title
stringlengths 1
312
| post_id
stringlengths 5
7
| over_18
bool 2
classes | subreddit
stringclasses 1
value | link_flair_text
stringlengths 2
63
⌀ | self_text
stringlengths 1
32k
⌀ |
|---|---|---|---|---|---|
ELI5: Predicative vs. Impredicative | 6v5qpu | false | explainlikeimfive | Mathematics | [deleted] |
ELI5: Euler angles and axis such as XYX, ZXZ, etc. | 6vcr7k | false | explainlikeimfive | Mathematics | null |
ELI5: negative percentages, is there any situation in math or science that uses them? | 6veasu | false | explainlikeimfive | Mathematics | null |
ELI5: How do we know that neural network can solve ANY problem? (is that a right sentence at all?) | 6vqpzc | false | explainlikeimfive | Mathematics | I've heard from a friend really good at math that it is proven that neural networks can solve any possible problems - that can be described in a formal way, I guess. ELI5 that whether it's a precise statement, and either way, what is the proof that NNs can, or can't solve such an endless range of problems. |
ELI5: Is it really more likely to be struck by lightening while drowning than winning the Powerball? I've never heard of that happening but I've heard of plenty of Powerball winners. | 6vrrux | false | explainlikeimfive | Mathematics | [removed] |
ELI5:Anything other than base 10. | 6vsyyo | false | explainlikeimfive | Mathematics | Stuff like, base 2, base 16 etc. |
ELI5: Why do two single digit integers that equal 10 when added together both have the same last number when squared? | 6w9djs | false | explainlikeimfive | Mathematics | 0^2 and 10^2 end in 0 1^2 and 9^2 end in 1 2^2 and 8^2 end in 4 3^2 and 7^2 end in 9 4^2 and 6^2 end in 6 5^2 ends in 5 So if >x + y = an integer ending in 0 >x^2 and y^2 have the same last number Is there a reason for this? It feels like this makes the unit of 10 mathematically unique. |
ELI5: How are lenght measures "measured"? How is it defined how long a centimeter should be on a ruler, for instance? What's the basis for comparison? | 6w9vh0 | false | explainlikeimfive | Mathematics | null |
ELI5:why are the outputs of deterministic hash functions difficult to reverse? If the hash function is deterministic, wouldn't this imply that there is no randomness involved and, therefore, the output is easily reversible? | 6wx8b1 | false | explainlikeimfive | Mathematics | From my understanding, hashing is a set of mathematical algorithms that will always produce an identical output given that the input is also identical. If the results are deterministic, wouldn't this imply that there is no randomness involved in the algorithm? If there is no randomness, then couldn't you simply reverse the algorithm and apply it to the output to determine the input? Hash functions such MD5 and SHA1 are public algorithms - couldn't you simply reverse them? |
How tf do u factor polynomials? | 6x4nf7 | false | explainlikeimfive | Mathematics | [removed] |
ELI5: Why is there a 30% a number will start with 1? | 68pvrd | false | explainlikeimfive | Mathematics | And how is any explanation not just another example? |
Quantum theory | 68ywmj | false | explainlikeimfive | Mathematics | [removed] |
ELI5: Why does the current century not line up with the calendar year? | 697kin | false | explainlikeimfive | Mathematics | For clarification: why do we live in the 21st century but the calendar year is 2000s, shouldn't we be in the 20th century? |
ELI5: If (a)^(1/2)*(b)^(1/2) = (ab)^(1/2), why isn't i^2 = [(-1)(-1)]^(1/2) = 1? | 69hw8q | false | explainlikeimfive | Mathematics | [removed] |
ELI5: Removing left recursion | 69tgm4 | false | explainlikeimfive | Mathematics | A > Br B > As | t I understand how there is left recursion but I'm confused on the process of removing it? |
Math problem help | 6aag4t | false | explainlikeimfive | Mathematics | [removed] |
ELI5:What are tensors and what are they used for? | 6aq8t1 | false | explainlikeimfive | Mathematics | null |
ELI5:Random natural processes | 6auxc4 | false | explainlikeimfive | Mathematics | Since processes are governed by natural laws, for how long of a period can a random process be accurately predicted? For example, wind driven waves are random due to fluctuation in transfer of energy from wind to surface. Then, ship motions are random, at lease in longer time periods. But, for short periods, is there a "mathematical" approach to "prove" how long of a duration the random process can be accurately predicted? |
