title string | question_body string | answer_body string | tags string | accepted int64 |
|---|---|---|---|---|
List of interesting math podcasts? | mathfactor is one I listen to. Does anyone else have a recommendation? | More or Less is a BBC Radio 4 programme about maths and statistics in the news, and there is a free podcast. It's presented by Tim Harford , the Undercover Economist from the Financial Times . | |soft-question|big-list|online-resources| | 0 |
How can you prove that the square root of two is irrational? | 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? | You use a proof by contradiction. Basically, you suppose that $\sqrt{2}$ can be written as $\dfrac{p}{q}$ . 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. Therefore, $2q^2$ has an odd number of factors of $2$ , which means it can't be equal to $p^2$ . | |elementary-number-theory|radicals|rationality-testing| | 1 |
What Does it Really Mean to Have Different Kinds of Infinities? | Can someone explain to me how there can be different kinds of infinities? I was reading " The man who loved only numbers " by Paul Hoffman and came across the concept of countable and uncountable infinities, but they're only words to me. Any help would be appreciated. | 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 langu... | |elementary-set-theory|intuition|infinity|faq| | 1 |
What Does it Really Mean to Have Different Kinds of Infinities? | Can someone explain to me how there can be different kinds of infinities? I was reading " The man who loved only numbers " by Paul Hoffman and came across the concept of countable and uncountable infinities, but they're only words to me. Any help would be appreciated. | 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 ... | |elementary-set-theory|intuition|infinity|faq| | 0 |
What is your favorite online graphing tool? | I'm looking for a nice, quick online graphing tool. The ability to link to, or embed the output would be handy, too. | 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. | |soft-question|math-software| | 0 |
How are we able to calculate specific numbers in the Fibonacci Sequence? | I was reading up on the Fibonacci Sequence, $1,1,2,3,5,8,13,\ldots $ when I 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 p... | Wikipedia has a closed-form function called " Binet's formula ". $$F\left(n\right) = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}$$ This is based on the Golden Ratio. | |combinatorics|generating-functions|fibonacci-numbers| | 1 |
How are we able to calculate specific numbers in the Fibonacci Sequence? | I was reading up on the Fibonacci Sequence, $1,1,2,3,5,8,13,\ldots $ when I 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 p... | The closed form calculation for Fibonacci sequences is known as Binet's Formula . | |combinatorics|generating-functions|fibonacci-numbers| | 0 |
How are we able to calculate specific numbers in the Fibonacci Sequence? | I was reading up on the Fibonacci Sequence, $1,1,2,3,5,8,13,\ldots $ when I 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 p... | 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 ) | |combinatorics|generating-functions|fibonacci-numbers| | 0 |
Is it true that $0.999999999\ldots=1$? | I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is? | 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. | |real-analysis|algebra-precalculus|real-numbers|decimal-expansion| | 0 |
Is it true that $0.999999999\ldots=1$? | I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is? | Indeed this is true. The underlying reason is that decimal numbers are not unique representations of the reals. (Technically, there does exist a bijection between the set of all decimal numbers and the reals, but it is not the natural/obvious one.) Here's a very simple proof: $$\begin{align} \frac13&=0.333\ldots&\hbox{... | |real-analysis|algebra-precalculus|real-numbers|decimal-expansion| | 0 |
What Does it Really Mean to Have Different Kinds of Infinities? | Can someone explain to me how there can be different kinds of infinities? I was reading " The man who loved only numbers " by Paul Hoffman and came across the concept of countable and uncountable infinities, but they're only words to me. Any help would be appreciated. | Hilbert's Hotel is a classic demonstration. | |elementary-set-theory|intuition|infinity|faq| | 0 |
Can you recommend a decent online or software calculator? | I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator. Can you recommend any? | Lots of people like to use InstaCalc which lets you do unit conversions and store intermediate calculations in variables. | |soft-question|big-list|math-software|computer-algebra-systems| | 0 |
