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Can someone explain to me how there can be different kinds of infinities? I was reading "the man who loved only numbers" and came across the concept of countable and uncountable infinities, but they're only words to me. Any help would be appreciated.
Different kinds of infinities?
[mathfactor][1] is one I listen to. Does anyone else have a recommendation? [1]: http://mathfactor.uark.edu/
List of interesting math podcasts?
I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grash what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
How can you prove that the square root of two is irrational?
What is your favorite online graphing tool?
I was reading up on the Fibonacci Sequence when I've noticed some were able to calculate specific numbers. So far I've only figured out creating an array and counting to the value, which is incredibly simple, but I reckon I can't find any formula for calculating a Fibonacci number based on it's position. Is there a way to do this? If so, how are we able to apply these formulas to arrays?
How are we able to calculate specific numbers in the Fibonacci Sequence?
Suppose no one every taught you the names for ordinary numbers. Then suppose that you and I agreed that we would trade one bushel of corn for each of my sheep. But there's a problem, we don't know how to count the bushels or the sheep! So what do we do? We form a "bijection" between the two sets. That's just fancy language for saying you pair things up by putting one bushel next to each of the sheep. When we're done we swap. We've just proved that the number of sheep is the same as the number of bushels without actually counting. We can try doing the same thing with infinite sets. So suppose you have the set of positive numbers and I have the set of rational numbers and you want to trade me one positive number for each of my rationals. Can you do so in a way that gets all of my rational numbers? Perhaps surprisingly the answer is yes! You make the rational numbers into a big square grid with the numerator and denominators as the two coordinates. Then you start placing your "bushels" along diagonals of increasing size, [see wikipedia][1]. This says that the rational numbers are "countable" that is you can find a clever way to count them off in the above fashion. The remarkable fact is that for the real numbers there's *no way at all* to count them off in this way. No matter how clever you are you won't be able to scam me out of all of my real numbers by placing a natural number next to each of them. The proof of that is Cantor's clever "[diagonal argument][2]." [1]: http://en.wikipedia.org/wiki/File:Pairing_natural.svg [2]: http://en.wikipedia.org/wiki/Cantor's_diagonal_argument
I'm told by smart people that `0.999... = 1` and I believe them but is there a proof that explains why?
Does .99999... = 1?
How many Natural (Integers) Numbers could you count? There are infinitely many, yet you can count them. It's called [Countable Set][1]. How many Real Numbers are there? Infinitely as well (Since at least every Natural Number is Areal Number). Yet you won't be able to count them (Intuitively, Let's say you name a number the first, then find the second, I can, for sure, find a number in between, their average which is Real Number as well).. It's called [Uncountable Set][2]. What you are after is how we define how big is a given set. Then you should look for [Cardinality][3]. [1]: http://en.wikipedia.org/wiki/Countable_set [2]: http://en.wikipedia.org/wiki/Uncountable_set [3]: http://en.wikipedia.org/wiki/Cardinality
Given a the semi-major axis radius and a flattening factor, is it possible to calculate the semi-minor radius?
How do you calculate the semi-minor axis for an elipsoid?
By matrix-defined, I mean <pre> < a b c > x < d e f > = | i j k | det(| a b c |) | d e f | </pre> (instead of the definition of the product of the magnitudes multiplied by the sign of their angle, in the direction orthogonal) If I try cross producting two vectors with no `k` component, I get one with only `k`, which is expected. But why?
Why is the matrix-defined Cross Product of two 3D vectors always orthogonal?
Well, I am not sure where you want to embed the graphs, but [Wolfram Alpha][1] is pretty handy for graphing. It has most of the features of Mathematica, can handle 3D functions, and fancy scaling and such. I highly recommend it. [1]: http://www.wolframalpha.com/input/?i=sin(x)
I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator. Can you recommend any?
Can you recommend a decent online or software calculator?
What I really don't like about all the above answers, is the underlying assumption that `1/3=0.3333...`, How do you know that?. It seems to me like assuming the something which is already known. A proof I really like is: 0.9999... x 10 = 9.999999... 0.9999... x 9 + 0.99999.... = 9.99999.... 0.9999... x 9 = 9.9999....-0.999999 = 9 0.9999... x 9 = 9 0.9999... = 1 The only thing I need to assume is, that `9.999.... - 0.9999... = 9` and this seems to me intuitive enough to take for granted. The proof is from an old highschool level math book of the Open University in Israel.
What does it mean when you refer to .99999...? Symbols don't mean anything in particular until you've *defined what you mean by them*. In this case the definition is that you're taking the limit of .9, .99, .999, .9999, etc. What does it mean to say that limit is 1? Well, it means that no matter how small a number x you pick, I can show you a point in that sequence such that all further numbers in the sequence are within distance x of 1. But certainly whatever number you chose your number is bigger than 10^-k for some k. So I can just pick my point to be the kth spot in the sequence. A more intuitive way of explaining the above argument is that the reason .99999... = 1 is that their difference is zero. So let's subtract 1.0000... -.99999... = .00000... = 0. That is, 1.0 -.9 = .1 1.00-.99 = .01 1.000-.999=.001, ... 1.000... -.99999... = .000... = 0.
What length of rope should be used to tie a cow to a fence post of a *circular* field so that the cow can only graze half of the grass within that field?
