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science1 | You use a proof by contradiction. Basically, you suppose that $\sqrt{2}$ can be written as $p/q$. Then you know that $2q^2 = p^2$. As squares of integers, both $q^2$ and $p^2$ have an even number of factors of two. $2q^2$ has an odd number of factors of 2, which means it can't be equal to $p^2$. | 0.286139 | 0 |
science2 | Suppose no one ever 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 integers and I have the set of rational numbers and you want to trade me one positive integer 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. 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." | 0.226851 | 0 |
science3 | The basic concept is thus: A 'countable' infinity is one where you can give each item in the set an integer and 'count' them (even though there are an infinite number of them) An 'uncountable' infinity defies this. You cannot assign an integer to each item in the set because you will miss items. The key to seeing this is using the 'diagonal slash' argument as originally put forward by Cantor. With a countable infinity, you can create a list of all the items in the set and assign each one a different natural number. This can be done with the naturals (obviously) and the complete range of integers (including negative numbers) and even the rational numbers (so including fractions). It cannot be done with the reals due to the diagonal slash argument: Create your list of all real numbers and assign each one an integer Create a real number with the rule that the first digit after the decimal point is different from the first digit of your first number, the second digit is different from the second digit of your second number, and so on for all digits Try and place this number in your list of all numbers... it can't be the first number, or the second or the third... and so on down the list. Reductio Ad Absurdium, your number does not exist in your countable list of all real numbers and must be added on to create a new list. The same process can then be done again to show the list still isn't complete. This shows a difference between two obviously infinite sets and leads to the somewhat scary conclusion that there are (at least) 2 different forms of infinity. | 0.551315 | 0 |
science5 | Well, I am not sure where you want to embed the graphs, but Wolfram Alpha 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. | 0.423106 | 99 |
science8 | The closed form calculation for Fibonacci sequences is known as Binet's Formula. | 0.480932 | 99 |
science9 | You can use Binet's formula, described at http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html (see also Wikipedia for a proof: http://en.wikipedia.org/wiki/Binet_formula#Closed_form_expression ) | 0.392118 | 99 |
science10 | Given (by long division): $\frac{1}{3} = 0.\bar{3}$ Multiply by 3: $3\times \left( \frac{1}{3} \right) = \left( 0.\bar{3} \right) \times 3$ Therefore: $\frac{3}{3} = 0.\bar{9}$ QED. | 0.343178 | 0 |
science12 | Hilbert's Hotel is a classic demonstration. | 0.438572 | 0 |
science13 | Lots of people like to use Instacalc which lets you do unit conversions and store intermediate calculations in variables. | 0.059678 | 99 |
science14 | Natural numbers The "counting" numbers. (That is, all integers, that are one or greater). Whole numbers The Natural numbers, and zero. Integers The Whole numbers, and the negatives of the Natural numbers. Rational numbers Any number that may be expressed by any integer A divided by any integer B, where B is not zero. Irrational numbers Any number that cannot be expressed as a rational number, but is not imaginary. All irrational numbers have an infinite decimal representation. Real numbers All of the Rational and Irrational numbers. Imaginary numbers All Real numbers, multiplied by the square root of negative one. Imaginary numbers are signified by the letter i. Complex numbers Numbers composed of the sum of a Real and an Imaginary number. This includes all Real and all Imaginary numbers. | 0.398044 | 99 |
science16 | You can visualise it by thinking about it in infinitesimals. The more $9's$ you have on the end of $0.999$, the closer you get to $1$. When you add an infinite number of $9's$ to the decimal expansion, you are infinitely close to $1$ (or an infinitesimal distance away). And this isn't a rigorous proof, just an aid to visualisation of the result. | 0.182492 | 0 |
science17 | What I really don't like about all the above answers, is the underlying assumption that $1/3=0.3333\ldots$ How do you know that? It seems to me like assuming the something which is already known. A proof I really like is: $$\begin{align} 0.9999\ldots × 10 &= 9.9999\ldots\\ 0.9999\ldots × (9+1) &= 9.9999\ldots\\ \text{by distribution rule: }\Space{15ex}{0ex}{0ex} \\ 0.9999\ldots × 9 + 0.9999\ldots × 1 &= 9.9999\ldots\\ 0.9999\ldots × 9 &= 9.9999\dots-0.9999\ldots\\ 0.9999\ldots × 9 &= 9\\ 0.9999\ldots &= 1 \end{align}$$ The only things I need to assume is, that $9.999\ldots - 0.999\ldots = 9$ and that $0.999\ldots × 10 = 9.999\ldots$ These seems to me intuitive enough to take for granted. The proof is from an old high school level math book of the Open University in Israel. | 0.175452 | 0 |
science18 | Google's calculator is very powerful: http://www.googleguide.com/help/calculator.html and your use history will be stored in your browser history. | 0.531551 | 99 |