What is a real number (also rational, decimal, integer, natural, cardinal, ordinal...)? | In mathematics, there seem to be a lot of different types of numbers. What exactly are: Real numbers Integers Rational numbers Decimals Complex numbers Natural numbers Cardinals Ordinals And as workmad3 points out, some more advanced types of numbers (I'd never heard of) Hyper-reals Quaternions Imaginary numbers Are th... | Natural numbers The "counting" numbers. (That is, all integers, that are one or greater). Whole numbers The Natural numbers, and zero. Integers The Whole numbers, and the negatives of the Natural numbers. Rational numbers Any number that may be expressed by any integer A divided by any integer B, where B is not zero. I... | |terminology|definition|number-systems| | 1 |
Is it true that $0.999999999\ldots=1$? | I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is? | .999... = 1 because .999... is a concise symbolic representation of "the limit of some variable as it approaches one." Therefore, .999... = 1 for the same reason the limit of x as x approaches 1 equals 1. | |real-analysis|algebra-precalculus|real-numbers|decimal-expansion| | 0 |
Why is the matrix-defined Cross Product of two 3D vectors always orthogonal? | By matrix-defined, I mean $$\left \times\left = \left| \begin{array}{ccc} i & j & k\\ a & b & c\\ d & e & f \end{array} \right|$$ ...instead of the definition of the product of the magnitudes multiplied by the sign of their angle, in the direction orthogonal) If I try cross producting two vectors with no $k$ component,... | The obvious but slightly trite answer is "because that's just how the cross-product works as an operation". If you're looking for an intuitive reason, the cross-product by definition produces a vector that is orthogonal to the two operand (input) vectors. You know that the base vectors $\mathbf{i}$, $\mathbf{j}$, and $... | |linear-algebra|matrices|inner-products|orthogonality|cross-product| | 0 |
Is it true that $0.999999999\ldots=1$? | I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is? | You can visualise it by thinking about it in infinitesimals. The more $9's$ you have on the end of $0.999$, the closer you get to $1$. When you add an infinite number of $9's$ to the decimal expansion, you are infinitely close to $1$ (or an infinitesimal distance away). And this isn't a rigorous proof, just an aid to v... | |real-analysis|algebra-precalculus|real-numbers|decimal-expansion| | 0 |
Can you recommend a decent online or software calculator? | I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator. Can you recommend any? | Google's calculator is very powerful: Google Guide Quick Reference: Google Calculator (Cheat Sheet) and your use history will be stored in your browser history. | |soft-question|big-list|math-software|computer-algebra-systems| | 0 |
Is it true that $0.999999999\ldots=1$? | I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is? | What does it mean when you refer to $.99999\ldots$? Symbols don't mean anything in particular until you've defined what you mean by them . In this case the definition is that you are taking the limit of $.9$, $.99$, $.999$, $.9999$, etc. What does it mean to say that limit is $1$? Well, it means that no matter how smal... | |real-analysis|algebra-precalculus|real-numbers|decimal-expansion| | 1 |
What Does it Really Mean to Have Different Kinds of Infinities? | Can someone explain to me how there can be different kinds of infinities? I was reading " The man who loved only numbers " by Paul Hoffman and came across the concept of countable and uncountable infinities, but they're only words to me. Any help would be appreciated. | A countably infinite set is a set for which you can list the elements: $a_1,a_2,a_3,\ldots$ . For example, the set of all integers is countably infinite since I can list its elements as follows: $$0,1,-1,2,-2,3,-3,\ldots .$$ So is the set of rational numbers, but this is more difficult to see. Let's start with the posi... | |elementary-set-theory|intuition|infinity|faq| | 0 |
Why is the matrix-defined Cross Product of two 3D vectors always orthogonal? | By matrix-defined, I mean $$\left \times\left = \left| \begin{array}{ccc} i & j & k\\ a & b & c\\ d & e & f \end{array} \right|$$ ...instead of the definition of the product of the magnitudes multiplied by the sign of their angle, in the direction orthogonal) If I try cross producting two vectors with no $k$ component,... | Assuming you know the definition of orthogonal as "a is orthogonal to b iff $a\cdot b=0$ then we could calculate $(a \times b)\cdot a = a_1(a_2b_3-a_3b_2)-a_2(a_1b_3-a_3b_1)-a_3(a_1b_2-a_2b_1)=0$ and $(a \times b)\cdot b-0$, so the cross product is orthogonal to both. As Nold mentioned, if the two vectors a and b lie i... | |linear-algebra|matrices|inner-products|orthogonality|cross-product| | 1 |
Can you recommend a decent online or software calculator? | I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator. Can you recommend any? | Essentially the most helpful is WolframAlpha , as Ami said, you can use your browser history here too. WolframAlpha can carry out complex equations can comparisons much like a TI Calculator. Additionally they have some areas where you can see the simplification of an equation paired with charts and graphs where possibl... | |soft-question|big-list|math-software|computer-algebra-systems| | 0 |