> A *countably infinite* set is a set for which you can list the elements a<sub>1</sub>,a<sub>2</sub>,a<sub>3</sub>,... For example, the set of all integers is countably infinite since I can list its elements as follows 0,1,-1,2,-2,3,-3,... So is the set of rational numbers, but this is more difficult to see. Let's start with the positive rationals. Can you see the pattern in this listing? 1/1,1/2,2/1,1/3,2/2,3/1,1/4,2/3,3/2,4/1,1/5,2/4,... (Hint: Add the numerator and denominator to see a different pattern.) This listing has lots of repeats, e.g. 1/1, 2/2 and 1/2, 2/4. That's ok since I can condense the listing by skipping over any repeats. 1/1,1/2,2/1,1/3,3/1,1/4,2/3,3/2,4/1,1/5,... Let's write q<sub>n</sub> for the n-th element of this list. Then 0,q<sub>1</sub>,-q<sub>1</sub>,q<sub>2</sub>,-q<sub>2</sub>,q<sub>3</sub>,-q<sub>3</sub>,... is a listing of all rational numbers. > A *countable set* is a set which is either finite or countably infinite; an *uncountable set* is a set which is not countable. Thus, an uncountable set is an infinite set which has no listing of all of its elements (as in the definition of countably infinite set). An example of an uncountable set is the set of all real numbers. To see this, you can use the *diagonal method*. Ask another question to see how this works...
As I've understood it, what I've learned as the dot product is just one of many possible "inner product spaces". Can someone explain this concept? When is it useful to define it as something other than the dot product?
What is an inner product space?
This is simply not clicking for me. I'm currently learning math during the summer vacation and I'm on the chapter for relations and functions. There are five properties for a relation: Reflexive - R -> R Symmetrical - R -> S ; S -> R Antisymmetrical - R -> S && (R -> R || S -> S) Asymmetrical - R -> S && !(R -> R || S -> S) Transitive - R -> S && S -> T then R -> T If that's not what you call the properties in English, then please let me know- I have to study it in German, unfortunately, and these are my rough translations. Continuing on, I just don't know what to do with this information practically. The examples of the book are horrible: 1) "Is the same age as" is apparently reflexive, symmetrical and transitive. 2) "Is related to" is also apparently reflexive, symmetrical and transitive. 3) "Is older than" is asymmetric, antisymetric and transitive. There are more useless examples like this. I have no idea how it comes to these conclusions because we're talking about a literal statement. I was hoping perhaps for some real mathematical examples, but the book falls short on those. I would greatly appreciate it if somebody could explain the above example and perhaps give me a better use for Relations other than... that. Also, how can a relation be a- and antisymmetrical at the same time? Don't they cancel each other out?
How do the Properties of Relations work?
Given two points in 2D space how would you calculate an angle from p1 to p2? How would this change in 3D space?
What is an elliptic curve, and how are they used in cryptography?
You can see that there are infinitely many natural numbers 1, 2, 3, ..., and infinitely many real numbers, such as 0, 1, pi, etc. But are these two infinities the same? Well, suppose you have two sets of objects, e.g. people and horses, and you want to know if the number of objects in one set is the same as in the other. The simplest way is to find a way of corresponding the objects one-to-one. For instance, if you see a parade of people riding horses, you will know that there are as many people as there are horses, because there is such a one-to-one correspondence. We say that an set with infinitely many things is "countable," if we can find a one-to-one correspondence between the things in this set and the natural numbers. E.g., the integers are countable: 1 <-> 0, 2 <-> -1, 3 <-> 1, 4 <-> -2, 5 <-> 2, etc, gives such a correspondence. However, the set of real numbers is NOT countable! This was proven for the first time by Georg Cantor. Here is a proof using the so-called [diagonal argument][1]. [1]: http://en.wikipedia.org/wiki/Cantor's_diagonal_argument
What length of rope should be used to tie a cow to an **exterior fence post** of a *circular* field so that the cow can only graze half of the grass within that field? ***updated:*** To be clear: the cow should be tied to a post on the exterior of the field, not a post at the center of the field.
I've recently started reading about Quaternions, and I keep reading that for example they're used in computer graphics and mechanics calculations to calculate movement and rotation, but without real explanations of the benefits of using them. I'm wondering what exactly can be done with Quaternions that can't be done as easily (or easier) using more tradition approaches, such as with Vectors?
Real world uses of Quaternions?
This is simply not clicking for me. I'm currently learning math during the summer vacation and I'm on the chapter for relations and functions. There are five properties for a relation: Reflexive - R -> R Symmetrical - R -> S ; S -> R Antisymmetrical - R -> S && (R -> R || S -> S) Asymmetrical - R -> S && !(R -> R || S -> S) Transitive - if R -> S && S -> T, then R -> T If that's not what you call the properties in English, then please let me know- I have to study it in German, unfortunately, and these are my rough translations. Continuing on, I just don't know what to do with this information practically. The examples of the book are horrible: 1) "Is the same age as" is apparently reflexive, symmetrical and transitive. 2) "Is related to" is also apparently reflexive, symmetrical and transitive. 3) "Is older than" is asymmetric, antisymetric and transitive. There are more useless examples like this. I have no idea how it comes to these conclusions because we're talking about a literal statement. I was hoping perhaps for some real mathematical examples, but the book falls short on those. I would greatly appreciate it if somebody could explain the above example and perhaps give me a better use for Relations other than... that. Also, how can a relation be a- and antisymmetrical at the same time? Don't they cancel each other out?