ELI5:Pareto Principle.....please explain why it makes sense? | 6axa6g | false | explainlikeimfive | Mathematics | just wondering what the general concensus is on investing, do you use the 80-20 rule and why/ why not is it agood idea |
ELI5: perfect numbers | 6b67a1 | false | explainlikeimfive | Mathematics | null |
ELI5: What is Clifford Algebra? | 6bc0cp | false | explainlikeimfive | Mathematics | null |
ELI5:Why divide by (n-1) for standard deviation? | 6bf7mv | false | explainlikeimfive | Mathematics | Why do you divide by (# of data points -1) when calculating the standard deviation? |
ELI5: Why there's still Gradian(grad) on scientific calculators if nobody use it? | 6bhvjp | false | explainlikeimfive | Mathematics | French milatary used it to point cannons decades ago but I don't know any other use for it now. |
ELI5: Why is Multiplication called an elementary operation? | 6bibgh | false | explainlikeimfive | Mathematics | Wikipedia says that multiplication and addition are both elementary operations, which sort of implies they are different operations, but it also says that multiplication is "repeated addition." Is multiplication truly its own operation, or is it just a more efficient way to add things together? |
ELI5: Hey Reddit, wanna tell me how this abacus works? | 6bmbjb | false | explainlikeimfive | Mathematics | [removed] |
ELI5:Bass diffusion model | 6bn4t2 | false | explainlikeimfive | Mathematics | What is it? What's its application to the product lifecycle? Thanks! |
ELI5: Is it possible that every single combination of cards in a deck has occurred at least once in human history? | 6bpkqc | false | explainlikeimfive | Mathematics | [removed] |
ELI5: what is the practical application of imaginary numbers? | 6cj5oc | false | explainlikeimfive | Mathematics | Had a thought about imaginary numbers brought on by my 3 year old son talking about his imaginary friend who is apparently a literal number. Couldn't find a good way to explain the concept to my kids, looking for some help. |
ELI5: why we delete the multiplication symbol before a bracket ? | 6cn5vp | false | explainlikeimfive | Mathematics | I would like to know the history of deleting the multiplication symbol before a bracket or between two letters. It's just about simplification or does it come frome something else ? This question come to me after I read somewhere that before 1900's the equation 6/2(2+1) was egual to 1 (for some reason about the bracket whose I don't uderstand) and now it's egual to 9 . Please don't argue about the result of this equation in this thread. Thank you |
ELI5: why is BODMAS used in mathematical problems? why not just solve the problem the way its shown? | 6coebo | false | explainlikeimfive | Mathematics | null |
ELI5: How does Euler's formula actually work? | 6ctn4n | false | explainlikeimfive | Mathematics | null |
ELI5: Why shouldn't I average logarithmic values? | 6d49o6 | false | explainlikeimfive | Mathematics | When averaging powers expressed in dB, such as dBm or dBW values... or anything logarithmic for that matter, I have been told not to average the logarithmic value, but to average the linear value and log it. Knowing that they produce different values, what makes the average of the logarithmic values less accurate than the average of the linear values? |
ELI5: Gambler's Fallacy. After a fair coin is tossed & lands on "heads" 500 times the chances of it landing heads again are 50/50. | 6dskxu | false | explainlikeimfive | Mathematics | In older post asking the same question someone explained it like this: "The fallacy lies in being confused about when you should start counting." That as independent events the chances are 50/50. But they're not independent events when you are you're taking into account the staggering unlikeliness of a fair coin landing on same side 500 times... The chances of a coin landing on heads AGAIN in that series isn't 50/50, is it? Because that seems to be what ppl argue with this... If so I can't wrap my head around it. I'm 5!! |
ELI5: Why can't a computer not calculate every number of pie? | 6e69gk | false | explainlikeimfive | Mathematics | null |
ELI5: How to calculate the average of a sample of times? | 6ea1qd | false | explainlikeimfive | Mathematics | [removed] |
ELI5: Commutativity of implication (also called the law of permutation) | 6efa05 | false | explainlikeimfive | Mathematics | [deleted] |
ELI5: why do we use (hour:minute.seconds) to write out time? | 6ehwvo | false | explainlikeimfive | Mathematics | [removed] |