Is it true that $0.999999999\ldots=1$? | I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is? | Suppose this was not the case, i.e. $0.9999... \neq 1$ . Then $0.9999... (I hope we agree on that). But between two distinct real numbers, there's always another one in between, say $x=\frac{0.9999... +1}{2}$ , hence $0.9999... . The decimal representation of $x$ must have a digit somewhere that is not $9$ (otherwise $... | |real-analysis|algebra-precalculus|real-numbers|decimal-expansion| | 0 |
The cow in the field problem (intersecting circular areas) | What length of rope should be used to tie a cow to an exterior fence post of a circular field so that the cow can only graze half of the grass within that field? updated: To be clear: the cow should be tied to a post on the exterior of the field, not a post at the center of the field. | So, the area of the field is $\pi r^2$ and you want the cow to be able to graze an area equal to half of that. All you need to do is set up the equation ($r_1$ is the radius of the field, $r_2$ is the length of the rope desired): $$\frac{(\pi r_1^2)}{2} = \pi r_2^2$$ You can then simplify it down: $$\frac{r_1^2 }{2} =r... | |geometry| | 0 |
The cow in the field problem (intersecting circular areas) | 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. | Let the total area of the field = $A$. We know $A = \pi R^2$ where $R$ = the radius of the field. We want the cow to be able to graze half the area, so we solve for a length of rope $r$ such that $\pi r^2 = A / 2$. This gives: $\pi r^2 = \pi R^2 / 2$, hence $r = R / \sqrt(2)$. In words, the length of the cow's rope sho... | |geometry| | 0 |
Calculating an Angle from $2$ points in space | Given two points $p_1$ , $p_2$ around the origin $(0,0)$ in $2D$ space, how would you calculate the angle from $p_1$ to $p_2$ ? How would this change in $3D$ space? | I will assume that you mean the angle of the line from $p_1$ to $p_2$ with respect to the $x$ -axis This is the best I can do given the information you have provided. In any case, the official mathsy way would be to find the dot product between the two, and divide by the magnitude of $p_1-p_2$ and take the arccossine. ... | |linear-algebra|geometry| | 0 |
Real life usage of Benford's Law | I recently discovered Benford's Law. I find it very fascinating. I'm wondering what are some of the real life uses of Benford's law. Specific examples would be great. | Forensic accountancy is a popular use, and is actually admissible as evidence in the USA. | |soft-question|big-list|statistics|applications| | 1 |
What is an elliptic curve, and how are they used in cryptography? | I hear a lot about Elliptic Curve Cryptography these days, but I'm still not quite sure what they are or how they relate to crypto... | Here is a super nice powerpoint on the subject! http://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf | |cryptography|elliptic-curves| | 1 |
How do the Properties of Relations work? | 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 - $R \rightarrow R$ Symmetrical - $R \rightarrow S$ ; $S \rightarrow R$ Antisymmetrical - $R \rightarrow S$ && ( $R \rightarr... | 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 ... | |elementary-set-theory|relations| | 0 |
What is a real number (also rational, decimal, integer, natural, cardinal, ordinal...)? | In mathematics, there seem to be a lot of different types of numbers. What exactly are: Real numbers Integers Rational numbers Decimals Complex numbers Natural numbers Cardinals Ordinals And as workmad3 points out, some more advanced types of numbers (I'd never heard of) Hyper-reals Quaternions Imaginary numbers Are th... | 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 z... | |terminology|definition|number-systems| | 0 |
List of Interesting Math Blogs | I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills. I'll start with my entries: Division By Zero Tanya Khovanova’s Math Blog | Gil Kalai's blog is pretty awesome. Mostly covers combinatorics. | |soft-question|big-list|online-resources| | 0 |
Understanding Dot and Cross Product | What purposes do the Dot and Cross products serve? Do you have any clear examples of when you would use them? | 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. | |linear-algebra|inner-products|cross-product| | 0 |
Understanding Dot and Cross Product | What purposes do the Dot and Cross products serve? Do you have any clear examples of when you would use them? | When you deal with vectors, sometimes you say to yourself, "Darn I wish there was a function that..." was zero when two vectors are perpendicular, letting me test perpendicularness." Dot Product would let me find the angle between two vectors." Dot Product (actually gives the cosine of the angle between two normalized ... | |linear-algebra|inner-products|cross-product| | 1 |