science19 | You might as well start at the 'lowest': Integers are all the whole numbers, Natural numbers (N) are the set of positive integers, {1, 2, ...} (0 optional), Rational numbers (Q) are any number that can be represented a/b with a and b being Integers (|b| < 0. Real numbers are all the Rational numbers and all the others. Cardinality is the number of elements in a set. An Ordinal is a well ordered set. | 0.531828 | 99 |
science20 | What does it mean when you refer to $.99999\ldots$? Symbols don't mean anything in particular until you've defined what you mean by them. In this case the definition is that you are 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 choose your number is bigger than $10^{-k}$ for some $k$. So I can just pick my point to be the $k$th spot in the sequence. A more intuitive way of explaining the above argument is that the reason $.99999\ldots = 1$ is that their difference is zero. So let's subtract $1.0000\ldots -.99999\ldots = .00000\ldots = 0$. That is, $1.0 -.9 = .1$ $1.00-.99 = .01$ $1.000-.999=.001$, $\ldots$ $1.000\ldots -.99999\ldots = .000\ldots = 0$ | 0.634401 | 0 |
science25 | Suppose this was not the case, i.e. $0.9999... \neq 1$. Then $0.9999... < 1$ (I hope we agree on that). But between two distinct real numbers, there's always another one in between, say $x=\frac{0.9999... +1}{2}$, hence $0.9999... < x < 1$. The decimal representation of $x$ must have a digit somewhere that is not $9$ (otherwise $x = 0.9999...$). But that means it's actually smaller, $x < 0.9999...$, contradicting the definition of $x$. Thus, the assumption that there's a number between $0.9999...$ and $1$ is false, hence they're equal. | 0.322959 | 0 |
science26 | I've heard of it being as a rough check to see if accounting numbers were being made up | 0.361789 | 99 |
science27 | Assuming this is relative to the origin (as John pointed out): Given two position vectors $\vec p_1$ and $\vec p_2$, their dot product is: $$\vec p_1\cdot \vec p_2 = |\vec p_1| \cdot |\vec p_2| \cdot \cos \theta$$ Solving for $\theta$, we get: $$\theta = \arccos\left(\frac{\vec p_1 \cdot \vec p_2}{|\vec p_1| \cdot |\vec p_2|}\right)$$ In a 2D space this equals: $$v = \arccos\left(\frac{x_1x_2 + y_1y_2}{\sqrt{(x_1^2+y_1^2) \cdot (x_2^2+y_2^2)}}\right)$$ And extended for 3D space: $$v = \arccos\left(\frac{x_1x_2 + y_1y_2 + z_1z_2}{\sqrt{(x_1^2+y_1^2+z_1^2) \cdot (x_2^2+y_2^2+z_2^2)}}\right)$$ | 0.228263 | 99 |
science28 | There are several important kinds of relations, each of which satisfy a different collection of properties: Equivalence relations: These are reflexive, symmetric, and transitive. Essentially they're relations that "behave like equality." The most important elementary one is "equivalence modulo m," where say 1 = 6 = 11 modulo 5. Partial orderings: These are reflexive, transitive, and (anti-symmetric or maybe asymmetric, I'm having trouble parsing your logical statements). Essentially they're relations that "behave like less than or equal to." An important elementary example is "divides" where we say a|b if the ratio b/a is an integer. Note that 2|6 but 6 does not divide 2. However if 2|4 and 4|8 certainly 2|8. | 0.293714 | 99 |
science30 | 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. | 0.092105 | 0 |
science31 | Forensic accountancy is a popular use, and is actually admissible as evidence in the USA. | 0.433701 | 99 |
science32 | Here is a super nice powerpoint on the subject! http://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf | 0.430863 | 99 |
science33 | An inner product space is a vector space for which the inner product is defined. The inner product is also known as the 'dot product' for 2D or 3D Euclidean space. An arbitrary number of inner products can be defined according to three rules, though most are a lot less intuitive/practical than the Euclidean (dot) product. Side note: It may seem slightly esoteric, but as a physicist the obvious application of inner product spaces are Hilbert spaces used in quantum mechanics. The inner product of an eigenfunction with a wavefunction in Hilbert space gives the corresponding eigenvalue. | 0.493685 | 99 |
science34 | Asymmetric means simply "not symmetric". So in the binary case, it is NOT the case that if a is related to b, b is related to a. Antisymmetric means that if a is related to b, and b is related to a, a = b. To explain your third example: "is older than" is asymmetric because if Alice is older than Bob, Bob is NOT older than Alice. "is older than" is antisymmetric since if Alice is older than Bob, and Bob is older than Alice, Alice must be Bob because someone must be older (and if this is not the case, Alice simply has two names..). "is older than" is transitive since if Alice is older than Bob, and Bob is older than Charlie, Alice is also older than Charlie. So asymmetric and antisymmetric don't cancel out because the first means it's sort of a one-way relation, whereas the second means, loosely, that if it you reverse the operands and both statements are true, the operands must be the same. | 0.42583 | 99 |
science35 | I believe they are used in quantum physics as well, because rotation with quaternions models Spinors extremely well (due to the lovely property that you need to rotate a point in quaternionic space around 2 full revolutions to get back to your 'original', which is exactly what happens with spin-1/2 particles). They are also, as you said, used in computer graphics a lot for several reasons: they are much more space efficient to store than rotation matrices (4 floats rather than 16) They are much easier to interpolate than euler angle rotations (spherical interpolation or normalised liner interpolation) They avoid gimbal lock It's much cooler to say that your rotation is described as 'a great circle on the surface of a unit 4 dimensional hypersphere' :) I think there are other uses, but a lot of them have been superseded by more general Vectors. | 0.312261 | 99 |
