 + 16: -- The (x-2) in the parentheses tells you the x-value for the vertex -- The 16 tells you the y-value for the vertex. Or, if the equation is in the form: y = -2x^2 + 8x + 8: -- You can find the x-value of the vertex using the formula: x = -b/(2a) where a is the coefficient for x^2 and b is the coefficient of x . -- Once you know the x-value for the vertex, you can substitute it into the quadratic and calculate the y-value for the vertex. Hope this helps."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "I got lost 1:30 and on! Can anyone explain what he is doing in a nutshell (especially the part where he is completing the square)???????!!!!",
        "A": "I suggest watching polynomials and rational expressions/equations before quadratics. If that helps."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "At 2:35 Sal is starting to add a four to the equation but the actual number is negative four...I don't understand how it helps to solve the original equation because the actual one is x^2-4x-4",
        "A": "He added 4 but also subtracted 4. So essentially, he added 0 or didn t add anything to change the original equation. But doing so allows us to complete the square using the +4. Basically, we just rewriting the -4 a little bit. -4 is the same as +4-4-4 So instead of x\u00c2\u00b2-4x -4, we replace -4 with +4-4-4, we get x\u00c2\u00b2-4x+4-4-4 Hope that explains how we could do that."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "At 7:18, this may sound stupid, but how did you get 16? Or did you multiply 2 (which is x) and 8 to get 16 (which is y)?",
        "A": "Sal found out that the value of x was 2. Then, Sal replaced the x in the formula with 2 instead, and solved the rest from there."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "I don't understand where that -8 came from @2:58. If he factored out -2 at the very beginning, which gave us the original -4, then why would he keep the -8 and still bring down the -2?",
        "A": "The second minus 4 came from the original equation (when a -2 was factored out of an 8), the first minus 4 came from the opposite of adding 4 to complete the square. When you put the two minus 4s together, -4-4= -8 (same signs add and keep the sign)"
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "What does Sal mean by \"more and more positive\"and \"more and more negative\"?\nThis is at around 3:38 .\nThanks.",
        "A": "the higher/larger the numbers (more positive) that are substituted for x, the larger the resulting positive number will be. However, because it is being multiplied by a negative, it will become a very large negative number. Thus, as x gets more and more positive, the resulting term becomes more and more negative."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "The only thing I don't understand is that around 6:59-7:20 how he found that sixteen was y. For example, if I only solved the equation only using the formula x=-b/2a how could I figure out that y was sixteen? It wasn't very clear....",
        "A": "I always forget this step too :) The quick and dirty vertex formula gives you only the x value of the vertex. To find the point of the vertex, you need both the x and the y values. So you need to plug the x value you discovered into the parabola s function. (Which makes sense, because what the function really does is tell you, for every given x, what the y value will be) So essentially, Sal skipped over this: y = -2x^2 + 8x + 8 y = -2*(2^2) + 8*2 +8 y = -2*4 + 16 + 8 y = -8 + 16 + 8 y = 16"
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "At 1:59, Sal says that if he had positive 4 it would be a perfect square. Why isn't -4 a perfect square then?",
        "A": "It is because it is negative. There is no real number times itself that will create a -4. 2*2 = +4 and (-2)*(-2) = +4 You can t have -2*(2) and get a perfect square because you aren t using the same number."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "At 3:05...why did we keep the -2 in the equation? It seems like in previous videos we factor out the number from both sides. And as as result the number does not reappear",
        "A": "In this case, if you divide both sides by -2, the left side becomes: -y/2 which is not useful. You need to keep the left side as just y which is the reason the -2 is just kept on the right side of the equation. Hope this helps."
    },
    {
        "video_name": "dfoXtodyiIA",
        "Q": "At 7:14 in the video, how did Sal get that when x=2, y=16? How you find what \"y\" is?",
        "A": "Sal did y=2x. Y= -2 ( x2 - 4x + 4 -4-4 Y= -2(x-2)2-8) Y=2x-2+16 Always X= 8 over 2"
    },
    {
        "video_name": "4F6cFLnAAFc",
        "Q": "at 7:11, couldn't you write r^10th power?",
        "A": "No, but it could be r^(10/3) if you want to make the mixed fraction 3 1/3 an improper fraction."
    },
    {
        "video_name": "4F6cFLnAAFc",
        "Q": "at 5:04 Sal writes \" r to the 3 AND 1/3 \".\nr should be r^3+1/3 which equals to r^4/3 and not r^3*1/3 which equals to r^1...\n\nWhat is going on?\nI can solve this problem ending up with the solution on the right (see at 7:20) but can't understand this step.",
        "A": "the thing is that if you remember from exponent rules, when there is a number to a power, to a power, such as (2^3)^4, you multiply the exponents, so the examples answer would be 2^12. Therefore, (r^3)^1/3 would be r^3*1/3, which is r^3/3, or r^1, or just r."
    },
    {
        "video_name": "4F6cFLnAAFc",
        "Q": "At 4:48 you say that r^(4/3) is the same as r^1(1/3) ..... how?!",
        "A": "Because \u00e2\u0081\u00b4\u00e2\u0081\u0084\u00e2\u0082\u0083= 1\u00e2\u0085\u0093. You can make the fraction on the left mixed to prove this. Hope this helps!"
    },
    {
        "video_name": "4F6cFLnAAFc",
        "Q": "At 5:38: Why is (r^4/3)*r^2=R^(3+1/3). I thought (r^4/3)*r^2=r^(8/3) ?? Sorry I found it: It is (r^4/3)(r^6/3) = r^10/3 which is correct.",
        "A": "x^a\u00e2\u0080\u00a2x^b = x^(a + b)"
    },
    {
        "video_name": "4F6cFLnAAFc",
        "Q": "I don't understand why the square root and cubed root equals ^1/2 and ^1/3 respectively? Like, where does the logic come from? As seen at 7:13, shouldn't it just be the square root of 8 which is 2 and the cubed root of, well, whatever number r represents? I have trouble grasping concepts I don't know how and why it works. Thanks!",
        "A": "Well first, the square root of 8 is NOT 2. The square root of 4 is 2, the square root of 8 is 2*sqrt(2)."
    },
    {
        "video_name": "b0U1NxbRU4w",
        "Q": "At 2:59 Sal said he already used theta. Couldn't he use theta again to denote that angle? It feels as if he used x to purposely prove that it was a right triangle when he could of used theta and come out to a different conclusion.",
        "A": "If he used theta for both angles, then that would mean both angles were congruent, which they weren t. He used a different symbol (x) to show that the angles weren t congruent."
    },
    {
        "video_name": "b0U1NxbRU4w",
        "Q": "At 3:42, isn't x just theta? Please explain if not.",
        "A": "At 3:42 angle x could be described as theta except for that we already used theta, so we can t use theta again. If we used theta again it would mean that the two angles are equal. Hope this helps!"
    },
    {
        "video_name": "b0U1NxbRU4w",
        "Q": "At 1:20 What does he mean by theta? ~_~",
        "A": "Theta is an ancient greek alphabet to show a variable of an angle :)"
    },
    {
        "video_name": "b0U1NxbRU4w",
        "Q": "At 3:42 , how does Sal figure out x = 90 - theta ?",
        "A": "The triangle has 180 total degrees in it. So, you could write 180= 2(theta) + x + x, or just 180 = 2 (theta) + 2x. By dividing both sides by 2, you get 90 = theta + x. Now, you subtract theta from both sides, and you get 90- theta = x."
    },
    {
        "video_name": "6OFwfxmhtE8",
        "Q": "At 1:50\nSal canceled the 5th power from the right side out by, raising it again to the 5th power, and you left with 3^2 how does raising the 5th power to a 5th power cancel it all out?",
        "A": "He does not raise the 5th power to the 5th power, he raises the 5th power to the 1/5 power, so 5 * 1/5 = 1. If you have a 5h root (same as to 1/5 power) the opposite operation is to the fifth power just as squares and square roots are opposite operations"
    },
    {
        "video_name": "6OFwfxmhtE8",
        "Q": "At 3:52, Do we have to solve 3 to the 2/5ths power, or can we leave it because we already solved for a.",
        "A": "No, because we re only solving for the exponent of 3."
    },
    {
        "video_name": "6OFwfxmhtE8",
        "Q": "At 2:08 Sal goes from\n3^5a = 3^2\nto\n5a = 2\nWhat operation is being performed on each side?",
        "A": "It isn t necessarily an operation, but just using logical reasoning. For 3^5a to equal 3^2, the exponents must be the same (or equal). If the exponents turn out to be different, non-equivalent exponent values, there is no way for the 2 sides of the original equation to be equal to each other. In other words, 3^2 = 3^2; but 3^3 does not equal 3^2. So, you are hunting for the value of a that makes the 2 exponents equal to each other. Thus, solve 5a = 2."
    },
    {
        "video_name": "6OFwfxmhtE8",
        "Q": "how do you 'undo the fifth root to raise it to the fifth power'? 0:28 I'm very confused on this part",
        "A": "Exponents and roots are sorta like opposites. If you square 3 it become 9, but taking the square root of 9 makes it 3. If you multiply the cube root of a number, x, by itself three times (raising it to the third power) it becomes x. The cube root of 8 is 2, while 2*2*2 or 2^3 is 8."
    },
    {
        "video_name": "xO4n3LfmVag",
        "Q": "It's shape four! 3:18 tells it! So easy!",
        "A": "I know its easy"
    },
    {
        "video_name": "LSIgi9krswU",
        "Q": "At 1:10, why does it 'Jump Up'? Slightly confused...",
        "A": "It jumps up because the graph is discontinued at that point. The value of the function is higher than the surrounding x s so it has the effect of jumping up."
    },
    {
        "video_name": "LSIgi9krswU",
        "Q": "at 1:20 why is it counter intuitive to construct an interval around B?",
        "A": "Because our intuition (from looking at the graph) would tell us that it would be a relative min, but it is a relative max instead. So it is counter intuitive or contrary to our intuition."
    },
    {
        "video_name": "NuccqpiUHrk",
        "Q": "at 0:10 couldnt you have just canceled out y^3 and the cube root symbol?",
        "A": "If you broke it into three cube root parts like Sal eventually does."
    },
    {
        "video_name": "NuccqpiUHrk",
        "Q": "when the video was at 0:38, how did he get the 1/3 power",
        "A": "nroot(x) = x^(1/n) 3rt(125) = 125^(1/3)"
    },
    {
        "video_name": "E1j8W64NQ0Y",
        "Q": "What I don't understand is why at 4:06 Sal decided to multiply each side by 8A and why he inverted the fractions, a lot of people have asked this before but no one has provided an answer, normally Sal always explains things very thoroughly but this question has me lost. Can anyone please help?\n\nEDIT: Nevermind, I looked through it again and I realized that the reason he inverted it is that the A's cancel out, like in 8A/2A, the two A's cancel each other out and you just get 4. Gosh I'm so stupid sometimes.",
        "A": "You multiply by 8A because it is the GCF for all of the denominators."
    },
    {
        "video_name": "E1j8W64NQ0Y",
        "Q": "I got lost around 4:20 when Sal started multiplying things by 8a. Why did he do this? It seems pretty arbitrary.",
        "A": "Isolating the unknown: 1/8 = 1/A + 1/2A *2A = *2A -------------------------- 2A/8 = 2A/A + 2A/2A 2A/8 = 2 + 1 *8 = *8 -------------------- 2A = 3 * 8 2A = 24 /2 = /2 -------------------- A = 12 Over time you gain more math skill and can solve the same exercise more quickly. That s what Sal did. He saw the variable A on one side and 8 on the other side, so he did everything in one step only. Hope this helps!"
    },
    {
        "video_name": "G-T_6hCdMQc",
        "Q": "At 3:34, why couldn't Sal just square the numerator and the denominator? Does this change the fraction in a way that multiplying does not?",
        "A": "Yes. If you square the top and square the bottom of a fraction, you are probably multiplying the top and bottom by different numbers. This will change the overall value of the fraction. To rationalize a fraction (so that there is no radical in the denominator), you can t just square everything for this reason. You have to multiply the top and the bottom by the same number instead. (like square root of 64 over square root of 64, for example)."
    },
    {
        "video_name": "-fKh6SNEPr4",
        "Q": "at 6:30 Sal says that the null space of A is only the zero vector, does that mean that the dim(ker(A))=1 or that the dim is NULL set?",
        "A": "No, imho, it is dim(N(A))=0 meaning dimension of the nullspace is R^0 => you have 1 solution (everything ^0 is 1) which is the zerovector."
    },
    {
        "video_name": "-fKh6SNEPr4",
        "Q": "At 5:07 doesn't Sal mean nth 0 instead of mth?",
        "A": "For Ax=0 , how do you get the first entry in the result (0) vector? the last entry?"
    },
    {
        "video_name": "i0VS3eEGjiQ",
        "Q": "at 5:40 when Sal plugs x into 2x, wouldn't the 2 distribute to each term ? Like\n1/2 EG*h = 2x and x = 1/2GB*h\n1/2EG*h = 2(1/2GB*h)\n1/2EG*h = 1GB*2h <---- when 2 is distributed to both 1/2GB and h.\n1/2EG = 1 GB*h <----- so when 1 h is canceled from each side, the right would still have 1 more h.",
        "A": "The Distributive Law only applies to cases of multiplication over addition. The distributive property would not apply in this case, as it is 1/2GB*h, not 1/2GB+h"
    },
    {
        "video_name": "i0VS3eEGjiQ",
        "Q": "At 3:34, why do we know that the area of the blue triangle is twice the area of the other one?",
        "A": "Watch the video Medians divide into smaller triangles of equal area which states that median generated triangles have the same area, so all 6 triangles have the same area.."
    },
    {
        "video_name": "i0VS3eEGjiQ",
        "Q": "At 5:52, how does Sal gets rid of \"h:? I dont understand. If we devide both sides by \"h\" why then we don't devide EG and GB by \"h\"?",
        "A": "At 5:52 Sal crosses h off in both sides to simplify the equation."
    },
    {
        "video_name": "i0VS3eEGjiQ",
        "Q": "what is an 0:01 triangle?",
        "A": "An arbitrary triangle is a random triangle it is equally likely to be any triangle"
    },
    {
        "video_name": "_btQus9HV_I",
        "Q": "At 10:20 How would you make a mixed number out of the fraction 60/10 in simplified form ?. I really need the right answer for this question.Thanks :-)",
        "A": "Remember to look at fractions as a division problem. The number on top is being divided by the number on the bottom. So 60 divided by 10. That s 6."
    },
    {
        "video_name": "XDjbNykJ28E",
        "Q": "This is so obvious; it is explanation A. You don't do ( at 1:06 ) 2x7 and 5x7 on the fraction.",
        "A": "No I think he s correct. In this video Sal Khan is proofing out in detail how to times a fraction and why you have to do it that way. The guy in the story did it wrong. Keep watching."
    },
    {
        "video_name": "v_W-aaB1irs",
        "Q": "At 6:30, Sal uses parenthesis to denote the vector. Isn't this the correct form <3,5>??",
        "A": "Yes, you are right."
    },
    {
        "video_name": "v_W-aaB1irs",
        "Q": "At 2:56, there is no need to calculate the magnitude at all! : ) We already know the components, and they're equal, if we don't take into consideration, the sign of the components. Is there any specific reason why Sal calculates the magnitude of the vectors? Is my assumption incorrect? (He doesn't calculate the magnitudes in the second problem...)",
        "A": "You are correct that all of the components of two vectors are equal up to their absolute value, then the magnitudes of the vectors are equal. Sal likely computed the magnitudes just to practice it."
    },
    {
        "video_name": "soIt2TwV6Xk",
        "Q": "a around 3:40 sal says that the angle is between - sqrt 3 over 2 and pi, what is -sqrt 3/2 because i searched it up and it seems to be the negative y-axis, not the positive. Please help",
        "A": "You re right. minus half of the root of 3 is negative, so we turn in the opposite direction."
    },
    {
        "video_name": "soIt2TwV6Xk",
        "Q": "At 4:30 why is 3pi/2 (270degrees) at the top of the cadrant ? I'm really confused because on the unit cercle I found on wikipedia for example, pi/2 is at the top of the cadrant and 3pi/2 is at the bottom ;-) Thanks !",
        "A": "It is - 3pi/2. This is measured clockwise around the circle, and you end up at the top of the y-axis. - 3pi/2 and +pi/2 are the location on the unit circle. You are thinking of + 3pi/2 which would be at the bottom of the circle on the y-axis. Note: +3pi/2 is the same as - pi/2. Hope this helps."
    },
    {
        "video_name": "soIt2TwV6Xk",
        "Q": "At 4:05 Sal is defining angle theta as the arc from 0 to the line in the second quadrant just past -pi in the clockwise direction. But several seconds earlier he said theta was the line between -pi and -3/2 pi, as defined in the original problem statement. Was his statement at 4:05 a misstatement?",
        "A": "I know it is all confusing. But from 3:05 he demonstrated it. Try to watch it again, maybe it will make sense. :) Or if you add 2pi to both, which is a full circle, so it won t essentially change the angle. We will get 1/2 pi < theta < pi"
    },
    {
        "video_name": "soIt2TwV6Xk",
        "Q": "I have a question. Around 2:28 why cos0 = can be negative or positive Sqr(3/4)?",
        "A": "The Pythagorean identity introduces extraneous values because it squares cos(t) then later we take the square root. We often use it only to get magnitudes, not signs. (E.g., if the cosine were -1/2, we would get the same answer with the Pyth. id.) (I think there s a better answer which I haven t yet figured out)."
    },
    {
        "video_name": "UmiZK6Hgm6c",
        "Q": "at 8:50, I thought a/2/2 was 2a/2 not a/4, am I mistaken?",
        "A": "Think about this it is a/2 / 2. Then you take the inverse of two and get a/2 * 1/2. Lastly, you multiply a and i and two and two and get a/4."
    },
    {
        "video_name": "UmiZK6Hgm6c",
        "Q": "At 4:27, what is Heron's Formula?",
        "A": "You can calculate the area of a triangle if you know the lengths of all three sides, using a formula that has been know for nearly 2000 years. It is called Heron s Formula after Hero of Alexandria formula: root of s(s-a)(s-b)(s-c) where a,b,c are the sides of the triangle and s is the semi-perimeter i.e a+b+c/2 hope that helps :)"
    },
    {
        "video_name": "UmiZK6Hgm6c",
        "Q": "At 8:25, what is soh cah tou?",
        "A": "soh cah toa is an acronym that helps you find sine, cosine, and tangent ex. say you have a triangle but you only have two side lengths given. depending on what side lengths they are, you can find the angle measurement. soh: sine = opposite/hypotenuse cah: cosine = adjacent/hypotenuse toa: tangent = opposite/adjacent"
    },
    {
        "video_name": "UmiZK6Hgm6c",
        "Q": "what does 'cah' and 'toa' mean in \"soh cah toa\"?(at 8:17)",
        "A": "cos = adjacent/hypotenuse tan = opposite/adjacent"
    },
    {
        "video_name": "lJdb4_Fuxvw",
        "Q": "at 1:40 does sal mean two halves?",
        "A": "2 halves is dividing the number line into fourths like, 1/4, 2/4, 3/4, and 4/4 a half is 1/2 2 of them is 1/2 + 1/2 or 2 times 1/2 still equals fourths. - Hope this helped, W"
    },
    {
        "video_name": "lJdb4_Fuxvw",
        "Q": "At 1:40, doesn't Sal mean the two halves?",
        "A": "He s dividing the halves into fourths"
    },
    {
        "video_name": "lJdb4_Fuxvw",
        "Q": "in the last Video at 0:01... what IS the last video?!?!?!?!",
        "A": "Multiplying fractions and whole numbers: two approaches."
    },
    {
        "video_name": "lJdb4_Fuxvw",
        "Q": "At 2:13, how does 3/4 go into 1/2? I thought 2/4 was equal to one half.",
        "A": "You are correct, that must have been a mistake 2/4 = 1/2 (2/4 \u00c3\u00b7 2/2 = 1/2)"
    },
    {
        "video_name": "lJdb4_Fuxvw",
        "Q": "at 1:40 what doesn't he say 2 halves?",
        "A": "Because he is a savage [\u00cc\u00b2\u00cc\u0085$\u00cc\u00b2\u00cc\u0085(\u00cc\u00b2\u00cc\u00855\u00cc\u00b2\u00cc\u0085)\u00cc\u00b2\u00cc\u0085$\u00cc\u00b2\u00cc\u0085]"
    },
    {
        "video_name": "lJdb4_Fuxvw",
        "Q": "At 2:58, why did Sal wrote 1/8 ?",
        "A": "Because he made those tiny marks on the number line from 0 to 1, dividing 1 up into 8 parts. Since 1 is divided into 8 parts, the first mark is 1/8."
    },
    {
        "video_name": "54ZFp4CwWEg",
        "Q": "At about 3:25 Sal distributes D across only 4y but doesn't put parenthesis around 4y(y - 2) so that D can be multiplied across the whole expression. Why doesn't he do that?",
        "A": "Sal doesn t want to distribute the D. He just wants to multiply it with the expression. You only distribute if you are adding or subtracting 2 terms. So if the problem were 4y + (y-2) Then he would distribute the D. Essentially whether you distribute in multiplication or not you will get the same answer."
    },
    {
        "video_name": "KjQ1KN5GgoE",
        "Q": "At 1:17 why would segment BO be 3 like segment CO and not 2 like segment AB?",
        "A": "Because segment BO and segment CO are both the radii of the circle, and all radii of a circle are equal lengths."
    },
    {
        "video_name": "KjQ1KN5GgoE",
        "Q": "At 1:25, how can 2 and 3 be added to get 5 as we don't know whether points A, B and O are on the same line (A-B-O) ?",
        "A": "Well if you want to get really technical, it didn t state that BO was a line either, or that AB was a line - they could all be curves. However, the video is proceeding based on defining the question that way, as one would take from visual inspection. I agree however, in the questions that go with this video, it should state: ABO is a line segment."
    },
    {
        "video_name": "0tXxFPHzFFI",
        "Q": "3:10, I don't quite understand how we can define that \u00ce\u00a3 a_n is \"smaller\" than \u00ce\u00a3 b_n? We compare the summations between them or what?",
        "A": "One of the assumptions established at the beginning is that each term in the a sequence is less than or equal to the corresponding term in the b sequence (that is, a_n is less than or equal to b_n). As a result, the sum of the terms in the a sequence has to be less than or equal to the sum of the terms in the b sequence."
    },
    {
        "video_name": "0tXxFPHzFFI",
        "Q": "At 0:38, Sal writes a[n],b[n] <=0.\nIs this just a given part of the problem i.e. is he saying \"Imagine two series, a[n] and b[n] where both of them are >=0...\"\nOR, did Sal use some information about the series to determine that they are nonnegative?",
        "A": "Sal is defining the properties of a[n] and b[n] so that the comparison test is valid and so that you can see the what is required of the series you wish to test in order to apply the Comparison Test - so your first thought is correct although this isn t a problem, it is (the basis of) a theorem."
    },
    {
        "video_name": "0tXxFPHzFFI",
        "Q": "at 0:49, how are an and bn greater than or EQUAL TO zero if n starts at 1? I don't understand what n is referring to in this sequence",
        "A": "In a_n and b_n, the n refers to the index and a_n >= 0 means that for ANY index n the corresponding term a >= 0."
    },
    {
        "video_name": "LSJuu4Qm2qQ",
        "Q": "Why do you have to take half of the coefficient,then square it? (at 4:49)",
        "A": "(y2-2y+4)+a+b)2 y2-2y+4-a-b+2= y+4+a+2"
    },
    {
        "video_name": "LSJuu4Qm2qQ",
        "Q": "@6:49...how did Sal know it was an ellipse?",
        "A": "The standard form of an ellipse is given by: ((x-h)^2)/a^2+((y-k)^2)/b^2=1. In this particular case we have that h=-3, k=1, a^2=4 and b^2=9."
    },
    {
        "video_name": "FK6-tZ5D7xM",
        "Q": "At 3:04, how is tan(\u00cf\u0080/4)=1? Isn\u00c2\u00b4\u00e1\u00ba\u0097 the point on the x-axis, where the y-axis equals the x-axis, less than 1?",
        "A": "The tangent of any number is equal to the sine of that number over the cosine of that number. Since on a 45\u00c2\u00b0 angle, sine and cosine are both equal to sqrt(2)/2, the tangent is equal to 1, because (sqrt(2)/2)/(sqrt(2)/2) = 1."
    },
    {
        "video_name": "FK6-tZ5D7xM",
        "Q": "At 4:05,why is Sal taking the x coordinate as 'square root of two over two'?",
        "A": "What s the tangent of pi/4? You can calculate it cause x=y and x^2+y^2=1."
    },
    {
        "video_name": "FK6-tZ5D7xM",
        "Q": "How did Sal process the pi/4 as 1 at 3:33?",
        "A": "He probably has it memorized. pi/4 is the same as 45 degrees. Since Tangent is the opposite side over the adjacent side, and a 45-45-90 triangle has to have equal sides, the tangent of 45 (or pi/4) is always 1."
    },
    {
        "video_name": "FK6-tZ5D7xM",
        "Q": "At 8:45 Sal says the tan of 3pi/4 is 1. Wouldn't it be negative 1, since the cos is negative square root of 2 over 2 and the sin is square root of 2 over 2. One negative and one positive make a negative, right? Thanks for any help. Oh, bye the way, my calculator says it's -1.",
        "A": "Perhaps you misheard Sal...I watched the video twice at that point, and Sal says that the tan of 3pi/4 is negative 1 both times, at 8:32 and 8:39. Sal isn t perfect though, I ve seen him make minor mistakes in a few videos. :^)"
    },
    {
        "video_name": "FK6-tZ5D7xM",
        "Q": "At 1:26, what is \"change in y\" over \"change in x\"?",
        "A": "It is the definition for the slope of a line. In this situation, it tells you the steepness of the ray created by angle theta."
    },
    {
        "video_name": "FK6-tZ5D7xM",
        "Q": "Why do you have to draw the dotted vertical lines at 6:48 ?",
        "A": "They are asymptotes. They are not actually part of the graph but are useful to have for reference"
    },
    {
        "video_name": "FK6-tZ5D7xM",
        "Q": "i have seen that the pi/4 and -pi/4 is undefined, but at 4:28 Sal says that it equals 1 and -1. can you help? i don't understand which is correct. thanks so much.",
        "A": "I think you are confusing pi/4 with pi/2"
    },
    {
        "video_name": "FK6-tZ5D7xM",
        "Q": "during 6:26 sal said the slope is getting closer to negative pi over 2 and from looking over the unit circle the bottom y axis shows that it is 3pi over 2?",
        "A": "-\u00cf\u0080/2 and 3\u00cf\u0080/2 are at the exact same position on the unit circle."
    },
    {
        "video_name": "FK6-tZ5D7xM",
        "Q": "At 7:13, why is the slope of the ray hugely negatve after it crosses pi/2?",
        "A": "It is so because it is a type of cubic function which repeat after some period"
    },
    {
        "video_name": "FK6-tZ5D7xM",
        "Q": "The explaination Sal gives at 2:50, does that mean that cos(x) is equal to sin(x)? else I don't see how the division of them gives 1.",
        "A": "At about 2:30 Sal makes it clear that he is talking about a 45 degree (pi/4) angle. And, yes, for 45 degrees, the tangent equals 1 because cos (45) = sin (45)."
    },
    {
        "video_name": "um2nlNVM_YM",
        "Q": "I get the concept but does anyone have helpful hints for remembering 7's so I do not have to skip count / break it down every time like @ 3:00 ?",
        "A": "Switch the numbers. Example: You know your 5 s and 6 s, But you wonder what 7* 5 is. Then switch it so it says 5* 7, Then it kinda seems easier. or Say you have to do 7* 4, but you know what 6* 4 is. Then just add 4."
    },
    {
        "video_name": "um2nlNVM_YM",
        "Q": "[Solved]At 4:37 is 3 X 7 the same as 3 X (5 + 2)?\nI'm assuming yes\u00f0\u009f\u0099\u0082.",
        "A": "The answer is the same."
    },
    {
        "video_name": "um2nlNVM_YM",
        "Q": "At 5:57 what does computing mean ?",
        "A": "It means the process or act of calculation"
    },
    {
        "video_name": "WmdN_iPhxAI",
        "Q": "At 7:57 Grant says the cross product is interpreted as the area of the parallelogram formed by the two vectors. I don't see how this corresponds to the area of the parallelogram without taking the magnitude of the cross product, because crossing the two vectors would result in another vector.",
        "A": "You are right. The geometric meaning of the cross product is that the magnitude of it is equal to the area of a parallelogram formed by two vectors. I think Grant meant the magnitude of the cross product here."
    },
    {
        "video_name": "UbUdyE1_b9g",
        "Q": "At 0:24,how come Sal didn't simplify 8/18? My homework problem hints say that the 1st step is to simplify the fractions.",
        "A": "Your homework problem is not entirely wrong.. but to subtract 2 terms they must have a common denominator And since both had a common denominator, there was no need to simplify the fraction at the beginning"
    },
    {
        "video_name": "UbUdyE1_b9g",
        "Q": "At 0:10 he said adding fractions is very similar to subtracting fractions WHAT DOES THAT MEAN!",
        "A": "When you have converted them so that they have the same denominator (same bottom number - same type of fraction) , then you just add the numerators (the top counting numbers) when you add, or subtract when you - subtract."
    },
    {
        "video_name": "UbUdyE1_b9g",
        "Q": "at 0:39 he dose not say what 4/5 - 1 would be ? what is it ?",
        "A": "To get 4/5 - 1, first you have to make the denominators (the number at the bottom of the fraction) the same. 1 is the same as 5/5, so that s easy. Now you have 4/5 - 5/5, and you work it like Sal showed. The answer is a negative number, but negative fractions are okay. So the answer would be - 1/5 (negative one-fifth)."
    },
    {
        "video_name": "d2cnQ5ahHgE",
        "Q": "I still don\u00e2\u0080\u0099t understand how you find the final answer. How did Sal get -10 at 3:52? Is that the only answer? Please help!",
        "A": "as long as its a number between the range of -14 and -7 the answer will work."
    },
    {
        "video_name": "d2cnQ5ahHgE",
        "Q": "At 5:13 when he says y<-7. Shouldn't the \"<\" be switched to \">\" since the 7 is negative?",
        "A": "No, he subtracted 7 from each side to get there. Addition and subtraction never cause you to flip the direction of an inequality. Only division and multiplication by a negative number flips the inequality."
    },
    {
        "video_name": "d2cnQ5ahHgE",
        "Q": "At about 1:55 in video, Sal says that you don't need to swap the inequality sign because your not dividing a negative but, the four is dividing the negative 56 so wouldn't you need to swap it because of -56. Thanks for your answer",
        "A": "what sal means is if you divide by a negative number you have to swap the inequality and sal divided by POSITIVE 4!"
    },
    {
        "video_name": "d2cnQ5ahHgE",
        "Q": "In 0:40 he says 3y - 2y equals y but wouldn't it be 1y?",
        "A": "1y is the same as y."
    },
    {
        "video_name": "d2cnQ5ahHgE",
        "Q": "At 1:31, he cancels out a few terms. So are inequalities like equalities in that doing something to both sides doesn't change anything (except dividing by a negative)?",
        "A": "Yes, you have it down very well even with flipping the inequality when dividing or multiplying by a negative."
    },
    {
        "video_name": "-nufZ41Kg5c",
        "Q": "At around 1:30, how what does y=3-x mean?",
        "A": "y=3-x is the formula for the line that we are going to reflect across. If we rearrange it we get y=-x+3, where -x gives us our slope, here -1, and 3 is the y-intercept, which we got by making -x = 0. Sal mentions these things in the video but he does it pretty quickly."
    },
    {
        "video_name": "omFelSZvaJc",
        "Q": "At 7:15, Sal says that 291 out of 312 is 31. But...shouldn't it be 21?",
        "A": "You were right, if it was 312 - 291 like you said, but if you watch again, you will see, that it s 322 - 291, so he was right."
    },
    {
        "video_name": "gjrGd9TjjnY",
        "Q": "at around 1:26 sal writes 2*5 as 2.5,I KNOW that it means the same as multiplication but can't people get confused thinking it as a decimal point.",
        "A": "Usually, you don t get confused by this. 2.5 and 2\u00c2\u00b75 are different for the eye."
    },
    {
        "video_name": "gjrGd9TjjnY",
        "Q": "At 4:14, why did Sal get -28 instead of 28? My math teacher taught me something different.",
        "A": "He s starting at -3 and he has to take away 25, so he gets -28."
    },
    {
        "video_name": "gjrGd9TjjnY",
        "Q": "At 1:32 when he wrote -1{(-7)+2*5} why do you need to put -7 in parentheses if there are already brackets?",
        "A": "The parentheses do not change the problem. It s always advisable to keep parentheses around your negative numbers when doing long operations until you re ready to use the negative number in order to avoid confusion."
    },
    {
        "video_name": "gjrGd9TjjnY",
        "Q": "At 1:55, he talks about distributing the -1. I had learned it that way, and I'm getting confused on if i should distribute it first, or multiply it the way he did.",
        "A": "either way is acceptable"
    },
    {
        "video_name": "gjrGd9TjjnY",
        "Q": "At 1:01, do the brakets count as parenthesis",
        "A": "Parentheses can look like: ( ); [ ]; { } - these are curly brackets though they don t look like it here. And, when you go into algebra, you will learn that the symbols used for absolute values also act as parentheses (these are straight bars: | | ). It is very common to alternate the style used (as shown in the video). This makes it easier to see where one set of parentheses start / stop vs. another set. Hope this helps."
    },
    {
        "video_name": "gjrGd9TjjnY",
        "Q": "A 1:30 why did he do the exponent before the rest of the parentheses and brackets, the exponents were outside the parentheses?",
        "A": "love the avatar"
    },
    {
        "video_name": "gjrGd9TjjnY",
        "Q": "at 1:41, could you erase the parenthese around the -7? does that matter?",
        "A": "you can erase the () brackets and work.. (you should still work the 2x5 first though) DO NOT erase the [] one"
    },
    {
        "video_name": "gjrGd9TjjnY",
        "Q": "4:05\ni don't get this part,\n-3 and -25 are two negative numbers, there is no sign between them (+ or -)\nalso, i thought two negative numbers would turn into a positive number...",
        "A": "He is adding -3 and -25. They give an answer of -28. Negative times Negative equals Positive. Negative divided by Negative equals Positive. Negative squared is Positive. With adding and subtracting, the answer can be Negative or Positive."
    },
    {
        "video_name": "gjrGd9TjjnY",
        "Q": "at 1:44 what is a brachet",
        "A": "this is a bracket; [ ] braces; { } parentheses; ( )"
    },
    {
        "video_name": "gjrGd9TjjnY",
        "Q": "At 1:24,\n-1 [(-7) +2 (3+2)]-(5)squared..............next steps...\n-1 [(-7) +2 (5)] -25..........next step?\n-1[(-7) + 10]-25 ? There is no sign between the 2 and the 5 (that is in brackets) and it there becomes a multiplication sigh between them? Why is this?\nThe same thing also happens at 4:00.\n??? Does it always become multiplication, if there is no sigh between number.... or does it have something to do with that the 5 is in brackets....?\n\nThanks.... Cheers.",
        "A": "why do u have to parenthesis then exponents then every thing else?"
    },
    {
        "video_name": "gjrGd9TjjnY",
        "Q": "At 1:09, why do we solve the exponent before finishing what's in the brackets?",
        "A": "(5)^2 = 5^2. they put the parentheses there so you didn t think is was negative 5^2. the same with (-7) vs. -(7). If you re talking about the brackets, you aren t technically supposed to do it, but it s rather common with standard arithmetic to not go left to right as long as it doesn t interfere. the 5^2 changing to 25 first doesn t change anything because no matter what it would ve happened before the subtractions. TL:DR - it s kinda just being informal as long as it works."
    },
    {
        "video_name": "mLRYCaYVWFc",
        "Q": "Sal said at 3:28 that the swimming pool has measurements of about 25m wide, 50 m long and 2 m deep, which gives a fluid volume of 2,500,000. By multiplying them I get 2,500. So where does he get the 1000 to multiply it and get 2,500,000? how much litres are in 1m cubed? perhaps it is 1000 liters in a meter cubed? if so, is that the rule or just for this example?",
        "A": "There are 1000 liters in a cubic meter. :)"
    },
    {
        "video_name": "qG3HnRccrQU",
        "Q": "What is an incremental side at 4:09?",
        "A": "The word incremental just means increasing. So if you have 3 sides and he uses the term each incremental side he means any side that is more than 3 sides"
    },
    {
        "video_name": "qG3HnRccrQU",
        "Q": "At 4:46, can't you make more than just 4 triangles in that hexagon? Isn't there supposed to be 6? Is that not what we were taught in 3rd grade?",
        "A": "Always use the smallest possible number of triangles."
    },
    {
        "video_name": "qG3HnRccrQU",
        "Q": "in part 2:29 when he says a irregular triangle isn't a regular pentagon that he draws?",
        "A": "He didn t say a irregular triangle, he said an irregular Pentagon. He said that because it was not like a regular looking pentagon."
    },
    {
        "video_name": "qG3HnRccrQU",
        "Q": "he did it wrong! one of the \"triangles\" at 7:00 is actually a quadrilateral",
        "A": "Yes, but he does fix it later on in the video, at 7:27. I was wondering about that too!"
    },
    {
        "video_name": "qG3HnRccrQU",
        "Q": "At 1:12, what if you were to cut the quadrateral into more triangles? Would the sum of the angles in quadralateral increase because each of the triangles are 180 degrees, or would the quadralateral still have the sum of 360 degrees?",
        "A": "yes. it still would be 360. Will if you cut it into half which would become two triangle s which equal 180 each so do 180*2 it would equal 360."
    },
    {
        "video_name": "qG3HnRccrQU",
        "Q": "I don't understand the formula concept 7:00 \u00e2\u009e\u00a1S-2\nSomeone could help me plz.",
        "A": "All right. S-2 means you subtract two from the number of sides,"
    },
    {
        "video_name": "qG3HnRccrQU",
        "Q": "Sal says at 0:05 that if triangle Z has angles of a, b, and c, then what would be the sum of the exterior angles be? Do they have to equal a certain number?",
        "A": "Yup. The sum of exterior angles in ANY polygon is always 360\u00cb\u009a."
    },
    {
        "video_name": "qG3HnRccrQU",
        "Q": "At 4:00,when you divide a polygon into triangles are you splitting it into the minimum amount of triangles?",
        "A": "If you try splitting a polygon into triangles (whose vertices are all vertices of the original polygon) in a couple different ways, you will become convinced that it will always produce the same number of triangles."
    },
    {
        "video_name": "qG3HnRccrQU",
        "Q": "At 5:00 What is a S-Sided Polygon?",
        "A": "S is a vague term for all the possible number of sides."
    },
    {
        "video_name": "qG3HnRccrQU",
        "Q": "In 7:41, isn't it 9 sides, not 10?\nTell me please :)",
        "A": "it is not 9 it is 10 mabye look again"
    },
    {
        "video_name": "qG3HnRccrQU",
        "Q": "At 4:30pm, So every kind of triangle no matter how much fat or thin it is it adds up to 180 degrees?",
        "A": "If a triangle didn t there are two reasons that comes to mind: 1.It is not a triangle, and I am confused 2. We have been wrong about triangles for quite a while"
    },
    {
        "video_name": "qG3HnRccrQU",
        "Q": "At 1:49 wouldn't you put parentheses so you could say b+z+(a+x)+(c+y)?",
        "A": "You could. You re still just using addition though though, so it will be the same regardless of which ones you add first. b+z+a+x+c+y is the same thing as b+z+(a+x)+(c+y)."
    },
    {
        "video_name": "ErHFQUhZ94I",
        "Q": "At 0:25 when Sal said that he likes to to put the bigger number on top, and my question is why should we do that?",
        "A": "it doesn t really matter unless you are subtracting"
    },
    {
        "video_name": "dviyBH3pOk8",
        "Q": "0:1 how can you find out how to divide the zelda triangle in tenths",
        "A": "You mean the triforce!"
    },
    {
        "video_name": "O3jvUZ8wvZs",
        "Q": "at 4:35, why do you multiply the 150 degrees with the pi radians/180 degrees instead of set it equal to it? and at 6:09 why do you bring the radian down to the denominator when you eliminate degrees?",
        "A": "1) pi/180 = 1. 180 deg = pi radians = a half circle. The number of half circles in 150 deg is 150/180 = 5/6. In radians, that s (5/6)pi = (150/180)*pi = 150(pi/180). 2) @ 6:09 he doesn t bring radians to the denominator. He s just doing (-45 deg)*(pi/180) = -(45/180)*pi = -pi/4 radians."
    },
    {
        "video_name": "O3jvUZ8wvZs",
        "Q": "At 1:25, Sal wrote that the circumference is 2pi radii, then at 1:54, Sal wrote 2pi radians:\nIs a radian equal to the radius. Or is it the same thing, just named differently? I mean, isn't the radian basically the radius bent around on the circumference. I'm really confused right now, so I would really appreciate it if someone would clearly explain this to me... the simpler the explanation, the better :) Thanks in advance!",
        "A": "You re right about the way you visualize the definition of a radian. It is the angle subtended by an arc equal in length to the radius. That is, one radian is the angle you would go through if you went one radius-worth of length on the circle. The radius, however, is a length measurement while the radian is an angle measurement."
    },
    {
        "video_name": "O3jvUZ8wvZs",
        "Q": "At 5:00 Sal is finally getting to the answer after explaining the idea behind how it works. The formula for what he is doing is:\n\nd\u00c2\u00b0 / 180\u00c2\u00b0 = r / \u00cf\u0080\n\nso to convert 150\u00c2\u00b0 (d\u00c2\u00b0) into radians (r)\n- set up the equation\n150\u00c2\u00b0 / 180\u00c2\u00b0 = r / \u00cf\u0080\n-cross multiply\n180\u00c2\u00b0r = 150\u00c2\u00b0\u00cf\u0080\n-divide by 180\u00c2\u00b0 to solve for radians (r)\nr = 5\u00cf\u0080/6\n* do not multiply your number by 3.14, leave it as \u00cf\u0080.\n* you want to find the fraction, not divide the number to get the decimal",
        "A": "Well, what he s really doing is showing the conversion factor from radians to degrees. Since 2pi = 360\u00c2\u00b0, pi = 180\u00c2\u00b0. From that, we get the conversion factor from degrees to radians, which is pi/180\u00c2\u00b0, and the conversion factor from radians to degrees, which is 180\u00c2\u00b0/pi."
    },
    {
        "video_name": "O3jvUZ8wvZs",
        "Q": "At 5:11, how did you have 5 as the numerator and 6 as the denominator? I'm really confused right there, and kinda a slow learner.",
        "A": "He s reducing the fraction. The fraction 150/180 can be reduced, since 150 and 180 have common factors. Their greatest common factor is 30. So, 150/30 = 5 and 180/30 = 6. Thus 150/180 reduces to 5/6. 150pi/180 reduces to 5pi/6."
    },
    {
        "video_name": "O3jvUZ8wvZs",
        "Q": "5:11 how did he get 30",
        "A": "Both 150 and 180 are multiples of 30."
    },
    {
        "video_name": "O3jvUZ8wvZs",
        "Q": "At around 6:40, why does Sal divide by 45 and not -45?",
        "A": "All he does is simplify the fraction -45/180 to -1/4. If he divided by -45, then he d get 1/-4, so the negative is still there."
    },
    {
        "video_name": "O3jvUZ8wvZs",
        "Q": "At 01:52, Sal says that 360 degrees is 2 * pi * radii, or 2 pi radians. What's the difference between them?",
        "A": "They re not equal. 2*pi*radius = 2*pi*r is the circumference, and 2*pi radians is (2*pi*r)/r = 2*pi = the angle of a full circle. He s not saying they re equal. He s comparing them."
    },
    {
        "video_name": "EXIUYib9d8M",
        "Q": "At 1:42, the sum of geometric sequence is a/(1-r), but this is only true for |r|<1, isn't it?\nApplying this to f(x) seems like a bit of a stretch to me.",
        "A": "Yes, you are correct. This series converges when |-x\u00c2\u00b3| < 1 \u00e2\u0087\u0094 |x| < 1."
    },
    {
        "video_name": "F5uiFXECnVY",
        "Q": "At 1:02, what does Sal mean by an outlier in the graph?",
        "A": "Something that doesn t fit in with the rest of the points"
    },
    {
        "video_name": "EC2mgUZyzoA",
        "Q": "Hi there I was wondering what exactly defines a column vector as Sal shows starting 6:15; also confusing for me because I don't understand how he was able to change linear equations into vectors....... PLEASE I WANNNA KNOW DX",
        "A": "A column vector is just a matrix with only one row. They re just another way of writing vectors."
    },
    {
        "video_name": "EC2mgUZyzoA",
        "Q": "At 6:09, Sal names 2 of the matrices column vectors. What do vectors have to do with matrices?",
        "A": "A column vector is just a matrix with only one row. All vectors can be thought of as matrices in this way."
    },
    {
        "video_name": "EC2mgUZyzoA",
        "Q": "At 6:31, was that supposed to be the dot product?",
        "A": "No. He is just rewriting the equation with the names of the matrices. Sal explains dot product at 4:06."
    },
    {
        "video_name": "EC2mgUZyzoA",
        "Q": "At 7:39, Sal says that it matters whether you multiply on the right side or the left side as it affects the outcome. He multiplies on the left side. What would happen if you multiplied on the right side? What would be the difference?",
        "A": "If you were to do this then you would get A \u00f0\u009d\u0093\u008d A\u00e2\u0081\u00bb\u00c2\u00b9 = \u00f0\u009d\u0092\u00b7 A\u00e2\u0081\u00bb\u00c2\u00b9. We know that A A\u00e2\u0081\u00bb\u00c2\u00b9 = \u00f0\u009d\u0090\u00bc but we can t say anything about A \u00f0\u009d\u0093\u008d A\u00e2\u0081\u00bb\u00c2\u00b9 because of the \u00f0\u009d\u0093\u008d in the middle. Also the dimensions don t work out. \u00f0\u009d\u0093\u008d is 2x1 but A\u00e2\u0081\u00bb\u00c2\u00b9 is 2x2. so you can t multiply A\u00e2\u0081\u00bb\u00c2\u00b9 on the right."
    },
    {
        "video_name": "Wm0zq-NqEFs",
        "Q": "At 1:59 why did i matter so much about the colors? Does it really matter to be so neat?",
        "A": "It also draws your attention to similar problems"
    },
    {
        "video_name": "Wm0zq-NqEFs",
        "Q": "Why couldn't you just use at 2:00 ? Why couldn't you just use 5+8+1=9?",
        "A": "what is 111,111,111 X 111,111,111"
    },
    {
        "video_name": "Wm0zq-NqEFs",
        "Q": "4:09 what ways can you add",
        "A": "Sagar, there are many ways to add ,but starters use their fingers or a counting chart"
    },
    {
        "video_name": "Wm0zq-NqEFs",
        "Q": "Why do we do carrying is it at all necessary on 0:25",
        "A": "Carrying is just easier, and yes, it is necessary. We might be in a timed test, and we can t say, Well, 70 plus 50 is 120, and 7 plus 5 is 12, and I have to add that up. You probably wouldn t finish in time. Hope this helps!"
    },
    {
        "video_name": "Wm0zq-NqEFs",
        "Q": "what was he saying at 0:37 ?",
        "A": "Try using the subtitles in english"
    },
    {
        "video_name": "pPnxPrhf6Ww",
        "Q": "At 6:24, Sal says the mode is the middle number. Does he mean to say the median?",
        "A": "Yes, it seems like it should be median . However, in this specific data set, the mode and the median are the same, which is probably what tripped him up while talking."
    },
    {
        "video_name": "pPnxPrhf6Ww",
        "Q": "In 2:28 what did you mean by \"it's almost like it is running out of digital link or something\"?",
        "A": "Sal s digital pen for writing was not writing the numbers cleanly for that color, as if the drawing tool he was using had run out of ink to write with."
    },
    {
        "video_name": "pPnxPrhf6Ww",
        "Q": "At 2:01, it's confusing.",
        "A": "At 2:01, Sal just realized he had left out the outlier at 19, (an outlier is a value way outside of the data cluster), so he had to add it to the total number of values to get an accurate mean (average)."
    },
    {
        "video_name": "SuB1gkto9LU",
        "Q": "At 1:33, Sal mentions that these problems are when dealing with two dimensions. How would 3d be different?",
        "A": "The main difference in 3 dimensions is that it is harder to see the no solutions or infinite solutions cases by just looking at the equations. You will usually need to go though a few steps before you realize things aren t going to work out."
    },
    {
        "video_name": "SuB1gkto9LU",
        "Q": "At 1:33, Sal mentions that these problems are when dealing with two dimensions. How would 3d be different?",
        "A": "A problem involving 3 divisions would have three variables and three equations."
    },
    {
        "video_name": "p7ycTWq6BFk",
        "Q": "At 3:27, if the numerator value is greater than the denominator value, how do you determine the horizontal asymptote then?",
        "A": "In that case (when the numerator s degree is smaller than the denominator s degree), the horizontal asymptote is always y = 0"
    },
    {
        "video_name": "LG6g7gxupHg",
        "Q": "3:50 Why is it that we are able to evaluate V(2)? Seems like the domain of the composition should be x> or = 5. Thanks!",
        "A": "Good question. After you compute V(d(t)), then simplify, you get 4t^3, and sqrt(t-5) is gone. Does that mean it no longer matters? I think it might - I m not sure though."
    },
    {
        "video_name": "vfMx9-rJXuo",
        "Q": "At 2:14 why did Sal forget that it was 2 fours not two twos?",
        "A": "No one is perfect"
    },
    {
        "video_name": "lxtlwnCV-HM",
        "Q": "At 2:17, Sal multiply the function (g) by -6... How come he did not multiply (h) by 3?",
        "A": "Sal never multiplied g(x) by -6. (h o g)(-6) means 1) find g(-6). This means you make the x in g(x) into -6 and find the output, which = 3 2) then use the 3 as the input to h(x). Thus, Sal is now finding h(3). This means the x in h(x) becomes 3 and you calculate the output, which = 7 Hope this helps."
    },
    {
        "video_name": "lxtlwnCV-HM",
        "Q": "2:41 it is not 36 - 33 scince 30 - 3 is not 33 it is supposed to be 36 - 27",
        "A": "When substituting into g(x), the second term becomes - 30, not + 30, so Sal adds -30 to -3 and gets -33."
    },
    {
        "video_name": "lxtlwnCV-HM",
        "Q": "At 1:07 what does it mean if h is \"of\" g? I have never heard the \"of\" term used before.",
        "A": "Ethan, h of g simply means the same as it would when you said h of x or h of 2. I hope that answers your question."
    },
    {
        "video_name": "ahdKdxsTj8E",
        "Q": "At 1:08 I am confused would that not change the answer by simplifying it like this,seeing as 5x sqaured equals 25x and 2x sqaured equals 4x so together that would be 29x,which is far less than if you simplified it like shown which would be 49x. Or does this have to do with this being an expression ?",
        "A": "You are changing 5x^2 into *(5^2)x*. In 5x^2, the exponent applies to the X, not the 5. You did the same thing with 2x^2. It doesn t become (2^2)x. The exponent is one the X, not on the 2. 5x^2 = 5 * x * x 2x^2 = 2 * x * x Hope this helps."
    },
    {
        "video_name": "ahdKdxsTj8E",
        "Q": "At 0:23 I don't understand why you get rid of the parenthesis?",
        "A": "Parenthesis doesn t matter in this equation. You can think the equation as 1*(5x^2+8x-3)+1*(2x^2-7x+13x) and when you multiply with 1 s neither of the equations change and you can get rid of the parenthesis."
    },
    {
        "video_name": "ahdKdxsTj8E",
        "Q": "At 0:43 Sal says \"... if this were a minus, we'd have to distribute the minus sign.\"\n\nIf it were a minus sign and it had to be distributed would the expression then look like this?\n\n5x\u00c2\u00b2 + 8x - 3 - 2x\u00c2\u00b2 + 7x - 13x",
        "A": "Yes, it would. You would then combine like terms to complete the problem."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "@2:00 The video says the yellow Angle is \"pi minus theta\". Is the angle that is being reference end where the yellow arrow is or the entire yellow line?",
        "A": "I think it is the arc from 0 to the line."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "I still don't understand why sin(a)=sin(180-a).\nIn the video, at 4:08 - How can you measure sin(180-a)? It's an obtuse angle.. (it's not even a right triangle). I can see why sin(a1)=sin(a2) where a2 is on second quadarant (yellow theta in video) and a1 is the green theta.\nThanks.",
        "A": "Hi, To understand why sin(a) = sin(180-a) look at the projection of the angle toward the y axis, instead of looking at the shape formed by the angle. In other words, from the tip of the angle drop an altitude toward the x axis. You ll see that that altitude has the same sign and magnitude either in the first or second quadrant. Different is the case of the cosine function there, because it switches sign (from the positive x side goes to the negative one)."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "At 1:52 Sal says that the angle in yellow is theta...bt isnt it vertically opposite to -theta and thus = -theta ?? if so, shouldnt the formula be pi-(-theta) = +theta??",
        "A": "It has magnitude theta. The actual angle of the yellow ray is the complement of the original theta, pi-theta. He uses theta to mean 1) the original theta in the 1st quadrant, or 2) an angle of the same magnitude. 3) any angle. It might be a little confusing, but when you get it I think it all makes sense."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "at 3:45, wouldnt cos of theta be positive because the x is positive? and the sin be negative because the y is negative? why are they both negative?",
        "A": "It s only the angle, or theta, that s negative. If you were to simplify the negative angles, then cos(-\u00ce\u00b8) = cos(\u00ce\u00b8) and sin(-\u00ce\u00b8) = -sin(\u00ce\u00b8)."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "At 6:40, why is the cos(theta) = -cos(pi - theta), not just equal to cos(pi-theta)?",
        "A": "Because pi - theta is the reflection of theta across the y axis, which makes x negative."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "At 1:53, why isn't the angle theta negative? Flipping the green line (terminal side) over the positive x-axis changes the direction of the angle, right? and so the angle is now going clockwise. right ?",
        "A": "This is because when working in radians you always presume the angle starts from the position of the x axis. They are positive if moving from the x-axis in an anti-clockwise direction and negative angles if moving in a clockwise direction. The theta is really to show you that the angle is the same, the actual angle in radians is (Pi - theta) as that shows the position from the x axis."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "At 6:48 where did Sal get cos theta = - cos pi minus theta?",
        "A": "The cosine of the angle is the x-coordinate of the point on the unit circle. Notice that the x-coordinate of a point at \u00ce\u00b8 radians is the opposite of the x-coordinate (with the same value, only opposite) of a point at \u00cf\u0080-\u00ce\u00b8 radians. Thus, the cosines or the angles are the same, only opposite."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "At 1:58 why he could said that the measure of half circle is PI?",
        "A": "The formula for the circumference of a circle = 2 pi r So, when working with the unit circle, r = 1. Thus, the circumference = 2 pi. If the entire circle = 2 pi, then half the distance around the circle must be pi. Hope this helps."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "What property is being used for angle when Sal makes statement \"this is theta, since this is theta\" @ 1:52 ? Thanks :)",
        "A": "There is a green theta and a blue minus theta. Those two angles are the same size, but the minus only indicates a different direction. I can t say what Sal intended, but one way of looking at this is that the yellow angle and the blue angle are vertical angles (a pair of opposite angles made by two intersecting lines) and they therefore equal each other."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "Starting at 1:29, why isn't the angle pi - 2(theta)?\nPi would be the whole top of the circle, and we are taking it out twice?",
        "A": "I had the same confusion when I first watched this. At around 4:00 Sal makes it clear that the \u00cf\u0080 - \u00ce\u00b8 angle does begin at the positive x axis, even though at 1:29 it seemed like it was beginning at the terminal ray that was defining \u00ce\u00b8. If it had been starting at the ray, then the angle would have been \u00cf\u0080 - 2(\u00ce\u00b8). But since it starts from the the x axis, then it is only \u00cf\u0080 - \u00ce\u00b8."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "At 5:30, is it okay to say that absolute value of the sin of negative theta is equal to the sin of theta?",
        "A": "No. For instance: If \u00ce\u00b8 is -45\u00cb\u009a, then sin(-\u00ce\u00b8) = \u00e2\u0088\u009a2/2, but the absolute value of this is not the sine of -45\u00cb\u009a."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "at 1:53, Sal says that the yellow angle is theta.\nbut the yellow angle should be -theta because it is vertically opposite to blue angle (-theta)\nthen why so ?",
        "A": "Since it is not in standard position, the orientation is unclear. But you could imagine the initial side of the yellow angle theta to be the yellow ray, and the terminal side to be the negative x-axis."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "Didn't get anything from 6:45 and onwards.",
        "A": "I would suggest you to watch the video again from the beginning, use a lot PAUSE and REPLY (partly or fully), with a pen and a paper to write down the notes, then move up and down the video, in order to understand the entire video."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "At 3:34, why is that point cos(), sin ()?",
        "A": "Cosine is defined as cos\u00ce\u00b8 = x/r and sine is defined as sin\u00ce\u00b8 = y/r. Since r = 1 in a unit circle, cos\u00ce\u00b8 = x/1 = x and sin\u00ce\u00b8 = y/1 = y."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "From 5:08 - 5:37, how does he make it that -sin(theta) = sin (-theta)? I understand that one length is negative and one is positive, but how are they so different? Can someone give me a mathematical explanation? Thanks !",
        "A": "You just have to accept what you can see on the unit circle with your own eyes. Sin is the sign of y, cos is the sign of x, wherever the angle puts them."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "I don't understand why the purple theta, placed at 2:43, is positive. Why did Sal measure that angle clockwise instead of counterclockwise? Thank you for taking the time to read and answer this.",
        "A": "He did measure it counterclockwise, and it was for the purpose of describing the sine and cosine identities for the angle \u00ce\u00b8. More specifically, the purple \u00ce\u00b8 is visually addressing the fact that the sine values (or y values of the coordinates that intersect the unit circle) of (-\u00ce\u00b8) and (\u00cf\u0080+\u00ce\u00b8)."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "At 7:20 and 7:35 why is sin and cos of theta plus pi equal to the negative sin and cosine of theta? I would think its positive sin or cos.",
        "A": "by adding pi to theta, the point on the unit circle is essentially rotated 180 degrees (pi radians) about the origin. that transforms the point (x,y) to (-x,-y). since sin(theta)=y and cos(theta)=x, sin(theta+pi)=-y and cos(theta+pi)=-x. you can test this by calculating the sin and cos of a random value theta and then calculating the sin and cos of pi + theta. hopefully this helps."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "At 3:30, Sal says that the horizontal value is sine of theta and that the vertical value is cosine of theta. Shouldn't it be opposite? In other words, shouldn't the x value be cosine of theta and the y value be sine of theta?",
        "A": "You are misunderstanding him. Like you are saying, the x value is the cosine and the y value is the sine. That is what he is saying, too."
    },
    {
        "video_name": "tzQ7arA917E",
        "Q": "At 4:46, I am comparing the blue and green rays. Is (cos(-theta)), sin(-theta)) also equal to (cos(theta), -sin(theta))?",
        "A": "Yes. Cos(-theta) = cos (theta) and sin(-theta) = - sin(theta) Well done for spotting that relationship. It is easier to see in the unit circle."
    },
    {
        "video_name": "0cHCpgQD_8k",
        "Q": "@4:51 If the median is six and the range is 4, isn't it impossible for the minimum to be less than 2? For example if the median AND the maximum were both 6? Yeah, these problems confuse me too.",
        "A": "You are correct gkniffen, it is impossible for the min to be less than 2, and Sal was wrong when he said that it can be 1 in the timestamp @4:51 you pointed out. You can have 2 as the min, because you can have all the slots from the median to the right, all be 6 s."
    },
    {
        "video_name": "OCNXS_m1HWU",
        "Q": "Sal why hadn't u taken Pink at 1:31 in B\\C.",
        "A": "B \\ C = { every element in B that is not in C } Pink is not in B to start with so it will not be in B \\ C Another way to look at it is that B \\ C is a subset of B which implies that any element of B \\ C is in B"
    },
    {
        "video_name": "OCNXS_m1HWU",
        "Q": "From 3:17, why set C is not in the \"all things in the universe\" ? (it has elements that are neither 0 or blue)",
        "A": "All things in the universe relates to whole expression in parenthesis, so C gets simplified before we finding this all things ."
    },
    {
        "video_name": "OCNXS_m1HWU",
        "Q": "From 3:17 why is C not in the \"all things of the universe\"?",
        "A": "Thanks a lot:)"
    },
    {
        "video_name": "ReEMqdZEEX0",
        "Q": "at 4:22 I still don't understand why you draw the hole there can someone explain in more detail ike what does he mean by that they cancel out",
        "A": "For any values of x other then x=-1 then (x+1)/(x+1) will equal 1 because the numerator and denominator have the same value so they cancel each other and the result is 1. For example if x=2 then (x+1)/(x+1)=(2+1)/(2+1)=3/3=1/1=1. But when x=-1 then (x+1)/(x+1)=(-1+1)/(-1+1)=0/0=undefind because of division by 0, so he uses the hole at (-1,1). The hole just means at x=-1 there is no y value."
    },
    {
        "video_name": "ReEMqdZEEX0",
        "Q": "at 7:12 why is the graph in the second and fourth quadrants? The function isn't negative so shouldn't it be in the first and third?",
        "A": "Do you know how to graph a function? Do that, and just let the y values be what they are. Whenever (x, f(x)) is in a quadrant, it s supposed to be there."
    },
    {
        "video_name": "ZsxMPW083mI",
        "Q": "at 4:30, you calculated the answer in radians. Why did you use radians and how do you know when to use radians or degrees?",
        "A": "When in doubt, assume radians. Almost all mathematicians use radians by default. You can tell the difference between radians and degrees by looking for the \u00c2\u00b0 symbol. If the numbers of an angle measure are followed by a \u00c2\u00b0, it will be degrees. Otherwise it will always be radians. I hope this helps!"
    },
    {
        "video_name": "ZsxMPW083mI",
        "Q": "At 5:23, may I use an integral of R(t) and D(t) to evaluate the total amount of water in the pipe and compare whether the amount of water is decreasing or increasing? I've used an integral and D(t) is greater than R(t), I just want to know whether I can use an integral? Thank you!",
        "A": "For the total amount use will have to use this formula R(t) - D(t). Then, integrate it and figure out whether the amount of water is increasing or decreasing."
    },
    {
        "video_name": "iulx0z1lz8M",
        "Q": "At 3:50 when you found the square root, what would you do if it wasn't a perfect square?",
        "A": "You would still have 2 answers, but both would have a square root radical within (which had been simplified as much as possible) For example, the answers of ( - 2 + 5 * sqrt3 ) / 7 AND ( - 2 - 5 * sqrt3 ) / 7"
    },
    {
        "video_name": "iulx0z1lz8M",
        "Q": "At 2:46 why didn't you multiply the 4 and -7 first?",
        "A": "First of all it is a negative four not a positive but the associative property of multiplication states that a multiplication problem (in our case -4*-7*9) can be solved in any order. (-4*-7)*9 is the same as (-4*9)*-7. Also, (-7*9)*-4 is the same. Hope it helped!"
    },
    {
        "video_name": "iulx0z1lz8M",
        "Q": "at 5:34 why do you use a ,b and c as a variable?",
        "A": "he used a,b, and c as examples to show the viewer the basic format of a quadratic formula."
    },
    {
        "video_name": "iulx0z1lz8M",
        "Q": "at 2:44 why do you multiply the -4 then 9 then -7?",
        "A": "Because in the formula, that part says -4ac or -4 (a value) (c value) - 4 (9) (-7) 252 Hope that helped."
    },
    {
        "video_name": "iulx0z1lz8M",
        "Q": "4:09 what do you mean by + or -?",
        "A": "By + or -. He means that either way you would still come out with 0 or the - and + can also be the solution set of the quadratic formula..."
    },
    {
        "video_name": "8y7xP4zz0UY",
        "Q": "At 0:59, Sal has to do long division to try to figure out the factors. Is there an easier/ less time consuming way of doing this? Like tricks for looking at numbers and being able to skip some parts of this lengthy process?",
        "A": "Sal could have skipped the first long division, or 1715/5, by simply factoring out 10 first. This gives him factors of 10 and 343, easily seen without long division. The 10 then factors down into the same 2 and 5 he started with. For the 343, however, I don t know of an easy way to factor that further without the long division."
    },
    {
        "video_name": "8y7xP4zz0UY",
        "Q": "what if the given number is a prime number?\n\n(at 00:33)",
        "A": "keep on dividing and use a decimal"
    },
    {
        "video_name": "8y7xP4zz0UY",
        "Q": "At 4:45, and pretty much most of the video, I did not understand how you were even supposed to get something like the cube root of 192",
        "A": "You can just use a calculator to get the cube root. 192 = 13.856 The cube root of 192 is about 13.9"
    },
    {
        "video_name": "DqeMQHomwAU",
        "Q": "5:38 pm There are 28 boys in the marching band. The boys are 7/10 of the students in the marching band. What is the total number of students in the marching band?",
        "A": "Set up the equation like this: 7/28 = 10/x where X represents the total number of students in the marching band. Start off by multiplying x on both sides to get 7x/28 = 10. Now multiple both sides by 28 to get 7x = 280. Now divide both sides by 7 to receive the final answer of x = 40."
    },
    {
        "video_name": "DqeMQHomwAU",
        "Q": "In 1:57 Sal said medium long division. What does that mean",
        "A": "I think he means division that is not super long but not super short. I hope this helps! :)"
    },
    {
        "video_name": "RHe9X2HDEjA",
        "Q": "at 1:23 is the answer anything between 147.5 and 152.5?",
        "A": "That s right."
    },
    {
        "video_name": "DaQlieZH1kk",
        "Q": "at 3:56, how could we get the same answer in a diffrent process?\ni refer to:2x5=10,and 5x2=10\nbut if its 2-digited,how can we get the same answer?",
        "A": "Oh, in that case, you have to just wrap your mind around it, or in other words you kind of have to deal with it."
    },
    {
        "video_name": "DaQlieZH1kk",
        "Q": "at 0:14 how do you solve the two numbers",
        "A": "Multiply the 7 by the 6 and then by the 3."
    },
    {
        "video_name": "CZ5ne_mX5_I",
        "Q": "At 0:39 why would you add the exponents and not just multiply 4 with as the -3 exponent (64) with the 4 with the 5 exponent (1024)? besides the answer being way to big?",
        "A": "Because 4^-3 is actually 1/64. Therefore 4^-3(1/64) * 4^5(1024) = 1024/64 = 16. When you do what Sal does, you get the same answer: 4^-3 * 4^5 = 4^-3+5 = 4^2 = 16"
    },
    {
        "video_name": "xC-fQ0KEzsM",
        "Q": "at 2:09, it says that they are good divisors, how would you know if a number is a good divisor? what are the traits and evidence?",
        "A": "What Sal means is that both 12 and 75 are divisible by 3. Technically, anything is divisible by 3, but he means you get a whole number when you do 12/3 or 75/3. Also, 3 is the GCF of 12 and 75, so it is the best choice to use for factoring in this case."
    },
    {
        "video_name": "xC-fQ0KEzsM",
        "Q": "at 2:21 where did u get 25?",
        "A": "75 is egual to 25 x 3"
    },
    {
        "video_name": "xC-fQ0KEzsM",
        "Q": "around 2:50, he just got rid of one of the three, is that possible? and their just okay with that?",
        "A": "What he s doing is he s taking one expression, (12+75), and factoring it. He splits this one expression into two: (3*4)+(3*25). Then, to simplify it, he combines these into 3(4+25). This is because there is a common factor in the two expressions:3. In the distributive property, anything attached to the outside of the parentheses means you have to multiply it by each term. So, 3(4+25)=(3*4)+(3*25). All in all, taking away one of the threes was necessary to simplify the expression. Does that make sense?"
    },
    {
        "video_name": "-jqWW22FMdI",
        "Q": "At 1:44 why is it 9pi squared?",
        "A": "The area of a circle equals pi times the radius squared The circumference of a circle equals 2 times pi times the radius"
    },
    {
        "video_name": "-jqWW22FMdI",
        "Q": "At 1:43 why do we get 9 pi/",
        "A": "In this example, you know that the radius of the circle is 3. The area of a circle is \u00cf\u0080r^2 and squaring a number means that you multiply it by itself. Therefore, the area is 3 x 3 x \u00cf\u0080 which is 9\u00cf\u0080."
    },
    {
        "video_name": "-jqWW22FMdI",
        "Q": "At 0:38 Sal mentions that from Circumference you can figure out Radius; and then from Radius you can figure out Area. What does he mean by that?",
        "A": "If you know the circumference (say 20pi), you can find the radius, because every circumference is just the radius times 2 times pi. We can work backwards and divide 20 pi by pi, to get 20, then divide that by 2 to get the radius of 10. You can find any area if you have the radius, because all areas of circles are pi times the radius squared. If our radius is 10, then 10x10 is 100, and times that by pi to get the area of 100 pi (or about 314 units squared)"
    },
    {
        "video_name": "BJSk1joCQsM",
        "Q": "a. If the ratio of the sides of two squares is 2:5, what is the ratio of their areas?\n\nb. If the ratio of the edges of two cubes is 2:5, what is the ratio of their surface areas?\n\nc. If the dimensions of a rectangular box are doubled what happens to its surface area?",
        "A": "A: area1 = 4x^2 (where x is the value of the side length of the first square divided by two which is equal to the second squares side length divided by 5). area2 = 25x^2 So the ratio is: 4x^2/25x^2 or just 4/25 B. SA1 = 24x^2 SA2 = 150x^2 ratio: 24x^2/150x^2 = 4/25 C. Let the dimensions of the rectangular box be x, y, and z. Then: SA = 2(xy + xz + yz) = 2xy + 2xz + 2yz DoubledSA = 8xy + 8xz + 8yz Therefore, the surface area increases by a factor of 4 when you double a rectangular box s dimensions."
    },
    {
        "video_name": "BJSk1joCQsM",
        "Q": "At 6:17 I didn't exactly understand what the rhombus part was.",
        "A": "a rhombus is kinda like a squished square. it has 4 sides that are all the same length. the sides that are across from each-other are parallel. the angles that are across from eachother are congruent. hope that helped\u00e2\u009d\u0080"
    },
    {
        "video_name": "jOZ98FDyl2E",
        "Q": "is 2:4 same as 4:2 ?",
        "A": "No... the order of the numbers in a ratio have meaning. For example, if your ratio represent the number of girls to boys, then 2:4 means 2 girls to 4 boys. If you use 4:2, then there are 4 girls to 2 boys unless you redefine the ratio as the number of boys to girls."
    },
    {
        "video_name": "jOZ98FDyl2E",
        "Q": "At 0:50 you divided 40 into 5, how can you tell which way to divide the numbers? Some problems I was trying to finish were supposed to be divided backwards. Like the smaller term was divided into the greater term like 5 divided by 40, 40 divided by 5 is different right?",
        "A": "The question asks for volume per fish... which means volume goes in the numerator and number of fish goes in the denominator. It becomes, for example, 40 litres / 5 fish = 8 Litres/Fish. 40 Litres is a volume and 5 fish is a number of fish."
    },
    {
        "video_name": "jOZ98FDyl2E",
        "Q": "At 0:53, Sal divides the numerator, (40), by the denominator, (5). What if the numerator is not divisible by the denominator? Then what do you divide the numerator by?",
        "A": "If the numerator and denominator have no common factors, you can t divide further. You can either leave the fraction as it is, or convert to a decimal."
    },
    {
        "video_name": "ahyWUV7AwKw",
        "Q": "At 4:55 some video games are violent and some are not... is this video just about the non violent ones... there are some sort of violent ones but they wouldn't influence bad things... how does that work?",
        "A": "This video is focusing only on violent video games. Of course, violent depends on your personal definition, but that doesn t matter here."
    },
    {
        "video_name": "ahyWUV7AwKw",
        "Q": "at 1:14 it says it's talking about stuff in violence. How is playing violent video games bad?",
        "A": "Well... if you play the right video games... then it won t destroy your life....and you won t be a sociopath in the real world... but some may think video games affect the way you act,,, and it is true... but it s affects for certain people... I play violent video games like very rarely... and I don t really have any evil thoughts... unless someone pisses me off really bad... but that s a different story... I just think Sal just chose a random question... and y all just need to calm down... :D"
    },
    {
        "video_name": "4fQeHtSdw80",
        "Q": "at 2:53 someone could say why not -60xy^2 ? how to answer him?",
        "A": "Because it isn t multiplication, it is addition. -30xy + -30xy = -60xy."
    },
    {
        "video_name": "IbI-l7mbKO4",
        "Q": "At 5:43, when Sal draws the rest of the parabola how does he know how wide to make it?",
        "A": "He s drawing it very roughly as an example. With an A value of -3 it would actually be much narrower (more steep). The larger the absolute value of A is, the steeper the sides of the parabola are. If A is a very small fraction like 1/1000, the parabola will be so wide (or shallow) that it looks almost like a line unless you see it graphed to very large/small x and y values."
    },
    {
        "video_name": "IbI-l7mbKO4",
        "Q": "At 3:38 how does Sal get x=4?\n-3 (x^2 + 16) = 0\n-3x - 48=0\n-3x^2=48\nX^2 = -16\nx= undefined\n\n\nWhere did I go wrong?",
        "A": "You used the wrong equation. He was only trying to figure out when (x-4)^2 was 0. This would give the x-value of the vertex. (x-4)^2=0 Take the square root of each side of the equation. x-4=0 Add 4 to each side of the equation. x=4 I hope this clarifies what Sal did!"
    },
    {
        "video_name": "IbI-l7mbKO4",
        "Q": "I don't understand where you get the 16's from at 2:07.. Why do you add 16 then immediately subtract 16? isn't that just more work?",
        "A": "You have to do that to make a perfect square so that it can be written into vertex form. Yes, it s more work, but that s just the way the math rules work haha. Hopefully, this clears things up a bit for you :)"
    },
    {
        "video_name": "IbI-l7mbKO4",
        "Q": "at 1:10, what happens if the equation is not divisible by 3",
        "A": "Lets consider the example -2x^2+24x-27 = -1*(2x^2-24x+27) = -1*2*(x^2-12x+27/2) = -2*( (x-6)^2-36+27/2) = -2*( (x-6)^2-45/2) Essentially I just algebraically manipulated such that I got rid of the A in A*X^2. Dont be daunted by the resulting fractions in the rest of the formula, the trick remains the same."
    },
    {
        "video_name": "IbI-l7mbKO4",
        "Q": "the way i learnt it the vertex form is a(x-h)^2 +k but during the video at the time 0:47 Sal says that the vertex form is y=a(x-b)^2+c so which one is correct??",
        "A": "The two forms are identical. The full version of the a(x-h)^2+k is y=a(x-h)^2+k. In the form given in the video, the h and the k are simply replaced with different letters."
    },
    {
        "video_name": "IbI-l7mbKO4",
        "Q": "at 1:50 why does sal put -8 as b when b is 24?",
        "A": "At 1:25 Sal Khan says 24/-3 is -8 and he obviously changed the symbol from negative (in -8) to positive by changing the addition to subtraction. Please comment if this helped you... I just need to know."
    },
    {
        "video_name": "IbI-l7mbKO4",
        "Q": "at 3:45 sal said that y= -3(x-4)^2-7 and in the next step he said y= -3(x-4)^2-21 how's that possible ?",
        "A": "So, did Sal start with -3((x-4)^2-7)? Then he would get y= -3(x-4)^2+21"
    },
    {
        "video_name": "IbI-l7mbKO4",
        "Q": "Why can't you use 3 as the common factor instead of -3? At around 1:15",
        "A": "We could but we will have 3(-x\u00c2\u00b2+8x-9). We want to get rid of the negative sign in front of x\u00c2\u00b2 to complete the square in the next step. To do that we would factor out the negative sign, we get 3(-1)(x\u00c2\u00b2-8x+9), which equals to -3(x\u00c2\u00b2-8x+9). So you see it is faster to faster to factor -3 instead."
    },
    {
        "video_name": "IbI-l7mbKO4",
        "Q": "at 3:49, what happens to -7 and why does he multiply it by -3?",
        "A": "He is distributing by multiplying three by everything in the parenthesis including -7."
    },
    {
        "video_name": "IbI-l7mbKO4",
        "Q": "6:08\n\nCan someone please explain what Sal meant when he said the graph would be at minimum 3? Thank you!",
        "A": "he means that the lowest point of the Parabola would be 3. which would mean the Parabola would be an upward opening one."
    },
    {
        "video_name": "IbI-l7mbKO4",
        "Q": "@5:45, Sal has drawn the graph for the equation, which shows a positive y-intercept. However, when I plug in 0 for x in the original equation y = -3x^2 + 24x -27, the y-intercept would be -27. What am I missing?",
        "A": "You are right - Sal just drew a rough guess of what the parabola looks like as, for this video, he was only interested in the vertex and whether the graph was concave up or concave down. But yes, the graph would intersect the y axis at -27. Good eye!"
    },
    {
        "video_name": "ZlhrXO1-osA",
        "Q": "at 1:09, can't you do a easier way for example the whole number 2 can be 2/1 and 4/3 remains the same. can you do 2/1 x 4/3. you would start off by multiplying 2 x 4 which equals 8 and 1 x 3 which equals 3. Put the two numbers together and you can get 8/3. is it ok if i do that instead would it be a right way because i am in sixth grade and we learned this is it the accurate way or is sal's way that is more accurate",
        "A": "Both ways are correct. Use the method that works for you. I think Sal chose his way to try and give people a picture of what is happening besides just the procedural steps."
    },
    {
        "video_name": "WNhxkpmVQYw",
        "Q": "at 6:25 sal says the light bulb will be off but wouldn't that be in pass 7 since pass 1 is multiples of 2?",
        "A": "They all start off, pass 1 is multiples of 1 when they all get switched on, pass 2 is multiples of 2. Hope this helps :)"
    },
    {
        "video_name": "Eq4mVCd-yyo",
        "Q": "at 1:39 do you NEED to borrow or how can you make this way more easier? can you all on khan academy help? any one? HELP?",
        "A": "I think you have to borrow."
    },
    {
        "video_name": "Eq4mVCd-yyo",
        "Q": "At 0:46, why do you have to line up the decimal points? Can't you just do the problem without lining up the decimal points?",
        "A": "It s the same things but with that is easier to understand"
    },
    {
        "video_name": "Eq4mVCd-yyo",
        "Q": "in 0:46 Why did he ad two zero's to the question ?",
        "A": "Sal added two zero s to the question to make the subtraction more understandable. 3.6=3.600, and in order to make it clear that the hundredths and thousandths of the number are equal to zero, he added the two zeros. If you know how to subtract decimals without adding the following zeros, you don t have to add them."
    },
    {
        "video_name": "iDQ1foxYf0o",
        "Q": "At 2:38, Sal reads the question: \"Three students attempt to define the geometric object called an angle.\" Are angles objects? Aren't they just a measure of an object?",
        "A": "No, you measure angles themselves. So first you define the angle, and then you take the measure of the angle."
    },
    {
        "video_name": "iDQ1foxYf0o",
        "Q": "From 6:43 to 7:08, why is the definition of distance between endpoints not enough for defining a line segment? They say there is 'more' to a line segment than how long it is. What does \"more\" include in order for the definition to be complete? The endpoints themselves? Color? Smell? etc. :)",
        "A": "Draw a line segment on some graph paper. Now, copy that line segment a few units above the original. These two line segments have the same lengths, but they are different line segments, so we cannot define a line segment by just its length."
    },
    {
        "video_name": "iDQ1foxYf0o",
        "Q": "at 7:17 isn't that a picture of a line",
        "A": "That is a line. The picture is of a line with 2 points, P and Q, that are on the line. They form a line segment."
    },
    {
        "video_name": "iDQ1foxYf0o",
        "Q": "At 7:48, what does equidistant mean?",
        "A": "Equidistant means the distances are equal."
    },
    {
        "video_name": "86NwKBcOlow",
        "Q": "At 2:28, why does the 2 in the denominator cancel with the 2 in the numerator?",
        "A": "At that point you have (5/2) times 2. To do that, we multiply the 5 and the 2 that s by itself and then we divide that answer by the 2 in the demonimator, right? (5/2) * 2 = (5 * 2) / 2 = 10 / 2 = 5. What about: 35 / 5? That s easy... 7. Take a look at 35. Isn t it the same as 5 * 7? So we could rewrite the problem as (5 * 7) / 5. The 5 s cancel out, leaving us with 7. Why? Because 5/5 = 1. So the problem becomes 7 * 1 which is 7."
    },
    {
        "video_name": "86NwKBcOlow",
        "Q": "At 1:53 in the video, can you explain how to answer the equations for y if y=10-5/2(0)=10. I didn't understand",
        "A": "In this question Sal is saying that y=10-5/2(0). He wrote the =10 at the end to say that y equals 10."
    },
    {
        "video_name": "86NwKBcOlow",
        "Q": "At 0:52, why doesn't he multiply 20 and -5x?",
        "A": "u dont multiply b/c u have to divide to get the result of y coordinate"
    },
    {
        "video_name": "86NwKBcOlow",
        "Q": "at 1:49 i thought that y should be 20?",
        "A": "The reason for this is when you have a unknown integers like x and y in this equation the easiest way to solve for one (y in this case) is to make it a zero so the equation becomes 5 times 0 + 2y= 20. You simplify this out so 5 times 0 is 0. So 0x+ 2y=20. Then you simplify 2y=20 Then you divide both sides by 2 So you end up with y=10. I hope that helps"
    },
    {
        "video_name": "86NwKBcOlow",
        "Q": "Im confused!! at 1:10-1:15.... I dont understand how 20-5x divided by 2 became 5/2??",
        "A": "(20-5x)/2= 20/2 - (5x)/2 <--- That was the distributive property 10-5/2x"
    },
    {
        "video_name": "86NwKBcOlow",
        "Q": "can someone explain why at 2:30, the 10- 5/2 times 2, the 2 is cancelled and then equal to 5?\nalso at 3:20 I do not understand why,10- 5/2 times 4 is equal to 0??where does the 1 and the 2 fit in??",
        "A": "A fraction written, say, 5/2 is the same as 5 divided by 2. And when you multiply a fraction, say 3/8, by a whole number, say 5, (3/8) * 5 * is times it is the same as (3*5)/8 So ((5/2) * 2) is ((5* 2)/2) or 10/2 which is five. ______________________ 10 - (5/2) * 4 is zero because = 10 - (5 * 4)/2 = 10 - 20/2 = 10 - 10 = 0"
    },
    {
        "video_name": "86NwKBcOlow",
        "Q": "At 00:23 can we not simplify the linear equation?!",
        "A": "It seems to me that he is doing it the easiest way. But if you think it s easier to introduce a fraction from the start, go ahead."
    },
    {
        "video_name": "86NwKBcOlow",
        "Q": "@ 0:34 does it matter if I got rid of the 2y instead? Does it matter?",
        "A": "I believe you will end up with the same graph if you solve for X instead of Y. One usually solves for Y in linear equations however, so that we can add X into the equation and get the Y position. The reason for this will become clear if you re doing word problems."
    },
    {
        "video_name": "86NwKBcOlow",
        "Q": "At 2:31, because there is a negative sign behind 5/2 then shouldn't he be dividing by -2? and why is the number in the parentheses the numerator? I though 5 was the numerator in what I think is - 5/2(I say I think because he divided by 2/2 instead of 2/-2). Can some someone please explain to me why he does this in the video? Thanks.",
        "A": "1) Remember, to make 2 into a fraction it becomes 2/1. That is why the 2 is in the numerator. 2) The minus on -5/2 can go to the top (the most common) or the bottom, but not to both. -5/2 * 2/1 = (-5*2)/(2*1) = -10/2 = -5 OR: 5/(-2) * 2/1 = (5*2)/(-2*1) = 10/(-2) = -5 OR in Sal s approach of dividing the 2 s, he could have done: 2/(-2) = -1. -1*5 = -5 Hope this helps."
    },
    {
        "video_name": "86NwKBcOlow",
        "Q": "2:00 So the coordinate is not (0,10)? There are two coordinates here? One is the x-coordinate and the other is the y-coordinate? So two coordinates make up the point (0,10) aka \"ordered pair?",
        "A": "yeah but when you say the coordinates of the point (A,B,C,D,or whatever point), you re referring to (0,10)... the 2 coordinates make up the coordinates of the point"
    },
    {
        "video_name": "86NwKBcOlow",
        "Q": "At 0:36, can you subtract 2y and still get the same answer?",
        "A": "Yes, but since we want to isolate y, doing so will take more steps to complete the task than subtracting 5x."
    },
    {
        "video_name": "86NwKBcOlow",
        "Q": "at 1:25 how did he write so much?",
        "A": "I have no clue what your talking about!"
    },
    {
        "video_name": "lDXaju6JoQ0",
        "Q": "At 1:09 will it also work if you borrow from the 1 and make it 0 and then go back and borrow from the nine to make 0 into 10?",
        "A": "Yes, I think so P.S. I don t know if my answer is right"
    },
    {
        "video_name": "WkmPDOq2WfA",
        "Q": "What's the song Vi plays at 4:26? (not spam, I just like the song.) :3",
        "A": "I believe that it isn t actually a song but two inverse chromatic scales, one going up and one going down, repeating."
    },
    {
        "video_name": "WkmPDOq2WfA",
        "Q": ":P Does anyone @ 2:30 , know where she got that fantastic bowl? It looks organic or something. Where would I find something like that?",
        "A": "I would look on Amazon.com"
    },
    {
        "video_name": "WkmPDOq2WfA",
        "Q": "starting at 0:37, Vi Heart used the word notation. What does that mean?",
        "A": "A notation is just a symbol for something such a letter."
    },
    {
        "video_name": "3v9w79NhsfI",
        "Q": "3:39 Why the sum of upward facing after rolling 7 dice is < or = to 30 even, if i assume that 6 is the maximum for a single dice roll then it should be 6 * 7 = 42, can some one explain please",
        "A": "Yes, 42 is the maximum possible (Y = number when rolling 7 dice). But you can ask What is the probability that Y is < or = 30?, or that Y is even ."
    },
    {
        "video_name": "3v9w79NhsfI",
        "Q": "At 1:24, could you define the variables Heads and Tails using the numbers 1 and 2, and then stating 0 as a value that cannot be an outcome.",
        "A": "You can define them however you want. 0 and 1 are nice if you end up calculating sums - 1 and 2 would interfere with each other without special care."
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "why is 6*x/3=2?? at about 1:05",
        "A": "6*x/3 = 2x, since the 3 cancels out the 6 leaving a 2."
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "I got lost in 3:27 were does the three come from?",
        "A": "If you re talking about the 3 that is a denominator... then that is because he is plugging the x value that was solved for into the original equation - so that he can check if he did the equation correctly... :)"
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "At 0:26 how is the LCM of the three numbers 6",
        "A": "To completely get rid of all fractions, you want to find the LCM. In this case, you want to find a number that will get rid of both 3 and 6. Yes 3 is the real LCM, however, to get rid of the 6 you need to find something higher that both numbers are multiples of. In this case it is 6."
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "1:06, he we got to 2?",
        "A": "6 * x/3 = (6x)/3 = 2x"
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "At 1:12, why did you divide 6 by 3? I thought it would be 6x over 3 since 6 over 1 multiplied by x over three equals 6x over 3.",
        "A": "6*(x/3) We don t really need the parentheses... 6*x/3 We can switch the 6 and the x. x*6/3 6 divided by 3 is 2. x*2 2x I hope you know why he divided 6 by 3!"
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "@ 3:55 Sal explains he multiplies 6 x 49/4. why is it not the same as (6/1)(49/4)?\nand how do you complete 6 x ((49/4)/(3))? Suggested video?",
        "A": "It is, 6/1 is same as 6. You complete 6 x [(49/4)/(3)] by working what is is in the parentheses first which is 49/4 divided by 3. Turn it into 49/4 divided by 3/1. Divide fractions by multiplying the opposite, so it is 49/4 x 1/3. That brings you to 49/3. Then multiply 49/3 by 6."
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "How to you get the - 48 at 4:06?",
        "A": "Sal is rewriting the pink/purple equation underneath in yellow. In doing so, he is multiplying that 6 on the left hand side with the two terms inside the parentheses. 6 * (-8) = -48."
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "At 4:31 where is three gone?",
        "A": "he didn t multiply he added"
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "at 1:09 how did he get the 2",
        "A": "6 times x divided by three can be calculated by first dividing 6 by 3 (which is 2) and then multiplying the result by x"
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "Is this sort of math for fifth grade? Because I'm watching this video and at the 3:16 time mark when Sal is doing the second part on the right I'm sort of confused.",
        "A": "I m not sure this is for fifth grade because I am in eighth grade and this is the math I am doing. You re school might be ahead of mine though. Good luck!"
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "At about 4:00, why is 49/4/3*6 = 49/2? I don't understand what Sal did there.",
        "A": "All Sal did to get to the number 49/2 is cancel out numbers to simplify the fraction. When you multiply 6 by the fraction you can divide 6 by the denominator 3 and 6/3 = 2. Therefore, 6*49/4/3 = 2*49/4. Now you can further simplify the fraction by doing the same to get rid of the number 2. 2/4 = 1/2, therefore 2*49/4 = 49/2. I hope this helps!"
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "At 4:14, how does Sal arrive at the answer 49/2?",
        "A": "It s a little hard to write the expression in text, but it looks something like: [(49/4)/3 + 1/6] * 6 He first multiplies the term (49/4)/3 by 6. Another way to write that is: (49/4) * 6 / 3 = (49/4) * 2 = 49 / 4 * 2 = 49 / 2 The second term is 1/6 * 6 = 1. Since he s been using the distributive property, he then adds to the two terms to get 49/2 + 1."
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "At 3:55,why does sal go straight to saying 6 over 4 is the same as 3/2 ? I don't understand his process in simplifying here....",
        "A": "Honestly, if you don t follow why 6/4 = 3/2 you probably need to slow down and go review some videos on reducing fractions. That is something you should be very familiar with at this point and if not, you are going to really struggle moving forward."
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "At 1:07 how did you get two, it should it be six, right?",
        "A": "at 1:07 he s multiplying 6(x/3). That s the same as 6/1 * x/3 which simplifies to 6x/3. 6/3 is 6 divided by 3 which is 2, which is why he got 2x."
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "at 5:00 why did he multiply 48 by 2 and then put it over 2? isn't that unnecessary?",
        "A": "Sal wanted to subtract the 48 from the 147/2. To subtract fractions, you need similar denominators. So it is not unnecessary."
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "I don't understand 0:59 where 6*x/3 is equal to 2x... help?",
        "A": "he is cross canceling 6/3=2 hope it helps"
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "Well, I didn't understand the part 4:07. How 49/4/3 is equal to 49/2 ???",
        "A": "But fellows what about this: (49/4)/3 isn t it = (49/4) times 1/3 ? And then wouldn t it result as 49/12 ? So, it is not that (49/4)/3 is equal to 49/2, lovevery, because it is not. But, yeah, (49/4)/3 [ or 49/12 ] times 6 is = 49/2 ."
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "At 3:16, Khan starts to check the problem. Could he have checked it without multiplying both sides by 6?",
        "A": "Yes, he could ve manipulated the equation in a different way. For instance, the left hand side of 49/4 - 8 he could ve changed to (49 - 32)/4 then to 17/4. The right hand side he could ve changed to 49/12 + 1/6 then changed to (49 + 2)/12 giving: 17/4 = 51/12. Then you multiply both sides by 4 and then multiply both sides by 12 to get 204 = 204 which is true so our work checks out."
    },
    {
        "video_name": "CJyVct57-9s",
        "Q": "How does 49/5/3 become 49/2? In the video at ~5:50.",
        "A": "49/4/3 is the same as 49 divided by 4 then divided by 3, which is the same as 49 divided by 12.Dividing by 12 and then multiplying by 6 is the same as dividing by 2.That s why he put it as 49/2. Hope I explained it well enough. :)"
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "how at 9:32 does he get to 2 to the 200th?",
        "A": "When you have an number with an exponent in parentheses such as (2^2) then raised to another exponent, such as (2^2)^100, the answer is the original exponent in parentheses times the other exponent. So, as 2 x 100=200, the answer is 2^200."
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "at 5:33 shouldn't he be adding 5^-100 and 5^102 instead of multiplying?",
        "A": "Thanks :P"
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "At 5:37 how does 5^-100 X 5^102 equal 5^2 ?",
        "A": "If you have the same base you can add the exponents. 102-100=2"
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "At 4:21 he is asking if you had 2^8 + 2^8 power you would just add the two set together wouldn't you?? That is the logical thing to me to do I would think. but I could be wrong!:( So what I am asking is what he said going to apply to all problems with exponents?",
        "A": "yes but remember the exponentiation should be same 2^ 8 + 2 ^ 8 could be written as 2(2^8) but we can t do the same with 2^ 8 + 2^7."
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "At 5:00 2^1 * 2^8= 2^9? Isn't it 4^9? Two times two is four not two. I don't understand.",
        "A": "it s addition so the 2 doesn t change :)"
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "Does the rule at 4:45 show up in problems very often?",
        "A": "Definitely. That rule is actually one of the most used when working with exponents, and is a good one to get down."
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "At 7:46, it just doesn't make sense. What's going on?",
        "A": "Well it is like 7^40 + 1/7^-5, and you took that 1/7^-5 and it is equal to just 7^5 (remember that exponents negatives turn upsidedown the fraction, and 7/1 it s just 7). From there, 7^40 + 7^5 is 7^45."
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "Would the exponent rule at around 2:00 work because of the commutative property?",
        "A": "Commutative Property is when moving around,commuting, numbers in an operation doesn t change it like saying ab = ba . I think exponent rule is for simplifying calculation and saving time . 2^2 x 2^3 = (2 x 2) x (2 x 2 x 2) we are multiplying 2 5 times so we can write it in exponentiation form as 2^5 and this number,the power of our solution expression here 5, is always the sum of the powers if the bases are same in an operation of multiplication.Hope this helps!"
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "I don't understand how this works \"4:45\" to \"5:00\". Can someone elaborate please?",
        "A": "See it like 2^8 + 2^8= (2*2*2*2*2*2*2*2) + (2*2*2*2*2*2*2*2). I mean, 8-times-2 plus 8-times-2 , which is 2 times 8-times-2 , like: (2^8) *2, that is: (2*2*2*2*2*2*2*2)*2, i mean, 9-times-2: 2^9"
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "I don't understand at 7:50... how does he turn the 7^-5 to 7^5? I mean, i know he can write 7^40 * 1/7^-5 but how it turns to 7^5? Someone help me please? D:",
        "A": "That s because it is the denominator of the fraction. To convert it to numerator, you have to change the value of the power to be the opposite of what it used to be. And of course this works vice versa. So 1/x^n = x^-n, where x the number and n the power."