What is an Inner Product Space? | I've learned that 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 ? | 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... | |linear-algebra|vector-spaces|inner-products| | 0 |
Is it true that $0.999999999\ldots=1$? | I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is? | \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} | |real-analysis|algebra-precalculus|real-numbers|decimal-expansion| | 0 |
What are some good ways to get children excited about math? | 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 ... | 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.) | |big-list|education| | 1 |
Why is the matrix-defined Cross Product of two 3D vectors always orthogonal? | By matrix-defined, I mean $$\left \times\left = \left| \begin{array}{ccc} i & j & k\\ a & b & c\\ d & e & f \end{array} \right|$$ ...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,... | Note that if you replace $i$, $j$, and $k$ with $m$, $n$, and $p$, the determinant becomes the dot-product of the vector $(m, n, p)$ with the cross-product of the two original vectors. If $(m, n, p) = (a, b, c)$ or $(m, n, p) = (d, e, f)$, the determinant is zero (any matrix with two identical rows has determinant zero... | |linear-algebra|matrices|inner-products|orthogonality|cross-product| | 0 |
What are the differences between rings, groups, and fields? | Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used? | 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 (... | |terminology|abstract-algebra| | 0 |
Online resources for learning Mathematics | Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning. As someone doing a non-maths degree in college I'd be interested in finding some resources for learning more maths online, mos... | Two good general references: Wikipedia MathWorld | |reference-request|online-resources| | 0 |
How do you calculate the semi-minor axis of an ellipsoid? | Given the semi-major axis and a flattening factor, is it possible to calculate the semi-minor axis? | Possibly something like this. Correct me if I'm wrong. $j$ = semi-major $n$ = semi-minor $e$ = eccentricity $n = \sqrt{(j\sqrt{1 - e^{2}}) \times (j(1 - e^{2}))}$ | |geometry| | 1 |
Online resources for learning Mathematics | Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning. As someone doing a non-maths degree in college I'd be interested in finding some resources for learning more maths online, mos... | 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 | |reference-request|online-resources| | 0 |
Online resources for learning Mathematics | Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning. As someone doing a non-maths degree in college I'd be interested in finding some resources for learning more maths online, mos... | A useful one for undergraduate level maths is Mathcentre . It has useful background material for people studying maths, or who need some maths background for other courses. | |reference-request|online-resources| | 0 |
List of Interesting Math Blogs | I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills. I'll start with my entries: Division By Zero Tanya Khovanova’s Math Blog | Not a pure math blog, but it's one of the most fascinating blogs in my RSS. Futility Closet | |soft-question|big-list|online-resources| | 0 |
What are some good ways to get children excited about math? | 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 ... | If they're fairly mathematically inclined anyway, then try to get them solving interesting problems with an obvious mathematical content, if they're less mathematically inclined try to find problems where the usage of maths isn't as explicit. Problems with a very mathematical bent can be found at places like NRich , th... | |big-list|education| | 0 |
What are some good ways to get children excited about math? | 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 ... | 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 pr... | |big-list|education| | 0 |
Faulty logic when summing large integers? | This is in relation to the Euler Problem $13$ from http://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. $... | 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 | |arithmetic|project-euler| | 0 |
Faulty logic when summing large integers? | This is in relation to the Euler Problem $13$ from http://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. $... | 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. | |arithmetic|project-euler| | 1 |
Real world uses of Quaternions? | 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... | You can view a real-world example of quaternions in computer graphics with the open source program known as NASA WorldWind (http://worldwind.arc.nasa.gov/java/). It uses a Quaternion object to represent rotation of various geometries. The class definition itself is located in the src/gov/nasa/worldwind/geom/Quaternion.... | |soft-question|big-list|linear-algebra|applications|quaternions| | 0 |