ELI5: Why does the harmonic series not converge? If we're basically adding zeroes as n gets really big, shouldn't it converge to a number eventually? | 55e9z5 | false | explainlikeimfive | Mathematics | null |
ELI5: Please explain statistical power | 55tlnp | false | explainlikeimfive | Mathematics | I'm taking a doctoral level statistics course and am having so much trouble understanding statistical power and 1-B (beta). Would love a clear, broken down explanation of this in general and in relation to rejecting a null hypothesis. |
ELI5:Fast Fourier transform | 55tm2r | false | explainlikeimfive | Mathematics | null |
ELI5:how many random objects are in a jar | 55za70 | false | explainlikeimfive | Mathematics | So our math teacher gave us this challenege to find out how many objects were in a certian jar.. The only thing is all the objects are different sizes.. My question is: is there any way actual solve this problem using math? Or is it just pure luck? (I understand how to find , or estimate, objects in a jar if they are all roughly the same size, but having multiple different objects i dont see how this is possible) |
ELI5: What is Rayleigh Distribution? | 55zh4y | false | explainlikeimfive | Mathematics | [deleted] |
ELI5: TREE(3) / Kruskal's tree theorem | 561ayl | false | explainlikeimfive | Mathematics | [deleted] |
ELI5: why are prime numbers important? | 56onay | false | explainlikeimfive | Mathematics | null |
ELI5:"lambda equivalence" | 56y1q1 | false | explainlikeimfive | Mathematics | A search turns up https://www.google.com/search?q="lambda+equivalence" |
ELI5: Why is sales tax, when converted to a percentage, 1.06 (for 6% sales tax ). Why isnt it .06 | 577s2u | false | explainlikeimfive | Mathematics | [removed] |
ELI5: eiuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhbuhb | 57gacm | false | explainlikeimfive | Mathematics | [removed] |
ELI5: What is a Quibit and what does it mean for the average everyday person? | 57pd5k | false | explainlikeimfive | Mathematics | [deleted] |
ELI5: Naming 3 dimensional structures. | 57qnt9 | false | explainlikeimfive | Mathematics | Pic related. |
ELI5: Girard's Theorem | 57qw24 | false | explainlikeimfive | Mathematics | Title says it all, need help understanding Girard's theorem in maths, regarding spherical triangles! |
ELI5: Why a ratio of proportionally increasing number is ever decreasing? | 57up1d | false | explainlikeimfive | Mathematics | [removed] |
ELI5: De Moivre's theorem | 57zm1j | false | explainlikeimfive | Mathematics | Would be great to see an accessible explaination. Thanks! |
ELI5: How does my childhood card trick work? | 586owt | false | explainlikeimfive | Mathematics | When I was a child, I tried to create my own magic card trick to show off to family during a camping trip. I discovered a pattern that gave me consistent results and I successfully tricked many people into believing it was an awesome sleight of hand. Almost every time afterward, I was asked how the trick is done. Truthfully, I have no idea. It was a trick I discovered accidentally. I have been meaning to post this question for years and I decided that finally, I'll do it today. I am not a mathematical person. In fact, I have struggled with understanding math all my life. I assume the answer contains mathematics so I turn to you for help! Thank you! Here is a video demonstration as it's easier to see than to put into words only. https://m.youtube.com/watch?v=aruixVQzfgs |
ELI5: How did the polls get it so wrong in the UK about BREXIT, and how does the same phenomenon apply to the US for this election? | 58d3b0 | false | explainlikeimfive | Mathematics | null |
ELI5: How are even numbers expressed in other base systems, especially 'odd number' base systems? | 58jbqv | false | explainlikeimfive | Mathematics | I understand that the definition of an even number is a number that can be divided (tada!) evenly into two groups, but does this change in other base systems or is this limited to an "even numbered" base system? For example, in base 10 you can "see" the even-ness in the "ones" column: 2,4,6,8,0,2,4,6,8,0,2,4,6,8,0 ... {repeating: 2,4,6,8,0} But in base 3, this is less evident, the first 15 base-3 "twos" column end in: 2,1,0,2,1,0,2,1,0,2,1,0,2,1,0, ... {repeating: 2,1,0} If I look at the number "10" (base 10) it is evenly divisible into 2 groups of 5. If I look at the number "101" (base 3, same number) it is evenly divisible into 2 groups of 12. So - it appears both numbers that are the same, are evenly divisible but is there a "trick" to obviously seeing an even number in other base systems? Can you prove simply that all odd-numbered base systems will express even-ness across all of possible numbers in the right-most column but even-numbered base systems will express even-ness across a repeating same subset of the possible numbers in the right-most column? [EDIT: Thanks everyone for such quick good answers. This wasn't, but I think it would make an interesting interview question (for a technical type of interview).] |