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "I don't get the problem starting from 5:44. Can i get some help?",
        "A": "2^9/2^10 Ok, the rule for division is to subtract the powers with the same base. When you subtract, you get 9-10=-1. 2^ - 1.Bring this to the denominator as a positive exponent, and you get 1/2^1 Hope this is what you expected =)"
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "At 6:29, what are level two exponents?",
        "A": "I think Sal is just referring to another video where he describes negative exponents. It isn t a name of a certain type of exponent."
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "at 5:10 ~ i don't rly get it\nHelp",
        "A": "You are taking 5^-100 and multiplying it by 5^102. Of course he does not write these numbers out because they would be unimaginably large. Then, you just add the exponents since you are multiplying the bases. So, you would do -100+102, or, using the commutative property of addition, 102-100, which is 2. So the answer you receive is 5^2."
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "6:28 why do you have to make the exponent negative when you didvide?",
        "A": "Maria, A negative exponent x\u00e2\u0081\u00bb\u00c2\u00b2 means 1/x\u00c2\u00b2 or 1/(x*x) If you had x\u00e2\u0081\u00b4/x\u00c2\u00b2 this would be (x*x*x*x)/(x*x) and two x s in the numerator and denominator cancel out leaving only x*x = x\u00c2\u00b2 You could also solve x\u00e2\u0081\u00b4/x\u00c2\u00b2 this way x\u00e2\u0081\u00b4/x\u00c2\u00b2 Convert the 1/x\u00c2\u00b2 to x\u00e2\u0081\u00bb\u00c2\u00b2 x\u00e2\u0081\u00b4* x\u00e2\u0081\u00bb\u00c2\u00b2 Now add the exponents x\u00e2\u0081\u00b4\u00e2\u0081\u00bb\u00c2\u00b2 and 4-2 = 2 so x\u00c2\u00b2 is your answer. Using the negative exponent and moving the variable to the numerator makes it easier to solve easily once you understand that x\u00e2\u0081\u00bb\u00c2\u00b2 means 1/x\u00c2\u00b2 I hope that helps make it click for you."
    },
    {
        "video_name": "kITJ6qH7jS0",
        "Q": "Solve this:\n-6^2+8:2=?\nMinus 6 to the square plus 8 divided by 2",
        "A": "-6^2+8/2 -36+4 -32, so -6^2+8/2 equals -32"
    },
    {
        "video_name": "cAAEMza4GvQ",
        "Q": "At 3:20 Grant mentions that a curl of 27 is \"quite positive\". What is this \"quite positive\" value relative to? I mean, I know 0 corresponds to no curl and the more positive the value, the greater the counterclockwise curl what is the range of how high the curl scale goes?",
        "A": "At around 5:30 Grant interprets curl values as the general direction and strength of rotation around a point. I m guessing higher (absolute) values may mean larger magnitude vectors overall going around in circles, and over time at a faster rate."
    },
    {
        "video_name": "N5xWYgeIWmU",
        "Q": "at 1:02 Sal said that we need to take away 2/10 but the subtitle in English is written as 2.10...pls make the changes",
        "A": "yup but it s at 1:01 not 1:02"
    },
    {
        "video_name": "EX9CdUAMpgE",
        "Q": "What does inequality mean at At 2:00",
        "A": "An inequality is the relationship between two numbers when they do not equal each other. For example, 3 > 4 is an inequality because 3 does not equal 4."
    },
    {
        "video_name": "6yrPU8rYOhs",
        "Q": "At 0:10 Sal says that all linear transformations can be written as matrix multiplication problems, but my linear algebra professor says that this is only the case when you're going from Rn to Rm. My professor says that, technically, the derivative and the integral are linear transformations that can't be written as matrix multiplication. Who's right?",
        "A": "Sal is right. Your professor is wrong (or at least the way you worded his words is). Yes, the derivative and the integral are transformations that can t be written as matrix multiplication, but they re not linear transformations because they don t preserve addition and scalar multiplication as is required. As Sal said, any linear transformation can be written as a matrix-vector product."
    },
    {
        "video_name": "6yrPU8rYOhs",
        "Q": "Sal at 14:50 says that the kernel is the set of all vectors x in R^2 where T(x)=0. Does x have to be in R^2 or could it be in any R^n?",
        "A": "In this example, x had to be in R^2 because the matrix that underwent transformation was a 2x2 matrix. In order to map an mxn matrix A (in this example, a 2x2 matrix) from a vector space V to a vector space W, we have to multiply the mxn matrix by an nx1 vector. The number of rows in the vector x must match the number of columns in the matrix A."
    },
    {
        "video_name": "YezsgIlcb68",
        "Q": "hmmmmm did he make a mistake in 7:07 ?\nhe wrote y' = (1+c2)*e^((-1/2)x) ? instead of e^((1/2)x)",
        "A": "I believe he did. When he was writing it, he stated it correctly, but wrote it down wrong. It should be a positive 1/2 in the exponent. It doesn t change the final answer to the particular solution though."
    },
    {
        "video_name": "aep6lwPqm6I",
        "Q": "Is it true that lambda should never be 0 because it does not have any geometrical meaning?\n\n(I get a little confused from 3:12-3:27. If x = 0, in the first equation, we don't know for sure if y = lambda. Then in the second equation, either y =0, or lambda = 0, or both = 0. If only lambda = 0, in order to satisfy the third equation, y can be equal to =1 or -1.)",
        "A": "Just saw he added a pop-up note to point this out, but the problem is that (at least for me) it is not displayed when I m watching it full-screen..."
    },
    {
        "video_name": "IE0HQoLrbX4",
        "Q": "6:02 PM- Is their an easier way to subtract decimals? Because I really didn't\nunderstand that",
        "A": "The only trick to subtracting decimals is lining up the decimal places, i.e. .41 - .21 ----- .20"
    },
    {
        "video_name": "73eSiVcg4tQ",
        "Q": "for 4:30. couldn't we solve the integral like this:\n\u00e2\u0088\u00absin (t^2) dt=1/2\u00e2\u0088\u00ab(1-cos(2t))dt and then use u-substitution?",
        "A": "You got the trig identity wrong. It must be sin^2(t) not sin(t^2)."
    },
    {
        "video_name": "73eSiVcg4tQ",
        "Q": "At 1:54, you calculate the value for sin9/13, but I heard you do not need to simplify on the AP test, so would I be okay in leaving it as sin9/13 or did that policy change or not apply to the calculator part?",
        "A": "It is always better to evaluate the answer where possible, so that means you would have to calculate the value for sin9/13."
    },
    {
        "video_name": "73eSiVcg4tQ",
        "Q": "At 4:30, Sal said it was hard to take the integral of sin(t)^2. Why?",
        "A": "When you integrate sin(x^2)dx you have to do a u substitution to integrate the sine function, so you set u= to the inside function which is x^2, so now you have the integral from 0 to t of sin(u)dx where u=x^2, now you need to have your integral all in one variable which would look like sin(u)du, so you need to relate du to dx, du=2xdx and herein lies the problem because you cannot introduce another x variable to the integral expression."
    },
    {
        "video_name": "-XRQovTI04Q",
        "Q": "at 3:50 when sal solved for the inequality. He said that when you multiply by the reciprocal you change the sign.\n\nsorry, but I was aware that the only time you switch the inequality was when multiplying or dividing by a negative number. can some one shed some light into his argument?",
        "A": "In this case, if (a/b) > (c/d) is true, the only way to keep the statement true after flipping the numerator and the denominator is to change the direction of the inequality. So, (b/a) < (d/c). For example, if (7/2) > 2 then (2/7) < (1/2). Or, if 30,000>3 then (1/30,000)<(1/3)."
    },
    {
        "video_name": "6OvhLPS7rj4",
        "Q": "6:33, we square to get positive values for all errors?",
        "A": "That s one of the reasons. Check out the top questions for this video, because there is already a lot of detailed discussion on why squaring the error is the agreed-upon strategy for finding the best fit."
    },
    {
        "video_name": "6OvhLPS7rj4",
        "Q": "4:1 please explan",
        "A": "At x(1) the y value on the line is m*x(1) + b."
    },
    {
        "video_name": "6OvhLPS7rj4",
        "Q": "I don't understand, why y1-(mx1+b) ?\nIt shouldn't be (mx1+b)-y1?\n3:52",
        "A": "We are looking for the distance, so either is fine, especially because we are going to square the result."
    },
    {
        "video_name": "6OvhLPS7rj4",
        "Q": "Why do you have to square the errors at around 5:30?",
        "A": "Probably to normalize out negative error values. We would want the sum of the errors to increase if there is an error in either direction - squaring has the effect of always producing a positive number so the sum of the errors is not actually decreased when a negative error value is added. That s just my gut intuition though. I hope Sal will touch on this in another video."
    },
    {
        "video_name": "Rcb7ZUTOQ1I",
        "Q": "At 1:12 and going on sal did not mention what pi was he said that pi r 2 was equal to 36 pi doesnt pi equal 16.2 what does he mean",
        "A": "Pi is not equal to 16.2, pi is equal to 3.14. When Sal mentioned 36pi, that means 36 times pi. Hope this helps!"
    },
    {
        "video_name": "Rcb7ZUTOQ1I",
        "Q": "Why is r a distance(1:04)",
        "A": "When we work with r in circles, we are referring the circle s radius. By definition, the radius of a circle is the DISTANCE from the center of the circle to its outside edge ."
    },
    {
        "video_name": "Rcb7ZUTOQ1I",
        "Q": "0:14 - 0:20 What is 36 pie equal to ...?",
        "A": "long answer 3.141592653589793"
    },
    {
        "video_name": "Rcb7ZUTOQ1I",
        "Q": "From 0:46 to 0:57, do we have to do that step?",
        "A": "The step was done so that we can get rid of a part of the equation and to isolate the variable. Isolating the variable is a necessary step in solving for any variable, such as x. It allows you to see things more clearly and quickly figure out what x is. I hope this helped! :D"
    },
    {
        "video_name": "Rcb7ZUTOQ1I",
        "Q": "on 0:05 how can it be 36pi, doesn't pi not change?",
        "A": "The 36 pi mentioned in the video is actually the area of the circle which is supposed to be multiplied by the diameter (2*r) in order to find the circumference.Is this helping?"
    },
    {
        "video_name": "99pD1-6ZpuM",
        "Q": "At 11:39, why would we have to redefine our boundaries in terms of u?",
        "A": "You don t have to, but it makes it a lot simpler, because you don t need to carry around all the a + b -t s around."
    },
    {
        "video_name": "P3IlneCNm8A",
        "Q": "at 5:40 ish, if you choose a for part b, is the answer just undefined?",
        "A": "something/0 is undefined."
    },
    {
        "video_name": "UKQ65tiIQ6o",
        "Q": "At 1:27 i don't get how he gets S over 2.",
        "A": "Since Sal cut the base in half, and the sides variable is S, half of S would be S/2."
    },
    {
        "video_name": "UKQ65tiIQ6o",
        "Q": "don't really understand the 30:60:90 and how 60 = (radical)3 if 30 equals 1",
        "A": "By Pythagorean theorem: a\u00c2\u00b2 + b\u00c2\u00b2 = c\u00c2\u00b2 (1)\u00c2\u00b2 + (\u00e2\u0088\u009a3)\u00c2\u00b2 = c\u00c2\u00b2 1 + 3 = c\u00c2\u00b2 4 = c\u00c2\u00b2 2 = c And, the angles work out to be 30:60:90 for this type of triangle."
    },
    {
        "video_name": "UKQ65tiIQ6o",
        "Q": "At 2:18 how did Sal get the square root of 3s/2 and the s/2?",
        "A": "He is using special right triangle 30-60-90 rules and in accordance with these rules, Sal merely plugged in what he was given into these rules and got the numbers that he did. Review the video for exact references to the special right triangle rules. Hope this helped."
    },
    {
        "video_name": "UKQ65tiIQ6o",
        "Q": "At 2:24 What is the highpotnus",
        "A": "the hypotenuse is the longest part of the triangle and opposite the right angle."
    },
    {
        "video_name": "UKQ65tiIQ6o",
        "Q": "what is he doing at 2:00 to 2:04 ?",
        "A": "he is explaining why we get root 3 s/2 as the side opposite to 60 degree angle."
    },
    {
        "video_name": "UKQ65tiIQ6o",
        "Q": "is the S/2 at 3:42 the equation?",
        "A": "No. That s just the base of half of the original triangle."
    },
    {
        "video_name": "UKQ65tiIQ6o",
        "Q": "What does Sal mean by s/2 at 1:24?",
        "A": "It has to do with the perpendicular line he drew near the beginning of the video that goes from the top corner of the triangle down to the middle of the base. It is called a perpendicular bisector because it divides the bottom of the triangle evenly into two pieces. Each pieces is half the length of the whole base. If the base is s, then half the base is s/2."
    },
    {
        "video_name": "UKQ65tiIQ6o",
        "Q": "At 0:55, how did Sal know that the angles were 60 degrees?",
        "A": "The 3 angles of any triangle add up to 180 degrees. In an equilateral triangle the 3 angles are equal. Since 180 divided by 3 is 60 - the angles of an equilateral triangle are always 60 degrees."
    },
    {
        "video_name": "UKQ65tiIQ6o",
        "Q": "At 0:49, how do we know that each angle of an equilateral triangle is 60 degrees?",
        "A": "The sum of a triangle s angles must be 180\u00c2\u00b0. Each angle is equal and 3 times 60\u00c2\u00b0 is 180\u00c2\u00b0."
    },
    {
        "video_name": "UKQ65tiIQ6o",
        "Q": "what exactly does bisect mean? 0:34",
        "A": "to split in half. btw homeschooltshunk plz do not use bisecting a frog again i just ate."
    },
    {
        "video_name": "UKQ65tiIQ6o",
        "Q": "He lost me at 2:20.",
        "A": "alaw, Just use this formula to find the area of an equilateral triangle: 3s\u00c2\u00b2 / 4 where s is any one side of an equilateral triangle. In the video, Sal was trying to prove how he got the formula. If you want to solve it quickly, skip 2:20. Just memorize the formula and you re good to go! Hope my answer helped"
    },
    {
        "video_name": "fp9DZYmiSC4",
        "Q": "I don't really understand what Sal did at 7:50 with the root of two. He said 'we can rationalise the denominator here', could anybody explain what he does?",
        "A": "He just multiplies by sqrt2/sqrt2. That s multiplying by 1, so it doesn t change anything, but it does get the square root out of the denominator. It s not mathematically different than converting 1/2 to 2/4."
    },
    {
        "video_name": "fp9DZYmiSC4",
        "Q": "at 1:40 i dont understand as to how cos theta is x co ordinate and sin theta the y co ordinate",
        "A": "Try watching the unit circle video again. The gist is: Cos=adj/hyp. On the unit circle the hypotenuse is always 1. The adjacent is going to be on the x axis. So, cos ends up being x/1, or x. The explanation for sin is similar."
    },
    {
        "video_name": "fp9DZYmiSC4",
        "Q": "I'm so confused at 9:08, how does he get these numbers 5pi/4? And pi/4 from earlier. It's kind of confusing as to how he gets all these, for me.",
        "A": "pi/4 is simply between 0 and pi/2. Likewise, 5pi/4 is between pi and 3pi/2. They are exact midpoints of the numbers on either side. Sal just memorized them, or did them in his head."
    },
    {
        "video_name": "fp9DZYmiSC4",
        "Q": "When Sal graphed the line at 4:10, why did he graph pi/2 and 3pi/2 where he did?",
        "A": "What s halfway between 0 and 1? halfway between 1 and 2?"
    },
    {
        "video_name": "fp9DZYmiSC4",
        "Q": "10:52, How can the Cosine of 5(pi)/4 be negative as well as the sine when Cosine is positive in the IV quadrant?",
        "A": "x changes sign at pi/2 and 3pi/2; y changes sign at pi and 2pi. So x/r and y/r change sign just like x and y"
    },
    {
        "video_name": "fp9DZYmiSC4",
        "Q": "At 3:37, is there another starting point for both the sine and cosine graph? is it possible to have the sine graph intercept the y-axis at (0,2)?",
        "A": "In the unit circle y=sin(theta). In order for y=2, sin(theta)=2. BUT the sine function can never be greater than 1, as sine is opposite over hypotenuse. Also, this is dealing with the original functions, when graphing y=sin(x) you can change aspects of the graph by using multipliers, but that is no longer the original function. (See Example: Amplitude and Period )"
    },
    {
        "video_name": "fp9DZYmiSC4",
        "Q": "At 7:02, wouldn't it be faster to apply the method of 45-45-90 triangles to figure out the lengths of the legs given the length of the hypotenuse?",
        "A": "This is where that property comes from."
    },
    {
        "video_name": "fp9DZYmiSC4",
        "Q": "At 6:25 couldn't Sal have used the 45-45-90 triangle rule rather than using the Pythagorean Theorem?",
        "A": "Yes, he definitely could have!"
    },
    {
        "video_name": "fp9DZYmiSC4",
        "Q": "I don't get that at 7:14 in the video, Sal says that a squared + a squared is equal to 1. a squared + b squared is equal to c squared so technically shouldn't the equation be a squared + a squared is equal to 1 squared?",
        "A": "1 squared is equal to one, so what he said is correct. As you probably know, 1 squared = 1 times 1 which is equal to 1."
    },
    {
        "video_name": "fp9DZYmiSC4",
        "Q": "Why do we take the principal root of both sides? Why they can't be negative? Because of first quadrant? 7:28",
        "A": "Yes because of the first quadrant, it can negative otherwise if you considering x and y values circle, although video what it trying find is distance A which is why it is positive"
    },
    {
        "video_name": "EV5dhv0A2wU",
        "Q": "At 3:10, how do we get dx = 2sin\theta ?\nIt cannot be d/dx x, because that would be 1.\nIf we replace with x=2sin\theta, then d(2sin\theta)/d(2sin\theta), which also equals 1.",
        "A": "Sal is taking the derivative with respect to theta so derivative of 2Sin t = 2 cos t dt. dx = 2 cos t dt is the same as saying dx/dt = 2 cos t"
    },
    {
        "video_name": "EV5dhv0A2wU",
        "Q": "At 1:50, how do you know if x is on the adjacent or opposite side to theta. I mean, can't you do the substitution starting with cos (theta)=x?",
        "A": "yes you could use cos (theta)=x and still get an equivalent expression, but we typically try to have it in terms of tan, sec, or sin just so the integral is a little bit nicer"
    },
    {
        "video_name": "EV5dhv0A2wU",
        "Q": "At 5:02, Sal says, \"...so let's just solve for x here.\" Shouldn't he have said, \"...solve for theta\", since he is looking for theta in terms of x ?",
        "A": "Yes, you are right, he should have said solve for theta ."
    },
    {
        "video_name": "EV5dhv0A2wU",
        "Q": "At 2:44, where does the 2cos(theta) come from?",
        "A": "That was cos = adjacent / hypotenuse cos = sqrt(4 - x^2)/2 2 * cos = sqrt(4 - x^2)"
    },
    {
        "video_name": "EV5dhv0A2wU",
        "Q": "At 3:14, Sal says that if x=2sin(theta) then dx=2cos(theta)d(theta), but I do not follow. What is the reason?",
        "A": "He is differentiating x with respect to \u00ce\u00b8. Before differentiating you have: x = 2\u00c2\u00b7sin(\u00ce\u00b8). After differentiating with respect to \u00ce\u00b8 you have: dx/d\u00ce\u00b8 = 2\u00c2\u00b7cos(\u00ce\u00b8) Finally, you solve for dx: dx = 2\u00c2\u00b7cos(\u00ce\u00b8)\u00c2\u00b7d\u00ce\u00b8"
    },
    {
        "video_name": "EV5dhv0A2wU",
        "Q": "At 2:10, Sal says, \"If we define this angle as theta...\". Why make this angle theta and not the other acute angle?",
        "A": "You can choose the other angle if you want, the procedure is the same and the end result is the same."
    },
    {
        "video_name": "5NjgcPri07E",
        "Q": "So just to clarify from 3:24 the triangles height is R?",
        "A": "for each small triangle in the first triangle, the height is R and the base is one of the sides. But we don t know the height of the first triangle, so we want to find the height of the smaller triangles inside it."
    },
    {
        "video_name": "5NjgcPri07E",
        "Q": "This video references a concept at 6:00 (Pythagorean Theorem) not addressed yet as if I follow the knowledge map. Should I be following another path to get to this concept?",
        "A": "In my opinion, it is better to follow the playlists in order on each subject than to use the knowledge map."
    },
    {
        "video_name": "wqnpSzEzq1w",
        "Q": "Shouldn't Sal say that he is going to create a series that goes on infinitely? When he says he is going to create a series that goes to infinity, doesn't that statement say that infinity is a defined value. I am referencing the statement at 0:26 - 0:46.",
        "A": "Where you do have a point, the terminology comes from the way we sum a series. If we were only taking the first 10 terms, we say sum to 10. As we are summing a series that goes on until n=infinity, we are summing to infinity."
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "At 3:27, -27/12 is converted to -9/4. Then at 3:57, it is simpiflied to -2 1/4. But I changed -27/12 right to a mixed number, getting -3 1/12. How are these numbers different? Did I do something wrong? Should I have waited to simplify before changing from an improper fraction?",
        "A": "If you use a calculator you can see that -27/12 = -2 1/4. Another way to think about it is 12 goes into 27 two times (12*2=24) and then the 3 left over 12 (3/12) is equal to 1/4 (3 goes into 3 one time and into 12 four times."
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "At ~1:00, why did Sal do this:\n2/3 > -4y - 25/3\n3 * 2/3 > 3(-4y - 25/3)\n2 > -12y - 25\n...\n27 > -12y\nwhen he could have done this:\n2/3 > -4y - 25/3\n2/3 + 25/3 > -4y - 25/3 + 25/3\n27/3 > -4y\n9 > -4y\nand end up with the answer sooner, because the inequality 9>-12y is effectively 27>-12y simplified.",
        "A": "He could have done it that way but I think he just wanted to get rid of the fractions, just so it was cleaner. But both ways work. It s just a matter of how good you are with fractions, because it sometimes is faster to just get rid of the fractions first. Just because there are less steps doesn t mean it s faster. :)"
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "Why at 3:00 did you swap the symbol? What does the negative number have to do with it?",
        "A": "You swap the symbol if you divide or multiply across the greater than, or less than sign"
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "With reference to 0:00, can someone please explain to me how we distinguish graphically what in this video are called \"mixed numbers\" and read eight-and-one-third (=25/3) from the standard multiplication which we would usually interpret as eight-times-one-third (= 8/3)? Graphically I'm confused, where I should see a mixed number I see a standard multiplication. Thank you.",
        "A": "When you see 1\u00e2\u0085\u0094, then know that it s a mixed number. If a number looks like 1(\u00e2\u0085\u0094), then it means standard multiplication. Hope this helps!"
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "At 3:26 sal says its (-9/4) when he divides (21/-12), but i thought is should be (9) over (-4) (9/-4), since the denominator was a negative",
        "A": "Mathematicians generally allow negative signs to drift to the top, they are equivalent fractions. It makes a difference if you are talking slope because -9/4 says to go down out right 4, but 9/-4 says to go up 9 and left 4, but they both end on the same line that goes through the origin."
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "At 3:20 I do not understand why Sal divided the fraction by 3. Can someone please explain?\nThank you!",
        "A": "In essence, he reduced the fraction by dividing both denominator and numerator by 3. So (27/3)/(-12/3) = 9/-4 or -9/4."
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "at like 1:39 to 1:45 when your were dividing the 3 by the negative 25/3 how did u get 25 im not understanding PLEASE HELP !",
        "A": "I just listened to the 1:39 one. He is distributing 3 x 25/3. Short cut way is the 3 being multiplied to the 3 in the denominator would cancel each other out. Leaving you the 25."
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "At 2:31 when sal adds +25 to both the sides,he did not change the ezuality sign but if we were to add and negetive number then do we have to swap the inequality or not?",
        "A": "no you only swap the inequality sign when multiplying or dividing by a negative"
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "Hi, at 3:29 Sal simplifies the fraction 27/-12 into -9/4 by dividing by 3. Shouldn't it be 9/-4? 27 divided by 3 is 9, -12 divided by 3 is -4. Clearly I'm missing something obvious about the rules of dividing negative fractions.",
        "A": "Remember, a positive divided by a negative = a negative. So, the whole fraction is negative. This is usually shown by placing the minus in front of the fraction."
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "At 0:20.... 8(1/3)=25 is that an error?",
        "A": "I think you mean is 8(1/3) = 25/3. If this is the case, it is not an error. The reason being is that we simply took our mixed number (8 and 1/3) and turned it into an improper fraction (25/3). This is done by multiplying the denominator (3) by the whole number (8) and adding our numerator (1). So, it would look like 8 x 3 + 1 = 25. Then, we put this number over our original denominator (3) to get 25/3. I hope this helps."
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "At 0:57, I get why he can multiply each side by 3, but can he also just add 25/3 to each side?",
        "A": "Yes, you are correct. It might not be as simple as the way he did it, but I d also prefer to do it this way: 2/3 > -4y -25/3 //+25/3 27/3 > -4y 9 > -4y //*(-1/4) y > -9/4"
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "At 1:05, why did Sal multiply both sides by 3?? What will I do if the problem doesn't have fractions?",
        "A": "He explains in the video that his purpose was to eliminate the fractions. All that means is that he wanted to make the problem easier to work with. We could solve the inequality with the fractions but it would be a little more complicated and we might be more likely to make a mistake when adding/subtracting/dividing the fractions. He picked 3 because it was the smallest number that would give us whole numbers as products. If there were no fractions, you would just move on to simplifying one side of the inequality."
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "What happens when the 25 on 1:56 is positive will you subtract 25?",
        "A": "Yes, you always do the opposite operation"
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "At 1:59: Is it technically correct to call an inequality an equation? Based on what we've learned so far, doesn't equation imply an equal sign?",
        "A": "Yes you re right. An inequality is not the same as an equation, so it s incorrect to call it one. Sal sometimes mistakenly describes inequalities as equations, most of the time a little pop up on the video corrects this, but looks like it hasn t been addressed yet in this case."
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "At 1:50, why do you have have all the constants on one side and all the variables on the other?",
        "A": "When solving an equation or an inequality, you always want the variables to be the only thing on one side of the equation. This is the case because in any equation, y is actually a number, but we don t know what it is. You want the variable by itself to figure out exactly what number it represents. At the end of this inequality, y is greater than or equal to -9/4"
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "When teacher writes (0:24) 8 1/3 he said and solve eight PLUS one third but without the sign of ad +\nI\u00c2\u00b4m used to see 8 1/3 like eight TIMES one third\nDoes it means that I have one problem with notation?\n\nThanks",
        "A": "You need a times sign in there for it to be 8 * 1/3. Let s say you have 9 cookies and you eat 2/3 of one cookie while you are waiting for someone to answer this question. Now you have 8 cookies plus 1/3 of a cookie. We write that 8 1/3. 8 1/3 is the same as 25/3. That is, if you broke up those 8 cookies into thirds plus the 1/3. This is the same as 3/3 + 3/3 + 3/3 + 3/3 + 3/3 + 3/3 +3/3 + 3/3 + 1/3 OR (8 * 3/3) + 1/3."
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "At around 1:09-1:25, Sal said that if you multiply or divide by a negative, you would have to switch the sign the other way. Why do you have to swap the inequality sign when you do the operation by a negative?",
        "A": "If you think about it in terms of a number line, it makes sense. Say you have the inequality a<b. This would be true if a is, for example, 4, and b is 5. Now multiply both sides by -1. Now you have a = -4 and b= -5. -4 is greater than -5, so the inequality has switched and a>b."
    },
    {
        "video_name": "y7QLay8wrW8",
        "Q": "At 1:43 how did -25/3 become just +25?",
        "A": "-25/3 became -25 when both sides got multiplied by 3. Then he added 25 to both sides, because he only wants to deal with one number on each side: -12y -25 +25 = -12y 2+25 = 27 This way he can solve for y."
    },
    {
        "video_name": "WWe7pZjc4s8",
        "Q": "At around 2:15 he states that the steps work whether you are using radians or degrees, but it doesn't seem reasonable that the polynomial will give you the same answer whether you use x = pi/2 or x=90. Is there some reason that you makes this approximation work specifically with radians?",
        "A": "That is only because you are dealing with an angle of 0, which is the same in radians and degrees. For any other angle, you need to use radians. At this level of study, degrees should not be used at all."
    },
    {
        "video_name": "WWe7pZjc4s8",
        "Q": "Why is -sin(x) the derivative of cos(x)? I mean, where does this information comes from? How can I calculate this?\n\nat 1:18",
        "A": "There are a variety of ways to prove this. I prefer direct computation using the definitions of the sine and cosine functions: Note that: sin x \u00e2\u0089\u00a1 \u00c2\u00bd*i*e^(-x*i) - \u00c2\u00bd*i*e^(x*i) cos x \u00e2\u0089\u00a1 \u00c2\u00bd e^(-x*i) + \u00c2\u00bde^(x*i) Thus: d/dx cos x = d/dx { \u00c2\u00bd e^(-x*i) + \u00c2\u00bde^(x*i)} d/dx cos x = d/dx {\u00c2\u00bd e^(-x*i)} + d/dx {\u00c2\u00bde^(x*i)} d/dx cos x = \u00c2\u00bd d/dx e^(-x*i) + \u00c2\u00bd d/dx e^(x*i) d/dx cos x = \u00c2\u00bd e^(-x*i) d/dx(-x*i) + \u00c2\u00bd e^(x*i) d/dx(x*i) d/dx cos x = \u00c2\u00bd e^(-x*i)(-i) + \u00c2\u00bd e^(x*i)(i) d/dx cos x = \u00e2\u0088\u0092 [\u00c2\u00bd i e^(-x*i) - \u00c2\u00bd i e^(x*i)] d/dx cos x = \u00e2\u0088\u0092 sin x"
    },
    {
        "video_name": "E78ji1uBtSA",
        "Q": "3:36 -Degenerate triangle! I've never heard of this before, but I guess it would have to: two angles of zero, and one of 180; why is is called \"degenerate\", though?",
        "A": "A degenerate case is a member of a certain class that is so different from the other members that it belongs to another, usually more simpler class. e.g. a triangle formed by three points all of which are on a straight line is better considered as a line segment. These are the limits of the definition of things. The degenerate triangle satisfies the definition of a triangle (it is a triangle) but it is better/simpler to consider it a line segment."
    },
    {
        "video_name": "E78ji1uBtSA",
        "Q": "at 2:49 i dont understand the 3,3,5",
        "A": "it means the sum of the biggest side should be smaller than the sum of other two sides"
    },
    {
        "video_name": "E78ji1uBtSA",
        "Q": "When he says degenerate triangle at 2:24, why is it called a degenerate triangle, why does it have that name?",
        "A": "A degenerate triangle is what you call 3 co-linear points - the figure doesn t actually have 3 sides (like a true triangle), but is just 3 points on the same line. In mathematics generally degenerate means belonging to a simpler case. In this case, the simpler case those 3 points belong to is that of a line segment."
    },
    {
        "video_name": "1TK8V_qmqrk",
        "Q": "A little after 3:00 Sal's looking for critical points and sets the derivative to 0. Do we always set the derivative to 0 to find critical points? Why not set the derivative to a negative number?",
        "A": "Critical points are where the derivative equals zero or does not exist. This is because only at these points can the function change sign on either side of it. So let s say a function has a positive slope to the left and a negative slope to the right the critical point would represent a maximum value. The opposite signs would represent a minimum value and if the slopes stay positive or negative across the critical point it is considered a shelf point."
    },
    {
        "video_name": "1TK8V_qmqrk",
        "Q": "At 4:12 it says Sal wrote \"\"x = 4\" but meant \"x = \u00c2\u00b14.\"\" but what is the point of writing \u00c2\u00b14?",
        "A": "x=4 means that the only answer is 4. +- 4 (I don t feel like looking up how to type that) means that there are two answers: x=4 and x= -4."
    },
    {
        "video_name": "1TK8V_qmqrk",
        "Q": "At 2:49 Sal said that if he put x=0 it will be undefined for y as well. Why we can't take 0 as critical point?",
        "A": "If you take it as a critical point the whole thing will be 0 which won t work."
    },
    {
        "video_name": "1TK8V_qmqrk",
        "Q": "New to calculus, but why on 4:25 does S prime prime = 2 + 1536x^-4? I get the second part but why 2?",
        "A": "The first term of S(x) is x^2. If we take the first derivative of this, from power rule, we get 2x^(2-1), or just 2x. If we take the derivative of the first derivative, or the second derivative, we get the slope of the line 2x, which is 2. That s why the second derivative of S(x) has the term 2 in it. I hope this helps!"
    },
    {
        "video_name": "1TK8V_qmqrk",
        "Q": "At 4:11, Sal says the square root of 16 is just 4. What happened to the plus and minus four?",
        "A": "You can check this by plugging the critical points(4,-4) in the second derivative [s (x)]. Since [s (x)]>0 at both the critical points there is a minimum at (4,-16) and it s concave up."
    },
    {
        "video_name": "1TK8V_qmqrk",
        "Q": "At 4:27: Wherefore do you take the second derivative of the original equation?",
        "A": "The second derivative was to verify that the point x=4, would indeed create the smallest sum. And not, for instance, the largest sum. The first derivative only showed us that x=4 was a critical point, not which kind."
    },
    {
        "video_name": "1TK8V_qmqrk",
        "Q": "At 2:00, why do we have to look for the critical values of the function when trying to minimize it?",
        "A": "the best way to show why is probably to draw a graph of the equation and then seeing where the critical points are on that graph."
    },
    {
        "video_name": "ko-cYG3d6ec",
        "Q": "At 05:06 can someone get all the angles for the entire star?",
        "A": "I could if I wanted... But I m to lazy. :)"
    },
    {
        "video_name": "ko-cYG3d6ec",
        "Q": "He said he got 50 at 7:17 but later he said he got 56 which was the answer.",
        "A": "actually, at 9:25 he fixed it. :)"
    },
    {
        "video_name": "ko-cYG3d6ec",
        "Q": "On 7:18, when solving the variable x, your answer came out as 50 degrees. 86-30=56\nSo x should equal 56 degrees.But you corrected yourself at the end of the video so it's all gravy. :)",
        "A": "Yeah! That confused me soo much when he first did it!!"
    },
    {
        "video_name": "ko-cYG3d6ec",
        "Q": "at 7:17 isn't the angle 56 degrees and not 50 degrees",
        "A": "You are correct. It should be 56 degrees. He catches this at 9:25 after solving it a different way."
    },
    {
        "video_name": "ko-cYG3d6ec",
        "Q": "Did anyone notice that at 7:19 in this video, Sal gets his subtraction incorrect, and that the angle he is talking about should actually be 56 degrees, not 50? I'm just saying.",
        "A": "Oh, he corrects it in the end. I was too hasty to condemn! Never Mind."
    },
    {
        "video_name": "m6c6dlmUT1c",
        "Q": "I am really confused when Khan mentions the \"acceleration of gravity\" at 4:22. He writes (t^2)/2 (which later turns into (g*t^2)/2 (g being the gravitational acceleration of 10)), but I do not understand the reasoning. Can somebody please help?",
        "A": "Sal left out a step. Probably to save space. He didn t write out the formula before he made the substitution g=10. The formula can be found in the kinematics videos. Always write out the formula in its original form before manipulating or substituting values. That practice will help you memorize the formula and prevent mistakes and confusion."
    },
    {
        "video_name": "m6c6dlmUT1c",
        "Q": "At 4:20, when Sal is creating an equation for y as a function of t, how does he get t^2/2? I understand where the 50 comes from and why we subtract instead of add, but I don't understand the origin of this term.",
        "A": "t^2/2 is for the term describing the distance traveled by the car due only to acceleration in the y direction. You can learn about these equation in the Kinematics section in Physics. But from calculus, since acceleration is the rate of change of velocity with respect to time, and velocity is the rate of change of distance with respect to time, acceleration is the second integral of distance with respect to time."
    },
    {
        "video_name": "m6c6dlmUT1c",
        "Q": "At 1:12 he says \"x = 10\"; what does this x (x axis) represent? I thought the x axis = time, but in the graph he gets x values of 10, 15, 20 and 25.",
        "A": "The x axis and y axis in this problem represent distance, which is measured in meters. The parameters of t represent time, which is measured in seconds."
    },
    {
        "video_name": "HGoZfzz6dU0",
        "Q": "Around 2:23, Sal said that there are other fields in math for factorials other than permutations and combinations. What are they?",
        "A": "Another example is in calculus when you calculate the taylor series of a function."
    },
    {
        "video_name": "1byR9UEQJN0",
        "Q": "Why do we write the negative of x + 4 at 2:20?",
        "A": "We are dividing the whole polynomial by x+4 Then that means that x+4 is a factor of the polynomial Therefore x+4=0 Hence by isolating the x term by subtracting 4, we get x=-4"
    },
    {
        "video_name": "1byR9UEQJN0",
        "Q": "why at 2:45 did Sal bring down the 3?",
        "A": "Because dividing the original 3x^3 by x will not change the coefficient of the 1st term (3x^3 / x = 3x^2). The other terms will change because of the 4. But the coefficient on the 1st term isn t impacted by the 4. Hope this helps."
    },
    {
        "video_name": "1byR9UEQJN0",
        "Q": "So can someone please tell me where he gets 3x squared from?\nat 4:36",
        "A": "You can do it backwards to help understand. Your remainder is -121, and your constant is 30. Then you have -8x and then 3x^2"
    },
    {
        "video_name": "1byR9UEQJN0",
        "Q": "Why do we write a negative 4 in this video at 2:37 time frame?",
        "A": "Synthetic division is used when we divide by a binomial that looks like X - C. The number you use in the synthetic division is then the C (ignore the X and the minus sign). Sal is dividing by X+4. This doesn t look like X-C. So, we change it into X - (-4) . We then have the right format to use synthetic division. Ignore the X and the 1st minus sign. C would then = -4. This is why Sal is using -4 as the divisor. Hope this helps."
    },
    {
        "video_name": "1byR9UEQJN0",
        "Q": "At 1:04, it was said that the divisor should be a first-degree polynomial. What if I have a second-degree polynomial, but it is factored into two binomial factors? Can I divide the dividend with the first factor of the divisor, then the second?",
        "A": "No. Even when factored it is not a first degree. You have to use long division."
    },
    {
        "video_name": "1byR9UEQJN0",
        "Q": "1:10. Why does the divisor have to be a first degree polynomial? Why can't it be a second or third degree polynomial, or larger?",
        "A": "It doesn t have to be first degree. But (a) dividing by first degree polynomials makes for simpler examples, and (b) usually when we use synthetic division, we re trying to factor a polynomial, in which case we re trying to break it into linear terms."
    },
    {
        "video_name": "1byR9UEQJN0",
        "Q": "at 1:10, when he says \"polynomial of the first degree,\" does he mean binomial?",
        "A": "No, a first degree polynomial is just a linear polynomial. This means that the highest exponent a variable is being raised to is 1."
    },
    {
        "video_name": "1byR9UEQJN0",
        "Q": "At 2:31, what if it was x-4, then do we keep it negative or change it into positive?",
        "A": "change it to positive. With synthetic division you always times the constant part by negative 1. Well in simple examples I ve seen that s the case."
    },
    {
        "video_name": "qzsR7cChujg",
        "Q": "On Part 0:46 you added 27 to both sides... WHY? What does that do and why are you doing that?\nCONFUSLED!",
        "A": "3x-27=24 If you are getting confused with that operation, then simply move -27 to the right hand side of the equality and then it becomes 3x=24+27 If you move negative integer to the other side, then it becomes a positive integer and vice-versa if you are moving positive integer. Hope that works!"
    },
    {
        "video_name": "qzsR7cChujg",
        "Q": "if the ratio is 5:3 and the total is 24, how is this figured?",
        "A": "I am assuming we are trying to find x. As you said we have a ratio 5:3, we can also represent this as 5x and 3x. Since the total is 24, we make an equation, 5x+3x=24 which is later 8x=24, and x=3. Hope this helps"
    },
    {
        "video_name": "qzsR7cChujg",
        "Q": "I Hate Math So Much..I'm Trying To Understand So maybe I can Like It ..At 1:01 In the video where did he get 17 from? I understand the 51 but what happened to 27? I really just need a easy step by step answer thankyou",
        "A": "Well it explained at 1:42 but if you multiplied those fractions 8 would go into them full"
    },
    {
        "video_name": "qzsR7cChujg",
        "Q": "What if you have x+2:x = 10:6. Can you still cross multiply?",
        "A": "Yes. => 10(x)=6(x+2) => 10x=6x+12 => 4x=12 => x=3."
    },
    {
        "video_name": "qzsR7cChujg",
        "Q": "At 3:50, where do they get the seven?",
        "A": "when you multiply 7(x+1) you get 7x and 7"
    },
    {
        "video_name": "qzsR7cChujg",
        "Q": "how does 3(x-9) turn into 3x-27? @ 0:25",
        "A": "distributive property,,, he put three into x-9... so the x turns into 3x.... if you multiply 3 and 9 you get 27"
    },
    {
        "video_name": "qzsR7cChujg",
        "Q": "at 3:52, why did Sal add 7 to the 7x? (7x + 7)",
        "A": "Distributive property. 7(x + 1) = 7(x) + 7(1). Therefore, 7(x + 1) = 7x + 7."
    },
    {
        "video_name": "qzsR7cChujg",
        "Q": "At 0:36 when you said it was 3x times 27 how did you get 27?",
        "A": "He said you get minus 27. This is because he does the distributive property."
    },
    {
        "video_name": "qzsR7cChujg",
        "Q": "at 0:34 how did Sal get 3x-27? Well just the 27.How did he get 27??",
        "A": "Quite Easy ! He did the cross multiplication. He multiplied the numerator of the left hand side with the denominator of the right hand side of the equality and hence that resulted to 3x-27. When multiplied like that it becomes - 3(x-9) 3x-27 Hope that works !"
    },
    {
        "video_name": "ik2CZqsAw28",
        "Q": "At 1:24:down a squiggle, up a squiggle, down a squiggle, up a squiggle.... How does she talk that FAST??",
        "A": "She already talks pretty fast in normal life, and she speeds it up so her voice is crazy fast."
    },
    {
        "video_name": "ik2CZqsAw28",
        "Q": "At 5:11, were those pipe cleaners Vi was using to make the little iterations?",
        "A": "i think... but she said friends jk... but yeah, i think they were pipe cleaners no joke: she did say friends!!!!"
    },
    {
        "video_name": "ik2CZqsAw28",
        "Q": "the girl at the end is very stylish with her Hilbert curve necklace. i want one.(5:25)",
        "A": "YOU ARE RIGHT!!!!"
    },
    {
        "video_name": "TvUCe6Bomy4",
        "Q": "At 1:05, Sal add's -12+(19/2) and he states that this sum is equal to 7/2. Isn't -12+(19/2)=(-5/2)?? Thank you for your time.",
        "A": "He is not adding negative 12 and nineteen halves , he is adding negative 12 and nineteen AND A HALF . -12 + (19 1/2) = -12 + 19 + (1/2) = 7 + 1/2 = 7 1/2 If it was nineteen halves , then you would be correct that the result should be -5/2."
    },
    {
        "video_name": "TvUCe6Bomy4",
        "Q": "i dont understand, is h negative or positive?\nand at 3:26, i dont get how -8 is greater than 7 1/2 ?",
        "A": "Absolute value means you use the positive version. So in this case, the absolute value of -8 is the same as +8, and positive 8 is greater than 7 1/2"
    },
    {
        "video_name": "xF_hJaXUNfE",
        "Q": "At 1:39 shouldn't he put another bracket after b(a+b ?\nBecause then wouldn't it mean that (a+b) times the blue (a+b) if he only puts 1 bracket?",
        "A": "Yes, you re right. Sal made a simple mistake, but I hope you got it... :)"
    },
    {
        "video_name": "xF_hJaXUNfE",
        "Q": "at 11:26 wouldnt it be 3 different combinations and not indentical combinations as it was mentioned as the color of b and a in each of the possibility of a^2b is different? if not then why is all the different permutations showed in the case of a^2b but not in a^3 or b^3?",
        "A": "Yes, you re right. He made a minor mistake there."
    },
    {
        "video_name": "xF_hJaXUNfE",
        "Q": "at 9:38 hwo do you get 3 for both (3 choose 1)a^2b and (3 choose 2)ab^2",
        "A": "(3 C 1) = 3!/(1!(3-1)!) = 3!/(1!(2!)) = 6/(1(2)) = 6/2 = 3 (3 C 2) = 3!/(2!(3-2)!) = 3!/(2!(1!)) = 6/(2(1)) = 6/2 = 3"
    },
    {
        "video_name": "tx2Niw7aJJ8",
        "Q": "at 0:05 what is rounding?",
        "A": "Rounding is a type of estimating. 42 is close to 40 = 4 tens. 29 is close to 30 = 3 tens. Do 4 tens times 3 tens in your head. 4 * 3 = 12 and tens place times tens place = hundreds place So our estimate is 12 hundred = 1200"
    },
    {
        "video_name": "tx2Niw7aJJ8",
        "Q": "Why are we trying to find out things about tickets that do not affect us in any way?\nTimestamp:\n0:53",
        "A": "If you needed to buy 11 candy bars, and each candy bar cost $4.99 each, would you rather figure it all out step by step, or would you say well, 11 is close to 10, and 4.99 is pretty much 5, so it ll be around $50 to pay for all these candy bars."
    },
    {
        "video_name": "ZJ39gHgz_Dk",
        "Q": "At 0:02,I don't understand the drawing please help me! :o",
        "A": "The yellow boxes contain 10 crayons each, and single crayons count as 1 crayon. That way the upper drawing shows 64 crayons: 6 yellow boxes with 10 crayons each and 4 single crayons = 64 crayons. The lower drawing shows 31 crayons: 3 yellow boxes with 10 crayons each and one single crayon = 31 crayons."
    },
    {
        "video_name": "ZJ39gHgz_Dk",
        "Q": "at 2:32 why did Sal put all those slashes ?",
        "A": "to shade in the box"
    },
    {
        "video_name": "mqsIJucBn6c",
        "Q": "At 1:31, if you have the closed captioning, why does it say \"and wee see that\". Shouldn't it be \" and we see that\"?",
        "A": "I agree with I M COOL he just made a simple mistake"
    },
    {
        "video_name": "b6WtwQddAcY",
        "Q": "Why for the Radical Equations 1 did he multiply the 3 and in this he is dividing by 5 (1:22) ?",
        "A": "Because in the first problem the 3 was divided on the right side of the equation and in the second the 5 is multiplied on the left. Remember that you must do the opposite operation to solve equations."
    },
    {
        "video_name": "Fc5t_5r_7IU",
        "Q": "At 7:14, Sal's explanation seems to imply that the standard error from the line can be seen as a fraction of the standard error from the mean. The SE of the line is the 'unexplained variation' and the SE of the mean is the 'explained variation'. But as far as I can tell, the SE of the line is measuring a completely different type of variation, right? How does the variation from the regression line in any way contribute or relate to the variation from the mean?",
        "A": "Be sure to watch the next video all the way to the end. Sal pulls it all together, and I think his explanation will answer your question."
    },
    {
        "video_name": "ynIq9IxbVso",
        "Q": "At 1:08, Sal says that T(v) = T(-v) defines the function as even. An odd function is defined as T(v) = -T(-v), right?",
        "A": "This is not super necessary to say but I just noticed that... Scott, you re right T(v)= -T(-v) but Gene is also right -T(v)=T(-v). Those two statements say the same thing. To convert from one to the other just multiply both sides by -1."
    },
    {
        "video_name": "6q8mk7z72AU",
        "Q": "Does anyone agree that at 2:28 there was a mistake in step 1?",
        "A": "I know it confused me at first"
    },
    {
        "video_name": "8TIben0bJpU",
        "Q": "Did anyone notice that at about 2:29, Sal writes P(TTHT) two times?",
        "A": "I did. I think he did it by accident."
    },
    {
        "video_name": "8TIben0bJpU",
        "Q": "(5:34)\nCan't you just do 4 choose 2?",
        "A": "That is what he is essentially doing. He is just reasoning through it for our sake."
    },
    {
        "video_name": "8TIben0bJpU",
        "Q": "Did Anyone Notice that he wrote P(TTHT) Twice at 2:40?",
        "A": "Yes. It was an error."
    },
    {
        "video_name": "8TIben0bJpU",
        "Q": "At 6:25 in the video sal points out that you aren't going to overcount with this method.if you wanted to overcount(like if you wanted to know the probability of getting specific coins in a specific order) how would you do that?",
        "A": "Example For three coins: 1 H H H 2 H H T 3 H T H 4 H T T 5 T H H 6 T H T 7 T T H 8 T T T n = 3 P(xxx) = 1/2^3 = 1/8 = 0.125 For four coins: 1 H H H H 2 H H H T 3 F H T H 4 H H T T 5 H T H H 6 H T H T 7 H T T H 8 H T T T 9 T H H H 10 T H H T 11 T H T H 12 T H T T 13 T T H H 14 T T H T 15 T T T H 16 T T T T n = 4 P(xxxx) = 1/2^4 = 1/16 = 0.0625 From either of the two table getting a unique P(x...) pattern occures 1/2^n times."
    },
    {
        "video_name": "8TIben0bJpU",
        "Q": "The majority of operations involved with probability involve multiplication, even after watching intro videos and working through the modules I'm having a hard time in general intuitively understanding why. Specifically at 1:39, why use multiplication? Couldn't you break it down into complicated addition at some point?",
        "A": "Multiplication is the same as repeated addition, so yes, if you wanted to you could break any multiplication down into a complicated addition problem, but I wouldn t recommend it."
    },
    {
        "video_name": "gbGghzo7yEM",
        "Q": "At 1:18, it said that the area of that particular circle at the top left corner is 9pi. I couldn't understand that why didn't it multiplied 9pi with 4 for finding the area of all the four circles and then subtract it? And these are also not full circles, only 1/4 of them.",
        "A": "The area of a circle with radius 3 is 9 pi. Thus, the area of 4 quarter circles (the white corners on the drawing) is also 9 pi, because you divide by four to get a quarter circle, then multiply by four to account for all 4 of them."
    },
    {
        "video_name": "gbGghzo7yEM",
        "Q": "At 0:47 why it is 10 times 10 instead of adding 10+10+10+10 ?",
        "A": "Because you re trying to find the area of the square not the perimeter. To get area, you multiply the length and the height together. When you multiply 10 by 10, you get 100 as your area. Hope this helps!"
    },
    {
        "video_name": "SjY5UytlTTs",
        "Q": "At about 2:30, he starts saying \"the magnitude of vector 'c' equals the square root of 'this business' squared plus 'this business' squared...\". But why is that true if the triangle formed by vectors a, b, and c isn't a right triangle? Because if it isn't, it seems like he's referring to unrelated lines as the same.",
        "A": "But it will always be a right triangle because the i and j vectors are perpendicular."
    },
    {
        "video_name": "SjY5UytlTTs",
        "Q": "From 5:08 : In the bottom vector diagram that sal have drawn, i have feeling that Vector A (In Orange) and Vertical component of Vector C (In Green) creates right angle triangle, so angle between them should be 90 degree. And since we know other angle is 30 degree, our theta should be 60 degree.\nCan anybody expain why this is wrong assumption ?",
        "A": "It is true that the two vectors you mention are perpendicular, but that only tells us that the angle between the Blue and Green vectors is 60 degrees and tells us nothing about the angle we are interested in."
    },
    {
        "video_name": "6nALFmvvgds",
        "Q": "At 0:48, why don't you make the denominators equal?",
        "A": "He i multiplying, so he does not need to. The denominators only need to be equal when adding or subtracting fractions."
    },
    {
        "video_name": "iTn3TTF0IXk",
        "Q": "At 4:30, why didn't Sal do (4xy)^2 (2x+3y)\ninstead of (4xy) (2x+3y)?\nHe started with two 4xy's but ended up only adding one to the problem. Why is that?",
        "A": "It was (4xy)(2x)+(4xy)(3y) 4xy is a common factor, he simply factored it out to get (4xy)(2x+3y). If you distribute this, you ll get the expression above. (4xy)\u00c2\u00b2 is only when you multiply (4xy) by (4xy), but it is not the case here."
    },
    {
        "video_name": "CfJzrmS9UfY",
        "Q": "At 2:35, the way she sketches that star is out of this world! It's perfect! I can't even do the supposedly simple 5-sided ones, and the way she drew that star, it was as if she'd been able to do that all her life! How'd she do it so flawlessly?\n\nThank you!",
        "A": "She does not draw it the same way the video suggests. For instance, she is drawing the star a lot slower than she shows, but then she speeds up the video. She is also not speaking while drawing the star."
    },
    {
        "video_name": "CfJzrmS9UfY",
        "Q": "What is vi saying right after 2:53?\nNormally after replaying the video 10+ times i can figure it out but not this time! :-)\nThanks for the great video",
        "A": "She says, Unless you count asterisk stars in which case 8 was two fours or four twos or two twos and a four, but ten is interesting because you can make it a composite in more than one way."
    },
    {
        "video_name": "CfJzrmS9UfY",
        "Q": "At 1:40, how does she get the dividing number when drawing the points without picking up her pencil? Like, where did the Q=3 come from and why?",
        "A": "she just picked a random number and went that far over before continuing."
    },
    {
        "video_name": "CfJzrmS9UfY",
        "Q": "also at 2:22 if you pause the video and look to the left the Q for one star is 1, but should be 4. please comment if you have an answer",
        "A": "Yes, the Q should be 4. If Q were 1, that would just be a regular nonagon."
    },
    {
        "video_name": "CfJzrmS9UfY",
        "Q": "At 2:35, she say one one-gon, and drew it on her paper, but wasn't what she drew a two-gon, because what she drew had both length and width?",
        "A": "Vi simply drew a dot as a 1gon. That is the correct thing to do."
    },
    {
        "video_name": "CfJzrmS9UfY",
        "Q": "At 3:43, how many lines are there in the circle?",
        "A": "There are 36 lines on the circle."
    },
    {
        "video_name": "CfJzrmS9UfY",
        "Q": "Wait she hasnt jumped out the window but cant say the same for every subject at 2:08\nDoesnt that mean she jumped or shouted out a window XD",
        "A": "probably both? the only class i would ever leave via window is sixth grade art class! NOT DRAWING PEOPLE!"
    },
    {
        "video_name": "CfJzrmS9UfY",
        "Q": "How does she make that at 3:45?\nIt's so cool!",
        "A": "she divides a circle into 36 equal bits. then she connects connects the dots in such a way to get 12 equilateral triangles."
    },
    {
        "video_name": "CfJzrmS9UfY",
        "Q": "at 0:24, what does \"soporific\" mean?",
        "A": "Inducing or tending to induce sleep."
    },
    {
        "video_name": "DNAnQBhGpRw",
        "Q": "@ 4:02, why do we divide by 7 instead of 8? I know he says it is the unbiased sample variance, but what exactly does that mean?",
        "A": "That means that he is using a better approximation for the variance of the population, given a normal distribution."
    },
    {
        "video_name": "jmbg-DKWuc4",
        "Q": "at 1:11 how did you get 9x squared(2) ?",
        "A": "Sal factored out a common factor of 5 using the distributive property. This divides the 2 terms by 5 to find out what remains inside the parentheses 45x^2 - 125 = 5 ( 45x^2/5 - 125/5) = 5 (9x^2 - 25) Hope this helps. If not, I suggest you back up and review the section on factoring out common factors."
    },
    {
        "video_name": "jmbg-DKWuc4",
        "Q": "At 1:17 you say 9x squared is a perfect square, how do you know it is a perfect square? Is it because it is 3x3x3, not 9x5, like 45 is?",
        "A": "9x^2 is a perfect square because 3 times 3 is 9 and x times x is x^2. So (3x)(3x) = 9x^2 45 isn t a perfect square so 45x^2 isn t a perfect square. Once 5 is factored out, we have 9x^2 which is a perfect square."
    },
    {
        "video_name": "jq6WFvMikBQ",
        "Q": "At 2:36 how did Sal know to pick (3,2)? Was it only because it was one unit away from a point on line A?",
        "A": "He arbitrarily chose (3,2). He even says at 3:30 that you could have picked any of an infinite amount of points."
    },
    {
        "video_name": "8OHEgD6YMBw",
        "Q": "At 2:56, what equation would describe the number of wild hogs?",
        "A": "It can be a function like this: W(y) = H * 3^(y/5) Where: y: is the number of years. H: the number of hogs that you start with (it must be a given information). W: the total number of hogs that tripled after y years."
    },
    {
        "video_name": "8OHEgD6YMBw",
        "Q": "At 0:42, how did Sal know to increase the function by 1.05?",
        "A": "The original weight increases by 5%. 5% = 0.05, but we are adding that to the original weight, so 1+0.05 = 1.05. If the original weight had been 100 and we increased it by 5%, we would be increasing it by 5% of 100, which is 5% * 100 = 5, so we add 100 + 5% * 100 = 1* 100 + 0.05 * 100 = (1+0.05) * 100 = 1.05 * 100 = 105."
    },
    {
        "video_name": "67jn5Zv-myg",
        "Q": "3:48 why did he subtract the exponents if its a multiplication question? isnt it if you divide you subtract and multiply you add?",
        "A": "He subtracted them because they are negative, and subtracting 12 is the same as adding -12. For example 6 - 5 is the same as 6 + (-5)."
    },
    {
        "video_name": "67jn5Zv-myg",
        "Q": "At 1:17 why do put 10 to the 8th power?",
        "A": "The 8 tells you to move the decimal 8 time to the right."
    },
    {
        "video_name": "67jn5Zv-myg",
        "Q": "At 2:24 (approximately) what is the explanation that you can add/combine exponents with the same base? What is the reason this is possible?",
        "A": "Suppose you have 2^4 x 2^3 then you can write this out as: 2^4 = 2 x 2 x 2 x 2 and 2^3 = 2 x 2 x 2 So 2^4 x 2^3 = 2 x 2 x 2 x 2 X 2 x 2 x 2 = 2^7 which is really the 4+3"
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "Is there any way to turn off subtitles? I could't see at 6:43 through 7:30.",
        "A": "No, but this is a problem they r trying to fix."
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "So at 3:02 hes saying that the arithmetic mean is different from the average in statistics?",
        "A": "I think he is basically saying that the word average is not as precise as mean , and that if you read through a high-quality statistics textbook, they will almost never use the word average at all. But I personally would say that 99.9% of the time anyone (including professional statisticians) say the word average , they mean arithmetic mean ."
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "At 2:05, is there an \"Average of Averages\"? An example being the average points a player scores per game over a given set of seasons?",
        "A": "Of course there is... you can do that. It s just using average on the results of other averages..."
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "At 2:33, what does Sal mean by \"Central Tendency\"? I'm confused on that part.",
        "A": "The measures of central tendency are used to work out and describe where the middle parts of your data are."
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "at 2:20 the average is the the mean right",
        "A": "Most likely. Usually, (aka. in casual talk) mean and average are considered the same thing. However, average could mean many forms of central tendency. P.S. Thanks Casey for the correction! :)"
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "at 7:09 Sal says that the median is greater than the first two and lesser than the last two\nbut what if your data set wen't something like 8, 2, 4, 7, 2,\nwhat would the median be?",
        "A": "You have to arrange the numbers from least to greatest - 2, 2, 4, 7, 8 and then take the middle number (4)."
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "at 6:22, what if the two middle numbers are way off like 5 and 100... How would you find the half of that?",
        "A": "Take the average of the two numbers. If it s 5 and 100, then add them and divide by 2 5+ 100 = 105 1/2 of 105 = 52 1/2 52 1/2 is exactly between 5 and 100"
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "At 2:43 he said central tendency. Is that the middle?",
        "A": "The mean, median and mode are all different ways of measuring central tendency. It depends on the data set which one is the better measure. For example symmetrical data would use the median, Scewed data would use the median."
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "At 7:36 Sal Khan says that Mode is the most common number. What if I have a set of 5, 1, 2, 1, 5, 1, 4, 7\nMy Mode is 1? It is not 1 and 5?\n\nThank you.",
        "A": "There are three ones and two fives. The mode of that set would be one. If there were three ones and three fives, the mode of that set would be one AND five. And a set with no multiples in it would have no mode. Mode is a bit tricky to understand at first, but once you get the hang of it, it is easy."
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "At 4:01 Sal said that you would have to do all the number over 6. Why?",
        "A": "You divide the sum of the values by six because there are six individual values so the mean is the measure of central tendency stating theoretically that if each unit were of equal value what the value would have to be if they were evenly distributed."
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "at 7:39 sal says mode is the number that repeats the most,but what if there was no repeating number like in 1,2,3,4,5,6,7,8,9?",
        "A": "Then if you are asked to give the mode, you simply write There is no mode . :)"
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "at 2:20 is average the mean",
        "A": "Usually, if someone says average , it is the (arithmetic) mean. But at 2:20 of the video, I think he wants to say that average could be interpreted more loosely as just some measure of a typical value , and that s why we have three different technical terms (mean, median, mode) that all might be what we might casually call an average ."
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "at 2:00, who actually came up with it?",
        "A": "I ve hard of this magical thing on the INTERNET yes really the INTERNET and guess what there is this thing called a SEARCH ENGINE you should try it."
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "at 8:52 I would like the videos to focus on one step at a time, because Khan Academy split up the practices so why can't the videos be?",
        "A": "so they go faster because some of us are fast learners and learn faster and you can always rewind the video and re-practice it!(:"
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "At 3:25, is there another way to get the mean?",
        "A": "Nope. There are easier ways than to do it by hand, for instance many computer programs can take a list of data and calculate the mean very quickly, but they are still performing this basic computation."
    },
    {
        "video_name": "h8EYEJ32oQ8",
        "Q": "At about 3:56, in stead of doing 4+3+1+6+1+7, why not 4+6+3+7+1+1? Because you get 10 twice, which is simpler.",
        "A": "Yes, that s a good idea."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "At 2:15, why did you multiply each side by x+1/1?",
        "A": "This was done to eliminate the fractions and get all instances of x out of the denominators. Since this is a proportion, the other option would be to cross multiply. You would get the same result. Hope this helps."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "In 5:38, I'm confused about the \"40 part,\" which basically made me lost for the rest of the problem.",
        "A": "Sal wrote 8(x + 5) this equals 8x + 40 BECAUSE 8 times x equals 8x and 8 times 5 equals 40. This is the way equations with parentheses are done; something you would understand if you had completed the Basic Arithmetic playlist before coming to Algebra."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "2:10- totally lost me there. Why do you multiply by 'x+1'? 4:10- Why do you multiply by 1/6? And do all linear equations deal with fractions??",
        "A": "Sal likes to do it this way."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "At 6:22 why does Khan subtract 8x from both sides and does not choose to subtract 3x from both sides? when we get to this level of the equation on any equation such as this one do we always take the x from the right side of the equation to keep keep going?",
        "A": "It doesn t matter which side you move x to. You just choose one. The answer is the same either way."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "why do the X's cancel out? 0:47 and at 3:14 why do you subtract the 7x, why not the x?? (the X from the \"x+2\" side)",
        "A": "If you mulitiply two of the same number, in this case X, then they both cancel out."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "Hi everybody!\nI'm having trouble doing linear equations with fractions. It's similar to the ones Sal showed with variables within the fraction at 0:13, but these problems also have fractions all by themselves. For example: 12(3x/2 - 1/3) = 12(x/4 + 2). I always manage to confuse myself on these. Does anybody have any tips for me? Thanks!",
        "A": "Distribute the 12. In the case of the fraction, put the 12 in fraction form. The begin to do the problem as sal did. Hope that helps!"
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "Where does the speaker get -1/5 from in 7:14 ? Can someone please explain, apart from that I understand. Thanks so much!",
        "A": "The left side has -5x remaining on it. To get the x by itself, you must divide both sides by -5 (which is the same thing as multiplying by -1/5)."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "At 6:23, why is the 8x being subtracted, shouldn't it be divided? I thought when you had to cancel out you had to do the reverse operation.",
        "A": "The video seems correct to me. In this case, we are doing the inverse operation: in (8x + 40), 8x is being added to 40. So, to get the 40 by itself, we should do subtraction, the opposite of addition. We do division later when we have -5x = 34: in (-5x), x is being multiplied by -5. So, to get the x by itself, we should do division, the opposite of multiplication."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "At 4:00 instead of x 1 / 6 you can just -6/-6 = x 5/-6 = -5/6 X= -5/6",
        "A": "There are many different ways to solve problems in mathematics."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "At 5:38 I get confused. Where did he get the 40?",
        "A": "Yeah I was completely thrown off when he said 40. But, I got it now."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "why do you cross multiply, like at 2:10?",
        "A": "it also simplifies the problem"
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "at 0:34, why is it not acceptable to multiply 3/x by 3 to cancel them out? I realise that it gives the wrong answer, and I suspect it has something to do with mixing the numbers and the variables, but I don't really understand.",
        "A": "Thank you! I can see that now, and don t know why this confused me yesterday. Guess that s learning for you."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "At \"2:05\" he says in order to get the x+1 out of the denominator, you multiply bouth sides of the equation by (x+1) over 1. I don't understand how he got this and I don't understand anything after that step either. Please help!",
        "A": "Think of any fraction as an unfinished division problem. Thus, X+2/X+1 is equal to X+2 divided by X+1. If you then take this expression, and MULTIPLY it but X+1, then the dividing X+1 is cancelled out by the multiplying X+1 . Hope that clears it up. In closing, any fraction is an unsolved division problem. So, if you have something that says X+6/X+2= 8, let s multiply the equation by X+2. So you end up with X+6= 8X+16. Good luck!"
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "Where did the 1/5 come from at 1:35?",
        "A": "It came from the 5x multiplying."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "from about 2:50 I think",
        "A": "I was adding what time I got the question from the question below"
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "What did he mean in 2:49 when he changed the 7(x+1) into 7x+7 ?\nI believe he explained it a little by advising to use the destributive property to solve it but i still didn't understand.",
        "A": "You are correct in thinking it is the distributive property. Here s the breakdown: 7(x+1) - the original value, you want to multiple both values inside by 7. 7*x + 7*1 - after the parenthesis is removed where each value multiplied by 7 7x + 7 - This is the final value You essentially multiple every value in the parenthesis by the outside value."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "Khan Academy,\nI still don't get the part when you have to multiply it by the whole expression. This is at 6:07.",
        "A": "well, you have to isolate the denominator so that is why he multiplies by x+2. hope this helps!(:"
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "At 0:46 he says that the two Xs will cancel out. Where did he get that from?",
        "A": "he wrote 3/X which means that 3 is being divided by X . Then, he wrote . X which means that 3 divided by x is now being MUTLIPLIED. If you divide 3 by X and then multiply it by X , you re really not doing anything, and you re just left with a 3 . (Multiplication and Division are opposites, just like Addition and Subtraction.) Hope that helped :)"
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "at 2:13 i understood why he put (x+1) but why is it over 1?",
        "A": "if it is over one it makes it (x+1). if you dont know, the whole equation is actually looks like this: (x+2) / (x+1) ------------- ((((((1)))))) *the parenthesis are the spaces."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "At 2:26, how does the \"(x+1)\" and \"x+1\" cancel out? (the problem on the left side)",
        "A": "(x+1) ------- = 1 (x+1) because anything divided by itself is 1. It may have been a little confusing in the video because Sal did not put parentheses around the x+1 in the denominator. Techically it would have been better to add the parentheses when he multiplied the (x+1)/1 by the equation. I hope that helps."
    },
    {
        "video_name": "9IUEk9fn2Vs",
        "Q": "At 2:47 Sal says \"Distributive property\", is there any others? What are they named?",
        "A": "Yes: associative property, commutative property."
    },
    {
        "video_name": "2G_Jr_XpnY4",
        "Q": "Around 5:11 Sal said that if your number is dividible by 6 that means it can be divided by 2 and 3. But can't it just be one of the numbers.",
        "A": "The factors of 6 are 2 and 3. So, for a number to be divisible by 6, both 2 and 3 need to work. If the number is only divisible by 2 or only divisible by 3, then it wouldn t be divisible by 6. For example: I can divide 8 by 2, but not by 6. I can divide 15 by 3, but not by 6. I can divide 18 by 2, 3 and thus also 6."
    },
    {
        "video_name": "2G_Jr_XpnY4",
        "Q": "Why does he write \"Not dw by 9\" at 5:42",
        "A": "It s just his handwriting. It should look like not div by 9."
    },
    {
        "video_name": "2G_Jr_XpnY4",
        "Q": "He said to find a number divisible by 3 we can add the didgits of the number and if that sum is divisible the number is (at about 1:00). Is three the only number you can do that with? And why does this work?",
        "A": "works with 9"
    },
    {
        "video_name": "2G_Jr_XpnY4",
        "Q": "At 3:00-3:30, Khan states that one finds the divisibility of 4 by seeing if the last 2 digits of a number are divisible by 4. What if you want to know if a 2 digit number is divisible by 4?\nFor example:If want to know if 62 is divisible by 4, how would I find this out?",
        "A": "If the tens digit is even you can just check if the second digit is 0,4 or 8 whereas if its odd you just check whether its 2 or 6"
    },
    {
        "video_name": "G7WyEp8gHs0",
        "Q": "At 2:24, Sal wants to write dy/dx in terms of x, so why doesn't he just subsitute arctan(x) for the y. After all, y does equal arctan(x).",
        "A": "Sal wants to show why the derivative of arctan(x) is 1/(1+x^2), and this method is the easiest way of doing so. Although there probably is a way to simplify cos^2(arctan(x)) to 1/(1+x^2) , I think Sal s way was simplest. And if you are wondering why it is so important to simplify it to the exact form of 1/(1+x^2), it is because that is the most commonly used form in textbooks, classrooms, and life."
    },
    {
        "video_name": "G7WyEp8gHs0",
        "Q": "4:30 Don't the two cos^2y's cancel out in the denominator? How is Sal then dividing the sin^2y by cos^2y? I'am not understanding that process.",
        "A": "No, the cos\u00c2\u00b2y in the denominator is being added to sin\u00c2\u00b2 y, so you cannot cancel out the other cos\u00c2\u00b2 y because you can only cancel factors, not terms."
    },
    {
        "video_name": "G7WyEp8gHs0",
        "Q": "At 2:11 why we do not directly use the identity (tanx)^2+1 = (secx)^2 , is it wrong to use it here ?",
        "A": "Sal really likes showing the derivation of things, but yes, you could have used the identity tan\u00c2\u00b2x + 1 = sec\u00c2\u00b2x and gotten the same result. Good eye!"
    },
    {
        "video_name": "G7WyEp8gHs0",
        "Q": "At 4:08 why do we just divide by cosy^2? Where is the cosy^2 coming from? Why are we allowed to do this?",
        "A": "We multiplied both numerator and denominator by 1/cos^2(y). Because we did this to both numerator and denominator, this is equivalent to simply multiplying by 1, which doesn t change the value."
    },
    {
        "video_name": "nMrJ6nbOQhQ",
        "Q": "At 7:50, when we are taking the integral of the function, shouldn't the result be also multiplied by \"u\", as the integral of \"du\" is just \"u+C\"?\n\nThank you!",
        "A": "That is built into the concept of an integral, so no, because you re already doing that."
    },
    {
        "video_name": "nMrJ6nbOQhQ",
        "Q": "At 4:04, why can't I just integrate sin^3(theta)*cos^2(theta) by substituting cos^2(theta)=(1-sin^2(theta)) and then distribute the sin^3(theta)?",
        "A": "If you do that you get sin^3(theta)-sin^5(theta), and then you would have to use reduction formulas, which takes more time and is also complex."
    },
    {
        "video_name": "yHo0CcDVHsk",
        "Q": "I don't understand how at 2:42 (around there) you said that if everything evaluavtes to zero, it means it will be on the y-axis. tecnically, x could still be zero, and that would only change the y-axis value. (in both cases the value of x, if input at 0 is going to stay zero, regardless of wether sin(0) is one or zero, it will always be on the y-axis)",
        "A": "He never said that it was going to be on the \u00f0\u009d\u0091\u00a6-axis, but of course a point (0, \u00f0\u009d\u0091\u00a6) will be on the \u00f0\u009d\u0091\u00a6-axis. What he did say is that the point (0, -2) is on the graph, as well as on the midline of the graph, and from that he concluded that \u00f0\u009d\u0091\u0093(\u00f0\u009d\u0091\u00a5) \u00e2\u0089\u00a0 3 cos (\u00f0\u009d\u0091\u0098\u00f0\u009d\u0091\u00a5) \u00e2\u0080\u0093 2. He could still have written it as \u00f0\u009d\u0091\u0093(\u00f0\u009d\u0091\u00a5) = 3 cos (\u00f0\u009d\u0091\u0098\u00f0\u009d\u0091\u00a5 + \u00f0\u009d\u0091\u009a) \u00e2\u0080\u0093 2, but, if avoidable, we don t want to include the modifier \u00f0\u009d\u0091\u009a, and therefore he instead chose to write it as \u00f0\u009d\u0091\u0093(\u00f0\u009d\u0091\u00a5) = 3 sin (\u00f0\u009d\u0091\u0098\u00f0\u009d\u0091\u00a5) \u00e2\u0080\u0093 2."
    },
    {
        "video_name": "yHo0CcDVHsk",
        "Q": "At 2:23, why is cos(0)=0and sin(0)=1? I am confused because when x is zero, doesn't that mean cosine is zero since the unit circle points are (cos,sin)? I can't really understand it. Cosine graphs start at 1 and sine graphs start at 0 so wouldn't it be a cosine graph?",
        "A": "Sal chose the sine graph because the behavior of the function around the origin closely resembles the behavior of sine around the origin. He could have used a cosine function to represent the graph, although he would have to apply a phase shift to it. When theta=0, the corresponding point on the unit circle is (1,0), so we know that x=1, y=0 and r=1. By definition of sine and cosine on the unit circle, cos(theta)=cos(0)=x/r=x=1 and sin(theta)=sin(0)=y/r=y=0."
    },
    {
        "video_name": "yHo0CcDVHsk",
        "Q": "at 1:33, Sal immediately writes the amplitude as 3. How does he know that it's positive three that not negative three?",
        "A": "Amplitudes are described by their absolute value. An amplitude of -3 or an amplitude of +3 will only flip the sinusoidal function over the x-axis. The actual height of the graph is unchanged."
    },
    {
        "video_name": "xKhU3iMKRiw",
        "Q": "At 0:14, the beginning of the video Sal said 7 times a, or 7 x a.\nWhy would I have to do that?\nAnd then also in step one how do you get the same answer in both equations?\nWould not sometimes the equations have different answers?",
        "A": "= sign means the value on the left-hand side of the sign is equal to the value on the right-hand side of the sign. So if a problem uses = , the values on both sides must be equal. If two values are different, you can use an inequality. For example, 4 and 7 are different. So you can represent that as 4 < 7 or 7 > 4."
    },
    {
        "video_name": "3OFH8OhpN08",
        "Q": "on 0:52 isn't it supposed to be 5 fifths",
        "A": "No, 3/5 is shaded in while 2/5 is not."
    },
    {
        "video_name": "KwR_ysrv3sg",
        "Q": "At 3:11, why does he neglect the 1/2 shown in the previous equation. He jumps from 1/2 (3sintheta)^2 to simply 9 (sintheta)^2. Why would it not be 9/2....",
        "A": "So first he sets up two different equations for the two different regions but then he discusses that both the regions have the same area hence he only uses one equation and multiplies it by 2 which cancels out the 1/2 in front and hence he just takes out the 9 as is and as there is no 1/2 out front, it remains as 9 and not 9/2. Hope that helps."
    },
    {
        "video_name": "KwR_ysrv3sg",
        "Q": "at 1:51, why did he choose pi/4?",
        "A": "Because the angle pi/4 is when the area stops being bounded by the first circle (r = 3 sin theta) and starts being bounded by the second circle (r = 3 cos theta). With the first integral, he is trying to measure the red area, which is bounded by the first circle (r = 3 sin theta) from angle 0 to pi/4."
    },
    {
        "video_name": "KwR_ysrv3sg",
        "Q": "At 3:05 what occurred with the 1/2 from the start it seems to have disappeared? It does not look like you brought it down, why is the answer not 9/2?",
        "A": "Because the area we are looking to find is symmetric about the line \u00cf\u0080/4, you can just find the area of half of the enclosed figure, and then multiply this by 2. So at 3:05 since the integral in red is equal to the integral in blue, he just multiplied the red integral by two. You could solve it the longer way with keeping the two distinct integrals, and you should get the same answer."
    },
    {
        "video_name": "XZRQhkii0h0",
        "Q": "at 1:08 what does he mean like 2 to the third power?",
        "A": "2 to the third power means 2*2*2 which is 8 if I am wrong please correct me"
    },
    {
        "video_name": "XZRQhkii0h0",
        "Q": "As sal says at 2:26 the numbers get large quickly. Is there a trick to finding the answer easily?",
        "A": "i don t know of any trick to to finding the answer quickly accept with 10 to the (insert power here) such as 10 to the 5th in which it is 10*10*10*10*10 in which you can simply count the zeroes. so the answer would be 100,000. other wise i think you just have to do the math."
    },
    {
        "video_name": "XZRQhkii0h0",
        "Q": "At 2:59 how on earth did Sal get to 625? Can someone help me with this?",
        "A": "5 * 5 = 25, 25 * 5 = 125, 125 * 5 = 625"
    },
    {
        "video_name": "XZRQhkii0h0",
        "Q": "1:09 you said that it just like 2x2x2x2x2 so do you just time sthe nuber youdid after you did part of it",
        "A": "so, I m not completely sure what your question means, but let me try to clarify... 2x2x2x2x2 would be the same as 2^5 because you re multiplying 2 by itself 5 times, it just allows you to shorten the multiplication problem at certain times, so you can write 2^15 instead of writing 2x2x2x2x2x2x2x2x2x2x2x2x2x2x2 Hope this helped, if you have any more questions, feel free to ask"
    },
    {
        "video_name": "XZRQhkii0h0",
        "Q": "Would it have not been easier to , at 3:00 to just have said 25 x 25?",
        "A": "Yes, but if there was less than four numbers being multiplied or an odd number he wouldn t have been able to say that because it would be confusing to some of the audience."
    },
    {
        "video_name": "5QY8yvZViO0",
        "Q": "For the question 2a at 1:44, why don't we have to split up the two portions of the graph to integrate them. In section R, g(x) is above the f(x) line however in section S the reverse is true. Wouldn't this affect the answer?",
        "A": "I never used the way he uses here to solve this kind of problem because I always have been splitting the functions into two intervals, but the way he uses is much easier and more efficient with the aid of calculator."
    },
    {
        "video_name": "5QY8yvZViO0",
        "Q": "At 1:24 how does he know which function f(x) is and which one g(x) is? how does he know which one to subtract from which in the integral? (as in, how does he know to do | f(x) - g(x) | as opposed to | g(x) - f(x) |?)",
        "A": "He doesn t have to know which function is above or below on what interval because he is taking absolute value of the difference between the two functions."
    },
    {
        "video_name": "rcBaqkGp7CA",
        "Q": "At 6:40 I don't understand how he found out that the height was b. What are the processes involved in finding the height. It just doesn't seem logical either.",
        "A": "There is a version of side b that was rotated 90 degrees. It is used as the altitude of the parallelogram."
    },
    {
        "video_name": "rcBaqkGp7CA",
        "Q": "Why is line b the height? It seems much smaller than line c. So how is it the height? 6:40-6:50",
        "A": "When Sal flipped the parallelogram so that b_ was the base, I found it helpful to flip the line _b that was on the triangle to the right of the parallelogram as well in my head. Since they are at a right angle to each other, it showed me that the hight of the parallelogram was in fact _b_."
    },
    {
        "video_name": "rcBaqkGp7CA",
        "Q": "At 6:40 is wrong the area of the paralelogram: the real base is c and the height is b. He's calculating b * b and that is the area of a square whos sides are all b.\nAt 9:55 you can see again how wrong is the area of the smaller rectangle of b and c.",
        "A": "It is right; Sal used b as the base and the other b perpendicular to it as the height. The diagram may be a little misleading but unlike what it looks like b and a is part of one line making a 180 degree angle. And even if c was used as the base the height wouldn t be b but a shorter line perpendicular to c ."
    },
    {
        "video_name": "rcBaqkGp7CA",
        "Q": "I do like this proof most of all, as I was not sure where it would end for all most 3/4 of the video. I was shortly a bit lost at 7:00. The top line of the second parallelogram was not in line with the triangle, so the height was not clear on the first view. One could draw a bit more carefully here.",
        "A": "Well, after a bit, it is understandable, but I agree that the height could have been drawn better."
    },
    {
        "video_name": "rcBaqkGp7CA",
        "Q": "How can a^2 + b^2 be equivalent to c^2 in the diagram at 10:12? It is drawn inaccurately, and would look more like a big square next to a little square.\n\nThanks to anyone who answers.\nJack.",
        "A": "The diagram is simply a proof of the Pythagorean Theorem. Also, the drawing isn t drawn to scale. For more proofs, search Pythagorean Theorem proofs. There are a whole bunch!"
    },
    {
        "video_name": "rcBaqkGp7CA",
        "Q": "i dont understand this ,how b to the power 2 is the volume in 6:00?",
        "A": "I think you mean area. When triangle abc is rotated 90 degrees clockwise, the new base (b) is facing diagonal. The new height (a) is on the same line as b (line b that is connected from the last parallelogram to the new right triangle) The base of triangle abc is b and that b also serves as the height of the parallelogram. B is also the base of the parallelogram and a parallelogram s area is bh. So b*b=b^2"
    },
    {
        "video_name": "rcBaqkGp7CA",
        "Q": "At 6:48 How can b be perpendicular height",
        "A": "It s the same thing that happened with side a. He rotated side b by 90 degrees, so it s going to be the height and base. - Avery @GhostFrog03"
    },
    {
        "video_name": "1LxhXqD3_CE",
        "Q": "At 3:49 Sal writes (c-c). What exactly does this mean? I'm confused about relationship between x and c.",
        "A": "Taylor series let us approximate non algebraic functions with a algebraic function at a particular point. (c-c) just means that he is finding the value of that function at that point c by plugging in c for x."
    },
    {
        "video_name": "1LxhXqD3_CE",
        "Q": "At 4:56 I am confused why Sal is only left with f'(c) after he expands f'(c)(x-c). You get f'(c)x - f'(c)c but then from there I just don't get it.",
        "A": "You re taking the derivative. Here, x is variable; c, f (c) are constants. If you expand f (c)(x - c), you get f (c)x - f (c)c The second term is two constants: its derivative is zero. The first term is (some constant) * x: its derivative is (the constant). Which is f (c). If f (c) gets in your way, you can rewrite it to remember it is a constant. For example, let a = f (c) Then d/dx ( ax ) = a"
    },
    {
        "video_name": "1LxhXqD3_CE",
        "Q": "at 6:40 why can't i write x^2-c^2 instead of (x-c)^2 for the second derivative and x^3- c^3 instead (x-c)^3 for the third derivative so on and so forth ?",
        "A": "Because (x-c)\u00c2\u00b2 is not equal to x\u00c2\u00b2-c\u00c2\u00b2. It is equal to x\u00c2\u00b2-2cx+c\u00c2\u00b2. Search for perfect squares to find lessons on this concept. Also, (x-c)\u00c2\u00b3 is not equal to x\u00c2\u00b3-c\u00c2\u00b3. It is equal to x\u00c2\u00b3-3cx\u00c2\u00b2+3c\u00c2\u00b2x-c\u00c2\u00b3. Search binomial theorem for lessons on expanding binomials of any degree."
    },
    {
        "video_name": "PQsgXNggV7Q",
        "Q": "At 1:13 why is a 1 under the 48?",
        "A": "Well any number divided by one is itself, so he is showing how multiplying a whole number by a fraction works in the same way as multiplying two fractions together."
    },
    {
        "video_name": "PQsgXNggV7Q",
        "Q": "At 0:57, how does multiplying the reciprocal get the correct answer?",
        "A": "When you are dividing by a fraction like 4/5 , you must flip the fraction (inverse) and so you must also turn the division into a multiplication (since multiplication and division are inverses of each other) Hope this helps!"
    },
    {
        "video_name": "x9lb_frpkH0",
        "Q": "Where did the nine at 4:20 come from?",
        "A": "Sal factors -2 from -18. -18/(-2) = +9 That s where the 9 comes from."
    },
    {
        "video_name": "x9lb_frpkH0",
        "Q": "at 6:51 why doesn't (x^2+9) mean that 3 and -3 are roots of the polynomial",
        "A": "Take another look at the video. You may be thinking of (x^2 MINUS 9) which would indicate roots of 3 and -3. But in this case it s (x^2 PLUS 9) --- and Sal explains that this would indicate imaginary roots, not real ones."
    },
    {
        "video_name": "x9lb_frpkH0",
        "Q": "At 6:37, why can't we take the square root of (x^2+9) as (x+3) instead of an imaginary number?",
        "A": "This because (x + 3) is not the square root of x\u00c2\u00b2 + 9. You can prove this to yourself by squaring (x + 3). (x + 3)\u00c2\u00b2 = x\u00c2\u00b7(x + 3) + 3\u00c2\u00b7(x + 3) = x\u00c2\u00b2 + 3x + 3x + 9 = x\u00c2\u00b2 + 6x + 9"
    },
    {
        "video_name": "x9lb_frpkH0",
        "Q": "At around 6:00, Sal factorized x2 - 2. can anybody explain how he did that?",
        "A": "Sal treated the ( x^2 - 2 ) as the difference of two squares. In general, ( a^2 - b^2 ) = ( a + b ) ( a - b ). We re used to seeing things like x^2 - 9 factored into ( x + 3 ) ( x - 3 ), but here Sal knew that he could use x and sqrt2."
    },
    {
        "video_name": "x9lb_frpkH0",
        "Q": "When he combined the (x+9)s together at 4:41, why are they not squared?",
        "A": "Sal was using the distributive property to move the single quantity of (x^2 + 9) to the outside of the parentheses [ in the same way that 3a+3b would be factored as 3(a+b) ]."
    },
    {
        "video_name": "x9lb_frpkH0",
        "Q": "At 7:20, couldn't x squared +9 be written as x+3 and then x could equal 0?",
        "A": "No... If we could change x^2 + 9 into (x+3)^2 to get x+3=0, then (x+3)*(x+3) would need to = x^2+9. It doesn t. (x+3)(x+3) = x^2 + 6x + 9. Hope this helps."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "What did Khan mean at time 1:42?",
        "A": "Are you referring to his statement the first degree coefficient ?"
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "Do the 'a' and the 'b' values have to be how it is. For example, at 3:04, it says 'a' is 1 and 'b' is 9. Can the 'a' be 9 and the 'b' be 1?",
        "A": "Order does not matter in this case because (x+1)(x+9) = (x+9)(x+1). Sal has a tendency to order numbers by their value, so since 1 is smaller, he tends to assign that to a because it comes before b."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 3:18, Sal mentions the last two videos, but I am not sure which videos the last two are. Could someone please post their links?",
        "A": "I think the last two could have been factoring polynomials 1 and 2. Just search factoring and they ll pop up."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 3:43, Is (x+1)(x+9) the answer or is it x=-1/x=1 or x=-9/x=9",
        "A": "If you are merely factoring, then (x+1)(x+9) is your final result; if you are solving for possible values of x , then x = -1 or x = -9 would be your results."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 14:49, can you simplify the problem -1(x-3)(x+8)? If so, would the answer be (-x+3)(-x-8)? Or would it be (-x+3)(x+8)?",
        "A": "All three solutions are correct. Many times you will be given multiple choice problems and the binomials you calculate aren t listed, but something similar is. Since two negatives multiplied together equals a positive, you could apply a negative 1 to each of the binomials being multiplied. If you have the product of two binomials (x+a)(x-b), then that is the same as (-1)(x+a)(x-b)(-1) or (-x-a)(-x+b)."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "I noticed that at 00:45, you represented the two binomials as (x+a)(x+b). Couldn't you also represent them as (a+b)(a+b) and make things simpler? The trinomial at the end would be: a^2+2ab+b^2. Isn't that simpler? Or am I mistaken?",
        "A": "Ok... but the (a+b) form is good for demonstrations, right?"
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "at 12:40 i dont understand how -x^2 turns into positive x^2(i saw the +24 go to -24 but still a bit confused.",
        "A": "Sal factored the -1 out, so that there was a -1 multiplied by all of the numbers inside the parentheses. :)"
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "Hold on. How can you multiply two negative numbers and get a positive one?\n8:14",
        "A": "Any two negative numbers multiplied together become a positive number"
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 2:10, why did Sal say that a = 1 and b = 9? Can't it also be a = 9 and b = 1?",
        "A": "Yes. It does not really matter which is which, the variables were just placeholders."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 3:06, could A be equal to 9 and B be equal to 1?",
        "A": "Yes - it doesn t matter which variable holds which value!"
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "at 5:55 that equation in the green (x^2+10x+5x+50)how would you factor that? Would I have to add the 5x and 10x then rework it? I have a test coming up on Friday, please help guys!!!And thank you so much khanacademy has brought my grade from an 81 to a 96!!! :D",
        "A": "Yes, add the 10x and 5x. Then you would factor it: (x+10)(x+5). To check, you can foil it and get your original equation. Foil= First, Outside, Inside, Last. Multiply in this order from both parentheses."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "9:15 the factored form of x^2 - 11x + 24 is (x-3)(x-8)\n\nNow, I'm curious as to whether or not it matters if I write it as\n(x-3)(x-8)\nOR\n(x-8)(x-3)\n\nIs the arrangement meaningless as no matter what they both add to -11 and can be multiplied to 24.",
        "A": "Because multiplication is commutative, (x-3)(x-8) = (x-8)(x-3)"
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 1:45, one of the numbers is underlined and called a coefficient, what is that? Are they all coefficients or just that one?",
        "A": "Coefficients are the numbers next to the variables (multiplied by the variables). It will be useful to know what the word means, so I would remember it."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "at 14:36 the original problem has a -5 why did he make it positive 5? could someone explain more",
        "A": "Look at about 13:10 in the video. Sal factors out a common factor of -1. Factoring out the -1 causes all the signs to flip to their opposites. He factored out the -1 so the leading term (-x^2) becomes positive. It is easier to factor the quadratic if it is x^2 rather than -x^2. Hope this helps."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 9:02, would it matter if you wrote (x-8)(x-3) instead of (x-3)(x-8)?",
        "A": "It doesn t matter because there is the commutative property of multiplication which allows us to order those terms however we want to."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "In 3:00 sal says a is 1 and b is 9\n\ncan't a be 9 and b be 1?\nis there a pattern or something? this is really confusing",
        "A": "Yes, you could reverse it. The only thing important is that a+b=10 and a*b=9 in the first example."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 14:49, can you do (-x-3)(x+8) instead of -1(x-3)(x+8)?",
        "A": "No. The distributive property is not being properly used. You are trying to multiply -1 by (x-3). Using the distributive property, you ll get -x+3 as a result. So it should be (-x+3)(x+8)."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 14:48, the equation -x^2 - 5x + 24 is factored as -1(x - 3)(x + 8). I want to know if the form (-x - 8)(x - 3) is correct too.",
        "A": "It is also correct, but, you could factorise (-x-8) as both have a common factor of -1."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 00:54, why does Sal use the expression (x+a)(x+b) to solve the problem? Is that a formula?",
        "A": "No, that s not a formula. That s just your basic form for what your answer should be after factoring. The basic goal is to have a + b equal to the middle x term, and have a x b equal to the last number term. So, by finding factors of 9: 1, 3, and 9, and then seeing which two add up to 10 ( 1 and 9) you can factor the equation to (x + 9)(x + 1)."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 0:59, can x(a) be equal to x(b)? If so, when could that happen?",
        "A": "That would be true if a were equal to b . However, unless we are specifically told that they are equal, we cannot assume that they are."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "I think there is a problem at 12:45 ...? I am getting -1(x-8)(x-3), because in order for the expression to be true you have to keep 8 a negative so that it turns positive when you redistribute. Am I wrong?",
        "A": "Perhaps you mean 14:45? If we used -8 and 3, then the linear term would end up being a positive (3 - 8 = -5, multiplied by -1 = 5). However this is not the case and since we ve already factorised the -1 out, we can freely factorise the quadratic."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "When would you factor out a -1, as shown at 13:05?",
        "A": "You would factor out the -1 right away to make the \u00c2\u00a8a\u00c2\u00a8 positive. This would give you -(x\u00c2\u00b2+5x-24), which is much easier to factor. The final factorization is -(x-3)(x+8) when you do this. If you didn\u00c2\u00b4t factor out the -1 beforehand, it probably wouldn\u00c2\u00b4t make a difference. You don\u00c2\u00b4t have to do it if you don\u00c2\u00b4t want to, but many people might find that it makes factorization alot easier. I hope that helps! :)"
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 3:12, I noticed that the was no more x^2. Is it because when you use the distributive property, you end up with x^2?",
        "A": "Yes, because when you factor x^2, it becomes two factors with an x in each of them."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "What's binomial at 00:41?",
        "A": "A binomial are just two terms that are together. For example: the expression 2x+3 is a binomial. You can identify the amount of terms because there is a addition sign between them, or it can be a subtraction sign. Hope this helped :)"
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 2:58 how is Sal so positive that a=1 and b=9? Couldn't a=9 and b=1 instead? Or does switching the numbers of a and b matter?",
        "A": "That s right. The order here is arbitrary. Sal made a choice, but your choice is equally valid."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "I'm unclear with the problem at 11:29. How does it figure that the difference would have to be -1?",
        "A": "Ha! Nvm, I figured it out. The formation of the problem was a little bit different so it threw me off."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "@ 16:19 how is math art? You can't draw things with it. And it sux lolz",
        "A": "Btw, I m sorry that you don t prefer math, but Math is an art because it s a subject of thinking through numbers and functions and ideas, ect. Like the Art of fighting or sciences , those don t draw either. Fyi, art isn t only associated with drawing..."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "@ 8:15 how can numbers jump? They don't have legs.",
        "A": "He didn t say the numbers jump. He said the numbers 8 & 3 jump out. It is an expressions that means they stand out, or they look like the right numbers."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 11:34 is it talking about the factors of 56? 8 x 7 = 56. Is that was a factor means?",
        "A": "That is one set of factors for 56, you could also have 4 * 14 or 2 * 28 Factors are simply a set of numbers whose product is the value you want."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "At 16:06, what does Sal mean by \"no cigar?\"",
        "A": "It is an American English expression (ca. 19th century) derived from when the prize for winning at some carnival game was a cigar. The phrase, close, but no cigar , was a polite way of saying, you lose. We now take it as meaning you were close to the right answer, but did not get the answer exactly right."
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "I still don't get why at 0:50 it is (x+a)(x+b)?????",
        "A": "a and b are just placeholders. It shows you the pattern. Both a and b will be replaced with numbers in an actual problem. x is also a placeholder, but in this type of a problem, it is left as a placeholder. If you multiply out (x+a)(x+b) you will get x^2 + (a+b)x + (a*b)x. If you were to replace a and b with your actual numbers the outcome will look like his initial problems: (x+1)(x+2) = x * x + 1x + 2x + 1 + 2 = x^2 + 3x + 3"
    },
    {
        "video_name": "eF6zYNzlZKQ",
        "Q": "at 7:18, how would I factor 32x^2 minus 2?",
        "A": "Start by factoring out a 2."