The cow in the field problem (intersecting circular areas) | 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. | 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... | |geometry| | 1 |
Is it true that $0.999999999\ldots=1$? | I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is? | One argument against this is that 0.99999999... is "somewhat" less than 1. How much exactly? 1 - 0.999999... = ε (0) If the above is true, the following also must be true: 9 × (1 - 0.999999...) = ε × 9 Let's calculate: 0.999... × 9 = ─────────── 8.1 81 81 . . . ─────────── 8.999... Thus: 9 - 8.999999... = 9ε (1) But: 8... | |real-analysis|algebra-precalculus|real-numbers|decimal-expansion| | 0 |
Is it true that $0.999999999\ldots=1$? | I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is? | Assuming: infinite decimals are series where the terms are the digits divided by the proper power of the base the infinite geometric series $a + a \cdot r + a \cdot r^2 + a \cdot r^3 + \cdots$ has sum $\dfrac{a}{1 - r}$ as long as $|r| $$0.99999\ldots = \frac{9}{10} + \frac{9}{10^2} + \frac{9}{10^3} + \cdots$$ This is ... | |real-analysis|algebra-precalculus|real-numbers|decimal-expansion| | 0 |
Chased by a lion and other pursuit-evasion problems | I am looking for a reference (book or article) that poses a problem that seems to be a classic, in that I've heard it posed many times, but that I've never seen written anywhere: that of the possibility of a man in a circular pen with a lion, each with some maximum speed, avoiding capture by that lion. References to pu... | Here is a book on this type of problem Paul J. Nahin, Chases and escapes: the mathematics of pursuit and evasion , Princeton University Press, 2007. it is also briefly mentioned in his other book "Euler's Fabulous Formula". | |reference-request|differential-games| | 0 |
Online resources for learning Mathematics | Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning. As someone doing a non-maths degree in college I'd be interested in finding some resources for learning more maths online, mos... | 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/ | |reference-request|online-resources| | 0 |
Why is $1$ not a prime number? | Why is $1$ not considered a prime number? Or, why is the definition of prime numbers given for integers greater than $1$? | One of the whole "points" of defining primes is to be able to uniquely and finitely prime factorize every natural number. If 1 was prime, then this would be more or less impossible. | |abstract-algebra|elementary-number-theory|ring-theory|prime-numbers|terminology| | 1 |
Real world uses of hyperbolic trigonometric functions | 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? | 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. | |soft-question|big-list|applications|hyperbolic-functions| | 0 |
How would you describe calculus in simple terms? | 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? | 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'. | |soft-question|calculus| | 0 |
How would you describe calculus in simple terms? | 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? | To be very brief and succinct: Calculus is the study of how quantities change Slightly more technically, it a subject based on infinitesimals . It may be pointing out the obvious, but the Wikipedia article does actually provide a pretty decent beginners introduction to the subject. You'll generally want to start with d... | |soft-question|calculus| | 0 |
What are some classic fallacious proofs? | If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already: Let $a = b$. Then $a^2 = ab$ $a^2 - b^2 = ab - b^2$ Factor to $(a-b)(a+b) = b(a-b)$ Then divide out $(a-b)$ to get $a+b = b$ Since $a = b$, then $b+b = b$ Therefore $... | Wikipedia has a long list of these: http://en.wikipedia.org/wiki/Mathematical_fallacy | |soft-question|big-list|fake-proofs| | 1 |
List of interesting math podcasts? | mathfactor is one I listen to. Does anyone else have a recommendation? | I listen to Math Mutation Podcast . The topics are interesting and understandable by a layman. | |soft-question|big-list|online-resources| | 0 |
How do the Properties of Relations work? | 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 - $R \rightarrow R$ Symmetrical - $R \rightarrow S$ ; $S \rightarrow R$ Antisymmetrical - $R \rightarrow S$ && ( $R \rightarr... | I'd like to change the notation of your definitions, since $R$, $S$ and $T$ would usually be used to stand for the relations themselves (and $x, y$ and $z$ would be more commonly chosen for the objects that might bear the relation to each other). Reflexive - For all $x: xRx$ Example reflexive relation: $xRy$ stands for... | |elementary-set-theory|relations| | 1 |