ELI5: the sixth question of the 1988 IMO. | 593xyu | false | explainlikeimfive | Mathematics | Let a and b be positive integers such that ab + 1 divides a2 + b2. Show that a2 + b2/ ab + 1 is the square of an integer. |
ELI5: Why would someone need to be quick at figuring out number sequences mentally? | 596402 | false | explainlikeimfive | Mathematics | [removed] |
ELI5: Propensity Score Matching Analysis | 598wkq | false | explainlikeimfive | Mathematics | Basically I'm trying to get the basics of PSM so I can follow this consulting session I am sitting in on Wednesday, so anything y'all can give me is appreciated. |
ELI5: How to calculate percentage? | 59fo8a | false | explainlikeimfive | Mathematics | [removed] |
ELI5:Why is the Rule of Threes the funniest? | 59g4po | false | explainlikeimfive | Mathematics | [deleted] |
ELI5: Calculus | 59ux9e | false | explainlikeimfive | Mathematics | I'm a complete beginner. I have this one book called "The Calculus by Baker" but I don't understand a word. Help? |
ELI5:can someone explain the concept of limits in mathematics? | 5a52v0 | false | explainlikeimfive | Mathematics | I can't quite get around the idea of limits yet. I don't know where to use it. I would really love an easy explanation. thanks! |
ELI5: Monty hall style problem I thought of that is irking my brain. | 5a7k2j | false | explainlikeimfive | Mathematics | Me and my buddies were talking about the monty hall problem when I proposed this question, to which it sparked heated debate as to what the answer ultimately is between us. 10 doors. Behind one contains a million dollars. You are given 6 chances to pick a door. What is the probability you get the million dollars? For example I can't think of any reason why any of these are wrong. There is a 60% chance you get the million dollars from the 6/10 fractional probability. By the same logic, there is a 90% chance you will fail to get the million dollars, as there is ultimately, only one door that contains it. There is a incremental increase in probability you get the million dollars for every door you pick. As the odds go from 1/10, to 1/9 all the way down to your last pick. Which is 1/5 or 20% 60% 10% or 20% What is the most correct answer? |
ELI5: In simple math equations like 6÷3+2×7, different orders of operations produce different results, why is one more "right" than another? | 6kmznk | false | explainlikeimfive | Mathematics | If different orders produce different answers in these situations, why do we choose or believe or enforce one over the other? What is the reasoning? Is it possible yo have entirely different mathematical systems (or internally consistent mathematics) that produce entirely different answers to the same set of problems? |
ELI5: What is Zorn's Lemma and what does it propose? | 6kup42 | false | explainlikeimfive | Mathematics | [removed] |
ELI5: What are plus minus sign used for? | 6kyz94 | false | explainlikeimfive | Mathematics | I don't seem to understand the concept. If an answer has a plus minus sign, what does that mean? The range of the answer? And how does the answer end up that way. Please help me I've been in hospital and I have trouble studying again at school and the teachers doesnt seem interested in helping me. |
ELI5: What's probability and how do we achieve the results? | 6l7e7u | false | explainlikeimfive | Mathematics | I understand probability is the likeliness of an event to occur given some condition(s). But what is it inherently? Why does data sets obey probability? Like, even for an artificially generated pseudo random numbers between 0 and 1, most of the time the chances of half of the number being less than 0.5 is 50%. Why so? Or why does naturally occurring datasets exhibit normal( or Gaussian) distribution? I am not a native English speaker so sorry if my question is poorly worded. |
ELI5: Normal Subgroup | 6manw3 | false | explainlikeimfive | Mathematics | null |
