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733,754 | Visually stunning math concepts which are easy to explain | Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are mathematically beautiful at the same time. Do you know of any other concepts like these? | 1,669 | soft-question, recreational-mathematics, education, big-list, visualization | https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain | I think if you look at this animation and think about it long enough, you'll understand: Why circles and right-angle triangles and angles are all related. Why sine is "opposite over hypotenuse" and so on. Why cosine is simply sine but offset by $\frac{\pi}{2}$ radians. | 1,171 | false | Q: Visually stunning math concepts which are easy to explain
Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are mathematically beautiful at the same time. Do you know of any oth... | 2026-01-26T11:29:09.672176 |
21,199 | Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? | In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $\textrm{d}y = f'(x)\textrm{d}x$ we are able to plug in values for $\textrm{d}x$ and calculate a... | 1,332 | calculus, analysis, math-history, nonstandard-analysis | https://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio | Historically, when Leibniz conceived of the notation, $\frac{dy}{dx}$ was supposed to be a quotient: it was the quotient of the "infinitesimal change in $y$ produced by the change in $x$" divided by the "infinitesimal change in $x$". However, the formulation of calculus with infinitesimals in the usual setting of the r... | 1,605 | true | Q: Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $\textrm{d}y = f'(x)\textrm{d}x$ we are ab... | 2026-01-26T11:29:10.897038 |
379,927 | If it took 10 minutes to saw a board into 2 pieces, how long will it take to saw another into 3 pieces? | So this is supposed to be really simple, and it's taken from the following picture: Text-only: It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long will it take for her to saw another board into $3$ pieces? I don't understand what's wrong with this question. I think the student... | 1,102 | arithmetic, word-problem | https://math.stackexchange.com/questions/379927/if-it-took-10-minutes-to-saw-a-board-into-2-pieces-how-long-will-it-take-to-saw | Haha! The student probably has a more reasonable interpretation of the question. Of course, cutting one thing into two pieces requires only one cut! Cutting something into three pieces requires two cuts! ------------------------------- 0 cuts/1 piece/0 minutes ---------------|--------------- 1 cut/2 pieces/10 minutes -... | 922 | true | Q: If it took 10 minutes to saw a board into 2 pieces, how long will it take to saw another into 3 pieces?
So this is supposed to be really simple, and it's taken from the following picture: Text-only: It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long will it take for her t... | 2026-01-26T11:29:12.138434 |
71,874 | Can I use my powers for good? | I hesitate to ask this question, but I read a lot of the career advice from MathOverflow and math.stackexchange, and I couldn't find anything similar. Four years after the PhD, I am pretty sure that I am going to leave academia soon. I do enjoy teaching and research, but the alpha-maleness, massive egos and pressure to... | 920 | soft-question, career-development | https://math.stackexchange.com/questions/71874/can-i-use-my-powers-for-good | If you are in the US, there are several thousand institutions of higher learning, and at many of them there is very little "pressure to publish". At others, the "pressure to publish" can be met by publishing a textbook or some work of scholarship that does not require proofs of interesting (original) results. High scho... | 313 | false | Q: Can I use my powers for good?
I hesitate to ask this question, but I read a lot of the career advice from MathOverflow and math.stackexchange, and I couldn't find anything similar. Four years after the PhD, I am pretty sure that I am going to leave academia soon. I do enjoy teaching and research, but the alpha-male... | 2026-01-26T11:29:13.460058 |
12,906 | The staircase paradox, or why $\pi\ne4$ | What is wrong with this proof? Is $\pi=4?$ | 915 | geometry, analysis, convergence-divergence, pi, fake-proofs | https://math.stackexchange.com/questions/12906/the-staircase-paradox-or-why-pi-ne4 | This question is usually posed as the length of the diagonal of a unit square. You start going from one corner to the opposite one following the perimeter and observe the length is $2$, then take shorter and shorter stair-steps and the length is $2$ but your path approaches the diagonal. So $\sqrt{2}=2$. In both cases,... | 574 | true | Q: The staircase paradox, or why $\pi\ne4$
What is wrong with this proof? Is $\pi=4?$
A: This question is usually posed as the length of the diagonal of a unit square. You start going from one corner to the opposite one following the perimeter and observe the length is $2$, then take shorter and shorter stair-steps a... | 2026-01-26T11:29:15.052144 |
358,423 | A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals? | Background. If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$, where $\dim$ denotes the Krull dimension. If $R$ is Noetherian, we have equality. Every proof of this fact I'm aware of uses quite a bit of commutative algebra and non-trivial theorems such as Krull's intersection theorem. It is ... | 891 | ring-theory, commutative-algebra, noetherian, krull-dimension, dimension-theory-algebra | https://math.stackexchange.com/questions/358423/a-proof-of-dimrt-dimr1-without-prime-ideals | 0 | false | Q: A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals?