What is an elliptic curve, and how are they used in cryptography? | I hear a lot about Elliptic Curve Cryptography these days, but I'm still not quite sure what they are or how they relate to crypto... | The technical definition is a nonsingular projective curve of genus 1, which is an abelian variety under the group law: basially, this means that you draw the line through two points on the curve -- which can be embedded in the projective plane -- and find where that line intersects the curve again (and call that the n... | |cryptography|elliptic-curves| | 0 |
Why is $x^0 = 1$ except when $x = 0$? | 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. | 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 i... | |definition|exponentiation| | 1 |
Why is $x^0 = 1$ except when $x = 0$? | 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. | $$0^x = 0, \quad x^0=1$$ 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 $x^0$. | |definition|exponentiation| | 0 |
Why is $x^0 = 1$ except when $x = 0$? | 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. | 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 $x^a \cdot x^b = x^{a+b}$ is an excellent formula that makes a lot of sense (multiplying $a$ times and then multiplying $b$ times ... | |definition|exponentiation| | 0 |
Why is $x^0 = 1$ except when $x = 0$? | 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. | Exponents are only "basically" defined under the natural numbers above zero. By this I mean, defined as "iterated multiplication" the same way multiplication is defined as iterated addition. The property $a^0 = 1$ only arises when we look at generalizing multiplication to the integers. We do this by: \begin{align} a^4 ... | |definition|exponentiation| | 0 |
How do I cut a square in half? | I have a square that's $10\mathrm{m} \times 10\mathrm{m}$. 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 $50\mathrm{m}^{2}$, so I can use my calculat... | 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.... | |geometry| | 0 |
How would you describe calculus in simple terms? | 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? | One of the greatest achievements of human civilization is Newton's laws of motions. The first law says that unless a force is acting then the velocity (not the position!) of objects stay constant, while the second law says that forces act by causing an acceleration (though heavy objects require more force to accellerat... | |soft-question|calculus| | 0 |
Why is $1$ not a prime number? | Why is $1$ not considered a prime number? Or, why is the definition of prime numbers given for integers greater than $1$? | The main point of talking about prime numbers is Euclid's theorem that every positive integer can be written uniquely as a product of primes. As Justin remarks, this would break horribly if $1$ were considered prime, for example we could factor $2$ as $2\times1\times1\times1\times1\times1$. Instead we say that $1$ is n... | |abstract-algebra|elementary-number-theory|ring-theory|prime-numbers|terminology| | 0 |
Are there any functions that are (always) continuous yet not differentiable? Or vice-versa? | 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... | 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 differen... | |real-analysis|continuity| | 1 |
Real world uses of Quaternions? | 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... | 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 orientati... | |soft-question|big-list|linear-algebra|applications|quaternions| | 0 |
Do complex numbers really exist? | Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meani... | Quantum mechanics, and hence physics and everything around us, fundamentally involves complex numbers. | |soft-question|complex-numbers|education|philosophy| | 0 |
Do complex numbers really exist? | Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meani... | I'll start by pointing out that a whole host of things that people think of as 'real' are on shakier ground than imaginary numbers. Given that quantum mechanics predicts a fundamental limit to how granular reality is, the whole concept of real numbers is on very shakey ground, yet people accept those as fine. I'd there... | |soft-question|complex-numbers|education|philosophy| | 0 |
Do complex numbers really exist? | Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meani... | 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 number 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 w... | |soft-question|complex-numbers|education|philosophy| | 0 |
How can you prove that a function has no closed form integral? | In the past, I've come across statements along the lines of "function $f(x)$ 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 whi... | 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 soluti... | |real-analysis|calculus|integration|faq|differential-algebra| | 1 |