    },
    {
        "video_name": "-J-6VGxld54",
        "Q": "I watched this twice and I still don't really get it. At around 1:20 and on I get kind of lost at what he means and/or how he got the different answers. :-)",
        "A": "well... see when you add a number that is high in something say 24, you will have to add that number again if its a rectangle. When you add you will get a higher perimeter because your not multiplying. I hope this helped you ;)"
    },
    {
        "video_name": "ZmVOR6n_fzY",
        "Q": "when he says f=(x-3) squared (1:50) doesn't that equal X squared -9?",
        "A": "It does but we won t be opening the bracket as our parent function is x^2."
    },
    {
        "video_name": "Z8j5RDOibV4",
        "Q": "Why is it -15i? (@ 2:23)\n(+ 3i) * (5i) is + 15i",
        "A": "It s just -15, not -15. i is the square root of -1. So i*i=i^2=-1. If we reorder 3i*5i to 3*5*i*i we get 15*-1 which is -15."
    },
    {
        "video_name": "Z8j5RDOibV4",
        "Q": "At 1:16, Sal says he's multiplying by 1. How does multiplying by 1 get rid of the 7-5i? Wouldn't you just end up with the original fraction?",
        "A": "that s why it works, the rule is whatever you multiply the numerator by you have to do the same to the denominator. Sal uses 7 + 5i / 7+5i, which is equivalent to 1. He uses 7+5i becuase it is the conjugate of the original denominator. You could say he ended up with the original fraction, as they are equivalent to each other. I hope this helped."
    },
    {
        "video_name": "Z8j5RDOibV4",
        "Q": "At 3:11, why do those two numbers ( both are 35i) cancel out?",
        "A": "+35i - 35i = 0"
    },
    {
        "video_name": "Z8j5RDOibV4",
        "Q": "at 2:19 how does I times I equal -1 ??",
        "A": "i isn t a variable. i is defined as the square root of -1. By definition if you multiply a square root by itself you get the number that it s the square root of. For example 3 is the square root of 9 and 3 times 3 is 9. i times i = -1."
    },
    {
        "video_name": "Z8j5RDOibV4",
        "Q": "if when you simplify the numerator and denominator at 3:11 if the denominator cancels out is the numerator your answer?",
        "A": "No, if the denominator completely cancels out, with both real and imaginary numbers, it is undefined as the denominator would then be zero. If the denominator were to equal one, the answer would be the numerator."
    },
    {
        "video_name": "DGZNaKnbQo0",
        "Q": "At 2:30, what did Sal mean when he said \"the arithmetic mean\"? I don't know what mean means.",
        "A": "Arithmetic mean is another name for average ."
    },
    {
        "video_name": "DGZNaKnbQo0",
        "Q": "If you had 6 people in the graph how would you do the median then? time is 1:40.",
        "A": "Then you would find the 2 medians, and find their mean."
    },
    {
        "video_name": "aiwy2fNF_ZQ",
        "Q": "I should understand this, but at 1:04 how does Sal get [1/3] / [ 1 + x ] ?? More specifically I don't understand how he got the 1/3 on top. Any help would be great!",
        "A": "consider a simpler situation: 1 divided by 12. 1/ (3.4) = 1/3 (4) = (1/3) /4 ( same as 1/3 divided by 4)"
    },
    {
        "video_name": "9VqL6POWIoc",
        "Q": "There is something that always confuses me: the natural logarithm function \"grows\" less and less for big values of x (as seen at 2:01 on the video)... But why, then, doesn't it have a limit value, since its derivative seems to be zero at infinity?",
        "A": "Well ln(x) is the inverse function of e^x. That means that the range of ln(x) is the same as the domain of e^x. Since the domain of e^x is the set of real numbers, that means the range of ln(x) is also the set of real numbers. Or in other words: you can put anything into e^x => you can get out anything out of ln(x)."
    },
    {
        "video_name": "HLGCq2h4ihY",
        "Q": "At 0:42, apparently 99.061 X 10 is 990.61. But wouldn't that be 990.610? If you know, please help.",
        "A": "990.61 = 990.610. Mathematicians tend to omit that zero as it has no value, and also makes life easier by typing less numbers!"
    },
    {
        "video_name": "zlRKO21qEpQ",
        "Q": "At 0:54, why doesn't Sal write the equation as (4^\u00e2\u0088\u009a5)(4^\u00e2\u0088\u009aa^4) (4^\u00e2\u0088\u009ab^4) (4^\u00e2\u0088\u009ab^3)? This results in a answer of ab(4^\u00e2\u0088\u009a5b^3), which is wrong. But why wouldn't it work? In previous videos, Sal has re-written equations in their smaller multiples. For example, 2^\u00e2\u0088\u009a20 would equal to (2^\u00e2\u0088\u009a4)(2^\u00e2\u0088\u009a5), resulting in 2(2^\u00e2\u0088\u009a5). Using this method, 4^\u00e2\u0088\u009ab^12 would be equivalent to (4^\u00e2\u0088\u009ab^4)(4^\u00e2\u0088\u009ab^3). He does this in the next video too, but why not here? I am confused.",
        "A": "He doesn t need to and it wouldn t work because by turning b^12 into b^3 and b^4 you create an extra ^1/4."
    },
    {
        "video_name": "zlRKO21qEpQ",
        "Q": "At 0:22 Sal says \"In general the Nth root of something is the same thing as that something to the one over N power.\" Shouldn't he have also said \"As long as N is not equal to zero\" as well? Or am I mistaken?",
        "A": "I believe that n not equaling 0 is implied. If someone was to use 0, it would lead to n^1/0, or a number to an indefinite power."
    },
    {
        "video_name": "zlRKO21qEpQ",
        "Q": "When Sal says that the 4th root of a^4 is a at about 2:05, how come it is not the absolute value of a? In other videos, if I remember correctly, I think he said that the square root of x^2 is the absolute value of x.",
        "A": "Because the exponent is an even number (4), so a^4 will always be a positive number."
    },
    {
        "video_name": "3szFVS5p_7A",
        "Q": "At 2:33, shouldn't it become 11 ?",
        "A": "No..Well...It should if he was going to give 10 from the twenty to the ones place but he is only giving 9 tothe ones place. Since there was already 1 to begin with you result with 10 total after you add 9 to the original 1."
    },
    {
        "video_name": "3szFVS5p_7A",
        "Q": "At 3:27 he takes 1 directly from the tens place and puts it in the tenths place. He says its equal to 13/10 but wouldn't it be equal to 103/10 since its drawn from the tens to the tenths place?",
        "A": "but it would be eaqual to 10 though"
    },
    {
        "video_name": "3szFVS5p_7A",
        "Q": "at 0:53 when sal says we give the 1 to the tenths place won't that look like 20.13, and that looks like the decimal= 13/100 so the value is different, is it not? 21.3 does not equal 20.13",
        "A": "its not different, because its not 13/100, it stays as 13/10, thus the name, regrouping with decimals."
    },
    {
        "video_name": "3szFVS5p_7A",
        "Q": "at 0:55 why did you regroup to the 13? My teacher did not teach me this way, she did nothing you did in the video.",
        "A": "I think that the point in this video is to get really comfortable with the process of regrouping. The more thoroughly you practice regrouping, the more likely you will be to remember how to do it in real life when you need to. Exchanging money is one of the real life times that this is used."
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "At 4:12, how does he get the 6 into the equation?",
        "A": "(3+\u00ce\u0094x)^2 =(3+\u00ce\u0094x)(3+\u00ce\u0094x) =3(3+\u00ce\u0094x) + \u00ce\u0094x(3+\u00ce\u0094x) =9+3\u00ce\u0094x + 3\u00ce\u0094x+(\u00ce\u0094x)^2 =9+6\u00ce\u0094x+(\u00ce\u0094x)^2"
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "5:55. At this point, Sal is writing out an equation with dx on the bottom. If he were to take the limit as dx approaches 0 of that equation, he would get infinity, not six. What is the explanation for why the original equation would give the wrong answer?",
        "A": "You wouldn t get infinity. Let s take a look at that equation again. It was: slope = (6*delta_x+9delta_x)^2)/ delta_x Always divide the numerator by the denominator when you can. This give you: slope = 6 + delta_x Now when delta_x approaches zero, the slope is zero."
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "at around 7:00 when hes writing the slope from yellow to blue why does (delta X)^2 change to just delta X?",
        "A": "The equation goes from 6\u00ce\u0094x+\u00ce\u0094x^2/\u00ce\u0094x to 6+\u00ce\u0094x. The division by \u00ce\u0094x cancels the like term multiplied to the 6 in the numerator and lowers the power of the other \u00ce\u0094x term."
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "Why suddenly at 7:35 does he start reducing delta x?",
        "A": "Because delta x can be take common from the numerator and then by simple operation on fractions, he reduces the delta x"
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "At 4:04 in the video, Sal says that 6h+h^2/h simplifies to 6+h^2. I thought it was supposed to be 6+h, can you explain this?",
        "A": "At 4:04, Sal is finding f(x + delta-x). At 6:18, he simplifies (6h + h^2)/h to 6+ h as you would expect."
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "At 6:29, for the slope formula 6+ deltaX, does deltaX have to be a positive number? Or can it be negative too?",
        "A": "When calculating slope, you can define delta X to be positive or negative, but when limits are taken, the slopes should be the same regardless of sign of delta X."
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "Also at 6:25 he ends up with simplified slope of 6 + delta x.\n\nhe only had 1 delta x in the denominator, and cancelled out delta x and (delta x)^2.\n\nwhy is the slope not 6 + (delta x)^2 ?",
        "A": "[(6*delta-x) + (delta-x)^2] / delta-x = [(6*delta-x)/delta-x] + [(delta-x)^2/delta-x] = 6 + delta-x Or, [(6*delta-x) + (delta-x)^2] / delta-x = [delta-x(6 + delta-x)] / delta-x = 6 + delta-x"
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "At about 6:20, shouldn't it simplify to \"6+ (delta x) squared\"?",
        "A": "No because it s 6 * (delta)x / (delta)x = 6, and (delta)x^2 / (delta)x, which simplifies to ( (delta)x * (delta)x ) / (delta)x, which is equal to (delta)x. You have to divide both sides of the + sign by (delta)x, if you re going to simplify."
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "at ~4:20, how do you get 9 + 6 delta x + (delta x)^2 ?\n\nWhere does the 6 delta x come from? (3 + delta x) shouldnt it be just 9 + (delta x)^2 ?",
        "A": "(a + b)^2 = a^2 + 2ab + b^2 (3 + delta-x)^2 = 3^2 + 2*3*delta-x + delta-x^2 = 9 + 6delta-x + delta-x^2 (a + b)(a - b) = a^2 - b^2"
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "At 6:25, Sal simplifies (6(delta)x + (delta)x^2)/(delta)x to 6+(delta)x. However, later he substitutes (delta)x for 0 to get a slope of 6. If he'd put that in the original expression it would have been division by 0. Why does this work? Aren't you not allowed to divide by 0?",
        "A": "That s not exactly what he does. He makes dx very very very small, but not zero. Close to zero, but not zero. 6+0.0000000000000000000000001 is pretty close to 6, and the smaller you make dx the less significant it gets. It will not become exactly zero, but it will become so small that we can ignore it."
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "at 3:50 you kinda just do a crap ton of math without explaining it enough you completely lost me there and i cant continue with the video simply because i do not wish to further confuse myself",
        "A": "I don t want to be mean but if you are struggling to keep up with what he is doing there, you need to go back and review your algebra before trying to learn calculus."
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "How does Sal get 6 delta x @ 4:15?",
        "A": "For f(x) = x^2 He just plugs in 3 + delta_x into the function and multiplies it out: (3 + delta_x)^2 = 9 + 6 * delta_x + delta_x^2)"
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "can someone explain what he means at 4:16?",
        "A": "He took the 3 + \u00e2\u0088\u0086x and squared it. (3 + \u00e2\u0088\u0086x)^2 = (3 + \u00e2\u0088\u0086x) (3 + \u00e2\u0088\u0086x) = 9 + 3\u00e2\u0088\u0086x + 3\u00e2\u0088\u0086x + (\u00e2\u0088\u0086x)^2 = 9 + 6\u00e2\u0088\u0086x + \u00e2\u0088\u0086x^2"
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "I don't understand what he did at 3:57. 9+6deltaX + (deltaX)^2?",
        "A": "Given the function y = x^2. Arbitrarily choosing a value for x = 3, results in y = 9. Coosing a second value of x = 3 + \u00ce\u0094x, results in y = (3 + \u00ce\u0094x)^2 \u00e2\u0086\u0092 9 + 6\u00e2\u0080\u00a2\u00ce\u0094x + (\u00ce\u0094x)^2"
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "at 4:21\n\nat x= 3+ delta x, shouldn't the y value be (3 + delta x)^2?",
        "A": "That s exactly what it is! Don t forget your past math! Think FOIL: expand (3 + delta x)^2: (3 + delta x)(3 + delta x) = 9 (First) + 3 delta x (Out) + 3 delta x (in) + (delta x)^2 (Last) He did (3 + delta x)^2, but he just expanded it"
    },
    {
        "video_name": "IePCHjMeFkE",
        "Q": "At 6:27 why is it slope = 6 + \u00ce\u0094X and not 6+ \u00ce\u0094x^2 (delta x squared)?",
        "A": "Straight up factoring. Both the 6\u00e2\u0088\u0086x and the (\u00e2\u0088\u0086x^2) have common factors of \u00e2\u0088\u0086x, the denominator divides both the values by what it is (conveniently \u00e2\u0088\u0086x), meaning that the 6\u00e2\u0088\u0086x / \u00e2\u0088\u0086x = 6, and teh (\u00e2\u0088\u0086x^2)/\u00e2\u0088\u0086x = \u00e2\u0088\u0086x, leaving just 6 + \u00e2\u0088\u0086x on the top, and nothing on the bottom."
    },
    {
        "video_name": "3uO_uPb9H8w",
        "Q": "at 4:52 why is the differential \"3/2.c.e^3/2.x\" not \"((3/2.x)-1).c.e^3/2.x\"?",
        "A": "Hi, Munya, I ll do it step by step: y = d/dx( c*e^(3/2*x) ) = c * d/dx( e^(3/2*x) ) ||| constant can go in front of the equation = c * d/dx( e^(3/2*x) * d/dx ( 3/2*x ) ||| chain rule: (f(g(x))) = f (g(x))*g (x) = c * e^(3/2*x) * 3/2 ||| differentiate = 3/2 * c * e^(3/2*x) Hope this helps :)"
    },
    {
        "video_name": "y2RHWA7t0WM",
        "Q": "At 4:33, Sal said that a pound and ounce are units of weight. I think they are units of mass and not weight.",
        "A": "Mass is a measurement of the amount of matter something contains, while Weight is the measurement of the pull of gravity on an object. The Mass of an object doesn t change when an object s location changes. Weight, on the other hand does change with location. Therefore if you take a pound or an ounce of something to mars the mass will stay the same but the weight will be different."
    },
    {
        "video_name": "y2RHWA7t0WM",
        "Q": "At 1:09, he said he would talk about a a foot, but he didn't, at least not how I was hoping :). So, I've got a question. Is a foot called a foot because it's about as big as a foot, or did it originate some other way? I know it seems like a silly question, but i want to know. And, I also heard that an inch is about as big as a fingernail. Is it true that many of the U.S. customary units of length are based on things like you r finger or your foot?",
        "A": "Don t worry, penguinsrule=^.^=. When I was little, I wondered a lot about this. Of course our foots grow so its not that certain, since there are different foot sizes. From what I ve seen is that in historical tradition, the Romans made this length the standard for the foot, which would be 12 inches. Hope this helps! :-)"
    },
    {
        "video_name": "lFQ4kMcODzU",
        "Q": "around 3:31 in the video, that max value looks like a sudden change and i thought that derivatives do not exist at sudden change?",
        "A": "You are right in that the derivative does not exist for x=x2. However, x2 is still a critical point. Different sources yield different results, but the definition of critical point I learned in my AP calculus class is a point on the graph at which the derivative is 0 or undefined. (Notice that the function must actually have a value). Thus x2 is indeed a critical point."
    },
    {
        "video_name": "lFQ4kMcODzU",
        "Q": "At 3:07, why did Sal say that the slope is undefined?",
        "A": "Three conditions that make the slope undefined: vertical tangent, discontinuity, and sharp point . The point Sal discusses at 3:07 is a sharp point, hence its slope is undefined."
    },
    {
        "video_name": "lFQ4kMcODzU",
        "Q": "At 5:41, I am simply confused. Can you please help me understand the difference between \"local minima/maxima\" and \"global minima/maxima\" ? And why isn't the point, where X=0, satisfying the condition of the minima ?",
        "A": "You were right. Thank you."
    },
    {
        "video_name": "Xtcylk5r0g8",
        "Q": "Why at 1:40, does Sal box the negitive 12c and when he boxes in the 4c he doesn't box the positive. Why is that?",
        "A": "Because +4c is the same thing as 4c. You really have to box the negatives more importantly than the positives. It lets you know if its negative or not."
    },
    {
        "video_name": "U44my48zgFE",
        "Q": "At 0:59 how did Sal call 1/9, ninth? It should still be one ninth.",
        "A": "Your answer is reasonable, no I mean very reasonable.When he calls It a ninth It s singular, and I wonder why his name is Megalograptus?"
    },
    {
        "video_name": "U44my48zgFE",
        "Q": "In 2:15-3:47,I don't understand about the sock thingimajjig",
        "A": "It was trying to say that since the sock was not divided into equal parts, you cannot make a fraction out of it."
    },
    {
        "video_name": "RgfKNIkpFWc",
        "Q": "At 4:02 Sal evaluates the function at the point -3 is approaching the value of 1, but there is a discontinuity, a point that looks like it's at -3, 4. Why is the limit not 4?",
        "A": "The limit, by definition, is a point that function is approaching as the x-values approach a specific value. In this case function s output at the x = -3 and lim( x\u00e2\u0086\u0092 -3) of f(x) will be different values. Just like in other cases when we have point discontinuity."
    },
    {
        "video_name": "RgfKNIkpFWc",
        "Q": "Of the three examples in this section, sal states the next to last function does not exist (at 3:25) where the limit rests along a vertical asymptote and the last function is undefined (at 4:50) where the interior limit at x is defined but that its value when graphed on the outer function rests along an asymptote. Are the terms DOES NOT EXIST and UNDEFINED being used interchangeably in anticipation of some section on imaginary numbers, or do they mean different things within this context?",
        "A": "In the first graph in your question (03:25) we are looking at it in search of the limit. In the second graph (04:50) we are looking at it from a normal function input mode. On a verticle asymptote we say: Limit-wise - doesnt exsist function-wise - undefined"
    },
    {
        "video_name": "RgfKNIkpFWc",
        "Q": "At 4:51 is there is a difference between undefined and \"does not exist\"? I am not sure whether or not Mr. Khan is simply interchanging the two terms together or if there was a mistake.",
        "A": "Does not exists and undefined usually mean slightly different things. Does not exists means that there is a definition for something which does not lead to a mathematical object in a specific case. Undefined means that there is no definition for a specific case. So, a point discontinuity, like the one Sal describes, is actually undefined, while, for example, limit near the point of asymptote would not exist."
    },
    {
        "video_name": "RgfKNIkpFWc",
        "Q": "Why do we take the limits of composite functions like that at 1:02?",
        "A": "The function along that interval is a constant function. The y-value and the limit as x approaches a point along that interval is constant along the interval."
    },
    {
        "video_name": "RgfKNIkpFWc",
        "Q": "What is the dot in the functions at 2:17 and 4:27?",
        "A": "the white dot means the function is not defined there. the blue dot means the fuction is defined. in 2:17 when x = 0 then y = 2 and in 4:27 when x = -3 then y = 4"
    },
    {
        "video_name": "RkJBZALhXRA",
        "Q": "In 2:31 Sal does 2^0 power inside parenthesis isn't that wrong?",
        "A": "No because it would be the same even if it was outside. Plus you are multiplying it with -1/8"
    },
    {
        "video_name": "RkJBZALhXRA",
        "Q": "At 2:33 you said that 2 to the zeroth power times -1/8 = -1/8, but it actually equals 0",
        "A": "Remember, anything to the power of zero = 1. So 2^0 = 1, and 1*-1/8 = -1/8."
    },
    {
        "video_name": "RkJBZALhXRA",
        "Q": "2:30 how is 2 to the zero power is what ? it should be -1/8(2 to zero power)= -1/8(0)=0\nnote (2 to zero power) is = 2*0=0 is this correct or what ?",
        "A": "Any number to the zero power equals one. 14^0=1, (-3)^0=1, (1/2)^0=1, 20675^0=1. So for the example in your question, -1/8(2 to zero power) = -1/8(1) = -1/8. Does this answer your question?"
    },
    {
        "video_name": "iK0y39rjBgQ",
        "Q": "4:19 why would we ever have to use this",
        "A": "It just helps you understand how to write large numbers in expanded form."
    },
    {
        "video_name": "iK0y39rjBgQ",
        "Q": "at 3:45 he asked if the poker chip example helped......it didn't",
        "A": "Just remember the main fact: A number in standard form, like 12345, can be written because of place value. ( 1 in the 10000s place, 2 in the 1000s place, etc....) . You can review Sal s other place value videos, if they help. Hope you found this helpful."
    },
    {
        "video_name": "VbgYvbtiirQ",
        "Q": "Sal, how did you get s^3 at 2:32? This area was unclear for me.",
        "A": "He got s^3, because s^2 is s*s, so if you multiply another s it will simplify to s^3"
    },
    {
        "video_name": "VbgYvbtiirQ",
        "Q": "2:12 Why is dividing by seconds the same as multiplying 1/s?",
        "A": "Because dividing is the same as multiplying by the reciprocal."
    },
    {
        "video_name": "kJIqoTw_Og8",
        "Q": "At 3:25, shouldn't Sal NOT check the answer? Because I calculated the slope of Charles's chart, and it turned out to be -7/5. And the problem stated that Tammy and Charles did not go to work at the same speed, so since Charles's slope is -7/5, Tammy's slope shouldn't be -7/5, so therefore, the second choice shouldn't be the answer. Am I right? I am a bit confused by this.",
        "A": "One strategy to learn with word problems is read carefully and reread multiple times. The problem states: They walked at constants speeds, but not necessarily at the same speed. The phrase not necessarily at the same speed does not say they don t go at the same speed. It just means, they don t have to be going at the same speed. This means the speeds could be the same or different. Hope this helps."
    },
    {
        "video_name": "kJIqoTw_Og8",
        "Q": "At 3:16, Sal chooses both the options (2 & 3), does it mean that 7/5 = 5/7, meters/second?",
        "A": "No, it s actually negative. And -7/5 is completely different from 5/7."
    },
    {
        "video_name": "iWgW2nD4wGQ",
        "Q": "I don't know how to find the exact point. 1:47 even sal says \"around here\"",
        "A": "This is just finding inflection points graphically. To find the exact point you need the equation of the function and find the 2nd derivative. That will be explained in a later video. This seems to be just to get you comfortable with the concept of an inflection point."
    },
    {
        "video_name": "wToSIQJ2o_8",
        "Q": "At 4:04, wouldn't be 10-sqrt(3) because the t starts at 10 not 0?",
        "A": "When t=0 (at the start), the y-value is 10. Eventually the car hits the ground (its y-value is 0) at t = sqrt 10 (Sal corrected this from sqrt3). The HORIZONTAL distance travelled (along the X-axis) therefore is the x-value at the end minus the x-value at the beginning which is [10 + 5(sqrt10)] - [10+5(0)] which equals 5sqrt10."
    },
    {
        "video_name": "wToSIQJ2o_8",
        "Q": "At around 5:40, Sal talks about substituting. Is this similar to solving a system of equations by substitution? Or am I misunderstanding something?",
        "A": "The objective here is to get rid of the parameter t while the objective in a linear system is to get a 1 point answer. Its just that substitution can be used to do both."
    },
    {
        "video_name": "wToSIQJ2o_8",
        "Q": "at 4:19 What does Sal mean by saying \" went back in time \" ? Time Travel ? just curious, i know this got nothing to do with maths",
        "A": "it seems most likely to me that he was referring to setting variable t to negative values... thus going back through time or whatnot."
    },
    {
        "video_name": "wToSIQJ2o_8",
        "Q": "At 6:52 he says that there is 4/5 x, where did that come from?",
        "A": "Use FOIL to multiply (x/5 - 2)^2 = (x/5 - 2)(x/5 - 2) = x^2/25 - 2x/5 - 2x/5 + 4 = x^2/25 - 4x/5 + 4 Hope this helps."
    },
    {
        "video_name": "wToSIQJ2o_8",
        "Q": "I do not understand when he simplifies the equation at 6:40? How does this work?",
        "A": "I suspect you are wondering about when Sal squared (x/5-2) as he use to do all the steps in earlier videos. What he did was multiply: (x/5-2)*(x/5-2) (shown in the video as (x/5-2)^2) and got: (x^2/25 - 4/5x + 4)"
    },
    {
        "video_name": "zNxEVaRbdus",
        "Q": "At 4:07, couldn't he just simplify and write\n 1/5(i) instead of -3/5 + 4/5 (i)",
        "A": "In the expression: -3/5 + 4/5 (i), the 2 terms are unlike terms. We can only combine like terms. Since the first term has no i , it is unlike the 2nd term. If the expression had been: -3/5(i) + 4/5 (i), then you could combine them into 1/5 (i) Hope this helps."
    },
    {
        "video_name": "_H4jllna_ec",
        "Q": "at 3:08 how do you know its 240 degrees? and how can you solve equations using these methods",
        "A": "Because all circles equal360 degrees, so we know one part equals 120 so we subtract 120 from 360 to get 240 degrees"
    },
    {
        "video_name": "dEAk0BHBYCM",
        "Q": "hey Sal, I have a question. This is at time 3:30 when you are setting up the diagram and drawing all the rectangles. Why is it that for each rectangle they are labelled 1, 1.5, 2, 2.5..??",
        "A": "Those are the x coordinates on the graph."
    },
    {
        "video_name": "dEAk0BHBYCM",
        "Q": "At 0:20 seconds, how do you know that you need four rectangles?",
        "A": "We can choose how many rectangles we like. The approximation will improve with more rectangles. The best approximation comes with infinite rectangles. This can be done by using infinitesimal width on the rectangles. I hope this was a little helpful!"
    },
    {
        "video_name": "gAV9kwvoD6s",
        "Q": "at 0:45, wouldn't 0.070 be bigger because it is in the hundredths place?",
        "A": "I think you are confusing hundreds with hundredths. 0.070 is 7/100, which means you have 7 parts out of 100 parts where 100 parts will create 1 whole number. Sal is comparing that to 1/10. If you change 1/10 to hundredths you get 10/100. 10 out of 100 parts is larger than 7 out of 100 parts."
    },
    {
        "video_name": "gAV9kwvoD6s",
        "Q": "AT 1:50 are there any millionths in decimals?",
        "A": "Yes this is easy now that you showed us how thank you"
    },
    {
        "video_name": "gAV9kwvoD6s",
        "Q": "At 2:03 Sal said \"Dousands\" instead of 'Thousands?",
        "A": "It s just a mistake. No worries."
    },
    {
        "video_name": "gAV9kwvoD6s",
        "Q": "At 1:37 you said 0.093 but it is 0.0093.",
        "A": "The other time is when he said 0 but meant 10"
    },
    {
        "video_name": "wKPyzrEVqaQ",
        "Q": "At 3:26, why isn't 1 included but 0 is?",
        "A": "We include 0 because the graph indicates positive velocity at that point, which means the particle is moving to the right. We exclude 1 because the graph indicates velocity is zero at that point, so the particle can t be moving to the right (it s not moving at all)."
    },
    {
        "video_name": "Efoeqb6tC88",
        "Q": "At 3:15, how is the distance equal to 4? Since that's on the imaginary axis, shouldn't it be equal to 4i? Then x^2 = 49 +16i^2, and since i^2 = -1 it would be x = \u00e2\u0088\u009a33 not \u00e2\u0088\u009a65.",
        "A": "Thats a good question. I think that since we are working with the complex plane the letter i simply indicates the vertical direction rather than representing the square root of -1. So for example (2 + 4i) and (3 + 6i) represent the points (2,4) and (3,6) on the complex plane, and the distance between (2 + 4i) and (3 + 6i) on the complex plane would be the same as the distance between (2,4) and (3,6) on the real plane. Hope this helps."
    },
    {
        "video_name": "J2Qz-7ZWDAE",
        "Q": "2:54 he talks about A, B, and D being non collinear. than does that mean that A, B, and C are collinear?",
        "A": "Yes, A B and C are collinear, which defines a set of points that lie on the same straight line. Because this is what collinear means, non-collinear defines a set of points that do NOT lie on the same line. Therefore, a plane is defined as three points that do not lie on the same line. I hope this helps! Please reply to this if you have any further questions!"
    },
    {
        "video_name": "J2Qz-7ZWDAE",
        "Q": "at 2:45 you set the point somewhere else not on the line. Can you do that? Why? Why are you able to point a point wherever. Can you put a point outside of the plane?",
        "A": "Point D is just an another reference point so we could specifically define a certain plane on the space."
    },
    {
        "video_name": "J2Qz-7ZWDAE",
        "Q": "At 2:18, why do two points connected by a line always make a segment?",
        "A": "Draw two points on a piece of paper. Line up a straightedge on them and draw a straight line between them. It forms a line segment. How could it not?"
    },
    {
        "video_name": "J2Qz-7ZWDAE",
        "Q": "At 3:15, can C and D be on the same line?",
        "A": "Yes. Any two points define a line. So they must be on the same line."
    },
    {
        "video_name": "J2Qz-7ZWDAE",
        "Q": "at 2:55 he says non colinier, what does that mean",
        "A": "It means not in a line . If something is collinear then it means all on the same line . Hope this helps! \u00e2\u0080\u0094CT-2/002-24"
    },
    {
        "video_name": "J2Qz-7ZWDAE",
        "Q": "At 3:29, when you name the planes, why couldnt you use plane AJD?",
        "A": "If you look at the drawing. the points are in the plane are A, B, W and J. Point D is below the plane. The definition of plane, is given by three points that are not collinear three. In the example, less ABW, any combination of the first four points serves you. ABJ, AJB, BAJ, BJA, JAB, JBA, AWJ, AJW, JAW, JWA, WAJ, WJA, BWJ, BJW, JBW, JWB, WBJ, WJB. ABW, being collinear would form infinite planes, which rotate around the line form. AJD, form another plane, which is totally different from what you are asking."
    },
    {
        "video_name": "J2Qz-7ZWDAE",
        "Q": "At 2:23 he says collinear what does that mean?",
        "A": "Collinear means they re part of the same line."
    },
    {
        "video_name": "J2Qz-7ZWDAE",
        "Q": "At around 0:20, Sal mentions that planes go on and on in every direction. I still don't fully understand the concept though, I mean, why do we draw the shape if it's infinite? If it goes on forever, then wouldn't these two dimensional planes take up everything?",
        "A": "1.) If we don t draw the shape, how will we know it s there? 2.) Yes, planes take up everything, but only in their two-dimensional space. When you think about planes in three dimensions like this, there are actually many planes and they don t take up all the space but rather just all the space in their second dimension. Does this answer your questions?"
    },
    {
        "video_name": "J2Qz-7ZWDAE",
        "Q": "At 2:53 it says non-collinear. What does that mean?",
        "A": "Non-collinear means that those points are not on the same line."
    },
    {
        "video_name": "cwnke_pjX90",
        "Q": "At 2:20 would 400 doubled not be 8000? If so Sal made a mistake in saying that it was 16000.",
        "A": "He corrected himself 4 seconds later at 2:24"
    },
    {
        "video_name": "naVFTNWVlQU",
        "Q": "At 3:10, why is it -6 plus or minus 5 instead of 5 plus or minus -6?",
        "A": "You started with x+6=\u00c2\u00b15, so subtracting 6 from both sides yields x=-6\u00c2\u00b15. If you started with x+6=+5 or x+6=-5 and subtracted 6 from both sides, you d get x=-6+5 or x=-6-5. And you could recombine those last two possibility back to just x=-6\u00c2\u00b15. So it s the same thing either way: the \u00c2\u00b1 sign is attached to the 5."
    },
    {
        "video_name": "-U53eHKCLcg",
        "Q": "At 0:40, why does he place two bares on each side of u? What do they represent?",
        "A": "Putting bars around u means the magnitude (or length) of u. You ll see them a lot with vectors."
    },
    {
        "video_name": "-U53eHKCLcg",
        "Q": "At 0:54 why is change in X is Squared ?",
        "A": "The magnitude of vector u is just the distance or the length. The distance is square root of change in x + change in y. This is derived from the Pythagorean theorem, a\u00c2\u00b2+b\u00c2\u00b2 = c\u00c2\u00b2 ==> c = \u00e2\u0088\u009a(a\u00c2\u00b2+b\u00c2\u00b2). a and b are the two legs of the right triangle and c is the hypotenuse. In this case, c is the magnitude of vector u while a and b are changes in x and y."
    },
    {
        "video_name": "JfuisfEdnjk",
        "Q": "(3:02) why did we pick (x+a)to work with and not (x-a)",
        "A": "Probably because x+a is the first factor and x-a is the second factor"
    },
    {
        "video_name": "LpLnmuAyNWg",
        "Q": "At 1:10 why does Sal say Lumber line?",
        "A": "my number line starts with 5.1 and ends with 5.2. What are the nine missing numbers in between?"
    },
    {
        "video_name": "LpLnmuAyNWg",
        "Q": "at 1:09 what's a lumber line? Did you mean number line?",
        "A": "Yes, that is what he meant."
    },
    {
        "video_name": "1BRXCIvt9e0",
        "Q": "At 5:55, when the final answer is given as 1/8*w to the negative fifteen over two power, can we simplify the 'w' to 1/(8w to the seventh power root w)?",
        "A": "You could. You can even take it a step further and rationalize the denominator to get \u00e2\u0088\u009a(w) / [8w^8]"
    },
    {
        "video_name": "1BRXCIvt9e0",
        "Q": "At 5:38, why is 4^(-3/2) equal to 1/8?",
        "A": "1) Deal with the negative exponent. This means we need the reciprocal of the original value to get a positive exponent: 4^(-3/2) = 1 / 4^(3/2) 2) Next, let s change to a radical and an exponent. The denominator tells us we need the square root. The numerator is the remaining exponent. 1 / 4^(3/2) = 1 / sqrt(4)^3 3) Simplify the square root: 1 / sqrt(4)^3 = 1 / 2^3 4) Do the exponent: 1 / 2^3 = 1 / (2*2*2) = 1/8 Hope this helps."
    },
    {
        "video_name": "_QKR8I5VE-w",
        "Q": "It's confusing at 0:50 where it says you can read it as third street, second avenue. The coordinate (2, 3) is not the same as (3, 2).",
        "A": "right, so you have to go x then y"
    },
    {
        "video_name": "4WChwIGKdaA",
        "Q": "It doesn't effect the outcome, but around 7:20 when Sal factors out the 1/(1-p) shouldn't it have made the rest [M^(1-p)]-1 ?",
        "A": "At 7:20, he factored 1/(1-p) out from the 1st term only. For the second term, he simplified 1^(1-p) to just 1 in the numerator because one to any power (any p-value) is still just 1. Because the simplified second term 1/(1-p) is just a constant, and will be the same regardless of the value of m, it does not affect whether the limit converges or diverges. So to determine convergence/divergence, he only uses the first term. I hope this helps."
    },
    {
        "video_name": "uzyd_mIJaoc",
        "Q": "At around 2:00 in the video what happened to the 2",
        "A": "The 2 didn t disappear. It s just that when you write y you have to remember that it stands for 1 y. What is 2 dogs minus a dog? Just a dog. So 2y minus 1 y is just a y. That is, 2y - y = y"
    },
    {
        "video_name": "uzyd_mIJaoc",
        "Q": "At 2:01, how is \"2y minus y is just y\"? Why \"y\"?",
        "A": "Any time when you see a variable by itself (just x, just a, just z, etc.) you can think of it as 1x, or 1a, or 1z. When multiplying, you are essentially just adding. In this case, 2y - y would be 1y + 1y - 1y, which equals 1y, or just y ."
    },
    {
        "video_name": "uzyd_mIJaoc",
        "Q": "at 1:57 of the video how did you get the number 3",
        "A": "Sal subtracts y from both sides of the equation. This results in y = 3"
    },
    {
        "video_name": "uzyd_mIJaoc",
        "Q": "At 2:50 how does he get x =3-4?",
        "A": "He substituted 3 for y into the equation x=y-4. I hope this clarifies that part of the video!"
    },
    {
        "video_name": "P6_sK8hRWCA",
        "Q": "at:8:25 how ho you combine like a term",
        "A": "When combining like terms, you find two items that have the same variable and you essentially add or subtract them together. So in the phrase 2x+2x+2x+3=15, you would add all of the 2x s together and get 6x+3=15."
    },
    {
        "video_name": "P6_sK8hRWCA",
        "Q": "i don't under stand at 1:20 Please help :(",
        "A": "Are you asking about how Sal said that the variable a can represent anything? What Sal meant was that if their is a question like 7b + 4b the variable b would not have to represent a specific object that begins with b. A certain amount of boxes could be the variable n. Also in a problem (lets say 6c + 2c) if the same variable is used that means its the same object. 6c & 2c both represent the same object. Hope this helped!"
    },
    {
        "video_name": "ljROszF_rqs",
        "Q": "why can't she be in the hole?(3:22)",
        "A": "In this case He is saying that she won t go into debt, the graph goes into the negative Y quardinate but for this example we are just not looking at those values."
    },
    {
        "video_name": "adPgapI-h3g",
        "Q": "Couldn't Sal have said a5 instead of 5a at 2:40 just like he did with -ax instead of -xa?",
        "A": "You are correct, but the more popular way of saying terms is saying the coefficient before the variable(s)."
    },
    {
        "video_name": "adPgapI-h3g",
        "Q": "At 4:06 Sal multiplies everything by -1. You can do this? I mean, it doesn't affect the result? I made the second equation on my own and I ended up with all negatives.",
        "A": "One of the properties of equality lets you multiply the entire equation by the same value. You create a new equivalent equation. For example: - 8 - 2 = -4 - 6 The 2 sides are currently equal to each other. Multiply both sides by -1 and you get: 8 + 2 = 4 + 6 The 2 sides are still equal to each other. Hope this helps."
    },
    {
        "video_name": "adPgapI-h3g",
        "Q": "at 1:34 couldn't he have just subtracted the three from both sides to make answer\nx= 2 over a-b",
        "A": "No... it doesn t work. The term is +3x. You are treating it as just +3 3x - 3 does not = 0 They are unlike terms, you can t do the subtraction. Hope this helps."
    },
    {
        "video_name": "adPgapI-h3g",
        "Q": "At 4:09 would you have to multiply the equation by -1 or does it not matter? If you didn't would it affect the answer and give you an incorrect one?",
        "A": "Both are the same. After *(-1) you get a simpler, and simpler looking (and much easier to use), result."
    },
    {
        "video_name": "adPgapI-h3g",
        "Q": "What lets you just multiply both sides by a -1? He does it at 4:27.",
        "A": "The properties of equality let us multiply the equation by any value as long as we multiply the entire equation by the same value. This creates an equivalent equation to the original. This is why Sal can multiply the equation by -1."
    },
    {
        "video_name": "adPgapI-h3g",
        "Q": "at 4:17 he multiplies the entire equation by -1 is that necessary to get the right answer?",
        "A": "Sal did not have to do it at this point, but it was a good opportunity to do at this time. If he put it off until later, he would have factored out an x to get x(-a-b) = -8 - 5a. then dividing would give (- 8 - 5a)/(-a - b), but then you should not leave it like this will all negatives, so you would have to factor out -1, so -1(8+5a)/(-1(a+b) and then they would cancel to get same answer. To resolve it earlier, he could have added ax and 8 near the beginning to get the x values positive."
    },
    {
        "video_name": "adPgapI-h3g",
        "Q": "At 2:49, he puts the x's on the left-hand side. I put them on the right-hand side. I got the same answer as he did. So am I right as well?",
        "A": "Yes, you are right as well."
    },
    {
        "video_name": "adPgapI-h3g",
        "Q": "in 5:09 why do we have to take out the X and put it in front of the equal sign instead of including it?",
        "A": "We factor out the X or un-distribute it to get (a+b) so we can simplify it with the (a+b) in the denominator to 1. Which leaves us X*1 or X."
    },
    {
        "video_name": "0eDwckZOffc",
        "Q": "5:10 So what's the definition of parallel lines then? Two lines are parallel if the corresponding angles are equal? That's it?...",
        "A": "It has to do with a transversal postulate or something like that. But the definition of parallel lines is 2 lines that never intersect. I guess that the statement Two lines are parallel if the corresponding angles are equal, is another thing that you can use to find parallel angles."
    },
    {
        "video_name": "0eDwckZOffc",
        "Q": "AT 5:15 on the transversal does the whole thing have to add up to 180 degrees?",
        "A": "What do you mean by the whole thing? Like, all of the angles in the transversal?"
    },
    {
        "video_name": "0eDwckZOffc",
        "Q": "At 0:33, isn't an obtuse angle w/ a measure of 60 degrees an impossibility?",
        "A": "yes Obtuse angles is a angle over 90 degrees"
    },
    {
        "video_name": "0eDwckZOffc",
        "Q": "Isnt the question at 3:40 worng because the 50 degree angle and the 120 degree angle dont add up to 180 degrees, and are these alternate exterior angles? Please help!!!!!!!!!!!!!!!!!",
        "A": "He wrote 50 and 120 to illustrate that you can prove that the first two lines he drew for that example are not parallel. Opposite exterior angles are equal for parallel lines, but when he determined the angle supplementary to 120 was 60 degrees he also proved that the first two lines must not be parallel because those opposite exterior angles were not equal."
    },
    {
        "video_name": "0eDwckZOffc",
        "Q": "what is an adjacent angle at 2:33",
        "A": "Adjacent angles are angles that are like neighbors, they share a side, or ray."
    },
    {
        "video_name": "0eDwckZOffc",
        "Q": "At about 4:24, he says that ALL corresponding lines have to have supplementry angles, but I dont understand as some angles seem not to?",
        "A": "In the video, the angle that is marked with an asterisk as *50 is not actually 50 degrees! It is actually 60, because the adjacent supplementary angle is 120."
    },
    {
        "video_name": "0eDwckZOffc",
        "Q": "At 0:20 why does he call the lines parallel, I'm confused. Is there a lesson on parellel lines?",
        "A": "Parallel lines are lines that never touch. Ever. everr"
    },
    {
        "video_name": "vN0aL-_vIKM",
        "Q": "At 3:30, to simplify the equation even more, can't you divide the entire equation by 2 to get the end result of x^3+x^2+2?",
        "A": "If this were an equation, then yes, you could divide both sides by 2. As an expression, the best we can do with what you noticed is to factor out the 2, and write it as 2(x^3 + x^2 + 2), because we want the expression we get at the end to be exactly equal to the original expression"
    },
    {
        "video_name": "vN0aL-_vIKM",
        "Q": "At 1:24, he puts 2x^3 at the beginning. Do we always put an exponent without a match there? Is it some kind of order they need to be put in?",
        "A": "The order doesn t matter but generally, you put the numbers with the highest exponent at the beginning and then the second highest etc. Hope this helps"
    },
    {
        "video_name": "vN0aL-_vIKM",
        "Q": "So you can add a negative x and a positive x as long as they have the same exponents, like at 1:35?",
        "A": "exactly. ax^2 + bx^2 = (a+b)x^2 Lets say a=2 and b=3: 2x^2 + 3x^2 = (2+3)x^2 = 5x^2 We cannot do this trick when the exponents differ."
    },
    {
        "video_name": "vN0aL-_vIKM",
        "Q": "at 2:28 you mention the commutative proper of addition AND subtraction without mentioning that there is no commutative property of subtraction unless you consider all subtraction as addition of negative numbers. So, for viewers, you can commute a -x as long as you keep the (-) associated with it's term and coefficient.",
        "A": "*Nods."
    },
    {
        "video_name": "vN0aL-_vIKM",
        "Q": "At minute 3:30 when you see the 8x cancel out the -8x why isn't x still there ? is the x a \"Negative ?\"",
        "A": "Start from 8x - 8x, factor out an 8(x-x). It does not matter what x is, if you subtract any number (or variable) minus itself, it is 0, so 8(0) = 0. The point of doing opposites is to eliminate things either by adding to be zero, dividing to be one, or some other form of elimination."
    },
    {
        "video_name": "vN0aL-_vIKM",
        "Q": "After you've simplified and added everything together, is it necessary to rearrange the terms in a certain way?\nExample: @ 3:30, Sy's final answer is 2x(to the 3rd power)+ 2x(to the 2nd power)+ 4\nSo is it necessary to switch the term around one last time, so it becomes, lets just say, 4+ 2x(to the 2nd) + 2x(to the 3rd)?",
        "A": "The answers would be the same so from a correct answer perspective I don t think it d be wrong. However, I think it s a good idea to order consecutively for other people looking at the information. For example, imagine how chaotic the equation could be if there were 50 exponents and they were in a random order."
    },
    {
        "video_name": "vN0aL-_vIKM",
        "Q": "why at 0:45 seconds cant you add X(2)+X(3)\n\n( ) means exponents",
        "A": "Because they are not like terms. If x was 2, then x^2 + x^3 would be 12. But if x was 1, then x^2 + x^3 would be 2. So because you don t know what x is, you can t add unlike terms."
    },
    {
        "video_name": "vN0aL-_vIKM",
        "Q": "at 3:19 when sal talks about being redundant what exactly does he mean?",
        "A": "Redundant means repetitive (If you say the same thing over and over, you re being redundant.) Hope that helps."
    },
    {
        "video_name": "EQoNfxToez0",
        "Q": "3:15 it talks about a negative slope. I'm wondering is -1/1 = -1 the same as 1/-1 = -1 ?",
        "A": "Yes it is. Becuase it doesn t change."
    },
    {
        "video_name": "EQoNfxToez0",
        "Q": "In 4:25 he says that the slope is negative,that's right because he does it on the opposite side(right side)...what if he does it on the side he did the other 2 slopes(left side)?Wouldn't it be positive then??Please help!I really didn't understand so confused right now.",
        "A": "It depends, left negative right positive and up positive and down negative. The both have to be opposite too."
    },
    {
        "video_name": "EQoNfxToez0",
        "Q": "0:55, it says if you increase x by 1 then you increase y by one right?",
        "A": "yes a slope of 1 is the same as 1/1 which says a change of y (1) divided by a change in x (1)"
    },
    {
        "video_name": "xPUm5SUVzTE",
        "Q": "At 0:39 Sal says 'Decision Tree', why? Shouldn't it be a Tree Diagram??",
        "A": "Sal is specifically talking about a probability (sometimes called decision) tree diagram, which is used to calculate dependent probabilities. There are other types of tree diagrams, such as the tree diagram used to map out all of the possible combinations in a sample space."
    },
    {
        "video_name": "xPUm5SUVzTE",
        "Q": "At 5:30, how do you multiply 0.60x0.60?",
        "A": "0.6x0.6, lets turn it into 6x6 = 36. Now how many 0 s do we need in front of the 36? Well, since theres two digits behind the . in the product 0.6x0.6, we say that we need two digits behind the . in our answer: 0.36. So, just remember to count the digits behind the . and use that many digits behind the . in your answer as well. This only works with multiplication."
    },
    {
        "video_name": "xPUm5SUVzTE",
        "Q": "At 3:30, Sal says that P(HH/Fair)=0.50*0.50...I don't understand it, because in the last video he said that the probability of an intersection (Fair+HH, for example) is equal to the product of the probability (Fair) and the dependent probability (HH/Fair). Why is P(HH/Fair)=0.50*0.50? Can someone help me?",
        "A": "because if the coin is fair ,the head probability is 50% which is 0.5 .they wanna u to find two heads so it might be 0.5*0.5 but there r 3 of the coins in unfair ,they already told u the probability which is 0.6 ,i don not have enough time to tell u ,my class gonna over if u have some question i am willing to tell u"
    },
    {
        "video_name": "xPUm5SUVzTE",
        "Q": "at 7:51 why does he add not multiply",
        "A": "it is a or question. If it was an and question you multiply"
    },
    {
        "video_name": "ALhv3Rlydig",
        "Q": "at 1:35 , why is it that right angles have a square indicator and acute and obtuse have a curved indicator.",
        "A": "The reason I think a right angle has a sqaure is to show that, that type of angle is actually a right angle."
    },
    {
        "video_name": "ALhv3Rlydig",
        "Q": "at 3:39 what is the rest of the angle",
        "A": "At 3:39 he finishes talking about obtuse angles."
    },
    {
        "video_name": "ALhv3Rlydig",
        "Q": "Why is Sal so hesitant to use the word 'perpendicular' when describing a right angle (for example, around 2:10 in this video, but also frequently in other videos)? It seems like he stumbles over his words when trying to describe the angle, when he could easily say, \"the angle is formed by 2 lines/line segments/rays which are perpendicular.\" Does 'perpendicular' only apply to lines? Is it geometrically incorrect to use that word to describe 2 line segments/rays?",
        "A": "He is trying to describe what it means to be a right angle. If you don t know what a right angle is, you are not going to know what perpendicular is, so he had to find another way to describe it."
    },
    {
        "video_name": "ALhv3Rlydig",
        "Q": "At 4:53,is there any thing similar with obtuse,acute and right angles?\nIt doesn't seem like it.",
        "A": "They re all angles. They re all less than 180 degrees."
    },
    {
        "video_name": "ALhv3Rlydig",
        "Q": "at 4:32 is an obtuse angle always have to be bigger than 90 degrees",
        "A": "Yes, an obtuse angle is always greater than 90 degrees."
    },
    {
        "video_name": "ALhv3Rlydig",
        "Q": "AT 0:32 can you please specify what a right angle is? It is kind of hard for me to move on when there is something I haven't learned yet. It confuses me.",
        "A": "A right angle is one that looks like a half open book, and like a L. Imagine that every time you are struggling. Hope that helps!"
    },
    {
        "video_name": "ALhv3Rlydig",
        "Q": "At 3:46, he only says that obtuse angles is larger than right angles, than isn't straight angles also obtuse angles?",
        "A": "No straight angles are larger than obtuse angles and thus are also larger than right angles."
    },
    {
        "video_name": "ALhv3Rlydig",
        "Q": "When he says \"all of these would have to be right angles\" at 2:11, that's because 90 (or right) is the only angle that divides 360\u00c2\u00b0 by 4, is that correct?",
        "A": "Yes, you are correct."
    },
    {
        "video_name": "ALhv3Rlydig",
        "Q": "At 1:16, the word should have been perpindicular",
        "A": "What? Perpendicular is the right spelling... let me Google it... Yes, perpendicular is right, not perpindicular. If you were referring to Sal s lines, a horizontal line is one that goes sideways like -----> and a vertical line is one that goes up/down like | | v so he is right in saying the first one is horizontal, the second is vertical and that they are at a right angle. They are perpendicular, but perhaps it is easier to understand at a right angle than perpendicular for younger people."
    },
    {
        "video_name": "ALhv3Rlydig",
        "Q": "At 1:37 Salman says half of a box but isn't it a quarter?",
        "A": "I think he was referring to the notation for the right angle which looks like an inverted L which is essentially half a box as it will become a complete box only when the L and inverted L are put together."
    },
    {
        "video_name": "UdHgWK13x5I",
        "Q": "At 7:37, Sal concludes that the transpose of the product of two matrices is simply the matrices themselves transposed and then multiplied in reverse order. Would this technique work for products of more than two matrices? For example, if we have (ABC)^T, would that equal (C^T) (B^T) (A^T)?",
        "A": "Great question, Let us use the fact that matrix multiplication is associative, that is (AB)C=A(BC). Then we can write (ABC)^T=((AB)C)^T. AB is just a matrix so we can use the rule we developed for the transpose of the product to two matrices to get ((AB)C)^T=(C^T)(AB)^T=(C^T)(B^T)(A^T). That is the beauty of having properties like associative. It might be hard to believe at times but math really does try to make things easy when it can."
    },
    {
        "video_name": "hrIPO8mQqtw",
        "Q": "Hello.\nI got really confused at 7:48. When Khan is integrating the lower bound of in the integral in the x direction (dx) he gets x = sqrt(y), makes sense. But my question is how did he get the function y = x^2 (at 2:13) from z = xy^2 in the first place, or is that just arbitrarily given?",
        "A": "He didn t derive y=x^2 from z=xy^2; they are not dependent on each other. For this video, he changed y from a constant (straight line) to simple function whose shape is familiar. He could have chosen any function as the y boundary."
    },
    {
        "video_name": "hrIPO8mQqtw",
        "Q": "At 5:50, Sal says after teachers set up integral then it's easy.I never thought about it also I have only a taste of high school mathematics I know what a double integral, but what is a single integral for?",
        "A": "To find the area under a function."
    },
    {
        "video_name": "O8vB1eInP_8",
        "Q": "Hello,\nat 2:14 why do you choose to assign U as 0 and F as 1. Why did you not assign U as 1 and F as 0?",
        "A": "i would say that it is completely arbitrary and does not really change much. It is just typical to assign 1 to success and 0 to failure. So it is really up to you what you consider a success :-)"
    },
    {
        "video_name": "O8vB1eInP_8",
        "Q": "(2:15 - 3:50) So if you assigned the unfavorable as 1 and favorable as 0, you'd end up with a different mean...? How do you know what number to assign to each variable?",
        "A": "In fact, you could choose -1 for unfavorable and +1 for favorable. That way, a 0 mean would represent a neutral overall favorability rating, a negative number would yield a negative mean sentiment and a positive number would yield a positive mean sentiment. However, choosing a 0 to represent one of the values simplifies the math."
    },
    {
        "video_name": "LoKEPEPaNm4",
        "Q": "at 0:42, shouldn't it be for every ONE y, x increases by 2.5?",
        "A": "No it doesn t since x is constant and y depends on x."
    },
    {
        "video_name": "ceDV0QBpcMA",
        "Q": "Please correct me if I'm wrong:\nError at 1:51, I see another possibility:\n\nhaving x + 16 on the bottom right hand corner, and having 3x + 5 on the bottom left hand corner.",
        "A": "Base angles of an isosceles triangle are equal, which gives the possibility that x+16 equals 3x + 5"
    },
    {
        "video_name": "0lSTXtwPuOU",
        "Q": "2:02 The second option is divided into 3 equal sections in terms of area. Isn't it?",
        "A": "yes. lets said that the width is 9 and the length is 15. the rectangle on the top is `5 by 9. the triangles are 10 by 9. the rectangle is 45 and the triangles are 45 too because you would have to divide them both by 2. HOPES THIS HELPS if not comet please!"
    },
    {
        "video_name": "0lSTXtwPuOU",
        "Q": "My teacher told me it is impossible to divide a rectangle into halves using the diagonal fold because she even showed us proof of a real rectangular paper and we all tried to fold it in half along the diagonal edges and at 2:04 Sal clicks on the one divided by the diagonal in half. Somebody please explain this to me! Thanks!",
        "A": "The reason for this Sean, is that folding a piece of paper into half is another way of finding a line of symmetry. The diagonal line that Sal chooses is not a line of symmetry, it is a line that divides the area of the rectangle into two."
    },
    {
        "video_name": "9JX2s90_RNQ",
        "Q": "At 03:57: Why is dx equal to 5 sec^2*delta*d*delta?",
        "A": "Since x=5tan(x), its derivative - which is dx - will be 5sec^2(x). The derivative of tan(x)=sec^2(x)."
    },
    {
        "video_name": "9JX2s90_RNQ",
        "Q": "At 10:00 why is this -pi/2, but not 3pi/2?",
        "A": "L = lim 50\u00e2\u0080\u00a2atan(n/5) n\u00e2\u0086\u0092-\u00e2\u0088\u009e Observe that you evaluating n\u00e2\u0086\u0092-\u00e2\u0088\u009e, this implies that your angle is moving in a negative direction, converging on -pi/2. L = lim 50\u00e2\u0080\u00a2atan(n/5) n\u00e2\u0086\u0092+\u00e2\u0088\u009e Now in this case our angle is now moving in a positive direct, which would now converge to 3\u00e2\u0080\u00a2pi/2."
    },
    {
        "video_name": "9JX2s90_RNQ",
        "Q": "at 9:51, how do you conclude that arctan of negative infinity is -pi/2?",
        "A": "Remember that the tan or arctan of an angle on the unit circle represents its slope. A slope of negative infinity is a vertical line drawn downwards from the origin ( if using the center of the unit circle as the reference point). This angle is 270 degrees which is expressed as 3pi/2 or -pi/2 depending on which way you approach it. In this case, we approached it from the negative direction."
    },
    {
        "video_name": "9JX2s90_RNQ",
        "Q": "(time = 0:34) When Sal said \"indefinite integral\" and the error fixing thing showed up saying \"but meant 'definite integral,' \" why would it be definite integral? Shouldn't it be improper integral, since it includes infinity as a bound? Or does definite integral include all bounded integrals, even ones with infinity?",
        "A": "Your last point is correct. By definition, an improper integral is a definite integral, even if it doesn t converge."
    },
    {
        "video_name": "EtefJ85R1OQ",
        "Q": "At 2:50min, Sal concludes that (a*r^(t+1)) / (a*r^t) equals r^(t+1-t). How did he do that ? How did he get rid of the denominator a*r^t ? Thanks",
        "A": "What s (x^7)/(x^6)? What s (r^(t+1))/(r^t)? What s a/a? What s (a*(r^(t+1)))/(a*(r^t))?"
    },
    {
        "video_name": "XPf8LMu7QSw",
        "Q": "At roughly 7:52, I am trying to follow along, and when he counts down to 7.125, it seems like he actually ends up 6 and a bit, is this an error in the video, or am I missing something?",
        "A": "It s an error, there s a notification now that tells you it is."
    },
    {
        "video_name": "XPf8LMu7QSw",
        "Q": "At 1:30 How did Sal factor out so fast? Can somebody tell me how do I do that? It's the fastest way so I wanna know :)",
        "A": "If you want to factor a quadratic in the form: ax\u00c2\u00b2 + bx + c where a = 1 The factorization will always be: (x + m)(x + n) Where (m, n) satisfies the conditions: mn = c m + n = b"
    },
    {
        "video_name": "XPf8LMu7QSw",
        "Q": "Why did he add x^2 to both side instead of 2x^2 at 0:58",
        "A": "Sal wanted to get rid of the -x^2 on the left, and put it on the right. To get rid of a -x^2 , you have to add x^2 to both sides. If he had added 2x^2 to both sides, then he would have been left with an x^2 on the left, which he didn t want."
    },
    {
        "video_name": "V8C15AfL6Gk",
        "Q": "At 0:33 did Sal say that 3/2 is the same as three 1/2's?",
        "A": "No. he said 3 over 2, is the same thing as 3 divided by 2"
    },
    {
        "video_name": "V8C15AfL6Gk",
        "Q": "At 1:49 how is 3/2 the exact same as 1 and 1/2. I think he should explain why he gets the answer more specifically, or its probably im not understanding because im still in 4th grade.",
        "A": "3/2 is 1 1/2 because 1 1/2 is basically 1+1/2. 1 is also 2/2, and 2/2+1/2=3/2. Do you see what I mean? I hope I helped you!"
    },
    {
        "video_name": "V8C15AfL6Gk",
        "Q": "at 0:43 how do num lines come to play",
        "A": "It s to help you visualize 3/2 better, so it s more than just a fraction; it s something you can imagine/see in you head."
    },
    {
        "video_name": "CBlSsy1eslw",
        "Q": "at 1:50, sal cancels the x+1 on the right hand side, why do we still multiply the -1 by x+1?\nis it because it is a different term?",
        "A": "Yes. We are multiplying -1 by (x-1) and (x+1) because they are different terms couldn t be combined from the start. If they could be though, then we wouldn t have to do FOIL and multiply twice because then it would be one term from being combined. You can tell that they are different terms because they are separated by a subtraction sign."
    },
    {
        "video_name": "_E9fG8BYcBo",
        "Q": "At 4:44 could a fraction really have a numerator of 0?",
        "A": "Sure thing! 0 is the same as 0/2, 0/17, or even 0/100000"
    },
    {
        "video_name": "_E9fG8BYcBo",
        "Q": "At 4:47 why do they put in O/9. is that even a fraction!",
        "A": "it is it just equals 0 because 9 divided by zero is zero"
    },
    {
        "video_name": "O1-9f1g0OsI",
        "Q": "At 3:13, Sal says that AE isn't equal to EA. How is this possible? With the Commutative Property of Multiplication, shouldn't they be the same?",
        "A": "The commutative property applies to the multiplication of scalars, or regular numbers. Matrix multiplication, however, is not commutative. You can try it yourself by finding AE and EA, and seeing that they re not the same thing. Sometimes they ll give the same result even if switched, but there s no guarantee it ll be right for any 2 matrices."
    },
    {
        "video_name": "iFpyKTuC7jY",
        "Q": "at 1:18 those arrows should be pointing inwards correct? That would denote a negative divergence as particles are coming into the surface.",
        "A": "He was just drawing the direction that mass/stuff has to travel in to be considered outward flux (basically the normal vector, he drew the normal vector direction when z=0). If you wanted to see the direction that mass traveled in when it crossed the surface you would need to draw the vector field F, which would be difficult to do by hand in this example. Also remember divergence is a scalar field, not a vector field (it has no direction)."
    },
    {
        "video_name": "iFpyKTuC7jY",
        "Q": "At 1:18 why didn't he draw the arrows the other way?. In that case the flux (out) would be equal to the flux (in) and there would be 0 flux, right?",
        "A": "He was just drawing the direction that mass/stuff has to travel in to be considered outward flux (he basically drew examples of the normal vector). It he were to draw the actual direction that mass travels in when it passes through the surface, he would need to draw the vector field F, which would be difficult to do by hand in this example. Also remember divergence is a scalar field, not a vector field (it has no direction)."
    },
    {
        "video_name": "cYw4MFWsB6c",
        "Q": "At 1:22 how is 2+(n-1)=(n+1)? That doesn't make any sense to me.",
        "A": "2 + \u00f0\u009d\u0091\u009b \u00e2\u0080\u0093 1 We can combine the 2 and the -1 to get 2 \u00e2\u0080\u0093 1 = 1. Thus: 2 + \u00f0\u009d\u0091\u009b \u00e2\u0080\u0093 1 = \u00f0\u009d\u0091\u009b + 1 Proficiency in this type of algebra is expected for studying arithmetic series. I would recommend looking at the solving one step equations playlist and the combining like terms video."
    },
    {
        "video_name": "r2PogGDl8_U",
        "Q": "5:48\nWhy is ty\u00e2\u0080\u00a2ty rewritten as (y\u00e2\u0080\u00a2y)t^2? Why is the t squared?",
        "A": "Because t is a scalar"
    },
    {
        "video_name": "r2PogGDl8_U",
        "Q": "Why is there 2(x*y)t at 6:05 instead of 2(x*ty) or 2(x*t*y) or 2t(x*y) ? It's bugging me for some time now and I can't figure out if t is meant to be multiplied by 2 or not.",
        "A": "Keeping in mind that the 2 and the t are scalar and the x and y are vectors and the symbol * is the dot product, the four terms you describe are all exactly equivalent. The entire term (x*y)t (which is equivalent to t(x*y) and (tx*y) and (x*ty)) is multiplied by 2 because there were 2 equivalent terms in the previous step, and in this step those two terms were added together."
    },
    {
        "video_name": "ssz6TElXEOM",
        "Q": "i get confused between product and removing something outside and the differentiating.for example at 2:09 why is 1/ln b got outside instead of applying product rrule",
        "A": "The key thing to know is that in this problem b is a constant and so you wouldn t need to use the product rule. So you can think of 1 / ln b just as you would any number. It s important to notice that you are differentiating with respect to x and so any other variables you see are probably constant values (there are exceptions which we aren t dealing with yet in this section)."
    },
    {
        "video_name": "ssz6TElXEOM",
        "Q": "At 1:20, Sal defined log of x with the base b as the lnx/lnb, but in school, they taught us that log of x with the base b is log x with base 10/log b with base 10. I know that e!=10, so can anybody help me?",
        "A": "The change of base formula you learned works for any base; you learned it for log base ten, but the same formula will work for any other base including e which is used for the natural log. If you change all the tens in what you learned to e then it should match up exactly to what Sal did. change of base formula log_b (x) = log_k (x)/log_k(b) You can change the base to any number you like; you learn log base 10 because most calculators only work for log base 10 and natural logs."
    },
    {
        "video_name": "ssz6TElXEOM",
        "Q": "At 1:17 why is (1/lnb)*lnx = logb(x)? And why can we put 1/lnb outside of the derivative?",
        "A": "Part one of your question is the change of base formula for logarithms. We can put 1/lnb outside the derivative because 1/lnb is a constant, recall d/dx[kx]=kd/dx[x]"
    },
    {
        "video_name": "ssz6TElXEOM",
        "Q": "At 0:25, Sal talks about a 'log with an arbitrary base'. Is that a log that's neither a natural (base-e) log nor a common (base-10) log?",
        "A": "arbitrary, adj. In Mathematics: undetermined; not assigned a specific value It can be e or 10, but also any other positive number."
    },
    {
        "video_name": "-cApVwKR1Ps",
        "Q": "At 2:30 why do we need to change the LHS to ((s^2 +a^2)/s^2)y ?",
        "A": "He factored it so that we only have one y. Makes things easier later."
    },
    {
        "video_name": "-cApVwKR1Ps",
        "Q": "at 3:11 i still don't understand why there is still exist a boundaries from [0,infinity]. Can anyone explain this? thanks",
        "A": "I also have the same question initially. I think it s because when applying int(u v)=uv-int(uv ), u(t) and v(t) should have the boundaries from int(u v), otherwise it does not make sense that the equation generates an expression with t but does not follow the original boundary. The final results should not have t, because t has a boundary of [0, infinity]."
    },
    {
        "video_name": "-cApVwKR1Ps",
        "Q": "Why did he at 3:06 evaluate the right-side of the equation?",
        "A": "It s simplifies the equation to a much more simpler form. Good a time as any, given how hairy the equation was. He didn t have to, really, but it helps everything."
    },
    {
        "video_name": "-cApVwKR1Ps",
        "Q": "pretty sure a small mistake was made at 2:04 when you factored the -e^-st out and re-wrote the terms inside the brackets, you forgot the 'a' before the cos (the term was (a/s^2)e^(-st)cos(at)\n\nif this is not a mistake please explain where that a went\n\n\nedit: it was caught at 7:30",
        "A": "6:50 there you go"
    },
    {
        "video_name": "-cApVwKR1Ps",
        "Q": "I'm not able to get the reason behind why Sal at 3:16 put the boundary condition. Even though the term to be integrated was taken to the LHS while the other din't need to be integrated. I'm a bit confused with that part..",
        "A": "Because all parts need to be evaluated at the boundary condition after the integration. Sal just evaluated those parts which he already integrated to simplify the calculation. The final results would be the same."
    },
    {
        "video_name": "-cApVwKR1Ps",
        "Q": "I don't understand how to get from y+(a^2/s^2)y to (s^2+a^2/s^2)y on the left hand side of the equation. Around time 2:22",
        "A": "Inside the brackets: after =y [ 1 + (a^2/s^2) ] just replace the 1 with s^2/s^2 and you can see you now have a common denominator = y [ (s^2/s^2) + (a^2/s^2) ]. Now you can add the numerators and get =y [ (s^2 + a^2) / s^2 ]."
    },
    {
        "video_name": "TvSKeTFsaj4",
        "Q": "At 1:51 when he says to \"Solve for x\" why can't you just divide 150 by 0.25 to get x? That's much simpler.",
        "A": "He can and he does that, see 2:30!"
    },
    {
        "video_name": "TvSKeTFsaj4",
        "Q": "At 1:31 how did you make 25% 0.25?",
        "A": "percent means per 100. So, 25% = 25/100 which i in decimals 0.25."
    },
    {
        "video_name": "TvSKeTFsaj4",
        "Q": "I dont understand how you moved the decimal point for problem #2? time:3:45",
        "A": "Well it is actually like this. since you are doing 150/.25 the .25 will be on the outside and the 150 on the inside. Since you cant have a decimal on the outside of the division problem then you try to make it a whole number by moving the decimal place two times to make it 25 but if you do one thing to one side you have to make it equal by doing that same thing on the other side so you move the decimal place of 150 so the problem is now 25/15000. So it is equaled out then you just divide like normal!!!!:)"
    },
    {
        "video_name": "TvSKeTFsaj4",
        "Q": "6:00 what does he mean by the base",
        "A": "In percent problems, the base is the original value or the number that you are trying to apply the percent to. For example: Find 40% of 120 The base is the 120. Hope this helps."
    },
    {
        "video_name": "TvSKeTFsaj4",
        "Q": "it would be 600 right? at 0:10, it says 25%. that means 1/4 of the whole number.",
        "A": "Yes, that s right! 150 is 25% or 1/4 of 600."
    },
    {
        "video_name": "TvSKeTFsaj4",
        "Q": "At 3:00 could we shift it farther than 2 places in the decimal? answer ASAP",
        "A": "You could, but then you just get bigger numbers to use in the division, making the division harder to do. Just convert 0.25 into a whole number. You just needs to move the decimal point 2 places. You might want to review decimal division."
    },
    {
        "video_name": "TvSKeTFsaj4",
        "Q": "At 3:42, Sal divides 150 by 0.25, and gets 600. Why? Why would he not get a quarter of 150 instead of what it is a quarter of? Please help me, because I am really confused. Thanks!",
        "A": "He is finding out what number 150 is 25% of, not what 25% of 150 is. and he is also dividing 15000 by 25 instead of 150 by 0.25 because he shifted the decimal two digits to the right to make the problem simpler. Hope this helps!"
    },
    {
        "video_name": "TvSKeTFsaj4",
        "Q": "If you look at 3:45 and pause it you see that the lime green color makes a face!XD",
        "A": "lol yas it looks like a pup"
    },
    {
        "video_name": "TvSKeTFsaj4",
        "Q": "At 3:06 why did he multiply it by 100?",
        "A": "In order to divide it he needed to move to decimal place so that the divisor (the .25 on the left) became a whole number. So, since you have to be fair in math, he did the same to 150."
    },
    {
        "video_name": "thDrJvWNI8M",
        "Q": "Sal 0:44 : \"subtracting a negative five\" no its adding a negative five",
        "A": "No Sal is right, inside the parentheses is x + 5 which is the same thing as x - (-5) which is why he moved 5 units to the left."
    },
    {
        "video_name": "k_wJsio68D4",
        "Q": "2:20\nHow can something plus something equal the original something? pi /= 0. 0 should be the only exception to this rule. Can someone explain?",
        "A": "He wrote that in a somewhat confusing way, here is what it means: tan (\u00ce\u00b8) = tan (\u00ce\u00b8+\u00cf\u0080) \u00e2\u0086\u0090 This is correct It is not: tan (\u00ce\u00b8) = tan(\u00ce\u00b8) + \u00cf\u0080 \u00e2\u0086\u0090 This is wrong"
    },
    {
        "video_name": "k_wJsio68D4",
        "Q": "at 5:18\nhow is tan(pi-theta) = sin(theta) / -cos(theta) if cos(pi-theta) = cos(pi+theta) NOT -cos(theta)?",
        "A": "To see why cos(\u00cf\u0080-\u00ce\u00b8)= -cos(\u00ce\u00b8), draw the angles \u00cf\u0080-\u00ce\u00b8 and \u00ce\u00b8 on the unit circle. As it happens, it is also true that cos(\u00cf\u0080-\u00ce\u00b8)=cos(\u00cf\u0080+\u00ce\u00b8). There are multiple related identities here, and the existence of one identity does not preclude the existence of others."
    },
    {
        "video_name": "k_wJsio68D4",
        "Q": "what if pi was subtracted from theta. would the tangent still not change, because the slope doesnt change ? (at 6:20)",
        "A": "yes, tan(theta)=tan(theta-pi) is a true statement."
    },
    {
        "video_name": "0zExhHh7_Ic",
        "Q": "At 3:30 Sal finds the slope with the assumption (x, lnx) is further from the origin than P. Why can he do that if he doesn't know for sure?",
        "A": "The formula works if x is lower or greater than 1, remember that the formula for the slope is m = \u00e2\u0088\u0086y/\u00e2\u0088\u0086x, so if you change the order of the points, all that will happen is that both the numerator and the denominator will switch signs, and since both will switch signs, the division will remain the same sign."
    },
    {
        "video_name": "0zExhHh7_Ic",
        "Q": "At 2:22, how did you decide that point Q would be there? Could it be anywhere because all that was given is (x, lnx), therefore, anywhere on the graph?",
        "A": "Point Q is chosen arbitrarily. In other words, point q is chosen anywhere on the graph y=ln(x)"
    },
    {
        "video_name": "0zExhHh7_Ic",
        "Q": "At about 1:25 how do you know the line runs to negative infinity without crossing the y axis. You seem to make this assumption, but I am unclear as to why this just seems to be a given. Could someone explain why this is just assumed automatically when the rough line is drawn? I am assuming it has to do with the log, but can someone explain it more succinctly?",
        "A": "You don t need the graph to know whether the line passes through the y-axis. Just set x=0 and see if there is a real solution."
    },
    {
        "video_name": "0zExhHh7_Ic",
        "Q": "0:55 I don't get it, what am I missing. How is e^1 = 0 :S?\n\nln(0) is undefined, it is impossible?",
        "A": "Yes, log (0) is undefined for all bases. In the video Sal misspoke. He meant to say that e\u00e2\u0081\u00b0 = 1"
    },
    {
        "video_name": "W3flX500w5g",
        "Q": "At 4:27 he wrote hours/page which should be page/hour cause the first way its 55 hours/page it should be 55 pages/hour XD",
        "A": "No, Sal is correct. He flipped the fraction to change division to multiplication He started with dividing by: 55 pages / 1 hour To multiply it becomes: 1 hour / 55 pages"
    },
    {
        "video_name": "QVxqgxVtKbs",
        "Q": "At 2:47 Sal starts talking about the square root of 3 over 2. Where did that come from? Why?",
        "A": "It comes from the basic properties of 30\u00c2\u00b0-60\u00c2\u00b0-90\u00c2\u00b0 triangles. The ratios of their lengths are: 1 : \u00e2\u0088\u009a3 : 2, Since (in that triangle), the hypotenuse had a length of s, then the side opposite the 30\u00c2\u00b0 angle must have a length of \u00c2\u00bd s. The side opposite the 60\u00c2\u00b0 angle must have a length of (\u00e2\u0088\u009a3)(\u00c2\u00bd s) = (s\u00e2\u0088\u009a3)/2"
    },
    {
        "video_name": "znT4W6n3WkI",
        "Q": "At 5:37, why does Sal take the antiderivative of both sides? Is the equation still equal?",
        "A": "Yeah, it s equal. Your trig equation and relativity equation in your bio are right too. And I ve always believed .9999... is equal to 1 too, I sometimes discuss it with my fellow calculus students! And also that the limit of x/0 as x approaches infinity is infinity, not undefined - so that would mean you can divide by 0 :)"
    },
    {
        "video_name": "b2C9I8HuCe4",
        "Q": "at 2:10 how do you spell first ....?",
        "A": "You mean the word after first ? That word is quartile . You can read the words he is saying by clicking the CC box in the lower right corner of the video."
    },
    {
        "video_name": "b2C9I8HuCe4",
        "Q": "At 2:10, can you just call it the LQ= Lower Quartile and the UQ= Upper Quartile",
        "A": "That s an alternate term, but in my experience the terms First Quartile and Third Quartile are more common."
    },
    {
        "video_name": "b2C9I8HuCe4",
        "Q": "At 1:27, he says that the middle line inside the box is the median of the entire set of data. So I assume that there is no use of the mean or mode in a box plot.",
        "A": "There is always use for the mean and mode on a box plot, just probably not in that question. In other questions you get, you might actually have to find the mean or mode."
    },
    {
        "video_name": "b2C9I8HuCe4",
        "Q": "Starting at 2:51 Sal says, \"So even though you might have trees that are as old as 50, the median of the forest is actually closer to the lower end of our entire... spectrum of all the ages.\" Earlier he said the median was 21 and the range was 42. Since 21 is exactly half of 42, shouldn't the median look more like it's located in the center?\nPlease, someone reply - even if it's only to say that you don't have an answer.\nThanks.\n-Tom",
        "A": "No it shouldn t. The line should be over there because the whole thing goes to 50. It s supposed to look like it it s off because it is. The line represents the median of the trees that are less than 21 i guess. Yes, i m confusing, but hopefully your mind may be able to process this. lol ;)"
    },
    {
        "video_name": "b2C9I8HuCe4",
        "Q": "At 2:06 Sal begins to talk about naming the quartiles (1Q, 2Q, 3Q and 4Q), but in a test I did they ask about the interquartile range. Does anyone know what this is? Thanks",
        "A": "Thanks Ada. Now I understand why I got the question wrong. :)"
    },
    {
        "video_name": "G7QiIkYfeME",
        "Q": "At 2:56, when Sal is writing the second \"and\" should it not be \"Twenty Thousand and five hunderd seven thousandths\"? Would not the second \"and\" be interpeted as a second decimal?",
        "A": "Yes, you re right"
    },
    {
        "video_name": "G7QiIkYfeME",
        "Q": "When Sal says twenty thousand and five-tenths and seven thousandths at 1:17, he said another \"and\". But my teacher said that if you say \"and\" it means that there is a decimal. why did Sal say and twice?",
        "A": "And does not always mean there is a decimal. When you say, buy 10 potatoes and 5 carrots it does not mean a decimal and I know you are taking about math but when Sal said that and he meant like the example.20,000 and 5/10 and 7/1000 means there is 20,000 with a 5/10 with a 7/10000. I know that is a tad bit confusing but just try to think of the definition of and DEF: along or together with hope this helps \u263b/ /\u258c / \\"
    },
    {
        "video_name": "4RBkIJPG6Yo",
        "Q": "At the inception of the video (0:06), why does Grant say partial derivative with respect to x? Shouldn't it instead be with respect to a?",
        "A": "No. Watch the videos on partial derivatives."
    },
    {
        "video_name": "4RBkIJPG6Yo",
        "Q": "6:17, he mentions that other authors divide by the abs. val. of vector v to ensure that scaling vector v doesn't affect the directional derivative. But since we're taking 2 nudges instead of 1 nudge of v, should we divide by 2 to compensate for the \"double\" nudge?",
        "A": "By 2 times the magnitude of vector V i suppose."
    },
    {
        "video_name": "4RBkIJPG6Yo",
        "Q": "At 4:17, in the formal definition of directional derivative, why we do only divide by \"h\" instead of \"h*v\"?",
        "A": "Watch the end of the video; he notes that some authors do in fact divide by h*|v|"
    },
    {
        "video_name": "LhMEqsL_M5o",
        "Q": "Around 2:00 Sal said milli ment 1,000 souldn`t it mean 1,000,000?",
        "A": "Milli means 1 thousandth not 1 thousand."
    },
    {
        "video_name": "LhMEqsL_M5o",
        "Q": "At about 1:45 it said that in the metric system you can use the prefixes such as centi,deci. So is there a such thing as a centiliter?",
        "A": "Yes we can add many prefixes such as milli, centi, deci, deca, hecto and kilo."
    },
    {
        "video_name": "LhMEqsL_M5o",
        "Q": "At 1:40 what does a liter mean?",
        "A": "Ali, A liter is a unit of volume and is used to measure things like soda, water, and juice. A liter is equal to 1000 milliliters, and 1000 liters make a kiloliter. Hope this helps!"
    },
    {
        "video_name": "LhMEqsL_M5o",
        "Q": "At 3:24 Sal states stuff about a kiloliter. Is it like a kilogram?",
        "A": "technically yes."
    },
    {
        "video_name": "1DgDt6gCUKs",
        "Q": "at 0:50 do we have to place the variable terms according to alphabetical order and then place singular numbers? ( 5X,7X,8Z,1)",
        "A": "No, you don t have to put them in alphabetical order, Sal was just grouping (or combining) the like terms or (variables). You group and combine the like terms or variables because You can t for example subtract 3y from 6x because might be two totally different numbers. Hope this helps :)"
    },
    {
        "video_name": "1DgDt6gCUKs",
        "Q": "I was taught that when there is no expression beside 2 numbers (example 3 4) you have to x them together, why did you add at 3:36?",
        "A": "He added them together because there was a plus sign between them. He just didn t write them until then. Look at the expression before it to see where the plus signs came from."
    },
    {
        "video_name": "d-2Lcp0QKfI",
        "Q": "At 3:21 Sal factors out a -3f. 6f^2 divided by -3f is -2f but -3f divided by -3f is not f.",
        "A": "He corrects himself a few seconds later."
    },
    {
        "video_name": "d-2Lcp0QKfI",
        "Q": "@5:16 I ended up with an answer of 2 (-3f-11)(2f-1)...I factored out a 2 from the original equation instead of a -2 as Sal did. Am I wrong?",
        "A": "It is standard to have a positive co-efficient on your variable, so you should factor out a -1 from (-3f-11)"
    },
    {
        "video_name": "d-2Lcp0QKfI",
        "Q": "What happened to the 6f^2 at 3:28?",
        "A": "-12f\u00c2\u00b2 - 38f + 22 = -2\u00e2\u0088\u00996f\u00c2\u00b2 - 2\u00e2\u0088\u009919f - 2(-11) = Sal doesn t show this -2(6f\u00c2\u00b2 + 19f - 11) = -2[6f\u00c2\u00b2 - 3f + 22f - 11] = break up 19f to prepare -2[(6f\u00c2\u00b2 - 3f) + (22f - 11)] = now factor by grouping -2[3f\u00e2\u0088\u00992f + 3f(-1) + 11\u00e2\u0088\u00992f + 11(-1)] = Sal doesn t show this -2[3f(2f - 1) + 11(2f - 1)] = factor out the GCFs -2(3f + 11)(2f - 1)"
    },
    {
        "video_name": "d-2Lcp0QKfI",
        "Q": "3:45 wouldn't it be 2f^2 instead of just 2f? From the very beginning, when he factored out the -12f^2, it became 6f^2 (keeping the exponent) but in 3:45 he drops the exponent..? Somebody explain?",
        "A": "First he factors out just a -2, so -12f^2 becomes 6f^2 (since -2*6f^2 = -12f^2). Then he factors out a 3f, so 6f^2 becomes 2f (since 3f*2f = 6f^2)."
    },
    {
        "video_name": "uCBm8iDyg1s",
        "Q": "Sal at 2:25 in the video you say \"lets see if 4 goes\", and after you did the maths it did go but you could have tried 5.64\nx 5= 325",
        "A": "At 5:08, the final answer is 151 but if you add 1 in front of 5.64, you would have gotten 15.64 which is incorrect."
    },
    {
        "video_name": "uCBm8iDyg1s",
        "Q": "If you have 10 cats and 15 dogs would you do this 10:15=1:5 because I put 10:15 in my phone and I gave my 1:5",
        "A": "Because you are trying to find the unit rate so for example 3 apples and 27 grapes What is the unit rate? You would find the greatest common factor then divide so you answer would be 1:9"
    },
    {
        "video_name": "uCBm8iDyg1s",
        "Q": "AT 3:00 he said that 2-2= 1 but that is not true right ?",
        "A": "He put in a note correcting that. I don t know if that was there when you watched, but yes, he did mess up, and he fixed it."
    },
    {
        "video_name": "uCBm8iDyg1s",
        "Q": "Is there a faster way to find the right number, in case of mistakes like he did at 3:00 ?",
        "A": "There is a faster way, but it takes a bit more work. Instead of wondering how many times does 65 go into 98, try calculating how many times 65 goes into 981. Then you re left with 65 at the end, which you know is going to be one in the ones place."
    },
    {
        "video_name": "uCBm8iDyg1s",
        "Q": "At 0:56 he said 198 instead of 98.",
        "A": "You re right, though! \u00f0\u009f\u0098\u0089"
    },
    {
        "video_name": "uCBm8iDyg1s",
        "Q": "I am getting a bit confused at 2:13. I don't really understand what happened",
        "A": "He s rounding to the nearest ten to make it easer"
    },
    {
        "video_name": "uCBm8iDyg1s",
        "Q": "at 3:00 sal said 2 minus 2 is one is this a mistake or am i VERY dumb?",
        "A": "This is a mistake. You are not dumb."
    },
    {
        "video_name": "JBqYTuA_WgE",
        "Q": "@ 2:23. I was expecting you to take the cube root of y-12, but instead you cubed both sides. what I did was I took the cube root of y-12 and added 6 to both sides and was left with a 3x on the right hand side, then I solved for x by dividing both sides by three. I want to know why you took the cube of both sides instead of only the cube root of y-12. I also want to know whether my method was right",
        "A": "1) You need to solve for X. To get the X out of the radical, you have to cube both sides. 2) Remember, you must do the same operation to both sides of the equation to create an equivalent equation. If you apply a cube root to (y-12), you also MUST apply the cube root to the right side. So, you would have cuberoot [ cuberoot (3x-6)] = 9throot (3x-6). It sounds like you did the cube root only to the left side. Hope this helps."
    },
    {
        "video_name": "KXZ6w91DioU",
        "Q": "What is the RSH Postulate that Sal mentions at 5:23?",
        "A": "It s a special case of the SSA, in which you have a side next to a right angle, and this alone makes the only possible triangle outcome to be by completing the triangle with the possible hypotenuse. Therefore, contrary to the SSA, which is ambiguous, the RSH is a postulate. If you still have any doubt, you can refer to Sal s video titled More on why SSA is not a postulate , where he explains it."
    },
    {
        "video_name": "KXZ6w91DioU",
        "Q": "At 5:20, did Sal have to use the RSH postulate? I interpreted it as two 45-45-90 triangles and because the corresponding angles and sides on each triangle were congruent, then both triangles would have to be congruent by the AAA concurrency postulate.",
        "A": "Congruent triangles means that the sides and angles are equal. If you know only that the angles are equal you can only say that two triangles are similar."
    },
    {
        "video_name": "KXZ6w91DioU",
        "Q": "At 5:32 what is the RSH postulate?",
        "A": "Is the RHS postulate the same as the HL postulate?"
    },
    {
        "video_name": "KXZ6w91DioU",
        "Q": "At 4:32, Sal knows that angle ACM is 90 degrees, yet, you don't know CM is a perpendicular bisector, so then how can you prove that angle ACM is 90 degrees",
        "A": "It does not need to be on the perpendicular bisector to be a right angle. It simply needs to be perpendicular to line segment B A which is assured by the fact that it is taking the shortest route from C to any point on line segment B A. If B C and A C were not of equal length you could still form a right angle by dropping a line but it would not prove that C is on the perpendicular bisector."
    },
    {
        "video_name": "KXZ6w91DioU",
        "Q": "At about 11:30 n the vid, when Sal is showing the circumcircle around the triangle, could you find the circumcenter of the triangle by drawing a circle around the triangle that intersects with all points and use the radius to determine the circumcenter? Thanks!",
        "A": "That sounds like it should work because it is the same distance from all three vertices."
    },
    {
        "video_name": "KXZ6w91DioU",
        "Q": "Hey Sal, my teacher said that an altitude can only be drawn if it is horizontal or vertical, no exceptions. Is this true? Because at 4:00 you draw a diagonal.",
        "A": "You could change the orientation of the triangle by rotating it and it would be horizontal, and you could rotate it in another way to make it vertical. Your teacher is right for only certain cases."
    },
    {
        "video_name": "4_xhiP6g2ow",
        "Q": "How did sal get those three red lines at 1:16?",
        "A": "He drew the perpendicular bisectors of each of the sides of the triangle. A perpendicular bisector is a line at a 90 degree angle to another line that cuts that line in half."
    },
    {
        "video_name": "4_xhiP6g2ow",
        "Q": "Could someone explain to me how you can have 'o' as the circumcenter of a triangle it is not in? ( at about 9:52 the triangle i am confused about is drawn.)\nThanks for your help!",
        "A": "That is because the circumcenter doesn t have to be inside the triangle in all cases. In fact, in acute triangles it is always inside the triangle; in right triangles, it is always on the triangle, and in obtuse triangles, the circumcenter is always outside the triangle!"
    },
    {
        "video_name": "4_xhiP6g2ow",
        "Q": "At 7:30 how come the circumcenter of the triangle isn't even in the center?",
        "A": "circumcenter does not mean that it is the center...it is just a math term (:"
    },
    {
        "video_name": "4_xhiP6g2ow",
        "Q": "At 1:08, it doesn't seem like the three lines actually intersect on a point,...it looks more like Sal drew the line at a point at which he thought it would make an intersection. So can we just assume that it is a correct intersection?\n\nThanks",
        "A": "Just pretend he did it correctly. if he did the whole thing properly his point would still be proven."
    },
    {
        "video_name": "PSDQdwL1T0M",
        "Q": "At 1:51, how did you get y = 2x - 1?",
        "A": "Sal started with the equation: 14x - 7y = 7 He then subtracted 14x from both sides, creating: -7y = -14x + 7 Then, he divide the equation by -7. I ll write it out so you can see the steps: -7y/(-7) = -14x/(-7) + 7/(-7) y = 2x - 1 Hope this helps."
    },
    {
        "video_name": "OhUkMQtBGmE",
        "Q": "At 5:01, the program says that the y-intercept is 52,847, but when looking at the table given, the income for x=0 is 53807! Why is this?",
        "A": "Thinker, The line is only a close estimate of the actual data. So the equation for the line will be a little off from the actual data points."
    },
    {
        "video_name": "asyIsn59Lnc",
        "Q": "At 2:55 isn't the height (z) of the region not always z=1-x^2 ? sometimes it is z=1-x^2 and sometimes it is the plane y=2-z?",
        "A": "Look at the xz plane, then you should notice that the region of xz plane (which is a region limited by -1<x<1 and 0<z<1-x^2) goes up in the y axis direction limited by y=2-z . I hope this solves your answer because i don t know to write english properly, :P Saludos-"
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "at 1:26 what does he mean by \"nine of the way through it? Please help",
        "A": "Take the distance from 36 to 49. that difference is 13 (49-36=13) Next take the distance from 36 to 45. that difference 9. So when you are going from 36 to 49, you will arrive at 45 at when you have completed nine 16ths of the trip."
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "At 0:10 Sal says that he assumes that he should not use a calculator, so instead he just does paper and pencil. So how would you type that or solve that on a calculator?",
        "A": "Type in 45 and hit the Square Root symbol."
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "at 0:53, he calls 6 the \"square root\" and then corrects himself to \"principal root\" -what is the difference?",
        "A": "When asked for a square root, the answer could be positive or negative Square root of 25: 5 or -5; If you multiplied -5 by itself, you would get 25. When asked for a principal square root, the answer must be POSITIVE"
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "Can somebody help me? I don't quite understand what Sal Khan did in 1:20 when he was comparing and approximating the space between 6 and 7.\nHow did he get the fraction 9/13?",
        "A": "The question is Approximate the square root of 45 Sal sees that 45 is between the squares of 6 and 7 because 45 is between 36 and 49. He takes the difference between both the squares of 6 and 7 and sees that the total is 13, and he makes it his denominator. He took the difference of 45 and 36 because 36 is between the square of 6 and 7. He got 9 and he put 9/13 and approximated to the hundreds place. I hope this helps you!"
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "What is a principal square root Sal mentions at 0:20?",
        "A": "The principle square root: Have you ever thought of it this way? Lets say you have \u00e2\u0088\u009a25. Now you would obviously automatically know that it is 5, right? But what about -5? You also know that -5 squared is 25 because a negative times a negative is a positive. Therefore, we call the positive square root (5) the principle square root to avoid confusion with the negative one. Hope this helped! :)"
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "@ 1:23 how did he get 13",
        "A": "13 is the difference between 49 and 36. 45 is closer to 49 than to 36 so it is a good approximation that sqrt45 is greater than 6.5 or half way between 6-squared and 7-squared."
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "At around 1:39, what did he mean by linear relationship?",
        "A": "It means a relationship through lines or linear equations."
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "At 1:27, where did you get the 9 to divide over 13?",
        "A": "There are 13 integers between 36 and 49 (which are the closest perfect square numbers). The number 45 is 9 integers greater than 36, so thats how you get 9/13. You want to find the closest perfect squares and then find how close the number is with the closest perfect squares."
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "How did Sal get 6.7 at 1:54",
        "A": "He said it was a guess at 1:55. (and it s also between 6 and 7)"
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "At 1:30, I understand that the fraction 9/13 that 13 came from the difference between 36 and 49, but for the 9 I know it came from 45 being 9 integers greater than 36, but why didn't he use 4 from 49 being 4 integers greater that 45?",
        "A": "He s trying to figure out (estimate) what decimal value to add to the 6 to get sqrt(45) . If you used 4/13, then you would have to take away (subtract) that value from 7 to estimate the value of sqrt(45). It s a lot easier to add the decimal. Hope this helps."
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "At 1:27, how did he get 9?",
        "A": "sqrt(36) < sqrt(45) < sqrt(49) \u00e2\u0094\u009c 45 - 36 \u00e2\u0094\u00bc 49 - 45 \u00e2\u0094\u00a4 \u00e2\u0094\u009c 49 - 36 \u00e2\u0094\u00a4 \u00e2\u0094\u009c 9 \u00e2\u0094\u00bc 4 \u00e2\u0094\u00a4 \u00e2\u0094\u009c 13 \u00e2\u0094\u00a4"
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "At 6:36, why does he say that 6.71^2 is a little bit over the square root of 45? Shouldn't that be just 45?",
        "A": "6.71^2=45.0241"
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "At 0:51, what are principal roots?",
        "A": "The non-negative (positive) square root of a number is called its Principal Square Root. For example : The square root 4 is equal to 2. And as we know to is a positive number. Hope this helped you re understanding on Principal square roots !! :)"
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "How did Sal get 6.7? 1:51",
        "A": "Oh, and why 9/13? I don t get 1:18 to 2:05"
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "At about 1:30, Sal says, \"It's 9 of the way through it.\" What does that mean??",
        "A": "He means, as there is a gap of 13 between the 6^2 (36) and the 7^2 (49), that the 45 is is nine numbers away from the 36 number... Here... 36 45 49 |----|----|----|----|----|----|----|----|----|----|----|----|----| |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ | | 9 Hope that helps... ;)"
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "I understood where he got 13 as the demoninator, but what does \"this is nine (9) of the way through\" mean (about 1:20 into the video) in order to determine that 9 is the nominator?",
        "A": "Firstly, the top number is called the numerator not nominator. He got that because there was a difference of nine between 36 and 45."