Let's say I know 'X' is a Gaussian Variable. Moreover, I know 'Y' is a Gaussian Variable and Y=X+Z. Let's say all of the are Independent. How can I prove Z is a Gaussian Variable? It's easy to show the other way around (X, Z Orthogonal and Normal hence create a Gaussian Vector hence any Linear Combination of the two is a Gaussian Variable). Thanks
Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
What purposes do the Dot and Cross products serve? Do you have any clear examples of when you would use them?
I'm talking in the range of 10-12 years old, but this question isn't limited to only that range. Do you have any advice on cool things to show kids that might spark their interest in spending more time with math? The difficulty for some to learn math can be pretty overwhelming. Do you have any teaching techniques that you find valuable?
What are some good ways to get children excited about math?
Given two points, around an origin (0,0), in 2D space how would you calculate an angle from p1 to p2. How would this change in 3D space?
How to accurately calculate erf(x) with a computer?
I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
This is in relation to the Euler Problem 13 from www.ProjectEuler.net. Work out the first ten digits of the sum of the following one-hundred 50-digit numbers. 37107287533902102798797998220837590246510135740250 Now, this was my thinking: I can freely discard the last fourty digits and leave the last ten. 0135740250 And then simply sum those. This would be large enough to be stored in a 64-bit data-type and a lot easier to compute. However, my answer isn't being accepted, so I'm forced to question my logic. However, I don't see a problem. The last fourty digits will never make a difference because they are at least a magnitude of 10 larger than the preceding values and therefore never carry backwards into smaller areas. Is this not correct?
Faulty logic when summing large integers?
Since you're asking for a reference, perhaps this will do? Wolfram Mathworld says the problem was listed by Rado in 1925. The reference is on the problem description page, [here][1]. [1]: http://mathworld.wolfram.com/LionandManProblem.html
It is 10 o'clock, and I have a box. Inside the box is a ball marked 1. At 10:30, I will remove the ball marked 1, and add two balls, labeled 2 and 3. At 10:45, I will remove the balls labeled 2 and 3, and add 4 balls, marked 4, 5, 6, and 7. 7.5 minutes before 11, I will remove the balls labeled 4, 5, and 6, and add 8 balls, labeled 8, 9, 10, 11, 12, 13, 14, and 15. This pattern continues. Each time I reach the halfway point between my previous action and 11 o'clock, I add some balls, and remove some other balls. Each time I remove one more ball than I removed last time, but add twice as many balls as I added last time. The result is that as it gets closer and closer to 11, the number of balls in the box continues to increase. Yet every ball that I put in was eventually removed. So just how many balls will be in the box when the clock strikes 11? 0, or infinitely many? What's going on here?
paradox: increasing sequence that goes to 0?
The argument against this is that 0.99999999... is "somewhat" less than 1. How much exactly? 1 - 0.999999... = Ξ΅ If the above is true, the following also must be true: 9 Γ— (1 - 0.999999...) = Ξ΅ Γ— 9 Let's calculate: 0.999... Γ— 9 = ----------- 8.1 81 81 . . . ----------- 8.999... Thus: 9 - 8.999999... = 9Ξ΅ (1) But: 8.999999... = 8 + 0.99999... (2) Indeed: 8.00000000... + 0.99999999... = --------------- 8.99999999... Now let's see what we can deduce from `(1)` and `(2)`. 9 - 8.999999... = 9Ξ΅ 9 - 8.999999... = 9 - (8 + 0.99999...) = = 9 - 8 - (1 - Ξ΅) = 1 - 1 + Ξ΅ = Ξ΅. Thus: 9Ξ΅ = Ξ΅ 10Ξ΅ = 0 Ξ΅ = 0 1 - 0.999999... = Ξ΅ = 0 Quod erat dimostrandum. Pardon my unicode.
I keep seeing this symbol βˆ‡ around and I know enough to understand that it represents the term "gradient." But what is a gradient? When would I want to use one mathematically?
What are gradients and how would I use them?
The argument against this is that 0.99999999... is "somewhat" less than 1. How much exactly? 1 - 0.999999... = Ξ΅ If the above is true, the following also must be true: 9 Γ— (1 - 0.999999...) = Ξ΅ Γ— 9 Let's calculate: 0.999... Γ— 9 = ─────────── 8.1 81 81 . . . ─────────── 8.999... Thus: 9 - 8.999999... = 9Ξ΅ (1) But: 8.999999... = 8 + 0.99999... (2) Indeed: 8.00000000... + 0.99999999... = ──────────────── 8.99999999... Now let's see what we can deduce from `(1)` and `(2)`. 9 - 8.999999... = 9Ξ΅ 9 - 8.999999... = 9 - (8 + 0.99999...) = = 9 - 8 - (1 - Ξ΅) = 1 - 1 + Ξ΅ = Ξ΅. Thus: 9Ξ΅ = Ξ΅ 10Ξ΅ = 0 Ξ΅ = 0 1 - 0.999999... = Ξ΅ = 0 Quod erat dimostrandum. Pardon my unicode.