science36 | The natural numbers can be defined by Peano's Axioms (sometimes called the Peano Postulates): Zero is a number. If n is a number, the successor of n is a number. zero is not the successor of a number. Two numbers of which the successors are equal are themselves equal. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S. (This definition includes 0 in the natural numbers; altering rules 1, 3, and 5 to refer to one instead of zero excludes 0 from the natural numbers. Whether or not 0 is a natural number varies in various texts.) The whole numbers are the natural numbers with the additive identity element called 0. The integers are the whole numbers and their additive inverses. The rational numbers are numbers that can be expressed as a ratio of an integer to a non-zero integer. The real numbers are the set of numbers that are limits of Cauchy sequences of rational numbers. The irrational numbers are the real numbers that are not rational numbers. The complex numbers are the numbers that can be expressed as a + b * i where a and b are real numbers and i behaves like a real number under addition/multiplication/distribution/etc., with the added rule that i2 = -1. The imaginary numbers are sometimes defined to be the "pure imaginary" numbers--complex numbers for which the "real part" a = 0, sometimes with the added restriction that b is not zero--and are sometimes defined to be the non-real complex numbers. The algebraic numbers are numbers that are solutions to polynomial equations with integer coefficients. The transcendental numbers are complex numbers (sometimes limited to real numbers) that are not algebraic. | 0.426351 | 99 |
science39 | The dot product can be used to find the length of a vector or the angle between two vectors. The cross product is used to find a vector which is perpendicular to the plane spanned by two vectors. | 0.501837 | 1 |
science41 | As for the utility of inner product spaces: They're vector spaces where notions like the length of a vector and the angle between two vectors are available. In this way, they generalize $\mathbb R^n$ but preserve some of its additional structure that comes on top of it being a vector space. Familiar friends like Cauchy-Schwarz, the parallelogram rule, and orthogonality all work in inner product spaces. (Note that there is a more general class of spaces, normed spaces, where notions of length make sense always, but an inner product cannot necessarily be defined.) The dot product is the standard inner product on $\mathbb R^n$. In general, any symmetric, positive definite matrix will give you an inner product on $\mathbb C^n$. And you can have inner products on infinite dimensional vector spaces, like $$ \langle \, f, \, g \, \rangle = \int_a^b \ f(x)\overline{g(x)} \, dx$$ for $f, g$ square-integrable functions on $[a,b]$. This becomes useful, for example, in applications like Fourier series where you want a basis of orthonormal functions for some function space (it's not just the trigonometric functions that work). | 0.115618 | 99 |
science42 | \begin{align} x &= 0.999... \\ 10x &= 9.999... \\ &= 9 + 0.999... \\ &= 9 + x \\ 10x - x &= (9 + x) - x \\ (10 - 1)x &= 9 + (x - x) \\ 9x &= 9 \\ x &= 1 \end{align} | 0.317285 | 0 |
science43 | Graph theory! It's essentially connecting the dots, but with theorems working wonders behind the scenes for when they're old enough. Simple exercises like asking how many colors you need to color the faces or vertices of a graph are often fun (so I hear). (Also, most people won't believe the 4-color theorem.) | 0.414826 | 3 |
science45 | You're right to think that the definitions are very similar. The main difference between groups and rings is that rings have two binary operations (usually called addition and multiplication) instead of just one binary operation. If you forget about multiplication, then a ring becomes a group with respect to addition (the identity is 0 and inverses are negatives). This group is always commutative! If you forget about addition, then a ring does not become a group with respect to multiplication. The binary operation of multiplication is associative and it does have an identity 1, but some elements like 0 do not have inverses. (This structure is called a monoid.) A commutative ring is a field when all nonzero elements have multiplicative inverses. In this case, if you forget about addition and remove 0, the remaining elements do form a group under multiplication. This group is again commutative. A division ring is a (not necessarily commutative) ring in which all nonzero elements have multiplicative inverses. Again, if you forget about addition and remove 0, the remaining elements do form a group under multiplication. This group is not necessarily commutative. An example of a division ring which is not a field are the quaternions. | 0.250455 | 1 |
science46 | Two good general references: Wikipedia MathWorld | 0.483034 | 99 |
science48 | Khan Academy, http://www.khanacademy.org/ You'll find tons of explanatory videos from various branches of mathematics; plus, each subject is explained pretty good, and the videos are easy to follow | 0.519485 | 99 |
science50 | Not always pure math, but I think John Baez' This Week in Mathematical Physics contains a lot of really interesting math reads. I should add Terry Tao's What's new. It's a very active math blog (both in posts and comments) and definitely covers some cutting edge math, even if it can be way over my head. | 0.120629 | 0 |