    },
    {
        "video_name": "EFVrAk61xjE",
        "Q": "at 1:26 what exactly does he mean by nine of the way through it does he mean nine goes into this number a certain amount of times",
        "A": "No, he means there are 13 numbers between 36 and 49, and 45 is the 9th number. Like if you start counting from 36, 45 will be the ninth number out of a total of 13."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "at 2:10 Cant you just do 7 x 3 x 2 ?",
        "A": "thats what its showing you"
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "At 2:21 he said that the 1/2 and the 4 cancel out, leaving 4 as 2. How and why is that?",
        "A": "You multiply 1/2 by 4/1 and you will get 2 as a answer.Multiply the numerators you get 4. Multiply the denominators you get 2. So 4 is in the numerator and 2 is in the denominator. 4/2=2. You can try it out but instead multiply 0.5 by 4 (if you need to, do it on a calculator) and you will still get 2."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "At 1:42, where did the 1/2 come from?? I thought that height was 3?",
        "A": "1/2 base*height... so 1/2 of 3 times height"
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "At 1:43 in the video, what does it mean by one half times the base times the height? I get the rest after, but I don't get this part. Can you also do b.h.w divide by 2.2?",
        "A": "Well, you probably already know that with a triangle, it will always be half of a rectangle. So, multiplying base by height would get you the rectangle s area. By multiplying by 1/2 or dividing by 2, you get the triangle s area. Same goes for triangular prisms and rectangular prisms. To get the volume of a triangular prism, multiply the base, the height, and the width, then divide it by 2. Hope this helps!"
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "At 3:00 Is volume the same formula as surface area for cubes?",
        "A": "The volume as explained at 3:00 is the surface area (A) of one face, times the depth (d): A*d. In this example, 9 units^2 * 3 units = 27 units^3 The surface area of a cube is the surface area (A) of one face, times the total number of faces (6): 6A. In this example, 9 units^2 * 6 = 72 units^2 Given side s_, the volume of a cube is _s^3 and the surface area is 6s^2. They will only give the same number when s = 6, but the units will still be different."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "Around 3:15, wouldn't it be width instead of depth?",
        "A": "These terms are imprecise and a little bit arbitrary. I wouldn t worry about it."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "so the hight \"h=3\" when your multiplying is equal to one half O.o 1:40",
        "A": "Yeah - the area of a triangle is always one-half of the area of a rectangle of the same base and height. For example, draw two right triangles and put them together with the long sides touching. Each of them is one-half of the total rectangle that you formed. It works the same way for other triangles, though you can t put them together like that."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "at school, we divide the 1/2 after we multiply all the other sides so at 2:06 this problem would be:3x7x4 and when you have the answer, 84, we divide it in half and that is our answer. is this correct?",
        "A": "Multiplication is both associative and commutative, so you can multiply in any order, using any grouping that you like."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "at 1:48 I don't understand why Sal used V = 1/2 instead of multiplying the height and the base ??",
        "A": "The volume of the prism is the area of the base times the height. Sal defined the base as the triangle. And the area of the triangular base is 1/2 * 7 * 3."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "At 1:40, does the moon actually spin or does it just orbit around the Earth, which is spinning around the Sun.",
        "A": "The moon is tidally locked with earth, so the period of rotation is equal to the period of revolution."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "a 1:47 he has used 1/2 to multiply but Volume = L * B * H then why has he multiplied 1/2?",
        "A": "You are thinking of the volume of a cube or rectangular prism, the volume of a triangular prism is V=1/2L*B*H."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "At 1:56, Khan is saying that you have to multiply by 1/2. Can someone explain to me why you multiply it by 1/2? I do not comprehend that. Thank you.",
        "A": "it is saying multiply by 1/2 instead of dividing by 2. the formula for a triangle is B*H/2 so dividing by 2 is equal to multiplying by one half."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "During 3:14, Sal says to get the area you have to do 3*3*3, length times width times height,but when he found the area for the prism He did one-half times W times H times L. Does it vary for each shape?",
        "A": "For the volume of a prism (a shape that has the same shape the entire length, e.g. not a cone or pyramid), the volume is the area of the base times the length. Since the base in the second example was a triangle, the area of a triangle is 1/2 * L * W. If the base was a circle, in the case of a cylinder, the volume would be pi * radius * radius (area of a circle) * height."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "at 1:52 in the video why dose Sal Kan use 1/2 to find the area of the triangle",
        "A": "Because to find the area of a triangle, you have to divide it by 2 or multiply it by 1/2. Think of a triangle as half of a square, rectangle, or any other quadrilateral. The formulas for the area of a quadrilateral and a triangle are the same except with a triangle, you divide by two."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "At 1:16 and at 3:43 what does he mean by \"This Dimention\"?",
        "A": "The dimension means length, width and height. or it could mean diffrent sides."
    },
    {
        "video_name": "ZqzAOZ9pP9Q",
        "Q": "at 2:15 what happened",
        "A": "He multiplied all the numbers he had in the column above it by each other."
    },
    {
        "video_name": "QfsK8ZyC_bE",
        "Q": "at 1:11 how did you get 2,400 bags?",
        "A": "The 2400 comes from multiply: 2 * 1200."
    },
    {
        "video_name": "HY-8ydAbiik",
        "Q": "PLEASE HELP !!\nI don't understand this place notation\ndoesn't 5 go into to 200 a 40 time ?\nso why Sal said \"it doesn't go any multiples of 100\" at 1:52\nwhy he wants to see how many multiples of 100 instead of how many times does 5 go into 200",
        "A": "He is breaking up the problem into easier chunks . if 200=2x100 and 100=5x20 then 200=2x5x20 so 200\u00c3\u00b75=2x20"
    },
    {
        "video_name": "HY-8ydAbiik",
        "Q": "At 0:25, can we put 25 underneath of 28 instead of adding a 0 before 25? Thank you.",
        "A": "That is perfectly fine. Sal just put the 0 there to make it easier for other people to understand."
    },
    {
        "video_name": "S0Fd2Tg2v7M",
        "Q": "At 10:14, When Sal says the sum of the distances to the two foci will be equal to 2a, will it always be 2a, or when the ellipse is tall and skinny rather than short and fat, will it be 2b? If that's the case, could we say that the sum of the distances to the two foci from any point in the ellipse will be 2 times the major radius or the semi-major axis, which is just the same thing as the major axis?",
        "A": "It will always be 2\u00f0\u009d\u0091\u008e. Your statement is correct, this is double the semi-major axis, i.e., the major axis."
    },
    {
        "video_name": "S0Fd2Tg2v7M",
        "Q": "Why was f=(the square root of a^2-b^2)? Really Don't Understand.(at 2:44)",
        "A": "trace segment from foci 1 and 2 to (0,b). see that, because of symmetry, segment1=segment2, and segm1+segm2 = 2a, then, let f = the distance to a foci, by pythagoras b^2+f^2=a^2"
    },
    {
        "video_name": "S0Fd2Tg2v7M",
        "Q": "I got lost at 14:14, how do I get the vertex points of a hyperbola? why are they at 2*a?",
        "A": "Sal misspoke slightly. The vertices are at plus and minus a, not 2*a. He immediately says they re at 3 and -3, and a is 3 (a^2 = 9)."
    },
    {
        "video_name": "dBVoIUASFS0",
        "Q": "At 4:10,how many zeroes were there?",
        "A": "16 zeros are there"
    },
    {
        "video_name": "dBVoIUASFS0",
        "Q": "how did she do on 3:53",
        "A": "She filmed lots of little videos and put them together, so it looked like she was splitting the numbers. She also used a kind of stop-motion animation to make 2 move after she left."
    },
    {
        "video_name": "dBVoIUASFS0",
        "Q": "at 0:35 there are the numbers vi uses when she splits 2 up, and then puts it there like she got it out of nowhere",
        "A": "She filmed lots of little videos and put them together, so it looked like she was splitting the numbers."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "@13:29 why did he write 2, because that's a bit confusing, why not just write xy sq?",
        "A": "Without the 2, the expression would be xy + xz + xz You have two (xz) s. So it can be simplified to 2xz. To most people, that is the less confusing, simpler way to write that expression."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At 8:09 why didn't you add the exponents? I remember my teacher saying you had to add the exponents or was it multiply? I'm not sure. But, I do remember him saying you had to do something with the exponents.",
        "A": "When you add, it always stays the same.Usually, an exponent only changes when you multiply."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At 10:50 Sal put -3ab next, but I thought you were supposed to add or subtract all the like terms first, then insert the remaining terms. Does it even matter what order you simplify it in?",
        "A": "Sal was going left to right because it feels more natural to most people and helps to keep everything aligned. -3ab^2 does not have any like terms so he simply dropped it down next. Order does not matter as long as your numbers come out right."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At around 8:10, why did Sal write it 4x and then 7x? isn't the 7 larger, and therefore first?",
        "A": "He didn t need to write it in standard form until he had added the two equations together.He was taking 4x from the equation on the left and then adding it to 7x from the equation on the right. It didn t matter which order he placed it, as 7x+4x and 4x+7x equal the same thing, no matter the order in which they are placed."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "Sal said at 13:30 it was a polynomial, but where is the exponents",
        "A": "Polynomial just means many terms . Not all polynomial have visible exponents. Binomials, like 3x+4, have exponents of 1 and 0, which are not usually written, but the binomials do have multiple terms, so could technically be classified as polynomials."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At 1:16, why doesn't x^-1/2 +1 qualify as a polynomial?",
        "A": "Polynomials can not have negative exponents. See 1:41"
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At 0:03 Sal said polynomial; what's a polynomial? Isn't a polynomial like an algebraic expression? Anyone have a simpler way to solve a polynomial?",
        "A": "a polynomials is an equation with a lot of monomials in it, try going to algebra nation i think they will help you understand it better :)"
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "In \"Terms coefficients and exponents in polynomials\" at 1:25 Sal calls 7 to be a coefficient for the reason that we can see 7 as 7x^0. But here at 4:50 when he comes to -5, he says that there is no coefficient here. Which one is right? May be both of them!",
        "A": "They are in fact both right. Sort of. It is just a constant, but sometimes it is useful to treat it as a coeffecient on an x to the 0 term. Most of the time it isn t called a coeffecient since there is no x term, (x^0 = 1) and is thus a constant. So, it s a constant, until you need to use it like a coefficient."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "I dont understand what a polynomial is. At 0:15 all he talks about what is not a polynomial. Can I have some help?",
        "A": "These exponents have to be non-negative integers (0,1,2,...) so the expression X^2+2x^(1/3) for instance is not a polynomial."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "In the video at 14:35 Sal talks about finding the area of different shapes with polynomials, and says that for the second example the C for the width is a typo and says that the answer is 4ab + ac. But shouldn't ac be corrected as aa? Since that green figure is a square, and all sides of a square have the same measurements.",
        "A": "That s quite debatable since Sal actually did say that They drew it a bit strange. I do somehow agree that it s like a square so it should be a^2 but then, what s the c for? as I said, quite debatable."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At 4:52 he says \"-5\" is a constant but not a coefficient. But in the previous video he says that even that last constant is also a co-efficient of \"x^0.\" Is it or is it not officially a coefficient?",
        "A": "Technically, -5 is a constant because it has no variable after it. He only said it was a coefficient because you could express -5 as -5x^0 (anything to the zeroeth power is equal to 1 and -5(1) = -5). I guess you could call a constant a special type of coefficient."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At 2:22 Sal states that degrees are relevant to the value of the greatest exponent. Does that mean that the degrees can be infinitely large? Such as the 57th Degree or a 78th Degree? Or is there a limit to the number of degrees a polynomial can have before it isn't considered a polynomial anymore?",
        "A": "Polynomials by definition are expressions of a finite length that contain constants and variables raised to non-negative integer powers. As long as all of the terms in a statement have positive exponents with no radical expressions, then it can be considered a polynomial. Degree and order are still determined by the highest power exponent used."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "at 12:02 what are the factors of this: x^2-5x-6?",
        "A": "x^2 - 5x - 6 = (x - 6)(x + 1) x = 6, -1"
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "what is a monomial?\nat 6:04, how does Sal find the standard form?",
        "A": "A monomial is a polynomial containing only one term. So basically, you have at least one variable, and you have no addition or subtraction. He finds the standard form by simply rearranging the terms from highest to lowest degrees (aka from highest to lowest exponent), and then the constant terms. That s why 7x^5 is put first since 5 is the highest exponent of the whole polynomial."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At 2:07 what is he talking about? Its so confusing.",
        "A": "At 2:07 he is talking about finding the highest exponent which means finding the highest value the variable (x or y or whatever) is raised to."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At 6:46 , why didn't Sal put x^2 before 7x? I'm just asking because he had talked about wanting to put everything in standard form with the highest power first, but in this case, he put them the other way around.",
        "A": "Just to show you not to expect everything in standard form, so be on the look out as the highest degree term might not be the leading term. Once you spot that, you can rewrite in standard form."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "12:35 : No need to factor a \"b\" out of every term?",
        "A": "You do not need to factor it unless otherwise instructed."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "at 1:05, when he said that a fraction is not a polynomial, does that mean a decimal isn't either.",
        "A": "the thing is that you don t simplify fractions into decimals unless your asked to but yes decimals are not a polynomial"
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "at 15:13, when you started solving the third figure, why did you subtract?",
        "A": "Because each of the white squares (the x^2 terms) represent an area that was cut out of the pink figure."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "@ 15:22 when he comes to the construct 2xy-2x^2 , I understand in this case he is doing it to reflect the sensible and logical way to determine the area of the shape, but by standard form should it not be -2x^2+2xy. I just want to be sure I am understanding standard form of polynomials completely.",
        "A": "Thanks susan."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "at 13:27, Sal says it could be 2xy. Shouldn't it be 2(xy)?\n2xy= 2*x*y while 2(xy) = (2x)(2y)\nAm I over thinking it?",
        "A": "You re probably confusing how multiplication distributes over addition: 2(x + y) = 2x + 2y but the same thing doesn t happen with multiplication. Multiplication is associative: 2xy = (2x)y = 2(xy) all are the same thing."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At 2:21 just like Ram's question about the 2nd degree polynomial, meaning the same for a binomial, does that explanation go the same for a 3rd degree polynomial referring to a Trinomial?",
        "A": "No. Degree has do do with exponent, not how many terms there are. A trinomial has 3 terms, but there can be a 10nth degree trinomial. The degree is the highest exponent."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At 8:19, I don't understand his adding process. He seems to just automatically make -2x^2 + 1x^2 into 1x^2, but I must have not learned that yet, because I am not sure how or why he would do that.",
        "A": "He has -1x^2, not 1x^2. You just didn t see the negative."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At 8:38, why are you putting the expression in standard form with the greastest degree first?",
        "A": "Just makes it easier to understand and read the equation than to read them all mixed"
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "At 13:07, why wasn't it xy+2xz^2(instead of xy+2xz) like 15:00 where is was 2xy-2x^2?",
        "A": "SuperBadNiceMan, In the one that was xy+2xz there were two areas that had a side of x and another side of z. So each of those areas has an area of xz. Because there are two areas of this size, you have 2xy In the other problem where the solution was 2xy-2x\u00c2\u00b2, you had two cut out areas that were squares having sides of x. The area of each of those squares is x*x or x\u00c2\u00b2 and because there are two of them, you have 2x\u00c2\u00b2. Because they are cut outs you want to subtract them, so you have -2x\u00c2\u00b2 I hope that helps."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "at 10:39 is 3ab squared a constant",
        "A": "No. 3ab as a whole is not a constant, because a and b are both variables. Separately, so 3 and a and b, 3 would be a constant, but a and b wouldn t."
    },
    {
        "video_name": "ZgFXL6SEUiI",
        "Q": "In the Polynomial introduction video at 15:30 you are explaining how a polynomial can be created using the diagram, specifically the very last example that is pink. I was wondering, can this also be expressed as...\n\nab+a^2b\nor\na^2b^2-ab\n\nJust curious if the same correct answer can be derived from these polynomials as well.",
        "A": "Your 2 versions are not the same as 3ab. Sal is calculating the area of each shape. In the last example, he broke that shape into 3 rectangles where each rectangle has a length=a and width=b. Area = length * width = ab. To get total area, you need to add the area of the 3 rectangles: ab+ab+ab = 3ab"
    },
    {
        "video_name": "HPRFmu7JsKU",
        "Q": "@0:38 is that a Chinese character?",
        "A": "That was a really rough looking graph of a horizontally opening hyperbola. All the extra lines were the axes and asymptotes. Hope this helps, Joseph"
    },
    {
        "video_name": "HPRFmu7JsKU",
        "Q": "At 9:25, why does -a^2 squared equal -2a^2xf+a^4. Why isn't it just a^4. It was that simple for the xf at the beginning. They became x^2f^2 when squared.",
        "A": "He was taking the TERM xf-a^2 and squaring it were as xf is a FACTOR."
    },
    {
        "video_name": "HPRFmu7JsKU",
        "Q": "I've proved the ellipse's foci by myself before this video, but the proofs were quite different. I definitely did more than just add the d's instead of subtracting them. I then attempted to prove it the way shown in the video, but the proof fell apart after 9:07. When squaring both sides, I got (fx-a^2)^2 which is the same thing as (a^2-fx)^2, meaning an ellipse is a hyperbola?",
        "A": "There is an error in your reasoning. When you got (fx-a^2)^2, that is not the same as (a^2-fx)^2, because you did not follow PEMDAS (Parenthesis, Exponents, Multiplcation and Division, Addition and Subtraction). You cannot say fx - a^2 is equal to a^2 - fx, because subtraction is not commutative, i.e., a - b does NOT equal b - a hope this helps"
    },
    {
        "video_name": "HPRFmu7JsKU",
        "Q": "At 4:42, wouldn't it be most proper to write d1 - d2 = +-2a ? Or does this not matter? This difference may not be positive so the video seems wrong to me.",
        "A": "I think the way u r writing is also correct. Sal mentions in the video that we should take the absolute value of d1 -d2 i.e. |d1 - d2| = 2a . So the video isn t wrong. He has clarified more about it in the previous video."
    },
    {
        "video_name": "HPRFmu7JsKU",
        "Q": "At ~ 5:30 when he squares the two radicals, why isn't it +/- the values beneath the radical?",
        "A": "Because he took the distance, and distance isn t negative."
    },
    {
        "video_name": "HPRFmu7JsKU",
        "Q": "We've been told the definition of a hyperbola, is the locus of all points whose difference in distance to two focal points, will always be = 2a.\nSo, if we were to call those distances D1, and D2, then we could say:\nAbsolute (D1 - D2) = 2a.\n\nBut at 4:18 sal simply writes D1 - D2 = 2a. He doesn't take the absolute value. Taking the absolute value would result in a different equation with a different solution wouldn't it?",
        "A": "you re right. Sal just focused on one of the branches, so he didn t use the absolute value. hope this helps"
    },
    {
        "video_name": "HPRFmu7JsKU",
        "Q": "13:55 Why is f^2-a^2=b^2?",
        "A": "one way to think about it is: Both the equation of a hyperbola( the one with the b^2), and the equation that we have near the end of the proof equal one. We could make make a new equation with the equation we found on one side and the original(the b^2 one)on the other side. Then you could solve for b^2."
    },
    {
        "video_name": "HPRFmu7JsKU",
        "Q": "Why does he multiply by 2 and get 4a at 6:30?",
        "A": "Because when you square a binomial a+b, you get a^2 + 2ab + b^2."
    },
    {
        "video_name": "vl9o9XEfXtw",
        "Q": "at 2:00, what if the equation is not devisable by a number",
        "A": "then go straight to factoring a*c= a number where the sum of 2 of the factors equal b"
    },
    {
        "video_name": "vl9o9XEfXtw",
        "Q": "At 1:42 he says 9x times 4x equals 36x but why would he multiply 9x times 4x instead of just leaving it x+4?",
        "A": "I think it s to make it simpler."
    },
    {
        "video_name": "vl9o9XEfXtw",
        "Q": "At 5:23, is it possible, say if we were graphing this box in 3-dimensional space, that x = -9 can be a solution if, the negative simply denotes location relative to an origin? Is that possible?",
        "A": "No, the box would not be closed otherwise. If all three points were negative, then it could be possible"
    },
    {
        "video_name": "vl9o9XEfXtw",
        "Q": "Couldn't you just divide both sides by 9 in 1:28 , so it could be much easier later on?",
        "A": "It is a possible way, good thinking! However, he just wants to teach us to do it a specific way because most problems like these, you can t always divide both sides by the coefficient, in some cases dividing both sides by 9 will give you the wrong answer, although we probably haven t gotten that far. THE TIME WILL COME SOMEDAY BE PREPARED"
    },
    {
        "video_name": "vl9o9XEfXtw",
        "Q": "Around 3:30, Sal divides 0 by 9. How is this possible?",
        "A": "0 divided by 9 is 0. This is because we are seeing how many times 0 can go into 9. 0 can go into 9 0 times. Also, another way to do it is to make the problem into a fraction: 0/9, which can be simplified into 0. No matter which way you go about it, 0 divided by 9 is 0 and it is totally possible to find the answer to that. :)"
    },
    {
        "video_name": "vl9o9XEfXtw",
        "Q": "At around 1:28 why does Sal multiply 9 and x first instead of distributing the x to the binomial? My original thought was to distribute, which resulted in 405=x^2+4x*9, then order of operations when resulted in 405=x^2+36x, which results in the wrong answer. I also tried distributing the 9, but that wasnt even a quadratic. Any help would be appreciated.",
        "A": "Well in the order of operations u would have to open the paranthesis so that would mean distributing the 9 and x. Also for me who is really good at factoring I recognized that this problem was going to in involve factoring. Hope this helps you to understand better!"
    },
    {
        "video_name": "ScvuRb6vsz4",
        "Q": "At 7:45, Sal writes 1 mL / 0.9 grams. I thought it was forbidden (at the very least not recommended) to write fractions with decimals in either the numerator or denominator. Wouldn't the best practice be to re-write the expression as 10 ml / 9 gram?",
        "A": "He doesn t mean it as a fraction as you have probably come across them, it s just another way of writing a divide sign. You wouldn t leave it like that as an answer but it s okay as part of a calculation."
    },
    {
        "video_name": "ScvuRb6vsz4",
        "Q": "Why this video is 10:44 that is like my 1/6 hour",
        "A": "There are a few videos on here that are a little less than 2/6 of an hour, so there..."
    },
    {
        "video_name": "ScvuRb6vsz4",
        "Q": "Hi guys in min 7:01 why did he do 11/1000 * 1/.9 Isn't 1 gram = 1000 so shouldn't it be 11/1000 * 1000 ?? because based on that I didn't understand please tell me thanks.",
        "A": "the 1000 in 11/1000 comes from converting mg to grams. The 1/.9 comes from the information specific to this problem: the solution is a 0.9g/mL solution."
    },
    {
        "video_name": "ScvuRb6vsz4",
        "Q": "I am wondering at 9:29 shouldn't 0.9 be multiplied by 1,000 not 100?",
        "A": "No. He didn t do it or say it wrong. 9 X 1000 would be 9000, but 0.9 times 1000 is 900. Think about it as 90% of 1000."
    },
    {
        "video_name": "ScvuRb6vsz4",
        "Q": "At 10:00 I don't understand why when Sal said he would round 0.88856 to 0.889? Could someone help please?",
        "A": "What he did was take the number 0.88856 and round it to 0.889 ( because the 5 in 0.88856 means you round up rather then down) so that we didn t have a huge number that is messy and long to work with. Does that answer your question?"
    },
    {
        "video_name": "ScvuRb6vsz4",
        "Q": "I don't get how it gets more confusing every minute and ,at 5:25 I didn't know how 5*2.2=11 how?\ncan someone explain it",
        "A": "it equals 11 because 5*2=10 so 5*.2=1 because .2+.2+.2+.2+.2 which is 1"
    },
    {
        "video_name": "vEZea0EThus",
        "Q": "in 4:25, sal said -4^2 power was negative 16. i think that that should equal positive 16 because it said earlier in the video that a negative number by a negative number that it was going to equal a positive number. so, why did he say that -4^2 equals negative 16.",
        "A": "Good point. However, when you really think it through and follow the order of operations, you can see why it equals negative sixteen. Another way to write -4^2 is -1*4^2. In the order of operations, exponents go before multiplication. So,you have 4^2 which is 16. Multiply negative one by that and you get 16 . Hope that helps!"
    },
    {
        "video_name": "vEZea0EThus",
        "Q": "At 3:13 Sal uses the word ambiguous, what does it mean?",
        "A": "The word ambiguous means to have more than one meaning (multiple meanings). Here s the official definition from Merriam-Webster s: (1) Able to be understood in more than one way; having more than one possible meaning. (2) Not expressed or understood clearly. I hoped this helped! :D"
    },
    {
        "video_name": "vEZea0EThus",
        "Q": "At 4:15, I am still confused on the difference between (-4)^2 and -4^2.",
        "A": "-4^2 = -(4^2) = -(4\u00c3\u00974) = -(16) = -16 (-4)^2 = (-4)\u00c3\u0097(-4) = (+16) = 16 So the difference is when you apply the negative sign: before/during the multiplication or after. -4^2 is like saying -1\u00c3\u00974^2 = -1\u00c3\u009716 = -16 Going the other way, (-4)^2 is saying: (-1\u00c3\u00974)^2 = (-1\u00c3\u00974)\u00c3\u0097(-1\u00c3\u00974) = (-4)\u00c3\u0097(-4) = 16 since the negatives .. euh .. negate."
    },
    {
        "video_name": "XFI7GK4Nv-s",
        "Q": "At 0:48, the transcript says \"beta.\" Isn't it supposed to be theta?",
        "A": "If, on further review, it appears to be, you may want to report it."
    },
    {
        "video_name": "XFI7GK4Nv-s",
        "Q": "At 1:45, Sal says radisese. This is an error that he repeats and it would be good if he stopped.",
        "A": "What is the volume of a cylinder with base radius 3 and height 6?"
    },
    {
        "video_name": "l3iXON1xEC4",
        "Q": "At 4:03, Sal writes something that could be considered an equation or a function. Could someone please explain why y=4x-10 could be both?",
        "A": "Because they are two different names for the same thing... the difference is in viewpoint and intent. An equation is a relationship whose primary use is solving for a value. A function is a mathematical machine whose purpose is to transform input variables into output according to the function rule... like a toaster transforms bread into toast using the toaster rule. In the end, y= and f(x)= mean exactly the same thing... I remember that frustration clearly."
    },
    {
        "video_name": "l3iXON1xEC4",
        "Q": "At 1:35, Sal says that x + 3 is not a function. But maybe those values could be inputs and outputs as in f(x) = x + 3.",
        "A": "x+3=10 is not a function. If you simplify it, you get: x = 7. When your graph this equation, it creates a vertical line. Every point on the line has an x-value = 3, with a different y-value. This means one input (x) will create multiple different outputs(y). Thus, it can t be a function. If the equation had been: y = x+3, then it is a function. Hope this helps."
    },
    {
        "video_name": "l3iXON1xEC4",
        "Q": "why do you only get one output if you put like three inputs? do you combine them?\nlike at 3:35 he sort of talks about only having one output and at 3:42 he talks about not having both foods",
        "A": "Someone can eat the same food on multiple days, but they can t eat two foods on the same day, because then it wouldn t be a function. The definition of a function is that with each input, or day, there can only be one output, or food. You can only have each input once, but you can have the same output multiple times. In a one-to-one function, you can only have each input AND output once, but that s different than a regular function."
    },
    {
        "video_name": "l3iXON1xEC4",
        "Q": "at 2:47 i dont understand it",
        "A": "he is telling that when input is 3 as a value of x then the output that f(x) will give is 3. he is picking random values as input and output to suit his tiny graph."
    },
    {
        "video_name": "l3iXON1xEC4",
        "Q": "What is algebra ll mentioned at 0:06?",
        "A": "Algebra ll that Jesse mentioned at 0:06 is the class after High School Geometry"
    },
    {
        "video_name": "4PCktDZJH8E",
        "Q": "At 5:28 you mention how the transformation is a1+a2+b1+b2....it is confusing and so to clarify are we supposed to take (a1+b1) as x1 in the green example and (a2+b2) as the x2 value? Or is it (a1+a2) is x1 and (b1+b2) is x2? It just seems really confusing putting the terms together while working through the example. it should be a1+b1+a2+b2 to make it clear to first time learners.",
        "A": "The transformation is T([x1,x2]) = [x1+x2, 3x1]. So if we just took the transformation of a then it would be T(a) = [a1+a2, 3a1]. a1=x1, a2=x2. In that part of the video he is taking the transformation of both vectors a and b and then adding them. So it is. x1 = a1, b1 x2 = a2, b2....so x1 + x2 = (a1+b1+a2+b2)"
    },
    {
        "video_name": "4PCktDZJH8E",
        "Q": "At 13:14, isn't c^2 just another scalar, like c? If so, why would c^2 cause the transformation to be nonlinear?",
        "A": "Note that c^2 is not the same scalar as c though. For it to be a linear transformation, for all possible c s, we need T(c v)=cT(v), where the c s have to be exactly the same. Otherwise, I could do something like T(3v)=9T(v), but 9 =/=3, so it is not a linear transformation."
    },
    {
        "video_name": "4PCktDZJH8E",
        "Q": "At 4:23, Sal said that component of a vector is scalar..but component of a vector also have their direction (like component along x axis or so)..right? so in that way component of a vector should also be vector, i think...! Well, m confused..plz help..and sorry for the silly out of context question.... :)",
        "A": "Well, strictly speaking component of a vector, that is just what is written inside a vector cell is a scalar, it has no information in which cell it was written. What you are talking about is vector decomposition, i.e. representing a vector as a sum of axis-aligned vectors, consider example, given vector v = (3, 4, 5) it has scalar components - just numbers 3, 4, and 5 it could also be decomposed into a sum, like this vx = (3, 0, 0) vy = (0, 4, 0) vz = (0, 0, 5) v = vx + vy + vz look up vector basis"
    },
    {
        "video_name": "6WMZ7J0wwMI",
        "Q": "Sal, you forgot the -1 it is at 1:50. Please clarify what that equals.",
        "A": "You can think of negative exponents as reciprocal powers If 3^2 = 9 then 3^-2 = 1/9 So if 3^1 = 3, that means 3^-1 = 1/3 Review the video again to see this pattern - it is very useful."
    },
    {
        "video_name": "6WMZ7J0wwMI",
        "Q": "At 7:10, wouldn't it be greater than 1,000,000 because you also have to add the people who give out the letters?",
        "A": "Anyway, (I just have to mention this, sorry :-D), by n=9 we would surely not get 1 bilion people because there would be many people who got the letter multiple times."
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "2:25 How far \"around\" point would one look when looking to define the local min/max. IE what's stopping me from declaring the local minimum at the point on the graph next to the local maximum where the value starts dipping towards -infinity?",
        "A": "It s not a question of how far you look, but rather the characteristics of the curve at that point: the first derivative is 0, and the second is positive for a min or negative for a max. There is no minimum size of the local region where this point has to be largest or smallest to be considered a local max or min."
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "Sal says that we are locating maximum value in this graph at 1:15 but why did he said the minimum value is infinity, where the function curve isn't defined.",
        "A": "We re talking about the function right now. When you re doing problems you do the function and then after you check endpoints."
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "At about 6:00, Sal said that critical point are at places where f'(x) is 0 or undefined. Can a critical point be at a discontinuity? f'(x) is undefined at that point (a discontinuity), so does that mean it's a critical point? Thanks.",
        "A": "A critical point can be at a discontinuity, but f(x) must be defined at that point."
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "Is it correct for me to say that at 4:10 in the video, I could also write it as f'(x2)=infinity?",
        "A": "No. Since the tangent line is not well defined at x2, you can t say anything about it, including that it is equal to infinity. Now, in the situation of a vertical asymptote, you could say that the limit of f (x) as x approaches x2 is infinity, but even in that case, you can t say that f (x2)=infinity. Keep Asking Questions!"
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "at 4:14: Why is f'(x) not defined?",
        "A": "Because you cannot take the derivative of a sharp corner or vertical shape. more of the graph of undefined slope."
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "When Sal started talking about local maxima at 2:23, could the global maximum be considered a local maximum as well?",
        "A": "a global maximum has all the traits of a local maximum, so yes"
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "0:30, can a global maximum also be a local maximum?",
        "A": "Yes. For example, in a parabola that opens downward, its local max is also its global max."
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "At 4:43 you explain about endpoints. If the endpoint of a function is its global minimum or maximum is it still considered a critical point?",
        "A": "How would that happen? Remember, Infinity goes on forever, so you can t really reach the bottom or end of the function, and thus, since there will always be a higher or lower point, an endpoint can t be a critical point. I hope this clarifies endpoints and critical points!"
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "At 5:30, why does Sal say that local minima and maxima cannot be at endpoints? In general, why do local minima and maxima not exist at enpoints,",
        "A": "For a local min/max to exist, you must be able to approach the point from both sides on an open interval, and you cannot approach an endpoint from both sides. Thus, endpoints can only be absolute extrema."
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "At 6:00, he says that critical numbers are both where the min/max is 0 or undefined. However, my cal teacher said undefined points aren't critical numbers. Is that only when they're asymptotes? Is it only the local max/min that are critical numbers?",
        "A": "You are misunderstanding. It is not the points themselves that would be undefined, but the DERIVATIVE of the function at those points that would undefined. If the point is not defined in the original function, it is not part of the domain and cannot be a critical point. So, if the DERIVATIVE of a function is either 0 or is undefined at some point AND the original function is defined at that same point, THEN that point is a critical point. Critical points are often BUT NOT ALWAYS extrema."
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "At 0:55, why is f(x0) greater than or equal to f(x)? Why wouldn't it be f(x0) greater than f(x)?",
        "A": "Because you can have f(x)=f(x0)"
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "At 07:07 what makes X\u001f1 a local minimum but not X3? He says a local minimum is a point that is lower than the points around it and both X1 and X3 are lower than the points around them so I'm a bit confused.",
        "A": "x\u00e2\u0082\u0081 is a local minimum because if you move to the left or the right of it, you re moving up. x\u00e2\u0082\u0083 isn t a local minimum because if you move to the left of it, you re moving up, but if you move to the right of it you re moving down. At x\u00e2\u0082\u0083, it plateaus, but it doesn t switch direction vertically."
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "at 2:47 why can't the point of global maximum be the point of local maximum?",
        "A": "The word global in global maximum implies that it is the highest point for the entire function. Since (x\u00e2\u0082\u0082, f(x\u00e2\u0082\u0082)) isn t the highest point of the whole function, it cannot be the global maximum."
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "@2:46 sal wrote local max at a smaller point than global maximum?someone explain..",
        "A": "The local maximum (or minimum) is just saying that compared to the points immediate right and left the point is highest (lowest), not compared with the whole graph/function. So while the point x2 is not higher than the global maximum, it is a peak of the function. Moving either to the left or right from x2 would be a downhill motion."
    },
    {
        "video_name": "lDY9JcFaRd4",
        "Q": "At 0:44 when he states it is a global max, is that term synonymous with absolute max?",
        "A": "Yes, in this context absolute and global are synonymous. Likewise, local and relative are synonymous in this context."
    },
    {
        "video_name": "x_aax6LSRtE",
        "Q": "At 3:34 the narrator says 'sub tans'. What does that mean?",
        "A": "The word is subtend . It means extend under or be opposite to ."
    },
    {
        "video_name": "x_aax6LSRtE",
        "Q": "I Dun understand some thing ,, isn't b the circumcenter of triangle ? So how in 5:44 , You considered the b center of circle.and distance from b to B which is vertex of triangle is a radius ?! Please,I need an instant answer , u used this then in similarity to say c over 2R",
        "A": "b represents the length of side AC. The circumcenter wasn t denoted by any letter, but R was used to represent the length from the circumcenter to a point on the circle such as B."
    },
    {
        "video_name": "x_aax6LSRtE",
        "Q": "At about 2:30, Do you have to draw the diameter to it to point B?",
        "A": "We need to use point B because earlier in the video, Sal drew the perpendicular BE coming from vertex B. In the triangle Sal drew, angle B is greater than 90 degrees, so this is the only perpendicular that lies fully inside the circle, and it would have made things a bit messy to have used another vertex. But if the triangle is the right shape, then any vertex can be used."
    },
    {
        "video_name": "x_aax6LSRtE",
        "Q": "At 7:19 I don't understand how Sal changes 2[ABC] / b/a into 2[ABC] / ab.All of his videos have been pretty straight forward until this point.Can anyone please help me with this?",
        "A": "Look at it as two fractions being divided by another. The numerator is 2ABC / b and the denominator is a / 1, then since they are being divided, you would instead multiply by the reciprocal. Therefore you get [2ABC / b] * [1/a] = 2ABC / ba = 2ABC / ab. A good numeric example is taking [1/2] / 2. Split a half in half and you get 1/4."
    },
    {
        "video_name": "x_aax6LSRtE",
        "Q": "2:57 Like Alex I would like to know where it is proven that if hypotenuse of a triangle inscribed in the circle is the diameter of the circle then it is a right triangle. And does subtended mean?",
        "A": "For an angle to subtend an arc means that is a circle didn t go a full 360 and only went the measure of the angle given, what would the arc look like."
    },
    {
        "video_name": "x_aax6LSRtE",
        "Q": "At 0:33 , Sal says \"I didn't want to make it look like any particular triangle\"\nBut, that is a obtuse scalene triangle. Why does he do that?",
        "A": "It s an arbitrary decision. The proof works for any triangle. He chose an obtuse scalene triangle simply to show that any triangle will work, not only an equilateral or isosceles triangle."
    },
    {
        "video_name": "x_aax6LSRtE",
        "Q": "At 7:40, isn't ab * c equal to ac+bc, not abc?",
        "A": "oh, now i got it silly me :P"
    },
    {
        "video_name": "8a1a5A3CfdQ",
        "Q": "At 1:00 Sal said 'geometric series'. I though series it when you sum up all the numbers. Did he wanted to say 'geometric sequence'?",
        "A": "Yes, he should have said sequence and there is a correction made at 2:40"
    },
    {
        "video_name": "8a1a5A3CfdQ",
        "Q": "at 0:59, when sal said geometric series, did he mean geometric sequence?",
        "A": "Geometric sequences are geometric progression . These are a type of series in which each number follow a specific pattern (like in arithmetic progression) . The pattern for Geometric progression(GP) is that the ratio of two consecutive terms always remain CONSTANT. for example: 2, 4, 8, 16, 32,...... in this GP the ratio of any two consecutive term(2 and4 OR 4and 8) remains constant =2."
    },
    {
        "video_name": "8a1a5A3CfdQ",
        "Q": "At 0:42, why couldn't the numbers be divided by 1/2",
        "A": "168 divided by 1/2 would be 336 which is not the second term in the sequence. The numbers can be divided by 2, to arrive at the formula shown at 2:45"
    },
    {
        "video_name": "8a1a5A3CfdQ",
        "Q": "At 0:11, Sal says \"G defines a sequence where N is the term of the sequence.\" Can functions have any letter? or are some letters (like the mysterious 'i') never used?",
        "A": "You can always use any letter (or language) you wish, but there are many conventions that help us communicate clearly among ourselves, and if I don t follow those conventions, others might have a hard time understanding what I m trying to say. The mysterious i itself is actually used in many different ways, depending on the context, so it s important to keep the context in mind when choosing our alphabet and our vocabulary."
    },
    {
        "video_name": "8a1a5A3CfdQ",
        "Q": "At 2:55, how does Sal get 168/(2^(n-1))?",
        "A": "(1/2)^(n-1) is the same as (1^(n-1))/(2^(n-1)) because you distribute the exponent to the numerator and denominator, right? 1^(n-1) is just 1, so you get 1/(2^(n-1)). then you multiply that to 168 to get 168/(2^(n-1))"
    },
    {
        "video_name": "JPQ8cfOsYxo",
        "Q": "at 7:51,what is AIME?",
        "A": "American Invitational Mathematics Examination"
    },
    {
        "video_name": "WSpF5uvApLA",
        "Q": "At 2:35, why didn't Sal say what the inconsistent system is? Is it dependent or independent?",
        "A": "I believe the designations of dependent or independent are used only when the system has a solution. Since inconsistent systems have not solution, they are not classified as dependent or independent."
    },
    {
        "video_name": "WSpF5uvApLA",
        "Q": "around 2:42 I was confused about whether inconsistent graphs are dependent or independent or none?",
        "A": "inconsistent graphs are neither dependent or independent. Those terms only apply to consistent graphs."
    },
    {
        "video_name": "WSpF5uvApLA",
        "Q": "At 3:30, couldn't Khan have used substitution instead?",
        "A": "Yes but there are many substitution"
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "Hey guys. I'm not really sure, but a lot of you seem to be from the USA, maybe the word is a US thing, but at 7:39, he says \"parentheses\". What are they? Are they brackets?",
        "A": "Yes, parentheses are these two symbols: ( ) Brackets are characters (or symbols) that group things. They are: - Square Brackets [] - Curly Brackets {} - Parentheses () - Chevrons <> You have more names for each of them. Look it up in Wikipedia for Bracket to get more info. Hope this helps."
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "I need help on 0:39 how he gets 3x-3=2x+6.",
        "A": "it is simple the equation is 3(x-1) = 2(x+3) now .. 3(x-1) means that 3 is multiplied with each term within the parenthesis so when you multiply the portion within the parenthesis you get 3x - 3 and you remove the parenthesis same goes for 2(x + 3) = 2x + 6 ... that is multiply the expression within the parenthesis with 2. hope you understand ... :-)"
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "At 3:43, why did Sal decide to subtract 6z from both sides instead of 9/16z?",
        "A": "Oh.. I got it. I have an answer for myself. You can t take the 9/16z because that is the only thing on the left side."
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "in 8:43,why 2x-3=15+3 is 18? cuz 15 not have any sign that with be 14.",
        "A": "Just watched the video and now I understand what you are saying. It really is 2x-3+3=15+3 because you are trying to get rid of the -3 in 2x-3. So, when you add 3 to both sides, -3 in 2x-3 is canceled out, and you end up with 2x, so 2x=18, now you divide by 2 and get x=9. Let me know if you still don t understand."
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "Why is it that when you times a fraction by a number, you need to times the numerator and not the denominator? Like at 3:11 Sal times z/16 by 9 and gets 9/16 z, why does he do that?",
        "A": "Sal is able to multiply these two numbers, because 9 can be converted into an improper fraction. How many 1 s make up 9? 9. So that can be turned into 9/1. Sal did multiply the denominator, but by one(because of the denominator of 9/1)."
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "At 3:06, for equation number 2, why does he multiply by 9?",
        "A": "He multiplies by 9 in order to get rid of the 9 in the denominator since 1/9 * 9 = 1."
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "I've always wondered why you only multiply the number to the numerator it cancel the number out. For example, at 3:17, you didn't multiply the denominator by 16. Why?",
        "A": "so he is also multiplying and dividing by 9 .the numerator one is kept on RHS and denominator is taken to LHS.(left hand side) of the equation."
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "2:02 \"This is being adding this equation to the equation three is equal to three\"? Adding equation to equation? What is he talking about? He completely lost me there.",
        "A": "You CAN add equations to equations. Why? Because you are allowed to do the same thing to both sides of an equation. Let s say I have these two equations: a = b and c = d. I can add them together and get a+c = b+d. I can do that because c is the same as d, so when I add c to one side and d to the other, I am actually doing the same thing to both sides. This technique becomes useful when you start to solve equations that have more than one variable, as you will see."
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "At 5:46,When Sal introduces the word problem,another question could be\"Logan and Izzy had the same number of stickers.After Izzy gave him 72 stickers Logan has 3times as much as Izzy what was the equal number?\"",
        "A": "you would have to divid to figure out what the equal number is."
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "At 3:08 can someone tell me how z16 multiplied by 9 becomes 9/16z?!?!?!?!?!?!?!?!?!?",
        "A": "a/b * c = (a * c)/b You may be confused by the way Sal has written it, but in that form the 9/16 is the coefficient of z. Before the multiplication, the coefficient of z was 1/16. Instead of z/16, Sal could have written 1/16 * z. Keep in mind that this is z/1, and so it s (1 * z) / (16 * 1) not 1 / (16 * z)"
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "at 4:34 i got really confused",
        "A": "why were you confused"
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "I NEED HELP. At 1:15 Sal subtracted 2x on both sides why didnt he subtract 3x on bothsides",
        "A": "Yes. I understand. so the x should always or should be usually on the left side...Thanks for answering my question"
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "At 2:57, how could x=9? Please answer as soon as possible! This is urgent!",
        "A": "x minus 1 equals 8 and 8 times 3 is 24. Do the same with the other equation and it matches yup with each other."
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "How is 9(z/16)=9/16*z? At around 3:30",
        "A": "think of 9(z/16) as nine times z sixteenths 9/16*z = z times nine sixteenths it s all the same... if you didn t catch that, think of it this way: 9/1 times z/1 times 1/16... you can change this order- it wont change the result(COMMUTATIVE PROPERTY) hope this helped :)"
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "At 5:28, are you sure -32/87 cannot be reduced?",
        "A": "Yes. If you look at the prime factorization, you will see that 32 and 87 do not share anything in common, and so must already be as reduced as it will be. Prime factorization of 32 is 2\u00e2\u0080\u00a22\u00e2\u0080\u00a22\u00e2\u0080\u00a22\u00e2\u0080\u00a22. Prime Factorization of 29 is 3\u00e2\u0080\u00a229. 29 is a Prime Number. 2\u00e2\u0080\u00a22\u00e2\u0080\u00a22\u00e2\u0080\u00a22\u00e2\u0080\u00a22 ------------- 3\u00e2\u0080\u00a229 nothing cancels out so -32/87 is fully reduced."
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "I really don't get how at 3:11, Sal says that 9 times z/16 is 9/16 z. Can you guys explain why this is? I've always been a little fuzzy in this area of algebra... Thanks! :D",
        "A": "If you use the distributive property, 9(z\u00c3\u00b716) should be equal to 9z\u00c3\u00b7144. You have to multiply z and 16 by 9."
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "At 3:15 Why is 9 over 16 and not 9z over 16?",
        "A": "(the fraction 9 over 16) times the variable z , is the same as (9 times the variable z) over 16."
    },
    {
        "video_name": "xMsG9hvqzbY",
        "Q": "At 3:06 Why is it 9/16 z?",
        "A": "There, we are multiplying both the sides by 9, to remove the 9 in the denominator in the right hand side of the = sign. And as you know, we can t just multiply 9 to 2 (3z+1) / 9 just to one side of the equation, we are multiplying both sides with the inverse of 1/9 (i.e. 9) , so that the value of the equation isn t changed."
    },
    {
        "video_name": "OelluIKIkCY",
        "Q": "5:40 ishh why is the lenghth? of the the slice bottom function minus top function?? (y+1-(y-1)^2. Everything up to this point makes sense but now Im lost. . . I thought you do top minus bottom/ right most - left most",
        "A": "Tilt your head 90 degrees to the right. Which is upper function now? If we calculate in terms of y then the function output is x, and x=y+1 returns higher x es then x=(y-1)^2, thus x=y+1 is the top function."
    },
    {
        "video_name": "OelluIKIkCY",
        "Q": "After the 5:00 mark, what does he do (y+1 - (y-1)^2) instead of the other way around? is x=(y-1)^2 not the larger radius, and he should therefore subract the smaller y+1 function to get the area between the two?",
        "A": "We are integrating with respect to y, so the functions are in terms of y. To visualize this, tilt your head to the right (as Sal said) and look at the graph. Then you ll see y+1 is on top while (y-1)^2 is on the bottom. So in this case, y+1 is the larger radius."
    },
    {
        "video_name": "OelluIKIkCY",
        "Q": "at 3:05, how is it y+2 for the radius?",
        "A": "In other words, it is because the shape is not rotated around the x axis but rather around a line that is two units below the x-axis. Therefore to get the radius, you have to add those two units to the value of y, so, (y + 2) is the radius"
    },
    {
        "video_name": "OelluIKIkCY",
        "Q": "At 3:52 shouldn't the radius be the absolute value of y +2?",
        "A": "There s no need, since we know y will be greater than -2 for all of the function values for y. Therefore y + 2 will already be positive, since it s really y - (-2), the difference in the two functions. Thus we can take the difference without having to introduce absolute value. Hope that helps!"
    },
    {
        "video_name": "OelluIKIkCY",
        "Q": "At 4:15 - Out of curiosity, if I swapped the upper and lower functions, would it just result in a negative volume?",
        "A": "I m guessing it would be like putting an upper bound on the bottom of the integral which results in putting a negative sign infront of the integral to make the upper bound go on top; so I think basically yes"
    },
    {
        "video_name": "vPYl4NM6DaM",
        "Q": "At 2:58, could answer C be written as \"f is differentiable over the interval [2,4)\"?",
        "A": "*EDIT* I m not sure, but I think you re technically correct. They probably just wanted to be clear."
    },
    {
        "video_name": "H9h-Xg7zjPU",
        "Q": "At 1:57 Sal says that option 2 is especially strange. I know it is not the right answer to this question but, is it possible for a function to be inversely proportional to the square of its input. If so, can I get some examples?",
        "A": "Option 2 describes a function that s inversely proportional to the square of its output, not its input, but yes - both are possible. For option 2: S(t) = k / S^2 { which means the same as S(t) = k / [S(t)]^2 } Multiplying both sides by [S(t)]^2: [S(t)]^3 = k S(t) = k^(1/3) Or, in other words, S(t) is constant for all t. An example of a function that is inversely proportional to the square of its input would be S(t) = k / t^2 { which is actually option 3! }"
    },
    {
        "video_name": "4Bc5-HRop5Y",
        "Q": "A little after 7:00, you've found the vertex. However, there was so much work required to convert the equation from standard to vertex form. What's the benefit of doing that equation transformation instead of just using the standard form and x = -b/2a (followed by substituting it back in to find y)?",
        "A": "the conversion to vertex form after finding the vertex is actually pretty easy. Here s how it goes:- y=(x+the negative inverse of the x value)^2 +the y value as it is for example: If the vertex is (1,2), then the vertex form of the equation will be y=(x-1)^2 +2"
    },
    {
        "video_name": "4Bc5-HRop5Y",
        "Q": "At 4:26 why does he take 1/2 of it?",
        "A": "Sal is using a method called Completing the square . it involves an equation (M*1/2x)^2. in his example if you take M which is the Middle term(5) and divide by 2 you would get 5/2. then square it and you get 16/4. so that is why he adds and subtracts 25/4. this wont change the answer at all because it still =0. so your adding a fancy way of saying +0"
    },
    {
        "video_name": "4Bc5-HRop5Y",
        "Q": "@7:35 hi,\n\nstill unclear on the rationality of how the vertex point can be found from the completing the square equation as Sal demonstrates above. So, I understand the point at which the perfect square is zero is the x point of the vertex...the y point for the vertex is at the end of the equation...but why is this the case? Why is this the point at which the equation is at its lowest point.\n\nThanks.",
        "A": "The way I think about it is this: As I see you understand how the x point of the vertex came to be found, you know that it is 5/2. So when you plug the value into the function, and work it out like any other function from there, you will come to f(5/2)= -1/4 Assuming f(x)=y"
    },
    {
        "video_name": "4Bc5-HRop5Y",
        "Q": "At 8:19 Before watching I assumed answer was 2.5. Now I'm not sure that my assumption was fair. My\nquestion is:\nIf a parabola crosses the x axis is it reasonable to assume that the axis of symmetry is half way between the crossing points?",
        "A": "Yes, the x-value for the vertex will be half way between the 2 x-intercepts. Your answer of 2.5 and Sal s answer of 5/2 are the same."
    },
    {
        "video_name": "cfr-yZxTH8Y",
        "Q": "at 0:30 why is there no decimal for 518? Im new to this",
        "A": "there is a decimal its just not written its right after the (8) in 518 or 518.0 and the you can add zeros as long as you want. (:"
    },
    {
        "video_name": "cfr-yZxTH8Y",
        "Q": "in 1:20 do you have to put the zeros? I'm just making sure.",
        "A": "Also think of it this way: 540.0 =540.0000000000"
    },
    {
        "video_name": "cfr-yZxTH8Y",
        "Q": "At 2:41, Is it just me, or is the transcript wrong? I mean, it's not a huge deal but...",
        "A": "It is wrong, but no big deal, right, well even though two words are connected."
    },
    {
        "video_name": "cfr-yZxTH8Y",
        "Q": "0:10 why does the decimal have a 0 in front of it why isn't there just a decimal with no 0 in the front?",
        "A": "there doesn t have to be one it s just easier to understand ."
    },
    {
        "video_name": "cfr-yZxTH8Y",
        "Q": "At 0:52, why do you have to move it over to the right?",
        "A": "Because he is showing how the decimals work."
    },
    {
        "video_name": "cfr-yZxTH8Y",
        "Q": "at 0:55-1:31 why do you need to move the decimal to the right?",
        "A": "because thats where you put the number of groups of what your dividing"
    },
    {
        "video_name": "rRTXKQpblEc",
        "Q": "at 1:45 why would you flip the triangle if you know it is going to be the same answer?",
        "A": "He did that to show that a triangle can be rotated to form a parallelogram or other quadrilateral, which has twice the area of the original triangle. Using this, he derived the formula for the area of a triangle. Hope this helps answer your question."
    },
    {
        "video_name": "rRTXKQpblEc",
        "Q": "At 1:54 is it posable to do that with any triangle?",
        "A": "Yes try it with some different ones"
    },
    {
        "video_name": "rRTXKQpblEc",
        "Q": "at 0:33 he connects it to the original shape but this is what confuses me my teacher says you can not just connect it you always have to do it the hard way. Why is this?",
        "A": "We may never know but maybe it s for a good reason"
    },
    {
        "video_name": "rRTXKQpblEc",
        "Q": "what is the green line at 3:22",
        "A": "The green line is the height of the obtuse triangle in relation to side b."
    },
    {
        "video_name": "3mimxluSVBo",
        "Q": "At 3:10 the video gets a little confusing, it would be helpful to have the B earnings explained further, I see that the - B has a coefficient of -1 right, but then very quickly we are told that B is 50 as well without it being shown in the work.",
        "A": "B is only 50 because of the given fact that abby earns nothing, because their two wages must add to 50$ under the first constraint"
    },
    {
        "video_name": "3mimxluSVBo",
        "Q": "At 1:45, they say abby takes place of the y axis and ben takes places of the x axis, but why can't it be switched around, like instead abby takes of the x axis?",
        "A": "We can switch it around, it doesn t matter. The only difference is that now \u00f0\u009d\u0090\u00b5 is a function of \u00f0\u009d\u0090\u00b4, so we would rewrite our equations as: \u00f0\u009d\u0090\u00b5 = \u00f0\u009d\u0090\u00b4 \u00e2\u0088\u0092 10 \u00f0\u009d\u0090\u00b5 = 50 \u00e2\u0088\u0092 \u00f0\u009d\u0090\u00b4 As we graph these two lines, we ll notice that they intersect at (30, 20), so Abby made $30 and Ben made $20."
    },
    {
        "video_name": "f-wz_ZzSDdg",
        "Q": "0:48, What is the proof for (a^2-b^2)=(a+b)(a-b)?",
        "A": "Well, if you do the multiplying (a+b)(a-b), you ll get: a*a+b*a+(-b)*a+(-b)*b That equals a^2+ab-ab+b^2 so if you simplify this (there s +ab and -ab) you get a^2-b^2. That s all!"
    },
    {
        "video_name": "f-wz_ZzSDdg",
        "Q": "Is the step at 3:00 neccesary?",
        "A": "Hi Shelby Richards, The step at 3:00 is not necessary, but it does make it easier to look at the equation and eliminate the common terms. Hope that helps! - JK"
    },
    {
        "video_name": "f-wz_ZzSDdg",
        "Q": "At 3:41. Why don't we say that A can't be equal to 2 since in the numerator their is (a-2) if a was 2 than (a-2) would be 0 making the numerator 0. Why would that not make the equation undefined? Thanks:)",
        "A": "Making the numerator equal to zero does not make the expression undefined. It is only when the denominator equals zero that the expression is undefined --- because DIVISION (not multiplication) by zero is never allowed."
    },
    {
        "video_name": "f-wz_ZzSDdg",
        "Q": "At 4:10, why wouldn't the two a's be able to cancel each other out? They're both positive and they're both only 1 a, so why couldn't those be canceled out? Please help!!",
        "A": "One is in the numerator, while the other is in the denominator. You can t cancel them."
    },
    {
        "video_name": "f-wz_ZzSDdg",
        "Q": "in 2:39 you can see that in the numerator that there is a (a - 2), woudnt you have to write a cannot be equal to 2 aswell? does the \"not equal part\" only occur for the denominator?",
        "A": "The domain only needs to exclude values that make the denominator zero. When the denominator is zero, the fraction is undefined. When the numerator is zero (and the denominator is not), the fraction is zero. And zero is a very well-defined quantity."
    },
    {
        "video_name": "0rlNHYHhrWs",
        "Q": "This question is not about the video! Why does he change ratios in the workabled problems? He says because the ratio of AB to BC is 2:1 then the ratio of AB to AC is 2:3. Where does he get the ratio of 2:3? I have asked a lot but can't seem to get an answer!",
        "A": "Here is why the ratio works out that way: If AB = 2 And BC=1 Then AC = AB + BC = 2 + 1 = 3 Thus AB/AC = 2/3"
    },
    {
        "video_name": "erZe85NrsK0",
        "Q": "At 1:10, when Sal splits the fourths, can we do 8 divide by four? I sort of don't get it.",
        "A": "When Sal is representing the mixed number, he makes three wholes and splits them into 8 parts each. To make 2 1\u0004, he took two wholes and one fourth of the last whole. If you notice, the 1/4 of the last whole is actually 2/8 of the whole. So, in order to get the same answer, we would have to divide the whole into eight parts. Then we would split the eight parts into 4 groups of 2 parts, or in other words, 4 groups each containing 2/8. This would be another way of thinking about it. Hope this helps!"
    },
    {
        "video_name": "erZe85NrsK0",
        "Q": "In 1:13 is that color really orange?",
        "A": "No. Maybe to you it is, but in truth it s not."
    },
    {
        "video_name": "erZe85NrsK0",
        "Q": "There is a mistake on 4:55. 1/2 isn't equal to 4/8 + 3/8 + 1/8. I know what you meant, but it's definitely incorrect and it's a bad habit to make sentences like this.",
        "A": "If you skip just to that point of the video, it does look like that, but if you watch through the entire video, it makes sense."
    },
    {
        "video_name": "NgYrsqoKXpM",
        "Q": "At 6:00 - why is it f(x) = 2(g(4x^2))? Why are we multiplying g(4x^2) by 2? I am just having a hard time seeing it... Sorry",
        "A": "f(x) = 2/(1+4x^2). It s easier to find the Maclaurin polynomial for a simpler series, 1/(1+x). So we found that Maclaurin expansion, and we called it g(x). Now, if g(x) = 1/(1+x), we have to transform function g(x) into f(x) so we know how to transform the expansion for g(x) to get the expansion for f(x). To go from 1/(1+x) to 2/(1+4x^2), we substitute 4x^2 for x, and multiply the function by 2. Thus, we want to find: 2*g(4x^2) to get f(x). That s why f(x) = 2(g(4x^2)). Hope that helps!"
    },
    {
        "video_name": "XwIs1nlDQ2I",
        "Q": "At 0:59, she mentioned that vectors are a point with only direction and magnitude. But does that mean that vectors can have more than one point?",
        "A": "Umm yes I think it is possible other points on the vector direction would be points on that vector. But don t take my word for it, even I am only just in my Physics class."
    },
    {
        "video_name": "XwIs1nlDQ2I",
        "Q": "What do oranges have to do with green beans?1:23",
        "A": "Both are healthy for you."
    },
    {
        "video_name": "dIGLhLMsy2U",
        "Q": "at 1:55 with the captions on why does it say hail our viewers just day when he is talking about geometric sequences",
        "A": "Well, that is because the caption application is not doing a good job of capturing what Sal is saying. It is trying to match conversational words with what Sal is saying, but he is saying math-y stuff, like a sub n and times r but these entities are not in its vocabulary. Probably needs tuning to better capture mathematical terminology."
    },
    {
        "video_name": "dIGLhLMsy2U",
        "Q": "At 5:08, Sal talks about explicit defining (correct me, if I'm wrong about that) and then writes {a n(subscript)} with n = 1 in subscript. What does that mean?",
        "A": "practically, n represents the term number in the sequence."
    },
    {
        "video_name": "dIGLhLMsy2U",
        "Q": "maybe I missed something but what is N factorial? @ 5:45",
        "A": "n represents the number n. n!=n * (n-1) * (n-2) *(n-3) ... (n-(n-1)) 5!=5 * (5-1) * (5-2) * (5-3) * (5-4) = 5*4*3*2*1"
    },
    {
        "video_name": "dIGLhLMsy2U",
        "Q": "At around 3:51, when Sal explains the other sequence, what type of sequence is that?",
        "A": "I don t believe it has a particular name, but it is an interesting sequence."
    },
    {
        "video_name": "dIGLhLMsy2U",
        "Q": "At 5:11, what kind of sequence would it be?",
        "A": "Its basically a geometric, since you are multiplying"
    },
    {
        "video_name": "dIGLhLMsy2U",
        "Q": "At 1:19 what does the equation that he wrote say I don't understand it.",
        "A": "At around 1:29 Sal writes the equation for the n-th term in the sequence. Is this the equation you are referring to? In this equation a-sub-n is just the notation used to refer to the n-th term in the sequence. a-sub-1 = a, a-sub-2 = ar, and so forth. Does that address your question?"
    },
    {
        "video_name": "sJmLjUj_h68",
        "Q": "at 1:20 how does the square equal 28?",
        "A": "Well, the question was asking for the perimeter of the square. A square s sides are all congruent. A formula for perimeter for * squares* is 4 s_ (with _s being the side length). The problem states that one side is equal to 7 m. So in the equation, replace s with 7. So the equation is now 4(7) = P (P is perimeter) 4 times 7 = 28 m, and that s how they got 28 m."
    },
    {
        "video_name": "kpG7xySkivg",
        "Q": "At 5:51 Sal mentions that the two yellow matrices sum up to Matrix B, hence so should their determinants. But instead there is a r1 for each of the three matrices, so shouldn't it really be adding two r1s which is not the r1 by itself as shown in Matrix B- there is two of each element vector, except the weird one. What is happening here?",
        "A": "I think the same. Perhaps what we should really add to have the B matix is a matrix filled with the 0 vector except for cr_i. The determinant of that matrix is cero."
    },
    {
        "video_name": "TXaWqy52dZY",
        "Q": "at 6:33 I don't understand",
        "A": "Just add a 0 on the end because 27 has been multiplied by 10 so 27 basically (it can be put like this) steals the 0 off of 10."
    },
    {
        "video_name": "Z0WsfO-RI8Y",
        "Q": "AT 2:49 ,What would happen if you divide the 2nd part in 5th what would happen",
        "A": "You could do that... then you could graph mixed numbers like 1 2/5"
    },
    {
        "video_name": "6EY0E3z-hsU",
        "Q": "I don't know if my questions were taken and aren't showing up, so I apologize if I'm asking them twice.\n-At 1:16, how is a^2 + b^2 = a^2 +2ab + b^2?\n-At 9:45, question 54, couldn't Marsha also be trying to draw a line parallel to P, intersecting line L?",
        "A": "At 1:16, (a+b)^2 = a^2 + 2ab +b^2. This is just an application of the distributive property. (a+b)*a + (a+b)*b = a^2 +ab+ab+b^2 = a^2 + 2ab + b^2 At 9:45, you can t draw a line parallel to p, because p is a point. Also, the construction is definitely trying to create a line parallel to line l going through point p."
    },
    {
        "video_name": "xUjRwepN7dk",
        "Q": "At 2:39, how come Sal has to break up the number then later at 3:36 divides the numbers by two and adds it up to get the answer? Can't you just break it up, add it in your head and get the answer?",
        "A": "He is simplifying it and showing the mental process for others that may not understand how to do it in their head quite yet."
    },
    {
        "video_name": "t8m0NalQtEk",
        "Q": "At 2:40 Khan does not seem to make much sense.Help?",
        "A": "If you re talking about the commas, then every three numbers a comma is placed, as a pause, because it notes when the number changes, to ones, to thousands, to millions, and so on. It might be a little confusing once you first start it, but it either sticks, or it doesn t, and doesn t make much difference either way. If not, I really don t know what you re talking about. I hope I have helped!"
    },
    {
        "video_name": "1kfq0aR3ASs",
        "Q": "At 8:44, What is a quintic trinomial?",
        "A": "Think about quin as in the amount of children inside the womb single, twin, triplets, quads, quintuplets. 1, 2, 3, 4, 5 quin is the greek word for one more than 4 otherwise known as 5"
    },
    {
        "video_name": "1kfq0aR3ASs",
        "Q": "I didn't get 0:58",
        "A": "The product of two numbers just means what you get when the two numbers are multiplied together. He s looking for two numbers a and b such that a + b (their sum) is -14 and ab (their product) is 40."
    },
    {
        "video_name": "1kfq0aR3ASs",
        "Q": "4:15\n(x-3)(x+4) is that also a good answer?",
        "A": "No. The way to check is to use FOIL and multiply the 2 binomials. You get: x^2 + 4x -3x -12 = x^2+x-12. The middle term has a different sign than the one being worked in the video (x^2-x-12)."
    },
    {
        "video_name": "1kfq0aR3ASs",
        "Q": "Take a look at 2:23.\nWhen you find two factors of 40 that have a sum of -14, in our case -4 and -10, how do you know which expression to put which in? For example, if you have -4 and -10\n( x + ? ) ( x + ? )\nSal does it as ( x - 4 ) ( x - 10 ) but does it matter?",
        "A": "It doesn t matter, ( x - 4 ) ( x - 10 ) is the same thing as ( x - 10 ) ( x - 4 ) . The point of factoring a quadratic is to discover what input values x will make the quadratic 0. It is hard to see when you have x\u00c2\u00b2-14x+40, but once we factor that, we get ( x - 10 ) ( x - 4 ), and we can see right away that if x=10 or x=4, the quadratic will be zero."
    },
    {
        "video_name": "1kfq0aR3ASs",
        "Q": "at 3:36 and 3:40 he says 3 plus negative 4 wich of course is negative 1 and 3 times negative 4 is negative 12 but I don't understand how is negative 1 what we were looking for?",
        "A": "If we expand this: (x+a) (x+b) = x^2 + (a+b)x + (ab) So, if we have to find out the value of a & b, we have to solve both, coefficient of x and that constant term i.e. (a+b) & (ab). For example, We have to factorise the following equation : x^2 - (12)x + 35 then, (a+b) = -12 and (ab) = 35 Is there any a & b that satisfies these two? Yes, if a = -7 & b = -5 a+b = -12 & ab = (-7)(-5) = 35 Therefore, We can write this as :- x^2 - 7x - 5x + 35 = x(x-7) - 5(x-7) => (x-5) (x-7)"
    },
    {
        "video_name": "u9v_bakOIcU",
        "Q": "At 5:24, Sal said that x couldn't be equal to negative 2 or negative 8, otherwise g(x) would be zero. However, in his answer, he only mentioned the condition that x not be equal to negative 8. Should he also have mentioned that x could not be equal to negative 2?",
        "A": "Sal didn t mention negative 2 because (x+2) it is still present in the denominator, so we can tell that function is undefined just looking at it. However, since we divided numerator and denominator by (x+8) we can t say by the look of it that original function is undefined at -8. Hence we put only one condition, second one is seen from the simplified function."
    },
    {
        "video_name": "u9v_bakOIcU",
        "Q": "Around the 2:15 mark Sal talks about this \"grouping\" technique. But instead of that couldn't I just create an area model?",
        "A": "There are several different techniques that you can use to factor the trinomial. If you are comfortable using an area model and can get the correct factors, then yes. You can use it."
    },
    {
        "video_name": "u9v_bakOIcU",
        "Q": "Is there a simpler/different way to break down the expression than to factor by grouping (as demonstrated at 1:39) ?",
        "A": "add and simplify"
    },
    {
        "video_name": "u9v_bakOIcU",
        "Q": "At 1:09, could I have simply divided the two functions without factoring?\nIf not, why?",
        "A": "If you use polynomial long division, you just end up with a different fraction that still needs to be reduced. So, you might as well factor 1st and reduce the fraction to start with. Then, you re done."
    },
    {
        "video_name": "u9v_bakOIcU",
        "Q": "At 3:41 couldn't he have used the quadratic formula to find the roots of f(x)",
        "A": "He said you could have"
    },
    {
        "video_name": "u9v_bakOIcU",
        "Q": "Why can't x not be equal to -8?\nAfter you simplify the equation at 5:00, it wouldn't matter, right?\nOnce you remove (x+8), then you can still have x=-8.\nI don't get it.",
        "A": "Both before and after cancelling (x+8) , x = -8 is OK. 0/0 = 1."
    },
    {
        "video_name": "u9v_bakOIcU",
        "Q": "At 5:09 how does Sal know that -8 and -2 wouldn't work out in the function. What steps could I take to figure out which specific conditions are required for the function. When I get two groups of numbers, (X+2) (X+8), if I solve this separate from the original problem, (X+2)(X+8)=0. Then I get X=-2 and -8. Is this correct?",
        "A": "Yes, that would be the correct approach to find the values that won t work."
    },
    {
        "video_name": "u9v_bakOIcU",
        "Q": "at 2:15 where did you get 16 from?",
        "A": "It s the product of the 2 in (2x-1) and the 8 in (x+8)"
    },
    {
        "video_name": "u9v_bakOIcU",
        "Q": "at 4:30, how did the negative 8 become a positive 8",
        "A": "Well, Sal factored out the - from (-x-8), which you will get x+8. To add on, Sal factored out the new equation (x+8) from 2x(x+8)-1(x+8). To think about how Sal factored out this equation, we can think this as ax+bx. We can factor x out to get x(a+b). So, Sal got the new equation (2x-1)(x+8)."
    },
    {
        "video_name": "3jBfLaLrk6I",
        "Q": "Why would you round down in 0:37 from 5?",
        "A": "You would round down if the number was less than 5 such as: 1594 That would round (to the nearest tenth) to 1590 since 4 < or equal to 5"
    },
    {
        "video_name": "3jBfLaLrk6I",
        "Q": "what would 23 be rounded to? At 0:49 he dose not say what to round it to ?",
        "A": "23 would be rounded down to 20 because the last digit, 3, is less than 5."
    },
    {
        "video_name": "3jBfLaLrk6I",
        "Q": "At 0:08 he said \"and\". Isn't \"and\" only said when there is a decimal in a number?",
        "A": "You are correct! Looks like Sal made a mistake."
    },
    {
        "video_name": "DaoJmvqU3FI",
        "Q": "at 2:28 how could you just subtract the to exponents 5 and 4 in the statement of 7/1*10\b",
        "A": "Because when you divide numbers with the same base, all you need to do is subtract the exponents. Then you just multiply that by seven to determine how many times larger 7*10^5 is than 1*10^4 (or just 10^4 if you want to simplify)."
    },
    {
        "video_name": "DaoJmvqU3FI",
        "Q": "2:20 why do we have to put the separate notations over each other to divide? Wouldn't it be easier to do the output of the two and divide them and then put it back in scientific notation? Or is it possible to do what I just described at all? Thanks in advance",
        "A": "no man just do it the way sal does it."
    },
    {
        "video_name": "dam16G6cZ8k",
        "Q": "At 9:05 , is that a local or absolute max?",
        "A": "Actually the global maximum depends on the interval in which it is to be checked. A plot of the functions depicts a maxima at the point and an infinite rise where x<0. Since you cannot make negative shoes, you must take the interval x>0. In this range the point is the global max. In x E R, there is no global maxima."
    },
    {
        "video_name": "dam16G6cZ8k",
        "Q": "I don't understand at 3:48 how you got that 3 before the first x. Or the other numbers.",
        "A": "Let s differentiate them one by one and then add them all together. When we differentiate a function via the power rule, we basically move down the exponent and reduce it by 1 -x^3: constant=-1, exponent=3, derivative=-1*3x^(3-1)=-3x^2 6x^2: constant=6, exponent=2, derivative=6*2x^(2-1)=12x^1=12x -5x: constant=-5, exponent=1, derivative=-5*1x^(1-1)=-5x^0=-5 we can add these derivatives together and get -3x^2+12x-5 if it s still hard we can do more examples!"
    },
    {
        "video_name": "dam16G6cZ8k",
        "Q": "at 7:00 why am i not able to simplify by dividing all terms by 6? all three numbers have 6 as a factor. that would simplify to 2 +-14. the only positive solution being 16.",
        "A": "\u00c2\u00b1sqrt(84)/6 is not equal to \u00c2\u00b114. If you were to simplify it, you would get \u00c2\u00b1sqrt(21)/3"
    },
    {
        "video_name": "n0DLSIOYBsQ",
        "Q": "1:50, I don't know if I am misunderstanding or making my own mistake in thought, but shouldn't the cos of theta be the green dotted line and sin of theta be the pink dotted line? Isn't cos of theta the base of the triangle in reference to the triangle the x-axis and the ray make?",
        "A": "He is connecting the pink dot to the Adj line of the Triangle, thus it is representing a coordinate on the x axis..the base. Cos0 = adj/hyp."
    },
    {
        "video_name": "n0DLSIOYBsQ",
        "Q": "At 4:48, the sine and cosine are labeled with absolute value. Why is that? Aren't they already positive?",
        "A": "Yes it may seem redundant, but using absolute value allows us to generalize into any quadrant. For instance, if we drew a triangle in quadrant III, both values would be negative, but the lengths (distance being a scalar value) must be positive (the absolute values of the sines and cosines)."
    },
    {
        "video_name": "n0DLSIOYBsQ",
        "Q": "At 2:50, how are you able to find the \"equation\" of a circle, I just have no idea, I thought you could only graph lines that actually go somewhere?",
        "A": "In general, the equation for any circle is ( x - h )^2 + ( y - k )^2 = r^2, where ( h , k ) is the center of the circle and r is its radius. In this particular case the center is at the origin ( 0 , 0 ) and the radius is 1. This is covered at length in Geometry. Use the Search box and type in equation of a circle to see the video."
    },
    {
        "video_name": "6JFzI1DDyyk",
        "Q": "Hi, I'm a bit confused at 6:00. He writes i = 1 under the sigma, but the first element in the data set is 2. Could someone please explain that (To be honest, I don't really understand the whole \"i=\" thing under the sigma)? Also, is it necessary to write \"i\"? Because I think I like \"n\" better.\nThanks, and sorry.",
        "A": "The first element in the data set is X_i, when i=1. i is the number of the element, not the element itself. You can use whatever letters you want, it s just convention, but that might be confusing if people need to check your work."
    },
    {
        "video_name": "6JFzI1DDyyk",
        "Q": "At 7:42, why does he say x sub 4 is equal to 4 when he writes 3 because it is the last term!",
        "A": "Slip of the tongue, probably. If you look at the data set, x sub 4 is equal to 3. (i.e. The 4th element of the data set is the number 3.)"
    },
    {
        "video_name": "6JFzI1DDyyk",
        "Q": "This is the first time I've done Maths in English and though I understand how to use symbols, I don't always know how to say them out loud. When Sal is listing the various x(i) terms (e.g. at 7:30) does he say: \"x of 1, x of 2, etc.\" or \"x sub 1, x sub 2, etc.\" ?",
        "A": "X sub 1 is probably the most common way to read it, although you wouldn t be misunderstood if you said x of 1 or even just x one ."
    },
    {
        "video_name": "6JFzI1DDyyk",
        "Q": "At 7:34, he says x of 4 = 4. doesn't he mean it equals 3 because 3 is fourth in the sequence (and there aren't even any 4's to begin with)? thanks!",
        "A": "You are correct. He meant x sub-4 (4th term) equals 3"
    },
    {
        "video_name": "6JFzI1DDyyk",
        "Q": "At 0:35 Sal says that, \"i=1.\" Why does i=1? Is \"i\" always equal to one when calculating the population mean?",
        "A": "i in the equation Sal uses represents a particular member of the population. He says i=1 because he wants to start looking from the first member of the population. Then he will look at member 2 (i=2), then member 3 (i=3) and so on until he reaches member N (i=N). N is the number of people in the population. If we are only concerned with members 5-10 and just want to ignore everyone else, below the E symbol, we would write i=5. Above the E, we would write i=10. Hope this helps!"
    },
    {
        "video_name": "6JFzI1DDyyk",
        "Q": "At 9:22 , I dont understand how do you know that sigma equal to 1 . Can you explain for me ?",
        "A": "Hi Giang, sigma means the sum of all the things that follow the sign. In this case it is the sum of all the (x - u)^2. Since all of these are 0.25, it means that Sigma is 0.25 + 0.25 + 0.25 + 0.25 = 1. Hope that is helpful."
    },
    {
        "video_name": "5EdbPz1ZVn0",
        "Q": "At 8:56, isn't it 36 instead of 35, when he did the multiples of 9?",
        "A": "Yeah, somebody earlier already caught that. Although, it isn t Sal s fault; it doesn t look like Sal made this problem up, like he usually does."
    },
    {
        "video_name": "5EdbPz1ZVn0",
        "Q": "At 8:57 isn't 9 times 4 = 36, not 35? :)",
        "A": "Yes, many people have been commenting about it. It was just a simple mistake though. :)"
    },
    {
        "video_name": "5EdbPz1ZVn0",
        "Q": "At 5:22, Sal said to \"break even.\" What does that mean?",
        "A": "It means that the profits from the lemonade is equal to the cost of running the lemonade stand."
    },
    {
        "video_name": "5EdbPz1ZVn0",
        "Q": "In the second problem when he starts graphing at 8:04 I wondered if you might have graphed it differently. To break even, you would need to be at 0. If they need to make 25.00 to break even, then that means they spent 25.00, so then the y-intercept of the line would be at -25.00, and the line would cross the x-axis at the break even point, about 56 (or 55.5....). Right?",
        "A": "I agree, y=profit, and they start with a $25 deficit."
    },
    {
        "video_name": "ZSESJ8TeGSI",
        "Q": "Wait at 4:48 he says that P(r)=x^2Q(r) etc.,why is there an x in the beginning when in the rest of the equation he switches it out with r?",
        "A": "It was just a slip-up. If P(x) = x^2 * Q(x) then P(r) = r^2 * Q(r). It doesn t actually affect the end result in any way because Q(r) = 0, so I guess Sal didn t notice it."
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "At 8:07, what did perpendicular mean, because it was said that when you have a right angle the two side are perpendicular I don't understand what two sides that are being referred to?",
        "A": "Perpendicular angles form a right (90 degree) angle. In a right triangle, the two lines that form the 90 degree angle are perpendicular to each other."
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "I don't understand how it's perpendicular... doesn't it have to be all the way a crossed (like a lowercase t) or it's just like an L at 3:40",
        "A": "No, for something to be perpendicular, it only has to have a right angle from the other line. So, if you re standing for a long time, and are wishing you could take a nap, you could say, I m tired of being perpendicular to the ground. Or, I can t wait to be parallel to the ground!"
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "When stating an angle, for example like angle DBA at 0:43, is it necessary to have the vertex of the angle (in this case point B) in the center? Is it possible that angle DBA could be stated as angle BDA?",
        "A": "Yes, it is necessary to have the vertex you want to find in the middle. DBA and BDA are two different angles so you can t state them as the same."
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "around 5:06 it kind of started to not make sense like why would we add angles?",
        "A": "you would add angles so that you can find whether two angles are complementary or suplementary"
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "3:13 is there a trick to remember the difference between complementary and supplementary angles?",
        "A": "The way I was taught to remember: It s always right to give someone a compliment! (compliment -> complement -> right (90 degrees)."
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "(at 0:14) Could \u00e2\u0088\u00a0ABC also be called \u00e2\u0088\u00a0CBA?",
        "A": "Yes, absolutely. As long as the vertex is the second point you name, you re good to go!"
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "At 2:31 when the video says there are angles called complementary angles are complementary angles only made up of 2 different angles?",
        "A": "Yes, complementary angles are when only two angles add up to give 90 degree."
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "At 5:08, is it possible to add 2 different angles?",
        "A": "yes kenya it is very very possible."
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "At 3:50, did it matter that ray BD was switched to ray DB?",
        "A": "DB isn t a ray. If you look closely, the line above it doesn t have an arrow on it, so it s just a line segment. It can be named either BD or DB. But when Sal writes ray BD, it must be written that way. Because when you re naming a ray, the first point (B, in this case) is always the endpoint. However, I don t think he really meant to differentiate between a line segment and a ray - it doesn t really matter in this case."
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "At 6:45, did Sal mean the measure of angles DBA and ABC?",
        "A": "Technically, you are correct."
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "At 2:31 when the video says there are angles called complementary angles are complementary angles only made up of 2 different angles?",
        "A": "Yes, it has to be two angles."
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "At 0:05, how can you tell the difference between the angle sign and the less-than sign?",
        "A": "Usually, the bottom line in the angle sign is flat; the lines in the less-than sign are pointed towards the middle. Less-than: < Angle: \u00e2\u0088\u00a0"
    },
    {
        "video_name": "BTnAlNSgNsY",
        "Q": "in all the 8:31 I did not get the video. I need help in how to find the complement and supplement of a angle can someone help me",
        "A": "Complementary angles are two angles that add up to 90 degrees. Supplementary angles are two angles that add up to 180 degrees. For example you are given a 52-degree angle. The complement of that angle is 38 degrees. The supplement of that angle is 128 degrees."
    },
    {
        "video_name": "tBRW-CyJE6k",
        "Q": "At 0:10, does he mean 3 times 2x? For that to be 3 times 2x, x would be a decimal, right?",
        "A": "No, it s three more than twice x or in other words, the number that s 3 integers bigger than 2x. So 3+2x. Not sure what you mean by x would be a decimal . x is any number."
    },
    {
        "video_name": "tBRW-CyJE6k",
        "Q": "At 0:31 what does it mean by \"one y\"?",
        "A": "It means one value for y , or one number for y . For instance, let s say you have a function where y = 2x. If you take 3 as your x, you will then have y = 2*3 = 6. In other words, when you used one x (here I used 3), I got one y, which was 6. Only one y for one particular x. If you graphed that function and looked at x = 3, you d find only one y that matches it."
    },
    {
        "video_name": "tBRW-CyJE6k",
        "Q": "At 0:35 minute, He said that y is a function of x, shouldn't it have been written as y=f(x)?",
        "A": "He could have use function notation and wrote y=f(x)=2x+3. But it is equally valid(though somewhat less formal) to just write y=2x+3. When an equation is written in explicit form(y=some expression in terms of x), it is generally understood that y is a function of x, regardless of whether or not the symbol f(x) is included."
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "At 3:23, could Sal also have applied the square-root to the squared terms, thus giving\ny - k = x - a + y - b?",
        "A": "It would have been nice to have a way of simplifying like that but unfortunately, \u00e2\u0088\u009a((x - a)^2+(y - b)^2) is not the same as x - a + y - b. Just like (x + 1)^2 is not the same as x^2 + 1^2."
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "2:36 Why is the distance (x-a) and not (a-x)? Same goes for (x-b), why not (b-x)?",
        "A": "Since the term (x-a) is squared, we get the same result with (a-x): (x-a)^2 = (a-x)^2: It really doesn t matter which we choose. But it s conventional to chose the order (x-a). All of the standard equations for conic sections use this order."
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "At 8:00, what happens if b=k? Wouldn't you get an undefined number?",
        "A": "b can t equal k because then the focus and the directrix would have the same y coordinate and actually be the same point on the coordinate plane. The focus and the directrix must have different y coordinates, which is why b would never equal k . Hope this helps!"
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "At 4:02 why wouldn't (y-k)^2 become y^2-k^2",
        "A": "You should review the distributive property. (y - k)^2 = (y - k) * (y - k) y * (y - k) + -k * (y - k) y^2 - yk - yk + k^2 y^2 - 2yk + k^2 Try it out with some integers, and you ll see this: (5 - 3)^2 = (5 - 3) * (5 - 3) It s pretty easy to see that this will be 4: 5 - 3 = 2. 2 * 2 = 4. 5* (5 - 3) + -3 * (5 - 3) 25 - 15 -15 + 9 25 - 30 + 9 = 4 If you try it the way you propose: (5 - 3)^2 = 5^2 - 3^2 = 25 - 9 = 16 Incorrect."
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "At 2:43, when Sal writes the distance formula for the line from point (x,y) to the focus at (a,b), why does he write square root((x-a)^2+(y-b)^2) instead of square root((a-x)^2+(b-y)^2)? Doing the latter would subtract away the portions of the a and b coordinates, thus giving the needed horizontal and vertical components of the line segment from point (x,y) to the focus, wouldn't it?",
        "A": "If you used square root((a-x)^2+(b-y)^2), you would get the exact same result. (a-x)^2 = a^2 -2ax + x^2 which is exactly the same as x^2 - 2ax + a^2 The same is true for (b-y)^2, you still get y^2 -2by+b^2"
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "how exactly did you get 2y in 3:54 when you expanded the equation.",
        "A": "Use FOIL to expand ( y - k )^2 [ written as ( y - k )( y - k ) ] First : + y^2 Outer : - ky Inner : - ky Last : + k^2 The sum of the Outer and Inner terms is - 2ky."
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "At 2:15, why does Sal square (y-k) and then take the square root? (y-k) will be positive anyway if he hadn't done that, because when you square the right hand side of the equation, (y-k) gets squared too.",
        "A": "He is saying that if you were to have k be more/greater than y, you would take the absolute value of y-k or squaring it and then square rooting it. Does that help??"
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "At 2:08, Sal squares the expression, then takes its square root which is equivalent to taking its absolute value. I understand that, but wouldn't it be less complicated just to take its absolute value?",
        "A": "Squaring y-k turns out to be helpful in changing the form of the equation."
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "1:43 What is K? I see that he says y=k, and I think y is the directrix. So, is K just another way of saying directrix?",
        "A": "Yes, k is the directrix."
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "At 3:26 why does Sal square both sides? I thought if you had a square root of (y-k)^2 then that would be the same as (y-k), since the square root is the same thing as (y-k)^1/2. Would one not just multiply ((y-k)^2)^1/2 like the exponent property says?",
        "A": "That is true for the left hand side, but on the right hand side, it s not quite the same. We have \u00e2\u0088\u009a((x-a)\u00c2\u00b2+(y-b)\u00c2\u00b2). We can t cancel this to (x-a)+(y-b) because it would be incorrect. So the best option to get rid of the radical sign is to square it, which you must do on both sides of the equation to not change its values."
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "at 7:53 sal said times be whoops times?! what?",
        "A": "He went to write times (b-k) but instead started writing times (k-b), so he said woops, erased the accident and started the sentence over with times."
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "AT 8:20, Couldn't Sal have factored out a 1/2 from that simplified equation?",
        "A": "If you factored out the 1/2 you would only have to put it back in again when you are using the formula as the vertex form of a quadratic."
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "On 4:00, Sal simplifies \" square(y-k)^2 = square(x-a)^2 + square(y-b)^2 \"\nto \" (y-k)^2 = (x-a)^2 + (y-b)^2 \"\nBut shouldn't the squares and powers cancel out, and get\n\" (y-k) = (x-a) + (y-b) \"\n?",
        "A": "Nope, as the first expression is square(y-k)^2 = square[(x-a)^2 + (y-b)^2] and not square(y-k)^2 = square(x-a)^2 + square(y-b)^2. Hope this helps! :)"
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "At 7:34, what happened to the (b-k)/(2)(b-k)? How did it just disappear?",
        "A": "It s not (b-k)^2 its b^2 - k^2 gives (b+k)(b-k)"
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "at 2:45 shouldn't it be (a-x) ^2 + (b-y)^2?",
        "A": "its the same thing. a-x and x-a are negatives of each other. square of a number is equal to the square of the negative of that number. eg: 5^2=25 (-5)^2 is also equal to 25"
    },
    {
        "video_name": "okXVhDMuGFg",
        "Q": "At 4:18 why does he choose to keep (x-a)^2 simplified? Is it because there are no other x terms which can be factored?",
        "A": "Yes, Sal knew there would be no other x terms that could be combined with (x-a)^2. However, he also knew the form he wanted to end up with ---he wanted the formula for a parabola to be easily recognisable. And, as you get more practice with these, you will get used to knowing when you can keep things simple (and thereby save yourself lots of work!)"
    },
    {
        "video_name": "9ylUcCOTH8Y",
        "Q": "@5:00 do the \"i\" and \"j\" need the little hat on them?",
        "A": "Commonly they are written with little hats, or little arrows. I ve seen a bunch of different notations used with vectors -- not sure there is any single standard?"
    },
    {
        "video_name": "9ylUcCOTH8Y",
        "Q": "at 4:57, is 'i' always horizontal and 'j' always vertical?",
        "A": "By convention i is the unit vector of the x-axis(horizontal) while j is the unit vector of the y-axis(vertical axis)."
    },
    {
        "video_name": "9ylUcCOTH8Y",
        "Q": "At 1:04, is he saying \"two tuple\"?",
        "A": "2-tuple (two-tuple, whatever) means a combination of two components in a vector. Some examples: 1-tuple: <1> 2-tuple: <4,4> 3-tuple: <5,6,8> 4-tuple: <1,3,6,7> 10-tuple: <1,6,8,4,3,67,24,6,1.-9> In an n-tuple, there are n components."
    },
    {
        "video_name": "9ylUcCOTH8Y",
        "Q": "At 1:05, Sal talks about 2-tuples. He talks about it many other times, but I never understood it. What exactly is a 2-tuple?",
        "A": "A tuple is just a list. The preceding number defines the length of the list. Hence, a 2-tuple is a list comprised of 2 numbers, a 3-tuple is a list comprised of 3 numbers, and so on."
    },
    {
        "video_name": "9ylUcCOTH8Y",
        "Q": "at 1:27 what did he mean?",
        "A": "He is just talking about the different ways of representing vectors. One way would be to use column vectors or you could use rectangular coordinates. Then he shows a third way of representing them - using unit vectors. Unit vectors are used because they are easier to type on one line."
    },
    {
        "video_name": "9ylUcCOTH8Y",
        "Q": "At 2:00 Sal uses i and j for unit vectors. Is i and j always used to represent unit vectors or can it be any letter?",
        "A": "i\u00cc\u0082 and j\u00cc\u0082 are typically the first 2 unit basis vectors. Meaning in R2, they d be: i\u00cc\u0082 = [1 0] j\u00cc\u0082 = [0 1] Usually i, j, and k are used for the main basis vectors of R3, but that s just a convention and you can use any letter(s) you want. Any unit vector (not just ones with a 1 and 0 s) will typically be denoted using the little hat symbol in place of the arrow above the letter. People like to use u\u00cc\u0082 pretty often, but the letter is chosen arbitrarily."
    },
    {
        "video_name": "9ylUcCOTH8Y",
        "Q": "At 4:03, why does Sal add 3 times vector j to 2 times vector i?",
        "A": "Those unit vectors i hat and j hat form the basis for R^2 since they are linearly independent. All vectors in R^2 can be represented as the sum of a scaled version of these two vectors."
    },
    {
        "video_name": "l5MrtV7ZN88",
        "Q": "at around 1:04, it should be the number of candies I have, not \"chocolates\", right? because some of those candies do not have chocolate outside.",
        "A": "Yes, well spotted :-) Sometimes - although rarely - the word chocolates is used for types of candy that don t have chocolate. But in this case you re right, Sal made a mistake."
    },
    {
        "video_name": "1j5WnqwMdCk",
        "Q": "At 4:47 - How are these two vectors a basis for R^3? A basis for R^3 must not only be linearly independent, but should span R^3 - which two vectors cannot! The augmented matrix of these vectors cannot have a pivot in each row. Am I missing something?",
        "A": "You are right, a basis for R3 would require 3 independent vectors - but the video does not say it is a basis for R3. In fact, it is instead only a basis of a 2 dimensional subspace within R3. In this case the subspace would just be the plane given by the span of the two vectors."
    },
    {
        "video_name": "1j5WnqwMdCk",
        "Q": "At 16:39 wouldn't you say Sal is pretty lucky that the reduced row echelon form only had 2 pivots? Or said in another way that the reduced row echelon form had all zeroes in the bottom row? Otherwise d couldn't have been represented in terms of B!",
        "A": "I wouldn t call it lucky but rather good planning on Sal s part for choosing d to be in the span of v1 and v2. Had he not planned ahead, as I suspect he did, then yes, Sal did get quite lucky that the row reduced form of the augmented matrix didn t end up with a pivot in the final row/column, as that would ve led to d being found outside the span of v1 and v2."
    },
    {
        "video_name": "1j5WnqwMdCk",
        "Q": "At 13:40 I don't understand how you row reduce. You should use the elementary row operations, right? There is no subtraction operation. You can add by a negative scalar, but -2+0 != 2-0.",
        "A": "At that time, he s replacing the second row with 2 times the first row minus the second row. This is actually 2 elementary row operations at the same time because it s multiplying the second row by -1 before adding 2 times the first row. You re right that the way he does this isn t kosher, but it s still valid row operations."
    },
    {
        "video_name": "qFFhdLlX220",
        "Q": "At 0:32 Sal uses the expression: principle root? what does it mean and how is it different than a square root?",
        "A": "principal root means only the positive square root whereas square root in general means plus or minus square root. the principal square root of 9 is 3, but if x^2=9, then x=3 or -3"
    },
    {
        "video_name": "qFFhdLlX220",
        "Q": "At 4:15, Sal used the term 'radical.' What does that mean?",
        "A": "It just means an expression that uses a root - square root, cube root. etc."
    },
    {
        "video_name": "qFFhdLlX220",
        "Q": "at 0:04 He says the principal root. Is the principal root the same thing as a square root, or what is the difference?",
        "A": "Remember, square roots have 2 possible answers. For example: 3*3 = 9 and -3(-3) = 9. Thus, sqrt(9) could be 3 or -3 The principal root is the positive root, and it is the default. So, if the problem just asks for sqrt(9), you should return the positive root. sqrt(9) = 3 To specify that you need to provide the negative root, there will be a minus sign in front of the square root. So, -sqrt(9) = -3 Hope this helps."
    },
    {
        "video_name": "qFFhdLlX220",
        "Q": "Can someone please briefly explain what just happened in this video.In 0:38 how did 500 change into 10*5?Any help please?",
        "A": "He states that 500 can be rewritten as 100 times 5, Then he says that it can be written as ten squared times 5. So he goes from 500 = 100 \u00e2\u0080\u00a2 5 to 10^2 \u00e2\u0080\u00a2 5 which allows the 10 to come out of the square root and the 5 to stay in. He does not say 10 \u00e2\u0080\u00a2 5 at this time stamp."
    },
    {
        "video_name": "XdsifNBa9kk",
        "Q": "At (2:40) he says that \"our change in x is 7.5 minus 6.5, change of x is still 0.5\". If it were the change of 7.5 minus 6.5, shouldn't change of x be 1? Did he mean to say x is 7.5 minus 7?",
        "A": "Yes, it s clear from the context that Sal misspoke there and intended to say 7.5 minus 7."
    },
    {
        "video_name": "XdsifNBa9kk",
        "Q": "At 7:15, isn't point-slope form y=m(x-h)+k?",
        "A": "Yes, but it s also very common to refer as point-slope form y - k = m(x - h) since the only step left is to add k to both sides."
    },
    {
        "video_name": "XdsifNBa9kk",
        "Q": "I have a question at 1:34. Why is the graph of the line y=f(x)? Isn't that equal to y=y?",
        "A": "It s just there to relate that the data that s being drawn over the y-axis is that labeled as f(x)."
    },
    {
        "video_name": "FtxmFlMLYRI",
        "Q": "I like the circle of numbers she does at 2:46, it's really cool!",
        "A": "I agree with your declaration. It gets old, though."