What are some classic fallacies?
I keep hearing about this weird type of math called calculus. I only have experience with geometry and algebra. Can you try to explain what it is to me?
How would you describe calculus in simple terms?
Is 1 classified as a prime number? And if so, why? If not, why not?
Is 1 a prime number?
The argument against this is that 0.99999999... is "somewhat" less than 1. How much exactly? 1 - 0.999999... = Ξ΅ (0) If the above is true, the following also must be true: 9 Γ— (1 - 0.999999...) = Ξ΅ Γ— 9 Let's calculate: 0.999... Γ— 9 = ─────────── 8.1 81 81 . . . ─────────── 8.999... Thus: 9 - 8.999999... = 9Ξ΅ (1) But: 8.999999... = 8 + 0.99999... (2) Indeed: 8.00000000... + 0.99999999... = ──────────────── 8.99999999... Now let's see what we can deduce from `(0)`, `(1)` and `(2)`. 9 - 8.999999... = 9Ξ΅ because of (2) 9 - 8.999999... = 9 - (8 + 0.99999...) = because of (1) = 9 - 8 - (1 - Ξ΅) because of (0) = 1 - 1 + Ξ΅ = Ξ΅. Thus: 9Ξ΅ = Ξ΅ 10Ξ΅ = 0 Ξ΅ = 0 1 - 0.999999... = Ξ΅ = 0 Quod erat dimostrandum. Pardon my unicode.
I covered hyperbolic triganomic functions in a recent maths course. However I was never presented with any reasons as to *why* (or even if) they are useful. Is there any good examples of their uses outside academia?
There came a time in mathematics when people encountered situations where they had to deal with really, really, really small things. Not just small like 0.01; but small as in *infinitesimally small*. Think of "the smallest positive number that is still greater than zero" and you'll realize what sort of problems mathematicians began encountering. Soon, this problem became more than just theoretical or abstract. It became very, very real. For example, velocity. We know that average velocity is the change in position per change in time (i.e., 5 miles per hour). But what about velocity **at a point in time**? What does it mean to be going 5 mph **at this moment**? One solution someone came up with was to say "it's the change in position divided by the change in time, where the change in time is an infinitesimally small amount of time". But how would you handle/calculate that? Another problem came about trying to find the area under a curve. The current accepted solution was to divide the curve into rectangles, and add together the area of the rectangles. However, in order to find the **exact** area under the curve, you'd need to divide it into rectangles that were **infinitesimally tiny**, and, therefore, add up an infinite amount of tiny rectangles -- to something that was finite (area). Calculus came about as the system of math dedicated to studying these infinitesimally small changes. In fact, I do believe some people describe calculus as "the study of continuous changes".
The Weyl equidistribution theorem states that $\{n \xi\}$ is uniformly distributed for $\xi$ irrational, modulo 1; it can be proved using a bit of ergodic theory or simply playing with trigonometric polynomials and using the fact they are dense in the space of all continuous functions modulo 1. In particular, one shows that if $f(x)$ is a continuous function with period 1, then $\int_0^1 f(x) dx = \lim \frac{1}{n} \sum_{i=0}^{N-1} f(t+i \alpha)$ by checking this (directly) for trigonometric polynomials via the geometric series. This is a very elementary and nice proof. Can the general form of Weyl's theorem - that if $p$ is an integer-valued, non-constant polynomial function, then $p(n \xi)$ is uniformly distributed modulo 1 (for $\xi$ irrational)?
How do you prove that $p(n \xi)$ for $\xi$ irrational is uniformly distributed modulo 1?
What are some classic fallacious proofs?
Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why 0^0 should be undefined.
Why is x^0 = 1 except when x = 0?
I was playing around with the squares and saw an interesting pattern in their differences. <pre> 0^2 = 0 + 1 1^2 = 1 + 3 2^2 = 4 + 5 3^2 = 9 + 7 4^2 = 16 + 9 5^2 = 25 + 11 6^2 = 36 etc. </pre> Also, in a very related question, which major Math Research Journal should I contact to publish my groundbreaking find in?
Why are the differences between consecutive squares equal to the sequence of odd numbers?
I have a square that's 10m x 10m. I want to cut it in half so that I have a square with half the area. But if I cut it from top to bottom or left to right, I don't get a square, I get a rectangle! I know the area of the small square is supposed to be 50, so I can use my calculator to find out how long a side should be: it's 7.07106781. But my teacher said I should be able to do this without a calculator. How am I supposed to get that number by hand?
how do i cut a square in half?
0^x = 0, x^0 = 1 both are true when x is not 0 what happens when x=0? undefined, because there is no way to chose one definition over the other. Some people define 0^0 = 1 in their books, like Knuth, because 0^x is less 'useful' than x^0.
This is a question of definition, the question is "why does it make sense to define x^0=1 except when x=0?" or "How is this definition better than other definitions?" The answer is that $x^a * x^b = x^{a+b}$ is an excellent formula that makes a lot of sense (multiplying a times and then multiplying b times is the same as multiplying a+b times) and which you can prove for a and b positive integers. So any sensible definition of x^a for numbers a which aren't positive integers *should still satisfy this identity*. In particular, $x^0 * x^b = x^{0+b} = x^b$; now *if x is not zero* then you can cancel x^b from both sides and get that x^0 = 1. But if x=0 then x^b is zero and so this argument doesn't tell you anything about what you should define x^0 to be. A similar argument should convince you that when x is not zero then x^-a should be defined as 1/x^a. An argument using the related identity (x^a)^b = x^(ab) should convince you that x^{1/n} is taking the nth root.