science53 | This really depend on how smart the kid is. I lean toward discrete math, elementary number theory related topics when talking to non-math people about math. They requires little background knowledge. There are some fun problems in discrete math, especially combinatorics. Simple probability is also nice. So are logic problem. Both topics can be used to formulate some simple puzzles. A simple number theory puzzle How many zeros are there in 20! I assume a bright 10 year old can solve it. | 0.545068 | 3 |
science54 | One place they are frequently used is in computer games when you want to smoothly transition from one rotation to another. An artist might have said "at this time I want the head oriented like this and at that time I want it like this". The computer needs to work out what happens in-between these poses. It's quite easy to find in-between poses using quaternions. If the two poses are reasonably similar, then you can get a half-way orientation simply by taking the average of the quaternions. You can find out more here. | 0.342764 | 99 |
science55 | First you are doing it in the wrong end, second, the statement in general is still not correct. for example: 9999999999 1000000001 Say if you want the first 2 digits, you will get 10 if you discard the last 2 digit and do the sum. The right answer is 11 | 0.304121 | 99 |
science56 | If you were supposed to find the last ten digits, you could just ignore the first 40 digits of each number. However you're supposed to find the first ten digits, so that doesn't work. And you can't just ignore the last digits of each number either because those can carry over. | 0.417022 | 99 |
science59 | The field is the smaller/left circle, centered at A. The cow is tied to the post at E. The larger/right circle is the grazing radius. Let the radius of the field be R and the length of the rope be L. The grazable area is the union of a segment of the circular field and a segment of the circle defined by the rope length. (A segment of a circle is a sector of a circle less the triangle defined by the center of the circle and the endpoints of the arc.) The area of a segment of a circle of radius $R$ with central angle $t$ is $\frac{1}{2}R^2(t-\sin(t))$, where $t$ is measured in radians. In order to express the grazable area in terms of $R$ and one angle, we consider the angles ∠CED and ∠CAD (which define the segments of the circles; call these α and β for convenience) and the triangle CEF. Let $\theta$ be ∠EFC. $2\theta$ is an inscribed angle for the central angle $\beta$ over the same arc, making $\beta = 4\theta$. The sum of angles in triangle CEF is $\theta + \pi/2 +\alpha/2=\pi$ or $\alpha =\pi-2\theta$. The grazable area is $\frac{1}{2}L^2(\alpha-\sin\alpha)+\frac{1}{2}R^2(\beta-\sin\beta)=R^2(\frac{1}{2}(L/R)^2((\pi-2\theta)-\sin(\pi-2\theta))+\frac{1}{2}(4\theta-\sin(4\theta)))$, where $a = CE = L/R=2\sin(\theta)$. We want that to be equal to half the area of the field, $\frac{1}{2}\pi R^2$. That is, the equality of areas is $$R^2(2(\sin(\theta))^2((\pi-2\theta)-\sin(\pi-2\theta))+\frac{1}{2}(4\theta-\sin(4\theta)))=R^2\frac{\pi}{2}$$ Simplifying: $$R^2(\pi+(2\theta-\pi)\cos(2\theta)-\sin(2\theta)=\frac{\pi}{2})$$ (The grazable area seems to be $\pi+\alpha\cos\alpha-\sin\alpha$; can this be seen easily?) The desired equality of areas is obtained for $\theta = \text{ca. } 0.618$ or $L=\text{ca. }1.159 R$ . | 0.510422 | 99 |
science61 | During one math class I had the same problem. the professor told us the answer can be a arbitrary integer, depend on how you remove the balls. Since you said every ball you put in was eventually removed, I will make it simple and assume you first number the balls to natural numbers, and then add and remove the balls with the smallest numbers first. In this case, the bin will be empty--every ball put in have been removed. If you define the sequence in the way such that a_n = amount of balls when you do the nth move, then this sequence never get to zero, because there is no n correspond to 11 o'clock. The increasing sequence doesn't define how many balls in the bin at 11. Therefore there is no paradox. | 0.585937 | 0 |
science65 | The following reddit post has a decent list of math resources: http://www.reddit.com/r/math/comments/bqbex/lets_list_all_the_useful_free_online_math/ One site I did not see it their list that I've found very helpful: http://betterexplained.com/ | 0.083195 | 99 |
science67 | If you take a rope, fix the two ends, and let it hang under the force of gravity, it will naturally form a hyperbolic cosine curve. | 0.243666 | 99 |
science68 | 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". | 0.194223 | 99 |
science70 | Calculus is a field which deals with two seemingly unrelated things. (1) the area beneath a graph and the x-axis. (2) the slope (or gradient) of a curve at different points. Part (1) is also called 'integration' and 'anti-differentiation', and part (2) is called 'differentiation'. | 0.095713 | 99 |
science74 | I listen to Math Mutation Podcast. The topics are interesting and understandable by a layman. | 0.016129 | 2 |
science77 | For non-zero bases and exponents, the relation $ x^a x^b = x^{a+b} $ holds. For this to make sense with an exponent of $ 0 $, $ x^0 $ needs to equal one. This gives you: $\displaystyle x^a \cdot 1 = x^a\cdot x^0 = x^{a+0} = x^a $ When the base is also zero, it's not possible to define a value for $0^0$ because there is no value that is consistent with all the necessary constraints. For example, $0^x = 0$ and $x^0 = 1$ for all positive $x$, and $0^0$ can't be consistent with both of these. Another way to see that $0^0$ can't have a reasonable definition is to look at the graph of $f(x,y) = x^y$ which is discontinuous around $(0,0)$. No chosen value for $0^0$ will avoid this discontinuity. | 0.15896 | 99 |