    },
    {
        "video_name": "FtxmFlMLYRI",
        "Q": "Is it just me or at 3:45 does an extra dot appear when she writes the extra 1?",
        "A": "its just fastforewarded"
    },
    {
        "video_name": "FtxmFlMLYRI",
        "Q": "Why did Vi use a Headphone Cord at 0:13?",
        "A": "Probably the closest thing she could reach, or else to catch people s attention (like she needed help there)"
    },
    {
        "video_name": "FtxmFlMLYRI",
        "Q": "is that \"string\" (0:20) part of a pair of headphones, or did my mind put something there that isn't?",
        "A": "Yes it s a pair of headphones because where are you going to get a string without damaging your sock from? It s simple."
    },
    {
        "video_name": "FtxmFlMLYRI",
        "Q": "at 1:19 what is a 4 dimensional object",
        "A": "The quarter of a sphere she was trying to do something with."
    },
    {
        "video_name": "FtxmFlMLYRI",
        "Q": "2:37 what's wrong with the alternate interior angle congruence theorem? I understand putting 1=1 as a theorem is stupid, but what's wrong with angle theorems?",
        "A": "amazing, amazing song love it <3"
    },
    {
        "video_name": "FtxmFlMLYRI",
        "Q": "At 2:21 she mentions that Tau equals two pi. How is Tau represented on a circle and why is it not used as often?",
        "A": "Because Pi has been used for a lot longer. Tau is relatively new compared to it."
    },
    {
        "video_name": "FtxmFlMLYRI",
        "Q": "(1:16)What was this about: 'Well Depending on dimension, 'cuase there's 1, 2, 3, then theres 4 and even more!'\n\nThanks in advance!",
        "A": "The dimensions that you are in. 1 is a line, 2 is a plane, 3 is 3-D, the field we exist in, and then 4 and even more planes exist, possibly, although no one and nothing yet exists in them, to the best of our knowledge. And check out Flatland, it shows a wonderful comparison of dimensions."
    },
    {
        "video_name": "FtxmFlMLYRI",
        "Q": "Starting at 3:18 it sounds more like a reverse song she is singing. Is it only me?",
        "A": "I think it was just all the new numbers not in the chorus."
    },
    {
        "video_name": "SwqOrUWzDY8",
        "Q": "if A to the fourth power times B to the fourth power =(AB)^4 then what do you mean in the other video's when you say A to the fourth power times B to the fourth power = 4x4?\n1:29",
        "A": "A^4 x B^4 is A^4B^4"
    },
    {
        "video_name": "SwqOrUWzDY8",
        "Q": "At 4:44 Sal drew a to the power b in a amount of c times. But below the A's he drew a yellow thing i seen it in math but what is it called and what is it used for.",
        "A": "I may be mistaken, but I believe that would simply be referred to as a bracket. It s much larger than the usual brackets we re familiar with, and it s pointing vertically, but it otherwise does the same thing - it implies a grouping."
    },
    {
        "video_name": "SwqOrUWzDY8",
        "Q": "At 4:20 he shows that (a^b)^c equals a^(c*b), using c groups of the base a^b. But what happens if c is negative, or a fraction?",
        "A": "Wrath, (a^b)^c equals a^(c*b) is still true when c is a fraction and/or negative."
    },
    {
        "video_name": "SwqOrUWzDY8",
        "Q": "At 4:45 Sal says that we have \"c\" numbers: but that would not work because that would mean that c must be more than 4. because he drew four.",
        "A": "What sal was trying to say is that there are c amount. C can be any number. I can see where it can be confusing, but it doesn t really matter how many a s there are."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "Does it matter if I subtract/add/multiply or divide 0:32? Do I in the end always get the right answer?",
        "A": "It actually does matter and it won t always be the same answer due to the coefficient and constant. They will have random values."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "Around 10:00 how do you go from 3(5/4) to 15/4??",
        "A": "3/1 x 5/4 = 15/4 3 x 5 = 15 1 x 4 = 4"
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "at 11:01 was he not suppose to divide by 7 not 1/7",
        "A": "He multiplied by 1/7 which is the same as dividing by 7."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "I noticed at 6:55 that Sal does something that I don't do - he sometimes multiplies one of the equations with a negative number just so that he can eliminate a variable by adding the two equations, while I don't care if I have to add or subtract the equations. This is just personal preference, right?",
        "A": "Yes, it is. When you subtract equations, you re really performing two steps at once. Sal chose to make each step explicit to avoid losing people."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "At 2:10 why did he multiply the equation by -5 instead of just 5.",
        "A": "The whole point behind it is that you want to cancel out (lose) one of the variables. If you have only one variable, you can solve the equation as usual. By multipliying by -5 Sal is able to add both equations and to cancel out the y, as he explains at 3:30, too. But you are right, you could also mulitply by +5 and subtract the two equations afterwards."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "why did he divide 10 by fifteen at 4:40?",
        "A": "He divided 15 by -10 because the Y needs to be alone. As I like to think about it, you re basically signing the divorce papers. Y can t be commited in a realtionship, so you need to help him get away from any one he might be attached to."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "At 1:40 in this video, Sal asks us ( the people ) a question. Except for one thing, we ( the people) cannot simply answer back! So why bother asking us a question? Sal might as well just explain it, not make us talk to a computer.",
        "A": "He is not asking you to talk to the computer, he is asking you to think pro-actively which can help with the learning process. He is asking you to take a look and see if you can spot the next step before he tells you what it is, which he did do."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "how did you get 15/4 when multiplying 3(5/4) in 10:08?",
        "A": "1) Change 3 into fraction form = 3/1 2) To multiply fractions, multiply numerator to numerator and denominator to denominator 3/1 * 5/4 = (3*5) / (1 * 4) = 15/4 Seems like you need to review how to work with fractions. Hope this helps."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "At \"10:32\" how does he get 7x= 35/4 ? Wouldn't it be 7x=20/4 because 5+15/4=20/4.",
        "A": "Remember that when you re adding fractions, you must have the same denominator. Thus, 5 + 15/4 is not equal to 20/4. 5 + 15/4 = 20/4 + 15/4 = 35/4, which is what Sal had."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "At 4:40, I noticed that Sal multiplied 3 and -5 and got 15. Wouldn't it actually be -15?",
        "A": "There is a slight mark above the arrow next to the fifteen, meant to be a negative sign."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "6:32...how do you know whether 7 or 5 is the negative number?",
        "A": "You can do it either way... Sal multiplied the bottom equation by -5 and the top by 7....but he could have multiplied the top by -7 and the bottom by 5. The idea is to multiply by something that will make one of the variables go away .Try it both ways to convince yourself."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "Why does Khan multiply the bottom equation with -5 at 6:50? Why not +5 and then subtract the two equations? Is there a reason for the -5? For example :\n\n7(5x+7y)=(15)7\n5(7x-3y)=(5)5\n\nSo then expanding it:\n\n35x+49y=105\n35x-15y=25\n\nThen subtracting the two to get the answer?",
        "A": "this amounts to the same thing, so if you are happy doing it your way, go ahead. Either way, you have to be careful with signs."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "Doesn't 3 5/4 equal 17/4 not 15/4 like Sal came to?\n@10:04 in the video",
        "A": "you are getting two operations confused. 3 5/4 is 17/4 which is equivalent to 3 + 5/4. But 3(5/4) is multiplication, if we divide 3 by one, we get (3/1)(5/4) and to do this we multiply numerators and denominators together to get 15/4. Try not to confuse a mixed number such as you have with the multiplication of two numbers."
    },
    {
        "video_name": "wYrxKGt_bLg",
        "Q": "Towards the end of the video, around 10:00, he is multiplying 3 by 5/4. Would the answer to that be 17/4, not 15/4? 3*4=12+5=17",
        "A": "Not really. What is important to realize is that 3 x 5/4 is NOT 3 5/4. In the second case we have three wholes and one 5/4. That is what you were calculating. However, 3 x 5/4 means we have 5/4 three times. It s as if I said that I have 5 pies (quarters) times three. So you have 15 pies (quarters). In the first case we have three pies and 5/4 of a pie and so in total we have 17/4 of a pie :)"
    },
    {
        "video_name": "4EJay-6Bioo",
        "Q": "At 1:47, Sal says \"You might say this is crazy . . .\" Yes, I agree, I am completely lost. WHY, would anyone in their right minds use the hexadecimal system to make their life a whole lot harder?! WHY?!",
        "A": "Big numbers require a lot of digits, with the hexadecimal system those same big numbers will require less digits and will be easy to read (though not easy to comprehend), this is especially useful in programming, when working with memory or RGB colors for example."
    },
    {
        "video_name": "4EJay-6Bioo",
        "Q": "At 0:43 , what is sal referencing? What is used more, binary or hexidecimal?",
        "A": "binary is used in computers today and programs binary is used as a 1 or a 0 in codes or as on or off in other uses."
    },
    {
        "video_name": "4EJay-6Bioo",
        "Q": "at 6:28 for the second practice question AF3 I got 563 and I don't see what I did",
        "A": "AF3 (in 16-base) = 10*16^2 + 15*16^1 + 3*16^0 = 2803 (in 10-base)"
    },
    {
        "video_name": "4EJay-6Bioo",
        "Q": "I don't understand why we have to use letters at 1:41",
        "A": "We use letters for unknown numbers let say like: X = 3+9 which x is therefor 12"
    },
    {
        "video_name": "ssY1dFl7d30",
        "Q": "what exactly was the y at 0:16",
        "A": "Y is only used as a variable. It has no specific value, it is just used to represent a number. It has no exact value. So basically, you do not need to know the value of y to complete what the question is asking you to do."
    },
    {
        "video_name": "BveA7JiRGS8",
        "Q": "At 3:45, why do we add $8000 to calculate the expected medical costs with the low deductible plan, instead of $9000? Can anyone explain it to me more clearly?",
        "A": "$8,000 is your annual insurance bill (what you pay the insurance company for the benefits of the $1000 deductible). Think of it as your membership fee, totally independent of any actual medical costs."
    },
    {
        "video_name": "OjF765iVuF8",
        "Q": "3:10 How do you know that ABC+A^2 is defined? They could have diferent dimensions",
        "A": "We re told that A, B, and C are square matrices. We also have that ABC is defined. So A, B, and C must all be the same dimension (if they weren t, one of those multiplications would be undefined). When we multiply square matrices of equal dimension, we get another matrix of the same dimension. So ABC must be the same dimension as A, and A^2 must be the same dimension as A. So ABC and A^2 must have the same dimension."
    },
    {
        "video_name": "OjF765iVuF8",
        "Q": "How do I square a matrix? (At 2:26, A is squared.)",
        "A": "If A is an nxn matrix (eg 2x2, or 3x3, or 10x10), then matrix multiply A by A"
    },
    {
        "video_name": "-aTy1ED1m5I",
        "Q": "Seeing that the line @ 3:50 is dotted, do we do the shading just under the line or do we actually connect the shading to the line?",
        "A": "Technically it should be getting infinitesimally close to an infinitesimally thin boundary line. Most teachers will accept shading like Sal s where it gives the gist of which side is included. Just don t cross the boundary line in your shading."
    },
    {
        "video_name": "-aTy1ED1m5I",
        "Q": "At 2:15, why did Sal count \"1,2,0, and then -3\"?",
        "A": "I think he simply made a mistake! ;)"
    },
    {
        "video_name": "-aTy1ED1m5I",
        "Q": "at 3:39 how do you know where to shade?",
        "A": "With inequalities, you shade either everything above the line, or everything below the line. If it is a greater than symbol > , then you shade above the line. If it is a less than symbol < , then you shade below the line. I hope this helps."
    },
    {
        "video_name": "-aTy1ED1m5I",
        "Q": "At 4:36 do you need a boundry line?",
        "A": "No because that point was just an example in the shaded region to support the equation."
    },
    {
        "video_name": "-aTy1ED1m5I",
        "Q": "at 4:50 is that multiplication sign ?",
        "A": "Yes, that dot represents the multiplication sign. x is not often used in algebra so you don t get confused with the commonly used variable x."
    },
    {
        "video_name": "Jp25LHI9wII",
        "Q": "I have a comment. At 3:42 in the video, Sal says \"You'll say, well, Sal, how do you know that?\", but the closed captions say: \"You'll say, well, how do you know that?\" Leaving out a very important name...",
        "A": "That is a mistake caused by the robot that creates the closed captions for Youtube."
    },
    {
        "video_name": "Jp25LHI9wII",
        "Q": "in 1:05 do you always multiply the stuff in the expressions?",
        "A": "When you are talking about expressions, then yes the way to distribute something directly outside the parentheses to all the terms inside is by multiplication. Later on when you get to equations with this kind of expression on one side, you may find it easier to move the outside by opposites instead of multiplying, so in that case, you may want to multiply the whole expression by 2 to get rid of the 1/2."
    },
    {
        "video_name": "bFtjG45-Udk",
        "Q": "At 0:28, he says that (x+9)^2 isn't equal to x^2 + 9^2. Why can't you use the distributive property? Why do you have to expand it out? I'm CONFUSED and need help ASAP!",
        "A": "You are not distributing the 2, you are squaring (x + 9). In other words you are trying to find (x + 9) (x + 9). (x + 9) (x + 9) = x\u00c2\u00b2 + 9x + 9x + 9\u00c2\u00b2 = x\u00c2\u00b2 + 18x + 81"
    },
    {
        "video_name": "bFtjG45-Udk",
        "Q": "at 4:11 will it still be correct if you put x squared plus b squared plus 2bx",
        "A": "Are you asking if the following is true? x^2 + 2bx + b^2 = x^2 + b^2 + 2bx If that s the question, the answer is yes, the two expressions are equal. It s customary to write the expression in the first form, but reordering the terms would not change the meaning of the expression."
    },
    {
        "video_name": "bFtjG45-Udk",
        "Q": "The question around 8:35 is essentially a Difference of Two Squares isn't it?",
        "A": "That s exactly what it is!"
    },
    {
        "video_name": "bFtjG45-Udk",
        "Q": "I don't understand the step in orange at 6:38. How did he turn the (4x^2 + y^2) = (4x^2)^2 + 2(4x^2)(y^2) + (y^2)^2 ??",
        "A": "(4x^2+y^2)^2 = (4x^2+y^2)*(4x^2+y^2) = (4x^2+y^2)*4x^2 + (4x^2+y^2)*y^2 = 4x^2*4x^2 + y^2*4x^2 + 4x^2*y^2 + y^2*y^2 =(4x^2)^2 + 2(4x^2)(y^2) + (y^2)^2"
    },
    {
        "video_name": "bFtjG45-Udk",
        "Q": "Around 4:50, can someone explain what Sal is doing here? I really don't understand how he got (3x)^2+2(3x)(-7)+(-7)^2. Thanks.",
        "A": "Try rewriting the expression as (a-b)^2. No matter what a and b replace, the expression will always be equal to a^2+2ab+b^2. (This pattern is called the difference of squares.) You could also expand Sal s example to (3x-7)(3x-7) and then use FOIL (first/outside/inside/last) to multiply it out. This gives 9x^2 -21x-21x+49. Notice you have -21x two times. (Which equals -42x). This is why Sal has a 2 in 2(3x)(-7)."
    },
    {
        "video_name": "bFtjG45-Udk",
        "Q": "At 9:26 Sal squares A, or 2x why does he do this when it doesn't show the square symbol?",
        "A": "2x in the first binomal is multiplied by the 2x in the second binomial. If you know FOIL then it s the F step."
    },
    {
        "video_name": "bFtjG45-Udk",
        "Q": "At 9:30, why is (2x)^2 equal to 4x^2?",
        "A": "Because (2x)^2 is the same as 2^2 * x^2. We know 2 squared is 4, so that simplifies to 4x^2. (2x)^2 is actually 2x * 2x which is 4x^2."
    },
    {
        "video_name": "bFtjG45-Udk",
        "Q": "At 0:28, Sal says that (x+9)^2 doesn't equal x^2 + 9^2. Why can't you use the distributive property? Why do you have to expand the binomial?",
        "A": "You can use the distributive property, but you must distribute to all terms. So: (x+9)^2 = (x+9)(x+9) <expand the exponent> x(x+9) + 9(x+9) <distributive property> x^2 + 9x + 9x + 81 <distributive property again> x^2 + 18x + 81 <combine terms> So, using the distributive property does not give the answer x^2 + 9^2. The correct answer is x^2 + 18x + 81 . You must expand because that is the definition of the exponent. Make sense?"
    },
    {
        "video_name": "bFtjG45-Udk",
        "Q": "At 7:06 , Why couldn't Sal add the y^4 with the y^2?\n\nPlease explain Guys,",
        "A": "You can only add terms together that have the same degree in the exponent, that is the same value of the exponent. You could add terms that all have y^4, or all have y^2."
    },
    {
        "video_name": "bFtjG45-Udk",
        "Q": "At 6:19, why is it that -7*-7 makes a positive 49 but when it is -21+(-21) it is negative? I am still a little lost on signs.",
        "A": "So multiplication and addition/subtraction are two different areas. Let s take the second part first. -21+(-21) is saying take negative 21 and add negative 21 to it. So you would start at -21 and go 21 spots to the left resulting in -42. Now, the first part is saying take -7 and multiply it by -7. The signs flip in this respect because you are using multiplication. -7*-7 = 49."
    },
    {
        "video_name": "bFtjG45-Udk",
        "Q": "QUESTION! Can someone please explain at 06:55 why Sal doesn't multiply the y^2 by 2? If one expands out the expression 2(4x^2)(y^2), would the answer not become 8x^2 2y^2?",
        "A": "Nick, You re mixing the distributive property which involved addition and multiplication with something just involving multiplication. If you had 2(4x\u00c2\u00b2+y\u00c2\u00b2) you would use the distributive property to get 8x\u00c2\u00b2 + 2y\u00c2\u00b2 But when you have 2 * 4x\u00c2\u00b2 * y\u00c2\u00b2 you just multiply the2*4 to get 8 and you have 8x\u00c2\u00b2y\u00c2\u00b2 I hope that helps make it click for you."
    },
    {
        "video_name": "bFtjG45-Udk",
        "Q": "At 5:17, how do you complete the square with 9x^2 - 42x + 49?",
        "A": "what do you mean ? I think that s the final form of the whole thing"
    },
    {
        "video_name": "hl58vTCqVIY",
        "Q": "What is the meaning of run and rise? How do you know +-2/3x=rise 2 for every 3 you run? I mean how do you know which is x and y? Forgotten.:)(at 3:00)",
        "A": "this is going to sound really dumb but hey whatever works right? x marks the spot on the ground (horizontal axis) and y goes up like this: Y (vertical axis). if it helps put your hands over your head and make a y. we learned this in third grade when we started coordinate planes. and x comes before y in the coordinate set, since it is in alphabetical order."
    },
    {
        "video_name": "hl58vTCqVIY",
        "Q": "At 1:21, can't the y = \u00c2\u00b1\u00e2\u0088\u009a4/9x\u00c2\u00b2+4 be simplified to y = \u00c2\u00b12/3x+2?",
        "A": "No, it can t be simplified like that. If we could apply your technique, then these 2 numeric examples should be equal. Consider: \u00e2\u0088\u009a9 + \u00e2\u0088\u009a16 and \u00e2\u0088\u009a(9+16) \u00e2\u0088\u009a9 + \u00e2\u0088\u009a16 = 3 + 4 = 7 \u00e2\u0088\u009a(9+16) = \u00e2\u0088\u009a(25) = 5 Hopefully you can see these do not create the same result. You technique doesn t work because we can t take the square roots of the individual terms inside the radical. Hope this helps."
    },
    {
        "video_name": "hl58vTCqVIY",
        "Q": "My math teacher taught something about drawing a box and using the diagonals of that box as asymptotes, but you give this equation at about 1:30 and in the first video. Can I use either method? Are there differences?",
        "A": "I learned it that way too, and just tried out Sal s method to study for a quiz that I have soon. Basically, the box is a shortcut to find the asymptotes, since you can use the vertices to create the box. Sal s method is good for when you don t want to graph the hyperbola to find the asymptotes. Both ways are completely valid. (I know this is a late answer, but I figured better late than never)"
    },
    {
        "video_name": "hl58vTCqVIY",
        "Q": "This may be a silly question but I am just beginning to understand this....do the asymptotes always pass through the origin? (like at 3:23)",
        "A": "Just like a circle, a hyperbola can be shifted. A normal or unshifted hyperbola: x^2/a^2 - y^2/b^2 = 1 A shifted hyperbola: (x-h)^2/a^2 - (y-k)^2/b^2 = 1 where h and k specify the amount of horizontal and vertical shift respectively. In Sal s examples so far, h and k have effectively been zero, so the asymptotes have gone through the origin."
    },
    {
        "video_name": "hl58vTCqVIY",
        "Q": "Are the asymptotes completely accurate? (@ 2:38) We got rid of the 4 that was originally added to to 4/9x^2 because it was \"insignificant\". Wouldn't getting rid of the four cause the asymptotes and the rest of the problem to not be completely accurate?",
        "A": "Since the asymptotes end up being virtually infinite, a 4 added to them won t make a great difference. With the 4 we would have suck a hard time calculating the asymptotes, that we ignore it. So no, they are not accurate, but we accept them in any case."
    },
    {
        "video_name": "hl58vTCqVIY",
        "Q": "At time 4:50 - he says that points (0, 2) and (0, -2) are on the hyperbola graph. Are these the vertices of the graph? I am trying to find out how to get the vertices from the equation.",
        "A": "yup, those are the vertices. in a vertical hyperbola, the vertices are at (0, b^2) and (0, -b^2) in a horizontal hyperbola, the vertices are at (a^2, 0) and (-a^2,0)"
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "Does it not work for whole numbers?\n4:33",
        "A": "It can output whole numbers. It just happens that we got a number very close to a whole number here."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "At 3:42, 3:43, Sal mentions inverse sine. Is this the same as cosecant?? (the reciprocal of sine) ??",
        "A": "No. Inverse sine (also referred to as arcsin or sin^-1 is the going backwards and is only inverse in the literary sense. Cosecant is inverse sine in the mathematical sense. Inverse sine, when you put in the ratio of the two sides (SOH-CAH-TOA), outputs the angle. Sine, when you put in the angle, outputs the ratio of the two sides. Cosecant is 1/sin. I hope this helps!"
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "At 3:42, why do we take the inverse sine instead of dividing both sides by sine to get rid of the sine of the right side?",
        "A": "You can t divide the sines of two different angles. On scene 3:42, our angles are theta and 40 degrees. Although dividing by sin(theta) would remove the sine from the right side, you would only be left dividing the sine of 40 degrees and the sine of theta on the left side. However, you can use inverse sine and remove the sines that way, because the inverse sine of the sine of 40 degrees is 40 degrees and the inverse sine of the sine of theta is theta."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "what did he do at 3:55? i didn't understand it?",
        "A": "on one side he had 4/3 sin40=sintheta Now in this part we are solving for theta so he had to get theta by itself to do that we had to use the inverse sine function so now on the left side we have sin^-1(4/3sin40)=theta"
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "I got lost at what sal did at 3:48, where did the sin on the left side go?",
        "A": "In order to isolate the angle theta, Sal used arcsin (or the inverse function of sin ). 4/3 * sin(40) = sin(theta) sin^-1( 4/3 * sin(40)) = sin^-1(sin(theta)) 58.99 = theta What may have confused you was how he used arcsin (sin^-1) on both sides but didn t explicitly show it being done on the right hand side. The arcsin of sin essentially cancels out, because arcsin is the inverse function of sin."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "I'm confused, at 3:26 Sal said multiply both sides by 11, but he divided. Why did he do that?",
        "A": "I didn t see any mention of 11 in the video. Are you sure you posted this question in the right place?"
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "At 1:33, does he mean Law of Sines, or does he actually mean to say Law of Cosines?",
        "A": "I think he said the Law of Cosines correctly. Watch the video again."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "Why is it 4/3 at 3:32 and not 1200?",
        "A": "Remember: Sal was multiplying by 40, but he was dividing by 30. He was not multiplying by both. And so if we multiply by 40 and divide by 30, that can be simplified to 4/3."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "At 3:44, why did you take the inverse sine of both sides? Does the inverse sine always cancel out the sine?",
        "A": "Yes this is just another pair of opposites that you need to know like add/subtract, divide/multiply/ square/square root, etc."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "When he says \"data\" when solving for the angle (at 3:18,) is it the same as using the variable x to indicate the unknown angle?",
        "A": "Yes. Theta is just a Greek alphabet. People still use it because the old guys used it."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "at 3:26, how does 40[(sin40degrees)/30] become 4/3? or is it 40/30*sin40degrees/30?",
        "A": "The number 10 divides evenly into both 40 (numerator) and 30 (denominator). He is simply expressing the fraction 40/30 in simpler terms: 4/3, just like you might do with 4 over 8 by saying that 4/8 is equal to 1/2."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "At 1:35 you said that the law of cosines wouldn't help you, but you have one angle and two sides? So why can't you use the law of cosines?",
        "A": "You could use the law of cosines twice. First you would solve 30\u00c2\u00b2 = 40\u00c2\u00b2 + x\u00c2\u00b2 - 2 * 40 * x * cos(40\u00c2\u00b0) and then you would solve x\u00c2\u00b2 = 40\u00c2\u00b2 + 30\u00c2\u00b2 - 2 * 40 * 30 * cos(\u00ce\u00b1). However, the law of sines is a better option in this case."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "At 3:27, Sia multiples sin40/30 by 40 and ends up with 4/3 * sin40. How did he end up with that answer, because I am not able to get that answer, no matter how hard I try",
        "A": "To make it easier to see, let s make the following substitution: \u00f0\u009d\u0091\u00a5 = sin 40\u00cb\u009a If we make this substitution, we are left with: 40\u00f0\u009d\u0091\u00a5/30 This simplifies to: 4\u00f0\u009d\u0091\u00a5/3 We can rewrite this as \u00f0\u009d\u0091\u00a5 times the fraction 4/3: (4/3)\u00f0\u009d\u0091\u00a5 Substituting back, we are left with: (4/3)sin 40\u00cb\u009a Comment if you have questions."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "@3:19 Can someone break down why exactly we multiply both sides times 40 and why this equals 4/3? Thanks!",
        "A": "We would like to solve for theta, and in order to do that, we have to multiply by 40 to get rid of the denominator. The denominator of the other fraction on the left side is 30, so 40 * (sin 40/30) = 4/3 sin 40"
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "Why does Sal use inverse sine, or sine^-1 when evaluating 4/3*sin 40 = sin theta at 3:45?",
        "A": "1. Sin(a) \u00c3\u00b7 A = sin(b) \u00c3\u00b7 B 2. Sin(a) = sin(b) \u00c3\u00b7 B * A 3. a = arcsin( sin(b) \u00c3\u00b7 B * A) Since the problem is asking for a specific angle, you have to pull the arcsin to get the specfic angle in this equation."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "Did Sal accidentally say law of cosines instead of law of sines? At 2:50 in the video.",
        "A": "Yes. That is why that little box pops up that says Sal said Law of Cosines but meant Law of Sines."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "Okay at 3:24, I know he's multiplying to cancel out the 40, but why did he cancel that number in particular? Would it work the same if he canceled out the 30 and multiplied the 40 by that instead?",
        "A": "Sal needs to solve for theta. So, he s trying to get sin(theta) by itself. This is why he chose to move the 40. If you moved the 30 instead, you are just putting more stuff on the same side as the theta. So, it doesn t really help you to isolate theta. hope this helps."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "At 2:52 , it is being said that the Law of Cosines could be used to figure out the angle theta , but the one used was Law of Sines. Is that a voice-over error or am I getting it all wrong?",
        "A": "It s a voice error, there s a little edit cited."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "At 2:51, does he really mean Law of Cosines or does he mean something else?",
        "A": "I think he meant to say Law of Sines there."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "On my calculator I put in exactly the same thing as Sal did at around 4:10. Why did I get 5.3367.... instead of 58.9869....? I chose the inverse of sine and everything. On my calculator it says asin when I choose sin-1 but I think that it's the same thing. There's got to be something I'm missing. Is Sal's calculator set to round the answer automatically or something?",
        "A": "Yeah, make sure your calculator is set to Degrees."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "what is theta @1:01",
        "A": "Theta is a Greek letter we use as a variable in geometry and trigonometry."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "(3:43) So when we are finding the angle with use inverse cosine?",
        "A": "Since this example covers an example of the law of sines(in this case of course), it requires solving for sin(theta) and using the inverse sine command to solve for theta."
    },
    {
        "video_name": "IJySBMtFlnQ",
        "Q": "Does it matter that much at 2:47 whether a capital \"A\" is used or a lowercase \"a\" is used in the low of sines.\n\nExample; sinA/a= sinB/b=sinC/c if the triangle angle are the capital 'A\"'s",
        "A": "no as long as the angle of sine is being divided by the length of the corresponding side"
    },
    {
        "video_name": "VhH2nEDCd68",
        "Q": "At 0.58 and 1:48, how did Sal immediately determine the slope in the equation?",
        "A": "The equation is in the slope-intercept form y=mx + b. m is the slope of the equation while b is the y-intercept"
    },
    {
        "video_name": "VhH2nEDCd68",
        "Q": "so on 2:05 he can clearly see that the intercept are whole numbers, but what if the lines intercept when half way in between? I'm having difficulties with understanding this.",
        "A": "It s certainly possible for the solution of two equations to be fractional, but that sort of problem would be a lot more difficult to solve with graphing. You would solve it with purely algebraic methods, like substitution."
    },
    {
        "video_name": "VhH2nEDCd68",
        "Q": "What does he mean at 2:00?",
        "A": "He is talking about the intersection point, where the two lines meet and cross paths. This point is (5, 2) and that means the solution is (5, 2)."
    },
    {
        "video_name": "Llt-KkHugRQ",
        "Q": "In 8:56\n\nSal says that he is going to do more examples of these type of problems :\nWhere are the examples ?",
        "A": "Most likely in the next video he does"
    },
    {
        "video_name": "6nUMiJfHLSA",
        "Q": "At 0:02 or 0:03, what does a -270\u00c2\u00b0 turn mean?",
        "A": "A -270 degree turn is a 270 degree angle below the +x axis. Meaning it goes clockwise from the positive part of the x axis 270 degrees. Clockwise is negative, counterclockwise is positive. Hope this helped :) Happy holidays!!"
    },
    {
        "video_name": "6nUMiJfHLSA",
        "Q": "At 4:35, Sal calls his new point, I \"prime\". Where does the term, Prime, come from, and what does it mean (specifically in this situation, I know what prime usually means).",
        "A": "Prime is basically used to denote a new point that has been shifted or modified from its original in some way (translation, rotation or reflection)"
    },
    {
        "video_name": "6nUMiJfHLSA",
        "Q": "Why, at 6:55, does Sal draw the orange line relating to P to the left, creating a right angle with the green line? Would it not still be a right angle if he had drawn it to the right? Why specifically left, and how would I know it would be left in the future?",
        "A": "This depends on the direction of rotation whether it s clock-wise or counter clock- wise. For this problem it s -270, which means that all points on the triangle should move -270 dehrees of rotation in clock-wise direction or 90 degrees counter clock-wise direction, and why 90!! it s because 90 has the same places for all points on the triangle as -270. Hope this heolps!"
    },
    {
        "video_name": "6nUMiJfHLSA",
        "Q": "on 6:00 sal wrote N' does the apostrophy mean prime",
        "A": "Yes. It denotes the image of N after the transformation."
    },
    {
        "video_name": "lkP-E2LUnjA",
        "Q": "At 2:46,i don't get the formula that he wrote\nalso, where did the 1/f come from??\nI also don't get how he used the formula to solve the problem, I sound confused but I just got lost.",
        "A": "The problem varies inversely, so the formula is x=k times 1/f. If you need help, there are other videos in this section."
    },
    {
        "video_name": "pJzmiywagfY",
        "Q": "At 8:26, Sal said that their are multiple vectors that are orthogonal to vector a and b. I thought there were only two vectors that were orthogonal to a vector. One that goes straight up, and one that goes straight down. What did he mean by multiple vectors?",
        "A": "You are right. They are all in only one of two directions. However, they can have an infinite number of magnitudes. : )"
    },
    {
        "video_name": "pJzmiywagfY",
        "Q": "At 1:14, Sal says that the cross product is only defined in R3. But isn't it also defined in R7?",
        "A": "Kind of. There is a definition for a seven-dimensional cross product that has some (but not all) of the properties of the three-dimensional cross product. It s beyond the introductory level that Sal is staying at in this playlist, but you can check out Wikipedia or StackExchange for more details."
    },
    {
        "video_name": "pJzmiywagfY",
        "Q": "at the beginning of the video, 0:36, he says that you can basically use the dot product method anytime, so why use the cross product method at all?",
        "A": "The dot product of two vectors is a scalar. The cross product of two vectors is another vector. The two operations do very different things."
    },
    {
        "video_name": "pJzmiywagfY",
        "Q": "At 12:45, what if its parallel?",
        "A": "If a and b are parallel, axb is 0."
    },
    {
        "video_name": "pJzmiywagfY",
        "Q": "Between 1:10 and 1:30 (dont wanna go back to check) he says the cross-product is orthogonal. What does that mean?",
        "A": "If we have vectors u and v, then u cross v is at a right angle to both u and v. Orthogonal means perpendicular, or at a right angle to."
    },
    {
        "video_name": "pJzmiywagfY",
        "Q": "At 15:30, Sal says we could have come up with the definition of cross product on our own, yet says the definition he gave also has other properties. Can we come up with that definition on our own if we are only given the condition that we want our vector to be orthogonal to a and b?\nI tried setting up x * a = 0 and x * b = 0 and solving for x but I don't know if I'm on the right track.",
        "A": "yes u can if u start muktiplying each term in the traditional old and complicated way u would notice that the vectors cum out 2 b perpendicular to each other and hence cancel out its long but it is possible"
    },
    {
        "video_name": "vAepSNDLZRM",
        "Q": "Where can I find the \"next video\" with another method for solving nonhomogenous d.e. mentioned in 5:47 ?",
        "A": "That would be the set of videos on Laplace Transforms. Check them out under the Differential Equations topic page."
    },
    {
        "video_name": "qoMimDT7D8w",
        "Q": "06:38 x=4 should it be removable discontinuity?",
        "A": "The vertical asymptote(s) can only be found once the equation is as simplified as possible. Removable discontinuities are found as part of the simplification process. If a factor like x=4 appears in both steps the vertical asymptote label is the stronger since it produces a vertical asymptote when graphed as Sal shows."
    },
    {
        "video_name": "qoMimDT7D8w",
        "Q": "At 2:13, why do we have constrain x is not equal to -4 to make it algebraically equivalent?",
        "A": "Before cancelling (x+4), there is division by zero, which is not defined in mathematics, at x= -4, so it s understood that the domain does not include x = -4. To keep the function the same, the domain must be the same after cancelling. So the note x != 4 is included, because just (x-6)/(x+6) by itself doesn t tell us that."
    },
    {
        "video_name": "6cMn1F6bNPU",
        "Q": "At 2:34, should the n term have a squared 'denominator' and also in the sum notation, is the squared missing?",
        "A": "Yes. You can infer this from the fact that $\frac{\\partial^2 f}{\\partial x_N}$ for N = 1 is an element of the series, explicitly being $\frac{\\partial^2 f}{\\partial x_1}$."
    },
    {
        "video_name": "VIVIegSt81k",
        "Q": "At 3:18 she says making trihexaflexagons are easy.\nut they're not exactaly easy! :)\ndo you find it easy, medium or hard?",
        "A": "Yhea I agree, I tried and failed but when I do make one it will make everyone (almost everyone) Jelly"
    },
    {
        "video_name": "VIVIegSt81k",
        "Q": "3:47, where did you get or how did you make those card structures?",
        "A": "With scissors and stencils"
    },
    {
        "video_name": "VIVIegSt81k",
        "Q": "Why did it turn white at 2:26?",
        "A": "Because he/she flipped it over to another side that is white"
    },
    {
        "video_name": "VIVIegSt81k",
        "Q": "From 2:05-3:03, how do you make that 6-flapped hexaflexagon? Since it's in fast-motion I couldn't see the steps very well. And once it got to the diagramming and such, it was still confusing.",
        "A": "Click on the gear at the bottom right of the video (the settings button), it will be near below the red line (the progress you made in the video). Then click speed , and you will see the speed is set to normal . Click 0.25 speed or 0.5 speed, and it will make your video go slower. This way the steps will be a little slower."
    },
    {
        "video_name": "VIVIegSt81k",
        "Q": "10:21 Can anyone tell me why it works",
        "A": "It works because it is a mobius strip, as said in other places in the questions"
    },
    {
        "video_name": "VIVIegSt81k",
        "Q": "what's wrong with my hexaflexagon if i can't do what she did at 1:12?",
        "A": "It takes practice, I find it easier to imagine it like folding a try-fold brochure, adjust and all that. Don t use the original edge of the paper as a guide, though."
    },
    {
        "video_name": "EuwKjpUpkhs",
        "Q": "At 4:33, where can you get a graphing calculator?",
        "A": "DuckDuck online graphic calculator . I got one into a browser tab that way."
    },
    {
        "video_name": "jCkhbKFZgLk",
        "Q": "What is he taking about at 2:50?",
        "A": "he is talking about unit vector whose magnitude isalways unity."
    },
    {
        "video_name": "jCkhbKFZgLk",
        "Q": "Is the u that he wrote in the bideo a special way to write a vector titled u?\nTime: 2:50",
        "A": "What the others said is true, it is basic for unit vector w/ magnitude of 1, although in statistics it is called mu, & represents the normal mean of a population if I remember correctly."
    },
    {
        "video_name": "JAdNNJynWM4",
        "Q": "How the 2.81 - 11.27 + 15.9 +12 - 12 Came here, I can't understand it! Anyone help me with that! Video: 1:41Min",
        "A": "The wording in your question is very confusing. You used decimal points. Sal has multiplication. Use the correct symbols to do the correct math. A raised dot is multiplication, not a decimal point. 2 * 81 - 11 *27 + 15 *9 + 12 - 12 Follow PEMDAS to do the math. If you are asking how Sal created: 2 * 81 - 11 *27 + 15 *9 + 12 - 12 It comes from 2x^4-11x^3+15x^2+4x-12 and using x = 3 2(3)^4 - 11(3)^3 + 15(3)^2 + 4(3) - 12 After calculating the exponents, you get: 2 * 81 - 11 *27 + 15 *9 + 12 - 12 Hope this helps."
    },
    {
        "video_name": "JAdNNJynWM4",
        "Q": "at 0:30, what is the symbol he uses for \"if and only if?\" I've seen it written as iff, but I've not see that symbol before.",
        "A": "The double implication: a<=>b iff ( a=>b and b=>a ). ( => means implies .) a if b means b implies a , or b only if a . a only if b means a implies b , or b if a ."
    },
    {
        "video_name": "VpuN8vCQ--M",
        "Q": "just a quick question, at 4:01, in the example. there are 100 fair coins being flipped and the expected value is 50. However, it says in the video that as n approaches infinity. Shouldn't it be as n approaches 100?",
        "A": "n is number of times you flip 100 coins . You misunderstood that part of variable assignment."
    },
    {
        "video_name": "dC1ojsMi1yU",
        "Q": "At 2:20, how come you can't simplify it saying 21 to the power of -96",
        "A": "You can only combine exponents if the bases match. 7 does not = 3. So, we have different bases. If the problem has been: 7^16 * 7^(-6), then you could add the exponents and get: 7^10 Also, multiplying 7*3 before dealing with the exponents violate PEMDAS rules. You have to complete the exponents before multiplying. Hope this helps."
    },
    {
        "video_name": "dC1ojsMi1yU",
        "Q": "Hey guys I have a questions about the answer at 2:01, which is 3^16/7^6, and the answer at 3:27, which is a^(-4)*8^14. For the first one, Sal initially changes/simplifies 3^16*7^(-6) to 3^16/7^6 but for the second one he stops at a^(-4)*8^14; however, isn't it true that a^(-4)*8^14 could be changed/simplified to 8^14/a^4? Essentially I'm wondering why he did that with the first one and not the second.\n\nThanks in advance for the help!",
        "A": "I m not sure why he stopped at: a^(-4)*8^14. It is preferred for answers to have positive exponents. So, your version: 8^14/a^4 is better than Sal s and is more complete."
    },
    {
        "video_name": "dC1ojsMi1yU",
        "Q": "at 1:58 sal says you can't simplify it anymore but you can simplify it more to 21^10",
        "A": "No you can t. In order to combine the exponents, you need a common base. 3 does not = 7. There is no common base. If the problem had been 3^16 * 3^(-6), it can become 3^10. And, your version violates PEMDAS rules. You multiplied before computing the exponents. Hope this helps."
    },
    {
        "video_name": "dC1ojsMi1yU",
        "Q": "At 2:33, in the top right corner, shouldn't it be 3^-8 not 3^8?",
        "A": "3^-8 is the same thing as 1/3^8 If you want to get rid of the negative power, you have to take the reciprocal and keep the absolute value of the power, if that makes sense?"
    },
    {
        "video_name": "dC1ojsMi1yU",
        "Q": "At 4:53 in the video why is four squared raised to the seventh power equal to four to the seventeenth power?",
        "A": "It was a mistake. Sal corrects himself at around 5:26"
    },
    {
        "video_name": "dC1ojsMi1yU",
        "Q": "Wouldn't 6 be negative in (2:34 minute place)",
        "A": "A negative exponent means to move the expression to the other part of the fraction. So when 7^(-6) moved to the denominator, the negative sign was dropped, leaving only 7^6. If we had started with 5 / 8^(-4), we could re-write it as 5 * 8^4."
    },
    {
        "video_name": "dC1ojsMi1yU",
        "Q": "At 6:00, The 2^-70/4^14 can be simplified as 2^-70/2^28 and now the 28th power can be subtracted from the -70th power cuz those have the same base but what if we have different bases? like 2^-x/3^x, can we simplify this even further?",
        "A": "You can simplify just a little bit. Because the bases are different, we re limited in what we can do. 1) Positive exponents are preferred over negative exponents. Change the negative exponent to a positive: 1 / [2^x * 3^x] 2) Because the exponents match (and only because the exponents match), we can take a further step: 1 / [2*3]^x 3) Then, we can multiply 2*3: 1 / 6^x Hope this helps."
    },
    {
        "video_name": "dC1ojsMi1yU",
        "Q": "At 1:50, how did Sal convert the number 3^16 x 7^-6 to its fraction form?",
        "A": "As he says at 1:45 Seven to the negative six is the same thing 1:47 as one over seven to the sixth, 1:49 so you could write it like three to the 16th. Generally a^(-b) = 1/(a^b) - provided a \u00e2\u0089\u00a0 0"
    },
    {
        "video_name": "dC1ojsMi1yU",
        "Q": "At 2:02, I don't get why he puts 3^16 and 7^6 as a fraction.",
        "A": "As you know Sal was trying to multiply the two integers. When we have a negative power, like 7^-6, when multiplying it by another number (3^16) we are in fact dividing it. We know 7^-6 is, in fact, 1/7^6, which is why we can write 3^16/7^6"
    },
    {
        "video_name": "dC1ojsMi1yU",
        "Q": "why is 2/2 =2 (This can be seen at 6:23)",
        "A": "You are completely ignoring the exponents at this point, it is 2^-70/2^28. That would be 1/2 multiplied 70 times divided by 2 multiplied 28 times."
    },
    {
        "video_name": "dC1ojsMi1yU",
        "Q": "At 2:02 is the 6 suppose to be negative or positive",
        "A": "The exponent on the 7 changes from a negative exponent to a positive exponent when you move it to the denominator. Remember, a negative exponent tells you to use the reciprocal to change to a positive exponent: 7^(-6) = 1/7^6 3^16 * 1/7^6 = 3^16/7^6 Hope this helps."
    },
    {
        "video_name": "dC1ojsMi1yU",
        "Q": "at 4:54 4^17 power is a mistake right? wait just watched farther into the vid. XD",
        "A": "Sal corrects his error at about 5:25 in the video."
    },
    {
        "video_name": "8_fT1BsTmt4",
        "Q": "I don't understand what is he doing in 0:53. Can someone explain me?",
        "A": "What he is doing is he is putting the number in expanded form."
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "at 6:30, why is the absolute value of z = rcos(phi) + rsin(phi), shouldn't we use pythagoras theorem (as done in the previous video)",
        "A": "That s not absolute value, that s value(or representation) of vector z. Here, r is the absolute value of z."
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "At 9:22, how can we tell that sqrt(7)/2 will be in the first quadrant?",
        "A": "The magnitude of z is \u00e2\u0088\u009a(7)/2. z itself is the complex number \u00e2\u0088\u009a(3)/2+i, and that s what lies in the first quadrant. We know this because both \u00e2\u0088\u009a(3)/2 and 1 (the coefficient of i) are positive, and the first quadrant is the quadrant where both coordinates are positive."
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "At 4:01, what symbol does Sal use to represent the argument?",
        "A": "\u00cf\u0086 If u mean this symbol it is the greek symbol Phi"
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "At 2:50, why did he name the angle \"phi\"? (This thing: \u00cf\u0086 )",
        "A": "He could have named it anything, theta, beta etc. Having said that, over time you will see that there are certain conventions that have been adopted, like f or g or h for function names, angle theta for trigonometry, etc. When dealing with complex numbers in polar form, phi is traditionally used."
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "at 12:10, what's the triangle he mentions, please? I can't hear clearly the numbers.",
        "A": "30-60-90 triangle"
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "8:54 could you explain how he got the 4?",
        "A": "He squared \u00e2\u0088\u009a3/2. \u00e2\u0088\u009a3/2 * \u00e2\u0088\u009a3/2 = 3/4"
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "At 9:19 (\u00e2\u0088\u009a3)/2 + i changes to \u00e2\u0088\u009a(3/4+4/4) . How i became 4/4 ?",
        "A": "Sal is moving everything inside the square root. And, he is looking for the absolute value. Remember: i = \u00e2\u0088\u009a(-1). The absolute value would force this to be sqrt(1) 2 = \u00e2\u0088\u009a(4) Thus (\u00e2\u0088\u009a3)/2 + i changes to \u00e2\u0088\u009a(3/4+1) To add fractions, you must have a common denominator. So, the 1 gets multiplied with 4/4 to create: \u00e2\u0088\u009a(3/4+4/4) Hope this helps."
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "Why is it that at 8:56 i^2 is written as +1? I thought that i^2 = -1",
        "A": "Sal is not taking the square of i. The number with which i is being multiplied is scaled. z = a+bi |z| = sqrt(a^2 + b^2) In this example, z = sqrt(3)/2 + i Here a = sqrt(3)/2 b = 1 |z| = sqrt()"
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "around the 9:00 mark...finding the magnitude of \u00e2\u0088\u009a3/2+i. Why would it be (\u00e2\u0088\u009a3/4 + 1)? If i\u00c2\u00b2 = -1, shouldn't it be (\u00e2\u0088\u009a3/4 - 1)?",
        "A": "For a complex number z = a + bi, the magnitude is given by \u00e2\u0088\u009a(a\u00c2\u00b2 + b\u00c2\u00b2). For \u00e2\u0088\u009a3/2+i, b = 1 (NOT i)"
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "at 3:17 how are r and z different? aren't they both the magnitude/length?",
        "A": "No, z without the abs value lines is the vector on the Argand diagram that represents the complex number, therefore z has both a magnitude and direction"
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "At \"6:38\", what is the name of the formula? I can't quite hear it.",
        "A": "Thanks v. much :)"
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "Why doesn't (i/2)^2 = -1/4? (10:12)",
        "A": "It is not (i/2)^2, but (1/2)^2. When Sal is performing that operation, he is trying to find r using a and b. Earlier, he derived a formula stating r = ( (a)^2 + (b)^2 )^(1/2); he was using this formula, and it doesn t involve i."
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "why must Z1 have an absolute value at 3:28 when he explains that the magnitude is denoted by abs(z1)?",
        "A": "The magnitude of a number is it s distance from 0. For example, the magnitude of the number 4 is 4, but the magnitude of -4 is also 4. Thus, the magnitude of any number is equal to its absolute value."
    },
    {
        "video_name": "FwuPXchH2rA",
        "Q": "At 12:30, Sal states that imaginary numbers in exponential form should be measured in radians. Why?",
        "A": "Degrees are a contrived unit, radians are not. From this point on, you will be usually dealing with radians not degrees. When you move on in math, you won t be having degrees at all, just radians. The reason for considering an exponent with an imaginary unit as an angle is because of this relationship: e^(i*x) = cos x + i sin x (this only works if x is in radians)"
    },
    {
        "video_name": "c02vjunQsJM",
        "Q": "From 3:59, is this solution called the Percentile Range??",
        "A": "Yes it is called percentile range"
    },
    {
        "video_name": "c02vjunQsJM",
        "Q": "Why did Sal add 8+4+3 together in the video at 3:33?",
        "A": "He wanted to show how many total pies had sixty or more cherries. As you can see from his histogram, the first five pies do not have sixty cherries. So, he only adds the 8 pies, 4 pies, and 3 pies that do have more than sixty cherries to find the total number of pies that have sixty or more cherries. I hope this answers your question!"
    },
    {
        "video_name": "c02vjunQsJM",
        "Q": "at 2:55 what is nuance",
        "A": "At 2:55 in a nuance question he means a more subtle and refined question."
    },
    {
        "video_name": "HXIj16mjfgk",
        "Q": "At 0:27, isn't 1 a common factor for all 4 terms?",
        "A": "(x+3)^-1/5+(x+3)^-6/5"
    },
    {
        "video_name": "HXIj16mjfgk",
        "Q": "At 0:53, the first two terms have a common factor. What if you don't in those two terms or at all have a common factor in the polynomial?",
        "A": "Then you cannot factorize the expression in this way. You may have to find if there is any other way to factorize it, or no any way at all."
    },
    {
        "video_name": "8Ld8Csu4sEs",
        "Q": "at about 3:30 he says sas, what does that stand for?",
        "A": "At 3:30, while writing SAS, he says Side, Angle, Side."
    },
    {
        "video_name": "8Ld8Csu4sEs",
        "Q": "at 1:40 Sall uses AAA. i learnt u only need AA. Am i a;lso right?",
        "A": "if you have AA you can easily get the third A too. So you basically only need AA."
    },
    {
        "video_name": "8Ld8Csu4sEs",
        "Q": "So from what I'm hear at 2:46, anything that is congruent is also similar but anything that is similar is not necessarily congruent. Is that correct?",
        "A": "Congruent is that it has the same things(measurement and angles). Similar is the same angles and both of the same shapes have proportional side measures. So yes, unless both triangle(or any other shape) is the same, it is congruent."
    },
    {
        "video_name": "8Ld8Csu4sEs",
        "Q": "at around 11:30 i was almost positive that SSA would allow the two triangles to be congruent. Can it ever be proven that a triangle is congruent to it's partner of the time using SSA?",
        "A": "No. The reason is that in some cases there are two possible lengths the third side of triangles could be."
    },
    {
        "video_name": "8Ld8Csu4sEs",
        "Q": "(0:11) the SSS stands for what?",
        "A": "SSS (Side-Side-Side) shows that if on two triangles, all the sides are congruent or the same length, the two triangles are congruent."
    },
    {
        "video_name": "8Ld8Csu4sEs",
        "Q": "At around 3:10, what if the big triangle and the little triangle both had all three tick marks on all three sides and had the angle domes on both? Are they then both congruent?",
        "A": "yes the triangles would be congruent by the marks but you have to look at the size of the triangle."
    },
    {
        "video_name": "8Ld8Csu4sEs",
        "Q": "SSA doesn't make sense to me. 13:15 he is trying to tell us that the triangles have the same angle but the one on the right looks like 90degrees and the one on the left looks like 80 degrees. I don't see how SSA could be anything but congruent.",
        "A": "Imagine a triangle has lengths 6 and 3 and an angle of 40 degrees (not included). You could pivot the 3 side until it was another triangle."
    },
    {
        "video_name": "8Ld8Csu4sEs",
        "Q": "At 10:21, what's the difference between ASA and AAS? Do you just pick whichever we want and that's it?",
        "A": "The difference is whether the side is between the two angles (ASA) or not (AAS). That s important. If two triangles have two congruent pairs of angles and one congruent pairs of sides, but in one triangle it s arranged like ASA and in the other it s like AAS, then the two triangles are not congruent."
    },
    {
        "video_name": "v-k5L0BPOmc",
        "Q": "At 3:33, Sal says that Area is joint variation. But, wouldn't it be direct variation, and something like volume is joint?",
        "A": "The idea is that the total area is dependent on two factors, length and width. The function is so simple that it looks like direct variation; if k was not 1 (as it is here) it would be more obvious that there is more going on."
    },
    {
        "video_name": "SLyM_-sQwMM",
        "Q": "1:40 1) I dont get why the change in x is 1 unit to the right, why couldnt it be one unit up.\n2) why is it called CHANGE in x/y nothing is changing.\n3) distance is just the radius right?\nThanks",
        "A": "1. because along x axis, we move in left and right direction and along y-axis we can move up and down"
    },
    {
        "video_name": "B0Z4s38YIgQ",
        "Q": "why is he multiplying by f(f+10) in 4:00 ?",
        "A": "Once he gets an equation, he uses algebra to solve for f. He wants to get f out of the denominator so he can isolate it."
    },
    {
        "video_name": "B0Z4s38YIgQ",
        "Q": "At 3:30, can't you just multiply by 60f to solve the equation? I'm confused :(",
        "A": "well,you can multiply by 60f and then you ll have: 60+(60f)/(f+10)=5f. The problem is that by multiplying by 60f you only get rid of the denominator of the first fraction(1/f).To solve it properly you need to get rid of all the denominators on the left side.To do this,you must multiply by the least common multiple and because we don t know the value of f we take it as f(f+10) as if the numbers were prime to each other."
    },
    {
        "video_name": "B0Z4s38YIgQ",
        "Q": "At 3:35, why did you multiply both sides of the equation? it just sounded random to me.",
        "A": "Because, whatever you do one side of an equation, you have to do to the other side. It is just basic algebra principles."
    },
    {
        "video_name": "B0Z4s38YIgQ",
        "Q": "3:37 I don't understand why Sal must multiply everything by f(f+10), and how he got the answer.",
        "A": "The reason you want f(f+10) and not just f or (f+10) is because f(f+10) is a common factor of both f and f+10. That way, when you do f(f+10) [1/(f) + 1/(f+10)] = 1/12, you can simplify the fraction into f+10 + f = 1/12, which will give you 2f + 10 = 1/12. Then you can solve for f."
    },
    {
        "video_name": "WAoaBTWKLoI",
        "Q": "At 1:42, would it be okay to say that cos \u00ce\u00b8 = sin 90-\u00ce\u00b8? So if x/3 = sin \u00ce\u00b8, then u = 90 - x/3, right?",
        "A": "Did you mean cos\u00ce\u00b8 = sin(90 - \u00ce\u00b8) ?"
    },
    {
        "video_name": "WAoaBTWKLoI",
        "Q": "At 2:21, Sal says that u=cos(theta)=sqrt(1-sin^2(theta)). I thought that sqrt(1-sin^2(theta)) = cos^2(theta)",
        "A": "You re forgetting to take the square root. As you know, sin\u00c2\u00b2\u00ce\u0098 + cos\u00c2\u00b2\u00ce\u0098 = 1. Then cos\u00c2\u00b2\u00ce\u0098 = 1 - sin\u00c2\u00b2\u00ce\u0098. Now take the square root of both sides of the equation to get Sal s result."
    },
    {
        "video_name": "P1DJxuG7U9A",
        "Q": "At 4:20 he starts to talk about how we can replace the original function for x!=5 with the new function, but why would we do that? Don't we need the original function and just to replace the x=5 function? If we replace the function for x!=5 then wouldn't the function be indeterminate at places the old one was not? Or, is this a special case where it works, does this work for all cases?",
        "A": "Yeah it works in all cases. Since the new function is equal to the old function. The new function is just simplified."
    },
    {
        "video_name": "P1DJxuG7U9A",
        "Q": "at 3:06 when u start simplifying the equation cant we apply limit since the beginning instead of applying it later? like i solved it by applying the limit while simplifying.",
        "A": "You could just solve for 5 as soon as you get the (x-5) out of the denominator. That s what he did in the video eventually."
    },
    {
        "video_name": "P1DJxuG7U9A",
        "Q": "At 4:07, why don't the (x-5) terms just cancel?",
        "A": "It does, and that s what he writes. x is not allowed to be 5 in the original equation, so that constraint must be kept after cancelling the x-5 term."
    },
    {
        "video_name": "QuZMXVJNLCo",
        "Q": "At 5:33, Sal stumbles over the plural for hypotenuse. Is there an official plural term?",
        "A": "It is actually hypotenuses."
    },
    {
        "video_name": "QuZMXVJNLCo",
        "Q": "How do angles have points of view at 4:49?",
        "A": "Sal means the line BC/BA, not the angle BC/BA"
    },
    {
        "video_name": "QuZMXVJNLCo",
        "Q": "@ 6:23 what is theta?",
        "A": "Theta is a variable for an angle."
    },
    {
        "video_name": "QuZMXVJNLCo",
        "Q": "At 5:28, Sal says he doesn't know the plural of hypotenuse. What is the plural of hypotenuse?",
        "A": "here is the plural form: hypotenuses"
    },
    {
        "video_name": "QuZMXVJNLCo",
        "Q": "at 6:45 what is an 'angle theta'?",
        "A": "unless you skipped the other video, Sal covered it before. Anyway, theta is simply a Greek letter commonly associated with an angle. Also, if you were wondering, you are usually not asked to find the value of angle theta."
    },
    {
        "video_name": "QuZMXVJNLCo",
        "Q": "How did Sal determine right away at 5:18 the adjacent/hypotenuse just by looking at the angle location?",
        "A": "About hypotenuse: You should be able to determine which side is hypotenuse right away by look into a right triangle. It is face to the right angle. For the other 2 angles, you have to know which one you refer to. Adjacent is the side of that angle (but not the hypotenuse). Opposite is the side that face to that angle."
    },
    {
        "video_name": "QuZMXVJNLCo",
        "Q": "At 4:31, shouldn't the side DF technically be FD?",
        "A": "No. Imagine the vertices were called Blue and Green - there is no natural order so BlueGreen is the same side as GreenBlue and FD is the same as DF"
    },
    {
        "video_name": "QuZMXVJNLCo",
        "Q": "At 4:50, Sal was talking about theta, What is theta? Is it the same as a degree sign or is it different?",
        "A": "Theta is a variable, like x or n, that is traditionally used to stand for an unknown degree measure of an angle."
    },
    {
        "video_name": "QuZMXVJNLCo",
        "Q": "Does the order matter when labeling sides of a triangle? Ex: At 4:00, Sal labels it as BC/CA, but at 4:10, he labels it as EF/DF, instead of EF/FD.",
        "A": "No, it doesn t matter. You should, however keep things consistent. For example, you shouldn t write AB at one point and at a later point write BA."
    },
    {
        "video_name": "QuZMXVJNLCo",
        "Q": "Why is Sal inconsistent in his naming of the triangle sides? For example, at 4:19, he calls the green side on the smaller side \"CA,\" going right to left. But he calls the similar side on the larger triangle \"DF,\" going left the right. I realize that it doesn't make any difference computationally, but shouldn't you name things consistently when you are trying to show they are proportional?",
        "A": "I think , we name them consistently but our understanding purpose only Sal done like that"
    },
    {
        "video_name": "h0FFEBHBufo",
        "Q": "Why, at 7:07, does Sal multiply both sides of one of the equations by 2? Doesn't this change the equation from what the parallelogram gave? And then he seemingly adds the two equations he got from the parallelogram? How and why does this work?",
        "A": "Sal used a method called elimination. This is a way of obtaining two variables from two separate equations, like the problem."
    },
    {
        "video_name": "5qfOViJda_g",
        "Q": "The answer is 100 0:02",
        "A": "5 to the power of 3, is the same as saying this. (5 x 5 x 5 = x), 5 times 5 is equal to 25, which times another five is equals to 125. Hopefully this makes sense!"
    },
    {
        "video_name": "-6EqUILZ1yw",
        "Q": "At 1:46 and 2:27, why does Sal approximate that h(3) is 650? We can clearly see that when x is 3, y is 650!",
        "A": "If you look at the graph closely, you will see h(3) is actually slightly above 650. The graph is not crossing directly at y=650. So, it is an approximation."
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "1:28 Is \"The # of molecules in a single sound wave\" still \"higher than the national debt?\" I mean, $17.74 trillion?!?!",
        "A": "Most likely. Although not all air waves have the same number of molecules in them."
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "Why did she break her viola at 1:46?",
        "A": "Because she went so tight on the string, that the vibration was so strong it actually had enough kinetic energy to break the string."
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "At 0:37, she mentions vibrations. How fast are the normal vibrations of a wave pattern to create the actual sound?",
        "A": "It depends on the note. A-440 has 440 waves per second. Other notes will have other frequencies. The range of human hearing goes approximately from 20 waves per second to 20,000 waves per second."
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "At 5:13 or so, she mentions the chromatic scale. What is it?",
        "A": "The chromatic scale is basically playing/singing every note in order from highest to lowest, or lowest to highest."
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "@ 1:45, she broke her viola string! Why would she do that just to make a video?",
        "A": "Maybe it was time to change strings and Vi decided that she could use the string for the video."
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "At 1:28, she says there is fluid in the cochlea. Why is there fluid?",
        "A": "That fluid is what the sounds travel through to our brains. Sounds actually travel better through liquids."
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "with the swing analogy at 0:45, wouldn't the swing have friction, causing her to go the exact speed of the push?",
        "A": "Yes, she would be going at the speed of the push, but that doesn t have any effect on the period required for one swing. If the push is faster, then she would end up going further up, meaning a larger distance, balancing everything out."
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "Wait. No piano is long enough for a 32 ft string! 9:17.",
        "A": "32 ft is the wavelength of the tone in air. In the string, however, the wavelength gets shorter."
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "In 0:15 and 1:15, who was playing the violin because I play it!",
        "A": "Vi Hart. The creator of this video. #Midnight Luna"
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "what does it say on the piano on 4:34 and 5:16",
        "A": "It has 1 at C and 1/2 at C an octave higher and 1/3 at G and 1/4 at C an octave above that and so on"
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "Since it's a viola used in the video, at 3:58, shouldn't the notes be in alto clef, not treble?",
        "A": "Nope, The violin or the viola use a treble clef while the cello and the double bass use a bass clef"
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "at 7:32, what program is she using?",
        "A": "she is using audactiy. Look in the top corner :) Thanks for your time.-Mark"
    },
    {
        "video_name": "i_0DXxNeaQ0",
        "Q": "At 5:45, she talks about how different cultures have created music with similar chords and tonality, even if their music had been created independently of each other. If this is the case, then what explains the use of semi tones? It seems to me that only a few cultures use them, and that the majority of music I hear is structured around the major or minor scale. Why do only a few cultures use semi tones?",
        "A": "well there are different answers possible, semi-tones are like hair color they vary for different reasons, one might be be genes or something that was to make it change"
    },
    {
        "video_name": "3uYb-RhM7lU",
        "Q": "3:01 He's subsituting L{ y'' }, but he says L{ y' }\nI know that L{ y' }= SL{y}(s) - y(0)\nbut if he's subsitutuing L{ y\" } should he use the formula\nL{ y\" } = S^2 L(y)(s) - S y(0) - y'(0)\nI hope I'm understanding the formula correctly. That part confused me and I can't seem to make sense of it.",
        "A": "At 3:01, he s not substituting at all. He is just giving a rule for how you evaluate the Laplace transform of derivatives. So, when Sal says later that L{y } = sL{y } - y (0), he is correct, because he is just adding a prime to every y. Sal then substitutes that into the equation. It would basically be doing the same thing as the formula you give; Sal is just doing it in two steps."
    },
    {
        "video_name": "_-02ze7tf08",
        "Q": "For the numerators at 1:00 , is it \u00ce\u00b4f(x) when the denominator is \u00ce\u00b4x and \u00ce\u00b4f(y) when the denominator is \u00ce\u00b4y ? Or does just having \u00ce\u00b4f mean something else?",
        "A": "\u00ce\u00b4f/\u00ce\u00b4x = \u00ce\u00b4f(x, y)/\u00ce\u00b4x = (f(x+dx, y)- f(x, y))/dx that s how I see it."
    },
    {
        "video_name": "O1R4H3Ca82E",
        "Q": "I have a question. On 2:27, What is a Furgle? 2 ft x 2ft? I heard it is made up but is it 2 ft x 2 ft?",
        "A": "A furgle is a made up unit of measurement to illustrate a concept. In the case of the video based on how it was measured, it s sides were 2ft, and since it was a square that gives it an area of 4 square feet. Looking at the big rectangle that was 12 square feet It took 3 furgles to measure it, so that s another way to determine the size of the furgle."
    },
    {
        "video_name": "O1R4H3Ca82E",
        "Q": "At 2:22 why do you call the dimension a furgle?",
        "A": "Well, It was probably a mistake. Contact them in you want to tell KA about it."
    },
    {
        "video_name": "O1R4H3Ca82E",
        "Q": "so in 3:38-3:43 4 square feet =1 square furgle?",
        "A": "Correct! We can see this easily by taking 12 square feet and 3 square furgles and reducing them. 12 square feet = 3 square furgle When we divide both sides of the equation by 3 (we are dividing both sides by 3 because 3 is the greatest common factor) 4 square feet = 1 square furgle"
    },
    {
        "video_name": "8bK-xfh8-rY",
        "Q": "At 1:40, Sal calls the spinner a \"fair spinner.\" What does that mean? What would an \"unfair spinner\" be?",
        "A": "A fair spinner is a spinner that has a equally likely chance of getting any possibility."
    },
    {
        "video_name": "8bK-xfh8-rY",
        "Q": "I'd like some more clarification on how he got the answer at 2:41",
        "A": "He got the answer by multiplying 210 by 4/7. The spinner spun 210 times, and there is a 4/7 chance that you are going to land on an elephant. If the spinner spun 7 times, lets say, then you would get to the answer by multiplying 7 by 4/7, which is 4. It s the same with 210. 210 x 4/7 is 120. If you are curious about how he divided 210 by 7, then multiplied 30 by 4, it is called a reduction."
    },
    {
        "video_name": "MpRUEGD2-_c",
        "Q": "at 4:50 , why did the -15 turn to +15 when he re- wrote the equation in slope intercept form?",
        "A": "He didn t change the -15 to +15. In fact he just rewrote the equation 6a + 3b = 15 in the slope intercept form, ie. 3b = -6a +15 or b = -2a +5. The -15 comes when the equation 2a +b = 5 was multiplied by -3."
    },
    {
        "video_name": "8I_KtODUekU",
        "Q": "At about 3:26 Sal writes that the integrand is -n^-1. How'd he get the negative out in front of the n? I'm sure this is a simple question, I'm just blanking on this subject. I understand why there's a negative on the exponent and why it went from -2 to -1.",
        "A": "So when integrating the format will be x^(n + 1)/(n + 1) which in this leads to n^-2 becoming n^(-2 + 1)/(-2 + 1) or n^-1/-1 which is simply -n^-1. So basically make sure when you re integrating this problem you don t forget about the denominator part of the problem!"
    },
    {
        "video_name": "8I_KtODUekU",
        "Q": "At 6:32 he states that. Can be 1.666 but it's upper bound is 1.664, is that right or am I missing something?",
        "A": "At one point he does say, 1.66 , but not 1.666 . So it s within range"
    },
    {
        "video_name": "muZmOiiukQE",
        "Q": "The question at 2:21 I don't understand it, why was it true? Point A is supposed to be representing the distance of 5 hrs not 4 hrs right?",
        "A": "Yes, you are right. Sal made a mistake. There is a correction box explaining the error."
    },
    {
        "video_name": "muZmOiiukQE",
        "Q": "Why at 2:47, did he say that the first option is correct when it was not. The vertical coordinate of point a was five hours when he said it was four hours. That would make it 320 kilometers instead of 400 kilometers.",
        "A": "You are correct. Sal made a mistake. There is a correction box that pops up to explain the error."
    },
    {
        "video_name": "muZmOiiukQE",
        "Q": "For the question at 2:20, isn't it supposed to the distance in 5 hours? Not 4? Why is it true?",
        "A": "Well spotted! It is indeed 400 km in 5 hours, not 4."
    },
    {
        "video_name": "muZmOiiukQE",
        "Q": "At 2:58, point A isn't supposed to be checked. The vertical coordinate of point A is supposed to be the distance driven in 5 hours, right?",
        "A": "Yes, you are correct. Sal made a mistake. There is a correction box explaining the error."
    },
    {
        "video_name": "muZmOiiukQE",
        "Q": "i dont understand 2:21 too.",
        "A": "He made a mistake. I suppose"
    },
    {
        "video_name": "kXULrUxGzmU",
        "Q": "At 5:08 when Sal is writing out the function g(x^2), wouldn't the sign in front of (x^4/2!) be positive now since it's been squared?",
        "A": "No, if we look at the first few terms here: cos(x) = 1 -x^2/2!+x^4/4!-... If you just make it cos(x^2) then you simply place an x^2 wherever you see an x. so: cos(x^2) = 1-(x^2)^2/2!+(x^2)^4/4!-... Ultimately, this does nothing to change the sign."
    },
    {
        "video_name": "FZ2APP6-grU",
        "Q": "Why does Sal write that you can buy 333 1/3 tiles at \"1:30\"? Isn't it 333.333333... repeating? So shouldn't it just simply be 333 tiles?",
        "A": "A repeating decimal is usually rounded to the hundredths place, not the ones place. This is to leave the fractional amount in the problem. Sal is not using decimals, so he leaves the fractional amount in the problem as 1/3. You have to use a little bit of intuition in deciding whether to round a decimal and to what place you round. For a fraction in an equation, though, you never round AFAIK."
    },
    {
        "video_name": "ArvnBba_ogI",
        "Q": "at around 1:42 i thought you are supposed to write 2\n1200m",
        "A": "Writing 1200 square meters is the same as writing 1200m^2. The m stands for meters, and the exponent 2 stands for squared."
    },
    {
        "video_name": "we829K4NOq8",
        "Q": "At 1:50 the w subspace of orthogonal compliment. The upside down T represents orthogonal compliment?",
        "A": "Yes, that upside down T does indicate the orthgonal complement of a subspace. So if V is a subspace, V ^ _|_ would be the orthogonal complement of V, and we sometimes read it as V perp , where the perp is short for perpendicular."
    },
    {
        "video_name": "SfWrVNyP9E8",
        "Q": "At 2:22 why can't the radius be 2-x^(1/2). Isn't the distance from the axis of rotation to the furthest point on the shell (the point on the function x^(1/2)), the value of the radius?",
        "A": "For any given x-value, the radius of the shell will be the space between the x value and the axis of rotation, which is at x=2. If x=1, the radius is 1, if x=.1, the radius is 1.9. Therefore, the radius is always 2-x. The x^(1/2) and x^2 only come into play when determining the height of the cylinder."
    },
    {
        "video_name": "SfWrVNyP9E8",
        "Q": "At 4:04 how did you find the interval from 0 to 1 when looking at volume?",
        "A": "Using the intersection points of two functions."
    },
    {
        "video_name": "WffVdYETdng",
        "Q": "@0:00 What is the relationship between the inscribed triangle and MJK ?",
        "A": "the inscribed angle on the circle and the angle MKJ i.e. both are equal to theta and have the same measure"
    },
    {
        "video_name": "iqeGTtyzQ1I",
        "Q": "At 0:25, why does Sal say that the point in the top of the triangle is a right angle?",
        "A": "In geometry you can t assume anything, but as this is marked as being a right angle we know for a fact that the diagonal line intercepts the hypotenuse of the overall large triangle at a right angle. Without this marking being given, we could not assume they were perpendicular to each other, but as it is included we know these two lines form a right triangle. :) Hope that helped!"
    },
    {
        "video_name": "iqeGTtyzQ1I",
        "Q": "At 2:17 why would you add the theta instead of the 32?",
        "A": "Unless I am mistaken, it is to cancel out the Theta so 32 can equal Theta instead of Theta still being part of the equation. Basically, IF my facts are strait, its kind of like canceling out a variable. Anyone, feel free to correct me."
    },
    {
        "video_name": "iqeGTtyzQ1I",
        "Q": "3:40 How do you know that those two angles are supplementary? How do you know that it forms a straight angle? Does that come from the fact that you are dealing with at triangle or the fact that one of the angles is a right angle?",
        "A": "Because that s clearly a line, and lines have straight angles, and in that line there is two adjacent angles. So, they are suplemmentary, and then, if one is 90\u00c2\u00ba, the other is 180\u00c2\u00ba - 90\u00c2\u00ba = 90\u00c2\u00ba Sorry for bad english, I m not american"
    },
    {
        "video_name": "iqeGTtyzQ1I",
        "Q": "I figured out the problem (the measure of \u00c3\u00b8) before Sal gave the answer, but I got confused when he started crossing out numbers (1:45). I would not have figured it out that way. How do you do it?",
        "A": "What Sal is doing when he is crossing of the numbers at 1:45 is he is just simplifying. The equation was 90- theta+90+32=180. He added 90+90=180, and by doing that he was simplifying the equation to where it was 180-theta+32=180."
    },
    {
        "video_name": "iqeGTtyzQ1I",
        "Q": "Before 4:02, why did Sal assume that the angle left of that Greek letter is also 90 degrees?",
        "A": "If a line segment is divided into two angles, and one of those angles is a right angle, then the other one must also be 90 degrees. This is the case with the top side of the triangle. The angle to the left of theta (a Greek letter often representing a variable angle) is a right angle."
    },
    {
        "video_name": "H4BstqvgBow",
        "Q": "At 3:45, Do you say the number literally one,one,zero,one? Or would it be okay to say one thousand, one hundred and one?",
        "A": "thousands, hundreds, millions, billions are terms used in based-10 number system. So we dont really say those when we deal with binary, at least we are not supposed to. But in your mind you can use those terms if they help."
    },
    {
        "video_name": "iw5eLJV0Sj4",
        "Q": "At 3:13, he says that the anti-derivative of 1 dx is x but the equation he is taking an anti-derivative of is the integral of 1*x dx. Why does he come up with x instead of 1/2*x^2?",
        "A": "If we look at the integrand, Sal wanted to cancel out 1/x * x with the white line. It seems like the line isn t crossing both factors, but that is what we should do to integrate by parts. I hope this was a little helpful!"
    },
    {
        "video_name": "iw5eLJV0Sj4",
        "Q": "At 2:42, How come that Sal taking the integral of a constant 1, shouldn't be it out of the integral sign?",
        "A": "No, it would be impossible to take it out, seeing as it is the only thing in the integrand. The antiderivative of 1 is x + C, so it is still important to the problem solution. If there is already a variable in the integrand, then yes, a constant can be moved to the outside, but in this case, you just take the integral of 1."
    },
    {
        "video_name": "iw5eLJV0Sj4",
        "Q": "@2:07\nIsn't the integral of 1 = x+ C? Because it is indefinite?",
        "A": "Yes, but you don t need to include it with integration by parts. You can, if you like, but you ll get the same answer in the end, only with more effort. So, it is customary to leave it off. One C per problem is sufficient for the types of integrals studied at this level."
    },
    {
        "video_name": "iw5eLJV0Sj4",
        "Q": "At 2:58 sal say the word integrand. What is the difference between an integral and an integrand?",
        "A": "An integral is the whole operator: \u00e2\u0088\u00ab f(x) dx An integrand is just the function you are integrating. So for \u00e2\u0088\u00ab 3x^2 dx, the integrand is 3x^2."
    },
    {
        "video_name": "tdwFdzVqito",
        "Q": "At 5:00, we could drop the perpendicular the other way; i.e. from the end of vector b to some point on vector a, giving adj(of a) = cos\u00ce\u00b8 ||b||. Does it matter which way you do it?",
        "A": "||a||||b||cost = ||a||*(||b||cost) = ||b||*(||a||cost)."
    },
    {
        "video_name": "tdwFdzVqito",
        "Q": "At 9:40 Sal says \"when a\u00e2\u0083\u0097 dot b\u00e2\u0083\u0097 is equal to 0, we say they're perpendicular\", is that really true in all cases? What if a\u00e2\u0083\u0097 or b\u00e2\u0083\u0097 is equal to the zero vector? Doesn't it just mean that they're orthagonal and CAN be perpendicular if a\u00e2\u0083\u0097 and b\u00e2\u0083\u0097 are NOT equal to the zero vector? Am i right?",
        "A": "Yes. I d say it this way: If a dot b is 0, then a and b are orthogonal,. If in addition neither a nor b is the 0 vector, then a and b are also perpendicular."
    },
    {
        "video_name": "tdwFdzVqito",
        "Q": "At 7:00, Sal explains how the dot product is the length of one of the vectors times the length of it's shadow on the other vector. Is this why perpendicular vectors are orthogonal.",
        "A": "Right, yes; the length of the shadow of a on b, with a and b at angle t with each other, is ||a||cost. When t = 90 the shadow is 0, and so is a dot b, which makes them orthogonal. Any vector a and the 0 vector are also orthogonal, because their dot product is also 0, but not perpendicular, because the zero vector has no direction. Orthogonal has a somewhat nebulous meaning which includes both perpendicular , and dot product = 0, among, I think, other things."
    },
    {
        "video_name": "vSijVSL3ChU",
        "Q": "at 6:05 where was Sal getting the extra 1,024?",
        "A": "he had 1024^t/10 -1 +1 so you can split this into 1024^t/10 -1 * 1024^1 using exponent rules. the 1024^1 can be simplified to 1024"
    },
    {
        "video_name": "vSijVSL3ChU",
        "Q": "What does 'home stretch' mean? (6:49)",
        "A": "He says that a lot It means that the end is so close you can see it Like in races the last few hundred meters is the homestretch you can see the finish line. Hope that helped!"
    },
    {
        "video_name": "vSijVSL3ChU",
        "Q": "at 0:40, where did you get t/10.",
        "A": "@ 1:05 in the video, you are told that this format was chosen arbitrarily. The whole purpose is to show you how to change the expression from one form to another form."
    },
    {
        "video_name": "vSijVSL3ChU",
        "Q": "at 2:02 why do you have to multiply 10 to t? don't you just have to divide? i was really confused on this part...",
        "A": "You want to get t into the t over 10 (t/10) form like it is on the right hand side, but if you only divide t by 10, then you change the whole value of the expression on the left hand side. So in order keep the value of the expression the same, you divide t by ten so you get the form t/10, and then you multiply t by 10 outside the fraction so that the expression t/10*10 is really just equivalent to t. Now the whole value of the expression on the left has not changed. Hope this helped."
    },
    {
        "video_name": "hmtJV49AWio",
        "Q": "At 1:48, why does he write (-2)(5) + (-2)(-3) and (-2)(5)-(-2)(3) instead of (-2\u00c3\u00975) + (-2\u00c3\u0097-3) and\n(-2\u00c3\u00975) - (-2\u00c3\u00973)? I know that (-2)(5) means that your multiplying negative 2 by 5, but is there any specific reason to write it that way vs. another?",
        "A": "The multiplication sign could also be mistaken for a variable, x."
    },
    {
        "video_name": "0xrvRKHoO2g",
        "Q": "At 2:44, is (y+5x)(y+6x) equal to 11xy or the whole quadratic?",
        "A": "The whole quadratic :)"
    },
    {
        "video_name": "Mci8Cuik_Gw",
        "Q": "@3:44 Sal wrote d((ln2)x)/dx and d(e^(ln2)x)/d((ln2)x) can someone explain what does this mean?",
        "A": "he s simply showing you how he derives d((ln2)x) in terms of dx and also d(e^(ln2)x) in terms of d((ln2)x)"
    },
    {
        "video_name": "Mci8Cuik_Gw",
        "Q": "At 3:20, why does Sal take the derivative of (ln2)x? Don't the brackets d/dx[ e^ln2x] Imply that we'd be taking the derivative of e^ln2? It seems like the derivative should be e^ln2.",
        "A": "f(x) = e^(ln(2)*x) f (x) = ln(2)*e^(ln(2)*x) The chain rule makes more sense when you substitute. g(x) = ln(2)*x g (x) = ln(2) h(x) = e^x h (x) = e^x f(x) = h(g(x)) f (x) = g (x) * h (g(x)) this line and the previous line are the definition of the chain rule f (x) = ln(2) * e^(ln(2)*x)"
    },
    {
        "video_name": "Mci8Cuik_Gw",
        "Q": "At 0:26 Sal wrote 2 as e^(ln 2) . Please explain .",
        "A": "They are inverse functions. the natural logarithm is defined as the function that has e^(ln(x))=x. (and vice versa, ln(e^x)=x. They are inverses of each other)"
    },
    {
        "video_name": "Mci8Cuik_Gw",
        "Q": "Around 2:35 I get confused when Sal starts using the Lebowitz notation in a way that is not '''d/dx\". Can someone explain what is happening or refer me to some material that will allow me to master the Lebowitz notation, because that seems to be significantly hampering my understanding of Derivates/Calculus.",
        "A": "Watch the two preceding videos on the chain rule (Chain rule intro & Chain rule definition) to understand a less confusing example of it s application. Here is what is going on at 2:35 in a notation you might be more comfortable with: f(x)=2^x and 2^x=(e^ln2)^x therefore f(x)=(e^ln2)^x d f(x)/d((ln2)x)=e^((ln2)x)) In plain language this is saying the derivative of the function f(x) with respect to the natural log of 2 times x"
    },
    {
        "video_name": "Mci8Cuik_Gw",
        "Q": "At 4:17 when Sal mentions that the derivative of e^(ln 2)(x) with respect to (ln 2)(x) is (e^ln 2)^(x) I dont understand why this is true. I thought that if you assumed (ln2) *(x) was x then the derivative would be e^x.",
        "A": "It s better to say (ln2) *(x) is u instead of (ln2) *(x) is x , in order to keep from getting confused. So let s say u = (ln2) *(x). So then we have the derivative of e^u with respect to u, and that is (like you said) e^u. But remember, u is (ln2) *(x). Does that make it clear? Just to put it into symbols: (d/d(something)) of e^(something) = e^(something), as long as all the something s are exactly the same!"
    },
    {
        "video_name": "Mci8Cuik_Gw",
        "Q": "At 0:50: How comes that e^ln2 equals 2?",
        "A": "Because ln2 equals the number that e has to be raised to to equal 2, so raising e to that number must equal 2."
    },
    {
        "video_name": "Mci8Cuik_Gw",
        "Q": "At 2:46 Sal said, e^(ln2)x is the derivative of e^(ln2)x w.r.t. (ln)x. I don't understand why?\nIn the previous video, \"Chain rule definition and example\" the example of sin(x) helped me to grasp but not here. Please explain the core reason.\n:) :)",
        "A": "This might help: d/dx(2^x) d/dx(e^(ln(2^x))) d/dx(e^(x\u00e2\u0080\u00a2ln(2))) e^(x\u00e2\u0080\u00a2ln(2))\u00e2\u0080\u00a2d/dx(x\u00e2\u0080\u00a2ln(2)) e^(x\u00e2\u0080\u00a2ln(2))\u00e2\u0080\u00a2ln(2)\u00e2\u0080\u00a2d/dx(x) e^(x\u00e2\u0080\u00a2ln(2))\u00e2\u0080\u00a2ln(2)\u00e2\u0080\u00a2dx/dx e^(x\u00e2\u0080\u00a2ln(2))\u00e2\u0080\u00a2ln(2) e^(ln(2^x))\u00e2\u0080\u00a2ln(2) 2^x\u00e2\u0080\u00a2ln(2) Rule: d/dx(n^x) \u00e2\u0087\u0094 n^x\u00e2\u0080\u00a2ln(n)"
    },
    {
        "video_name": "Mci8Cuik_Gw",
        "Q": "At 1:00, why did he convert 2^x to (e^ln 2)^x? Why should we write it in that form?",
        "A": "He did it that way to avoid using implicit differentiation. But, you don t have to do it like that. For example: y=2^x log\u00e2\u0082\u0082 y = log\u00e2\u0082\u0082(2^x) log\u00e2\u0082\u0082 y =x d(log\u00e2\u0082\u0082 y) = dx dy/(y ln 2)= dx Rearrange to: dy/dx = y ln 2 since y=2^x dy/dx = (2^x )(ln 2) Although in this simple case there was not much to be gained by using the fact that f(x) = e^(ln (f(x)), there are problems where trying to solve the derivative any other way is nightmarishly difficult. So, it is a good method to learn."
    },
    {
        "video_name": "Mci8Cuik_Gw",
        "Q": "3:20\nSal says \"the derivative of x(ln2) is just ln2\" why is that?\nThat threw me completely off",
        "A": "ln(2) is a constant, and the derivative of a constant times x is just the constant."
    },
    {
        "video_name": "Mci8Cuik_Gw",
        "Q": "I don't quite understand what is meant by the phrase, \"with respect to...\", especially at 3:00 when we go from respecting x to respecting ln2*x. Is there some video where this is explained, or should I have picked it up by context? How would the problem at 3:00 be different with respect to x instead of ln2*x?",
        "A": "In differentiation, the derivative of y with respect to x means the change in y as x changed. So you can think of it as the change in y depends on the change in x. In the video example, e^(Ln2)x, the derivative (change) of e^(Ln2)x depends on the change of (Ln2)x, but the derivative of (Ln2)x depends on the change of x. So with respect to something means depends on the change of something."
    },
    {
        "video_name": "Mci8Cuik_Gw",
        "Q": "At 4:25, Sal cut the e^In2 and said that this equals to 2 why so?",
        "A": "This result falls out of the definition of the natural log function. ln(x) is defined as the exponent of e that will produce x, so ln(2) is the exponent of e that will produce the number 2. As a result, if we use ln(2) as the exponent of e, we get 2."
    },
    {
        "video_name": "bsDu8VQKwxA",
        "Q": "at 0:47 isnt that at major arc?",
        "A": "The two angles are in fact vertical because they are both intersecting diameters. Your confusion, however, is justified. Perhaps you should understand the definition of diameter. A diameter is a line segment whose two points lie on the circumference of the circle and the bisected is the origin of the circle. Hope you find this helpul."
    },
    {
        "video_name": "bsDu8VQKwxA",
        "Q": "around 3:40, he says that even though the measure was 2 letters, it was the minor arc, wouldn't it be the major arc though?",
        "A": "The minor arc is the default. To specify that you want the major arc, you use 3 letters to identify it."
    },
    {
        "video_name": "bsDu8VQKwxA",
        "Q": "At 1:19, Sal says that (4k+159) is vertical to (2k+153). How can we tell if they are vertical if we are not given any diameter? Did we assume it was vertical since they are directly across from each other (\"eye-balling it\")?",
        "A": "Yeah!! Eyeballing it, as you say!"
    },
    {
        "video_name": "8Y-snjheI9M",
        "Q": "At 1:06, Sal says that he is breaking this up infinitely. Can anything be (mathematically speaking) infinitely small?",
        "A": "Yes, mathematically speaking. 1/\u00e2\u0088\u009e"
    },
    {
        "video_name": "8Y-snjheI9M",
        "Q": "At 1:45, is it really the change in **\"arc length\"**?",
        "A": "I think it would be better if he said If I sum up all these tiny contributions to the arc length, I will get the total arc length . In this context, I don t think it s clear to say change in arc length . The arc length is not changing . It s just a length that we want to find by dividing it up into a huge number of tiny pieces (that are like tiny hypotenuses), and then summing them up."
    },
    {
        "video_name": "8Y-snjheI9M",
        "Q": "At 4:10 it doesn't seem right that Sal can factor out (dx)^2 so that it creates (dy/dx)^2, which is a derivative squared. Why is it OK to do that?",
        "A": "It is okay to do that since he is really just rewriting the (dx)^2 + (dy)^2 part. Note that he is not changing the value of the original expression. He is just rewriting it in a different way so that we can algebraically manipulate it to be useful to us. If you were to distribute the (dx)^2 part to (1 + (dy/dx)^2), you would just get the same thing under the original square root, which is (dx)^2 + (dy)^2. All we are doing is manipulating it algebraically to make it so that it can be of some use"
    },
    {
        "video_name": "a601bUPEYUs",
        "Q": "At 0:40, when you take the anti-derivative of f'(x), wouldn't it be f(x) + C?",
        "A": "Here Sal is taking a definite integral. when we take definite integral between a and b then the anti-derivative of f (x) becomes f(b)-f(a) ( the constant cancels out). For indefinite integrals, anti derivative of f (x) has the the form f(x) + c."
    },
    {
        "video_name": "oOTFGdjhqqM",
        "Q": "At 3:20 could he have just used FOIL(First,Outer,Inner,Last)?",
        "A": "FOIL is a good method for multiplying binomials. However, it is limited to only binomials, which is why using the box method and the distributive property are much simpler. But yes, FOIL is basically an extended version of the distributive property named after the stuff we play with and try to flatten using our nails."
    },
    {
        "video_name": "oOTFGdjhqqM",
        "Q": "At 2:38, when Sal adds 2x and 3x and gets 5x, why isn't it 5x squared? Or is it because he's adding?",
        "A": "It is because he is adding. Exponents are repetitive multiplication. So, you need multiplication to create x^2: x*x = x^2 Hope this helps."
    },
    {
        "video_name": "bZ8YAHDTFJ8",
        "Q": "At 3:15, isn't it the antiderivative of 1 is x, why Sal left that out?",
        "A": "It s a bit hard to explain, but the 1 here isn t a isolated constant. The 1 is showing how many of sin(x) there are in the integrand. If the 1 was by itself, then it s antiderivative would be x + C, but here it is just one sin(x) , so it isn t integrated. If you want to prove it, try doing integration by parts with 1*sin(x), and you will find the answer is the same as what Sal got. :)"
    },
    {
        "video_name": "bZ8YAHDTFJ8",
        "Q": "At 1:20, I don't understand how \"X\" can be a derivative in the first place. You cannot derive plain old \"X\" from the power rule. \"1\", yes; but not \"X\".",
        "A": "Take the derivative with respect to X of (X^2)/2. According to the power rule, the exponent is multiplied by the base, and the exponent is lowered by 1. Thus, the derivative with respect to X of (X^2)/2 is just X."
    },
    {
        "video_name": "bZ8YAHDTFJ8",
        "Q": "At 1:46, isnt the antiderivative of cos(x), sin(x)+c ? Why didn't Sal added the constant ?\nThank-you for your answer",
        "A": "Technically, there is an arbitrary constant at this point, but it is never used in practice. The reason is that there is still an indefinite integral in the expression. When you perform that integration, you will get another arbitrary constant. The sum of two arbitrary constants is again an arbitrary constant, so the one at 1:46 would be unnecessary."
    },
    {
        "video_name": "NC6HXlrH53Y",
        "Q": "A question for the first problem, at 0:00\n\nAt 1:25, Sal says that the answer has to be rational, but this proves that it can be either rational OR irrational.\n\nThe numbers 5678 and 385 are rational.\nSo there's a and b, so a = ( m = 5678 and n = 2 )\nb = ( p = 385 and q = 5)\n5678 / 385 = 14.748051948051948 (which is Irrational isn't it?)\n\nSo the answer can be either Rational or Irrational, and not just rational right?",
        "A": "Why do you think that the decimal is irrational? First, it is shown as a ratio of two integers which is rational by definition. Second, it starts repeating at the hundredths digit, and all repeating numbers are rational. I do not understand what you are trying to argue."
    },
    {
        "video_name": "NC6HXlrH53Y",
        "Q": "if a and b are two rational numbers, e.g a= 22 and b= 7, then a/b = 22/7 which is pi, therefore will be an irrational number. so the answer @1:30 should be irrational or rational. someone please explain!",
        "A": "The important fact that you are missing is that 22/7 does not equal Pi. It is just an approximation for Pi. 22/7 = 3.142857.. where the 142857 repeats over and over again. Pi is a non-ending and non-repeating decimal. Pi can not be written as the ratio of 2 integers. 22/7 is rational Pi is irrational Hope this helps."
    },
    {
        "video_name": "NC6HXlrH53Y",
        "Q": "At 3:44, Sal said that the square root of 8, is an irrational number because it has a perfect square in it, but is not a perfect square in and of itself.\n\nI understand perfect squares to be a product of two equal integers. 8 is not a product of 2 equal integers... Can someone please explain what Sal is saying here??",
        "A": "8 has factors that can be written as 4 and 2. 4 is a perfect square. 2 is not a perfect square. So, 8 has a perfect square in it (4), but 8 is not a perfect square in and of itself because the remaining factor is not a perfect square. Hope this helps!"
    },
    {
        "video_name": "ry81_iSHt6E",
        "Q": "Can someone help me out so that I can understand how Sal calculated the variance until 4:08?",
        "A": "He uses the same old formulas, but he simplifies the calculation because these Bernoulli distributions are so simple - they have only one parameter - p. Mu = [sum i=1,n (x(i)]/n = [.6n*1 + .4n*0]/n (For any population size n, there are .6n favorable) = .6*1+.4*0 = .6 sigma^2 = [sum i=1,n (x(i)-mu)^2] / n (We can divide by n because we have the entire population.) = [.6n*(1-.6)^2 + .4n*(0-.6)^2] / n (For any population size n, there are .6n favorable) = [.6n*.4^2 + .4n*.6^2] / n = .6*.4^2 + .4*.6^2 = .24"
    },
    {
        "video_name": "8C5kAIKLcZo",
        "Q": "At 4:23, can you write 27/2 x 12 as 27x12/2?",
        "A": "You technically could. There aren t 6 inches in a foot, but it would work if you did it your way."
    },
    {
        "video_name": "8C5kAIKLcZo",
        "Q": "at 3:31 sal says 12 inches per yard does he mean 12 inches per foot?\nbecause i say that for every 1 foot there is 12 inches can anybody clear this out for me?",
        "A": "It tell you in the black box on the bottom right corner."
    },
    {
        "video_name": "8C5kAIKLcZo",
        "Q": "At 1:18 how come we don't convert it to inches right away?",
        "A": "At 1:11 he said Lets take this in baby steps meaning he will do it so they will undestand it. You can do it your way but Sal is just doing the long way in order to make sure he doesnt make any mistake :)"
    },
    {
        "video_name": "8C5kAIKLcZo",
        "Q": "At 0:17 Sal says that it \"complicates the issue\" to have a mixed number - why?",
        "A": "Good question why?"
    },
    {
        "video_name": "8C5kAIKLcZo",
        "Q": "Sal said \"12 inches per yard, but meant \"12 inches per foot.\" @3:31",
        "A": "Well, its easier because then you could change that mixed number into a improper fraction and proceed to the next step."
    },
    {
        "video_name": "8C5kAIKLcZo",
        "Q": "what does 9/2 have to do with anything 0:57",
        "A": "Because 9/2 is the improper fraction form of 4 1/2. basically... 9/2 = 4 1/2"
    },
    {
        "video_name": "uzOXGgAgmPs",
        "Q": "at 10:48, did Sal mean to integrate (1/2+1/2cos2s) ds?",
        "A": "yeah. (1+cos2s)/2 is a trigonometric identity for cos^2 (s)"
    },
    {
        "video_name": "Z3db5itCIiQ",
        "Q": "Why does rationalizing mean that the denominator can't have a square root? Sal says that it does at 0:22. If it was the square root of 4, wouldn't it still be rational?",
        "A": "A square root can be a rational - such as \u00e2\u0088\u009a4, as you pointed out. It doesn t have to be rational though and when it can t be simplified any further, it probably is not rational. After all, if you had \u00e2\u0088\u009a4 you wouldn t go through the same rationalizing process - you d just simplify it to be 2. Therefore an answer is generally not considered properly simplified when there is a square root in the denominator. Rationalizing is the name for removing it - independent of whether or not it is confirmed to be irrational."