I was playing around with the squares and saw an interesting pattern in their differences. <pre> 0^2 = 0 + 1 1^2 = 1 + 3 2^2 = 4 + 5 3^2 = 9 + 7 4^2 = 16 + 9 5^2 = 25 + 11 6^2 = 36 etc. </pre> (Also, in a very related question, which major Math Research Journal should I contact to publish my groundbreaking find in?)
By matrix-defined, I mean <pre> < a b c > x < d e f > = | i j k | det(| a b c |) | d e f | </pre> (instead of the definition of the product of the magnitudes multiplied by the sign of their angle, in the direction orthogonal) If I try cross producting two vectors with no `k` component, I get one with only `k`, which is expected. But why? As has been pointed out, I am asking why the algebraic definition lines up with the geometric definition.
It seems like functions that are continuous always seem to be differentiable, to me. I can't imagine one that is not. Are there any examples of functions that are continuous, yet not differentiable? The other way around seems a bit simpler -- a differentiable function is obviously always going to be continuous. But are there any that do not satisfy this?
Are there any functions that are (always) continuous yet not differentiable? Or vice-versa?
To understand the benefits of using quaternions you have to consider the alternatives. Here are few ways with a summary of the pros and cons: - Euler angles - Rotation matrices - Axis angle - Quaternions - Rotors (normalized Spinors) Euler angles are the best choice if you want a user to specify an orientation in a intuitive way. They are are also space efficient (three numbers). However, it is more difficult to linear interpolate values. Consider the case where you want to interpolate between 359 and 0 degrees. Linearly interpolating would cause a large rotation, even though the two orientations are almost the same. Writing shortest path interpolation, is easy for one axis, but non-trivial when considering the three Euler angles(for instance the shortest route between (240, 57, 145) and (35, -233, -270) is not immediately clear). Rotation matrices specify a new frame of reference using three normalized and orthogonal vectors (Right, Up, Out, which when multiplied become the new x, y, z). Rotation matrices are useful for operations like strafing (side ways movement), which only requires translating along the Right vector of the camera's rotation matrix. However, there is no clear method of interpolating between them. The are also expensive to normalize which is necessary to prevent scaling from being introduced. Axis angle, as the name suggests, are a way of specifying a rotation axis and angle to rotate around that axis. You can think of Euler angles, as three axis angle rotations, where the axises are the x, y, z axis respectively. Linearly interpolating the angle in a axis angle is pretty straight forward (if you remember to take the shortest path), however linearly interpolating between different axises is not. Quaternions are a way of specifying a rotation through a axis and the cosine of half the angle. They main advantage is I can pick any two quaternions and smoothly interpolate between them. Rotors are another way to perform rotations. Rotors are basically quaternions, but instead of thinking of them as 4D complex numbers, rotors are thought of as real 3D multivectors. This makes their visualization much more understandable (compared to quaternions), but requires fluency in geometric algebra to grasp their significance. Okay with that as the background I can discuss a real world example. Say you are writing a computer game where the characters are animated in 3ds Max. You need to export a animation of the character to play in your game, but cannot faithfully represent the interpolation used by the animation program, and thus have to sample. The animation is going to be represented as a list of rotations for each joint. How should we store the rotations? If I am going to sample every frame, not interpolate, and space is not an issue, I would probably store the rotations as rotation matrices. If space was issue, then Euler angles. That would also let me do things like only store one angle for joints like the knee that have only one degree of freedom. If I only sampled every 4 frames, and need to interpolate it depends on whether I am sure the frame-rate will hold. If I am positive that the frame-rate will hold I can use axis angle relative rotations to perform the interpolation. This is atypical. In most games the frame rate can drop past my sampling interval, which would require skipping an element in the list to maintain the correct playback speed. If I am unsure of what two orientations I need to interpolate between, then I would use quaternions or rotors. That is probably confusing, but hopefully it helps a little :).