ELI5: How does Latent Dirichlet Allocation work? | 6mnrma | false | explainlikeimfive | Mathematics | null |
ELI5: why do numbers like 1, 3, 5, and 10 appear to be less random than numbers like 4, 7, 8, and 9? | 6mo48s | false | explainlikeimfive | Mathematics | [removed] |
ELI5: Tropical mathematics - help me understand how to talk to my daughter about it. | 6mve1x | false | explainlikeimfive | Mathematics | My daughter is at a summer math camp at the same university where I graduated 25-ish years ago. I was a math major until I switched to Economics my senior year. I've been the go-to "help me with my math homework" guy for all my kids for a lot of years. Now one daughter is at this math camp and learned yesterday about Tropical math. I'd never hear of it. I've researched online but can't find any explanation that's making sense to me. I found why I'd never heard of it - it gained "popularity" after I graduated. It seems to be a way to do geometry in a different syntax, but I just don't get it. Can anyone explain it to me? |
ELI 5: How does a calculator(specifically a scientific calculator) work? | 6mz05y | false | explainlikeimfive | Mathematics | [removed] |
ELI5:How is "if p, then q" considered True when 'p' is False. | 6n3j58 | false | explainlikeimfive | Mathematics | I've been trying to wrap my head around this for a couple days. Can anyone come up with a concrete example? |
ELI5:Mathematicians,how to calculate probability? | 6n9z67 | false | explainlikeimfive | Mathematics | [deleted] |
if 1/3 = 0.333 and 2/3 = 0.666, how does 3/3 = 1 | 6ngo7a | false | explainlikeimfive | Mathematics | [removed] |
Eli5: Kronecker Delta | 6ni6ay | false | explainlikeimfive | Mathematics | null |
ELI5: When to solve for Present Value/Future Value in Time Value of Money problems | 6nnalk | false | explainlikeimfive | Mathematics | [removed] |
ELI5:In mathematics, can every possible curve be represented as a mathematical equation? | 6nolnt | false | explainlikeimfive | Mathematics | can any curve in the universe be written as "y is a function of x" ? |
ELI5: GAM statistical models: splines, smoothers and tensors? | 6nsyuq | false | explainlikeimfive | Mathematics | In Generalised Additive Models how do smoothers, splines and tensors work mathematically and what is their purpose? |
ELI5: How to understand Zorn's Lemma | 6nzwzm | false | explainlikeimfive | Mathematics | [removed] |
ELI5: Where would be the most equally homogeneous place to conduct a survey where education level matters? | 6oa6mj | false | explainlikeimfive | Mathematics | There was a link to a National Science Foundation study that determined that 26% of Americans (of a sample size of 2200) believe the Sun moves around the earth. I presented this to my wife with the witty quote of "someone within shouting distance of you thinks this". Her response, was that she doesn't work in an environment where people are less educated and as such, that statistic doesn't apply in every scenario. That of course is true. Which got me to think, "Where would be the most equally homogeneous place to conduct a survey where education level matters?" DMV: Not everyone drives Grocery store: Everyone has to eat, but more educated individuals may have the resources to grocery shop more frequently, affecting the probability that they will be questioned. I just started to think about this, and figured I'm come here straight away and see what reddit thinks. |
ELI5:What is quotienting all about? | 6plldi | false | explainlikeimfive | Mathematics | I'm starting to read abstract algebra, and this term quotient is coming about a lot. I think the primary motive is to form equivalence classes in different structures. I want to understand, how mathematicians go about doing this in different contexts? That is, what is the core commonality of quotienting? |
ELI5: This video | 6prhjk | false | explainlikeimfive | Mathematics | [removed] |
ELI5: How does x^ai (where a is any real number) actually work? | 6px9wl | false | explainlikeimfive | Mathematics | Follow-up from https://www.reddit.com/r/explainlikeimfive/comments/6pthg1/eli5_how_does_eulers_formula_actually_work/ Again, I got the taylor series expansion. But, what does it actually mean, in terms of laws of exponentiation and the exponential function, to raise a real number to an imaginary exponent? |
ELI5: What is the difference between derivative (gradient) and divergence(upside-down triangle)? | 6qkwjt | false | explainlikeimfive | Mathematics | null |