    },
    {
        "video_name": "Z3db5itCIiQ",
        "Q": "What Sal writes at 2:35- 2:38, can I rewrite that as \" x^2\u00e2\u0088\u009a2 \"?",
        "A": "Yes, you could write the answer as: 4\u00e2\u0088\u009a2 + x^2\u00e2\u0088\u009a2/2 Hope this helps."
    },
    {
        "video_name": "Z3db5itCIiQ",
        "Q": "At 1:10, why does Sal say to rewrite the expression as 2 times the principal square root of 2? Why is the negative version of the square root not allowed?",
        "A": "The square root symbol always means the principal square root, or the positive square root. If you want to specify both you must include the plus-or-minus sign."
    },
    {
        "video_name": "Z3db5itCIiQ",
        "Q": "At 1:40 2.sqrt of 2 turns into sqrt of 2. I don't get that. Can anyone help me please?",
        "A": "Sorry, I think I got it. We already devided the 2 out so we keep the remaining sqrt of 2. Am I right?"
    },
    {
        "video_name": "Z3db5itCIiQ",
        "Q": "at 2:30 he did square root of 2 divided by square root of 2 wouldn't that be 1?",
        "A": "It would, that s why he could do it. If he multiplied by an expression that wasn t actually 1, the original expression would change and he d get a different result. The point was to get rid of the square root in the bottom, so he multiplied both parts of the fraction with the same value, eventually getting a rational number in the denominator."
    },
    {
        "video_name": "Z3db5itCIiQ",
        "Q": "At about 2:15 Sal multiplies the fraction by the square root of 2 over the square root of 2. Isn't that the same as multiplying by one?",
        "A": "Yes, it is exactly the same as multiplying by one. And since anything multiplied by one is unchanged, you can multiply by one any time you like. If you multiply by \u00e2\u0088\u009a2 / \u00e2\u0088\u009a2 then you don t change the value of the expression, but you introduce a factor of \u00e2\u0088\u009a2 on both the top and bottom of the fraction -- and this is helpful in removing a square-root from the denominator, which is the aim of this video."
    },
    {
        "video_name": "Z3db5itCIiQ",
        "Q": "At 4:39 when Sal starts dividing everything by 2, he skips the first 8 and goes right to the 2 sqrt 2. Why does he not divide the first 8 by 2?",
        "A": "The 8 is part of one term: 8*2 sqrt(2) that is still in a factored form (the multiplication is incomplete). He could have simplified it into 16 sqrt(2). If you divide the 16 by 2, you are back to the 8. Sal just elected to cancel out the factor of 2 in that term. You get the same answer either way. Hope this helps."
    },
    {
        "video_name": "AqMT_zB9rP8",
        "Q": "In the answer at 2:00, shouldn't the consonant should be in the last place according to its degree",
        "A": "It is commonly written in descending order of power or degree, but you don t have to. There is no rule that you must write it in order (unless your teacher specifically requires you to do so, then you should)."
    },
    {
        "video_name": "AqMT_zB9rP8",
        "Q": "At 0:26, why did Sal wrote a number 1 in between of the quantities? What is the purpose of that?\n(4x\u00c2\u00b2y - 3xy + 25) - 1 (9y\u00c2\u00b2x + 7xy - 20)",
        "A": "Just to show that it is a negative one for all number in the parentheses instead of just the first number."
    },
    {
        "video_name": "AqMT_zB9rP8",
        "Q": "At around 1:30, he adds -3xy, and -7xy into -10xy. Could you add -3xy and -7yx, and still get-10xy because of the commutative property of multiplication?",
        "A": "Yes! Good observation -- the commutative property of multiplication does apply. Many students miss that concept."
    },
    {
        "video_name": "IKsi-DQU2zo",
        "Q": "Would it have been possible at 1:50 to square the bottom and square the top (a+2)? Or would that have made a different question?",
        "A": "You can t. That would change (a-2)/(a+2) completely. The reason why multiplying by (a+2)/(a+2) is okay because (a+2)/(a+2) is 1. Any number multiplying by 1 will have its value remain the same. If you are squaring it, it is the same multiplying by (a-2)/(a+2), which isn t 1."
    },
    {
        "video_name": "IKsi-DQU2zo",
        "Q": "at 3:00 why did Sal make it into a difference of square, couldn't he have just left it as\n(a+2)(a-2)(a-3)/(a+2)(a+2), then crossed out the (a+2)'s and simplified? (just curious)",
        "A": "No he couldn t, listen carefully around at 4:05 up to 4:21 to see why. The final answer is: a\u00c2\u00b2-a-1/(a+2)\u00c2\u00b2 Because if we factor a\u00c2\u00b2-a-1 it will not turn to (a+2)(a-2)(a-3). In fact if solve this using the quadratic equation you will see that it will factor into complex numbers."
    },
    {
        "video_name": "IKsi-DQU2zo",
        "Q": "@1:15 what happened to number 4 in the denominator?",
        "A": "Are you talking about the 4 in a^2 + 4a + 4 ? That polynomial was factoring into (a+2)(a+2) using factors for 4 that add to 4. If you don t understand this process, you need to review all the techniques for factoring. They are used in nearly every problem involving rational expressions. So, If you don t understand the factoring, you re going to be lost."
    },
    {
        "video_name": "XN48Ba9Ncr8",
        "Q": "at 0:23 when u said that if we could explain a relationship mathematically between the mystery box and the mass what did u mean by that.\n\nDevraj",
        "A": "He actually said that we can explain the relationship between both sides of the scale mathematically. Because the scale is balanced, we know that the masses on each side must be equal, so side A = side B. He then just replaces side A with the masses (and mystery box) that are on side A and and side B with the masses that are on side B."
    },
    {
        "video_name": "XN48Ba9Ncr8",
        "Q": "At 2:50,when Sal subtracts 3 from both sides,you can cancel out the +3-3 from the left-hand side to get ?=10-7,so you could see immediately that ?=7!Cool,or what?",
        "A": "Cool! It s really awesome that you can do that!"
    },
    {
        "video_name": "XN48Ba9Ncr8",
        "Q": "I don't understand 1:00",
        "A": "You don t know the mystery mass but when you add 3 and it adds to 10 it means the mystery mass is 7"
    },
    {
        "video_name": "XN48Ba9Ncr8",
        "Q": "I already know how to solve all kinds of equations but I have one question.\nWhy do we even have to remove the three squares from the left side?\n2:06.",
        "A": "An equation is not solved until you know the value of X. This means X must be on one side of the equation by itself, which is why Sal needs to bov the 3 squares to the other side."
    },
    {
        "video_name": "XN48Ba9Ncr8",
        "Q": "Why does sal pause at 2:02? He could keep going and still show the same operation taking place.",
        "A": "He pauses so we have a couple of seconds to think about what we might do. It s better to give a person a chance to figure something out before telling them. At least he still told us."
    },
    {
        "video_name": "XN48Ba9Ncr8",
        "Q": "At 3:06, can we really use question mark in algebra or this is just for beginners to understand?",
        "A": "I think it s just for beginners to understand. In algebra, we use alphabets like x and y for variables. I ve never seen a question mark being used as an unknown number in a math book."
    },
    {
        "video_name": "XN48Ba9Ncr8",
        "Q": "how do i solve 40:X=8:2 what steps do i take",
        "A": "Multiply 8 by whatever it takes result in 40. Multiply 2 by the same amount. The ratio will not have changed. Rewrite the equation replacing 8 and 2 by the two multiples."
    },
    {
        "video_name": "XN48Ba9Ncr8",
        "Q": "Around 0:54 cant you just subtract 3?",
        "A": "Ludvig, Yes, you can. And that is exactly what the video shows you at 2:47. The video is trying to teach the concept of why you subtract the same thing, in this case 3, from both sides of an equation."
    },
    {
        "video_name": "XN48Ba9Ncr8",
        "Q": "at 2:13 why do you have to remove the three things on the scale",
        "A": "The mystery mass plus the three things on the left side is equal to the whole mass of 10 things together on the right side. If you want to know the mass of the mystery box, you remove the three things from both sides of the scale so that they will remain balanced. After the three things are gone from both sides, the mystery box equals whatever is left over on the other side because the scale is still balanced; in this case, it is 7 things."
    },
    {
        "video_name": "XN48Ba9Ncr8",
        "Q": "\u00e2\u0099\u0089in 1:51, there is 3 1 kg blocks and on the other side there are 10 kg. so isn't the ? block 7? \u00e2\u0099\u0089",
        "A": "Well yes, it s quite simple math being 10 = 3 + X which can be easily done 10 - 3 = X"
    },
    {
        "video_name": "XN48Ba9Ncr8",
        "Q": "at 2:50, instead of removing 3 from each side, can you just flip the equation around from ? + 3 = 10 to 10 - 3 = ?.",
        "A": "Mathematically...yes...but not in real life."
    },
    {
        "video_name": "XN48Ba9Ncr8",
        "Q": "1:06 isnt it like obvious to you if it 7 should we keep going untill it comes out to 7kgs or do we just stop and put 7?",
        "A": "He is showing a different way to solve the unknown, so he has to show it, despite his knowledge."
    },
    {
        "video_name": "gS6s3MGZlRc",
        "Q": "Why does Sal put Absolute Value signs around the change in x and y? Isn't distance always positive? (Around 1:50-2:00)",
        "A": "Distance is always positive but change in x s and y s can be negative. E.g. change in x s between A(4;9) and B(1;9) is equal to 1-4=-3, but distance is 3 ( |-3|=3)."
    },
    {
        "video_name": "gS6s3MGZlRc",
        "Q": "At 1:24 what is the little number going down?",
        "A": "Subscript to indicate that which point it belongs to. In the example we had 2 points with the (x,y) coordinates. If we labelled them both (x,y) it will be confusing. So we assign either point to be point 1 (usually the first given point) and the other is point 2. So point 1 has the coordinate (x subscript 1, y subscript 1) and point 1 has the coordinate (x subscript 2, y subscript 2)"
    },
    {
        "video_name": "gS6s3MGZlRc",
        "Q": "At 1:37, what does the triangle thing represent?",
        "A": "The triangle thing is the greek letter delta , and it can be used to denote change in when you re doing math or science."
    },
    {
        "video_name": "T-aCweuimis",
        "Q": "@1:24 i don't understand how is it working, i mean why are we going 1 step towards x and 2 steps toward y??",
        "A": "The slope of the function is 2 (or 2/1). Slope is rise/run, so Sal is going 1 step for the run and 2 steps for the rise."
    },
    {
        "video_name": "T-aCweuimis",
        "Q": "Is there a video on how to apply and point out the points on a graph before this? Like on 1:18 I want to know how he does that. I get it but I would like to watch a video to clarify.",
        "A": "Watch the point slope videos."
    },
    {
        "video_name": "T-aCweuimis",
        "Q": "In the video the part around 2:00 if the x goes up by 1 and then y goes up by 3, how did Sal get if x goes up 2, y goes up by 6? It doesn't make sense to me because it should go up 1 and 3 each time, right?",
        "A": "2 over 6 is the same as 1 over three. 2 over 6 simplified is 1 over 3"
    },
    {
        "video_name": "gEGKvx1wlFg",
        "Q": "at 1:00 he says that 5 = 3+2 is the same as 3+2=5\nhow does that work?\ni mean, there's a commutative property for mult and adding\nbut an equal sign?",
        "A": "cuz it s the same thing just backwards same numbers same problem get it?"
    },
    {
        "video_name": "xEDnwEOOf7Y",
        "Q": "At 3:00, Sal says you need to subtract, but don't you need to add as well?",
        "A": "Nope.......Addition isn t needed in that problem"
    },
    {
        "video_name": "xEDnwEOOf7Y",
        "Q": "I am confused. At 4:09 how does 400-150= -550?",
        "A": "The 400 is negative, so it goes like so: -400-150=-550."
    },
    {
        "video_name": "EdyociU35u8",
        "Q": "did any one else see blood on her finger at 6:51?",
        "A": "I think that is probably just ink from the pen."
    },
    {
        "video_name": "EdyociU35u8",
        "Q": "did anyone else notice that when Vi was drawing the fire for the dragon at 6:27 the video got brighter?",
        "A": "Yes, I noticed."
    },
    {
        "video_name": "EdyociU35u8",
        "Q": "At 3:25, how did that happen? Vi used trapezoids to create that triangle, (which I forget the name of) but that triangle was made of triangles. I am so confused.",
        "A": "I think drawing it with trapezoids is like drawing it with triangles with the tops cut off. Same shape, different pattern."
    },
    {
        "video_name": "EdyociU35u8",
        "Q": "nice dragon @ 5:30 :)",
        "A": "It is math class so she has to throw the dragon."
    },
    {
        "video_name": "EdyociU35u8",
        "Q": "Why does Sierpinski's triangle appear everywhere? Reference: 3:22",
        "A": "Just the beauty of math"
    },
    {
        "video_name": "5iUh_CSjaSw",
        "Q": "1:22\nwhen you say 9.999999..... or .33333........ you can turn those numbers into a fraction. like .333333......... can be turned into 1/3",
        "A": ".33333..... and 1/3 are the same. If we didn t include the dots, they would be different. For example, .33333 does not equal 1/3, it is slightly less."
    },
    {
        "video_name": "5iUh_CSjaSw",
        "Q": "So, when Vi gave the analogy about choosing a random # on the number line at 2:44, what does she mean? If you are choosing randomly, doesn't the probability stay the same? E.X P(picking 5) = P(Picking 3.3252524534).",
        "A": "Yes, the probability will be the same for any specific real number. Since there are infinitely many real numbers, the chance of hitting any specific number is zero. What may be interesting, however, is that the chance of hitting a rational number is also zero, meaning that the chance of hitting an irrational number is 100%."
    },
    {
        "video_name": "5iUh_CSjaSw",
        "Q": "in the video at 0:20 seconds vi says pi isnt infinate but what if you did 3.14111111{infinity}",
        "A": "A number being infinite means that it is infinitely large, larger than any counting number (1, 2, 3, 4, 5, and so on). Pi is not infinite, because it is not larger than any counting number, it has a position between 3 and 4. 3.14111111... is also not infinite, because it is also between 3 and 4. Although pi has an infinite amount of digits after the decimal point, so does every other number; 4 can be written as 4.000000000... with an infinite number of zeros after it."
    },
    {
        "video_name": "5iUh_CSjaSw",
        "Q": "At 2:38 what is i and e ?",
        "A": "i is the square root of -1 I think, and I do not see an e."
    },
    {
        "video_name": "5iUh_CSjaSw",
        "Q": "I think I've found why the chance is 0 at 2:45\nNow, for every rational number, there are infinite irrational numbers between it's neighbors, so the chances are 1/infinity. However, if you go by a system of mathematics that says that 1/0=infinity, then we have 1/infinity=0 (Similar to how 6/3=2 and 6/2=3). Therefore, the chances are 0%",
        "A": "wait but shouldn t 1/infinity approach 0 (limits) so chances approach 0 but never exactly 0. Also, if any number can be represented as x/y with y >< 0 then shouldnt it be rational?"
    },
    {
        "video_name": "5iUh_CSjaSw",
        "Q": "[0:28] Are referring to Gaben when you mentioned that some cultures only count to 3?",
        "A": "To tell you the truth I don t know, but I don t think it is an opinion answer. It either is or it isn t. Maybe there is some sort of opinion argument regarding what infinite is or something like that. It s between 3 and 4, so it can t be infinite I believe is her point. But then how many points (numbers) are there between 3 and 4. Maybe that s infinite. It reminds me of fractals somehow, or at least the infinite part."
    },
    {
        "video_name": "5iUh_CSjaSw",
        "Q": "Just a thought: 1/3 is a repeating decimal, while Pi is not. Pi does not repeat in any form or fashion.\nAnd my question: Doesn't that make them different? Sure, they both go on forever, but doesn't Pi deserve a little bit of acknowledgement for that? And also, triangles and squares and so on do not use an irrational decimal so famous as Pi for for the circumference calculations (3:46). If you ask someone what the square root of 2 is, they say 1 or 2. No put down intended here.",
        "A": "\u00cf\u0080 does not deserve any acknowledgement for that, because there are an uncountably infinite set of irrational numbers, only one of which is \u00cf\u0080. And while \u00e2\u0088\u009a2 is less famous than pi, people still know more than that it is between 1 and 2. Most would say 1.414. And \u00cf\u0086 is at least as famous as \u00cf\u0080."
    },
    {
        "video_name": "5iUh_CSjaSw",
        "Q": "At 2:40 she says that the chance of getting a rational number when throwing a dart at a number line is exactly 0. That implies that the chance of getting exactly pi is (nonzero, so) greater than the chance of getting exactly five, am I correct?",
        "A": "Well, not really. Five and \u00cf\u0080 would both occupy exactly one point. (as well as every other single number in existence) Each would be equally impossible to hit. (an infinite number of points in a plane)"
    },
    {
        "video_name": "72xChzYjJdM",
        "Q": "At 2:25 Sal says that C is -1. But if we would add plus c to the other side of the equation so that we have sin(y)+2y + c = x^2 it results in c= 1 not -1. Why is -1 right if it is? Is there a rule a to analyse where to add c. The rest is clear to me only that constant adding looks strange.",
        "A": "If you think about it, moving C from one side to the other will require you to add or subtract it depending on its sign. Adding -1 to the other side will result in 1 vice versa subtracting the 1 to other side will result in -1. Both answers are correct."
    },
    {
        "video_name": "HJV_HY0Sh0s",
        "Q": "at 1:33 sal subtracted one-fourth x and one fourth x when he sould only be subracting one fourth x and one-fourth",
        "A": "you are very and absolutly correct"
    },
    {
        "video_name": "HJV_HY0Sh0s",
        "Q": "in 2:12 he says no solution. what does no solution",
        "A": "No solution means that the equation simply does not have a solution; it is impossible to solve."
    },
    {
        "video_name": "a5z-OEIfw3s",
        "Q": "what is abacabadabadabdpattern.? 1:40",
        "A": "If you take a small pattern that repeats in the fraction and label it A, and the next B, and then C, and so on, and then try to pronounce the letters as a word, you will get abacabadabacabaeabacaba... and so on."
    },
    {
        "video_name": "a5z-OEIfw3s",
        "Q": "what is an abacabadabacaba pattern and what did she say at 2:10",
        "A": "If you take a small pattern that repeats in the fraction and label it A, and the next B, and then C, and so on, and then try to pronounce the letters as a word, you will get abacabadabacabaeabacabadabacabafabacabadabacaba... and so on."
    },
    {
        "video_name": "a5z-OEIfw3s",
        "Q": "Isn't the fractal tree at 3:19 close to the number Wau(F) but its 1/2 and 1 instead of 1/2 and 2?",
        "A": "Yes, in this one, the answer is sqrt(2)/2 but she used 1s for the fraction. In the Wau video, she used 2s for the fraction, so the answer is 1."
    },
    {
        "video_name": "a5z-OEIfw3s",
        "Q": "At 1:28, Vi said that it could be \"pi to the i.\" Just curious: what is pi to the i?",
        "A": "Its 0.413292116+i*0.91059849921"
    },
    {
        "video_name": "0sCttufU-jQ",
        "Q": "At about 6:47, why does f(x) - L = e and not e/2?",
        "A": "Because delta = epsilon/2, but | f(x)-L | < 2*delta. So 2*epsilon/2 = epsilon."
    },
    {
        "video_name": "0sCttufU-jQ",
        "Q": "At 3:20-3:35, Sal says that x is not equal to 5 as if the expression |x-c| < \u00e2\u0088\u0082 exactly implies that x is within range of c but not equal to c. For the latter proposition, shouldn't there also be a 0 < at left-hand side of the inequality ? ( 0 < |x-c| < \u00e2\u0088\u0082 ). Because we are trying to approach c, not to get there, given the definition of limit.",
        "A": "Yes, it should be 0 < |x - c| < \u00ce\u00b4."
    },
    {
        "video_name": "0sCttufU-jQ",
        "Q": "I am having difficulty understanding the function which relates delta to epsilon.\n\nAt 6:16, Sal says that since '|2x - 10| < 2 delta' and since '|f(x) - L| < 2 delta' you can now relate delta to epsilon (delta = epsilon/2). He bases this upon the fact that '|f(x) - L| < epsilon'.\n\nThis would make sense if '|2x - 10|' and |f(x) - L| had been equal to '2 delta'. How can you then say that 'delta = epsilon/2' if the relationships that form the basis for this claim are only 'greater than' or 'less than'?",
        "A": "delta less than epsilon/2"
    },
    {
        "video_name": "0sCttufU-jQ",
        "Q": "at 4:11, why isn't C just 5 as it matches up with the other equation?",
        "A": "c is 5. That is what c is defined to be."
    },
    {
        "video_name": "SbpoyXTpC84",
        "Q": "Why does Sal put the denominator in parenthesis when he's using the calculator? 4:26",
        "A": "It s a good habit to make sure that the calculator divides the numerator by the entire denominator. For example, if you put in 10/ (5+5) you get 10/10 =1 If you put 10/5 +5 you get 2 + 5 = 7 (Does the fraction first then adds 5."
    },
    {
        "video_name": "SbpoyXTpC84",
        "Q": "I don't understand anything that Sal said after 5:28. Can someone please help?!\n\n(PS I am a 5th grader going to 6th grade who is trying to get ahead in math class.)\n\n\nThanks!",
        "A": "Dude, do u know the Permutation and Combination formulas? This is actually important in GRE TESTS. This is also focused in High Schools. Please revisit previous videos. This area of math requires some out of the box thinking. Good Luck!"
    },
    {
        "video_name": "SbpoyXTpC84",
        "Q": "At 2:28 Sal says that order would NOT matter. Would order matter in other problems?",
        "A": "In the previous video (ways to arrange colours) the order does matter because the game is about making a code. In this case you consider it a permutation, not a combination. Here, the cards can be reshuffled so the order does not have an effect."
    },
    {
        "video_name": "SbpoyXTpC84",
        "Q": "at 3:50 , mr.khan mentions that there are 9 choices to pick a card, then 9-1 i.e 8 choices then 7 so on...\nY is it 9*8*7*6.... ? shouldn't the total number of choices be 9+8+7+ ...so on ?\n\nis there a intuition behind it ?\n-tnx",
        "A": "The key to understanding it is in the verbiage for each . In the first position, there are 9 choices. For each of those 9, there are 8 choices for the second. That is a total of 72 choices for the first two positions. For each of those 72 choices, there are 7 choices for the third position, and so on."
    },
    {
        "video_name": "SbpoyXTpC84",
        "Q": "at 7:14 I don't get why\n(n)\n(k)\nequals n!/(n-k)!k!\ncould someone explain that to me...",
        "A": "It s just the definition. Just like when we write 5 x 3 the x means multiplication, when we write: (5) (3) the notation means 5 choose 3 . It s just a simpler and more clean way to write a combination."
    },
    {
        "video_name": "SbpoyXTpC84",
        "Q": "00:32 Why don't we have to worry that there are suits within the 36? If we pick a spade, the probability of picking another spade increases, why don't we have to worry about that affecting our result?",
        "A": "Do you perhaps mean that the chance to pick another spade decreases as there are less spades in the deck once one is removed (increasing the chance of picking anything but a spade)?"
    },
    {
        "video_name": "2pwnr_soZEU",
        "Q": "I am have some trouble with these practice problems... the video is nothing like the practice problems after it. At 8:00 there is usually at 2pi * n at the end with the practice problems but not this video. so like t = 365/2pi * (0.9273 + 2pi* n) Anyone now how to help me solve this?\n\nAt 2:30, how does Sal calculate the fraction 1.5/1.7 in his head so fast? I need to develop number sense like that!",
        "A": "He doesn t say, but I assume he was thinking in fractions. Think of 1.5 = 3/2 and 7.5 = 15/2. Dividing is the same as multiplying by the reciprocal so the 2 s cancel and then you have 3/15 = 1/5 = .2."
    },
    {
        "video_name": "2pwnr_soZEU",
        "Q": "at 5:45 khan uses his calculator he puts in cos^-1 -0.2 and gets 1.7722 I put the same thing in to the calculator that khan provides in the skill sinusoidal models word problems and i get 101.5370, please help",
        "A": "Put your calculator in radian mode. Your answer is the same angle measure, but in degrees."
    },
    {
        "video_name": "a2Ia_ZlUCaQ",
        "Q": "At 1:20, sal says \"just grows realy, really fast.\" This is a distraction. The function ln(x) goes to infinity as x goes to infinity, but it goes rather slowly. \"Fast\" is not the issue.",
        "A": "Sal s not talking about \u00f0\u009d\u0091\u00a6 = ln(\u00f0\u009d\u0091\u00a5), but \u00f0\u009d\u0091\u00a6 = 1 \u00e2\u0088\u0095 \u00f0\u009d\u0091\u00a5, which indeed grows really, really fast as \u00f0\u009d\u0091\u00a5 \u00e2\u0086\u0092 0\u00e2\u0081\u00ba. He never implies that this is the case with all functions that tend to infinity for certain values of \u00f0\u009d\u0091\u00a5."
    },
    {
        "video_name": "u2hLYcmI5y4",
        "Q": "Why is pie a \"slang\" term for pizza, in the annotation at 2:55?",
        "A": "A pizza can be called a pizza pie or, the slang term, just a pie."
    },
    {
        "video_name": "I9eLKDbc8og",
        "Q": "Why at 2:24 he said hairier what does that mean ( I don't mean really hairy)",
        "A": "It is a colloquial, or slang expression used to indicate that something is difficult, hazardous or problematic."
    },
    {
        "video_name": "I9eLKDbc8og",
        "Q": "At 1:52, if we were trying to find the perimeter, would we use x cubed, instead of x squared?",
        "A": "No, because cubed means volume and thats only for 3D shapes such as a cube. Squared is for 2D shapes such as a square. If we want the perimeter of a cube we to refer to a 2D shape so think of each face of the cube is a square. To find the perimeter of one of the squares we take the 4 sides and add them up, and ti get the area of the cube, we multiply it by 6 because there are six faces on a cube. Hope this helps!"
    },
    {
        "video_name": "I9eLKDbc8og",
        "Q": "at 0:28, Sal says 3 sides, instead of 6\nSorry! I know this isn't a question, but where can you give comments?",
        "A": "Use the tips and thanks section to give comments. However, this is a clarification. There is a submit a clarification button for all videos, post it there."
    },
    {
        "video_name": "I9eLKDbc8og",
        "Q": "At 3:19, how would it eaqual 37.5?",
        "A": "It comes from: 6 (2^2) + 6 (1.5^2) Do the exponents 1st: 6 (4) + 6 (2.25) Multiply: 24 + 13.5 Add: 37.5 Hope this helps."
    },
    {
        "video_name": "d9pO2z2qvXU",
        "Q": "At 0:33, I don't get why square root of 8 over 2 is irrational. I mean isn't it 4/2 which is 2?",
        "A": "sqrt(8)/2 is irrational, but sqrt(8/2) is rational"
    },
    {
        "video_name": "d9pO2z2qvXU",
        "Q": "2:18\nwhat about 2 root 7 dived by 7 root 7??",
        "A": "After simplifying that, the \u00e2\u0088\u009a7 s cross out and you would be left with 2 / 7 Which is a rational number"
    },
    {
        "video_name": "d9pO2z2qvXU",
        "Q": "At 1:35 , Sal said something about that line........ Could anybody give me more info on this ' line ' ?",
        "A": "What do you mean? An example of when to use that line is i have 2/3 or .6666666666666666666666666666666666666666, i could just right the line above the first six to make it go on forever."
    },
    {
        "video_name": "d9pO2z2qvXU",
        "Q": "at 0:18 in the video, what does Sal mean by perfect square?",
        "A": "perfect squares are based on squaring whole numbers (1)^1=1, (2)^2 = 4, (3)^2 = 9, then 16, 25, 36, 49, etc. So if you look at a times table, it is all the numbers along the diagonal going from left to right."
    },
    {
        "video_name": "d9pO2z2qvXU",
        "Q": "at 0:22, why is the square root of eight not going to be rational?",
        "A": "sqrt(8) = sqrt(4*2) = 2*sqrt(2) sqrt(2) - not rational"
    },
    {
        "video_name": "d9pO2z2qvXU",
        "Q": "At 0:51, if 5.0 is rational does that make the square root of 5 irrational? Sorry if this might sound like a stupid question.",
        "A": "5 is rational and sqrt(5) is irrational, but neither one determines the other. For example, 4 is rational, but sqrt(4) = 2, which is also rational. Hope that helped!"
    },
    {
        "video_name": "d9pO2z2qvXU",
        "Q": "At 2:01 he said that 7.7... is a rational number but how can it be a rational number if it keeps going on and on.",
        "A": "It can be represented as a fraction. 7.7... = 70/9."
    },
    {
        "video_name": "d9pO2z2qvXU",
        "Q": "From 0:26 to 0:38, why is Sal saying that the square root of 8 is irrational. He then says that when yo simplify (divide by 2) it's irrational. I don't get it.",
        "A": "Sqrt(8)/2 = sqrt(2) when completely simplified. The definition of a rational number is that it can be written as the ratio (fraction) of 2 integers. Sqrt(2) is not an integer. So, it can t be a rational number. Within the real numbers, if a number is not a rational number, then it is an irrational number. This is why Sal said it is an irrational number. All irrational numbers become non-ending and non-repeating decimals if you try to convert them to a decimal form. hope this helps."
    },
    {
        "video_name": "uBVhtGL9y88",
        "Q": "At 4:38, how did sal knew that he has to intersect the X axis at 3\u00cf\u0080/2. If you answer me that cos(3\u00cf\u0080/2) gives \"0\" then why sal only used 3\u00cf\u0080/2 ? \u00cf\u0080/2 is also gives zero !",
        "A": "You are right; cosx=0 at both x=pi/2 and x=3pi/2. However, in order for -2.5cos(x/3)=0, x/3 must equal either pi/2 or 3pi/2. This means x must be either 3pi/2 or 9pi/2. Notice that 3pi/2 is between 0 and 3pi and 9pi/2 is between 3pi and 6pi. In short, even though pi/2 is a solution to cosx=0, it is not a solution to -2.5cos(x/3)=0 since dividing pi/2 by 3 gives us pi/6, which is where cosine equals ((3)^.5)/2, not 0."
    },
    {
        "video_name": "VuJEidLhY1E",
        "Q": "at 3:00, give or take a few, didn't he say that 200*5=5000\nIsn't the correct answer 1000 bags?",
        "A": "1200 is the number of bags eaten all together. According to your theory each man and each woman would have to eat 1 bag each. Then 500 bags would have been eaten. However you do not know how many bags each person eats, only how many have been eaten at the end of the day, which is 1200. This is a fact. w stands for the amount of bags eaten per woman, which you thought was 1. m stands for the amount of bags eaten per man. By solving the equations you can find out, what numbers w and m represent."
    },
    {
        "video_name": "VuJEidLhY1E",
        "Q": "At 5:25 can you multiply the equation by anything?",
        "A": "yes but you will be unsuccessful in solving the problem he is doing that so he can cancel out something"
    },
    {
        "video_name": "VuJEidLhY1E",
        "Q": "At 4:56, why couldn't you just divide the top equation by -2?",
        "A": "Yes, that is another way of doing it. Both Sal s and your way will work to solve the equation."
    },
    {
        "video_name": "VuJEidLhY1E",
        "Q": "Is it just my laptop, or do the closed captions stop at 3:10?",
        "A": "It does that on my computer too..."
    },
    {
        "video_name": "VuJEidLhY1E",
        "Q": "At about 8:32, Sal subtracts 800 from 1100, what if you figure out what \"m\" was first, would you add, or subtract?",
        "A": "if you knew m, you wouldn t have to do anything. you don t have to multiply 400 by 2 and divide and do all that stuff. you just plug the number for m (in this case, 3) into the equation. so it would become 100(3)+400=1100, which is a complete equation... hope this helped!"
    },
    {
        "video_name": "VuJEidLhY1E",
        "Q": "what's the difference between 5:17 and 5:45",
        "A": "actually its 28.23456237856238976 seconds to be exact"
    },
    {
        "video_name": "VuJEidLhY1E",
        "Q": "at 3:00 sal said that \"if m was 5 bags per man then it would be 5000\"don't you think it should be 1000 as:-\n200*5=1000",
        "A": "Yes you are right, Sal made a small mistake in the video, but it had already been corrected. there is a small box in the lower right hand corner that corrects Sal in certain videos."
    },
    {
        "video_name": "t-RXwwBjxsc",
        "Q": "at 3:40\nHow is the derivative negative before x=3 and positive after x=3?\nI am looking at the orange graph of the derivative.",
        "A": "If the derivative (or any function for that matter) has a zero or a discontinuity at any point, it can change from negative to positive, or from positive to negative. Any point that meets those criteria is called a critical point, which makes it a possible extreme point on the graph of the original function. In fact, a critical point is an extreme point only if it switches signs."
    },
    {
        "video_name": "rBaCDC52NOY",
        "Q": "2:49 What is a protocol?",
        "A": "consider it a set of rules for how to interact (an internet protocol will tell two programs how to communicate, a diplomatic protocol may be to shake hands or bow when greeting)"
    },
    {
        "video_name": "rBaCDC52NOY",
        "Q": "(4:05)\nDid he say 104 people or 100?",
        "A": "he said 100 befor"
    },
    {
        "video_name": "rBaCDC52NOY",
        "Q": "@ 7:35 how do the logicians know whats on there fore head?",
        "A": "it think your right"
    },
    {
        "video_name": "rBaCDC52NOY",
        "Q": "t 6:02 only 10 of the people are pionted at",
        "A": "Yeah and have a sneaking suspicion that the perfect logicians arent truly perfect either"
    },
    {
        "video_name": "rBaCDC52NOY",
        "Q": "6:49 it seems that since everyone else has a blue forehead then you would figure out that you do too",
        "A": "it think your right"
    },
    {
        "video_name": "wRBPzExOH2A",
        "Q": "At 4:54 how can you reduce the amount when it's negative plus a negative. I thought it increases",
        "A": "Let s say you got a sum like this: -5 + (-7) etc. That is basically -5 - 7. -5 - 7 = -12. When you re adding a negative from a negative, you re making the answer smaller. When you re subtracting a negative from a negative, you re making the answer BIGGER. e.g. -5 - (-7) = -5 + 7 = 2 Hopefully, this helped you with your question."
    },
    {
        "video_name": "wRBPzExOH2A",
        "Q": "At 0:50 i tested that logic and its not true. Or am I getting it wrong? Like if i substitute q with -10 and n with -2 my expression will be -10-(-2) = -8. So I'm not getting +ve as Sai explained that it doesn't matter. Or how am i to approach his logic?",
        "A": "Now look, you substituted -10 for q and -2 for n. But the thing is that q on the number line is a positive number for you can t substitute with a negative."
    },
    {
        "video_name": "_01wqwsb66E",
        "Q": "1:58, cant you do 25 times 1 ?",
        "A": "I don t think you can have 25... That s because 25 wouldn t go into the other equation which was 10 times c times d squared because it is finding the factors. The way 5 , c and d squared went into that equation 25 wouldn t. Hope that helped :)"
    },
    {
        "video_name": "1PsNIzUJPkc",
        "Q": "at 10:35, Sal draws the vector <5,0> and <3,1> and says that the sum of the former and scalar multiples of the latter will form the solution set.\nI can see that this is in the form of y=mx+c (where c is <5,2>) but since we ARE talking about vector addition, how can the solution set lie along the green vector ?\nI envision that the sum when x2=1 will complete the third side of the triangle. Then when x2=2, the vector formed will have a larger slope etc. What am I missing ?\nThanks",
        "A": "Never mind, I realized that the parallel line to the null space doesn t represent the solution vectors in standard form, but rather the solution set as a whole."
    },
    {
        "video_name": "1PsNIzUJPkc",
        "Q": "At about 8:00 the vector x1,x2 is set equal to the vector b1,0 plus x2 times the vector 3,1. Does this imply the x1,x2 is equal to the augmented matrix specified earlier in the lecture?",
        "A": "The augmented matrix is used to solve for x1 and x2, it is not equal to x1 and x2. In the example, this gives x1 = b1 + 3x2, while x2 = x2 (free variable). Then the vector x is written in terms of its components. x = x1,x2 = (b1 + 3x2),x2 = (b1 + 3x2),(0 + 1x2) = b1,0 + 3x2,1x2 = b1,0 + x2(3,1) in your notation."
    },
    {
        "video_name": "1PsNIzUJPkc",
        "Q": "At 3:06 and 5:43 I'm really confused on what exactly is the matrix and what is the vector and by not knowing that it gets me confused throughout the whole video.Can someone help me understand.",
        "A": "He has a matrix A = [a1 a2] (a1 and a2 are column vectors (1, -1) and (-3, 3), a variable (column) vector x = (x1, x2), and a fixed column vector b = (b1, b2). and: Ax = b."
    },
    {
        "video_name": "iHXdzfF7UEs",
        "Q": "What does the n stand for in S square, at 7:28",
        "A": "A lower-case n typically stands for sample size (in this case, 6). An upper-case N would stand for the total size (in this case, about 300 million?)."
    },
    {
        "video_name": "iHXdzfF7UEs",
        "Q": "At 9:39, why is i = 1 put below the sigma?",
        "A": "Sigma notation always looks like that."
    },
    {
        "video_name": "iHXdzfF7UEs",
        "Q": "Under sigma is that a c = 1 at 9:38? He writes it very tiny, so I can't tell.",
        "A": "no, it is i=1"
    },
    {
        "video_name": "iHXdzfF7UEs",
        "Q": "Where did the \"-x{bar}\" at 9:59 come from? I've totally missed that.",
        "A": "X bar is for the sample mean whereas \u00c2\u00b5 is for population mean."
    },
    {
        "video_name": "iHXdzfF7UEs",
        "Q": "Does the formula at 10:26 work for normal variance calculations and not just sample calculations?",
        "A": "By normal, do you mean population ? If so, not quite. This formula is for samples, where we don t have the data of the entire population. If you DO have data for the entire population, then you can use almost the same formula, except you ll use n instead of n-1 in the denominator."
    },
    {
        "video_name": "QIU1daMN8fw",
        "Q": "At ~5:00, Sal lays out the conditions for invertibility. It seems possible that f of a (unique) x might map to 2 (or more) elements in Y. I don't see how that situation is covered by the stated conditions.",
        "A": "The definition of a function is that for any input x, there is a unique output. So it can t map to 2 or more elements in Y."
    },
    {
        "video_name": "gyqodNhM3EU",
        "Q": "At 3:49, the quantity is multiplied by 46, Why?",
        "A": "The Mega-Millions lottery consists of 5 pulls from a common pool of 56 balls + an additional MegaBall which is from it s own pool of 46 balls. The 46 represents this MegaBall."
    },
    {
        "video_name": "gyqodNhM3EU",
        "Q": "Did you know your chances of getting hit by lighting is 1/10000 chances!\nIts at 4:39",
        "A": "Well i do nowXD!!!"
    },
    {
        "video_name": "nZu7IZLhJRI",
        "Q": "Why are the yellow parentheses at 6:10 unclosed? Did Sal just forget?",
        "A": "Yes, you re right. He should ve closed the sqrt."
    },
    {
        "video_name": "nZu7IZLhJRI",
        "Q": "At about 04:30 of the video, in the 3rd line of the work: when he changes the signs of (c sq. + a sq. - b sq.), doesn't he also need to change the sign of \"4,\" which is the dominator of the fraction? Shouldn't he be multiplying by -1/-1?\n\nThanks for any help.",
        "A": "No. Multiplying by -1/-1 is the same thing as multiplying by 1/1 (since the minuses cancel out) and that would essantially not do anything to the other fraction. -1/1 times 5/1 is the same as (-1*5)/(1*1) = -5/1 = -5. NOT however equal to -5/-1 = 5. Hope that helps."
    },
    {
        "video_name": "nZu7IZLhJRI",
        "Q": "At 1:34, how does Sal get C^2/4 again? Is there a 1 between the a^2- and the (c^2+a^2-b^2/2c)^2 that I am not accounting for? Essentially what I am asking is did he change c^2/4(a^2-[c^2+a^2-b^2/2c]^2) to c^2/4{a^2-1(c^2+a^2-b^2/2c)^2}?",
        "A": "Yes. You re right. You can present any number or expression as itself multiplied by one."
    },
    {
        "video_name": "nZu7IZLhJRI",
        "Q": "What did he simplify at 8:24 and how?",
        "A": "Sal basically split the two fractions of denominator 4 into four fractions with each denominator 2. He then rearranged the formula to sqrt of{(a+b+c)/2} * [{(a+b+c)/2}-a] * [{(a+b+c)/2}-b] * [{(a+b+c)/2}-c] . So, if we define s={(a+b+c)/2}, the formula becomes sqrt of s * (s-a) * (s-b) * (s-c), which is nothing but Heron s formula."
    },
    {
        "video_name": "nZu7IZLhJRI",
        "Q": "at 2:02 how did you get 4c^2?",
        "A": "Then why would you have fraction squared still if the square was applied to denominator?"
    },
    {
        "video_name": "nZu7IZLhJRI",
        "Q": "At 9:45, WHERE does the 2b come from? Does it just drop out of thin air?",
        "A": "He changes (c+a-b)/2 to (a+b+c)/2-2b/2. This is true because: (c+a-b)/2 c+a-b=c+a-b+b-b=a+b+c-b-b=a+b+c-2b. (a+b+c-2b)/2 Separate the a+b+c and the -2b into separate fractions. (a+b+c)/2-2b/2 He does a similar process with the two other fractions after that. I hope this clarifies how the -2b appeared!"
    },
    {
        "video_name": "EgOyJJxuuP8",
        "Q": "At 0:24, I don't understand. We don't learn inches and feet in NZ ... can I please have some help?",
        "A": "Don t worry I find it difficult too! There are 12 inches in a foot, 3 feet in a yard ( so 36 inches in a yard), 1, 760 yards in a mile ( so 5280 feet in a mile and 63360 inches in a mile. Phew!) There are approximately 2.5 cm in an inch. 1 yard is about 0.9 meters. 1 mile is about 1.6 kilometres. Hope this helps!"
    },
    {
        "video_name": "EgOyJJxuuP8",
        "Q": "3:21 why is he multipying by 40?",
        "A": "Because he worked out that the scale of the drawing was 40:1, because the scale of the area was 1600:1, and the square root of 1600 is 40, so to find the real area, you need to multiply by 40."
    },
    {
        "video_name": "EgOyJJxuuP8",
        "Q": "at 3:47, he says the dimensions is 40x40, presuming it's a square, but it says rectangular up above. I know a square is a rectangle, but how could he be sure those were the dimensions?",
        "A": "He could do the reverse operation to check his work"
    },
    {
        "video_name": "HUn4XwV7o9I",
        "Q": "At 0:51 Khan draws a line on the graph; How do we know where to draw the line?",
        "A": "At 0:51, he was just drawing an example of a line. That was a rough and random line of a graph, not a literal graph."
    },
    {
        "video_name": "HUn4XwV7o9I",
        "Q": "where did he get -8 at 0:32 ?",
        "A": "HumanBoomBox, At 0:32 Sal multiplied the 2*-4 to get the -8"
    },
    {
        "video_name": "lUD7ti47IQc",
        "Q": "At 3:33, I believed that ds as a vector quantity was equal to the magnitude of your same partial parameters crossed with each other which is then multiplied by dA.",
        "A": "You can t cross the magnitude of one vector with the magnitude of another vector. Scalars don t have a cross product."
    },
    {
        "video_name": "dToPqCXHNDA",
        "Q": "I am confused. At 3:11, he moved the decimal one to the left, but dividing by 10 moves it one to the right, so did he multiply it?",
        "A": "When you divide the decimal goes to the left because it is getting smaller. When you multiply the decimal goes to the right because it is getting bigger. It is like a number line. i agree with Emma because every fraction can be multiplyed"
    },
    {
        "video_name": "gAbadNuQEjI",
        "Q": "What does he mean at 6:02?",
        "A": "A function that takes a 3D vector as input and outputs a 2D vector."
    },
    {
        "video_name": "gAbadNuQEjI",
        "Q": "At 0:36, Sal gives the example of apples and bananas, can we also consider the exchange rate between apples and bananas as a function! It's not exactly a question its more of a concept check!",
        "A": "That would be a bit different. It could be represented as a function of this form: e:F^2->R. Where e is the exchange rate function, F is the set of fruits, and R is the reals in one dimension."
    },
    {
        "video_name": "Kv2iHde7Xgw",
        "Q": "I'm slightly confused at 3:16 of the video. When i got to the stage of: log3x = log 8\ni divided log3 on both sides leaving:\n\nx = log8 / log3\n\nI didn't drop the log on both sides as he did in the video, and when i put it in the calculator, they're two different answers. Is dividing out by log3 agebraicly incorrect or is it just assumed you have to drop the log when they're on both sides of the equation in the above situation?\n\nThanks so much for the video though! It was super helpful :)",
        "A": "Yes, when you have the same log base on both sides of the equation (log base 10 or log base 2) they will cancel each other out. It s like if you had x+2=x+2. The x s will cancel out when you subtract the x s from each other. When solving logs, many of the same rules for solving equations apply, the only difference is in addition to solving for x, you have to condense the logs."
    },
    {
        "video_name": "Kv2iHde7Xgw",
        "Q": "I am pretty confused at 3:14. I don't understand why sal equaled 3x and 8, any explanations?\n-thnks",
        "A": "You want to get the x by itself so you can solve for it."
    },
    {
        "video_name": "Kv2iHde7Xgw",
        "Q": "At 1:40 Sal says that \"log(b) - log(c) = log (b/c)\" is partially derived from \"b*log(c) = log(c^b)\", but I don't see how these two are related. Can someone explain, please?",
        "A": "We know log (a) + log (b) = log (a*b) and we know b*log (a) = log (a^b) so if we take log (b/c) we can rewrite that as log (b*c\u00e2\u0081\u00bb\u00c2\u00b9) = log (b) + log (c\u00e2\u0081\u00bb\u00c2\u00b9) = log (b) + (-1)log (c) = log (b) - log (c)"
    },
    {
        "video_name": "Kv2iHde7Xgw",
        "Q": "2:51 why is Sal dividing 16/2 ? I thought he already used the 2 for the exponent of 4.\n4^2 = 16, so where does the 2 come from to divide 16 with?",
        "A": "It comes from the second term in the logarithm see log10^16-log10^2 instead of ending up with log10^14 you actually divide (its multiplication for addition) but I understand what you mean by you thought he already used the 2 actually that was only to square the 4 to give him the log10^16 see there was a -log10^2 next to it that s where you get your other 2 from the first 2 he used for the exponent of 4."
    },
    {
        "video_name": "Kv2iHde7Xgw",
        "Q": "At 3:10 Sal concludes that from the logarithmic equation he can jump to the equality 3x = 8. Doesn't he need first to fix the range of the x to make this conclusion (or domain of log_10(3x))?",
        "A": "Well, defining the range of x should have been the first thing to do in the beginning."
    },
    {
        "video_name": "Kv2iHde7Xgw",
        "Q": "I do not understand how Sel, cancel out the Log a 3:47. Can someone explain?",
        "A": "log_10(3\u00e2\u0080\u00a2x) = log_10(8) 10^(log_10(3\u00e2\u0080\u00a2x)) = 10^(log_10(8)) 3\u00e2\u0080\u00a2x = 8 x = 8/3"
    },
    {
        "video_name": "Kv2iHde7Xgw",
        "Q": "0:25 Is 10 the most common base so that we don't need to write? And why?",
        "A": "log_10 and log_e are very common so we write log for 10th base and ln for base e. The reason is just to save time."
    },
    {
        "video_name": "Q7htxHDN8LE",
        "Q": "At 0:02 Sal says six but aren't there only three called sine, cosine and tangent?",
        "A": "There s also cotangent, secant and cosecant. This is how they relate to the other three: cot = 1 / tan sec = 1 / cos csc = 1 / sin"
    },
    {
        "video_name": "Q7htxHDN8LE",
        "Q": "At 3:18, He starts to mention the other three trig ratios, CSC, SEC and COT. Are these ratios the same thing as cos-1, sin-1, and tan-1?",
        "A": "Nope - CSC(x) = 1/(sin(x)) .. So, If sin(30)=0.5, cosec(30)=2 (30 = 30 degrees) On the other hand, if we have an equation y=sin(x) .. then It can also be expressed in the inverse form as x=sin-1(y) ..(Notice how sin from one side moves to other side and becomes sin -1 or sin inverse )"
    },
    {
        "video_name": "Q7htxHDN8LE",
        "Q": "At 0:38 when Sal writes SOH CAH TOA he is able to use the triangle side measurements to start filling in the fractions. How would you do this if you were not given all the lengths of the triangle's sides?",
        "A": "Do you just want an expression for sin/cos/tan? Name the unknown sides a, b, c, or some other letters of your choice. If a is adjecant to you angle alpha and c is hypotenuse of the triangle (for example), you can write cos(alpha) = a/c, according to CAH."
    },
    {
        "video_name": "ugtUGhBSeE0",
        "Q": "What the? I didn't get this at all! Why is the center of the green circle moving? The blue curve is a circle, right? And at a certain point of the center of the green circle, the 2 circles (curves) seem to coincide...? somewhere around 1:42.",
        "A": "The blue curve is actually not a circle, otherwise the curvature would be the same everywhere and the green circle would stay the same. The two curves almost coincide when the red point ist at the top but there s still a noticable difference where the blue curve is close to the x-axis. The center of the green circle is moving because the curvature of the blue curve isn t the same everywhere, so the radius of the green circle changes. SInce its edge always touches the red point, its center has to move."
    },
    {
        "video_name": "RuEJzPQ_N10",
        "Q": "I think he makes a mistake at 4:30. \"Since 3/4 is less than e, the natural log of 3/4 is going to be a negative number\" A value has to be smaller than 1 for its natural logarith to be negative. ln(2) for example is positive ALTOUGH it is smaller than e.",
        "A": "Yes, that s right. x: (1, e)-->ln(x): (0, 1) x: (0, 1)-->ln(x)<0 Good job on figuring that out! However, 3/4 is in the interval (0, 1), so ln(3/4) is still negative."
    },
    {
        "video_name": "mu3HHCdYYtY",
        "Q": "at 0:20 the area is 12 square units right?",
        "A": "For 0:20 you re right, A=3*4=12!"
    },
    {
        "video_name": "CYjAgOTVfZk",
        "Q": "at 1:30, couldn't you simply say that by pythag. property, sec^2(x) = tan^2(x) + 1?",
        "A": "Yes you could simply say that, but I guess he prefers to proof things right away instead of just using them."
    },
    {
        "video_name": "CYjAgOTVfZk",
        "Q": "Would it ever be useful to use a \"difference of squares\" rather than trig substitution at 0:46",
        "A": "Most often, when you are taking the square root of a^2 - x^2, you will use trig substitution. Otherwise, you will use partial fraction decomposition to find the answer."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "Shouldn't the first term (at approx 9:30 in the video) be fhx^2 ?\nHe has written just fhx. But I imagine he got that by multiplying the first terms of each binomial (fx) , (hx) , which to me seems like it should be fhx^2 ?\nWhat am I missing ? Thanks.",
        "A": "You are correct a little box pops up on the side correcting the mistake. Good catch."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "At 9:35, wouldn't ' fx * hx ' be equal to ' (f*h)*x^2 ' (f times h times x-squared) instead of just ' fhx ' ?\nTake ' (2*3) * (2*4) ' , for example: that would be ' (3*4)*2^2 ' (3 times 4 times 2-squared), NOT ' (3*4)*2 ' , right?",
        "A": "Indeed it should be fhx^2. There s an annotation pointing out the error a few seconds later, though."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "at 9:26, sal said to not use \"i\" as a variable. Why is that?",
        "A": "Sal said that because i is the square root of negative one."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "At 6:33, how do you describe it as fun?",
        "A": "Math is fun, especially once you understand the concept fully. Like Sal said, you ll get the hang of it and it will be fun!"
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "at 6:12 how does he factor out x+7?",
        "A": "For a moment, let s pretend the equation was 4xa - 3a. You can factor out an a, correct? So then you have a(4x - 3). This is just the same exact thing if a = x + 7. You have 4x(x + 7) -3(x + 7). The two terms share a common factor of x + 7, so you can factor that out to get (4x - 3)(x + 7)."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "At 9:24, Sal doesn't like using \"i\" as a variable because it represents the imaginary unit right?",
        "A": "yes, i is the square root of -1, so it gets confusing as a variable."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "At 9:35 Sal multiplies fx by hx and gets fhx. Wouldn't that be fhx^2 because x*x=x^2 and f*h is fh",
        "A": "he apologizes in the next video"
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "At 13:30 shouldn't it be -x((x-10)(x-7)) instead of -x(x-10)(x-7)",
        "A": "Your 2 expressions are equal. The commutative property of multiplication tells us that the order of multiplying factors does not matter: 2 * 3 = 3 * 2. And, the associative property tells us that the grouping symbols for multiplication can be changed without impacting the result. These same properties apply to your expression. The extra parentheses in your 1st version are not needed."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "At 8:30, why was Sal's answer in reverse?",
        "A": "What do you mean by in reverse? Sal s answers are correct: (x+1)(6x+1) With commutative property of multiplication, they could also be written as: (6x+1)(x+1) The order of the factors does not matter. If this is not what you meant, then comment back."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "6:23, why is it called a quadratic even though it is a trinomial?",
        "A": "The quadratic refers to the degree of the polynomial. The highest power in the polynomial is x^2, which is commonly referred to as quadratic. The trinomial part refers to the number of terms in the equation. For example x^2 + 4x + 3 is quadratic and a trinomial because it has three terms and is of degree two. The expression x^2 + 3 is quadratic but is a binomial because it only has two terms. I hope this is correct and helpful to you."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "@6:20 or so, Sal changes 4x (x + 7) - 3x (x + 7) into (x + 7) (4x - 3). Where did that other (x + 7) go? (Sorry if this question is worded poorly)",
        "A": "He factored it out, which basically meant he discarded it. I know that seems a bit strange, but that s how it works."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "At 9:22, Sal uses \"j\" instead of \"i\" and says we will learn in the future why he doesn't like using \"i\". Why doesn't he like using \"i\"? If you put the dot on the \"i\" it shouldn't be confusing.",
        "A": "As you know from earlier parts of KA, we generally can t find the square root of negative numbers. But in math, i is and irrational number and equals the square root of -1, so mathematicians (and physicists and engineers too) avoid using i to denote an unknown so that there s no confusion."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "When you start the process of factoring out at 4:50, why do you that instead of foil? i am confused when to foil and when to factor in the way that you did.",
        "A": "Okay, so he isn t foiling because the two groups are connected by an + sign. For example: (3x + 6)(x^2 + 3) compared to... (3x + 6) + (x^2 + 3) In the first example you would foil, but in the second you would factor and then add like terms. Hope this helped."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "At 9:36 isn't it supposed to be fhx^2? Instead of fhx.",
        "A": "That part of the video has been updated and now has an icon that corrects the err. Nice catch."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "Does Sal mean \"a x c\" since the equation is ax^2+bx+c and does he mean simply \"b\" because 4+25= 29. I learned this in class but I wanted to make sure. So can someone clear this up you can find it in 1:00 to 1:42",
        "A": "Yes, when Sal uses a and b here, he s not using the same a s and b s that are in the equation ax^2+bx+c. Well noticed! Instead of using a and b for the factors, we could use d and e : d*e = a*c d+e = b"
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "I'm a bit confused by the use of a and b here. They seem to be different from the previous videos, because otherwise a*b should be equal to -21. But what exactly are a and b in this video? And why is a*b=4*-21=84 and a+b=25 (1:30)\n\nEdit: Found the answer, Sal explains it at 8:45",
        "A": "Because the quadratic used in this video has a leading coefficient other than 1 (in this case 4), when factoring by this method you first need to multiply that coefficient by the constant (-21) rather than simply trying to find a*b=-21."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "around 6:20, how do we factor out a (x + 7)???",
        "A": "thanx"
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "I understood everything up until 6:05 when he explains how to \"factor out the x+7.\" How do you go from 4x(x+7) x -3(x+7) to (x+8)(4x-3)?",
        "A": "Both sides have (x+7), so you can just combine those. Remember, you re not multiplying the groups, you re adding them. Since the 4x and the -3 are both outside the parentheses, you can combine them into their own parentheses."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "At the start of grouping and khan writes a^b he uses a and c. Why?\n6:45",
        "A": "Unfortunately, Sal chose the same variables to represent the numbers he is seeking as we use to define the coefficients of standard form for quadratics: Ax^2 + Bx + C Relabel Sal s a and b as m and n . The video will make more sense. You multiply A*C (the 2 coefficients from the quadratic) Then you find the values m*n = AC and m+n = B Hope this helps."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "at 6:27 why dont you combine the x+7 together to have x^2+7^2?",
        "A": "1) Sal is factoring. You are trying to multiply. Multiplication is the opposite of factoring. 2) There is only 1 factor of (x+7). I think you are trying to change (x+7) into (x+7)^2. But, that would mean the factors would be: (x+7)(x+7)(4x-3). Those factors are incorrect. Rewatch the video, you will see that the (x+7) is not squared. 3) (x+7)^2 does NOT equal x^2 + 7^2. To multiply binomials, you need to FOIL them."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "at 4:10 does the order of the 2 terms change the value of the expression?",
        "A": "The order of the terms, because there is an addition sign, doesn t matter. If there was a subtraction sign, you would have to put the negative sign in front and make the sign in the middle an addition sign. Example: 1-x -x+1 x^2-y^2 -y^2+x^2 I hope this helped!"
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "can someone do what sal does @ 9:14 with a, b, and c because im confused when he uses f, g, and h",
        "A": "In that part f, g, h and j are just used as variables. Sal uses a and b to help with factoring."
    },
    {
        "video_name": "X7B_tH4O-_s",
        "Q": "At 8:20, why factor out a 1?",
        "A": "The greatest common factor of x and 1 is 1. Explicitly factoring this out makes it easier to see the next step there we factor out an (x+1) from each of the two remaining terms to get (6x + 1)(x+1). You don t HAVE to write the second term as 1(x+1) but I think it makes the next step clearer."
    },
    {
        "video_name": "FJ7AMaR9miI",
        "Q": "At 3:45, why is the derivative of the curve the same as the slope of its tangent line?",
        "A": "Because the derivative was defined to be exactly that."
    },
    {
        "video_name": "pXo0bG4iAyg",
        "Q": "at 9:41, wouldn't it be to the 11th power since it's to the power of n-1 in the formula without the numbers?",
        "A": "No, because we found that to model this situation, we needed to use the alternate formula of 0.6^n (120) So, the exponent is n when we start counting at 0"
    },
    {
        "video_name": "pXo0bG4iAyg",
        "Q": "At 9:41 when Sal does the equation for the 12th bounce, wouldn't he bring 120(.06)to the eleventh power since n-1 (where n=12) would equal 11?",
        "A": "No: The 12th bounce is the 13th jump."
    },
    {
        "video_name": "pXo0bG4iAyg",
        "Q": "At 6:59, what is the next erm in the geometric sequence 16,-4,1,...",
        "A": "16, -4, 1 a(n) = a(0)\u00e2\u0080\u00a2r^n a(0) = 16 r = a(1)/a(0) = -4/16 = -1/4 a(n) = 16\u00e2\u0080\u00a2(-1/4)^n n a(n) 0 16\u00e2\u0080\u00a2(-1/4)^0 = 16\u00e2\u0080\u00a21 = 16 1 16\u00e2\u0080\u00a2(-1/4)^1 = 16\u00e2\u0080\u00a2(-1/4) = -4 2 16\u00e2\u0080\u00a2(-1/4)^2 = 16\u00e2\u0080\u00a21/16 = 1 3 16\u00e2\u0080\u00a2(-1/4)^3 = 16\u00e2\u0080\u00a2(-1/64) = -1/4"
    },
    {
        "video_name": "pXo0bG4iAyg",
        "Q": "At point 8:00 .is that process really wrong ???????cant i use the process of (n-1)????",
        "A": "thanx a lot!!!!!"
    },
    {
        "video_name": "pXo0bG4iAyg",
        "Q": "At 2:36, Sal says that -10/3 * -1/3 = 10/3, while it is actually 10/9. There is a correction in the small screen mode, but it should be reshot because you do not spot the mistake in full screen. Sal also corrects his mistake afterward, but still, someone can get confused.",
        "A": "At 2:23, as well as a few other times, yes I agree as I almost missed it and had to rewind."
    },
    {
        "video_name": "pXo0bG4iAyg",
        "Q": "Sal wrote 10/3, not 10/9 @ 2:46.",
        "A": "That has been found a while ago and a text box on bottom right stated that -"
    },
    {
        "video_name": "pXo0bG4iAyg",
        "Q": "How do I apply the word problem given (at 4:16), to a different geometric sequence problem that asks me \"What is the 4th term in the sequence -10, 100, -1000?\" I know my answer would be 10,000 since I am multiplying by -10. But what if I was asked what was the 20th term or so in the problem I typed here? Which does have any percentages.",
        "A": "So the formula is A(n) = a * r ^ n-1 where A(n) = required term a = first term r = common ratio n = Term required A (20) = (-10) * (-10) ^ 20-1 = -10 * (-10) ^ 19 = + 10^20"
    },
    {
        "video_name": "pXo0bG4iAyg",
        "Q": "at 6:03 should'nt sal put the end ) when he finished writing (120 ? it's multiplication",
        "A": "Yeah,I think so too"
    },
    {
        "video_name": "pXo0bG4iAyg",
        "Q": "at 5:45 Sal says 0.6 x 0.6 x 120 couldn't he just say 0.6 x 72. in fact couldn't he just multiply 0.6 times the previous answer. it would be a whole lot simpler and easier to understand",
        "A": "He could have, and your right: 0.6 * 0.6 * 120 is harder to understand 0.6 *72. However, the average beginner who s starting on geometric sequences will not understand how someone got 72. It s hard to make that leap. The Sal does it, he can understand every element in the equation, and how Sal got it. Hope this helps!!"
    },
    {
        "video_name": "d5lcGCbV5cM",
        "Q": "At 8:00 are where sal says uses the calculator to find out how many meter per second couldn't he have just shifted two decimal points",
        "A": "i don t think so, because he was dividing not multiplying"
    },
    {
        "video_name": "EzE53aPGbrQ",
        "Q": "At 5:23 he told that if it was another angle, it would not be a perfect cylinder but, it actually would still be a perfect cylinder.",
        "A": "Sal is not saying that a non 45\u00c2\u00ba angle would not yield a cylinder. What he is telling us is that the 45\u00c2\u00ba makes the dimensions very nice because it leads to the diameter equaling the height of this cylindrical section. On the actual AIME, everything must be done by hand (i.e. without calculator) and everything is an integer answer. The 45\u00c2\u00ba ensures that our final answer is indeed an integer."
    },
    {
        "video_name": "rfm0wQObxjk",
        "Q": "why can he assume that there is a nontrivial solution to the null-space, even if that is what we are looking for at 5:50",
        "A": "Because if there is a non-zero eigenvector, then there must be a nontrivial solution (what Sal explains at around 1:40 and 5:20), on the other hand if there is no non-zero eigenvector, then the matrix \u00ce\u00bbI-A has linearly independent columns, which (combined with the fact that it is square matrix) means that it is invertible and therefore det(\u00ce\u00bbI-A) = 0 has no solution, so the assumption that there is a nontrivial solution can t be harmful."
    },
    {
        "video_name": "6DpzCKJBsz0",
        "Q": "At 8:25 he says \"The third row minus two times the first row\", does he mean \"The third row minus two times the second row\"?",
        "A": "You are right. This was reported in tips and feedback 2 years ago (by me actually LOL). Wow! Time flies..."
    },
    {
        "video_name": "iI_2Piwn_og",
        "Q": "At 4:12, why did you flip the sign? Why wouldnt it just be 7x>21? I have a test tommorow and I have no idea why you do this... NEED HELP",
        "A": "Well, you need to flip the sign if you re multiplying/dividing by a negative number."
    },
    {
        "video_name": "iI_2Piwn_og",
        "Q": "At 7:44 what does he mean that 5x is less than 4",
        "A": "He means exactly that. When you subtract 3 from both sides in both equations, you find that 5x is greater than -10, and it is also less than 4. Maybe you could specify what you don t get? Hope I could help! :)"
    },
    {
        "video_name": "iI_2Piwn_og",
        "Q": "at 12:00 why dose he reverse the sign on 2x>-58 when he divides by 2 but dose not reverse the sign when dividing by 2 on the 2x<-68 equation ?",
        "A": "Only reason the sign should change is when you are dividing the other side by a negative number"
    },
    {
        "video_name": "iI_2Piwn_og",
        "Q": "At 4:10, If you have |7x| >= 21 and |7x| <= -21, wouldn't the absolute value for the 2nd inequality be \"no solution\" since the absolute value cannot be negative? I think I'm missing something. Please explain. Thank you.",
        "A": "Watch it again. In the two inequalities you mention, the absolute value bars are not still there. You would be correct, but the two possibilities are that 7x >=21 or 7x <= -21."
    },
    {
        "video_name": "iI_2Piwn_og",
        "Q": "what is \"interval notation\" as Sal uses at 2:00?",
        "A": "We use [ and ] and ( and ) to describe intervals. For 5<= x <= 10 we write [5,10] For 5<=x<10 we write [5,10) for 5<x<10 we write (5,10). for 5<x<=10 we write (5,10] As you can see ( represents an open interval in which the end point is not included and [ a closed interval where the end point is included."
    },
    {
        "video_name": "iI_2Piwn_og",
        "Q": "at 9:33 how can A be a negative?",
        "A": "It s an absolute value so that means that even if the number is negative it gets changed to a positive."
    },
    {
        "video_name": "iI_2Piwn_og",
        "Q": "what is the big difference between Equations and Inequalities? he corrects it at 8:29. it's always being corrected.",
        "A": "An equation means that it has an equal sign. When this happens, it shows that there is the answer, whereas with inequalities, it says that it is greater than this. For instance, say you had: |x-4| = 7 This would work out to be: x = 11 OR x = -3 If we made this problem an inequality, it would work out the same but the signs would be different: |x - 4| < 7 x < 11 AND x > -3"
    },
    {
        "video_name": "iI_2Piwn_og",
        "Q": "How come Sal says equation 11:35 when the problems are both inequalities!\nIt's not the two problems that are the equation, they have the word \"or\" in between.",
        "A": "Were you referring to how he said inequality instead of equation? Sometimes Sal makes mistakes. You can catch these mistakes, so it must not be a big deal. The videos he makes are still awesome, though, despite the minor errors."
    },
    {
        "video_name": "iI_2Piwn_og",
        "Q": "11:12 how does multiplying by 7 get rid of the denominator and why doesn't he multiply 2x by 7 as well?",
        "A": "If you look more closely on what he is doing, you ll get it. When he multiplies the inequality 2x/7+ 9 > 5/7 by 7, you ll have 14x/7 + 63 > 35/7 ( he does multiply the numerators by 7) when u simplify you ll end up with 2x + 63 > 5 ."
    },
    {
        "video_name": "iI_2Piwn_og",
        "Q": "At 4:00, could you start by saying 7x \u00e2\u0089\u00a5 21 or -(7x) \u00e2\u0089\u00a5 21? Would that give you the same answer since you have to flip the inequality when you divide by a negative? Thanks!",
        "A": "Yes, that is correct."
    },
    {
        "video_name": "iI_2Piwn_og",
        "Q": "at 1:21 rather than putting an empty circle on 12 why dont we dark circle 11 . it makes it simple to understand . please help me and thanks in advance!",
        "A": "There are numbers that sit between 12 and 11 on the number line that are solutions. For example: 11.5. So, if you start your graph at 11, you omit all those solutions. By putting the open circle on the 12, it says start here, but don t include the 12 in the solution set"
    },
    {
        "video_name": "k3aKKasOmIw",
        "Q": "At 2:36 why is the first instance of 23 regarded as less than the second? Is it \"less\" in terms of further to left, and not in amount?",
        "A": "It s not regarded as less than the second one. It s just that we have two of them. Switching them around doesn t change anything."
    },
    {
        "video_name": "k3aKKasOmIw",
        "Q": "At 1:49, why do you have to order the numbers from least to greatest?",
        "A": "okay... so you have to because it helps you find the median"
    },
    {
        "video_name": "k3aKKasOmIw",
        "Q": "I have a question with regard to the mode (3:33):\nI can understand it if there is one number which appears most, but what if you have 2 numbers that appears as much?\nFor example: 1,2,2,4,4,7,8\nWhat's the mode in this case?",
        "A": "Thanks. So if I understand correctly , in my example there are 2 modes : 2 and 4"
    },
    {
        "video_name": "k3aKKasOmIw",
        "Q": "At 3:40 for the mode what if no number repeats more then once?",
        "A": "If no number in a data set repeats, then there is no mode for that set."
    },
    {
        "video_name": "k3aKKasOmIw",
        "Q": "at 1:40, he calculates the mean with a calculator. now it is no problem for me to add all the numbers together. but im afraid i do not think i can divide such an odd number without using a calculator. i am not allowed to use a calculator",
        "A": "increase a zero in the numerator and make that value either greater than the denominator or equal, then use the table and cut the possible fractions and finally , you ll get the most closest answer, then reduce your complete answer by 10^-1 . that works fine for mee atleast."
    },
    {
        "video_name": "k3aKKasOmIw",
        "Q": "What if you're using time when it comes to medians.\nLike 2:41 2:55 2:58 3:09 3:31 3:40\nHow can you find the median using time like this?",
        "A": "You would need to convert to a 24hr clock and to decimal form to do the computations. Since the median is the average of 2 : 58 and 3 : 09, you d get: (2 \u00e2\u0081\u00b5\u00e2\u0081\u00b8\u00e2\u0081\u0084\u00e2\u0082\u0086\u00e2\u0082\u0080 + 3 \u00e2\u0081\u00b9\u00e2\u0081\u0084\u00e2\u0082\u0086\u00e2\u0082\u0080 )/ 2 = (6 \u00e2\u0081\u00b7\u00e2\u0081\u0084\u00e2\u0082\u0086\u00e2\u0082\u0080 ) / 2 = 3 \u00e2\u0081\u00b7\u00e2\u0081\u0084\u00e2\u0082\u0081\u00e2\u0082\u0082\u00e2\u0082\u0080 converting back to hours:min:sec = 3 : 03 : 30"
    },
    {
        "video_name": "k3aKKasOmIw",
        "Q": "At 0:13, what does the word \"Arithmetic\" mean?",
        "A": "The branch of mathematics dealing with the properties and manipulation of numbers. It s on dictionary.com"
    },
    {
        "video_name": "k3aKKasOmIw",
        "Q": "at 8:06, what is an easy way to remember the mean median & mode?",
        "A": "take notes thats my only option"
    },
    {
        "video_name": "XWq8bplP-_E",
        "Q": "1:18 Why not 2^2 * 3^3?",
        "A": "3 goes into 36 only twice, so 36 can only have 3^2 and not 3^3 in its prime factorization. If you attempt to divide 36 by 3^3, you will not get a whole number. (36/(3^3) = 4/3.)"
    },
    {
        "video_name": "Ht-YXje4R2g",
        "Q": "At 1:27, why is the ^3 only related to the 1/5 (1/5^3), and not (6^1/5)^3?\n(6^3)^(1/5^3)?",
        "A": "Exponents don t distribute like that. Remember that an exponent is a number multiplied by itself that number of times, x^a = x * x * x * .... * x with a x s. If we then take (x^a)^b then what we are doing is multiplying x^a by itself b times. But x^a is just x multiplied a times so (x^a)^b is x multiplied by itself a*b times or x^(ab). This relationship doesn t change when we end up with fractional exponents."
    },
    {
        "video_name": "5cK86VKoBPw",
        "Q": "At 3:39, what if that Stewart is another Stewart? Then it would be a function, correct?",
        "A": "It would be a function as long as its domain is properly defined (the Stewarts must be somehow indicated to be different persons) and each person has only one height associated with her/his name. And if the same person is measured at different heights in different moments of time, then different functions are required for each case."
    },
    {
        "video_name": "5cK86VKoBPw",
        "Q": "Dear Sal, I am a little confused here. Can 2 items in the independent variables be associated with the same value in the dependent variables list? Like at 3:00, LJ's and Joel's heights are both 5'6\". If this is okay, then, what will not be a function?",
        "A": "Yes! As long as a function does not produce 2 different values for a given independent variable, it is a function. BUT It would not be a function if Joel (or any one else) had 2 heights. Consider this function, f(x)=5. It outputs 5 no matter what x is."
    },
    {
        "video_name": "KNrEAJeRIdo",
        "Q": "at 2:04, why do you exclude 1/5 from the ratio? Doesn't it have to be (5*9)^n/(5*10)?",
        "A": "Hi, That s because he applies one of the laws of exponents where: a\u00cb\u00a3 * b\u00e2\u0081\u00bf * c\u00e2\u0081\u00bf = a\u00cb\u00a3 * (bc)\u00e2\u0081\u00bf. It s possible to gather numbers or variables under a common exponent if they are raised to the same exponent. Since 1/5 is not raised to the nth power, it was excluded, yielding the value of a in a * r\u00e2\u0081\u00bf. (the 10 in the denominator is raised to the nth power as well)."
    },
    {
        "video_name": "KNrEAJeRIdo",
        "Q": "How does that thing at 1:30 work? What allows you to bring the five to the other fraction and how does the 5/5 end up as 1/5?",
        "A": "Sal combined a few steps in that maneuver. First, we need to see that 5\u00e2\u0081\u00bf/5 can be broken apart into (1/5)*5\u00e2\u0081\u00bf. (This is just the same as saying x/5 is equal to (1/5)*x.) Second, we need to see that (9/10)\u00e2\u0081\u00bf is the same as 9\u00e2\u0081\u00bf/10\u00e2\u0081\u00bf. Now we can combine 5\u00e2\u0081\u00bf with the fraction, giving us 5\u00e2\u0081\u00bf*9\u00e2\u0081\u00bf in the numerator. And then, because everything in the fraction is raised to the same power, we can rewrite it as ((5*9)/10)\u00e2\u0081\u00bf. Combined with 1/5, this gives us the result in the video."
    },
    {
        "video_name": "KNrEAJeRIdo",
        "Q": "from 3:47 to 3:56 how did he move from 2^(n)/3^(n)3^(-1) to 3* 2^(n)/3^(n) ? Did he just multiply by 3 to cancel the 3^(-1) ?",
        "A": "Well, 1 / (3^(-1)) = 1 / (1 / 3) = 1 * 3 / 1 = 3. So: (2^n) / (3^n 3^(-1)) = (2^n) / (3^n) * 1 / (3^(-1)) = (2^n) / (3^n) * 1 / (1 / 3) = (2^n) / (3^n) * 3 = 3(2^n) / (3^n) Write it out on paper, without parentheses, and it may become clear."
    },
    {
        "video_name": "6dyWKD_JPhI",
        "Q": "I get almost all the video, but this one part confuses me. At 1:19, where did Sal get the 3 to minus the 10 with?",
        "A": "Point P is located at (-7,3) and the dilation needs to be 10 away from that. The 3 is the y-coordinate of the original point, so Sal subtracted the 10 from the positive 3 to get -7, the y-coordinate of the new point."
    },
    {
        "video_name": "6dyWKD_JPhI",
        "Q": "1:19 what? how did he do that?",
        "A": "He multiplied 6, which is the distance between P (center of dilation) and C, by the scale factor 1 2/3. 6 multiplied by 1 2/3 equals 10, so the image of point C is 10 units away from P. The image of point C didn t move at all in the X direction, because the initial point was already 0 units away (in the X direction) from point P, and 0 multiplied by 1 2/3 is just 0. Hope this helps!"
    },
    {
        "video_name": "TjOlZr_2uW4",
        "Q": "In 1:09, why would the base of the isosceles triangle be x/2?",
        "A": "The base of the isosceles triangle is not x/2, it is x. Whereas x/2 is the base of each of the right triangles."
    },
    {
        "video_name": "TjOlZr_2uW4",
        "Q": "At the time 1:53 you cut the base to find the hypotonuse on one half of the whole triangle. At 3:11, why was the answer that you got for the whole base then. Wouldn't ten be for one of the sides and then 20 for both of them combined?",
        "A": "Notice: Sal defined x in the very beginning as the entire base. He then used x/2 in the formula. This is why the answer = the entire base. If Sal had used x as just 1/2 the base, then he would have set up the formula as: x^2 + 12^2 = 13^2 Solving gives you: x^2 + 144 = 169 x^2 = 25 x = 5 You would then calculate the entire base by doing: 2x = 2(5) = 10 Hope this helps."
    },
    {
        "video_name": "TjOlZr_2uW4",
        "Q": "Why at 0:17 he said that these are right angles? There's nothing in the diagram that says so.",
        "A": "The altitude that is dropped is perpendicular to the base of the triangle. Altitudes are dropped from the vertex of the triangle and intersect with the base of the triangle to form right angles. I hope this helps you."
    },
    {
        "video_name": "TjOlZr_2uW4",
        "Q": "At 3:03, what's the difference between a square root and a principal root?",
        "A": "All numbers have two square roots - a positive one and a negative one. For example, the square roots of 4 are 2 and -2. The principal root is the positive square root. So the principal root of 4 is 2. I hope that helps!"
    },
    {
        "video_name": "t7NvlTgMsO8",
        "Q": "At 2:00, what is the difference by having your calculator mode in radians or degrees?",
        "A": "I m sorry, but understanding the distinction between radians and degrees is prerequisite for this level of study. You may need to review that subject before continuing with calculus. Angles are measured in radians only in this level of study. Degrees and other means of measuring angles are almost never used. So, you must have your calculator in radians mode. You will get the wrong answers in degrees."
    },
    {
        "video_name": "t7NvlTgMsO8",
        "Q": "At 1:20, why does Sal have to verify that he's in radian mode. What is the difference between radian mode and degree mode and how do you know when to use each?",
        "A": "Its like the difference between miles per hour and kilometers per hour, its just tells the calculator what unit of measure your providing in the trig functions. Example 3.14 radians is roughly equal to 180 degrees."
    },
    {
        "video_name": "t7NvlTgMsO8",
        "Q": "At 2:08 why is it incorrect to substitute zero into the equation to find the limit as the function approaches zero?",
        "A": "The limit of a function as x approaches a value is not necessarily the function value, unless the function is continuous. If it is continuous, which all polynomials and many of the functions you have seen so far are, then by definition, the function value does equal the limit."
    },
    {
        "video_name": "t7NvlTgMsO8",
        "Q": "From 1:30 is Sal just substituting values of x?",
        "A": "Yes. Sal is substituting the x values in the equation for 0.1 :) Replied by, Beyza Duymayan"
    },
    {
        "video_name": "t7NvlTgMsO8",
        "Q": "At 1:55, how do they know that .1 is 2.002",
        "A": "He plugged it into the calculator later on in the video. The calculator program uses something called Taylor-expansion or Maclaurin-expansion to calculate the answer."
    },
    {
        "video_name": "t7NvlTgMsO8",
        "Q": "At 1:20, what happens if you have your calculator in degrees?",
        "A": "it will different answer because both are different units. 1 rad = 180/pi deg; so the answer obtained will be different."
    },
    {
        "video_name": "t7NvlTgMsO8",
        "Q": "At 1:20, Sal verifies the calculator is in radian mode. I understand the difference between radians and degrees. How do I know which I should be using? What part of the problem tells me I should be using radians?",
        "A": "If you are in Calculus, you should completely stop using degrees. You just cannot reasonably work with degrees at this level of math. As far as I know, no calculus is ever done in degrees. It is always radians."
    },
    {
        "video_name": "t7NvlTgMsO8",
        "Q": "What's the 'range' thing at 2:50?",
        "A": "basically the range of values for which you need to find x or y values that are in that range"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 3:44, what is the purpose of the square in the formula for the hypotenuse, A square + B square = C square?",
        "A": "Thank you very much. :)"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 5:03, can the 3(a) and the 4(b) be switched around?",
        "A": "sure a and b can be switched around but c has to be the longest side of right triangle from the formula c^2=a^2+b^2"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 6:53, can the side with the question mark be C?",
        "A": "You can call each side whatever you want. But then of course you would have to change the formula accordingly."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "at5:14 what does he mean",
        "A": "his example was 4^2 + 3^2 = c^2 this is the same as 16 + 9 = c^2 and 16 + 9 = 25 the you d just have to find the cube root of 25 and you have the answer. did that help at all?"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "How come at 10:00 Sal put the last factor 3 as its own separate self and not together with the other factors?",
        "A": "What he is doing to simplify the radical is to find perfect squares so he can remove them from under the radical. He factors 108 and finds that it equals 2*2*3*3*3 so there are two 2s and three 3s. If you take 2x2x3x3 you get 4*9=36, a perfect square (sqrt=6). The remaining 3 must stay under the radical because it cannot be simplified any further."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 5:39, what is a principle root?",
        "A": "The principle root referenced at 5:39 means the main root of a square. For example, square roots of 81 could be both -9 and 9. This is because 9^2 = 81 and (-9)^2 = 81. However, the principle root refers to the positive one, which is 9."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 6:45, is this example a special case of the 30-60-90 triangle?",
        "A": "Yes, it is. In a 30-60-90 triangle, the sides have a ratio of x : sqrt(3) times x : 2x. So since the triangle s sides do have this ratio, it s a 30-60-90 triangle."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 10:15 how does Sal come up with sqr 36 and then 6 sqr 3? I'm confused.",
        "A": "Hello, well first you have to find the square root of 36. That is 6. Then because the square root of 36 is being multiplied by root 3 you would find the square of 36 (6) and then multiply it by root 3. Since six is simplified and has no root attached to it it ends up multiplying to 6 root 3"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 1:26, Do you ALWAYS need to know at least two sides of the right triangle to put the Pythagorean Theorem into effect? If not how do you find the length of the hypotenuse or the missing leg? If so how do you do it?",
        "A": "In order to use the Pythagorean theorem, you will need to be given 2 sides or at least the information to find the values for 2 sides."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 10:41 Sal says the square root of 108 would be a little bit larger than 6, but wouldn't the square root of 108 be a little more than 10, not a little more than 6, because 10 x 10 = 100, so the square root of 108 must be around 10.",
        "A": "Yes, you are correct! The square root of 108 is 10.4 to three significant figures which is indeed a bit larger than 10 rather than 6. \u00e2\u0088\u009a108 can be simplified to \u00e2\u0088\u009a36x3 or \u00e2\u0088\u009a36 x \u00e2\u0088\u009a3 and then becomes 6 x \u00e2\u0088\u009a3 which is simply written as 6\u00e2\u0088\u009a3 in surd form. That s probably why Sal said 6 instead at 10:41."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "8:18 I've noticed that Sal seems to have a bunch of shortcuts for arithmetic problems up his sleeve. I paused and solved that subtraction problem systematically from right to left, but he just sort of used intuition. Are there certain arithmetic tricks I can learn to speed things up?",
        "A": "yes there are I can help you"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 3:41, I learnt that in the Pythagorean Theorem, a right triangle's hypotenuse, equals the sum of a and b squared and divided by 2. Why do we calculate it as 'squared'?",
        "A": "That is because you don t get the right answer if you don t square them because \u00e2\u0088\u009a(a+b) is NOT equal to \u00e2\u0088\u009aa + \u00e2\u0088\u009ab. Thus c\u00c2\u00b2 = a\u00c2\u00b2 + b\u00c2\u00b2 We solve for c by square rooting: c = \u00e2\u0088\u009a(a\u00c2\u00b2+b\u00c2\u00b2) This IS NOT equal to c = (\u00e2\u0088\u009aa\u00c2\u00b2) + (\u00e2\u0088\u009ab\u00c2\u00b2) = a + b For example \u00e2\u0088\u009a(3\u00c2\u00b2 + 4\u00c2\u00b2) = \u00e2\u0088\u009a25 = 5 But, \u00e2\u0088\u009a3\u00c2\u00b2 + \u00e2\u0088\u009a4\u00c2\u00b2 = 3 + 4 = 7"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "Wait a second, so at 10:30 he said you can just leave the answer as the square root of 108. Does that mean you don't need to simplify to 6 times the square root of three?",
        "A": "Yes you can just leave it unsimplified. However, it s more common for it to be simplified. On a test teachers usually ask to simplify it, even though it s essentially the same answer."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 1:50, is the hypotenuse the longest side of the right triangle, or the side opposite of the right angle? Or is the longest side always opposite of the right angle? Or does the hypotenuse needs to be both the longest side, and the side opposite of the right angle?",
        "A": "The hypotenuse is always the side opposite the right angle. It is the longest side IF the triangle is a right triangle."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "In the last question he showed (solving for B squared (6squared + Bsquared = 12squared)) at 8:44, the square root of 108 is, on your calculator, 10.39230485. He says it's 6 square root 3.\n\nI understand how he got that completely, but is 10.3923085 wrong?",
        "A": "Thank you very much. Your answer has helped me with some of my test prep."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "Why would you want to square root both sides? (minute: 5:42)",
        "A": "its because of the properties of equality, if you do something to one side, you do it to the other"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 7:51, can I just take the positive root of both sides to solve for B?",
        "A": "Yes the negative root has no meaning for distance (cannot have a negative distance)."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 3:45, Why is it important to square the variables?",
        "A": "Pythagoras actually found the property by noting that if you squared the two sides, then that is equal to the square of the third side Lets take a basic Pythagorean Triple 3-4-5 If we just add the sides, 3+4 does not equal 5 But if we square 3 (9) and 4(16) and then add 9+16 = 25 and we square 5 (25) we get the same value We can prove the theorem using Geometry and that allows us to apply it with problems, so the importance of the squares are because that is what the theorem states"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 0:42, Sal said the triangle has to be a right triangle. Does this mean you cannot use the Pythagorean Theorem for other triangles? If so, why not?",
        "A": "The Pythagorean Theorem only works for right triangles. It is expanded later into the Law of Cosines (c^2=a^2+b^2-2abcos(C)) which works for any triangle."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At the second question at 6:26, is the length of the unknown side sqrt108? I paused the video(you hear me saying that a lot).",
        "A": "You ve gotten these right 4 in a row! Good job!"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 3:29 he writes side a as a squared, same thing with side b. Why did Sal do that?",
        "A": "The theorem states that if you multiply a*a and b*b will the theorem be valid. Take a right triangle with a=3 b=4 c=5 If you add 3+4=5, the point of the theorem is not proved If you square the numbers then add 9+16=25, the theorem is not valid"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 9:58, why does Sal write the square root of the last 3?",
        "A": "When you are simplifying square roots, you look for pairs in the prime factorization of the radicand. In this case: \u00e2\u0088\u009a(2 * 2 * 3 * 3 * 3) Here we have one pair of 2s and one pair of 3s with a leftover 3. So: \u00e2\u0088\u009a(2 * 2 * 3 * 3 * 3) = [\u00e2\u0088\u009a(2 * 2 * 3 * 3)] * \u00e2\u0088\u009a3 The square root of 2 * 2 is just 2 and the square root of 3 * 3 is just 3, so the square root of 2 * 2 * 3 * 3 is 2 * 3 = 6 [\u00e2\u0088\u009a(2 * 2 * 3 * 3)] * \u00e2\u0088\u009a3 = 6\u00e2\u0088\u009a3"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 5:44, it says that the hypotenuse was 5, but could it have been negative?",
        "A": "No; the hypotenuse couldn t have a negative number because there is no such thing as a negative distance in this and similar situations."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "I need to find a perimeter of a rectangle that has a ratio of 4:3 respectively, and a diagonal of 20 cm. How can I find its perimeter using the Pythagorean Theorem?",
        "A": "20^2=(4x)^2+(3x)^2 25x^2=400 x=4 The perimeter is 2(4x+3x) so 2*7*4=56"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "8:08\nWhy is it -36, and not just 36?",
        "A": "Sal is subtracting 36 from each side of the equation: 36+B^2=144 so as to get: B^2=144-36 which is B^2=108. That is why it is -36 (minus 36), not 36. Hope this helps! :)"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 10:20 how do you simplify sqrt 36 sqrt 3 into sqrt 6 sqrt 3?",
        "A": "The square root of 36 is 6."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "at 10:2 why does he do square root of 2X2X3X3 and put the last 3 in a seperate square root? confused.",
        "A": "Square roots of perfect squares can come out of the square root sign (2\u00e2\u0080\u00a22 and 3\u00e2\u0080\u00a23), and numbers that do not match (3) have to stay in, so he separates it out into perfect squares and non-perfect squares."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 0:43, why does the triangle have to be a right triangle in order for the Pythagorean Theorem to work?",
        "A": "That is how Pythagoras devised his theorem to work. It is actually a specific case of a more general result called the Law of Cosines, which you might learn later. The Law of Cosines works for any type of triangle."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 3:31, why are the three sides squared. please explain?",
        "A": "Because that is how Pythagorean theorem works. The are always squared when your talking about Pythagorean theorem."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "If the resulted number, such as 108 in 8:41, doesn't have a root or anything that makes it up?",
        "A": "If it wasn t a perfect square, you would put it under the radical sign, and that would be your answer. And if you could, you would simplify your answer. Hope I helped! :)"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 7:50, instead of doing what Sal did, couldn't you just do:\n\n12\u00c2\u00b2 - 6\u00c2\u00b2 = B\u00c2\u00b2?\nDoes that work?",
        "A": "Yes...you re omitting a couple steps, but as long as you have a good conceptual understanding then you re good."
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "Can someone explain to me how Sal simplifies square roots, like at 8:50? Thank you!",
        "A": "Prime factorization"
    },
    {
        "video_name": "AA6RfgP-AHU",
        "Q": "At 5:01, It says that a squared + b squared = c squared. Why square it at all? Wouldn't it be easier to just add them together and get the simpler version?",
        "A": "You would get a different answer. Let s say a = 3 and b = 4. Now let s solve for c... a + b = c 3 + 4 = 7 OR a^2 + b^2 = c^2 3^2 + 4^2 = c^2 9 + 16 = c^2 25 = c^2 5 = c The only way it works is if you square each term."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 7:21 I don't understand how they did the circles and squares can someone explain that?",
        "A": "Don t worry about the circles, Sal was just using them to highlight the 3 s in the equation on the left hand side of the screen. The squares are another way of trying to visualise what it means to divide 5 by 3 . If you use shapes (like squares) to represent the ones , then you can get some idea of what it means to divide 5 of these ones into 3 portions . He draws 2 lines through the 5 squares to divide the into 3 equal portions, and ends up with 1 and 2/3 for each portion."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "what were u talking about on 7:42 or 7:38",
        "A": "he had reduced the equation to be 5 =3x so he represents that as 5 blocks = x x x to make the equation work 1 2/3 blocks would equal an x"
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 7:50, I didn't get how he broke the 5 squares into groups...... can you explain how he did it?",
        "A": "You may wish to look at the videos on fractions in the arithmetic playlist to help you understand. 5 doesn t divide evenly into 3 parts, so you need to use fractions to represent it."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "Around 6:59, does it really matter if if you write 5/3 or 1 2/3?",
        "A": "yes"
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "1:33 if x +x is 2x\nthen what is X multiplied by x ?",
        "A": "X + X means one something plus another of that something, making two of that something or 2X (X being just any something). X TIMES X means one of that something times one of that something, making X squared, since it is an unknown number times itself."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "Hello, at 6:13 it said that the answer to the question was 5/3 or 1 2/3, what if we put this in decimal form? Would it still be a correct answer? Thanks and have a great day.",
        "A": "Well it would if you wrote it correctly as a repeating decimal"
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "I don't understand why he didn't cross out the other x's at 3:14>.",
        "A": "im sorry i miss read you question he didnt because thos are the ones that needed to be there for him to visualize his work and to check the answer be sure to always do that good luck on finding out other ways people think why!"
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "OK so at 8:53 you can add 2x and 3??? i thought you could only add like 2x and 3x cuz 3 has an x, was i wrong pls explain:/",
        "A": "I think you got the time wrong, because 8:53 is the end of the video. But In short, you can add terms that are the same kind so x can be added to other x, y can be added to other y, and numbers can be added to numbers. Once you know x s value, you can also treat it as a number. Hope this helps."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 2:41 Sal claims to minus 2x from both sides, why couldn't you just subtract 5x from each side. Since 5x is on the other side? Also why does he add 2 to -2 instead of 3 plus negative 3? At 4:40",
        "A": "You can indeed minus 5x instead. Using the example given: 2x + 3 = 5x -2 -3x + 3 = -2 ----->5x was subtracted from both sides. -3x = -5 --------> 3 was subtracted from both sides. -x = -5/3 -------->Dividing both sides by 3. (Note we now have -x not x, so we can times both sides by -1, or if you prefer add x to both sides and add 5/3 to both sides) x = 5/3 So essentially if you minus 5x, you are simply adding in an extra step to solving the equation. You would still arrive at the same answer either way."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 0:49, what does isolate mean? Is it a math word?",
        "A": "Isolate ain t a math word but it means to seperate or set a part from others pretty much what jrt.levi said ."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 2:50, why would you subtract 2x instead of dividing by 2 to get the x alone? Sorry if this is a dumb question but I am in 7th grade trying to teach myself and going over each bit helps :/",
        "A": "If you divide by 2, you would have to divide all the terms by 2 to keep both sides of the equation equal. Then you would have to divide the 5x by 2 which would give you 2.5x on the right side. You would have x by itself on the left side, but it wouldn t really help you because you d have to deal with the 2.5x on the other side. Your goal is to get all the x s together first. I hope that helps."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 2:11, why does he write the equation out as x+x+1+1+1=x+x+x+x+x-1-1? Would that just make the math simpler or would that do nothing but spread it out to see the equation easier?",
        "A": "It s just to see the equation easier."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "why do we at 3:50 add 2 to each side instead of dividing 3 by 3 and dividing 3x by 3 instead?",
        "A": "If you divide by three, in order to keep both sides equal, you have to divide EVERYTHING by 3, including the 2. You could do this, but it makes the work easier if you get rid of the two first. If you only divide parts of the two sides by three, then they are no longer equal. Think about it with numbers in place instead of variables 8 = 6 + 2..... is 8/3 = 6/3 + 2...that is 8/3 = 2+2? Now with dividing all parts 8 /3 = 6/3 + 2/3.... 2 2/3 = 2 + 2/3. Does that help?"