To understand the benefits of using quaternions you have to consider different ways to represent rotations. Here are few ways with a summary of the pros and cons: - Euler angles - Rotation matrices - Axis angle - Quaternions - Rotors (normalized Spinors) Euler angles are the best choice if you want a user to specify an orientation in a intuitive way. They are are also space efficient (three numbers). However, it is more difficult to linear interpolate values. Consider the case where you want to interpolate between 359 and 0 degrees. Linearly interpolating would cause a large rotation, even though the two orientations are almost the same. Writing shortest path interpolation, is easy for one axis, but non-trivial when considering the three Euler angles(for instance the shortest route between (240, 57, 145) and (35, -233, -270) is not immediately clear). Rotation matrices specify a new frame of reference using three normalized and orthogonal vectors (Right, Up, Out, which when multiplied become the new x, y, z). Rotation matrices are useful for operations like strafing (side ways movement), which only requires translating along the Right vector of the camera's rotation matrix. However, there is no clear method of interpolating between them. The are also expensive to normalize which is necessary to prevent scaling from being introduced. Axis angle, as the name suggests, are a way of specifying a rotation axis and angle to rotate around that axis. You can think of Euler angles, as three axis angle rotations, where the axises are the x, y, z axis respectively. Linearly interpolating the angle in a axis angle is pretty straight forward (if you remember to take the shortest path), however linearly interpolating between different axises is not. Quaternions are a way of specifying a rotation through a axis and the cosine of half the angle. They main advantage is I can pick any two quaternions and smoothly interpolate between them. Rotors are another way to perform rotations. Rotors are basically quaternions, but instead of thinking of them as 4D complex numbers, rotors are thought of as real 3D multivectors. This makes their visualization much more understandable (compared to quaternions), but requires fluency in geometric algebra to grasp their significance. Okay with that as the background I can discuss a real world example. Say you are writing a computer game where the characters are animated in 3ds Max. You need to export a animation of the character to play in your game, but cannot faithfully represent the interpolation used by the animation program, and thus have to sample. The animation is going to be represented as a list of rotations for each joint. How should we store the rotations? If I am going to sample every frame, not interpolate, and space is not an issue, I would probably store the rotations as rotation matrices. If space was issue, then Euler angles. That would also let me do things like only store one angle for joints like the knee that have only one degree of freedom. If I only sampled every 4 frames, and need to interpolate it depends on whether I am sure the frame-rate will hold. If I am positive that the frame-rate will hold I can use axis angle relative rotations to perform the interpolation. This is atypical. In most games the frame rate can drop past my sampling interval, which would require skipping an element in the list to maintain the correct playback speed. If I am unsure of what two orientations I need to interpolate between, then I would use quaternions or rotors. That is probably confusing, but hopefully it helps a little :).
Complex numbers involve the square root of negative one, and most non-mathematicians can't accept that such a number exists. What's the best way to explain that complex numbers really do exist, and are just as "valid" as real numbers?
Do complex numbers really exist?
I've come across statements in the past along the lines of "function f has no closed form integral", which I assume means that there is no combination of the operations: * addition/subtraction * multiplication/division * raising to powers and roots * trigonometric functions * exponential functions * logarithmic functions which when differentiated gives the function f. I've heard this said about the function f(x) = x^x for example. What sort of techniques are used to prove statements like this? What is this branch of mathematics called?
How can you prove that a function has no closed form integral?
The argument isn't worth having, as you disagree about what it means for something to 'exist'. There are many interesting mathematical objects which don't have an obvious physical counterpart. What does it mean for the Monster group to exist?
The concept of mathematical numbers and "existing" is a tricky one. What actually "exists"? Do negative numbers exist? Of course they do not. You can't have a negative amount of apples. Yet, the beauty of negative numbers is that when we define them (rigorously), then all of a sudden we can use them to solve problems we were never ever been able to solve before, or to solve them in a much simpler way. Imagine trying to do simple physics without the idea of negative numbers! But are they "real"? Do they "exist"? No, they don't. But they are just tools that help us solve real life problems. To go back to your question about complex numbers, I would say that the idea that they exist or not has no bearing onto whether or not they are actually useful in solving the problem of every day life, or making them many, many, many times more easy to solve. The math that makes your computer run and involves the tool that is complex numbers, for instance.
I've heard of some other paradoxes involving sets (ie, "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand. Why is "the set of all sets" a paradox? It seems like it would be fine, to me. There is nothing paradoxical about a set containing itself. Is it something that arises from the "rules of sets" that are involved in more rigorous set theory?
Why is "the set of all sets" a paradox?
I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $frac{4}{3}$ is so random! How could somebody guess something like this for the formula?
Why is the volume of a sphere $\frac{4}{3}\pi r^3$?
There are a few good answers to this question, depending on the audience. I've used all of these on occasion. **A way to solve polynomials** We came up with equations like `x - 5 = 0, what is x?`, and the naturals solved them (easily). Then we asked, "wait, what about `x + 5 = 0`?" So we invented negative numbers. Then we asked "wait, what about `2x = 1`?" So we invented rational numbers. Then we asked "wait, what about `x^2 = 2`?" so we invented irrational numbers. Finally, we asked, "wait, what about `x^2 = -1`?" This is the only question that was left, so we decided to invent the "imaginary" numbers to solve it. All the other numbers, at some point, didn't exist and didn't seem "real", but now they're fine. Now that we have imaginary numbers, we can solve every polynomial, so it makes sense that that's the last place to stop. **Pairs of numbers** This explanation goes the route of redefinition. Tell the listener to forget everything he knows about imaginary numbers. You're defining a new number system, only now there are always pairs of numbers. Why? For fun. Then go through explaining how addition/multiplication work. Try and find a good "realistic" use of pairs of numbers (many exist). Then, show that in this system, `(0,1) * (0,1) = (-1,0)`, in other words, we've defined a new system, under which it makes sense to say that `sqrt(-1) = i`, when `i=(0,1)`. And that's really all there is to imaginary numbers: a definition of a new number system, which makes sense to use in most places. And under that system, there **is** an answer to `sqrt(-1)`. **The historical explanation** Explain the history of the imaginary numbers. Showing that mathematicians *also* fought against them for a long time helps people understand the mathematical process, i.e., that it's all definitions in the end. I'm a little rusty, but I think there were certain equations that kept having parts of them which used `sqrt(-1)`, and the mathematicians kept throwing out the equations since there is no such thing. Then, one mathematician decided to just "roll with it", and kept working, and found out that all those square roots cancelled each other out. Amazingly, the answer that was left was the **correct answer** (he was working on finding roots of polynomials, I think). Which lead him to think that there *was* a valid reason to use `sqrt(-1)`, even if it took a long time to understand it.