ELI5 How was the numbering system structured before the concept of zero? | 6qmtmu | false | explainlikeimfive | Mathematics | Zero plays an important role in the number line in terms of being in the middle of negative and positive numbers. Additionally, it is important as a place holder i.e. tens, hundreds, and thousands. How did people conceptualize large numbers without zero? |
ELI5: Take away game | 6xb0l9 | false | explainlikeimfive | Mathematics | My math teacher and I played a take away game that I've never really heard of, but I'm willing to bet some of the community has. Say you have N amount of sticks, you and another player have to take away sticks from the pile but can only draw 1-3 sticks at a time. The player that draws the last stick(s) wins. No one has beaten him at it before so I'm wondering what's the strategy behind this game. |
ELI5: What is the investment needed to live off 70k per year after adjusting for inflation? | 6xdpq3 | false | explainlikeimfive | Mathematics | [removed] |
ELI5: The Equation in the movie 21 | 6xijq2 | false | explainlikeimfive | Mathematics | I just watched the movie 21 but don't understand the game show door thing. In the movie he picks door a, and then the teacher reveals that door c is a goat, so then they say that the 33% chance that door had now goes to door b. I don't understand why the 33% doesn't go equally to each door. sorry if I am dumb |
ELI5: in what ways are mathematics useful outside of the educational system? | 6xmhmo | false | explainlikeimfive | Mathematics | (excluding universities and and business that utilize monetary systems daily,such as basic cashier) how exactly is mathematics that useful? |
ELI5: Bayes Theorem for disease diagnostics | 6xsmns | false | explainlikeimfive | Mathematics | null |
ELI5: Do the betting odds on a football game today take into account the historical record of same matchup from previous seasons? | 6xue5y | false | explainlikeimfive | Mathematics | The Winnipeg Blue Bombers play the Saskatchewan Roughriders today in the Labour Day Classic. This matchup occurs every year, and I'm wondering if the fact that Saskatchewan has won 11 of the past 12 years is factored into today's odds, or if the odds are calculated solely on data from the current seaeon/performance. |
ELI5: Central Limit Theorem | 6xz307 | false | explainlikeimfive | Mathematics | null |
ELI5: What does my GRE textbook mean when it says "one billion means 1,000,000,000, or 10^9 (not 10^12, as in some countries)"? | 6y2q08 | false | explainlikeimfive | Mathematics | Do other places express scientific notation differently? Or am I completely missing something? Ever since I read it I have been trying to understand why there would be a time where 10^9 = 10^12. I cannot find anything about this online but I could be using poor search terms. |
ELI5: How does hypnosis to forget a number (AKA amnesia hypnosis) actually work? | 6y9vqy | false | explainlikeimfive | Mathematics | [removed] |
ELI5: Why is the standard deviation formula theway it is? | 6yhoqe | false | explainlikeimfive | Mathematics | I was taught that standard deviation was the mean distance from the mean. If this is so, why is the "n-1" included inside the square root? Shouldn't it be outside the square root? Also why is this value n-1? I looked it up and it said that this was to control for the sample size in comparison to the population, but then why n-1 as opposed to n-2 or .75n or something like that? |
How come we consider sqrt(-1) an imaginary number, but not abs(-1) an imaginary number? Or do we? Is it a whole different dimension os imaginary numbers? | 6z1ihl | false | explainlikeimfive | Mathematics | [deleted] |
ELI5: Why do we consider sqrt(-1) to be imaginary but not abs(x) = -1? Is it a different kind of imaginary number or is it not? | 6z1m3z | false | explainlikeimfive | Mathematics | I posted this question about 10 minutes ago and I made a FATAL error in my question. What I said previously is why |-1| isn't imaginary. Well duh! It's just 1! What I meant to say was if |x| = -1 was an imaginary number. I'm not sure if I should have deleted the comment or if there was a way to edit it but it has been corrected now. |
ELI5: Why are 7 and 43 considered arbitrary numbers but 1 and 100 not? | 6ziyvp | false | explainlikeimfive | Mathematics | null |
ELI5: TREE(3) and Kruskal's Tree Theorem | 6zmh5r | false | explainlikeimfive | Mathematics | I've read many explanations but I still have no idea how it works, and why TREE(3) is such a massive number. |