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "Hi, at 2:47 how do you now what do subtract or add to what side you have to do the operation on first. My second question is when you get your answer as a fraction can you turn it into a decimal? Thank you.",
        "A": "You have to first divide the numerator by the denominator to find the whole number then slide the decimal over the left two times."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "at 5:31 he said 5 = 3x. is that true",
        "A": "Yes. 2x +3 = 5x - 2 Subtracting 2x... 3 = 3x-2 Adding 2, 5 = 3x. ^ There. :) Hope this helps."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 2:16 why do you subtract two from both sides",
        "A": "It maintains the equality. You always have to keep it balanced to maintain it as true."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 2:53, why would he subtract 2x? Should't it be dividing instead because 2x and 5x means it's a multiplication problem.",
        "A": "The equation is 2x+3=5x-2. This can also be written as +2x+3=+5x-2. Yes it is 2*x and 5*x, but we are not trying to separate the x from the 2 or 5, so we don t want to divide. To move a term to the opposite side of the equality, we must do the opposite operation. Since 2x is currently positive, or added, we must do the negative, or subtract."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 2:15 in the video what x's is he talking about subtracting",
        "A": "He is talking about the 2x on the right side of the equation."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "How come at 6:34 minutes he didn't divide the 5 by 3 in 5/3, instead he turned it into a mixed number, why is that? How do you know when to stop reducing in an equation?",
        "A": "You stop reducing the fraction when the numerator and the denominator have no common factor to divide by."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "why did u do x+x and stuff at 1:07",
        "A": "so people could know it s basics"
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "In 3:01 of the video, are you supposed to remove the 2x so that you can get x on one side of the equation? And if so, why?",
        "A": "Because you want one side with a variable. When we solve an equation, we want to end up with x = some number. If you have an equation like 3x + 5= 2x +7, your main goal is to get ride of an x on one side so you only have one answer, which would then look something like x = 2. I ll write the steps on how I got that answer: Step 1: Subtract 2x from both sides. 3x+5\u00e2\u0088\u00922x=2x+7\u00e2\u0088\u00922x x+5=7 Step 2: Subtract 5 from both sides. x+5\u00e2\u0088\u00925=7\u00e2\u0088\u00925 x=2 Let me know if this helps or if you have anymore questions!"
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 5:44 he says to Divide the left side by 3, but instead just leaves it as a fraction 5 over 3, i don't understand why this happens. i would have just divided 5 by 3 and got 1.66 repeating..",
        "A": "Either way works. In a case like that when you would get a repeating decimal, it is better to leave it as a fraction because it very difficult to work with a repeating decimal and get an exact answer."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 1:29 Sal did the other set of 5x's in blue. When he did the 2x's in Pinkish-orange. Why did he do this? The x's did not change. Did he do this because he just wanted to change the color (to explain or something)?\n\nThanks in advance.",
        "A": "There wasn t any particular reason for it. He could have used the same color for both. I guess he just felt like changing colors."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 2:27, Sal says that as long as you use legitimate operations, the solution will be the same no matter how you do it. So what would an illegitimate operation be?",
        "A": "That is a great question. There are two ways of doing this one which is the normal way 4(3+5) = 4(8) =32 or you can use the distributive property...... 4x3=12+4x5=20 so 20+12=32. But later on when you use Algebraic expressions you have no choice but to use the distributive property. Sorry this answer is late but hope this helps."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "Why did he put stuff in boxes at 1:30",
        "A": "When he breaks down the equation on the right, the information he shows is in the same color as the box it corresponds to. 1+1+1 was done in purple because it relates.\\ to the 3 in the purple box."
    },
    {
        "video_name": "f15zA0PhSek",
        "Q": "At 4:34 he talks about adding plus 2 on both sides ,my ? is how and why.",
        "A": "When solving equations, you do the same thing to both sides. In order to get 3x by itself, Sal needed to get rid of the -2, so he added 2 to both sides. That left 3x by itself."
    },
    {
        "video_name": "m2mW2FQJgEE",
        "Q": "Shouldn't the W vector in the directional derivative equation around 4:00 be a unit vector?",
        "A": "yes w vector should be normalized. the problem is i think grant has never applied this or created a simulation of directional derivative and that is why he forgets to normalize the w"
    },
    {
        "video_name": "rGaM6pwqhB0",
        "Q": "at 3:29 we found two roots. r= -2+i and r=-2-i why we choose u=1 instead of u=-1?",
        "A": "It doesn t make a difference, since both constants are arbitrary. C1 cos -x = C1 cos x, and C2 sin -x = -C2 sin x. You can just use C2 instead of -C2, since it s arbitrary."
    },
    {
        "video_name": "rGaM6pwqhB0",
        "Q": "at \"3:55\", how come mu is equal to 1 instead of +-1?",
        "A": "Because we have that cos(x)=cos(-x) and -sin(x)=sin(-x). In the case of cos(x), the minus sign gets cancelled out automatically. But in the case of a negative x for sin(x), namely, sin(-x), this becomes -sin(x), and the negative sign gets absorbed by whatever constant C right in front of it."
    },
    {
        "video_name": "PQiXRrT_14o",
        "Q": "At 0:38, how did Sal tell that the graph of the function would be a downward sloping line just by looking at the function definition?",
        "A": "For example, we have a linear function, y = mx + b. IF the value of m is positive, it is an upward slope. IF it is negative, it will be a downward slope. Now, @0:38, the given is -0.125x + 4.75 where in the value of m is negative that s why Sal said that is a downward slope."
    },
    {
        "video_name": "PQiXRrT_14o",
        "Q": "At 1:02 in the video why did he write the equation out like that? PLEASE HELP ASAP!",
        "A": "At 1:02, in the video, he wrote out the equation like that in order to find the y coordinate at x= -10. He knew that at x=-10 he has to use the equation [-0.125x+4.75]. So in order to find the f(x) coordinate at that point, he replaced x in that equation with -10. If he had wanted to find the f(x) value at x= -5, he could have used the same equation, except replace the x with -5."
    },
    {
        "video_name": "4STGvuoBi5U",
        "Q": "At around 3:07 Sal mentioned perfect squares, what even is a perfect square?",
        "A": "A perfect square an anything raised to a power of 2. You should learn to recognize them in their exponent form and the form they take if you complete the exponent. In this video, Sal is teaching you what a perfect square looks like when you square a binomial. Here are examples: with exponent (and after exponent where possible) 5^2 = 25 x^2 (x^3)^2 = x^6 (x+2)^2 = x^2 + 4x + 4 (2x-3)^2 = 4x^2 - 12x + 9 Hope this helps."
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "I don't really get the whole thing about the repeating 3s at 8:48. Can anybody here explain it please?",
        "A": "You have to know how to change decimals into fractions. For example: 0.52 = 52/100 (divide top and bottom by 4) = 13/25. Another example: 0.5 = 5/10 (divide top and bottom by 5) = 1/2. This next one is a little tricky, but what you have to know is that every time that you see recurring 3 s, 6 s, and 9 s you have to divide by [9], not 10, not a 100, not 1000 etc. For example: 0.333333... = 3/9 (divide to and bottom by 3) = 1/3."
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "At 1:27, would you do the same with 60% of ___ is 15?",
        "A": "Your question is: 60% of ? = 15 The problem Sal was working was: 0.2% of 7 = ? So the unknown is not in the same place in the two problems. Here s one way to 60% of ? = 15: 60% of x = 15 is the same as 60/100 * x = 15 so we can multiply both sides by 100: 60 * x = 15 * 100 => 60 * x = 1500 then divide both sides by 60: x = 1500 / 60 so x = 25 60% of 25 = 15"
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "At 9:51, how do I figure out what % of 200=2?",
        "A": "It s 1% because in 100 the percentage of the number is the number with a % added to it, but in 200 you divide it by 2 and add a % so 2 divided by 2 is 1 add a percent 1%. I hope you find that helpful. Please vote up!"
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "7:02 I don't understand how that proof works.",
        "A": "All Sal did was substitute his answer into 20% of x = 4 . His answer was 20. So he plugged 20 into his equation, and solved it. 20%->.2 .2*20=4 It checks out."
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "At 4:08 Sal says \"So always do a reality check to see if you have the right answer.\" What is a reality check, and how do i do one?",
        "A": "A reality check is a way to make sure your final answer makes sense in the real world. For instance, if you are doing a calculation of how many people prefer basketball over baseball and your answer is over 100% you know something is wrong because it would be impossible for over 100% of people to prefer one thing over another."
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "When Sal puts the decimal place 2 to the left instead of saying it was .2 in 2:22 through 2:27, do you do that with all percentages?",
        "A": "Yes! To convert percents to decimals, divide by 100 or move decimal place 2 spaces to the left. To convert decimals to percents, multiply by 100 or move decimal place 2 spaces to the right."
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "Wait, at 2:10-2:47, I always thought that .2% also equals .20 as a percentage. Please Help!",
        "A": "100% in decimal form is unity, so .2% is .2/100=.002"
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "At 10:09, how do I figure what % of 75=7.5?",
        "A": "You divide 7.5 by 75, and 7.5/75 is 0.1, which equals 10%."
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "At 7:58 he says that 9% in decimal form would be 0.09, but wouldn't it be 0.9?",
        "A": "No he is correct 9% would be 0.09 in decimal form because 0.9 would be 90% here are som examples 1% = 0.01 25% =0.25 30% = 0.3 45% =0.45 99% =0.99 100% = 1"
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "4:59 my first thoughts were 8,but i was thinking to quick so i was wrong\n)|\n\nit was actualy 20",
        "A": "math is good for learning to make your mind go steadily, accurately, checking your own self, and getting to trust your own method. also read your own handwriting. stillness? nah..."
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "Can someone thoroughly explain what he was doing with the decimal points in the division problem at 6:14? I understand what you do with decimal points when calculating an answer, but what he did with the divisor and dividend does not make sense to me. Please explain it to me!",
        "A": "Hello Brynne, I believe that what Sal is communicating in this lesson is that the ratio of the divisor to dividend is kept the same as long as both the divisor and dividend are multiplied by the same number. For example, when dividing 7.5 / 2.5, one could just as easily multiply 7.5 by 10 to get 75, and 2.5 by 10 to get 25 - making the new problem 75/25. Note: In both cases the quotient is 3. Moving the decimal place like this sometimes makes the dividing process more convenient or apparent."
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "When Sal puts the decimal place 2 to the left instead of saying it was .2 in 2:22 through 2:27, do you do that with all percentages?",
        "A": "yes, all percentages cent is latin-ish for a hundred, so saying per cent means ratio. We find this number for every hundred we count. 5% says 5 per 100. Percent gives you a proportion."
    },
    {
        "video_name": "_SpE4hQ8D_o",
        "Q": "Why did he put .2 at 5:30 instead of of .20.?? If you use .20 to do the problem you would get a totally different answer, How is .20 and 2 the same?",
        "A": "Oh................................. I get it so like .3 is the same like 3 tenths which is 3 10s which equals 30. Thanks so much you explained it so simple."
    },
    {
        "video_name": "hKkBlcnU9pw",
        "Q": "At 3:10 he explains how the absolute value of -8 is equal to 8 because they are the same distance from zero, but when doing the practice concept the question -1___1 came up I chose equal and it said -1< 1 can someone explain this please?",
        "A": "In that particular problem, the answer would be -1 < 1 Now, if the problem was | -1 | ___ 1 , then the answer would be | -1 | = 1 The symbol used for absolute value is | |"
    },
    {
        "video_name": "hKkBlcnU9pw",
        "Q": "At 0:09, when Sal draws the lines on either sides of the numbers, what do they mean? Are they just to separate the numbers from each other so it doesn't become confusing?",
        "A": "When you see: | -2 | The bars are the symbol to indicate that you need to find the absolute value of the number. The absolute value of a number is its distance from zero. Thus: | -2 | = 2 because -2 is 2 units from zero. In the video, Sal is asking you to compare absolute values. Hope this helps."
    },
    {
        "video_name": "hKkBlcnU9pw",
        "Q": "in 3:17 how did the -9 turn into a positive 9",
        "A": "Because the absolute value symbol makes the negative a positive UNLESS the negative is outside of the absolute value symbol"
    },
    {
        "video_name": "hKkBlcnU9pw",
        "Q": "At 0:04 sal says carrying absolute value. Why?",
        "A": "because at 0:04 he is introducing the topic... :D"
    },
    {
        "video_name": "hKkBlcnU9pw",
        "Q": "At 3:00 Absolute value of a negative number is just the number but positive?",
        "A": "Yes. The absolute value of a number is the same number with a positive sign."
    },
    {
        "video_name": "hKkBlcnU9pw",
        "Q": "At 3:59, did Sal forget to add the \" | \" on both sides of the 2, or is that just how you do it? Or is it redundant?",
        "A": "The | symbols are not numbers (there is no addition). They are the symbols to indicate absolute value. | 2 | is read as the absolute value of 2 , it = 2. | -3 | is read as the absolute value of -3 , it = 3 Hope this"
    },
    {
        "video_name": "hKkBlcnU9pw",
        "Q": "At 2:30 Sal explains that the absolute value of both 9 and -9 was 9. Why is this?",
        "A": "-9_________0_________9 Both 9 and -9 are 9 units away from zero. Absolute value is distance, or how far away the value is from zero. It s always positive, so the absolute value of 9 and -9 is 9."
    },
    {
        "video_name": "LoaagZPWvpM",
        "Q": "At 2:10 how do you mark the slope in the co-ordinate plane in that slanting way? I mean, how could we infer the nature of that slope line, whether it is horizontal, vertical or slant in clockwise or slant in anti-clockwise??",
        "A": "In the column labeled dx/dy he has the slope value. He uses that value to plot the slope. A value of 0 is a horizontal slope, 1 is a slope increasing 1 , -1 is a slope decreasing. You could review slopes if necessary."
    },
    {
        "video_name": "i9j_VUMq5yg",
        "Q": "Why was there a chart at 0:05?",
        "A": "The reason why there is a chart of table in 0:05 is because you need to use it in the algebraic problem."
    },
    {
        "video_name": "i9j_VUMq5yg",
        "Q": "At 1:23, How do I know the difference between independent and dependent variables? Whats a easier way I can understand this concept?",
        "A": "the dependent variable depends upon the independent variable, while the independent does not depend on the dependent variable."
    },
    {
        "video_name": "0zCcFSO8ouE",
        "Q": "At 10:00, wouldn't flipping the graph over the x-axis work as well?",
        "A": "Well, in this select case, yes. But if you take some other function that isn t odd, f(-x) won t be the same as -f(x). In one case, we are flipping over the y-axis, and in the other, over the origin (respectively). You can draw a random line on a coordinate plane (not going through the origin, mind you) and try flipping it over the axes. The difference will be obvious :)"
    },
    {
        "video_name": "0zCcFSO8ouE",
        "Q": "In 2:12, why is sine of pi, one?",
        "A": "Because in the unit circle, the sin of any angle is the y-coordinate at that point, and when you have an angle of pi on the unit circle, the y-coordinate is 1!"
    },
    {
        "video_name": "0zCcFSO8ouE",
        "Q": "At 8:24 in the video, sal said that the difference between the graphs sin(x) and sin(-x) was that they were reflections of each other. When I put cos(x) and cos(-x) into my calculator, both graphs were identical. So why is there a reflection with sin but not with cosine?",
        "A": "sin(-x)=-sin(x) and that s why u need to reflect sine once over Oy and once over Ox to obtain the other half of the function. cos(x)=cos(-x) and hence u need to reflect it once over Ox Cos(x) is an even function like x^2 for example. If u plot x^2 u see is symmetrical of Ox like cos(x). Sin(x) is not symmetric of Ox."
    },
    {
        "video_name": "0zCcFSO8ouE",
        "Q": "At roughly 12:20 the period is discussed. I'm a little unclear on the equation I can apply to any and all problems to solve for the \"new period\" when there are transformations and whether amplitude, vertical shifts, and phase shifts should be included in this.",
        "A": "Suppose you have a function y = a sin (bx+c), then the period of the function will depend on b . since the period of y=sin x is 2pi , the period of the new function will be 2pi/b. This is because the graph will move b times faster , to put it simply. In general , if you have a function y = f(x) with a period =T then the period of the function y = a f(bx+c) will be = T/|b| the period will not be affected by a (the amplitude) or c (phase shift). Hope this helps"
    },
    {
        "video_name": "0zCcFSO8ouE",
        "Q": "At 8:25, couldn't it also have been reflected about the x-axis?",
        "A": "Yes. That is conceptually what you are doing."
    },
    {
        "video_name": "OP91XWBRI1w",
        "Q": "At 4:40 why did you include (x+5<0 and x-2<0)",
        "A": "becuase thats the factorization of the quadratic, and then up in the top right corner is where he proves the inverse is the same thing"
    },
    {
        "video_name": "OP91XWBRI1w",
        "Q": "At 4:15, I am still confused why he did a second inequality. Why do you have to do the \"or (x+5<0 and x-2<0)\" along with the one before it?",
        "A": "A * B > 0 A and B can both be positive. 2 * 2 > 0 or A and B can both be negative. -2 * -2 > 0 An x value has to be within one of the constrains set by the inequalities. A or B doesn t work because you could have one negative and one positive value for A as well as B. You say or not and because A and B don t have to meet both constraints, just one."
    },
    {
        "video_name": "VgqsM-XdBD0",
        "Q": "at time 3:34 in the video sal says that the lowest X coordinate value is 3 but isn't it 1?",
        "A": "Very good observation! I think you are right, and I think it would have helped if the Khan team had corrected this. Which vector has this x-value? (Thanks for the time-stamp: they really help with a question like this :-)"
    },
    {
        "video_name": "VgqsM-XdBD0",
        "Q": "@3:38 Why is the lowest x value 3 and not 1?",
        "A": "I wondered about that too, but now realise that although Sal didn t say it, he was only looking at the original vectors and the first problem ( a + b ). He wasn t including the answer to the second problem ( a - b )."
    },
    {
        "video_name": "bvcXEJbEzSs",
        "Q": "At 0:42, Sal did not write 2 to the power of 2, or 4. Is this a mistake?",
        "A": "Yes, it was a mistake."
    },
    {
        "video_name": "hV4pdjHCKuA",
        "Q": "at 2:47 we use a t-chart with 1-sided and 2-sided %'s given but in the Fundamentals of Engineering exam, the handbook gives the tail probability uptop instead. Please guide on how to use that in this problem or anywhere.\n\nThanks,\nAshar.",
        "A": "The t-distribution is symmetric. Hence, if you find the t-critical value for 1-tailed, 5%, it will be the same that for 2-tailed, 10%. The critical value for 1-tailed, 2.5% will be the same as that for 2-tailed, 5%. Se, if you re looking at a 1-tailed test, you use the significance level that s given. If you re looking at a 2-tailed test, you need to divide the significance level in half."
    },
    {
        "video_name": "HRVrbJzXxdM",
        "Q": "at 1:05 in this video, you compute (d + 2x)^2 should be d^2 + 4dx + 4x^2",
        "A": "This is a known problem in the video. A box pops up and states that there is an error and provides the correct info."
    },
    {
        "video_name": "Ye13MIPv6n0",
        "Q": "on 0:15 why does he have to lable it x can't he put a question mark at all",
        "A": "I think he used x because it is a variable, and variables are another part of algebra."
    },
    {
        "video_name": "Ye13MIPv6n0",
        "Q": "2:28 we are using 3 because there are 3 masses we do not know the value of ?",
        "A": "Yes. In order to solve equation we need to isolate variable. If equation is 3x = 9, you need to do something to both sides of equation to make x stand by itself. In this case dividing by 3 both sides does the trick. 3x/3=9/3 therefore x=3 If equation was 2x=8 than you would divide both sides by 2. If equation was x + 3 = 5 than you would subtract 3 from both sides."
    },
    {
        "video_name": "Ye13MIPv6n0",
        "Q": "At 2:27, wouldn't 3 of those small, yellow blocks equal 1 of those big, blue blocks?",
        "A": "Yes, if you watched all of the video than you would see the explanation."
    },
    {
        "video_name": "Ye13MIPv6n0",
        "Q": "At 0:35, what would have to be the number in order for it to be x (x) (x), or x^3, instead of 3x?",
        "A": "I don t think there s a way to represent the cube of a square or cube graphically. You d have to start with valued line and draw a cube. So the line equals x and the box is x^3."
    },
    {
        "video_name": "Ye13MIPv6n0",
        "Q": "At 2:27, wouldn't 3 of those small, yellow blocks equal 1 of those big, blue blocks?",
        "A": "Yes, it does."
    },
    {
        "video_name": "Ye13MIPv6n0",
        "Q": "wouldn't dividing 3 to both sides be easier than multiplying 1/3 at 2:20 . Or are they the same thing ?",
        "A": "It can be considered the same, Though personally dividing by three is easier."
    },
    {
        "video_name": "Ye13MIPv6n0",
        "Q": "At 2:16 he says that 1/3 of this total mass is equal to 1/3 of that total mass. If the unknown mass was 5x and not 3x would you have to multiply by 1/5? Or if it was 8x would you have to multiply by 1/8?",
        "A": "Yes, or instead you could divide by 5 or 8. In fact the division sign came from fractions..."
    },
    {
        "video_name": "ycBxbgjQns4",
        "Q": "Lets first think about subtracting with borrowing. The begining of the operation starts at the ones place, because we will need to borrow from the tenths place. Don't you think that starting this operation at the hundreths place (1:58) migth be confusing in posterior subtraction with borrowing exercises?",
        "A": "You try to learn as many different ways as you can to figure your math problems so you can find a way that you like to use that consistently and dependably gives the right answer. You figure the problem the way that works the best for you."
    },
    {
        "video_name": "ycBxbgjQns4",
        "Q": "at 0:38 is the word hundred a number and place value?",
        "A": "One hundred is a number, but hundred is a place value."
    },
    {
        "video_name": "ekDTjvF_CJw",
        "Q": "at 2:1 Sal said 4*3 is 12 but the 3 is in the tens place",
        "A": "When you are doing long multiplication you times the first bottom number by both top numbers so when he is saying 3 x 4 he actually means 4 times 30 but you get the idea"
    },
    {
        "video_name": "SE1ltVuE5yM",
        "Q": "At 8:25, why is the particle speeding up between 1<t<2? Shouldn't it be slowing down since the particle's velocity reaches 0 at t=2?",
        "A": "The particle velocity is -3 at time 2 (v(2)= -3) That is the minimum velocity of the particle, so the particle was accelerating (negatively) up to that point."
    },
    {
        "video_name": "NM8qTo361ic",
        "Q": "At 0:30 why did Sal add 3 decimal places to 16?",
        "A": "At 0:30 Sal added decimal places to 16 because 21 doesn t go into 16 at all, so to make 160 he put the already existing decimal places in place."
    },
    {
        "video_name": "NM8qTo361ic",
        "Q": "At 0:17 Sal means that in the fraction 16/21 - 21 (the denominator) is greater than 16 (the numerator).\nSimplified-\n16/21 = 21 > 16 - Means the answer will be less than one.\n16 \u00c3\u00b7 21 = 0.7619\n\nWhenever the denominator is greater than numerator the answer will ALWAYS BE less than one.\nExample -\n\n5/5 is a full - 1\n4/5 is 4 parts out a whole = 0.8\n\nI hope this helps you, or else please refer to videos about fractions.",
        "A": "Not a question...put this in tips and thanks. Thanks! ^_^"
    },
    {
        "video_name": "NM8qTo361ic",
        "Q": "What does the squiggly equal sign at 3:07 mean and what is it supposed to be called?",
        "A": "It means approximately not exact or about"
    },
    {
        "video_name": "NM8qTo361ic",
        "Q": "at 3:05 sal says we are rounding up to the nearest thousands doesnt he mean nearest hundereds because the 3 digit after the decimal place is a hundred?",
        "A": "I know right :("
    },
    {
        "video_name": "NM8qTo361ic",
        "Q": "what does he mean at 0:17",
        "A": "Like 0.9 it is under 1"
    },
    {
        "video_name": "bjVn4WGmNis",
        "Q": "I'm sorry , maybe I just missed it, but why at 0:34 and so on did he put the negative number in parentheses? Thanks.",
        "A": "Sometimes you just HAVE to use that. see this? -5+-8 you will get confused, right? but, it is better like this: -5+(-8)"
    },
    {
        "video_name": "_DkttKctXFY",
        "Q": "Why are we allowed to cancel out (x-3) from the denominator and numerator and later on we still substitute x for 3? Which previously could give us 0 in the denominator.\nTime on the video is 3:50",
        "A": "He had the difference of squares (x^2-9) which he factored into (x+3)(x-3). Since he had a (x-3) term in the denominator, he could cancel out that term in the numerator and denominator. That comes straight out of algebra. If you have difficulty with that, you might wish to review algebra on that point."
    },
    {
        "video_name": "EKvHQc3QEow",
        "Q": "I'm having a conceptual block about the term \"instantanenous\" which Sal uses first at 0:59 and then again a bit later in the video. Does anyone have a different interpretation of the one used by Sal, and if so, how did you come to it?",
        "A": "well when we talk about instantaneous velocity we talk of the speed (in a specified direction) at that very instant . We may say, it is the distance covered for an infinitesimally small amount of time."
    },
    {
        "video_name": "EKvHQc3QEow",
        "Q": "At 0:40 Sal said that Usain bolt is the fastest man who ever lived. His average speed is roughly 24 miles per hour. Cant he go faster if you calculated his actual speed?",
        "A": "This was mentioned in the video - Usain Bolt has a maximal instantaneous speed of approximately 30mph."
    },
    {
        "video_name": "EKvHQc3QEow",
        "Q": "at 8:36 sal says that dy &dx are both an infinitely change in y and x respectively which doesn't match up with the limit definition",
        "A": "Why should it? dy and dx apply to derivatives, not limits. Taking a derivative means finding the slope of a particular function, whereas taking a limit would involve finding what a function is or is approaching at a certain point. Derivatives and limits do very different things; don t confuse them."
    },
    {
        "video_name": "EKvHQc3QEow",
        "Q": "At 0:57 and many times throughout the video, he's talking about \"instantaneous rate out change of something\"... But change only happens through time and here were talking about something instantaneous, so shouldn't it be more \"the best constant approximation for rate of change\" (from 3Blue1Brown on Youtube).",
        "A": "One of the primary purposes of differential calculus is precisely this: to determine the exact (not approximate) instantaneous rate of change of a function. If the motion of an object (and many other physical possibilities) can be described by an explicit mathematical function, then the derivative of that function at any point yields the instantaneous rate of change. We can thank Newton, Leibniz, and other great mathematicians for discovering the mathematics which makes this possible. Good luck!"
    },
    {
        "video_name": "EKvHQc3QEow",
        "Q": "I have heard people say that what you learn from pre-algebra, algebra and trigonometry will be used in calculus more than we think. At 3:32 the use of the rise over the run is mentioned. How much of the classes materials be used in calculus?",
        "A": "Almost everything you learn from algebra/trig will be used in calculus. Not so much from geometry."
    },
    {
        "video_name": "EKvHQc3QEow",
        "Q": "wouldn't distance be x? 1:56",
        "A": "No we usually do it in terms of time. How far you moved in certain amount of time. Not how much time in certain amount of moving."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "At 2:20, how come sal canceled out the 2x and the square root of 5, also -2x and the square root of 5 instead of leaving them in the answer?",
        "A": "2\u00e2\u0088\u009a(5) - 2\u00e2\u0088\u009a(5) = 0"
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "At 1:40 (I don't know how to link video url at current time...) he puts them both into parentheses and simply factors a -1 out of the 2nd one. Is this mandatory?",
        "A": "Not mandatory, but simpler"
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "2:00, Sal says that -1 times -4 equals 5. Hmmmm? Am I missing some complex number magic? Because under normal circumstances -1 times -4 equals 4, which means that the answer would be x^2+3 which is what I worked out in my head. Am I missing something?",
        "A": "a box comes up and tells correction"
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "2:30 , why sal makes so mistakes ?\ndoesnt look good :/",
        "A": "Sal doesn t make many mistakes it was just 1 mistake in this video and it didn t ruin it all"
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "So the problem sal used was (g-f)(x), where does the (x) go into all of this? Also, at 1:48 why does he put a negative 1 to distribute itself among f(x)?? Why doesn't it just subtract f(x)?",
        "A": "To subtract any polynomial, you need to distribute the minus (of a -1). If you don t, the only part this is subtracted is the very first term. You aren t subtracting the entire function."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "hello, I am in algebra 2, and I would like to know why you put a positive five when you combined (-1) and (-4) at 2:00 minutes... I thought that (-1) n= -n\nIf someone could please respond, thank you",
        "A": "Yes I just looked over the same equation as you and I was confused on how Sal got +5 as the product of -1 and -4... Strange However, I tried plugging them into a graphing calculator (Use Desmos Graphing Calculator online, really helps), and I found that the equation that Sal came up with didn t fit the graph. So yeah, I am inclined to believe that he just made a silly mistake, and you are correct in that combining -1 and -4 should be 4."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "at 2:03 why does he add the -1 but multiply's everything else i thought you were supposed to multiply everything when it comes to distribution",
        "A": "Sal did not add a -1 . The minus sign in front of the parentheses is equivalent to having a minus 1 in front of the parentheses. You can distribute either the minus sign OR you can distribute a minus 1. Both achieve the same result of changing the sign of all items inside the parentheses. Hope this helps."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "at 2:04 Sal said negative one times negative four equals five but it actually is four",
        "A": "Lol, I noticed that the first time too."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "At 2:04, Sal says that negative one times negative four is positive 5. He made a mistake there, because the correct answer is positive four.",
        "A": "The video has a correct box that pops up in the lower right of the video window with the correct info and answer."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "at 1:07 Sal talks about the principle square root of 5. What is a principle root? Is there a vid on it.",
        "A": "The principal square root is the positive square root. There are 2 different numbers that can be squared to make 5, +sqrt(5) and -sqrt(5). If you squared either, they would make positive 5."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "At 2:00-20:4, If -1 is distributed among 2x times the square root of 5 -4 shouldn't it be positive four instead of positive 5, because wouldn't you be multiplying rather than whatever that was.",
        "A": "Yes, you are correct. You should see a little correction window pop-up indicating the proper result."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "2:03 Sal made an error",
        "A": "This is a known error and, as such, a pop-up box appears in the lower right-hand corner to say so."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "At 1:13 why do we have to subtract ?",
        "A": "Because we are subtracting functions, for that we need to use subtraction symbols."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "At 2:05, isn't (-1) times (-4) = 4 and not 5?",
        "A": "This is a known problem with the video. There is a box that pops up and states the error along with the correct info."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "at 2:20 is it supposed to be -(-2x times the square root of 5) or is it - 2x times the square root the 5?",
        "A": "After distributing the -1, there will be one minus sign in front of the 2x\u00e2\u0088\u009a(5). So: -2x\u00e2\u0088\u009a(5)"
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "at 2:08 I got -5 square root, sal got -2x-five square rooted. I think of the 2x and the 5 square root has two different numbers, but thats clearly not the case. can any1 explain me why that is?",
        "A": "Because they are multiplied they are treat as a single unit (called a term). However, Sal made a math error -- his answer is wrong. Here is the correct solution: f(x) = (2x\u00e2\u0088\u009a5) - 4 g(x)= x\u00c2\u00b2 + (2x\u00e2\u0088\u009a5) - 1 (g-f)(x) = g(x) - f(x) =x\u00c2\u00b2 + (2x\u00e2\u0088\u009a5) - 1 - {(2x\u00e2\u0088\u009a5) - 4} = x\u00c2\u00b2 + (2x\u00e2\u0088\u009a5) - 1 - (2x\u00e2\u0088\u009a5) + 4 = x\u00c2\u00b2 + { (2x\u00e2\u0088\u009a5) - (2x\u00e2\u0088\u009a5) } + (4 - 1) = x\u00c2\u00b2 + 0 + (3) = x\u00c2\u00b2 + 3"
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "At 2:13 how does the 4 become a 5 if you are distributing a negative one?",
        "A": "This is a known error in the video. A box pops up and tells you there is an error and provides the correct solution."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "Shouldn't be a +4 at 2:04? When multiplying -1 times -4. Thank you!!!!",
        "A": "Yes, it was an error."
    },
    {
        "video_name": "KvMyZY9upuA",
        "Q": "At 2:05, how does the -4 become a +5?",
        "A": "This is a known problem in the video. There is a box that pops up and tells you the error and the correct info."
    },
    {
        "video_name": "AjYil74WrVo",
        "Q": "At 1:24, what did he say?",
        "A": "He said, We are done. This means that the problem is over. I hope this helps!"
    },
    {
        "video_name": "8mAZYv5wIcE",
        "Q": "The matrice Khan constructs at 4:04 is shows a mapping from R^2 to R^3 but all the equations are those of lines. I understand the set of three linear equations initially presented as well as how he constructed the matrice at 4:04 but I do not understand the connection between the two upon analyzation.\nThank You.",
        "A": "This is not a mapping from R2 to R2. It is still R2, since you still have two columns. The three rows in the matrix represent three equations in a two-dimensional space."
    },
    {
        "video_name": "TdVDqWOR3dU",
        "Q": "at 2:05 he substitutes x with 3 but the function is defined between 0 < x < 3. In other words it's defined up to three but not three. So isn't that technically wrong?",
        "A": "in that particular function you mentioned,, it is continuos until x approaches 3. We needed to find the limit of f(x) of that function as x approached 3. That is only the idea of limit.. It is not defined at x =3, but we can find a value of f(x) so that it would have formed a continuos graph.. So we find the value of f(x) that would have been for x=3. Hope that was not complicated"
    },
    {
        "video_name": "TdVDqWOR3dU",
        "Q": "At 1:45 how we can understand that function is continuous?",
        "A": "It is not undefined for any positive argument. This means that there are no asymptotes or removable discontinuities, but proving continuity can be done in a variety of ways (for instance, noting that it is differentiable or noting that its inverse is differentiable etc.). Since differentiability is a stronger condition than continuity, all differentiable functions are also continuous over the differentiable interval."
    },
    {
        "video_name": "TdVDqWOR3dU",
        "Q": "At 1:47 and 3:16, what if the function is non continuous? What numbers do we plug in for x?",
        "A": "If there is a jump discontinuity, then the limit from the left side and the limit from the right side will not be equal so the overall limit does not exist. You still have to plug in the same x value in both equations and you will get different values so the overall limit does not exist. But if there is a removable discontinuity, both the limit from the left side and the limit from the right side will be equal so the overall limit exists. In any case, you have to plug in the same x value."
    },
    {
        "video_name": "6CFE60iP2Ug",
        "Q": "At 6:30 when Sal is finding the slope of the equation he gets 4.5/-8 as an answer. Then he says that he does not like a decimal in the numerator, so he multiplies the numerator and denominator by 2 to get -9/16. Why did he specifically choose the number 2? How did he decide on that number instead of another method to get 4.5 out of decimal form?",
        "A": "Because 2 is the smallest number you can multiply the numerator and denominator to get rid of the decimal form but, at the same time, still get a simplified fraction."
    },
    {
        "video_name": "f8B2dBQn3_U",
        "Q": "At 2:11, where is the 0/1000?",
        "A": "0/1000 is just zero, it has no value so you do not need to write it down. If you were to include all of the place values with 0, you would have to include 0/10000, 0/100000, 0/1000000... and so on forever!"
    },
    {
        "video_name": "f8B2dBQn3_U",
        "Q": "at 2:42 why does he have to list out all the fractions? isn't it easier just to look at the number and find out yourself?",
        "A": "because he s just so slow"
    },
    {
        "video_name": "f8B2dBQn3_U",
        "Q": "at 2:47, did Sal write down the 90/100? Or did I miss something?",
        "A": "9/10 is the same as 90/100 because 90/100 is just 9/10 x 10/10 (which is the same as multiplying by 1)."
    },
    {
        "video_name": "outcfkh69U0",
        "Q": "at 3:03 why do they use points 0,1,2,3,4 ? at school we always use -2,-1,0,1,2",
        "A": "Teachers always want you to think about negative numbers. They do exist."
    },
    {
        "video_name": "outcfkh69U0",
        "Q": "At 10:38, Sal uses a negative number to shift the parabola to the right. I thought it was a negative number to move to the left.",
        "A": "On a real number line, adding a+b can be thought of as moving right by b from a and subtracting a-b as moving left by b from a. Transformations of polynomials don t seem to play by that rule at first, BUT if you look more closely, you ll see that they do! Simply recall that the x-coordinate of the vertex of parabolas is given by -b/2a for quadratic equations in standard form, then the reason should be clear."
    },
    {
        "video_name": "outcfkh69U0",
        "Q": "At around 1:00, Sal said principle root means you take the positive value. Does that also means that principal root of 9 = the absolute value of square root of 9?",
        "A": "Yes it does. However, principle root is the scientific name."
    },
    {
        "video_name": "outcfkh69U0",
        "Q": "Did anyone notice at 7:13 that Sal put 16,4 at 16,5?",
        "A": "I Think he did that AFTER he said : well that would that would be fine too."
    },
    {
        "video_name": "6n0yxfWNFR4",
        "Q": "At 2:14 is it not 2-(-2) instead of 2-(-4)?",
        "A": "I would suggest deleting this question, so nobody would be tempted to answer it, as I was."
    },
    {
        "video_name": "6n0yxfWNFR4",
        "Q": "What did Sal mean by saying \"with respect to x\"? 2:42",
        "A": "X always is the input so without x there is no y"
    },
    {
        "video_name": "DhW9pz5Vfwo",
        "Q": "At 0:44, how do we know that b raised to the power 1 is equal to four. Is it a definite variable with its value",
        "A": "It is a constant that we deduced from the graph that is given to us."
    },
    {
        "video_name": "DhW9pz5Vfwo",
        "Q": "I am sorry, but I got lost after part 1:33 in the video.",
        "A": "I also had the same trouble at the start until the relationship between logarithms and exponentials helped me to solve problems. In this video Sal is explaining that b^x is the inverse of the function y= log x to the base b. So given a log problem it is helpful to be able to transfer it to an exponential problem and the reverse."
    },
    {
        "video_name": "TXGhMDbbnzQ",
        "Q": "@ 8:06 sal says 75 hours its suppose to be 7 hours",
        "A": "He immediately corrected himself."
    },
    {
        "video_name": "tbQ_7zvRoN4",
        "Q": "I think I get it. If I were to multiply out the polynomials found around 7:40, then I would end up finding each factor because I'm multiplying by all the powers that make up each factor. Which is why I need the ones. Is it possible to then rewrite this function using only the bases? Is their a function the gives me the sum of all the powers up to k of that number?",
        "A": "Yes, there is! Look up geometric series."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "at 2:33, why does he add the exponates and not multiply them?",
        "A": "maybe its the way you do it or if you didn t watch the whole video then watch the whole video. That might be the way to get the right answer."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "At 7:37 could I use adding or subtracting polynomials as well?",
        "A": "Addition and subtraction at this stage in the solving of the equation is not quite possible. It is multiplication equation, so in order to isolate or move variables, you must divide. If the equation was (3x+2)+(9x^2-6x+4) then you would be able to subtract a polynomial expression."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "At 5:40 if (x-3)(x+2) is a binomial wouldn't the result (x^2 - x - 6) be a trinomial since it has 3 terms? I'm confused now on whether the original expression is a trinominal or a binomial then.",
        "A": "(x-3)(x+2) is the factored form of the polynomial. This entire thing is a monomial (one term). Each factor, the (x-3) and the (x+2) are binomials, each contains 2 terms. If you multiply the binomials, you create the simplified version of the polynomial which is the trinomial: x^2 - x - 6, it has 3 terms."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "at 6:46 Sal squared the two x,s why not just add them together?",
        "A": "Because x * x is not 2x. It would be x^2, similar to squaring a number. 3 to the 2nd power would be 3^2, or 9, and not 3 * 2, which equals 6."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "At 9:10 why is it 27x cubed if its multiplying it would be 27x sqaure not cubed",
        "A": "You have 9x^2, and 3x. That is the same as: 9*x*x*3*x 27*x*x*x 27*x^3"
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "Why at 4:55 did you not a X squared to - X",
        "A": "I do not understand your question. Are you asking why he did not add x-squared to -x ? At 4:40 he already has (x - 3) x + (x - 3) 2 Then he uses the distributive property on each part. From 4:40-5:04 he writes x*x - 3*x + x*2 - 3*2. For review, the distributive property means that ( a - b ) c = ac - bc."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "Sal said \"green x\" (at 3:42) and \"green two\" (at 3:51) but meant to say \"dark yellow.\"",
        "A": "This made my day! XD"
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "When he gets the last part of the second problem after he finishes multiplying at 3:04, is that the end of the problem? Or do you have to combine like terms and find a real answer?",
        "A": "Michelle, At 3:04 Sal had 27x\u00e2\u0081\u00b5 - 18x\u00e2\u0081\u00b4 - 63x\u00c2\u00b3 The terms have different exponents, so they are not like terms and cannot by combined. And it is an expression, not an equation, so it does not have a solution. It is just 27x\u00e2\u0081\u00b5 - 18x\u00e2\u0081\u00b4 - 63x\u00c2\u00b3 There is nothing more to do. I hope that helps."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "At 3:34, can you also workout the binomials using the F-first O-outer I-inner L-last method?",
        "A": "Yes! The FOIL method is a way of using the distributive property, which is the same property that Sal used. He just wrote it in a different way. As long as both expressions are binomials, you can always use FOIL to multiply them."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "At 1:14, why don't you just add the exponents?",
        "A": "x^1 * x^1 is the same as x * x they both equal x^2"
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "at 4:00 you say AA and that's isn't what was happening.",
        "A": "I don t see where he says AA near 4:00. He talks about the distributive property and uses a in place of x-3 to try to show how he is using the distributive property."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "When Sal was doing the problem around 2:30, can you take the problem 9x to the third (3x squared - 2x + 7) and make it (9x to the third (+ or -) 3x squared ) (2x + 7) then FOIL the problem?",
        "A": "No, you definitely cannot. That would be like taking a ( b + c + d ) and making it into ( a + b ) ( c + d ). I hope you can see that those are not the same. If you cannot see that they are not the same, try plugging a=1, b=2, c=3, d=4 into both expressions and see what you get."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "Hello! At 2:32 how did you do that? Do you just multiply and then add the exponents and put the variable?",
        "A": "Yes, review the properties of exponents. Sal has a vid on it."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "At 7:37, how would you multiply this way if there were other variables like y or z?",
        "A": "Make them a separate column so to speak. He s really just lining up all the like terms vertically, you d do that if you had different variables too."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "At 7:46, what does he mean by \"degree\" in \"first degree\"?",
        "A": "First degree refers to the power of the variable. In that case, he is lining up all of the x s that have a power of 1, all the variables that have a power of 2, and so on. It is a clever way to make sure that you combine only like terms together."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "at 1:20, how come Sal multiplied -5 by 2x? They are not like terms!!!!",
        "A": "You cannot add or subtract terms that are not like terms. There is no such rule for multiplication and division."
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "At 2:52 couldn't you further simplify 27x^5-18x^4+63x^3 to 72x^4",
        "A": "No, all of them have different exponents for x. :)"
    },
    {
        "video_name": "fGThIRpWEE4",
        "Q": "At the end of the video (7:35 to 9:49) the question setup was a bit confusing. Could someone break it down for me?",
        "A": "It is the same principle as multiplying numbers, with the exception that in this case you work with independent term, first degree term and second degree term, so you proceed multiplying each term by each term in order following the same algorithm as Sal did."
    },
    {
        "video_name": "EBKNtjZAjXg",
        "Q": "At 7:22 or so, why did he use 30 degrees instead of 41 degrees?",
        "A": "Because the opposite side of the 41-degree angle is just a part of the line segment AC (there is another angle above 41\u00c2\u00ba without label). By using the 30-degree, the segment is now the adjacent side of that angle, so it can be expressed in that way too."
    },
    {
        "video_name": "EBKNtjZAjXg",
        "Q": "at 2:05 Sal said that we have no way of telling the 3 arced angle. is there some kind of rule for the angle in the middle (The one E creates) to tell that angle? E.G. triangle ADE has angles DAE, 30, ADE, unknown, and AED, the middle. Could we use the fact that the middle adds up to a certain number (E.G. 360) to figure out angle DEA. Since we know that angle DEC is 49 (180-(90+41)) then we know that the middle adds up to at least 98 (49*2)",
        "A": "Since you know mDEC = 49 then since DEA and DEC are supplementary (they form a line so add to 180), mDEA = 180 - 49 = 131."
    },
    {
        "video_name": "EBKNtjZAjXg",
        "Q": "At 7:45, why is the sin(CDA) = cos(30), don't they have to be complements?",
        "A": "sin(CDA) (= opp/hyp) = AC/AD (= adj/hyp) = cos(CAD)"
    },
    {
        "video_name": "EBKNtjZAjXg",
        "Q": "at 2:43 why did you only choose the 30 degree angle i mean you could have started with of the three and to find angle B's mesure we could have calculated it which would be 60",
        "A": "Sal started with the 30 degree angle most likely because: 1. it was the easier of the two angles to handle 2. neither the sin(41) nor the tan(41) actually pertained to AC/BC, and handling these first could introduce minor confusion on the student side. Sal was just trying to keep things as coherent as possible."
    },
    {
        "video_name": "EBKNtjZAjXg",
        "Q": "at 6:07, why didn't he just say,\"The only angle we see in the triangle that we see in a bucket is 30 degrees.The answer is cos(30).\"?",
        "A": "He wants to explain, I guess"
    },
    {
        "video_name": "1DcsREjyoiM",
        "Q": "Why, early in the equation solving, was the 5 taken out of the derivative, 1:40, and then multiplied back in later, 3:56, instead of performing the derivative of 5x^2y^2 when taking the derivative of everything else? Although I understand the the ending result and the concept this small point is a bit confusing to me in relation to the rest of the problem and the prior videos.",
        "A": "Novamwilliams2, I m not sure what to say besides it being a matter of taste. Sal could have chosen to do the product rule using 5x^2 and y^2 as his two functions that were being multiplied. In general I see people factor out any constants and then bring them back after all the calculus is done. I would not be surprised if Sal is not consistent with how he does this."
    },
    {
        "video_name": "1DcsREjyoiM",
        "Q": "At 1:40, Sal took the 5 out of the derivative. I understand that it can be taken out since it has nothing to do with x and y but what if I did not know that? Is there such a thing as a triple product rule for derivatives involving a multiplication between three terms?",
        "A": "This rule is only necessary when each function contains a variable. Since the first function is a constant, the rule is just overkill. You can just leave the five as a coefficient of the quadratic x and go from there or factor it out as Sal did. It still works all the same. I would just leave it in the expression and run with it."
    },
    {
        "video_name": "1DcsREjyoiM",
        "Q": "at 2:17 sal was finding the derivative of 5(x^2)(y^2)\nwhy is there + between 2xy^2 and 2yx^2?\nshould it not be just using chain rule as x^2(y^2), and then making it 2x(y^2) multiplied by the derivative of the inner function 2y? Please help!",
        "A": "It is the product rule, not the chain rule. That states d/dx (uv) = udv + vdu (where u and v are functions of x)"
    },
    {
        "video_name": "1DcsREjyoiM",
        "Q": "I think I need to go back to Algebra. That distribution at 3:17 was not F.O.I.L.\n\nCan anyone recommend the video that I missed that describes his method there?",
        "A": "Actually, I think it just looks more complicated than it actually is. He s not multiplying any binomials together there; he s just distributing the purple term to both terms of the green binomial, just like you might distribute 5(2x + 3y) = 10x + 15y."
    },
    {
        "video_name": "1DcsREjyoiM",
        "Q": "1:14 Sal derives two things in the second bracket. X^2 with respect to x and Y^2 with respect to why. Can you even do that? same bracket derived with respect to different values?",
        "A": "Sal derives y^2 with respect to x by the chain rule. Using the chain rule he first derives y^2 with respect to y and then y with respect to x. This is the basic tenet of implicit differentiation. It starts to look a bit hairy and magical when the thing you are deriving gets more complicated. Go back and look at the first lecture on implicit differentiation and that will probably help."
    },
    {
        "video_name": "1DcsREjyoiM",
        "Q": "1:40 - Sal takes the 5 out. I dont understand how this is possible.\n\nCan someone show me another example where this works?",
        "A": "He states at 1:46, that: d/dx {5(x^2)(y^2)} is the same as 5 d/dx {(x^2)(y^2)} Think of d/dx as a type of variable . Similarly, this can be seen with the Distributive property of multiplication, where: (Ax b) + (Ax c) = Ax (b + c) Hope this helps."
    },
    {
        "video_name": "1DcsREjyoiM",
        "Q": "Why does Sal keep distributing terms like this at 3:11? I keep thinking he should be using FOIL, but it seems like he is treating binomials as if they are a single term?",
        "A": "That s exactly what FOIL does, too: they both use the distributive property. a(b + c) = ab + ac (a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd (a + b + c)(d + e) = (a + b + c)d + (a + b + c)e = ad + bd + cd + ae + be + ce etc."
    },
    {
        "video_name": "k45QTFCHSVs",
        "Q": "How could you know that the triangles are similar? 6:58",
        "A": "@Rita: it is wrong because 1. You cannot prove triangles similar by proofs (ASA, AAS, etc); proofs prove triangle congruence. 2. AAA is not a form of triangle congruence. the two triangles are similar because Sal states that the hypotenuse of triangle AHG (10) is 2/3 of the length of the hypotenuse of triangle AED (15). Same with their bases; HG (6) is 2/3 of the length of HG (9). The same can also be said for their heights; AH (8) is again 2/3 of the length of AE (12). :)"
    },
    {
        "video_name": "k45QTFCHSVs",
        "Q": "How can it be that GH is the height of triangle AFG ?\n5:50",
        "A": "If you re wondering how the height of a triangle can be outside of the said triangle, the answer is thatthe height of an obtuse triangle is always outside of the triangle. Hope that helps !"
    },
    {
        "video_name": "k45QTFCHSVs",
        "Q": "At 7:00 I tried to find the length of FH by using the pythagorean theorem, already knowing the length of GH and FG. So GH is 6, as Sal showed, and I thought that CF=AD=15. FG is \"the shorter part\" of CF, so it should equal 1/3*15=5, shouldn't it? Now the problem occurs: according to pythagorean theorem, FG^2=GH^2+FH^2 and this is equal to 5^2=6^2+2^2 ! Where did I go wrong...?",
        "A": "The median lines aren t necessarily all the same length. You can t assume CF=AD."
    },
    {
        "video_name": "qLYYHWYr8xI",
        "Q": "At 2:04, why are we using median instead of mean to calculate IQR? What's the difference?",
        "A": "I think the nature and the purpose of this method is to use the median. So that we can easily identify the 4 quarter of a group of data. by looking at the box plot."
    },
    {
        "video_name": "qLYYHWYr8xI",
        "Q": "At 1:20, why does the median have to be 4 to the left, 4 to the right?",
        "A": "Because the median is the number in the middle, therefore if there are 9 numbers, the one in the middle will have 4 numbers to the right and 4 to the left."
    },
    {
        "video_name": "qLYYHWYr8xI",
        "Q": ".At 5:12 I have a question. If the median is ten then how do two tens join to find the median of the right and left set?",
        "A": "The median is in between the tens so you count them when you are trying to find the left and right sets median. (7, 9, 9, 10, 10, - 10, 11, 12, 12, 14)"
    },
    {
        "video_name": "qLYYHWYr8xI",
        "Q": "At 4:54 what if there are only 3 numbers on the left and right what would I do?",
        "A": "If you have only 3 numbers on each side, then you still choose the middle number on each side to be the median number."
    },
    {
        "video_name": "iUQR0enP7RQ",
        "Q": "At 0:48 he he says it is important what you do the order of multiplying in but if its multiplying the order shouldn't matter right? Doesn't 1 x 2= 2 and 2 x 1= 2?",
        "A": "In matrix multiplication, order does matter because the dimensions have to be ordered correctly. For example, you can multiply a 3x3 matrix with a 3x2 matrix because the center numbers- the number of columns in the first matrix and the number of rows in the second matrix-(3 and 3) are the same. However, you can t multiply a 3x2 matrix by a 3x3 matrix because the center numbers (3 and 2) are not the same."
    },
    {
        "video_name": "iUQR0enP7RQ",
        "Q": "what is that sign that sal uses at 0:53",
        "A": "Are you referring to the equal sign (=) that has an extraneous mark on it? I think this was just a writing mistake: it s not easy to write things out using the program that Sal is using. So: this is just the equal sign. It s unfortunate that Sal happened to make the same mistake on both symbols, here :-("
    },
    {
        "video_name": "iUQR0enP7RQ",
        "Q": "How to come up with this matrix at \"7:03\"? What's the relation with matrix A?",
        "A": "Sal doesn t go into it in this video, but there is a way to get the inverse without memorizing the formula. The idea is to set up an augmented matrix [A|I] and then use reducing to get to [I|A^-1]. Try checking out the videos on Reduced Row Echelon Form (that s where I d start at least) and then try it yourself with the generic matrix he gives here to see how to get this formula."
    },
    {
        "video_name": "iUQR0enP7RQ",
        "Q": "shouldnt it be -15/7??? at 12:00+",
        "A": "No, the first part of what he is doing is he is multiplying the first object from Matrix A -- in this case, 5/7 -- by the first object in Matrix B -- in this case, 3 -- and since both values are positive, the answer is positive. If one of these values negative, then the 5/7 would be negative; however this is not the case."
    },
    {
        "video_name": "iUQR0enP7RQ",
        "Q": "At 3:34, Sal mentioned column vectors. So, each column in a matrix is a vector?",
        "A": "Yes, a matrix made up of only one column is called a column vector while a matrix made up of one row is called a row vector. Therefore you can consider a single column in a matrix as a vector."
    },
    {
        "video_name": "iUQR0enP7RQ",
        "Q": "At 2:18 he says that an identity matrix is zeros with ones along the diagonal. Could you put the ones anywhere so long as you had one one in each column and row?\n\nfor example, would this work?\n[0010]\n|1000|\n|0100|\n[0001]",
        "A": "No. They must be in this (diagonal order). He proves it at 2:20 by showing you why it works. :)"
    },
    {
        "video_name": "17a443nL7Qw",
        "Q": "at 1:05 why less than or equal to?",
        "A": "You typically preserve the equality (in this case, the less than or equal to) from the original problem statement."
    },
    {
        "video_name": "17a443nL7Qw",
        "Q": "At 0:45 he says that x is less than or equal to negative 2, but he didn't switch the sign. Why did he not do that? Help plz!!!",
        "A": "He did not divide or multiply by a negative #,only then do you switch the sign."
    },
    {
        "video_name": "17a443nL7Qw",
        "Q": "at 1:13, why is the dot filled?",
        "A": "The little line under \u00e2\u0089\u00a5 means or equal to . Let s say, for example, x\u00e2\u0089\u00a53.That would mean x is greater than or equal to 3. So to say or equal to, you would have to fill in the dot so that it doesn t look like x>3, with the empty dot. Get my point?"
    },
    {
        "video_name": "17a443nL7Qw",
        "Q": "at 1:24 what does Sal mean by \"all the nubers lower that -2\"?\ni mean it could go all the way to -1,000,000,000,000 (etc).",
        "A": "Yes it can,we don t know what the number is"
    },
    {
        "video_name": "ClFrIg0PpnM",
        "Q": "by the way in the last part at 7:12 grant missed the transpose of the (x-x0) i believe correct me if i am wrong",
        "A": "You re correct :)"
    },
    {
        "video_name": "Kt7Dwr7BftE",
        "Q": "at 1:45, why did he put a 2 at the end of the number line when he didn't even use it",
        "A": "It also lets you see how big some numbers are in comparison to other numbers."
    },
    {
        "video_name": "Kt7Dwr7BftE",
        "Q": "At 8:59, who invented Khan Academy",
        "A": "The person who teaches you in the KA videos is the creator of Khan Academy. He is Salman Khan(Sal for short)"
    },
    {
        "video_name": "j5xis6Hlnds",
        "Q": "At 0:45 isn't X 4 less than 9. Maybe I'm just not understanding . . .\n\nThanks",
        "A": "Point A is 5 less than 9 as Sal said at 0:50. A is at the point (4, -3), and the centre of dilation is at (9, -9). Therefore, A is 5 away from 9, as 9 - 4 = 5. Maybe you were confused because one of A s coordinates is 4."
    },
    {
        "video_name": "-Zlq5tNl94M",
        "Q": "Why does he say he messed it up at 0:34, when he counted right?",
        "A": "He skipped the second rectangle at first :)"
    },
    {
        "video_name": "xCIHAjsZCE0",
        "Q": "At 5:00 in the night how is it so dark?",
        "A": "its in alaska"
    },
    {
        "video_name": "xCIHAjsZCE0",
        "Q": "At 7:12 why did Sal say Each child drank 2 cups instead of eating 2 Cupcakes?",
        "A": "I think it was by accident... Either that, or what the previous answers were saying. Good job noticing it! :)"
    },
    {
        "video_name": "xCIHAjsZCE0",
        "Q": "At 6:58, why does he take 200c - 300c?",
        "A": "This is because he is subtracting the second equation from the first equation. Here is the breakdown of what he did: 500a + 200c = 2900 500a + 300c = 3100 500a - 500a = 0 200c - 300c = -100 2900 - 3100 = -200 It was necessary to take 200c - 300c in order to find out what c is equal to. After that, take the value of c and back substitute to figure out what a is equal to."
    },
    {
        "video_name": "xCIHAjsZCE0",
        "Q": "At 7:11 Kal says that each child drinks two cups. Which is it, are we drinking cups or eating cupcakes? Ha!.",
        "A": "Daryl, It another example that shows Sal is human and imperfect and can make mistakes just like most of us. I just wish I made as few mistakes as Sal does."
    },
    {
        "video_name": "xCIHAjsZCE0",
        "Q": "at 4:00 I started to wonder\nhow does elimination help me?\nalso, what if you are doing something with the formula d+tr (or whatever it is) how to you get it to y=mx+b?\nthanks <3",
        "A": "If you are presented with an equation in the form ax + by = c if is usually easier to solve the system using elimination rather than converting to slope intercept form and trying to work with fractional coefficients."
    },
    {
        "video_name": "xCIHAjsZCE0",
        "Q": "at 0:44, how does the king know how many cupcakes both kids and adults eat?",
        "A": "I don t know. Maybe he keeps track?"
    },
    {
        "video_name": "xCIHAjsZCE0",
        "Q": "at 6:25, why do you subract the two equations and not add them?",
        "A": "The goal is to eliminate one of the variables, in this case it was a . Sal subtracted the two equations to cancel out the a s . If he added the equations, the result would still have two variables. I hope that helps"
    },
    {
        "video_name": "xCIHAjsZCE0",
        "Q": "at 2:27 do you add or subtract to get the answer for both of the equations?",
        "A": "For this equation he subtracted, but he subtracted to get rid of both the 500a s so he could solve for c. Hope this makes sense."
    },
    {
        "video_name": "NIazpCER9oM",
        "Q": "4:38, divisor (x-a) is 0, dividing 0 is not in the definition of division.",
        "A": "x-a is not in the denominator anywhere in the theorem. Because division by 0 is undefined, there is a removable discontinuity in q(x) at x=a; but since in the remainder theorem q(x) is multiplied by x-a, the reason for a discontinuity no longer exists."
    },
    {
        "video_name": "NIazpCER9oM",
        "Q": "At 1:21, why isn't the remainder placed over the divisor as it is in other videos? Is there a difference between the two answers or the way they're are written? Such as having added (6/(3x-1)) or having added 6.",
        "A": "It s because if you move (x - 1) from the right side of the formula to the left by dividing, you also have to divide 6 by that monomial. The equation then changes to: (3x^2 - 4x + 7)/(x - 1) = (3x - 1) + 6/(x - 1)"
    },
    {
        "video_name": "2mzuFKCuDg4",
        "Q": "At 0:14 he says the angle makes a complete line. Is there a special name for this type of angle or is it just called a line?",
        "A": "a straight angle"
    },
    {
        "video_name": "SjN3_xCJamA",
        "Q": "At 3:55, he makes an equation that says (x + 3)(x + 7) = x to the second power + 10x. Why is the x also following the 10? It's already been expressed in the x to the second power.",
        "A": "we are multiplying two binomials. so we can express it as x(x+7) + 3(x+7) = x^2 +7x + 3x + 21 = x^2 +10x +21"
    },
    {
        "video_name": "WR9qCSXJlyY",
        "Q": "At 5:18 Sal Says that you cannot add or subtract matrices with different dimensions. Why can't you just put a zero in all the missing places and then add all the others?",
        "A": "That s a very interesting idea! If it hasn t already been done, you could create this new operation on matrices. Who knows? It might turn out to be of great use, in some new branch of mathematics. My first question would be: Where would you put the missing places? There would be different ways of making two matrices conform, so the result of this new kind of addition would depend on the chosen method of padding (so to speak) with zeros."
    },
    {
        "video_name": "WR9qCSXJlyY",
        "Q": "From 4:08-4:20. how do you properly understand how the negative (-) and positive (+) work in adding and subtracting?",
        "A": "You simply add negative numbers instead of positive numbers(in the second matrix)"
    },
    {
        "video_name": "WR9qCSXJlyY",
        "Q": "At 4:20, why do you put negative for some of the numbers and not all?",
        "A": "The (-1) is multiplied by all the numbers in the second matrix: 0 + (-1)(-1) = 1, 1 + (-1)(3) = -2 3 + (-1)(0) = 3, 2 + (-1)(5) = -3 And the answer matches the first method he used."
    },
    {
        "video_name": "WR9qCSXJlyY",
        "Q": "3:27-3:28 Did Sal said negative one too many times?",
        "A": "He just repeated negative one. No biggie."
    },
    {
        "video_name": "WR9qCSXJlyY",
        "Q": "At about 2:10, Sal said that you can't just put a matrix in any order when multiplying and dividing. I understand why you wouldn't be able to put the matrices in any order while dividing, but since multiplying is simply repeated addition, wouldn't the order of two matrices not matter?",
        "A": "Actually, repeated addition of a matrix would be called scalar multiplication. For example, adding a matrix to itself 5 times would be the same as multiplying each element by 5. On the other hand, multiplying one matrix by another matrix is not the same as simply multiplying the corresponding elements. Check out the video on matrix multiplication. Indeed, matrix multiplication is not commutative."
    },
    {
        "video_name": "OJ-wajo6oa4",
        "Q": "On 0:17, is borrowing the same as regrouping?",
        "A": "ues, in subtraction they are the same......"
    },
    {
        "video_name": "n4EK92CSuBE",
        "Q": "At 2:10, why are you allowed to arbitrarily set (2/3)x^2 equal to sin^2(theta)?",
        "A": "Sal has noticed that 1-(2/3)x^2 looks like 1-sin^2(\u00ce\u00b8). So, in order to see if that can help him solve the problem, he sets them equal to each other. Here he skips a few steps. This appears to be what confused you. 1 - (2/3)x^2 = 1 - sin^2(\u00ce\u00b8) - set them equal -(2/3)x^2 = -sin^2(\u00ce\u00b8) - subtract one from both sides (2/3)x^2 = sin^2(\u00ce\u00b8) - multiply both sides by -1, and now we re caught up to where Sal started. I hope this helps!"
    },
    {
        "video_name": "n4EK92CSuBE",
        "Q": "At 0:54, is there a reason Sal chooses to use 1-sin\u00ce\u00b8 instead of 1-cos\u00ce\u00b8?",
        "A": "Sometimes it is a matter of preference. However, the existence of a negative sign can make one easier than the other. Remember that your sine and cosine derivatives are related, but they cycle a negative around. That also happens in integration."
    },
    {
        "video_name": "n4EK92CSuBE",
        "Q": "At around 2:10 why would we set sin^2(theta) equal to (2/3)x^2?",
        "A": "Sal wants to see if he can use the Pythagorean identity to help him substitute and solve the problem. He skips a few steps that show this, but he originally sets 1-(2/3)x^2 equal to 1-sin^2(x)."
    },
    {
        "video_name": "n4EK92CSuBE",
        "Q": "At 2:52, he solved for theta and found that it equaled arcsin times the sqrt of 2/ sqrt of 3 x. Where does the arcsin come from and how do I know to think of that?",
        "A": "arcsin is the inverse of sine therefore, sin(arcsin( anything ))= anything"
    },
    {
        "video_name": "n4EK92CSuBE",
        "Q": "how does he get the sin squared in his equation at 2:12",
        "A": "He s using trig substitution to solve the problem. By using algebra to make the the equation look like the trig identity and then substituting sin in for the x term to get 1-sin^theta"
    },
    {
        "video_name": "n4EK92CSuBE",
        "Q": "So, in the video at about 6:14, Khan takes the constant term 1/sqrt(2) out of the integral. What if he wanted to, say, rationalize that fraction with the integral? By multiplying in a sqrt(2)/sqrt(2)? The inside of the integral read as sqrt(2)*d(theta), but what would the outside look like?",
        "A": "For a \u00e2\u0088\u0088 \u00e2\u0084\u009d: d/dx(a\u00e2\u0080\u00a2x^n) \u00e2\u0086\u0092 a\u00e2\u0080\u00a2d/dx(x^n) \u00e2\u0086\u0092 a\u00e2\u0080\u00a2n\u00e2\u0080\u00a2x^(n - 1) \u00e2\u0088\u00ab(a\u00e2\u0080\u00a2x^n)dx \u00e2\u0086\u0092 a\u00e2\u0080\u00a2\u00e2\u0088\u00ab(x^n)dx \u00e2\u0086\u0092 a\u00e2\u0080\u00a21/(n + 1)\u00e2\u0080\u00a2x^(n + 1)"
    },
    {
        "video_name": "n4EK92CSuBE",
        "Q": "At 2:07, Sal takes (2/3)x^2 = sin^2(theta).\nBut shouldn't it be (2/3)x^2 = m.sin^2(theta) where m is any arbitrary constant?\nThe range of the sine function is only from -1 to 1.\nIn case our x is greater than, say, 3, take 6 for example, then (2/3)x^2 cannot equal sin^2(theta), since sin^2(theta) can never equal 24, since it can equal 1 at max.\nThis where the constant 'm' will come in handy.\nWhy is it okay just to use sin(theta) ?",
        "A": "You are wrong that the sine maps only to the range [-1,1]. While its true if you are dealing with the real numbers, it is not true if you are dealing with complex numbers. For example, sin(pi/2-i ln(2+sqrt(3)))=2"
    },
    {
        "video_name": "n4EK92CSuBE",
        "Q": "at 5:43 when Sal takes the square root of cos^2(x) why does he only take the positive form of it and not the +- form",
        "A": "Good question. The \u00c2\u00b1 really comes from \u00e2\u0088\u009a(x\u00c2\u00b2) = |x|. So, to answer your question, I m gonna ask it a different way: why doesn t it simplify to cos \u00ce\u00b8/|cos \u00ce\u00b8|? For us to use arcsine at all, we have to assume theta is be -\u00cf\u0080/2 < \u00ce\u00b8 < \u00cf\u0080/2, that s the domain of arcsine. (We can t use -\u00cf\u0080/2 or \u00cf\u0080/2 because then you ll have a division by zero problem) Graph cosine and look at what happens for \u00ce\u00b8 in that interval."
    },
    {
        "video_name": "n4EK92CSuBE",
        "Q": "how do you solve for theta like sal did in 2:46 ?",
        "A": "inverse trigonometric functions sin (arcsin(x))=x cos (arccos(x))=x tan(arctan(x))=x csc(arccsc(x))=x sec(arcsec(x))=x cot(arccot(x))=x"
    },
    {
        "video_name": "n4EK92CSuBE",
        "Q": "At 3:29 isn't it supposed to be \u00e2\u0088\u009a(3/2)*-cos\u00ce\u0098, or I somehow missed something out? Isn't dx/d\u00ce\u0098 sin\u00ce\u0098 = -cos\u00ce\u0098 and vice-versa??",
        "A": "d/dx[sin(x)] = cos(x) d/dx[cos(x)] = -sin(x)"
    },
    {
        "video_name": "n4EK92CSuBE",
        "Q": "at 2:45, how does he know that theta equals that value? Like what are the steps he used to solve for theta, I don't understand.",
        "A": "He took the inverse sine of both sides. You take arcsin to get rid of the sine just like you would square to get rid of a root."
    },
    {
        "video_name": "z6lL83wl31E",
        "Q": "I can't help but notice that the time of this video is 3:14 (pi). Did Vi Hart do this on purpose? Does anyone else notice this? Thanks in advance.",
        "A": "Vi supports Tau and dislikes Pi, so I doubt it was intentional. It is a funny coincidence!"
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "In 0:55, why do you evaluate something in the parentheses before you tackle the exponents?",
        "A": "its just the order of operations. You will get it wrong if you don t do it that way."
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "At 2:00, How Would You Know When To Use The Parantheses Of 5 And 4 For Multiplication Instead Of Just Using Them Normally? How Would You Know Which Parantheses To Do First? I Would Appreciate If You Reply Back To Me. Thanks!",
        "A": "If there are parenthses, then you do all the parenthesis at once, then solve for the rest. For example, 5 x (3 + 2) + 1 x (9 - 1) , you would first do the parenthesis, which would look like this: 5 x 5 + 1 x 8. Then use order of operations to get: 25 + 8, which equals 33."
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "at 2:35 where does the 5x4 go in the order of operations? It's not technically multiplication",
        "A": "It is multplication. These are the different ways of writing a multiplication expression: 1.a x 2 2.a2 3.a(2) 4.(a)(2) 5.a * 2 these methods work for both numbers and variables, except we rarely use method 2 for variables."
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "At 1:17 Sal mentions multiplication and division. For multiplication and division and also addition and subtraction, does it matter which order you multiply or divide? Do you multiply before you divide? Do you add before you subtract?",
        "A": "The order is from left to right, but it doesn t really matter."
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "At 4:04, why did Sal subtract 28 - 11 first instead of adding 11 + 44 first? I thought the order of operations says to do addition before subtraction?",
        "A": "you can do either one first he just chose to subbtract first."
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "Since when does 60/2=8? 2:50",
        "A": "He said 16, not 60. You can see on the board 6 + 10 : 2. If you do addition first you ll get 6 + 10 = 16 and if you do division after you ll get 16 / 2 = 8. Sal said 16 . He also said that it is wrong. Order of operation tell us to do division first 10 / 2 = 5 and than do addition 6 + 5 = 11 8 is wrong answer. 11 is right."
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "4:09 there first minus is there 28-11+44 as per pemdas first want to do addition then subraction but u did first 28-11 then u done adition tell me sir reply",
        "A": "Addition and Subtraction are actually on the same level. When you get problems like these, evaluate from left to right."
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "at 1:14 why do we need parenthesis",
        "A": "There are 3 sets of parentheses in the problem -- which ones are you referring to? The 1st 2 sets are used to show multiplication. The 3rd set is used to group a set of work that must be done 1st."
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "At 4:07, Why did Sal do 28 - 11? As in the order of operation (BODMAS), Addition should come first, right?",
        "A": "Addition and subtraction are done in the order you find them from left to right. The same goes for multiplication and division. Example 5 - 3 + 2 = 4"
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "at 1:57 i realized that the (5)(4) is 5X4. is that another why? or what?",
        "A": "Yes, they both mean the same thing. You will see it a lot in the future."
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "At 2:22, Sal said that we multiply 5 and 4. But why do we multiply them even though they're in different parentheses?\n\nAny thoughts?",
        "A": "When there are two parentheses that are right next to each other, and already simplified as well as possible, (i.e. (5)(4) ) we can take away the parentheses, because there is nothing else to do inside of them, and we get 5 x 4. Since there is nothing in front of 5x4 that is higher up in PEMDAS we can multiply them, and get 20! Hope that helped :)"
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "Hi, in school I learned a trick called PEMDAS , so at 4:09 wouldn't it be 28-55 instead of 28-11+44 whereas you could've just used PEMDAS to solve the expression. Thanks",
        "A": "It s the same thing, so it doesn t matter. It doesn t matter which way round you do the addition and subtraction because you will get the same answer. As long as you do brackets then powers first, then multiplying and dividing, then adding and subtracting, it doesn t matter which order. You can switch multiplying and dividing and it should still work (in the UK we re taught to divide then multiply) or switch adding and subtracting."
    },
    {
        "video_name": "GiSpzFKI5_w",
        "Q": "At 3:30 what does it mean when it says to simplify",
        "A": "it means to make a fraction such so it does the top number and the bottom number do not share a factor. for example, take the fraction 2/4. both the top and the bottom share the factor of 2, so if we divide the top and the bottom by two, you end up with the fraction 1/2, which is equal to 2/4 but is simplified."
    },
    {
        "video_name": "L70UJVjc-bc",
        "Q": "at 2:00 where did he get the one",
        "A": "It s part of the exponential growth equation: y=a(1+r)^t"
    },
    {
        "video_name": "vxQvL_WhBGU",
        "Q": "At 4:26, did he intend to change the domain of the x-integral from [0,2] to [0,3]? If so, by what reasoning did he do so?",
        "A": "The domain for x was always [0,3]. That was what was stated at the very beginning."
    },
    {
        "video_name": "vxQvL_WhBGU",
        "Q": "on 2:54 Sal does dz first. What happens if you do dx first rather than dz first for triple integral? would I still get the same result?",
        "A": "yes because changing the order of integration doesn t affect the final result. It is a matter of convenience as to which order you choose to integrate in. Hope this helps"
    },
    {
        "video_name": "X1E7I7_r3Cw",
        "Q": "At 7:47-7:49 why didn't Pythagoras run right through the bean field? True he was scared of beans but he was going to get killed.",
        "A": "As he proclaimed, he was more afraid of beans than death."
    },
    {
        "video_name": "X1E7I7_r3Cw",
        "Q": "Does this \"irrationality D:\" attitude of Pythagoras at 6:57 remind anyone of this?\nA long long time ago\nLong before the Superbowl\nAnd things liiike lemonade\nThe Hellenic Republic was full of smarts\nAnd a question resting on the Grecian hearts was\nWhat is the circumference of a circle?\nBut they were set on rational numbers\nAnd it ranks among their biggest blunders\nThey worked on it for yeeears\nAnd confirmed one of their biggest fears\nI don't know if they cried\nWhen irrationality was reeealized",
        "A": "...But something deep inside them died the day, they discovered, Pi...."
    },
    {
        "video_name": "X1E7I7_r3Cw",
        "Q": "3:44 to 3:52 What the blocks is she playing with?",
        "A": "Vi used the piece of her paper she cut while doing stopmotion"
    },
    {
        "video_name": "X1E7I7_r3Cw",
        "Q": "3:37 does anyone know any other whole nunber thing that works other than 3,4,5?",
        "A": "You re looking for Pythagorean triples . Three whole number sides that give you a right triangle, like 3,4,5 or 5,12,13 or 8,15,17. There are infinitely many. One easy way to generate them is to square any odd number (say 7 which gives you 49) and then the two consecutive numbers that add up to 49 (24 and 25 in this case) give you the other two sides. So, 7,24,25 is also a right triangle."
    },
    {
        "video_name": "X1E7I7_r3Cw",
        "Q": "In 4:47, b square is even. Why does that mean that b is even?",
        "A": "Good question. An odd number squared is always odd. - 3*3=9 - 5*5=25 - 17*17=289 An even number squared is always even. - 2*2=4 - 4*4=16 - 24*24=576 You can reverse these two rules to say that the square of an even number (assuming it has a square) is even and the square of an odd number is odd"
    },
    {
        "video_name": "X1E7I7_r3Cw",
        "Q": "at 6:46, why does someone keep saying \" unless it doesnt exist\" .",
        "A": "That person has just proven that things wouldn t make sense unless the ratio of two whole numbers equalling \u00e2\u0088\u009a2 does not exist."
    },
    {
        "video_name": "X1E7I7_r3Cw",
        "Q": "At 4:21, aren't the ratios supposed to be the other way around?\n\nsqrt(2) \u00e2\u0089\u0088 14:10 \u00e2\u0089\u0088 1414:1000 \u00e2\u0089\u0088 141,421,356:100,000,000 \u00e2\u0089\u0088 ....",
        "A": "Yes they are."
    },
    {
        "video_name": "X1E7I7_r3Cw",
        "Q": "6:32 but it can be odd because 6\u0002 is odd and so is 10\u0002. can somebody help?",
        "A": "there are ratios modern ratios say 6:2 is the probability of say in baseball 6 points at inning 6 and 2 points at inning 2 now 6:2 is the probability of the 6 point team winning"
    },
    {
        "video_name": "X1E7I7_r3Cw",
        "Q": "At 00:03 what does she mean by dirt on him?",
        "A": "No no no! The interactive transcript said it turned out he was just too good."
    },
    {
        "video_name": "X1E7I7_r3Cw",
        "Q": "At 8:09 she says he might not have liked them metephorically. What does she mean by that?",
        "A": "to abstain from beans meant not to vote because back then you voted by putting beans in a jar."
    },
    {
        "video_name": "X1E7I7_r3Cw",
        "Q": "At 0:16 Why was Pythoragras scared of beans ?",
        "A": "Thanks WallAvi for the anwser i will look it up on the website"
    },
    {
        "video_name": "iPwrDWQ7hPc",
        "Q": "At 11:53, we see that 4 choose 0 is equal to 4 choose 4, and 4 choose 1 equals 4 choose 3. Can we generalize this and say that n choose k equals n choose (n-k)?",
        "A": "Yes we can. There are (at least) two ways to see this. n choose k = n!/(k!\u00c2\u00b7(n-k)!), and n choose (n-k) = n!/((n-k)!\u00c2\u00b7(n - (n-k))!) = n!/((n-k)!\u00c2\u00b7k!), which is the same thing. Or we can note that Pascal s triangle is symmetrical and n choose k is given by the kth entry on the nth row."
    },
    {
        "video_name": "iPwrDWQ7hPc",
        "Q": "at 0:53 can we make (a+b) to the 2nd power be 2a+2b",
        "A": "No. 2(a+b) will create 2a + 2b. The exponent of 2 on (a+b)^2 is not the same as doing 2(a+b). You need to use FOIL to multiply the 2 binomials. (a+b)(a+b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2"
    },
    {
        "video_name": "iPwrDWQ7hPc",
        "Q": "Sal, can you explain more clearer how you got n/k!(n-k)!, I understand its a formula we must know to solve the binomial theorem but can you explain how for example at 9:56 you did the n/k!(n-k)!. want to know how did you get 4x3x2x1 ?",
        "A": "the product of n natural numbers from 1 to n is denoted n! and is read as n factorial. So n!=1x2x3x4.......(n-1).n. therefore 4! = nx(n-1)x(n-2)x(n-3). => 4 x 3 x 2 x 1."
    },
    {
        "video_name": "iPwrDWQ7hPc",
        "Q": "At around 6:00 Sal writes the formula. Isn't the sum of numbers between 0 and n if n=4 infinite? This would make the answer be infinite every time n>0. Enlighten me please.",
        "A": "That notation simply means that the equation he wrote right to sigma, gets added to itself when k varies from 0 to 4. So first time in that equation we are gonna write 0 in place of k and then add the same equation but this time we use 1 in place of k and this continues till k reaches the given number, that is 4 in this case."
    },
    {
        "video_name": "iPwrDWQ7hPc",
        "Q": "at 1:20 why is it ab+ab how did it come to that",
        "A": "FOIL method. (a+b)*(a+b)= a*(a+b)+b*(a+b)= a^2 +ab+ba+b^2 and ab=ba so it is a^2+2ab+b^2"
    },
    {
        "video_name": "mXsn-YYUN4Y",
        "Q": "What is the value of e at 4:07?",
        "A": "e is an irrational number like \u00cf\u0080 (meaning it cannot be written as the ratio of 2 integers and thus in its decimal form it will go on forever without any pattern). The first few digits of e are: e \u00e2\u0089\u0088 2.718"
    },
    {
        "video_name": "mXsn-YYUN4Y",
        "Q": "At 2:05 Sal uses his calculator and chooses \"e\" and raises that to a power of 2.1. What is \"e\" and where did that irrational come from?",
        "A": "e is a constant, just like PI is a constant. e is something like: 2.71828.... e is the base of natural logarithm. I believe that e stands for Euler s number."
    },
    {
        "video_name": "mXsn-YYUN4Y",
        "Q": "7:44 The solution is not 2.06 because the values we got are 7.8 and 7.6, so they're in distance of 0.2 apart. He should have been more precise, it's probably 2.055 that would have gotten a more precise answer that would get the solution to be 0.01 apart (I'm guessing, we'd have to verity).",
        "A": "It s x, not y, to the nearest 100th. 2.06 is the closest in hundredths for x. Do you think it s closer to 2.07? 2.05? (The closest I got was 2.05855, with the difference in y s about 3 thousandths)"
    },
    {
        "video_name": "mXsn-YYUN4Y",
        "Q": "At 6:48, Sal says that 2.06 is definitely going to work. How does he know right off the bat that that will be the estimate? Because that is the closest value hundredths away, does that mean that that is how he can estimate it? To get the 2 answers within .01 of each other, how do we know that the answer is not going to be 2.065, or 2.062354, or anything really? It seems too precise for that to be the answer...",
        "A": "He knows that 2.05 is too small and that 2.07 is too large, and that 2.06 is much closer than either. So he has a reason to think that 2.06 is as close as he can get in hundredths."
    },
    {
        "video_name": "ZTbTYEMvo10",
        "Q": "At 2:57, grant said red vectors are super long but they should be super small right ?",
        "A": "I agree, I think the colors are wrong - the ones near the origin should be blue / small vectors."
    },
    {
        "video_name": "u8JFdwmBvvQ",
        "Q": "at 2:10 can the circle area be less than the area of the sector?",
        "A": "think about this: a circle is a pizza sliced into piece(s). The piece(s) largest area can only be the pizza. That s like what a sector is : a slice of a pizza"
    },
    {
        "video_name": "u8JFdwmBvvQ",
        "Q": "In 0:56, where did he get 360?",
        "A": "That is how many degrees there are in a circle."
    },
    {
        "video_name": "-eAzhBZgq28",
        "Q": "@13:00 Sal is talking about the existance of more than one inverse functions. So for f(x) there exists f-1(y) and g-1(y) that will get us back to x (if that makes sense). But aren't f-1 and g-1 the same functions, something like \"y= x - 2\" and \"x - 1 - 1\"? Or are there 'really' different functions that map every value of x to the same value of y?",
        "A": "He only assumes that there are two distinct inverses in order to show that that leads to the conclusion that they re the same (a contradiction), and thus proves that really there s only one inverse for any function."
    },
    {
        "video_name": "-eAzhBZgq28",
        "Q": "i dont get it, you read the composition thing from right to left or what? 14:00if you were to read it normally then he is wrong isnt he?for example the composition of f with g should be Ix and not Iy, because we first do X -> Y then Y -> X it's Ix right?\n\nIm starting to thin you read it like parenthesis? like f(x) etc?",
        "A": "(fog)(y) = f(g(y)). (gof)(x) = g(f(x)). So e.g. for fog you first do g(y), then f(x=g(y))."
    },
    {
        "video_name": "-eAzhBZgq28",
        "Q": "Also he says at 9:51 that y is a member of the set capital Y, what does he mean by the set?",
        "A": "Any collection of things {1, 2, 3} , or {you, me, Sal}. or whatever."
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "What's the line on top of 36? On 0:27",
        "A": "The line is just there to indicate that 36 repeats forever. That s why Sal wrote that 36 with the line over it equals 0.363636..... They mean the same thing just expressed differently."
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "At 2:06 it says we divide both sides by 99, but they are leaving it as a fraction, Is it supposed to be left as a fraction or is it supposed to be divided?",
        "A": "Well, since the title of this video is Converting repeating decimals to fractions (part 2 of 2) you should probably leave it as a fraction. Besides, if you divided your answer, you would be left with the original decimal number that you were trying to convert into a fraction. For example, if you converted 0.3333333 to 1/3 and divided 1/3 then you would be left with 0.333333333."
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "I don't get 5:45",
        "A": "U"
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "i don't get 7:30 to the end of the video. Can someone explain to me what sal is doing in the last two minutes?\n(He uses a different method.)",
        "A": "All Sal is doing in the last several minutes is converting the repeating decimal into a fraction. He shows that you can, instead of creating a infinitely long decimal, use long division to get a mixed number. That makes it a lot easier to use in many different things; multiplication, addition, subtraction, division, etc., etc."
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "At around 5:18 in the video Sal multiplies the Numerator and Denominator by 10. Why is it that we can multiply the right side of the equations by 10 and not the left side but still get a valid equation?",
        "A": "Oh, haha, that makes sense. I appreciate the response Danielle. :)"
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "at 1:25 you multiply the 0.3636... by 100 which is 36.36....., but you don't multiply the x(0.36) by 10?",
        "A": "You would want to multiply by 10^2=100 instead of 10^1=10, because now the repeating group is 2 digits (the 36) instead of just 1 digit. If you are not convinced of this, try multiplying by 10 instead of 100, and you will see that when you subtract the value of x from the value of 10x in the next step, you will encounter a difficulty because the repeating parts of the decimals will fail to cancel out. Have a blessed, wonderful day!"
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "At 5:00, how does multiplying 70.7 times 10 make the fraction correct? This is might be stupid, but I just dont get it.",
        "A": "Miriam, 70.7/900 can t be a fraction because it contains a decimal, 70.7 By multiplying 70.7/99 by ten, it changes to 707/990, and since there s no decimal anymore, the fraction is acceptable. Hope that helps!"
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "At 4:09, how does 100x - x =99X?",
        "A": "If you have 100 of something (x in this case), then take away 1 of those x, you are left with 99 of x."
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "At 3:55, wasn't he supposed to put 10x instead of x alone?",
        "A": "The repeating decimal has two repeating digits. In order to make the repeating portions of the decimal cancel each other out, he had to use 1x. If he used 10x, he would have had another repeating decimal on his hands, as 71.4141... - 7.1414... = 64.2727... Thus, 1x was the proper choice."
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "2:55 Wait a second! If he says that in 0.714 only the 14 part is repeating, then why is he multiplying with a 100 instead of a 10? I don't get this part. From the looks of it you could even argue that the repetition starts from 0.714. Or am I wrong here? He would have got just 99x = 7.07 right?\n\nCan someone please explain this to me?",
        "A": "The trick is to get the repeating pattern directly below itself. If you only multiply with 10 you get: 7.14141414... (positive 10x) 0.71414141... (minus 1x) 6.42727272... (10x-1x) Which is just another repeating pattern I am not sure what you mean by your last question about the 99x, but 99x=70.7"
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "@3:45 isn't \"14\" the number that is mark to be infinitely repeating? why Sal left 41 after the decimal: 71.414141 instead of 7. 141414.... why? ;_; shouldn't the digits after the decimal point be identical?",
        "A": "He multiplied the decimal by 10 which we know from elementary school just shifts the decimal 1 place value to the right."
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "At 3:18, what does Sal mean with 'the repeating pattern can be right under itself'?",
        "A": "i think it means that he is dividing the repeating decimal out"
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "At 8:15 I was completely lost: why/how he did that?\nWho ever respond to this, thank you very much!",
        "A": "Well, it s a little odd how he did that, but he s just subtracting one number from another, and setting up the borrowing ahead of time. It s just: 3254 -2997 ------ 257 So he sees the first subtraction is 7 from 4, so he has to borrow, and that reduces the 5 in the tens place to a 4, and so on."
    },
    {
        "video_name": "Ihws0d-WLzU",
        "Q": "Why at 2:04 did he subtract \"x\" one time? what is the point? Help!",
        "A": "Here we have two equations. The first one (100x=36.363636...) was made by multiplying x=0,363636... by 100 so we can get rid of fractional part after subtracting lower equation from upper one. We can do it cuz this way we subtract THE SAME QUANTITY (since x = 0,363636.. ) from the both sides of upper equation, which is valid step to do. This way we lose one x one the left side and all the fractional part on the right hand side. Hope it helps!"
    },
    {
        "video_name": "dNnItWLuHH4",
        "Q": "At 2:20, I didn't understand how he got 101 tenths.",
        "A": "Imagine you wanted to buy some candy and the shop keeper had no change. The candy costs $20.10 (twenty dollars and ten cents) and you have a twenty dollar note and a dime. You go to the bank and they give you: a 10 dollar note 100 ten cent coins (dimes) plus you already have your original ten cents Now you can buy your candy with 101 dimes!"
    },
    {
        "video_name": "dNnItWLuHH4",
        "Q": "at 1:10 how is it 2500? when the question is20.1?",
        "A": "The question was 0.25 and it s 25 hundredths."
    },
    {
        "video_name": "GDppV18XDCs",
        "Q": "i was lost at 6:46. When solving the quadratic is 1/4 and 4 the only answer? what does he mean both of the constrains has to have the same sign?\n\nHow did he get 1/4 and 4 > 0, then 1/4 and 4 < 0?",
        "A": "Both (p-1/4) and (p-4) must be the same sign so when you multiply them together it will be a positive number (aka > 0). He breaks it into two scenarios: if they re both + or both -. If both (p-1/4) and (p-4) are positive they will be >0 and if they are both negative they ll be <0. Then he just makes sure the variable is alone on the one side and uses logic to deduce that the answer is less than 1/4"
    },
    {
        "video_name": "GDppV18XDCs",
        "Q": "Around 8:34 in the video, Sal says that the equation can be simplified to p>4 because if something is greater than 4 it must be greater than 1/4. Wouldn't the equation be simplified to p>1/4 because if it is p>4 than p cannot be 2/4 which is greater than 1/4 but not greater than 4.",
        "A": "No, that part of the solution cannot be simplified to p > \u00c2\u00bc You want a summary statement that works all the time, and the solutions must satisfy both p > \u00c2\u00bc and p > 4 There are two parts of the constraint he has to consider: the positive case where the raw solutions are p > \u00c2\u00bc and p > 4 Using your 2/4, that cannot be a solution because it is not greater than 4. Giving p > \u00c2\u00bc as the summary solution would say that you could have 2/4 as a solution."
    },
    {
        "video_name": "GDppV18XDCs",
        "Q": "At 7:54 why is it for both negative you need to flip the inequality sign?",
        "A": "Because a negative times a negative will always be positive, which fill in (p-0.25)(p-4)>0."
    },
    {
        "video_name": "GDppV18XDCs",
        "Q": "At around 4:14 when you are taking away 2 1/4 p from both sides and you also move some of the parts of the left, why didnt you keep the squared on the one, on the left side of the greater than sign.",
        "A": "The reason is that 1\u00c2\u00b2 means you want to square the 1, in other words that you will multiply 1 x 1 ---and that result is just 1 There wasn t any reason to keep it as 1\u00c2\u00b2 when its value is so simple."
    },
    {
        "video_name": "GDppV18XDCs",
        "Q": "At 3:38, where does he get -2p from for the inequality??",
        "A": "Sal was taking the product of two binomials ( 1 - p ) * ( 1 - p ). The -2p came from the sum of the products of the 2 inner terms and the 2 outer terms."
    },
    {
        "video_name": "GDppV18XDCs",
        "Q": "At 3:31, where did he get the -2p from?",
        "A": "Recall FOIL for multiplying binomials. (1 - p)\u00c2\u00b2 can be written as (1 - p)(1 - p), now use FOIL F, 1 times 1 = 1 O, 1 times -p = -p I, -p times 1 = -p L, -p times -p = p\u00c2\u00b2 Now we have 1 - p - p + p\u00c2\u00b2 Combine like terms: 1 - p - p + p\u00c2\u00b2 = 1 - (p + p) + p\u00c2\u00b2 = 1 - p(1 + 1) + p\u00c2\u00b2 = 1 - p(2) + p\u00c2\u00b2 = 1 -2p + p\u00c2\u00b2."
    },
    {
        "video_name": "GDppV18XDCs",
        "Q": "At 7:05, what does he mean by both positive, and what does he mean by both negative?",
        "A": "Listen again carefully from about 6:30 to 6:45. Sal explains that if the product of the two factors is greater than zero (in other words, positive), then the only way that can happen is if both are positive or both are negative."
    },
    {
        "video_name": "4tsjCND2ZfM",
        "Q": "At 1:53, what is a \"gelatinous cranberry cylinder\"? Dose anyone know how she made it?",
        "A": "You can buy a can of cranberry in the store. Personally, I don t like it but thats beside the point. When she refers to it as a gelatinous cranberry cylinder , she is just giving it a fancy, mathematical name. Hope that helps! Ben Doucette"
    },
    {
        "video_name": "4tsjCND2ZfM",
        "Q": "did anyone see on 2:02 the pi and the tau?",
        "A": "Yes. The pi was half ot the tau ."
    },
    {
        "video_name": "4tsjCND2ZfM",
        "Q": "At 0:44, shouldn't she say \"inseparably\" instead of \"in-separately?\" It's not in my Web browser's dictionary unless I add the hyphen (which I did)? The video is AWESOME, like all Vi Hart videos are, but I just noticed that.",
        "A": "Vi-isms are in a dictionary class all by themselves. snakesnakesnakesnake star flakes mathemusician matherole Christmath etc. Prediction: Some day you will see Vi-isms in your browser s dictionary. A word we use today Chortle was made up by Lewis Caroll by combing Chuckle and Snort."
    },
    {
        "video_name": "4tsjCND2ZfM",
        "Q": "Who is the other person at 0:13?",
        "A": "A French person. Apparently a cook of some sort. Vi Hart lives in Canada, and many French live there too, so I m not surprised that she could get a French cook to help her."
    },
    {
        "video_name": "4tsjCND2ZfM",
        "Q": "at 0:55 it says they can be made in the fourth dememsion but can they be made in the fifth dememsion.",
        "A": "You only need 4 dimensions. Any extra dimensions have no effect."
    },
    {
        "video_name": "iivtjjdSu9I",
        "Q": "i got confused on the part 2:30",
        "A": "Yep you are right izaac"
    },
    {
        "video_name": "AuD2TX-90Cc",
        "Q": "At 0:57 Sal says, \"And instead of the decimal we'll write and.\" Do you always have to say \"and\" for a decimal when writing it out or saying it?",
        "A": "Without the and , one hundred sixty seven hundredths could mean 160.07 or 100.67 or 1.67 That s one hundred (and) sixty and seven hundredths or one hundred, and sixty-seven hundredths, or one hundred and sixty-seven hundredths. The extra (and) is the way it is said in English English."
    },
    {
        "video_name": "AuD2TX-90Cc",
        "Q": "So at 0:03 to 0.06\nwhat did it mean that you would give away the answer ? How would you give away the answer if you haven't even write it in word from yet?>",
        "A": "He would give it away by saying it in WORDS."
    },
    {
        "video_name": "R-6CAr_zEEk",
        "Q": "At 6:17, Sal did CB/CA=CD/CE, but can you do CB/BA=CD/DE?",
        "A": "Yes you can, the reflexive property dictates as such."
    },
    {
        "video_name": "R-6CAr_zEEk",
        "Q": "At 6:02, Sal starts talking about the ratios and their equivalents. Does this change for right triangles?",
        "A": "Sal used ratios and equivalents to calculate DE,the triangles in this video are similar triangles,which means their angles are the same.It also holds for right triangles ."
    },
    {
        "video_name": "R-6CAr_zEEk",
        "Q": "Throughout the video(like at 6:15), you use the fractions and cross multiplication to find the unknown. I have trouble knowing what goes where in those. I know how to cross-multiply after that. I just have trouble with the first step. Any advice?",
        "A": "When Sal puts down a fraction, say for example bc/dc, the letters are referring to line segments inside of the diagram. You have to find which sides correspond (are on the same side of the triangle) to each other and form a fraction or formula. Then you take a side you don t know and place the corresponding side on top of it. replace the segments with the lengths you do know, and begin cross multiplying. The answer will give you the missing length."