Complex numbers involve the square root of negative one, and most non-mathematicians can't accept that such a number exists. What's the best way to explain *to a non-mathematician* that complex numbers really do exist, and are just as "valid" as real numbers?
Calculus is the mathematics of change. In algebra, almost nothing ever changes. Here's a comparison of some algebra vs. calc problems: algebra: car A is driving at 50 kph. How far has it gone after 6 hours? calc: car B starts at 10 mph and begins accelerating at the rate of 10 kph^2 (kilometers per hour per hour). How far has car B gone after 6 hours? Note how the algebra problem nothing changes, where in the calc problem, the speed of the car is constantly changing. calc: If a ball is rolling in a straight line at 10 fps with a diameter of 1 foot and Q is a the point at the top of the ball when t=0, how fast is point Q moving at time t=4 relative to the ground? The speed of the point in relation to the ground is never the same (its zero when its at the bottom, 20fps when it's at the top. Calculus lets you figure out how fast it's going exactly at a specific moment. There are two main branches of calculus, differential and integral. These problems pertain to differential calculus as they concern how something is changing. Integral calculus deals with how much something has changed, the opposite of differential calculus. The number of cubic feet of oxygen circulated by Jeff lungs per hour at time t follows the equation f(t) = t^3 + sin(t) How many cubic feet of oxygen do Jeff's lungs cycle per day? To find out how much something has changed when its rate of change isn't constant requires integral calculus. (the equation is purely hypothetical unless Jeff happens to be the size of a beluga whale).
Does this give you any ideas? ![alt text][1] [1]: http://farm5.static.flickr.com/4119/4813793520_0d7fa53b99_o.png
Consider the task of generating random points uniformly distributed within a circle of a given radius $r$ that is centered at the origin. Assume that we are given a random number generator $R$ that generates a floating point number uniformly distributed in the range $[0, 1)$. Consider the following procedure: 1. Generate a random point $p = (x, y)$ within a square of side $2r$ centered at the origin. This can be easily achieved by: a. Using the random number generator $R$ to generate two random numbers $x$ and $y$, where $x, y \in [0, 1)$, and then transforming $x$ and $y$ to the range $[0, r)$ (by multiplying each by $r$). b. Flipping a fair coin to decide whether to reflect $p$ around the $x$-axis. c. Flipping another fair coin to decide whether to reflect $p$ around the $y$-axis. 2. Now, if $p$ happens to fall outside the given circle, discard $p$ and generate another point. Repeat the procedure until $p$ falls within the circle. Is the previous procedure correct? That is, are the random points generated by it **uniformly distributed** within the given circle? How can one formally [dis]prove it? ---------- **Background Info** The task was actually given in [Ruby Quiz - Random Points within a Circle (#234)][1]. If you're interested, you can [check my solution][2] in which I've implemented the procedure described above. I would like to know whether the procedure is mathematically correct or not, but I couldn't figure out how to formally [dis]prove it. Note that the actual task was to generate random points uniformly distributed within a circle of a given radius *and position*, but I intentionally left that out in the question because the generated points can be easily translated to their correct positions relative to the given center. [1]: http://rubyquiz.strd6.com/quizzes/234-random-points-within-a-circle [2]: http://gist.github.com/447554
Will this Procedure Generate Random Points Uniformly Distributed within a Given Circle? Proof?
We will will first consider the most common definition of i, as the square root of -1. When you first hear this, it sounds crazy. 0 squared is 0; a positive times a positive is positive and a negative times a negative is positive too. So there doesn't actually appear to be any number that we can square to get -1. A mathematician would collectively term 0, negative numbers and positive numbers as the real numbers. They would also define the term complex numbers as a group of numbers that includes these real numbers. So while we have shown that no real number can square to get -1, we haven't even defined complex numbers at this point, so we can't rule out that one might have this property. At this point, it makes sense to ask what does a mathematician mean by a number? It certainly isn't what most people associate it with - as an abstract representation some kind of real world quantity. We need to understand that it isn't uncommon for one word to have different meanings for different groups of people - after all words mean whatever we make them mean. Most people only need to real world quantities, so they find it convenient to call those numbers. On the other hand, mathematicians explore a variety of different number systems. Indeed some, such as complex numbers, are useful for solving problems that are actually about real numbers. So mathematicians define i as a number that obeys most of the normal algebraic laws. They also defined i*i to equal -1. From this we can derive all of the standard results about [complex number][1]s. As for whether they are real - it depends on what you want to know. Obviously, they don't correspond to quantities of physical objects. On the other hand, complex numbers can useful for representing [resistance in an electric circuit][2]. Ultimately, they are an idea and while ideas don't exist physically, saying they don't exist at all is inaccurate. [1]: http://en.wikipedia.org/wiki/Complex_number [2]: http://en.wikipedia.org/wiki/Electrical_impedance
I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to *why* (or even if) they are useful. Is there any good examples of their uses outside academia?