Do complex numbers really exist? | Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meani... | There are geometric interpretations of imaginary numbers where they are thought of as parallelograms with a front and back, or oriented parallelograms. That interpretation requires geometric algebra but only uses real numbers. Here is a link: http://en.wikipedia.org/wiki/Geometric_algebra#Complex_numbers That doesn't h... | |soft-question|complex-numbers|education|philosophy| | 0 |
Why is $1$ not a prime number? | Why is $1$ not considered a prime number? Or, why is the definition of prime numbers given for integers greater than $1$? | 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... | |abstract-algebra|elementary-number-theory|ring-theory|prime-numbers|terminology| | 0 |
Do complex numbers really exist? | Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meani... | 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. | |soft-question|complex-numbers|education|philosophy| | 0 |
Why is the volume of a sphere $\frac{4}{3}\pi r^3$? | 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? | 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... | |geometry|volume|solid-geometry|spheres| | 0 |
How would you describe calculus in simple terms? | 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? | 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 ... | |soft-question|calculus| | 0 |
How do I cut a square in half? | I have a square that's $10\mathrm{m} \times 10\mathrm{m}$. 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 $50\mathrm{m}^{2}$, so I can use my calculat... | Does this give you any ideas? | |geometry| | 1 |
Do complex numbers really exist? | Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meani... | 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 $... | |soft-question|complex-numbers|education|philosophy| | 0 |
Why is "the set of all sets" a paradox, in layman's terms? | I've heard of some other paradoxes involving sets (i.e., "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 contain... | 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 i... | |paradoxes|logic|set-theory| | 0 |
What are the differences between rings, groups, and fields? | Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used? | I won't explain what a ring or a group is, because that's already been done, but I'll add something else. One reason groups and rings feel similar is that they are both "algebraic structures" in the sense of universal algebra. So for instance, the operation of quotienting via a normal subgroup (for a group) and a two-s... | |terminology|abstract-algebra| | 0 |
What are some good ways to get children excited about math? | 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 ... | I have found most people liked math at some point, but something happened in their learning process that made them feel so stupid, they became disenfranchised with mathematics. What tends to happen is students are presented with some mathematically result they are expected to memorize by route, which takes all the joy ... | |big-list|education| | 0 |
Do complex numbers really exist? | Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meani... | 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. | |soft-question|complex-numbers|education|philosophy| | 0 |
Is it true that $0.999999999\ldots=1$? | I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is? | If you take two real numbers x and y then there per definition of the real number z for which x or x > z > y is true. For x = 0.99999... and y = 1 you can't find a z and therefore 0.99999... = 1 . | |real-analysis|algebra-precalculus|real-numbers|decimal-expansion| | 0 |
Recasting points from one vector space to another | 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 th... | 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. | |linear-algebra|vector-spaces| | 0 |
How does the wheel paradox work? | I keep looking at this picture and its driving me crazy. How can the smaller circle travel the same distance when its circumference is less than the entire wheel? | If the two circles are fixed, then they will be traveling the same difference, but at different velocities. In fact, the ratio of the radii is equal to the ratio of the velocities a point on either circle will be traveling. If you tried to repeat this by putting two different-sized circles on a track and making them sp... | |geometry| | 0 |
Distribution of primes? | Do primes become more or less frequent as you go further out on the number line? That is, are there more or fewer primes between $1$ and $1{,}000{,}000$ than between $1{,}000{,}000$ and $2{,}000{,}000$? A proof or pointer to a proof would be appreciated. | 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 n... | |number-theory|prime-numbers| | 1 |
Why is "the set of all sets" a paradox, in layman's terms? | I've heard of some other paradoxes involving sets (i.e., "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 contain... | 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.... | |paradoxes|logic|set-theory| | 0 |