science78 | $$0^x = 0, \quad x^0=1$$ both are true when $x>0$. What happens when $x=0$? It is 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 $x^0$. | 0.153071 | 99 |
science80 | A nice thing to notice. It's basically because $(n + 1) ^ 2 - n ^ 2 = 2n + 1$ | 0.318766 | 99 |
science82 | Take a pair of compasses and draw an arc between two opposite corners, centred at another corner; then draw a diagonal that bisects the arc. If you now draw two lines from the point of intersection, parallel to the sides of the square, the biggest of the resulting squares will have half the area of the original square. Here's an illustration: | 0.554383 | 99 |
science83 | Andrea Feretti's MathOnline page. | 0.388951 | 99 |
science86 | It's easy to find a function which is continuous but not differentiable at a single point, e.g. $f(x) = |x|$ is continuous but not differentiable at $0$. Moreover, there are functions which are continuous but nowhere differentiable, such as the Weierstrass function. On the other hand, continuity follows from differentiability, so there are no differentiable functions which aren't also continuous. If a function is differentiable at $x$, then the limit $(f(x+h)-f(x))/h$ must exist (and be finite) as $h$ tends to 0, which means $f(x+h)$ must tend to $f(x)$ as $h$ tends to 0, which means $f$ is continuous at $x$. | 0.357398 | 99 |
science87 | 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. | 0.043591 | 99 |
science88 | Quantum mechanics, and hence physics and everything around us, fundamentally involves complex numbers. | 0.304768 | 1 |
science89 | 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? | 0.398186 | 1 |
science92 | It is a theorem of Liouville, reproven later with purely algebraic methods, that for rational functions $f$ and $g$, $g$ non-constant, the antiderivative of $$f(x)\exp(g(x)) \, \mathrm dx$$ can be expressed in terms of elementary functions if and only if there exists some rational function $h$ such that it is a solution of $$f = h' + hg'$$ $e^{x^2}$ is another classic example of such a function with no elementary antiderivative. I don't know how much math you've had, but some of this paper might be comprehensible in its broad strokes: https://ksda.ccny.cuny.edu/PostedPapers/liouv06.pdf Liouville's original paper: Liouville, J. "Suite du Mémoire sur la classification des Transcendantes, et sur l'impossibilité d'exprimer les racines de certaines équations en fonction finie explicite des coefficients." J. Math. Pure Appl. 3, 523-546, 1838. Michael Spivak's book on Calculus also has a section with a discussion of this. | 0.355915 | 3 |
science96 | This was first "guessed at" by Archimedes, by what fraction of the volume of a cylinder a sphere is. That is, a sphere that is contained within that cylinder. From that, he figured out $4/3$. You can do it these days with the tools of calculus. One way would to use the [Disk Method], over the graph of a semicircle. Another would be to use spherical coordinates and calculate the 3D Integral. All of these methods should give you $\frac{4}{3} \pi r^3$ Note that it's not just $r^3$ because it's "3D"; but also because it's the only way the dimensional analysis would work; that is, the units would have to be in $\text{distance}^3$. But that might be what you said, but just clarifying. | 0.151127 | 1 |
science97 | It's important to understand that this is not something that can be proved: it's a definition. We choose not to regard 1 as a prime number, simply because it makes writing lots of theorems much easier. Noah gives the best example in his answer: Euclid's theorem that every positive integer can be written uniquely as a product of primes. If 1 is defined to be a prime number, then we'd have to change that theorem to: "every positive integer can be written uniquely as a product of primes, except for infinite multiplications by 1". So we choose to go with the easier path of defining 1 to not be a prime. | 0.398876 | 0 |
science98 | Let $|S|$ be the cardinality of $S$. We know that $|S| < |2^S|$, which can be proven with generalized Cantor's diagonal argument. Theorem The set of all sets does not exist. Proof Let $S$ be the set of all sets, then $|S| < |2^S|$. But $2^S$ is a subset of $S$. Therefore $|2^S| \leq |S|$. A contradiction. Therefore the set of all sets does not exist. | 0.240856 | 3 |
science99 | You may be interested to read the MathOverflow question "Demystifying Complex Numbers," here. A teacher is asking how to motivate complex numbers to students taking complex analysis. | 0.343456 | 1 |
science100 | Pappus's centroid theorem (second theorem) says that the volume of a solid formed by revolving a region about an axis is the product of the area of the region and the distance traveled by the centroid of the region when it is revolved. A sphere can be formed by revolving a semicircle about is diameter edge. The area of the semicircle is $\frac{1}{2}\pi r^2$. The centroid of the semicircle can be found by intersecting two lines that both divide the area of the semicircle into two equal parts. One such line is perpendicular to the diameter edge through the center of the semicircle (this is a line of symmetry of the semicircle). Another such line is parallel to the diameter edge, $\frac{4r}{3\pi}$ away from it (verification of this is left as an exercise for the reader). When revolved about the diameter edge of the semicircle, the centroid travels $2\pi\cdot\frac{4r}{3\pi} = \frac{8}{3}\cdot r$, so the volume of the sphere is $\frac{1}{2}\pi r^2\cdot\frac{8}{3}\cdot r = \frac{4}{3}\pi r^3$. | 0.513128 | 1 |