I have a collection of 3D points in the standard x, y, z vector space. Now I pick one of the points `p` as a new origin and two other points `a` and `b` such that `a - p` and `b - p` form two vectors of a new vector space. The third vector of the space I will call `x` and calculate that as the cross product of the first two vectors. Now I would like to recast or reevaluate each of the points in my collection in terms of the new vector space. How do I do that? (Also, if 'recasting' not the right term here, please correct me.)
To understand the benefits of using quaternions you have to consider different ways to represent rotations. Here are few ways with a summary of the pros and cons: - Euler angles - Rotation matrices - Axis angle - Quaternions - Rotors (normalized Spinors) Euler angles are the best choice if you want a user to specify an orientation in a intuitive way. They are are also space efficient (three numbers). However, it is more difficult to linear interpolate values. Consider the case where you want to interpolate between 359 and 0 degrees. Linearly interpolating would cause a large rotation, even though the two orientations are almost the same. Writing shortest path interpolation, is easy for one axis, but non-trivial when considering the three Euler angles(for instance the shortest route between (240, 57, 145) and (35, -233, -270) is not immediately clear). Rotation matrices specify a new frame of reference using three normalized and orthogonal vectors (Right, Up, Out, which when multiplied become the new x, y, z). Rotation matrices are useful for operations like strafing (side ways movement), which only requires translating along the Right vector of the camera's rotation matrix. However, there is no clear method of interpolating between them. The are also expensive to normalize which is necessary to prevent scaling from being introduced. Axis angle, as the name suggests, are a way of specifying a rotation axis and angle to rotate around that axis. You can think of Euler angles, as three axis angle rotations, where the axises are the x, y, z axis respectively. Linearly interpolating the angle in a axis angle is pretty straight forward (if you remember to take the shortest path), however linearly interpolating between different axises is not. Quaternions are a way of specifying a rotation through a axis and the cosine of half the angle. They main advantage is I can pick any two quaternions and smoothly interpolate between them. Rotors are another way to perform rotations. Rotors are basically quaternions, but instead of thinking of them as 4D complex numbers, rotors are thought of as real 3D multivectors. This makes their visualization much more understandable (compared to quaternions), but requires fluency in geometric algebra to grasp their significance. Okay with that as the background I can discuss a real world example. > Say you are writing a computer game > where the characters are animated in > 3ds Max. You need to export a > animation of the character to play in > your game, but cannot faithfully > represent the interpolation used by > the animation program, and thus have > to sample. The animation is going to > be represented as a list of rotations > for each joint. How should we store > the rotations? > > If I am going to sample every frame, > not interpolate, and space is not an > issue, I would probably store the > rotations as rotation matrices. If > space was issue, then Euler angles. > That would also let me do things like > only store one angle for joints like > the knee that have only one degree of > freedom. > > If I only sampled every 4 frames, and > need to interpolate it depends on > whether I am sure the frame-rate will > hold. If I am positive that the > frame-rate will hold I can use axis > angle relative rotations to perform > the interpolation. This is atypical. > In most games the frame rate can drop > past my sampling interval, which would > require skipping an element in the > list to maintain the correct playback > speed. If I am unsure of what two > orientations I need to interpolate > between, then I would use quaternions > or rotors.
Basic definitions: a tiling of d-dimensional Euclidean space is a decomposition of that space into polyhedra such that there is no overlap between their interiors, and every point in the space is contained in some one of the polyhedra. A vertex-uniform tiling is a tiling such that each vertex figure is the same: each vertex is contained in the same number of k-faces, etc: the view of the tiling is the same from every vertex. A vertex-transitive tiling is one such that for every two vertices in the tiling, there exists an element of the symmetry group taking one to the other. Clearly all vertex-transitive tilings are vertex-uniform. For n=2, these notions coincide. However, Grunbaum, in his book on tilings, mentions but does not explain that for n >= 3, there exist vertex uniform tilings that are not vertex transitive. Can someone provide an example of such a tiling, or a reference that explains this?
I've heard of some other paradoxes involving sets (ie, "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand. Why is "the set of all sets" a paradox? It seems like it would be fine, to me. There is nothing paradoxical about a set containing itself. Is it something that arises from the "rules of sets" that are involved in more rigorous set theory? EDIT: I don't really have much of a background with Set Theory, so an answer that deals with common english would be much appreciated =)
I'm going to use $P(a,b)$ for probability of x = a, and y = b. $P(a,b) = P[x = |a|]P[y = |b|] * 1/4$ for any a, b inside the square. Therefore you are picking a coordinate uniformly inside the square. Now the problem reduced to the following: Pick a element uniformly in a set A, then discard ones that are not B (B is a subset of A). Is it the same as pick a element uniformly in the set $A \cap B = B$? I think this is true, I don't know if the following is a formal proof(haven't done probability in a while). please point out my mistakes. P(B) is the event that a element in B is picked. P(A) is the event that a element in A is picked. Pick a x uniformly, we have b \in B is P(x=b|B), it's obviously uniform. The conditional probability P(B|A) = P(A \cap B)/P(A) = P(B)/P(A) = P(B) P(A) = 1, w/e you pick are in A. P(x=b| (B|A)) = P(x=b|B), which is uniform. Answer to your question: It is uniformly distributed.