    },
    {
        "video_name": "R-6CAr_zEEk",
        "Q": "Around 0:10 Sal said tranversals. What are tranversals?",
        "A": "A transversal is any line that passes through two other lines on the same plane."
    },
    {
        "video_name": "R-6CAr_zEEk",
        "Q": "At 7:30 you say that 6 2/5ths MINUS 4 2/25ths = 2 2/5ths. Did you mean to say that 6 2/5ths MINUS 4 = 2 2/5? Because 6 2/5 MINUS 4 2/5 = 32/5 MINUS 22/5 equals 10/5. Simplified 10/5 equals 2.",
        "A": "no he said it the right way"
    },
    {
        "video_name": "R-6CAr_zEEk",
        "Q": "My question refers to the second example, at 6:15.\n\nInstead of the ratio CB/CA = CD/CE, I used CB/CD = CA/CE and came up with the same answer.\n\nWould this method work on all similar problems? Or did I just get lucky?",
        "A": "That will work on similar problems due to the interchange means property of proportions (if a/b=c/d then a/c=b/d) . There are a lot of different ways to write proportions."
    },
    {
        "video_name": "R-6CAr_zEEk",
        "Q": "At 4:14, how come Khan didn't convert the fraction into a decimal?",
        "A": "Its probably simpler to have as a fraction."
    },
    {
        "video_name": "H9BWRYJNIv4",
        "Q": "At 6:45, should Sal have said & written +/- a sub1n times the DETERMINANT of A sub 1n instead of just a sub1n times A sub1n ?",
        "A": "It should indeed be the determinant of A_1n, not just A_1n, since the formula would make any sense that way (It wouldn t be a pure number in the end)"
    },
    {
        "video_name": "H9BWRYJNIv4",
        "Q": "Why did u make the ex. matrix have at 9:44 have straight lines like | | rather than the matrix border like [ ]?",
        "A": "He s trying to find the determinant of the matrix, which is written with the || lines instead of the [] brackets for the matrix. If you go back to the video on the determinants of 2x2 matrices, Sal talks about the various notations used for the determinant."
    },
    {
        "video_name": "H9BWRYJNIv4",
        "Q": "How does the formula beginning at 5:03 translate into the formula for computing determinants for a 2*2 matrix given there are no submatrices?",
        "A": "All recursive formulas require a base case, where the problem has become simple enough to solve without performing any more recursion. In this situation, the base case is the 2x2 determinant, which we have defined previously."
    },
    {
        "video_name": "H9BWRYJNIv4",
        "Q": "9:18 Do you stop at a_1ndet(A_1n), or do you do the same for the rest of the rows?",
        "A": "You only need to do this for one of the rows."
    },
    {
        "video_name": "3cnIa0fYJkY",
        "Q": "At 4:35, why does Sal multiply the matrices this way? I thought you could multiply them straight across. Can you?",
        "A": "No, you don t multiply matrices entry-by-entry. You add matrices entry-by-entry. Check the section on matrix multiplication for the exact process."
    },
    {
        "video_name": "3cnIa0fYJkY",
        "Q": "Can you divide a matrix by a matrix? If so, and if it follows standard division, than Matrix I has to be equal to 1. Is this correct? at 3:17?",
        "A": "Matrix arithmetic doesn t have division, but it has inverses, which is really the same thing. A matrix multiplied by its inverse (if it has one) gives an identity matrix."
    },
    {
        "video_name": "3cnIa0fYJkY",
        "Q": "So, what's the identity matrix of a non square matrix i.e. :\n| 1 2 |\n| 3 4 |\n| 5 6 |\nat 6:53",
        "A": "They do have an identity matrix (two in fact) and it is square. It needs to have 1s going diagonally, 0s in all other positions, and obey such dimensions that multiplication is possible (is defined). In your case: [ 1 0 0 ] [ 0 1 0 ] x [ Your 3x2 Matrix ] [ 0 0 1 ] ...or: [Your 3x2 Matrix] x [ 1 0 ] [ 0 1 ]"
    },
    {
        "video_name": "6xPkG2pA-TU",
        "Q": "At 3:28 why do you multiply by P(B)?",
        "A": "We can use the formula P(A I B) = P( A intersect B) / P(B) and then transpose for P( A intersect B)"
    },
    {
        "video_name": "apkT6rVE8qo",
        "Q": "At 3:00, what is a principle ( or principal?) root? I have dealt with square roots many times, but is there a difference between a principle root and a square root? Thanks!",
        "A": "The root that matches the radicand in sign(+or-)."
    },
    {
        "video_name": "mKsKU0BAiRM",
        "Q": "At 0:35, what does Sal mean by \"one's place?\"",
        "A": "The one s place is the position a digit has in a number. For example, in the number 493, the digit 3 is in the ones place. the digit 9 is in the tens place, and the digit 4 is in the hundreds place. Think about it as in the number 493, you have 4 groups of 100, 9 groups of 10, and 3 groups of 1 (100+100+100+100+10+10+10+10+10+10+10+10+10+10+1+1+1) = 493"
    },
    {
        "video_name": "yiH6GoscimY",
        "Q": "Around 2:43 when he references line AB I was wondering if it can also be written as BA or does it have to stay as AB?",
        "A": "It can also be written as BA"
    },
    {
        "video_name": "D5lZ3thuEeA",
        "Q": "At 2:35, Sal said that an acute triangle is a triangle with all sides less than 90 degrees. When it came to explaining the right and obtuse triangle, he said ONE of the angles was 90 degrees (in the right triangle's case) and ONE of the angles was over 90 degrees (in the obtuse case). Is this true for the triangles, or did he mean only ONE angle is less than 90 degrees for the acute?",
        "A": "A triangle can have 2 or 3 acute angles. But, if it had 2or3 right or obtuse angles. it would not be a triangle at all!"
    },
    {
        "video_name": "D5lZ3thuEeA",
        "Q": "Can a triangle be 90 degrees? In 10:32 p.m",
        "A": "actually you can. watch in the next post"
    },
    {
        "video_name": "D5lZ3thuEeA",
        "Q": "at 3:14 the acute triangle looks like a equilateral",
        "A": "an acute triangle can be equelatitial"
    },
    {
        "video_name": "D5lZ3thuEeA",
        "Q": "At 2:37, Sal says \"all of the angles are less than 90 degrees\". Later, at 3:21 he says \"notice they all add up to 180 degrees\". So just to make sure that I understand this concept... In order for a triangle to be classified as Acute, it must meet 2 criteria. 1: The lengths are less than 90 degrees and 2: All add up to 180 degrees.",
        "A": "Only 1. All add up to 180 degrees has nothing to do with acute angles - that s true for any triangle. Ever. Isosceles, equilateral, scalene, acute, obtuse, right; the angles will always add up to 180 degrees. Try it with any triangle anywhere."
    },
    {
        "video_name": "D5lZ3thuEeA",
        "Q": "Is the right scalene triangle Sal mentioned at 5:34 an example of a Pythagorean triple?",
        "A": "Yes, it is because 3 squared plus 4 squared equals 5 squared."
    },
    {
        "video_name": "D5lZ3thuEeA",
        "Q": "at about 4:28, Sal says that obtuse triangles only have one angle, is that always true?",
        "A": "Yes if Sal says it it is right"
    },
    {
        "video_name": "D5lZ3thuEeA",
        "Q": "At 1:59, Salman Khan said that Equilateral Triangles can be Isosceles Triangles. Is he correct?",
        "A": "Yes, an Isosceles triangle has to have 2 equal sides and the angles opposite the equal sides must also be equal. Every equilateral triangle meets this definition since all the sides and angles are equal. However not all isosceles triangles are equilateral. Hope this helps"
    },
    {
        "video_name": "D5lZ3thuEeA",
        "Q": "At 5:50, Salman Khan said Equilateral Triangles can be Acute Triangles. Is he correct?",
        "A": "Yes! A triangle s angles have to add up to 180 degrees. An acute triangle s angles all have to be less then 90. So for an acute triangle to be equilateral all the angles would have to be equal, less than 90, and add up to 180. The example of this is a triangle with 60 degree angles since it is less than 90, equal to each other (making it equilateral), and add up to 180 (60+60+60=180). This triangle would be both equilateral and acute."
    },
    {
        "video_name": "KblYjo1Ijws",
        "Q": "At 4:10, why can't the interval be within -infinity < x < 0 and 0 < x < 5/2?",
        "A": "Careful with the word and, it implies that both are simultaneously true. The statement you gave says any real number less than 5/2 excluding 0. If the word or is used, it would read x is either less than 0 or x is between 0 and 5/2, but not both. You might have to say this out loud to grasp the concept I m getting at, but it is very helpful to understand when watching the video."
    },
    {
        "video_name": "bRwJ-QCz9XU",
        "Q": "At 0:55, why does he try to find the excluded values when there is only one answer to the equation?",
        "A": "In general when you see an equation like this, it s not always easy to see what the solution is, but it is relatively straight-forward to find the values for which the fractions are undefined by having a zero in the denominator. This step is necessary because if you proceeded to solve this type of equation and the solution was indeed one of the excluded values, then this equation doesn t have a solution because the equation is undefined at this specific value."
    },
    {
        "video_name": "bRwJ-QCz9XU",
        "Q": "at 0:55,why did he not exclude 0?",
        "A": "Since we cannot divide by 0, we exclude values for p that would cause the denominator on either side to become 0. For the left, we say that p cannot equal 1, and for the right we exclude -3. We don t exclude a value of 0 for p, since the resulting denominators are non-zero, but in working through the problem we see that the solution is something else anyway."
    },
    {
        "video_name": "bRwJ-QCz9XU",
        "Q": "At 3:13 I Have no idea how you got (-1 ). The guy in the video please answer and what is you name please. If anybody knows please tell.",
        "A": "Once he gets to -p = -17 he needs to get rid of the negative sign, so he multiplies both sides by -1 to get p by itself. His name is Salman Amin Sal Khan."
    },
    {
        "video_name": "BKGx8GMVu88",
        "Q": "8:56 what 1 doing inside of a bracket. example: 1( 1 + 0.100/2)^2",
        "A": "when you invest $1 at 10%, do you get back 0.10, or do you get back (1 + 0.1)?"
    },
    {
        "video_name": "BKGx8GMVu88",
        "Q": "2:19 where did he get 75 percent from ?",
        "A": "Its not 75% . Its 0.75 . 50% of 1.5 is 0.75 (1.5*50)/100 =0.75"
    },
    {
        "video_name": "BKGx8GMVu88",
        "Q": "At 9:20, shouldn't the yearly computation be 365.25 to account for leap years?",
        "A": "Good question, it s small details like that which show a very bright mind! Leap years do affect the way interest is calculated and sometimes it is even explicitly addressed in a contract. To break it down logically it would work like this : non leap year = 365 days = 365 * i = $365 leap year = 366 days = 366 * i = $366 average long run = 365.25"
    },
    {
        "video_name": "7S1MLJOG-5A",
        "Q": "1:02 - 1:35 how did he get 12 by multiplying 8 x 2 because it is 16, did he divide somewhere with the 1/2? how did he do that?",
        "A": "Half of 8 is 4 4 times 3 is 12"
    },
    {
        "video_name": "7S1MLJOG-5A",
        "Q": "In 1:12, I get the first part of the equation but cannot understand how he got the second part of the equation. Can anyone help me?",
        "A": "The area of a triangle is given by the formula (1/2)bh and he explains why that makes sense. Yes, he does make a mistake, which he later corrects. h = 3 and not 4 like he initially has it."
    },
    {
        "video_name": "yAH3722GrP8",
        "Q": "at 3:21 why and how do you switch the order and signs of the terms?",
        "A": "-a+b is the same thing as b-a. Sal was simply moving the negative term to the right, turning an expression with a negative number into an easier to understand subtraction expression."
    },
    {
        "video_name": "yAH3722GrP8",
        "Q": "At 5:56, can you simplify the square root of 2 x squared minus square roots of 6 x squared?",
        "A": "You can t really simplify the sum of the radicals anymore, but you can write sqrt(2) * x^2 + sqrt(6) * x^2 as x^2(sqrt(2) + sqrt(6)), which is factorized."
    },
    {
        "video_name": "yAH3722GrP8",
        "Q": "In 1:33 of the Addition and Subtraction of Polynomials video Sal mentions that a Polynomial cannot have fractional or negative exponents in it. Yet, in this video, they have. Which one is it? Or is it a binomial that different from a polynomial? Finally, is it that important to know these terms further into Math or are they just fancy words to showoff at parties?",
        "A": "This is just a matter of what you call things. If a function has only positive integer exponents of x and only real coefficients of x, then it is a polynomial. Otherwise, it is some other kind of function. So, it isn t that you cannot have fractional or negative exponents in a function, it is just that the name polynomial cannot be used to describe the function if you have fractional or negative exponents of x."
    },
    {
        "video_name": "2EwTHdg-xgw",
        "Q": "At 1:34, Sal talks about an exponential model. What is that?",
        "A": "Exponential function, in this case, is a function that y-values are decreasing but never touches x-axis. It keeps decreasing but NEVER touches x-axis."
    },
    {
        "video_name": "wbAxarp_Ug4",
        "Q": "Im confused at 4:27 please help:)",
        "A": "Sal is saying that when the denominator is smaller you divide the whole into bigger pieces and when the denominator of the fraction is bigger you divide the whole into smaller pieces. So take Sal s example of 3/4 and 3/9. Draw a square and divide it into 4 pieces and then draw another square the same size and divide it into 9 pieces. Now see. Is 3 pieces of the 4 bigger or 3 pieces of 9? 3 pieces of 4 is bigger."
    },
    {
        "video_name": "wbAxarp_Ug4",
        "Q": "At 3:59, Why does Sal put a dot on the smaller fraction? Wouldn't you just make the sign point to the bigger number?",
        "A": "I guess he would rather make a dot. But yes, I think the other way might work better."
    },
    {
        "video_name": "hoa1RBk4dTo",
        "Q": "At 3:14 he says it would make a trapazoid, but i dont get how if your cutting from the top down, you get a trapeziod",
        "A": "It just where the knife meets the rectangular pyramid. The way it meets,it turns into a 2-Dimensional figure,a trapezoid."
    },
    {
        "video_name": "hoa1RBk4dTo",
        "Q": "At 8:52,how is high school geometry way different from elementary geometry?Shouldn't they be the same?",
        "A": "High school geometry builds on a lot of the topics that you learned in elementary school in order to be able to solve more complex problems. In high school, you will often learn proofs of topics that you studied throughout middle school and elementary school, as well as more trigonometry topics dealing with cosine, sine and tangent."
    },
    {
        "video_name": "bRZmfc1YFsQ",
        "Q": "At 0:58, can n be imaginary?",
        "A": "Yes, it can. But, of course, working with complex exponents is a bit difficult, although the power rule still applies. Thus, d/dx 5x^(3i) = 15\u00f0\u009d\u0091\u0096x^(-1+3\u00f0\u009d\u0091\u0096)"
    },
    {
        "video_name": "kZzoVCmUyKg",
        "Q": "In 4:31 of this video he said that he has a candy bar and divides into 25 equal parts. So why 25 equal parts? Why not 2 equal parts?",
        "A": "It s an example. Besides, 25 parts is a lot more possibilities than 2."
    },
    {
        "video_name": "kZzoVCmUyKg",
        "Q": "at 7:28, how do you divide fractions?",
        "A": "Dividing and multiplying fractions is really easy, Ex: 4/10 divided by 2/5 Two 10ths = one 5th so if we have four 10ths it would equal two 5ths another way to say it is 1/2 ----- this is just a basic problem"
    },
    {
        "video_name": "kZzoVCmUyKg",
        "Q": "9:35, how do you do fractions without the pictures?",
        "A": "Do you mean like 1, 1/2, 1/3, 1/4, 1/5 and so on?"
    },
    {
        "video_name": "kZzoVCmUyKg",
        "Q": "How can you use the fruits as a whole, because the fruits are not equal and he said at 4:41ish that the whole point is to be equal when dividing a fraction.",
        "A": "Sal assigned each fruit as 1 whole instead of using their mass. He tried to simplify it a bit. Think of it this way. If we used only apples and each apple counted as 1 whole; ho many wholes do we have if we have 5 apples? We would consider that 5 wholes. But in reality each apple is not exacly the same size and shape as the others. We consider each one a whole to make it simpler to understand."
    },
    {
        "video_name": "kZzoVCmUyKg",
        "Q": "In 3:40 of this video he ask : \"what fraction of my fruit is yellow\"? why not ask : \"What fraction of my fruit is blue\"?",
        "A": "because otherwise it would be 1/5 and that s boring"
    },
    {
        "video_name": "RFn-IGlayAg",
        "Q": "8:00 How many times more intense than the smallest detectable earthquake is an earthquake measuring 5 on the Richter Scale in Columbia in 1906?",
        "A": "So I would say the Columbia Earthquake is 2.5 times more intense"
    },
    {
        "video_name": "RFn-IGlayAg",
        "Q": "At 7:27, why is the base 10? if base was greater, the differences between two earthquakes would be different right?",
        "A": "Correct, the differences would be greater with a larger base. I believe that the Richter scale and most logarithmic scales use base 10 for sake of simplicity, since multiplying numbers by ten or powers of ten is much easier."
    },
    {
        "video_name": "s4cLM0l1gd4",
        "Q": "At 0:29, doesn't Sal mean the middle not between? If the point is ONLY between, then it could be anywhere from the maximum to the minimum.",
        "A": "He could have been more specific and said middle or half-way between, since otherwise between could mean anywhere from the maximum to the minimum (as you said). I guess he assumed we would understand what he meant."
    },
    {
        "video_name": "s4cLM0l1gd4",
        "Q": "I'm not sure that Sal's definition of \"period\" at 3:11 is sufficient. After all, there are periodic functions (more complex ones than sine and cosine) in which you could return to the same y value and the same slope and still not have completed the cycle. An example would be a wave that alternates two short oscillations with one long one.\n\nSo my question is, what is the technical definition of a period?",
        "A": "A period is the length of the smallest interval that contains exactly ONE copy of the repeating pattern of a periodic function (any function that repeats its values in regular intervals, or periods ). A period can start anywhere and end anywhere, but the length is always the same."
    },
    {
        "video_name": "s4cLM0l1gd4",
        "Q": "At 2:41 he says 1 - 3 = -1, its -2.",
        "A": "This is a known error in the video. At 2:43 in the video, a box pops up and tells you that Sal meant to say -2 rather than -1."
    },
    {
        "video_name": "s4cLM0l1gd4",
        "Q": "At 4:07, Sal starts explaining what the period is, and he says that it is 2, but isn't the period of this graph 2pi?",
        "A": "No, the period of this particular graph is just 2. It takes 2 units for the function to go through a full cycle, so the period is 2."
    },
    {
        "video_name": "s4cLM0l1gd4",
        "Q": "@3:00pm does value of period ,amplitude and midline have units?",
        "A": "There was no indication of units on the x or y axis, nor did Sal mention what exactly the graph was intended to represent, so I do not think so."
    },
    {
        "video_name": "utQi1ZhF__Q",
        "Q": "at 6:25 Sal said I encourage you to test it out on the original equation. what did he meant by that. How can I test the solution on the original equation ?",
        "A": "I think that he meant solving it as separable equation."
    },
    {
        "video_name": "utQi1ZhF__Q",
        "Q": "In the first example of this video (when he started solving at about 1:00), the differential equation is a separable differential equation. So, how is the name of this equation, should I say 'separable' or 'exact'?\n\nThank you!!",
        "A": "You say, that this is an exact equation, because My=Nx, and separable, because you can algebraically sweep x and y to different sides of the equation. Two different definitions of two properties, and both match."
    },
    {
        "video_name": "mJ1P4A-KA8k",
        "Q": "At 1:10, why is 6 to the eighth power not equal 6 times 8?",
        "A": "the exponent, or power, indicates how many times you multiply a number by itself. So 6 to the 8th is really 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6. ... 6 multiplied by itself 8 times. Hope that helps."
    },
    {
        "video_name": "4n7TIvRHuDs",
        "Q": "At 2:26 why one and five are combined while negative seven is separate? f(x)= 1,5 OR -7\nIs there something between those two numbers in common? For example, why not to say just 1 or 5 or -7 instead of 1,5 or -7.",
        "A": "Sal is just using a common list form used in English. When you write a list, you typically don t but the word and or or between every list item. You list the items separated by commas then us the the conjunction ahead of the last item in the list. The prior line is what the range looks like in math language."
    },
    {
        "video_name": "g_dStt4st2I",
        "Q": "at 2:24 Sal said that \"all the points on this circle are equidistant from c\" and then later for point p as well but that doesn't seem to be correct because not all the points in a circle are equidistant from the centre only all the points on the CIRCUMFERENCE are. right?",
        "A": "Listen again carefully to the video at 2:24. Sal says that all the points on the BLUE circle are equidistant from C and that all the points on the YELLOW circle are equidistant from P. It s only the 2 points of intersection of the blue and yellow circles that are equidistant from both C and P (because the 2 circles have the same radius)."
    },
    {
        "video_name": "g_dStt4st2I",
        "Q": "In 3:48 what does circle centered mean?",
        "A": "He s referencing where the center of the circle is; it s centered at point (a, b), say."
    },
    {
        "video_name": "g_dStt4st2I",
        "Q": "At 1:54 in the video, Sal makes the two circles (centered at the center of the original circle and at point P) big. Is there a specific size for the circles for that specific distance or does any size circle work for any distance?",
        "A": "The circle you use in this process does not have to have a definite circumference, however must be larger than the original circle in the diagram."
    },
    {
        "video_name": "8vFhNhL-zm8",
        "Q": "so at 4:25 what's basically happening is that an angle at the center of the circle is double the angle made anywhere else on the circle, and conversely the angle anywhere else on the circle would be double the angle at the angle at the center.",
        "A": "Yes you are correct. It is a theorem, the inscribed angle is half of the central angle if and only if they are on the same arc."
    },
    {
        "video_name": "8vFhNhL-zm8",
        "Q": "At 4:05, can you just skip the step in which Sal finds the measure of the arc and go straight to finding the solution?",
        "A": "Yes, you can. The important fact in the problem is to know that for any arc the inscribed angle is half the central angle."
    },
    {
        "video_name": "8vFhNhL-zm8",
        "Q": "At 1:14. How did he figure out it was a right angle from it just being perpendicular?",
        "A": "All perpendicular line segments form 90\u00c2\u00b0 angles. It s the definition of perpendicular."
    },
    {
        "video_name": "FRlTh5HQORA",
        "Q": "Typically, on the AP Statistics exam, when they ask you to provide a 5-number summary of a set of data, can the minimum and maximum be outliers in your answer? So, at 8:03, would your 5 number summary be 6, (Q1), (Q2), (Q3), and 19? Or would it still be 1, (Q1), (Q2), (Q3), and 19? Which is better?",
        "A": "It should say in the question. If it doesn t say anything then include the outliers."
    },
    {
        "video_name": "61ecnr8m04U",
        "Q": "why does Sal, at 2:24, talk about the slope being positive one? isnt it rise over run, so that would make it 0/1=0?",
        "A": "Yes, the slope of the BOTTOM curve is 1 but that s not what he s graphing. He s saying the slope of the TOP curve is 1 and that s reflected in the BOTTOM curve which has a constant value of 1 (for the interval he is talking about)."
    },
    {
        "video_name": "61ecnr8m04U",
        "Q": "At 0:48, why is d/dx being used to represent \"the derivative of\"?",
        "A": "d stands for infinitesmally small change . dy/dx means the ratio of an infinitesmally small change in y to an infinitesmally small change in x. That s the definition of a derivative."
    },
    {
        "video_name": "61ecnr8m04U",
        "Q": "At 3:00; instead of having an absolute value where the slope is 1 & -2 could you also have a parabola?",
        "A": "First of all, in technicality, where the slope is 1 & -2, it is actually neither 1 nor -2 (indicated by the empty circles). Even if the slope were one and negative two, you could not use a parabola. Using a parabola would imply a changing slope. The function shown in the video had constant slopes of 1 and -2. When the derivative value is constant over a segment, that means that the slope of of the anti-derivative over the same segment is constant as well. tl;dr: no"
    },
    {
        "video_name": "pIMfRbznxKA",
        "Q": "At 6:01, why does Khan use k? Wouldn't the diagram be\nr => T (r) => k => H (k)\nWhy does Khan go from T (r) straight to H (k)? What happens to the K? Do we just ignore the K? Why??\nThanks.",
        "A": "The k is not lost. It is just assigned the value of T(r). Sal 1st uses the function T(r). It has an input of r and its output is T(r). Remember, k is just a variable to represent the input to the function H(k). Sal then makes k = T(r) . The input value for function H(k) will be the output from function T(r). It s like saying find the value of function H(k) if k=2. You find H(2). But, Sal is finding H(T(r)). Hope this helps."
    },
    {
        "video_name": "gRKZaojKeP0",
        "Q": "On 7:00 they metion about the angles wouldnt the 70 degree and the angle right next to it have to equal 180 because then the whole thing wouldnt make sense right?",
        "A": "the 70 and 110 degrees are supplementary angles . when they are supplementary that means they add up to 180 . so 110 + 70 = 180."
    },
    {
        "video_name": "gRKZaojKeP0",
        "Q": "At 4:75 to 4:59 he says a lot of \"thats\" which that is the top yellow angle?",
        "A": "I meant 4:53 to 5:00. Oops :)"
    },
    {
        "video_name": "gRKZaojKeP0",
        "Q": "4:40 What does he mean by congruent angles?",
        "A": "When he says congruent angles, that means that the angles are the same measurement."
    },
    {
        "video_name": "Jpbm5YgciqI",
        "Q": "At 0:59, how can you tell which point is which when they're all so close together?",
        "A": "There is a shape to the dots...they are organized from lower left to upper right."
    },
    {
        "video_name": "Jpbm5YgciqI",
        "Q": "At 1:25 what would be his score",
        "A": "Can you clarify what you mean about the score at the time @1:25?"
    },
    {
        "video_name": "Jpbm5YgciqI",
        "Q": "Can anybody really really descriptively explain what a linear function is ? I started to get confused at 0:14. Can any body help me ??",
        "A": "A linear function is a function in degree 1, i.e. it has no term with x to the power of anything > 1. So linear functions can t have x^2, x^3, so on, types of terms. They can only have terms with x^1 or x^0. Also they cant have negative degree terms."
    },
    {
        "video_name": "PC_FoyewoIs",
        "Q": "At 4:01, how does that happen, it just DOES NOT MAKE SENSE?",
        "A": "Calm down Think about it"
    },
    {
        "video_name": "PC_FoyewoIs",
        "Q": "At 0:57, you have to enter x first, then y right?",
        "A": "yes. The Z coordinate would be third on a 3-D plane."
    },
    {
        "video_name": "j9FDoYNxZlw",
        "Q": "At 2:50 the constant x's power one meant the derivative was one, but I remember in one of the first videos covered that the derivative of any constant is zero. What made it one here?",
        "A": "The derivative of a constant is 0, but x is not a constant. It s a variable. So we apply power rule, which works for functions of the form x^n."
    },
    {
        "video_name": "499MvHFrqUU",
        "Q": "from 2:11 on I am completely confused. And yes I have watched it multiple times.",
        "A": "Factor the terms. The gcf, in this situation, is 10uv. You put all the rest in parentheses. Does that help?"
    },
    {
        "video_name": "LohXZt7QzCM",
        "Q": "at 2:21 to the end of video i was lost plz someone explain",
        "A": "Sal moves the decimal point two spaces to the left because he is dividing by 100."
    },
    {
        "video_name": "LohXZt7QzCM",
        "Q": "At 3:50 I don\u00c2\u00b4t understand why 1,098 turns into 1098. The decimal just gets lost. Can someone explain this to me ?",
        "A": "Hello Daniel! Sal only removed the comma, which makes it 1098. The decimal isn t lost, it s behind 8. since ur supposed to divide 1,098 by 100 u would move the decimal two spaces to the left which gives u the decimal 10.98 does that make sense?"
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "I am confused where the plane intersects the 3D cone, in the hyperbola. (9:45 in the video)",
        "A": "In order to picture a hyperbola you have to imagine there being two cones, one on top of the other, touching at their points (sort of like this |><| ). If you then slice this double-cone with a vertical plane you ll end up with a hyperbola. Does that help?"
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "At 00:24 (about) Sal was writing down the names of the conic sections, could anybody please tell me the reasons for those particular names???\n\n\nThanks",
        "A": "circle ellipse parabola and hyperbola ha ha"
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "On 3:20, Sal drew asympototes, aren't they suppose to be drawn in dotted lines?",
        "A": "My teacher always instructs to draw them in dotted lines, because otherwise you might misinterpret them as the axes of the coordinate plane (which are drawn as continuous lines). Though there is no official rule about this. I guess it boils down to personal preference or the preference of your maths teacher."
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "06:00 That is also a Minkowskiy space, isn't it?",
        "A": "Yes, it is a Minkowski space."
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "Is there a way to find when (what angle measure) a plane must be tilted to to \"skew an ellipse until it pops and turns into a parabola\" (8:55)?",
        "A": "Yes. The angle, t (theta) = s , of the side of a cone with the horizontal. Then if s < t < 90 it s still a parabola, and 0 < t < s it s an ellipse."
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "At 3:21 what is an Asymptote if you were to explain it to someone?",
        "A": "A line in which no points rest and cannot rest depending if holes exist (Entirely different topic so I won t get into detail). Asymptotes can be crossed in some situations."
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "At 3:21 Sal mentions asymptotes. What are they used for?",
        "A": "Asymptotes help you think about the limit of where a curve is headed. You can look at a parabola and say, I see that it is curving, but because of where the asymptote is, I can tell that it will never get past this particular line."
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "around 4:00, is the graph of y=tan(x) hyperbola?",
        "A": "No, y=tan(x) is a tangent function."
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "Hey i couldn't see his yellow drawing at top left @ 10:11",
        "A": "It s OK. It was pretty messy.What part of it could you not see?"
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "what does he mean it pops (9:06 - 9:16)?",
        "A": "that it s no longer shown as a circle, it s just an open U shape"
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "at 3:27 what are the blue lines?",
        "A": "The blue lines are the asymptotes. As x approaches negative infinity or positive infinity the hyperbola will come closer and closer to these asymptotes but never reach them."
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "At 3:41, what is the exact term for a \"vertical hyperbola\"",
        "A": "vertical hyperbola dum dum"
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "Is the mathematical definition of a cone what is drawn in the video at 4:45, with 2 openings? Or is this just a useful construction for showing the difference between a hyperbola and a parabola?",
        "A": "Technically it s a double cone, but it s just referred to as a cone."
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "at 10:05, there is a verticle plane. so, the circle will be ... (yellow one). why it becomes like that shape?",
        "A": "Because it intersects both cones, you are left with a shape called a hyperbola. It expands in two separate directions becuase the cones it intersects are also expanding in both directions."
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "At 8:50, how can there be a parabola when the cone is infinite?",
        "A": "The cone is infinite, and so is the plane and so is the parabola. The key to the parabola is that the plane has to be exactly parallel to a side of the cone. Tilt one way and you get an ellipse, and the other way you get a hyperbola."
    },
    {
        "video_name": "0A7RR0oy2ho",
        "Q": "At 8:12 wouldn't the parabola have a bottom edge so therefore it wouldn't be a parabola?",
        "A": "It has an edge, but it is on the top part of it. Parabolas don t necessarily have their inflection (I don t recall the right term, but I think that s the one) on their bottom part. What differentiates this parabola (the inflection is at the top) is that if it was plotted on a plane, it would probably has a formula like this: y = -a*(x^2) + b*x + c, with a>0 (a being a real number), and b and c any real numbers."
    },
    {
        "video_name": "IReD6c_njOY",
        "Q": "At 4:44 Sal says that 2 * pi/2 is pi, so why does he then take the cosine of pi/2? Is that a mistake, or am I misunderstanding something?",
        "A": "Hi, He takes 3cos(2*\u00cf\u0080/2) = 3cos(\u00cf\u0080) = 3*(-1) = -3."
    },
    {
        "video_name": "IReD6c_njOY",
        "Q": "You said cos(pi/2) is minus 1 at 4:48 when you should have said cos(pi) is minus 1.",
        "A": "Hi said 2 times \u00cf\u0080/2 equal \u00cf\u0080, so cos(\u00cf\u0080) = -1"
    },
    {
        "video_name": "IReD6c_njOY",
        "Q": "At about 9:50 in your video did you mean that it has to be in the form:\nx = 3 cos(anything*t)\ny = 2 sin(anything*t)\n\nBecause you said:\nx = 3 cos(anything*t)\ny = 2 sin(anything*t)\n\nCheers Stefan",
        "A": "He definitely meant y = 2 sin {anything}t. Otherwise, this parametric equation would follow the path of a circle, not an ellipse."
    },
    {
        "video_name": "IReD6c_njOY",
        "Q": "Couldn't you take a normal equation and turn it into a parametric equation if you were given the coefficient of t, kinda like you can find the specific antiderivative with an initial condition? (see 10:00)",
        "A": "You are correct. Notice Sal says just with this information when he says you can t."
    },
    {
        "video_name": "IReD6c_njOY",
        "Q": "at around 9:50, i believe there is a mistake -- shouldn't it be 3 Cos of anything *t\nand 3 SIN of anything *t",
        "A": "I think it should be 2 sin of anything *t not 3 sin of anything *t, Cobyswat."
    },
    {
        "video_name": "IReD6c_njOY",
        "Q": "At 3:00, how can you put any number before t in the equation sin^2t + cos^2t = 1? For instance, if you put a 2 (as Sal does in this problem), wouldn't you have to multiply 1 by 2 as well? If not, how come?",
        "A": "So, is the reason why you can plug in any number because if sin^2t + cos^2t =1, then it is almost like they are canceling each other out. And this means that whatever number you plug in to multiply by, as long as it s the same number for sin and cos, they would still cancel each other out? Can someone provide an example?"
    },
    {
        "video_name": "IReD6c_njOY",
        "Q": "At around 5:30, how would I know which direction the graph is moving if the point jumps from (3,0) to (-3,0)?",
        "A": "Good question. You don t. Since the equations were basically the same in the video except the second set were based on 2t rather than 2 then Sal knew that the second graph was going in the same direction as the first but just twice as fast"
    },
    {
        "video_name": "IReD6c_njOY",
        "Q": "Around 07:30, how can you see which direction the ellipse is going when you go one half circle at a time? If you only have those coordinates, it might aswell go the other way couldn't it?",
        "A": "You are right in thinking the direction cannot be determined from what Sal has said. Rather, in this case, we know from the previous example that the path is anticlockwise. If he hadn t done that example first (the one with 3cos(t) and 2sin(t)) then we couldn t know the direction at all. Instead we would have to plot a point based on another t value, like pi/4."
    },
    {
        "video_name": "lxjmR4pYIVU",
        "Q": "I'm a bit confused at what Sal is doing at 1:48. Why does he bracket a single 2 with both 6 and 8?",
        "A": "Because three 2s are the same as 8 and one 2 and one 3 is the same as 6 if you multiply them."
    },
    {
        "video_name": "lxjmR4pYIVU",
        "Q": "At 2:11, I can't understand, I mean, I now know the prime factorization of eight, but I can't understand 'to be divisible by eight, you have to have at least three 2's in the prime factorization.' What does that mean? Why do you need at least three 2's to be divisible by eight? How does all of this work?",
        "A": "The prime factors of 8 = 2*2*2. So if you find the prime factors of another number, and it also has prime factors of 2*2*2, you know you can divide by 8 (because 2*2*2 IS 8, just in a different form). For example, prime factors of 72 = 2*2*2*3*3. The factors of 2*2*2 = 8, so 72 can be divided evenly by 8. 72 / 8 = 9. Notice, the 9 is the 3*3 in the prime factors. Hope this helps."
    },
    {
        "video_name": "lxjmR4pYIVU",
        "Q": "at 4:36, how do you add the fractions? won't it become a improper fraction?",
        "A": "If it becomes an improper fraction, you can simply make it a mixed fraction later. Remember that when adding or subtracting fractions, the denominators need to be the same. 1/7 + 14/7 = 15/7 which is 2 1/7. Hope this helped!"
    },
    {
        "video_name": "lxjmR4pYIVU",
        "Q": "Does 3/3 really equal EXACTLY 1?\n( 3:48 ) and how about 9/9 because I heard It was .99999999999999999.... on and on for \u00e2\u0088\u009e!",
        "A": "yah, cause 3 out of 3 would equal a whole."
    },
    {
        "video_name": "lxjmR4pYIVU",
        "Q": "at 3:45 how is 3 over 3 one?",
        "A": "3/3 is 3 thirds, which is the same as 1 imagine cutting a cake into three pieces (thirds). now, eat the three pieces. how much cake did you eat? 3 * 1/3 or 3/3 or 1 cake. hope this helps"
    },
    {
        "video_name": "qdf2CuMGdKs",
        "Q": "I realize that I still have some difficulties in understanding the meaning of position vectors.\n\nStarting at 15:30 Sal explains that the vector [3 0] - r (\"the pink vector\") is a member of N(A) although it is lying on the solution set of Ax = b (\"that pink vector is not in standard position but it is going to be a member in our null space).\n\nDoes this mean that every vector is part of both the solution set and the null space? And would this mean that the solution set and the null space are equal?",
        "A": "The pink vector is still part of the null space because shifting a vector from standard position does not change it at all. You could also think of it like this: The solution set to Ax=0 is contained as a subsection to the solution set of Ax=b. I hope this cleared things up for you."
    },
    {
        "video_name": "qdf2CuMGdKs",
        "Q": "At 14:37, Sal defines r as a member of the column space. Isn't that a mistake - did he not mean that r is a member of the row space?",
        "A": "yea he did, he misspoke, he never actually graphed the column space which in this case would be 3 6"
    },
    {
        "video_name": "qdf2CuMGdKs",
        "Q": "Can you do the dot product on two column vectors? (See 16:56) Shouldn't Sal be using a transpose of one of them or something?",
        "A": "The dot product can be used on any vectors, as long as they are of complementary sizes."
    },
    {
        "video_name": "DuYgVVU_BwY",
        "Q": "At 9:03, Sal says that that the arch of the graph will never touch the y axis. So would would y axis be called an asymptote? I mean, in hyperbolas and such we have asymptotes that can't ever touch certain lines. So i was just wondering if logs used the same terminology.\nThanks. Sylvia",
        "A": "Yes, it is an asymptote. Logs all have the y-axis as an asymptote unless translated. However, there is no limit to the range; there are no horizontal asymptotes in log functions."
    },
    {
        "video_name": "DuYgVVU_BwY",
        "Q": "I want to graph the function f(x)= -1/2 log2 (x+3)\nI am having trouble figuring out points to plot, x and y coordinates, so I can graph it. Like he begins to do at 2:38.\nPlease help show the mathematical steps in determining coordinates for this more complex logarithmic function.\nThanks!",
        "A": "x+3 tells you where the function lies on the x axis. in this case, 3 units to the left at 3 .. the -1/2 tells you the vertical stretch of the line ( i.e 2 means the line is steaper( like a V, 1/2 would be wider like a U. (but only one half of it ..)"
    },
    {
        "video_name": "DuYgVVU_BwY",
        "Q": "At 6:35 , sal says that the domain : x > 0 , but here we are putting the value of y and then finding the value for x , so how the domain is x > 0 ?\nDomain should be in the form of y i guess .\nPlease help me with this .",
        "A": "The domain is always the values of x - that is literally the definition of domain. The range is the values of Y"
    },
    {
        "video_name": "DuYgVVU_BwY",
        "Q": "at 3:58\nDoes anything to the 0th power equal 1 or 0. Some people say 1...some say 0.",
        "A": "Anything to the zero power always equals 1."
    },
    {
        "video_name": "DuYgVVU_BwY",
        "Q": "At 5:01 how do you use a calculator? Like if the problem was log2(x)=y then how can I find out what power to raise 2 to for a result of 3?",
        "A": "Some calculators let you choose a base, but most don t. You would have to learn how to convert from one base to another. Sal has vids on that."
    },
    {
        "video_name": "DuYgVVU_BwY",
        "Q": "At 6:33, what does Sal mean by that the domain has to be \"x>0?\"?",
        "A": "He means that the value you input into the equation cannot be 0 or negative. This is because there will be no solution (no possible y value) for those values of x."
    },
    {
        "video_name": "DuYgVVU_BwY",
        "Q": "At 6:21 it is said that there is no value of y for which we get a negative x.Is this applicable to all logarithmic functions?",
        "A": "Yes. If you have any function which looks like y = log (x) then x > 0, otherwise you get imaginary numbers."
    },
    {
        "video_name": "OmLl6pkvV-I",
        "Q": "At 1:25, there was a mistake, the question said:The bar graph shows how may( SUPPOSED TO BE many)! just telling you guys! Have a great day!",
        "A": "At 1:25, that was probably a typo. When people do the question, they ll probably know it s many and not may."
    },
    {
        "video_name": "ZRRG386v6DI",
        "Q": "At about 1:00, can there be an orthogonal version of a single vector? Doesn't a vector have to be orthogonal to something? ... or have I misunderstood the concept?",
        "A": "The dot product can t be 0 if there s no dot product."
    },
    {
        "video_name": "NehkLV77ITk",
        "Q": "0:50, the question \"is4792 divisible by three\"i wanted to ask if it is possible to just add up the last two numbers and check if they are divisible by three e.g. 9+2=11.And we know that 11 is not divisible by three . I'm really beginning to think that it also works that way.",
        "A": "On 0:57-1:20, they explain how to do that test"
    },
    {
        "video_name": "NehkLV77ITk",
        "Q": "At 4:17, Sal said, \"If you add two things that are divisible by 3, then the whole thing (sum) is going to be divisible by 3.\" If that confused anyone, here is is the proof:\nif numbers A and B are divisible by 3, then we need to prove (A + B) is divisible by 3.\nRewrite A = 3 * A'\nRewrite B = 3 * A'\nTherefore, A + B = 3 * A' + 3 * B'\n= 3 * (A' + B')\n\n3 * (A' + B') is clearly divisible by 3, so (A + B) is divisible by 3.",
        "A": "edit: Rewrite B = 3 * B"
    },
    {
        "video_name": "NehkLV77ITk",
        "Q": "I'm confused at 2:11",
        "A": "The 4 in the hundreds place is equal to 400. so that 4 equals 4 x (1 + 99). 400 = 4(100) = 4(1 + 99)."
    },
    {
        "video_name": "NehkLV77ITk",
        "Q": "sal says is 4792 divisible by 3 this is an emergencey! at 0:06 why say that?",
        "A": "For a number to be divisible by 3, the sum of the digits must be divisible by 3. Notice that the sum of the digits of 4792 is 4+7+9+2 which equals 22. However, 22 is not divisible by 3. So 4792 is not divisible by 3."
    },
    {
        "video_name": "NehkLV77ITk",
        "Q": "Is there really someone wholl come up to you and ask you- I dunno, could 3488 be divided by 3( Its mentioned at 00:10)",
        "A": "Not very likely to happen. But best to be prepared. BTW this was also mentioned at the very start from 0:01 but with a different number. (4792) And how do you write in code?!"
    },
    {
        "video_name": "NehkLV77ITk",
        "Q": "At 2:53, why is it just plus 8? Why doesn't Sal write something like 5+3 or 9-1? Three and nine are divisible by 3.",
        "A": "here goes my first answer, he could actually do the same for 8, it would have been: 8[1+0]. get it 4[1+99]=4*100=400 9[1+9] =9*10 =90 8[1+0] =8*1 =8 =498"
    },
    {
        "video_name": "NehkLV77ITk",
        "Q": "At 1:00 Sal says that in order to find out if 386,802 is divisible by 3, you add 3+8+6+8+0+2. On a math test, is it necessary to write the zero in your work?",
        "A": "Not really, because 0 plus what ever equals the number. It doesn t really change anything."
    },
    {
        "video_name": "NehkLV77ITk",
        "Q": "At 2:10 Sal says 400 equals 4(1+99), but why can't you just write it as 4(2+98)or 4(3+97). And the video doesn't prove that it works because he's just writing it as 1+99, but you could also write it as 2+98. Why is that?",
        "A": "Because 99/3=33, thus 99 is a perfect example of a number divisible by 3."
    },
    {
        "video_name": "qb0QSP7Sfz4",
        "Q": "At 0:44, she mentioned one tenth, is there such a thing as an \"oneth place\" in decimals?",
        "A": "There isn t really a word oneth but lets start by thinking what a oneth would represent. Well, if a tenth is one divided by ten or 1/10 and a hundredth is one divided by one hundred or 1/100, then a oneth would be one divided by one or 1/1. Since 1/1 is equivalent to 1 then a oneth is equivalent to a one. So, to answer your question, there isn t a oneth place because oneths are equivalent to ones."
    },
    {
        "video_name": "k68CPfcehTE",
        "Q": "At 0:08 Sal says he likes to have the bigger number on top, but you can have them the other way round can't you.",
        "A": "You can, but you end up with an extra step if you do it that way ."
    },
    {
        "video_name": "k68CPfcehTE",
        "Q": "0:10 why write 32 again?",
        "A": "he does this because he is writing the numbers in a different arrangement. This has two purposes: first, this is easier for some people to visualize, second (and more importantly) because this is the way to do long multiplication on paper. Hope this helps!"
    },
    {
        "video_name": "PXKHyT__B2k",
        "Q": "If you compare the \"How many department stores have exactly 7 watches\" (2:08) to the tiger question at 2:31 i think Khal messed up.\nThe answer should have been (on the tiger one) 3 because of 2,3, and 4 zoos having more than 24 tigers. How did Khal get 9? I do not get it, can someone explain that to me?",
        "A": "You are counting only the rows, not the number of zoos in each row. Remember, each number on the right represents a zoo. Thus... The row labeled 2: has 3 zoos with 25, 28, and 29 tigers. The row labeled 3: has 5 zoos with 30, 31, 32, 38 and 38 tigers The row labeled 4: has 1 zoo with 40 tigers. If you count all these, there are 9 zoos with more than 24 tigers. Hope this helps."
    },
    {
        "video_name": "aHzd-u35LuA",
        "Q": "around 3:06, he says that the equation equals theta. does that mean the inverse tangent and normal tangent just cancel each other out??\nthanks in advance for replying!",
        "A": "tan(\u00ce\u00b8) = 324/54 Yes, inverse tangent does cancel out tangent, another way to think about it is that inverse tangent undoes tangent. To isolate the theta in the above equation, you undo the tangent with an inverse tangent, but remember, you have to do it to both sides. \u00ce\u00b8 = tan^-1(324/54)"
    },
    {
        "video_name": "aHzd-u35LuA",
        "Q": "At 4:17, it was 80.53 degrees",
        "A": "The question required rounding off the answer to 2 decimal places. As the value the calculator gave was around 80.537, rounding off to 2 decimal places would give you 80.54 degrees"
    },
    {
        "video_name": "aHzd-u35LuA",
        "Q": "At 8:37 pm. The test I just took today asked me this. \"angle A is 30(degrees), using the link below (a trigonometric chart). What is the Cos of A?\" Now on the chart I can see the sine, cosine, tangent, etc, as they relate to the degrees including 30 degrees. But the cosine that I see on the chart for 30 degrees is not the answer. Please help.",
        "A": "cos 30\u00c2\u00b0 = \u00c2\u00bd\u00e2\u0088\u009a3 This can also be expressed as \u00e2\u0088\u009a\u00c2\u00be and (\u00e2\u0088\u009a3)/2 For reference, sin 30\u00c2\u00b0 = \u00c2\u00bd tan 30\u00c2\u00b0 = \u00e2\u0085\u0093\u00e2\u0088\u009a3 = 1/\u00e2\u0088\u009a3"
    },
    {
        "video_name": "aHzd-u35LuA",
        "Q": "But.. at 2:22 what is inverse tangent",
        "A": "It is the inverse function of tangent. In other words, if you know the tangent of an angle, the inverse tangent would give you the angle. NOTE: Although inverse tangent is typically symbolized as tan\u00e2\u0081\u00bb\u00c2\u00b9 x, it is NOT the reciprocal of tan x. The same goes for the inverse functions of the other trig functions."
    },
    {
        "video_name": "aHzd-u35LuA",
        "Q": "at 2:36 does sal multiply both sides by tan^-1 like we do in linear equations?",
        "A": "Tan^(-1) is called inverse tangent or arc tangent (arctan). It does not mean and is not the same as 1/tan. When you take tan^-1 of tan(\u00ce\u00b8) they cancel out leaving you with \u00ce\u00b8 alone. This is how you find the measurement of an angle in the right triangle using SOH CAH TOA. Similarly, there are sin^-1 (arcsin)and cos^-1 (arccos) for sine and cosine."
    },
    {
        "video_name": "aHzd-u35LuA",
        "Q": "Um, I don't want to be nit-pickey, but 4:18 in the video, Sal says 80.54, but the calculator said 80.53. Did I miss something? Did he say to round up?",
        "A": "If you listen carefully in the video, he says that the problem asks us to round up."
    },
    {
        "video_name": "aHzd-u35LuA",
        "Q": "At 4:18, Sal says that theta is 80.54 degrees, but the calculator shows 80.53.. So was that a mistake or was there just a reason behind it?",
        "A": "Sal s answer is correct. The problem asked for the answer to be rounded to 2 decimal places. The calculator has 80.537677792. The 3rd decimal place is above 5, so you need to round up. You didn t. You just truncation the number rather than rounded. Hope this clarifies things."
    },
    {
        "video_name": "acfjqWTwee0",
        "Q": "At 3:41 Sal talks about using a random digit table. In his example, he took two consecutive digits and then moved to the next two consecutive digits to get 59, 83, 35, 59, 37, .... Would it have been legitimate to have just moved one digit at a time rather than two to get 59, 98, 83, 35, 59, 93, 37, ...?",
        "A": "I had the same in mind. The answer is: You have to assign each of your cases a random number to pick that has the same number of digits than your largest case. Let s say we have 1, 2, 3 .. 25 cases, then we assign the following random numbers: 01, 02, 03, .. 24, 25."
    },
    {
        "video_name": "acfjqWTwee0",
        "Q": "At 3:32, How is choosing people with numbers assigned from 01 to 80 random? The person which is assigned the number 80 will have the least chance of being chosen because there are so many other 70's, 60's.... Shouldn't a random sample have everything in it equally likely?",
        "A": "it is equal because the person who is number 76 has the same amount of chance as being chosen with a person who has number 80. They are just numerical place holders to facilitate the generating process."
    },
    {
        "video_name": "d8lP5tR2R3Q",
        "Q": "At 5:40 why does he say that 0*12=0 when before in one of the last couple videos he says it is undefined.",
        "A": "Oh now I understand."
    },
    {
        "video_name": "d8lP5tR2R3Q",
        "Q": "at 1:43 you talk about properties what are those??",
        "A": "In your standard algebra a*b would always equal b*a for all numbers a and b. This follows from the commutative property . However, it is also possible to create other number systems where this is not the case."
    },
    {
        "video_name": "d8lP5tR2R3Q",
        "Q": "at 0:55 sal said that a negetive number times a negetive number makes a positive number i don't get that.",
        "A": "i guess its like the opposite of adding is multiply so yeah"
    },
    {
        "video_name": "d8lP5tR2R3Q",
        "Q": "at 1:40, Sal guesses Transitive and Commutative Properties. Isn't it the Associative Property?",
        "A": "It is the commutative property."
    },
    {
        "video_name": "d8lP5tR2R3Q",
        "Q": "from 1:54 to 2:54 what does he mean",
        "A": "He was explaining the rules of multiplying and dividing negative numbers:D"
    },
    {
        "video_name": "LSaaKau63Gg",
        "Q": "(on 1:05 ) how do you know the amount you can borrow from the other number?",
        "A": "Always borrow one from that digit. That is borrow one 10, or one 100 or one of whatever the place value is."
    },
    {
        "video_name": "pCLGpRaJfJ8",
        "Q": "At 1:10, Sal says that angle SIE is a supplementary angle. How did Sal know that LE was a diameter?",
        "A": "LE isn t a diameter... It s just a straight line segment and <SIE and <LIS add up to 180 degrees which makes them supplementary. A segment doesn t have to be the diameter to be straight. A diameter has to pass through the center of the circle to be a diameter. In this case that means that to be a diameter a line segment would have to pass through point O."
    },
    {
        "video_name": "QE2uR6Z-NcU",
        "Q": "@ :10:10 Can someone give a Actual Real Life example of that.?? of Mutally exclusive thank you",
        "A": "When you flip a coin, heads and tails are mutually exclusive."
    },
    {
        "video_name": "QE2uR6Z-NcU",
        "Q": "anyone noticed the error @ 7:00 (12+13)/29-5 !=20/29!",
        "A": "If I know what you mean, I believe it was actually (12 + 13 - 5)/29 = 20/29. :)"
    },
    {
        "video_name": "QE2uR6Z-NcU",
        "Q": "At 5:00 the question P(Y or cube) was asked could you subtract the green spheres and get the same answer",
        "A": "Yes. P(not green sphere) = P(yellow or cube)"
    },
    {
        "video_name": "4Bx06GFyhUA",
        "Q": "At 2:39 why the negative sign in: [ - ( x - 3 )^2 - 8 ] was distributed to the -8 too? Because I learned that for example 2(x+1)(x+2) = 2x+2 + x+2. Which means only distributing the first number to the first expression without the second.",
        "A": "The expression you wrote above is not the same as the one that Sal wrote. Sal s blue parentheses enclose 2 terms : (x-3)^2 and -8. The negative sign is outside of those blue parentheses and is therefore distributed to each of the terms inside. Your expression has moved the negative sign inside the parentheses (brackets) so that, as written above, the negative sign will only affect the first term. [ By the way, 2(x+1)(x+2) = (2x+2) times (x+2), not plus. }"
    },
    {
        "video_name": "4Bx06GFyhUA",
        "Q": "At 1:00 Sal begins using completing the square method in order to find the lowest maximum value.\nCan I use X= -b/2a where X gives the lowest maximum value and b and a are the coefficients of x and x^2 respectively.\nThanks.\nPS I did get the correct answer using this method. He showed it in finding the x coordinate of finding the vertex of a parabola.",
        "A": "By completing the square, Sal finds a quadratic (x - something)^2 that is shifted up or down by some other thing. X=-b/2a does not take into consideration the c term or constant. The quadratic shows explicitly the min or max value of this function."
    },
    {
        "video_name": "oBREri10ZHk",
        "Q": "What does Sal mean, from 1:07 to 1:17, about each of the four total parts not being exactly 25%? If I had a set of data, would the four parts vary in percentage depending on the number of data I have in a set?",
        "A": "Sal means that the areas in the box plot aren t exactly looking like 25%, although he should have said that instead."
    },
    {
        "video_name": "oBREri10ZHk",
        "Q": "At 1:25, what are the 2nd and 4th quartiles?",
        "A": "The second quartile is 10, and the fourth is 16"
    },
    {
        "video_name": "oBREri10ZHk",
        "Q": "Regarding the last statement at 6:32, can we say that if this statement is true (exactly half are older than 13 (the median)), then we know for sure that there is an even number of students. Otherwise (if the statement is false), then we know that there is an odd number of students. Right?",
        "A": "That sums it up good."
    },
    {
        "video_name": "zFksanIexHI",
        "Q": "As a general rule, is it better to use decimals or fractions when doing problems like this? My personal preference is fractions, but could someone please point out the merits of using decimals? For example, if you look at 1:52, that table gets a bit messy. Is it better to use fractions in general or are there some cases where decimals are good?",
        "A": "It s sometimes hard to immediately see the value of a fraction, where decimals show you very clearly the magnitude of the number, so decimals will be more useful in the real world in situations where you need to know what exactly a number is. I find fractions easier to deal with when manipulating expressions, though, and you always know you ll have the exact value when dealing with fractions where decimals sometimes have repeating numbers and so forth. Basically, it really depends on what you re trying to do."
    },
    {
        "video_name": "oajrmwCALmc",
        "Q": "At 2:18 Sal says, \"Now to find the median of the seven number\". What do you do when the upper and lower half are even?",
        "A": "You take the average of the two middle numbers. For example if the numbers were: 1,2,3, 5, 6, 7. The median would be the average of 3 and 5 which is 4. (3+5)/2 If the numbers were 1, 2, 3, 4, 5, 6, the median would be the average of 3 and 4 which is 3.5 or 3 1/2 (3+4)/2 Hope this helps."
    },
    {
        "video_name": "-cR6FzM1zNE",
        "Q": "At 3:09, why can't Sal use the fact that the antiderivative of (cosx)^2 = 1/2(x+sinx*cosx)? I worked the problem out this way initially, and unless I made some unrelated mistake, the two answers aren't the same. I got 2pi as the final answer.",
        "A": "(cos(x))^2 = 1/2(x+sin(x)*cos(x)) should yield the same answer, as sin(x)*cos(x)=(1/2)sin(2x)"
    },
    {
        "video_name": "za0QJRZ-yQ4",
        "Q": "At 3:25, Sal gives the formula for calculating the vertex of a parabola,\n-b/a.2\n\nIs this formula -b/a.2 or -b/a.c since in the example c = 2.",
        "A": "-b/2a not -b/ac"
    },
    {
        "video_name": "za0QJRZ-yQ4",
        "Q": "In 1:39-1:42, Sal says that a quadratic has a parabolic shape. I have been learning about quadratics lately and I wonder: why do they have a parabolic shape? I just started to learn about parabolas in physics two weeks ago (so I know a little about them), but I can't seem to understand how the two connect.",
        "A": "The parabolic curve derives from the 2nd degree term inside of the quadratic. Polynomials with different degrees have even more curves!! And as for seeing the parabola in physics, did the equation/formula used to get the parabola have a 2nd-degree term in it? Gargamel"
    },
    {
        "video_name": "za0QJRZ-yQ4",
        "Q": "At 2:39 why does Sal select the numbers -2,-1,0,1 for his x values? I have often seen these numbers selected for use when needing to input values for x. Is it for simplicity?",
        "A": "Those are convenient numbers. They are relatively small, with some positive and some negative."
    },
    {
        "video_name": "za0QJRZ-yQ4",
        "Q": "Somewhere around 3:04, why isn't the equation 3x+6x-2. Why is the equation 3-6-2?",
        "A": "You are solving f(x) for a particular value of x, so this isn t the equation in general, just what it equals at a particular value of x. f(x) = 3x\u00c2\u00b2 + 6x - 2 f(-1) = 3(-1)\u00c2\u00b2 + 6(-1) -2 f(-1) = 3(1) + (-6) - 2 f(-1) = 3 - 6 - 2 = -5"
    },
    {
        "video_name": "za0QJRZ-yQ4",
        "Q": "At 0:53, Mr. Khan mentions about imaginary and complex numbers. What are some examples of imaginary and complex numbers, and how are they used in equations and in everyday life? In addition, in what aspect of mathematics are these numbers covered?",
        "A": "Very little mathematics beyond arithmetic is used in everyday life if you don t work in a field involving mathematics. Imaginary and complex numbers have a lot of applications in physics, like electrical engineering. They are introduced algebra 2 or precalculus."
    },
    {
        "video_name": "za0QJRZ-yQ4",
        "Q": "At 7:30, Sal says the domain is all real numbers >= -5. But why is this? The parabola keeps going on to the left with more negative numbers? Don't those numbers count as well?\n\nThanks in advance!",
        "A": "It is the range, not the domain, that is f(x) \u00e2\u0089\u00a5 \u00e2\u0088\u00925. The domain is all real numbers (which is always the case for polynomials)."
    },
    {
        "video_name": "za0QJRZ-yQ4",
        "Q": "How do you get the x values to put in the table, like you said -2. -1, 0, 1, 2 as x values? at 2:25",
        "A": "You choose them at random. In most cases, it s favourable to use small numbers to make it easier on oneself. Of course, you should also choose negative AND positive numbers to be able to draw the parabola in both directions. However, the numbers you choose are your own choice. :) Hope this helped."
    },
    {
        "video_name": "za0QJRZ-yQ4",
        "Q": "At 1:05 Sal is talking about real numbers. Is \"e\" a real number?",
        "A": "No, e is not a real number. e is an imaginary number, like i . They do not belong to the number line. They have their own plane which is called the complex plane ."
    },
    {
        "video_name": "za0QJRZ-yQ4",
        "Q": "At 3:14, I don't get how -5 is the vertex. For example, if you have -5 as x and insert it you would get:\n3(-5)^ + 6(-5) - 2\n3(-25) + -30 - 2\n-75 - 30 - 2\n-107\nSo that's why i'm confused.",
        "A": "-5 is the y-coordinate of the vertex. Also, your arithmetic is wrong here. (-5)^2 = 25, not -25."
    },
    {
        "video_name": "za0QJRZ-yQ4",
        "Q": "In the first video, Sal used different notations to state that the domain was all real numbers. Why didn't he at 0:40? Does it make a difference?",
        "A": "Nope! As long as other people understand the point you are trying to make, it doesn t really matter. This is perhaps the clearest way to express all real numbers , as its quite literally spelled out for you."
    },
    {
        "video_name": "r3M68V9Joac",
        "Q": "do we regroup by taking away 1 from 17 0:33- 0:78",
        "A": "This will help you regroup. Regrouping fractions means taking away 1 from the whole number and adding it to the fraction to make it an improper fraction helping you add or subtract."
    },
    {
        "video_name": "tVDslyeLefU",
        "Q": "Around 1:25, Sal says \"the power of three\". What douse that mean?",
        "A": "The power of... term is meaning if you re multiplying something like 10x10x10 or 10 to the third power. It s simply saying you are multiplying this number by another number however many times."
    },
    {
        "video_name": "tVDslyeLefU",
        "Q": "At 1:30 i don\u00c2\u00b4t get what that means",
        "A": "At that point, it is the volume of 1 single cube with the side 1/4 ft."
    },
    {
        "video_name": "tVDslyeLefU",
        "Q": "At 0:05 Sal says the unit cubes are \"1/4 ft by 1/4 ft by 1/4 ft.\" But in previous videos he said a unit cube is \"1 by 1 by 1.\" Which is it?",
        "A": "This video uses fractional cubes, (look at the title) regular unit cubes are 1 by 1 by 1. I may be wrong, but I think I am correct. :)"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "At 1:45, Sal khan said 'For every two apples, we have five pieces of fruit.' What did he mean by that? I don't get it.",
        "A": "Sal wants to find the ratio of apples to fruit The number of apples = 4 Fruit in this case means the total of all apples and oranges (apples + oranges = 4+6 = 10 pieces of fruit). Thus, the ratio of apples to fruit = 4 / 10 Reduce the fraction, and you get: 2 / 5 Hope this helps."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "At 2:44, is a ratio is basically a fraction?",
        "A": "yes it is. It confused me to."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "So when we write the time we say something like 8:20. Is that the same as 8 hours for 20 minutes?",
        "A": "No, it means 8 hours and 20 minutes after noon if it says p.m., and 8 hours 20 minutes after midnight if it says a.m."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "Can you only use fractions in the problem used in the video which is 2:5. 2, representing the number of apples we have, and 5, representing the number of fruit.",
        "A": "Yup. Totally correct."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "for a ratio, the opposite of it, such as 2:3 & 3:2 is the same. but when you turn each of these equivalents into fractions, they are completely different! I don't get it! someone explain this to me.",
        "A": "No, 3:2 and 2:3. Is nearly (not completely same) the same because if 2:3 is comparing apple to number of oranges 3:2 is comparing oranges to apples Take it as a fraction 2:3 is 2/3 but not 3/2( same as 3:2) Hope you understood that they are different, the opposite in one way :) :) good question!"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "Are fractions and ratios the same thing?Are they different?Is 2:5the same as 2/5?",
        "A": "Well... fractions and ratios are not the same thing, but 2:5 can mean the same thing as 2/5."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "I saw him box 2 apples and 3 oranges, but it was 2:5. Isn't it supposed to be 2 apples for every 5 oranges?",
        "A": "It is 2 apples for every 5 pieces of fruit, not for every 5 oranges."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "How do I solve this question.\nThe ratio of selling price to cost price is 4:3. How much is the profit if the selling price is K1, 200?",
        "A": "Start by finding the cost price. You can do this using a proportion: 4/3 = 200/x Cross multiply, then solve for X Once you have X (the cost), subtract: selling price (200) - cost (x) and you will know the profit. Hope this helps."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "I get how he got 2:5- 8:20 (sorry if it says part of the video, it should be the ratio 8 to 20), dividing by their LCM. But in the picture it shows 2 apples for every 3 ORANGES not five, like the ratio says. So would the picture be accurate or not?",
        "A": "It shows 2 apples for every 3 oranges, but Sal is not trying to find the ratio of apples to oranges, 2:5 is just the simplified version of apples to all of the fruit. Imagine if the picture showed 8 apples for every 20 pieces of total fruit; then, you could simplify it down to 2 apples for every 5 pieces of fruit (including the two apples)"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "How did he get 8:20",
        "A": "8 apples. 8 apples + 12 oranges = 20 pieces of fruits. ratio apples to fruit is 8/20"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "I didn't quite understand from 2:33 on, can i have some help explaining.",
        "A": "Well it s just like a division promplem but sometimes on worksheets it would tell u write it in colon form and that s what u write"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "Why is the ratio 8:20 and not 2:3 ?",
        "A": "You have to make sure to simplify your answer or it could be counted wrong in many cases. 8:20 simplified/divided by 4 is 2:5! :) Hope it helps :)"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "how did you get 8:20",
        "A": "The total number of fruits is 20. The total number of apples is 8. Boom! You get 8:20 ;) Hope it helps!"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "At 2:07 (and a little before) what is he talking about?\n\nThanks in advance!",
        "A": "It s just a fraction. It is basically telling us how much more or less is the numerator than the denominator, as all fractions do. Basically like Sal says, 2/5, it means 0.4 of how many fruits there are. All ratios are fractions but not always the other way around."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "at 3:00 , the fraction for oranges and the total amount of fruits is 3/5 right",
        "A": "Yes... the ratio for oranges to total fruit would be 3/5"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "Well, at 2:39, what does it mean when it says \"2/5 of the fruit are apples\"?",
        "A": "This sentence 2/5 of the fruit are apples translates into: 2/5 x the number of fruit = the number of apples We know there are 20 pieces of fruit. So, if we do: 2/5 * 20, it will tells us the number of apples."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "Solve 5:x-3=25:10",
        "A": "You have one fraction = another fraction. This is a proportion. Cross multiply to solve. 5 / (x-3) = 25 / 10 5(10) = 25(x-3) 50 = 25x - 75 I ll let you finish from here. If you get stuck, comment back."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "At 0:52, Sal said 8 to 20, or 8 : 20. Anyway, I've heard people express it as 8 is to 20. So is saying is to and to the same thing?",
        "A": "so ratios are"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "at 2:47, how did he get 2/5 when it was 2/3?",
        "A": "i like to eat, i like to eat, apples and bananas ,apples and bananas i like to oat, i like to oat, oppols and banoonoos (so on and so on with different vowels) This is a song i grew up with."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "is 3:5 same as 3/5",
        "A": "Yes, 3:5 same as 3/5."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "At 0:55, shouldn't the ratio be 8 : 12 because there are 8 apples and 12 oranges?",
        "A": "It will not be 8:12 because he is comparing apples to all the fruit as a whole. There s 20 fruits in total, (8+12). Therefore, there s 8 apples, but 20 fruit, which makes the ratio 8:20."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "I understand that it's # of apples to # of fruit, but the 2:5 or 8:20 makes no sense to me, could you help make some sense?",
        "A": "2x4=8 5x4=20 So, 2:5 = 8:20"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "I still don't understand the difference between the regular fractions and fractions that are from ratio. Ex. 1/2 When I picture 1/2, I think of a circle divided into two parts where one piece is shaded. However with 1:2 or 1/2 as a ratio, I think of 3 units (comparing as a circle divided into 3 parts) where a part is shaded into one colour and two other parts is shaded into another coulour.\n\nDoes someone have an explanation that will help me integrate the two ideas? Thank you",
        "A": "A ratio is a different way of displaying the fraction, because when you are reading an equation, this; 21/8 could be read as a division question instead of a fraction. So if it helps, you can instead picture this; 7:3, as this 7/3 if it helps, but it is so that you don t misinterpret the question"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "how can i change to the simplify 6:9 ?",
        "A": "im guessing you are asking how to simplify 6:9. well first to simplify we need to find what number can I divide both numbers by(not including 1) SO THAT THERE IS NO REMAINDER? well 2 doesn t work. 3 does work! so if we divide 6 by 3 we get 2. if we divide 9 by 3 we get 3. so 6:9 is 2:3 simplified. we can also right our final ratio like this;::: 2:3 2 to 3 two to three 2/3 Hope This Helps!! =)"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "Could this be a example. 9:36 could also be 3:12 or 1:4 please help meeeeee?",
        "A": "Yes, the ratio of 9 : 36 can be (and should be) reduced to 1 : 4. Ratios are fractions, and like fractions, they should be fully reduced."
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "Can you have a fraction ratio, where you have a number like 2 1/2:5? Like you cut a apple in half? Just something I want to think about.",
        "A": "No you can not unless you are looking for the unit rate"
    },
    {
        "video_name": "UK-_qEDtvYo",
        "Q": "at 1:35 what are you dividing?",
        "A": "you are dividing the number of apples by the total number of fruit in each group. There are 2 apples and 3 oranges in each group. So the total number of fruit is 5. Therefore the number of apples divided by the total number of fruit is: 2/5 2:5 2 to 5 It can be written in any of these ways."
    },
    {
        "video_name": "eb-GHXCqkhQ",
        "Q": "So, 3:4 and 6:8 are equivalent ratios?",
        "A": "Yes, because when you simplify 6:8 by dividing by 2 you get a ratio of 3:4."
    },
    {
        "video_name": "J1HAY8E3gms",
        "Q": "how do u get the ratio does 3:2 refer to the 2x and 3x in the problem",
        "A": "Remember that if we have a ratio of 3 to 2, we can still write this ratio as 3x to 2x because multiplying the numerator and denominator by x is equivalent to multiplying the fraction by 1. This does not change the fraction s value."
    },
    {
        "video_name": "J1HAY8E3gms",
        "Q": "At 4:11, why is it 6x squared? Why do you make it squared.",
        "A": "3x * 2x = 3 * x * 2 * x = 3 * 2 * x * x = 6x^2"
    },
    {
        "video_name": "J1HAY8E3gms",
        "Q": "At 4:32,how is 21x+16x=37x?Why it is not 37x^2",
        "A": "Danilo, The reason that 37x is correct and 37x^2 is not is because at that point in the problem (video time 4:32) we are combining like terms. So the two terms of 21x + 16x add together, if we were multiplying then it would be x^2 but then 32 would be wrong."
    },
    {
        "video_name": "xMz9WFvox9g",
        "Q": "At 2:19 he said rectangle instead of square",
        "A": "A square is a rectangle with 4 equal sides."
    },
    {
        "video_name": "xMz9WFvox9g",
        "Q": "At 0:31 What is the meaning of \"unit length\"?",
        "A": "A unit length is just what we say to mean, any unit of [whatever we re measuring, length or volume or whatever]. It s just what we say instead of specifying that we re using meters or feet or inches or miles. The math works the same way as it would it anyone said [inches, meters, or whatever]"
    },
    {
        "video_name": "xMz9WFvox9g",
        "Q": "at 4:22 is there a 4th dimention?",
        "A": "in theory, yes there is, just dont count on finding a 4th dimension in this universe"
    },
    {
        "video_name": "xMz9WFvox9g",
        "Q": "At 4:28 he drew the three dimension cube. Why didn't he draw the back line?",
        "A": "It can appear more confusing than it has to"
    },
    {
        "video_name": "xMz9WFvox9g",
        "Q": "I've been learning this topic. How do you find the specific measurement if it doesn't tell you how much the base is and only tells you the base of the bigger figure on a composite figure? I've tried dividing the base of the larger piece into however much of the piece the smaller one takes up. It doesn't seem to work. Can someone help me? 0:00",
        "A": "You can try using the Pythagorean theorem or trigonometric ratios. They tend to be extremely helpful and is very basic and easy when coming to triangles. These methods are simple and will work for any grade. In other words this method is as easy as adding 1+1"
    },
    {
        "video_name": "QOTjdgmNqlg",
        "Q": "at 14:08, why it become r, rather than r^T?",
        "A": "No, w is linear combination of C(A^t), so you transpose A and every row is transposed again = r^t becomes (r^t)^t = r (transposition of transposition of r is just r)."
    },
    {
        "video_name": "QOTjdgmNqlg",
        "Q": "at 16:00 is every member of N(A) also orthogonal to every member of the column space?",
        "A": "every member of N(A) also orthogonal to every member of the column space of A transpose."
    },
    {
        "video_name": "QOTjdgmNqlg",
        "Q": "@ 22:00, why is \"N(B^t)\" the \"left nullspace\", when \"B^t\" is the same matrix as \"A\", and so therefore if N(A) = {x | Ax=0}, then N(B^t) must be {x | (B^t)x=0}?",
        "A": "N(A^t) is represented by span of vectors, which satisfy the equation: (A^t)x=0 , lets take transpose from both sides of this equation to obtain A instead of A^t : ((A^t)x)^t=0^t -> (x^t )*A=0. So, as you can see, vector x is on the left side for now, therefore we call N(A^t) the left Null Space."
    },
    {
        "video_name": "mtMNvnm71Z0",
        "Q": "At 1:01 why does Sal halve (1/2) to find the area of the triangle?why does he multiply 1/2 by dimensions 8 and 12?why does just not multiply 12 by 8 to know the area?\nWAITING FOR THE ANSWER. Thanks",
        "A": "Because when you do 8*12 you are getting a square and half of a square is a triangle. Also answer this question, How much swiss cheese does it take to fill my bathtub that is 6 ft. long 3 ft. wide and 2 ft. deep?"
    },
    {
        "video_name": "W7H-VcaSSu8",
        "Q": "At 5:06, Sal's equation is S(t) = -0.25t +2. Is S(t) the dependent variable or just S?",
        "A": "S(t) is a function not a dependent variable. A function is kind of a box which takes a number and throws a different number back. The name of the box is S and the number you put in is t ( which could be any number ). Let us say that t is 4 and we throw 4 in the S box, inside the box, we multiply 4 * -0.25 which equals -1 , then we add 2 to it. If we see it in the equation form it will look like this: S(4) = -0.25 * (4) + 2"
    },
    {
        "video_name": "W7H-VcaSSu8",
        "Q": "At 0:19 what exactly does denote mean in math terms?",
        "A": "It comes from L. nota (letter) and means among other things to be the outward or visible mark or sign of, to indicate as letters indicate a word."
    },
    {
        "video_name": "W7H-VcaSSu8",
        "Q": "why is it negative 0.75 @ 2:10. Why couldn't it just be 0.75?",
        "A": "you could if you subtracted 1.25 from 2 instead of 2 from 1.25 then you would subtract 3 from 0 and you would still get the same slope of -0.25"
    },
    {
        "video_name": "liRNTieIU_k",
        "Q": "At 2:13 and 2:19, he says that the 5 and 3 can be either negative or positive, why is this so? Wouldn't it be just positive 5 and positive 3 since those are the square roots of 25 and 9?\n\nPlease answer fast! I need it for a math class (2nd grader) in 3 days!",
        "A": "What is 5 \u00e2\u00a8\u0089 5? It s 25. What is 3 \u00e2\u00a8\u0089 3? It s 9. What is -5 \u00e2\u00a8\u0089 -5? It s 25 as well! What is -3 \u00e2\u00a8\u0089 -3? It s 9 as well! When he means what you mean, where \u00e2\u0088\u009a25 or \u00e2\u0088\u009a9 are only +5 and +3, respectively, he will say principal square root . Hope this helps! \u00e2\u0080\u0094CT-2/002-24"
    },
    {
        "video_name": "liRNTieIU_k",
        "Q": "You made a mistake at 2:25. B=+- not b+_",
        "A": "The plus with the minus under it is the plus or minus sign which indicates a positive and negative solution."
    },
    {
        "video_name": "liRNTieIU_k",
        "Q": "at 0:54 how did he get the formula (ax+b) squared? How do you know when or when nt to use that formula?",
        "A": "I THINK he used it because the a and the b values both appear to be perfect squares (25 and 9) so when you factor they will be in that form. (5x-3)^2"
    },
    {
        "video_name": "liRNTieIU_k",
        "Q": "at 1:57 what did he do to get the numbers",
        "A": "The numbers are in the original problem stated in upper left. Sal just copied it over to the right side of the screen."
    },
    {
        "video_name": "liRNTieIU_k",
        "Q": "At 1:04, is this what the FOIL method is?",
        "A": "Yes, First Outer Inner Last. First terms of the two groups, then the ones farthest apart, then the closest ones, then the last ones. (x+b)(x+c) First: x*x Outer: x*c Inner: b*x Last: b*c So then it s (x*x)+(x*c)+(b*x)+(b*c)= (x^2)+cx+bx+bc. cx and bx will usually combine into one term though."
    },
    {
        "video_name": "liRNTieIU_k",
        "Q": "at around [3:25] Sal says both of them work out. my question is when will one not work out and the other work?",
        "A": "in a perfect square trinomial it will always work i think hope that helped"
    },
    {
        "video_name": "liRNTieIU_k",
        "Q": "A 3:09,how did he get a2x2+2abx+b2? That just made me get confused....",
        "A": "Hi Brittany, he got that by simply multiplying the two terms (ax+b) * (ax+b). FOIL is a useful acronym to remember here. First * First: ax * ax = a^2x^2 Outside * Outside: ax * b = abx Inside * Inside: b * ax = abx Last * Last: b * b = b^2 This yields: a^2x^2 + abx + abx + b^2 You then add the two middle terms as if they were both simple x s and you get: a^2x^2 + 2abx + b^2. Hope this helps."
    },
    {
        "video_name": "Tn9U8hLu9aI",
        "Q": "Do you have to do the transformations with one point in common, like in 0:40? Or can you do them on any point?",
        "A": "I m not quite sure what you mean, but I ll try to answer your question. -You can do the transformations anywhere, but it s much easier if you do have one point of each pentagon match up. -Transformations shouldn t change the angles or the length of the sides, so as long as the two pentagons were congruent in the first place they will have all five points the same. -Sorry that I was the first to answer this and was a year late... -.-;"
    },
    {
        "video_name": "meibWcbGqt4",
        "Q": "At 2:30 Isn't being a square also redundant?\nI mean, if it is in R^n, shouldn't it has always n basis vectors? If not, could you show me a case where this doesn't happen?\n\nBecause, if it's a n-dimensional space, then there are n diferent basis vectors (and n coeficients) to describe it, aren't there?",
        "A": "No, for example a plane in R^3 has 2 basis vectors. a*(1,1,0)+b*(0,0,1)"
    },
    {
        "video_name": "ccS5Fy5yLjk",
        "Q": "At 5:30, just do 56 divided by 8, then add three zeros. Then you should get 7000.",
        "A": "yep! true :) :) :) :) :) :) :) :) :) :) :) :) :) :) :) :) :) :)"
    },
    {
        "video_name": "yvddTWa9ptU",
        "Q": "At around 0:50 why does Sal say a sub 2\u00e2\u0080\u00a6 shouldn't the sequence start at a sub 1?\nHe also states at around 2:35 that the index is one less than the exponent, what is the first index then, if there isn't one then shouldn't the index start with one making the exponents two points higher than the index?",
        "A": "Note the notation at the beginning of the video: loosely translated into lay English, it would read the sum of a sub n from n equals two to infinity is equal to negative eight fifths etc. Specifically, the starting index is n=2, which would logically lead to starting the sequence at a sub 2."
    },
    {
        "video_name": "yvddTWa9ptU",
        "Q": "How come at 0:46 sal starts with a2(a sub 2) instead of a1(a sub 1)",
        "A": "This is simply a matter of what we re given as part of the problem. The notation underneath the sigma symbol tells us the starting point, and in this case it says n=2, so we begin counting at 2."
    },
    {
        "video_name": "H6ZNLD1AeM8",
        "Q": "Once we establish that the slope of the secant line over [a, 0] is greater than a line with a slope of 1, how do we extrapolate that to mean that f(-a) is less than that (around 4:00, part I)? Is it implied that f(-a) atually means \"the slope of the line tangent to f(-a)?\" Or am I forgetting something that says if the slope is greater, then the value of the function is always greater?",
        "A": "The actual numeric value of f(-a) is less than 1 (as Sal points out between 0:45 - 1:00). He later shows that the value of the slope (average rate of change) between points (-a,f(-a)) and (0,1)) is greater than 1. (That s the 1-f(-a)/-a, part). So he is ultimately comparing the two values, not two slopes. It was confusing to me too, at first. :)"
    },
    {
        "video_name": "OZtqz_xw0SQ",
        "Q": "At 1:35, why do you divide the original equation by 2?",
        "A": "In order to simplify the equation and to also have a non-negative leading coefficient."
    },
    {
        "video_name": "OZtqz_xw0SQ",
        "Q": "At 1:40 why did Sal divided the equation by -2? Could he also have simplified the original equation?",
        "A": "he just took out a common factor, simplifying and also taking away the negative from the leading coefficient. He could do the division because 0 divided by anything is still 0."
    },
    {
        "video_name": "OZtqz_xw0SQ",
        "Q": "at 0:29 can someone provide me a link to the problems that Sal is talking about please? thanks!",
        "A": "I don t know specifically, but he mentioned the physics content. You can get to Physics by selecting it off the subjects menu. Or, use the search bar."
    },
    {
        "video_name": "OZtqz_xw0SQ",
        "Q": "At 5:11, what did Sal do to evaluate 20(5/2) so quickly?",
        "A": "You times 20 by the numerator (5) and divide by the denominator (2), which is: 20*5/2 = 100/2 = 50"
    },
    {
        "video_name": "Xqfcy1rqMbI",
        "Q": "At 2:39, who are Larry, Curly, and Moe? Are they the Three Stooges?",
        "A": "Yes, that is correct. The original three stooges were Larry, Shemp, and Moe, but Shemp quit, and was replaced by Curly. Later on, I think Shemp got switched back in."
    },
    {
        "video_name": "Xqfcy1rqMbI",
        "Q": "At 1:38 why is it 2^8 instead of 3*8??",
        "A": "because sal is going to flip a coin 8 times and every flip will have 2 outcomes, either heads or tails...that s why 2(no of outcomes)^ 8(no. of flips) to calculate total outcomes"
    },
    {
        "video_name": "Xqfcy1rqMbI",
        "Q": "From about 0:39 to 0:42 Sal mentions \"...equally probable events...\" I don't ever remember Sal defining what an event is exactly. Although I have a general idea what he means in the context of say this problem, could someone give a quick answer to that in the perspective of all basic probability? Thanks.",
        "A": "An event is an outcome. When Sal talks about all equally probable events he s talking about all the possible outcomes something can have while meeting the requirements or conditions stated. That s why the numerator is different than the denominator at 1:17. The numerator is all possible outcomes that can have 3 heads and the denominator is all possible outcomes that can have any combination of Heads of Tails . Hope it helps."
    },
    {
        "video_name": "Xqfcy1rqMbI",
        "Q": "At 1:50, why do you multiply 2 by the number of flips?",
        "A": "To calculate the total number of possible outcomes. For every flip there are 2 possibilities - heads or tails. So to find out the total number of possible outcomes for two flips we need to calculate 2*2 or 2\u00c2\u00b2. For eight flips this means 2^8."
    },
    {
        "video_name": "Xqfcy1rqMbI",
        "Q": "6:37 Doesn't he mean there's 56 ways of putting 8 flips into 3 heads? Is the number of ways the same for vice versa or am I missing something?",
        "A": "If you flip a coin 8 times, there are 56 different ways to have those 8 flips result an outcome of 3 heads. Here are some of them: H=Heads, T=Tails HHHTTTTT TTTTTHHH HTHTHTTT TTHHHTTT . . . . . etc. There are 56 ways to arrange (combine) 3Hs and 5Ts."
    },
    {
        "video_name": "Xqfcy1rqMbI",
        "Q": "Starting at 2:55, Sal was comparing the combinations of 3 heads in eight flips to sitting 3 people in a car from a group of 8. I have a problem making this mental leap. In my head, each flip has two possible outcomes. But a person does not have this 50/50 of being H or T. Thus, it is not really C(8,3). Can someone help to explain?",
        "A": "The idea is that he can pick a person to sit in the car, or not. This is binary: either the person goes in the car, or they do not. There is no third option. That s why he said pick 3 people : either a person is chosen, or they are not. Again, there is no third option. Just like there is no third option between heads or tails (disregarding the very unlikely physical possibility of the coin landing on its side)."
    },
    {
        "video_name": "jxA8MffVmPs",
        "Q": "At 0:06 how is shouldn't Sal say 3 is worth 300?",
        "A": "Watch the video... He does."
    },
    {
        "video_name": "jxA8MffVmPs",
        "Q": "I love when you explain how to do things, like in the way you did at 1:28 :)",
        "A": "I do not know how to watch the video because I am on a iPad that is very pretected so if you can tell me how I would thank you"
    },
    {
        "video_name": "jxA8MffVmPs",
        "Q": "at 2:16 sal doesnt mension 4 thousands but he mensions all the other place values... can someone please tell me why?!",
        "A": "He did not say the thousands place because he was looking for the place of the 3, and the 3 happens to be in the hundreds place, so he stopped there."
    },
    {
        "video_name": "LwhJVURumAA",
        "Q": "At 6:30 Sal talks about looking up the equation on khan academy of the sum of n^2 but where would we look it up at? Is it in the form of a video or a FAQ?",
        "A": "You can find the explanation for that formula used at 6:30 in the videos Finding the sum of n squares part1 and part2"
    },
    {
        "video_name": "LwhJVURumAA",
        "Q": "Waait... where does Sal get the 7/6 at @4:10?",
        "A": "He gets it from the equation at 3:50. You can derive it yourself or find derivations for it."
    },
    {
        "video_name": "LwhJVURumAA",
        "Q": "@ 3:45, Sal gives an example as a sum of fractions. Why? Shouldn't it be 1^2 +2^2+....+7^2?",
        "A": "He is using a formula to find the sum, rather than finding each value and adding them one by one."
    },
    {
        "video_name": "KT32CsdEZEY",
        "Q": "At 1:44 it is said that you should multiply both sides by the reciprocal. Reciprocals only exist for rational numbers that are nonzero. What would happen if the rational number happens to be zero?\n\nA very subtle counterexample.\n\nCan anyone see how to reword the statement to make it true?",
        "A": "You are right, this rule only works if multiplier is not equal to zero. Because everything multiplied by zero equals zero so you always get rational number as a result."
    },
    {
        "video_name": "UdLYAqN0gNY",
        "Q": "At 0:38, how did you determine that the y intercept for y=g(x) was 3. I understand how you found it for the first equation because the y intercept was 4, but how did you find it for g(x)?",
        "A": "You look for the point on the y-axis that is (0, #). for that one it happened to be 3"
    },
    {
        "video_name": "UdLYAqN0gNY",
        "Q": "Why did Sal refer to the x-solutions as \"roots\" @2:31 ?",
        "A": "A root is the solution to an equation. When a quadratic function is set equal to zero, it is simply a quadratic equation that can be solved. A zero is the value of a variable that makes a function equal to zero. For polynomials like quadratics, and many other types of functions (especially in two dimensional space), the root and zero are basically synonymous - they are both the x-values where the graph would intercept the x-axis (where the y is zero)."
    },
    {
        "video_name": "TqslX-bUTD8",
        "Q": "at 5:30, when Grant compute the mixed partial derivative, he doesn't multiply sqrt(3) by 6: (Df/Dyx)=6x = 6*sqrt(3) = sqrt(108), so FxFy-Fxy^2 = -108\nI understand that in this case, the magnitude of the mixed partial Derivative didn't matter since the other term =0, but is there another reason he didn't compute or might've he forgotten?",
        "A": "No. Chiarandini s right. All results are -108 except for one that s 36>0. I think Grant s very stressed doing these videos! That s too bad because he s doing an awesome job otherwise."
    },
    {
        "video_name": "ao9cx8JlJIU",
        "Q": "At 1:12 on the video. You simplify the expression for the Total price from P+0.25P to 1.25P. I do not follow what you do - divide by P? What is th emath theory to explain that? Thank you!",
        "A": "P = 1 times p p = 1 p p + 0.25p does equal 1 p + 0.25 p Any thing equals one of itself. Hope that helps! ;-)"
    },
    {
        "video_name": "ey_b3aPsRl8",
        "Q": "In the previous video when Sal states the domain he just writes \"Domain= and then the number\" but in this video when he states the domain @0:45 he says the \"Domain is equal to the \"set\" of and then the number. What's the reason behind the change?",
        "A": "Its the same thing, maybe because the previous vid had 2 values that make the expression undefined."
    },
    {
        "video_name": "ey_b3aPsRl8",
        "Q": "at 2:13 why do we multiply by -1 and then multiply by -1 AGAIN? why not just once? and home we don't do this to the denominator? please and thank yoU!",
        "A": "You cannot change the expression, meaning you can modify it, but it must be equivalent, this is why you have to multiply 2 times by -1, because (-1)(-1) = 1, so you are not changing the expression. If you multiplied only once by -1 then you would be changing the expression, you would be making it negative, and that is not valid. Sal decided to to this on the numerator, but he could have done so in the denominator if he wanted to, the result would have been the same either way."
    },
    {
        "video_name": "ey_b3aPsRl8",
        "Q": "3:00 how come you can just multiply by -1?",
        "A": "Not just multiplying by -1. Using this equivalence to put the factors in a form to allow dividing them out. x - 6 = (-1)(-x + 6) = (-1)(6 - x)"
    },
    {
        "video_name": "ey_b3aPsRl8",
        "Q": "At 2:16, why do you have to multiply by -1 twice? What do you do with the negative one that is left out (2:47)?",
        "A": "@ 2:16, he did that so that there s no net change (If he just multiplied by -1 that would be changing the expression). @ 2:47 both -1 s remain in effect - they both have an effect on the answer. (x^2-36)/(6-x) = ((x+6)(x-6))/(6-x) = ((x+6)(x-6)(-1)(-1))/(6-x) = ((x+6)(-x+6)(-1))/(6-x) = ((x+6)(6-x)(-1))/(6-x) = ((6-x)/(6-x))((x+6)(-1)) = (x+6)(-1) = -(x+6)."
    },
    {
        "video_name": "TV5kDqiJ1Os",
        "Q": "At 0:58, how did he just immediately go, bam, it's 75? I could really use skills like that, so could somebody explain please?",
        "A": "75 goes into 150 twice, and 150 goes into 300 twice."
    },
    {
        "video_name": "TV5kDqiJ1Os",
        "Q": "At 0:24, why did he divide everything by 4 instead of dividing only 4x^2 and 0 by 4?",
        "A": "If he divided only 4x^2 that would of changed the quadratic. To make it say the same you need to dived every term by 4. Someone tell me if I m wrong."
    },
    {
        "video_name": "TV5kDqiJ1Os",
        "Q": "How did you get 25 at 3:09",
        "A": "We have the identity of (x+a)^2 i.e x^2+2ax+a^2. Sal have just factorize 10x .i.e 2*5*x. So we get the value of a i.e. 5 . And Sal has just put the value of a in a^2 . By this we get the value 25."
    },
    {
        "video_name": "TV5kDqiJ1Os",
        "Q": "At 1:21, Sal just divided everything by four. However, when do we put the four on the outside of parenthesis?",
        "A": "If you divide everything on both sides of the equation by 4, it can just go away. If he only factored it out on the left side, he would have to leave it there outside of parenthesis."
    },
    {
        "video_name": "TV5kDqiJ1Os",
        "Q": "@ 3:12 How come you don't subtract 25 too. Isn't that what you usually do?",
        "A": "If you subtracted 25 then you would be doing different things to each side of the equation. You have to add 25 to each side in order to get the perfect square."
    },
    {
        "video_name": "TV5kDqiJ1Os",
        "Q": "Hi Sal,\nThese video's are very helpful. however id just like to know:\nwhen you get to the pint in your working similar to what we see at 3:50 but you don't have equal brackets... how would you workout the answer? What happens to the constant on the other side of your equation?\nThanks\nCBR",
        "A": "I don t really know what you re talking about, but here s what he does from 3:50 onwards. (x+5)^2=100. He square roots everything => (x+5)=Positive or negative 10. Solving => x= -15 or 5."
    },
    {
        "video_name": "E_Hwhp74Rhc",
        "Q": "At 5:15, he mentions that \"taking the cross section that is parallel to the xy-plane will give a smaller radius\" what's the point of taking the cross section? what's the point of getting a smaller radius? If it's a sphere, shouldn't the radius always be the same?",
        "A": "We have t being used to determine x and y, so we need something else to determine z, so that is why we take different cross sections at different values of z. The cross section is getting a smaller radius because some of the radius is being used for the z-component, so not as much can be used for the x and y components."
    },
    {
        "video_name": "aASUZqJCHHA",
        "Q": "At 1:00, why is the Celsius scale called the Celsius scale and why is the Fahrenheit scale called the Fahrenheit scale?",
        "A": "Fahrenheit (symbol \u00c2\u00b0F) is a temperature scale based on one proposed in 1724 by the physicist Daniel Gabriel Fahrenheit. Eighteen years later In 1742, Swedish astronomer Anders Celsius (1701\u00e2\u0080\u00931744) created a temperature scale which was the reverse of the scale now known by the name Celsius : 0 represented the boiling point of water, while 100 represented the freezing point of water."
    },
    {
        "video_name": "hr_mTd-oJ-M",
        "Q": "3:22 i dont understand",
        "A": "don t understand what?"
    },
    {
        "video_name": "hr_mTd-oJ-M",
        "Q": "At 1:20 it is a very good pictorial representation on multiplying fractions",
        "A": "When you multiply two fractions you do numerator times numerator and denominator times denominator and then you get a new fraction. If the numerator is bigger than the you need to divide it. If they are both even divide both by two Example: 3/4 x 2/5=6/20. The numerator is not bigger but you still divide both by two to get 3/10."
    },
    {
        "video_name": "tvXRaZbIjO8",
        "Q": "at 0:22, does that work? I don't understand how it would work. can anyone help?",
        "A": "Yes, if there is a parenthesis after a subtraction you MUST distribute the negative. This is because you are subtracting everything in the parenthesis. Remember subtracting a negative is the same as adding a positive. Essentially on the last term you have \u00e2\u0080\u0093(\u00e2\u0080\u009318i) which is why it simplifies to + 18i"
    },
    {
        "video_name": "tvXRaZbIjO8",
        "Q": "1:01,how do we know that the numbers with the 'i' are imaginary?",
        "A": "That s how we define it. How do we know that the numbers with the - are negative?"
    },
    {
        "video_name": "Lh7NMBPFVZw",
        "Q": "At 4:14 he puts 11< square root 123 < 144\nIt should be 11< square root 123 < 12",
        "A": "If you watch the next few seconds he actually corrects that and 144 is greater than the square root of 123 anyway."
    },
    {
        "video_name": "Lh7NMBPFVZw",
        "Q": "At 0:45 how did Sal get the 5?",
        "A": "5 x 5 = 25 and 6 x 6 = 36. So the square root of 32 will be between 5 and 6."
    }
]