science102 | Does this give you any ideas? | 0.105908 | 99 |
science103 | 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 numbers. 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. Ultimately, they are an idea and while ideas don't exist physically, saying they don't exist at all is inaccurate. | 0.130895 | 1 |
science104 | An informal explanation is Russel's Paradox. The wiki page is informative, here's the relevant quote: Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all squares. That set is not itself a square, and therefore is not a member of the set of all squares. So it is "normal". On the other hand, if we take the complementary set that contains all non-squares, that set is itself not a square and so should be one of its own members. It is "abnormal". Now we consider the set of all normal sets, R. Attempting to determine whether R is normal or abnormal is impossible: If R were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if it were abnormal, it would not be contained in the set of normal sets (itself), and therefore be normal. This leads to the conclusion that R is both normal and abnormal: Russell's paradox. | 0.321981 | 3 |
science107 | Dan's algebraic justification is correct, but you may get more intuition about why this is happening from the above picture. Each time you want to enlarge the square by one unit, you have to add an extra row, an extra column, and one more square to fill in the corner. These correspond directly to the $n+n+1=2n+1$ that Dan mentioned. And of course, $2n+1$ is how odd numbers look. Looking at it from the other direction, you can use the same idea to convince yourself of Noah's claim that $1+3+5+\dots+(2n-1) = n^2$. Imagine the left-hand side as representing the upper sequence of green and orange squares. You add on first 1 tile, then 3, then 5, and so on to construct larger and larger squares. At each step, you are adding one row (n) and one column (another n), then removing that one tile in the top right corner where they overlap (for a total of 2n-1). | 0.553257 | 99 |
science109 | No number does "really exist" the way trees or atoms exist. In physics people however have found use for complex numbers just as they have found use for real numbers. | 0.384838 | 1 |
science110 | If you take two real numbers x and y then there per definition of the real number z for which x < z < y or x > z > y is true. For x = 0.99999... and y = 1 you can't find a z and therefore 0.99999... = 1. | 0.316788 | 0 |
science111 | You are changing your basis vectors, call your new ones $i$, $j$, and $k$ where $i$ is defined from $a-p$, $j$ from $b-p$, and $k$ the cross product. Now recall that your basis vectors should be unit, so take the length of your three vectors and divide the vectors by their length, making $i$, $j$, and $k$ unit. Now $a = d(a,p)*i$, $b = d(b,p)*j$. | 0.354265 | 99 |
science112 | What you are describing is an Affine Transformation, which is a linear transformation followed by a translation. We know this because any straight lines in your original vector space is also going to be a straight line in your transformed vector space. | 0.171082 | 99 |
science114 | From the Wikipedia article about the prime number theorem: Roughly speaking, the prime number theorem states that if a random number nearby some large number N is selected, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N. For example, near N = 10,000, about one in nine numbers is prime, whereas near N = 1,000,000,000, only one in every 21 numbers is prime. In other words, the average gap between prime numbers near N is roughly ln(N). | 0.338671 | 99 |
science115 | The "set of all sets" is not so much a paradox in itself as something that inevitably leads to a contradiction, namely the well-known (and referenced in the question) Russell's paradox. Given any set and a predicate applying to sets, the set of all things satisfying the predicate should be a subset of the original set. If the "set of all sets" were to exist, because self-containment and non-self-containment are valid predicates, the set of all sets not containing themselves would have to exist as a set in order for our set theory to be consistent. But this "set of all sets" cannot exist in a consistent set theory because of the Russel paradox. So the non-existence of the "set of all sets" is a consequence of the fact that presuming it's existence would lead to the contradiction described by Russel's paradox. This is in fact the origin of Russel's paradox. In his work "The Basic Laws of Arithmetic", Gottlob Frege had taken as a postulate the existence of this "set of all sets". In a letter to Frege, Bertrand Russell essentially blew away the basis of Frege's entire work by describing the paradox and proving that this postulate could not be a part of a consistent set theory. | 0.55237 | 3 |
science117 | The distinction between constructive mathematics and traditional mathematics has nothing to do with Russell's Paradox. Constructive mathematics simply requires working with one less basic postulate that many mathematicians have believed to be sensible and on which some proofs are based, namely the Axiom of Choice | 0.521533 | 99 |
science118 | Personally, I always use Python's console. It has history and allows all kinds of math operations. It is available for Linux, Windows, Mac, ChromeOS, Android, and others. | 0.002688 | 99 |