Background. If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$, where $\dim$ denotes the Krull dimension. If $R$ is Noetherian, we have equality. Every proof of this fact I'm aware of uses quite a bit of commutative algebra and non-t... | 2026-01-26T11:29:16.459675 | |
8,337 | Different ways to prove $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$ (the Basel problem) | As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$ However, Euler was Euler and he gave other proofs. I believe many of you know some nice proofs of this, can you please share it with us? | 889 | sequences-and-series, fourier-analysis, big-list, faq, euler-sums | https://math.stackexchange.com/questions/8337/different-ways-to-prove-sum-k-1-infty-frac1k2-frac-pi26-the-b | OK, here's my favorite. I thought of this after reading a proof from the book "Proofs from the book" by Aigner & Ziegler, but later I found more or less the same proof as mine in a paper published a few years earlier by Josef Hofbauer. On Robin's list, the proof most similar to this is number 9 (EDIT: ...which is actua... | 402 | true | Q: Different ways to prove $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$ (the Basel problem)
As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$ However, Euler was Euler and he gave other proofs... | 2026-01-26T11:29:17.804728 |
44,704 | How to study math to really understand it and have a healthy lifestyle with free time? | Here's my issue I faced; I worked really hard studying Math, so because of that, I started to realised that I understand things better. However, that comes at a big cost: In the last few years, I had practically zero physical exercise, I've gained $30$ kg, I've spent countless hours studying at night, constantly had sl... | 867 | soft-question, advice | https://math.stackexchange.com/questions/44704/how-to-study-math-to-really-understand-it-and-have-a-healthy-lifestyle-with-free | In my view the central question that you should ask yourself is what is the end goal of your studies. As an example, American college life as depicted in film is hedonistic and certainly not centered on actual studies. Your example is the complete opposite - you describe yourself as an ascetic devoted to scholarship. M... | 289 | true | Q: How to study math to really understand it and have a healthy lifestyle with free time?
Here's my issue I faced; I worked really hard studying Math, so because of that, I started to realised that I understand things better. However, that comes at a big cost: In the last few years, I had practically zero physical exe... | 2026-01-26T11:29:19.320024 |
668 | What's an intuitive way to think about the determinant? | In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a $2\times 2$ matrix by the formula. Our teacher showed us how to compute... | 849 | linear-algebra, matrices, determinant, intuition | https://math.stackexchange.com/questions/668/whats-an-intuitive-way-to-think-about-the-determinant | Your trouble with determinants is pretty common. They’re a hard thing to teach well, too, for two main reasons that I can see: the formulas you learn for computing them are messy and complicated, and there’s no “natural” way to interpret the value of the determinant, the way it’s easy to interpret the derivatives you d... | 518 | true | Q: What's an intuitive way to think about the determinant?
In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a $2\time... | 2026-01-26T11:29:21.065070 |
216,343 | Does $\pi$ contain all possible number combinations? | $\pi$ Pi Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever love, the date, time and manner of your death, and the answers to all the g... | 810 | elementary-number-theory, irrational-numbers, pi | https://math.stackexchange.com/questions/216343/does-pi-contain-all-possible-number-combinations | It is not true that an infinite, non-repeating decimal must contain ‘every possible number combination’. The decimal $0.011000111100000111111\dots$ is an easy counterexample. However, if the decimal expansion of $\pi$ contains every possible finite string of digits, which seems quite likely, then the rest of the statem... | 1,020 | true | Q: Does $\pi$ contain all possible number combinations?