Aren't constructive math proofs more "sound"? | Since constructive mathematics allows us to avoid things like Russell's Paradox, then why don't they replace traditional proofs? How do we know the "regular" kind of mathematics are free of paradox without a proof construction? | 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 | |proof-theory|constructive-mathematics| | 0 |
Can you recommend a decent online or software calculator? | 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? | I use R these days. It was built to be a calculator and does the job well. Its syntax might be a bit strange but it allows you to do a lot with little typing. | |soft-question|big-list|math-software|computer-algebra-systems| | 0 |
Aren't constructive math proofs more "sound"? | Since constructive mathematics allows us to avoid things like Russell's Paradox, then why don't they replace traditional proofs? How do we know the "regular" kind of mathematics are free of paradox without a proof construction? | A whole bunch of things in mathematics are inherently nonconstructive. For instance, invariant theory--recall the famous quote by Gordan that Hilbert's mathematics was "theology." (A quote which, I believe, was in jest.) The Hahn-Banach theorem, a fundamental tool in functional analysis (and a great tool for proving al... | |proof-theory|constructive-mathematics| | 0 |
Is there possibly a largest prime number? | Prime numbers are numbers with no factors other than one and itself. Factors of a number are always lower or equal to than a given number; so, the larger the number is, the larger the pool of "possible factors" that number might have. So the larger the number, it seems like the less likely the number is to be a prime. ... | 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 ari... | |number-theory|prime-numbers| | 1 |
Is there possibly a largest prime number? | Prime numbers are numbers with no factors other than one and itself. Factors of a number are always lower or equal to than a given number; so, the larger the number is, the larger the pool of "possible factors" that number might have. So the larger the number, it seems like the less likely the number is to be a prime. ... | 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 | |number-theory|prime-numbers| | 0 |
How can you prove that a function has no closed form integral? | In the past, I've come across statements along the lines of "function $f(x)$ 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 whi... | Brian Conrad explains this in the following: Impossibility theorems on integration in elementary terms (archived PDF ) | |real-analysis|calculus|integration|faq|differential-algebra| | 0 |
Will this procedure generate random points uniformly distributed within a given circle? Proof? | 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: Generate ... | Yes this will work; it's called rejection sampling . Even better is to generate a point in polar coordinates though: pick θ from [0, 2π) and r 2 from [0, R 2 ] (ie. multiply R by the square-root of a random number in [0, 1] - without the square-root it is non-uniform). | |algorithms|probability-theory| | 0 |
Mathematical subjects you wish you learned earlier | I am learning geometric algebra, and it is incredible how much it helps me understand other branches of mathematics. I wish I had been exposed to it earlier. Additionally I feel the same way about enumerative combinatorics. What are some less popular mathematical subjects that you think should be more popular? | I don't really think that graph theory is a "less popular mathematical subject," but I certainly wish I had been exposed to it earlier. | |soft-question|learning| | 0 |
Mathematical subjects you wish you learned earlier | I am learning geometric algebra, and it is incredible how much it helps me understand other branches of mathematics. I wish I had been exposed to it earlier. Additionally I feel the same way about enumerative combinatorics. What are some less popular mathematical subjects that you think should be more popular? | 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 mathemati... | |soft-question|learning| | 0 |
Do complex numbers really exist? | Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meani... | Are real numbers "real" ? It's not even computationally possible to compare two real numbers for equality ! Interestingly enough, it is shown in Abstract Algebra courses that the idea of complex numbers arises naturally from the idea of real numbers - you could not say, for instance, that the real numbers are valid but... | |soft-question|complex-numbers|education|philosophy| | 0 |
How does the wheel paradox work? | I keep looking at this picture and its driving me crazy. How can the smaller circle travel the same distance when its circumference is less than the entire wheel? | 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 rea... | |geometry| | 1 |
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