science121 | Euclid's famous proof is as follows: Suppose there is a finite number of primes. Let $x$ be the product of all of these primes. Then look at $x+1$. It is clear that $x$ is coprime to $x+1$. Therefore, no nontrivial factor of $x$ is a factor of $x+1$, but every prime is a factor of $x$. By the fundamental theorem of arithmetic, $x+1$ admits a prime factorization, and by the above remark, none of these prime factors can be a factor of $x$, but $x$ is the product of all primes. This is a contradiction. | 0.207636 | 99 |
science122 | According to XKCD, we have the following Haiku: Top Prime's Divisors' Product (Plus one)'s factors are...? Q.E.D B@%&$ I wonder if we can edit it to make it correct | 0.292489 | 99 |
science123 | The existence of universal set is incompatible with the Zermelo–Fraenkel axioms of set theory. However, there are alternative set theories which admit a universal set. One such theory is Quine's New Foundations. | 0.52001 | 3 |
science126 | Brian Conrad explains this in the following: Impossibility theorems on integration in elementary terms (archived PDF) | 0.257542 | 3 |
science129 | Though I'm sure it's not unpopular, I don't think many people learn it early: Group Theory. It's a real nice area with a lot of cool math and some neat applications (like cryptography). | 0.39437 | 2 |
science131 | Theory of computation, information theory and logic/foundation of mathematics are very interesting topics. I wish I knew them earlier. They are not unpopular(almost every university have a bunch of ToC people in CS depatment...) , but many math major I know have never touched them. They show you the limits of mathematics, computation and communication. Logic shows there are things can't be proved from a set of axioms even if it's true--Godel's incompleteness theorem. There are other interesting theorems in foundation of mathematics. Like the independence of continuum hypothesis to ZFC. Theory of computation showed me things that's not computable. Problems that takes exponential time, exponential space, no matter what kind of algorithm you come up with. Information theory proves the minimum amount of information required to reconstruct some other information. It pops up in unexpected places. There is a proof of there are infinite number of primes by information theory (Sorry I can't find it, I can only tell you it exists. I might find it later). | 0.161069 | 2 |
science135 | That picture confuses things by making it look as though the red line is being "unwound" from the circle like paper towel being unwound from a roll. Our brains pick up on that, since it is a real-world example. Both circles complete a single revolution, and both travel the same distance from left to right. If these really were rolls of paper towel, the smaller roll would have to spin faster (and therefore complete more than one full revolution) in order to lay out the same length of paper towel as the larger roll. Alternatively, if the two rolls were spinning at the same rate, the free end of the strip of towel left behind by the smaller roll would also move to the right. In short, the image is a kind of optical/mental illusion, and you're not going crazy :) | 0.079366 | 99 |
science136 | I don't know how I missed that one, indeed: http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables Thanks Kaestur Hakarl! | 0.428347 | 99 |
science137 | On a map using the Mercator projection, the relationship between the latitude L of a point and its y coordinate on the map is given by $y = \operatorname{arctanh}(\sin(L))$, where $\operatorname{arctanh}$ is the inverse of the hyperbolic tangent function. | 0.204543 | 99 |
science138 | To the degree that anything actually "exists" in math, yes complex number exist. Once you accept that groups, rings and fields exist, and that isomorphism of rings makes sense, complex numbers can be recognized as (isomorphic to) the subring (which happens to be a field) of the ring of $2 \times 2$ real matrices. Generators of this subring are the following matrices $ \begin{vmatrix} 1 & 0 \\ 0 & 1 \\ \end{vmatrix} \;$ and $\; \begin{vmatrix} 0 & -1 \\ 1 & 0 \\ \end{vmatrix} $ which correspond to $1$ and $i$ in the normal notation of complex numbers. As people tend to accept that matrices exist, this may be a convincing argument. | 0.450636 | 1 |
science139 | I will assume the polygon has no intersections between the edges except at corners. Call the point $(x_0, y_0)$. First we determine whether we are on a line - this is simple using substitution and range checking. For the range checking, suppose we have a segment $(x_1, y_1)$, $(x_2, y_2)$. We check that $x_1\leq x_0\leq x_2$ or $x_2\leq x_0\leq x_1$ and do the same for $y$. Now, if we aren't on a line, we draw a ray from the point and count how many lines you intersect. If the line doesn't intersect a corner or contain a non-degenerate segment of an edge, then an odd number of intersections should mean you are inside and an even number mean outside. If we intersect with a corner, it is more difficult. Clearly, the ray could leave with it only intersecting a corner, but it could also pass through the corner without entering the polygon. One solution is to pick the ray so that it doesn't intersect a corner or go along a line - this should always be possible. We can also use cross product to determine whether we are crossing the lines at their corner or not (just check if the adjacent vertices are on the same side of the line or not). | 0.547764 | 99 |
science140 | When I was that age, I discovered Raymond Smullyan's classic logic puzzle books in the library (such as What is the name of this book?), and really got into it. I remember my amazement when I first understood how a complicated logic puzzle could become trivial, just symbolic manipulation really, with the right notation. | 0.093327 | 3 |
science141 | Nearly always the direct proof is easier to understand, shorter, and more helpful! | 0.296861 | 3 |
science144 | Speaking from my own experience with elementary mathematics, yours is a very hard question to answer because there is little in the elementary math curriculum worth getting excited about. Before students are going to get excited about math, the math curriculum has to be changed so that the process of doing mathematics is made to be engaging and thought provoking - not tedious. I think every teacher of elementary mathematics needs to watch this video about how math education can and should be reformed. | 0.457412 | 3 |
science147 | In the sciences (as opposed to in mathematics) people are often a bit vague about exactly what assumptions they are making about how "well-behaved" things are. The reason for this is that ultimately these theories are made to be put to the test, so why bother worrying about exactly which properties you're assuming when what you care about is functions coming up in real life which are probably going to satisfy all of your assumptions. This is particularly ubiquitous in physics where it is extremely common to make heuristic assumptions about well-behavior. Even in mathematics we do this sometimes. When people say something is true for n sufficiently large, they often won't bother writing down exactly how large is sufficiently large as long as it's clear from context how to work it out. Similarly, in an economics paper you could read through the argument and figure out exactly what assumptions they need, but it makes it easier to read to just say "well-behaved." | 0.048579 | 99 |
science150 | All of three dimensional space can be filled up with an infinite curve. | 0.165938 | 0 |
science152 | The two envelopes problem is a good one. See also: Card doubling paradox and: https://mathoverflow.net/questions/9037 | 0.286537 | 99 |
science153 | The Monty Hall problem fits the bill pretty well. Almost everyone, including most mathematicians, answered it wrong on their first try, and some took a lot of convincing before they agreed with the correct answer. It's also very easy to explain it to people. | 0.30647 | 0 |
science155 | There are true statements in arithmetic which are unprovable. Even more remarkably there are explicit polynomial equations where it's unprovable whether or not they have integer solutions with ZFC! (We need ZFC + consistency of ZFC) | 0.111392 | 0 |
science159 | Polya's "How To Solve It" | 0.440328 | 0 |
science160 | Every simple closed curve that you can draw by hand will pass through the corners of some square. The question was asked by Toeplitz in 1911, and has only been partially answered in 1989 by Stromquist. As of now, the answer is only known to be positive, for the curves that can be drawn by hand. (i.e. the curves that are piecewise the graph of a continuous function) I find the result beyond my intuition. For details, see http://www.webpages.uidaho.edu/~markn/squares/ (the figure is also borrowed from this site) | 0.438214 | 0 |
science162 | This is an extremely broad question, especially given the wide variety of mathy people here, but I'll bite. HSM Coxeter's Introduction to Geometry is a book that was very important to the development of my interest in mathematics and inclination towards its geometric aspects. | 0.565642 | 0 |
science163 | Nicolas Bourbaki's Éléments de mathématique (specifically Topologie Générale and Algèbre). | 0.084904 | 0 |
science166 | Journey Through Genius A brilliant combination of interesting storytelling and large amounts of actual Mathematics. It took my love of Maths to a whole other level. | 0.337066 | 0 |
science169 | Euclid's Elements Newton's Principia Mathematica Ideally in the original languages of Ancient Greek and Latin respectively! No, just kidding. But they are true classics that any accomplished mathematician should read at some point during their career. Not because they'll teach you something you don't already know, but they provide a unique insight into the mind of these giants. | 0.574064 | 0 |
science171 | Purplemath has a list for math lessons and tutoring. It's a list with various links and short reviews referring to tutoring and instructional resources. | 0.079149 | 99 |
science175 | John D Cook writes The Endeavor One of the MathWorks blogs: Loren on the Art of Matlab ... a few more: eon Peter Cameron's Blog Walking Randomly Todd and Vishal's Blog (Check their blogrolls for more) | 0.128631 | 0 |
science176 | No, this is not a proof of your statement. If you look very closely, you will see that the hypotenuse of the triangles aren't straight. You can also verify this algebraically by calculating the internal angles of the triangle using trigonometry. The other way to demonstrate the non-straightness of the hypotenuse is in the fact that the red and blue triangles aren't similar, which can be seen by the difference in the ratios of their width and height. For the blue triangle, this is 2/5 and for the red triangle this is 3/8. As they are both right angled triangles, these ratios would be the same for similar triangles and the fact that they aren't means the interior angles are different. | 0.08178 | 0 |
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