$\pi$ Pi Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever love, the date, ti... | 2026-01-26T11:29:22.546435 |
637,728 | Splitting a sandwich and not feeling deceived | This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an hour to an hour, and then just grins at me, and whispers whole day: "You will never solve me..." Please save me from this ... | 697 | game-theory, fair-division | https://math.stackexchange.com/questions/637728/splitting-a-sandwich-and-not-feeling-deceived | For more than two, the moving knife is a nice solution. Somebody takes a knife and moves it slowly across the sandwich. Any player may say "cut". At that moment, the sandwich is cut and the piece given to the one who said "cut". As he has said that is an acceptable piece, he believes he has at least $\frac 1n$ of the s... | 288 | false | Q: Splitting a sandwich and not feeling deceived
This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an hour to an hour, and then just grins at me, and whispers whole day: "You... | 2026-01-26T11:29:24.183639 |
323,334 | What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) | I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and found that I, too, lament the uninspiring quality of my elementary math education. I want to make a book that dis... | 660 | soft-question, education, big-list | https://math.stackexchange.com/questions/323334/what-was-the-first-bit-of-mathematics-that-made-you-realize-that-math-is-beautif | This wasn't the first, but it's definitely awesome: This is a proof of the Pythagorean theorem, and it uses no words! | 315 | false | Q: What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)
I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and... | 2026-01-26T11:29:25.878151 |
952,466 | Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not? | Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ is connected. Question: Assume $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with th... | 656 | general-topology, metric-spaces, examples-counterexamples, connectedness | https://math.stackexchange.com/questions/952466/is-there-a-bijection-of-mathbbrn-with-itself-such-that-the-forward-map-is | 0 | false | Q: Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ i... | 2026-01-26T11:29:27.339132 | |
1,681,993 | Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real? | So we all know that the continued fraction containing all $1$s... $$ x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}}. $$ yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as $x = 1 + \dfrac{1}{x}$, solving the resulting quadratic equation and assuming that a continued fraction that only contains... | 651 | complex-numbers, recursion, continued-fractions | https://math.stackexchange.com/questions/1681993/why-is-1-frac11-frac11-ldots-not-real | You're attempting to take a limit. $$x_{n+1} = 1-\frac{1}{x_n}$$ This recurrence actually never converges, from any real starting point. Indeed, $$x_2 = 1-\frac{1}{x_1}; \\ x_3 = 1-\frac{1}{1-1/x_1} = 1-\frac{x_1}{x_1-1} = \frac{1}{1-x_1}; \\ x_4 = x_1$$ So the sequence is periodic with period 3. Therefore it converges... | 558 | true | Q: Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?
So we all know that the continued fraction containing all $1$s... $$ x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}}. $$ yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as $x = 1 + \dfrac{1}{x}$, solving the resulting quadratic equa... | 2026-01-26T11:29:28.865987 |
111,440 | Examples of patterns that eventually fail | Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up to $20×2^{58}$, then it must always be true?"; and "the sequence of number of edges on a complete graph starts $0,1,3,6... | 643 | big-list, examples-counterexamples | https://math.stackexchange.com/questions/111440/examples-of-patterns-that-eventually-fail | I'll translate an entry in the blog Gaussianos ("Gaussians") about Polya's conjecture, titled: A BELIEF IS NOT A PROOF. We'll say a number is of even kind if in its prime factorization, an even number of primes appear. For example $6 = 2\cdot 3$ is a number of even kind. And we'll say a number is of odd kind if the num... | 436 | false | Q: Examples of patterns that eventually fail
Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up to $20×2^{58}$, then it must always be true?"; and "the sequence of numbe... | 2026-01-26T11:29:30.384720 |
2,755 | Why can you turn clothing right-side-out? | My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids are easily amused, and so am I). So he learned that you ca... | 640 | general-topology, algebraic-topology | https://math.stackexchange.com/questions/2755/why-can-you-turn-clothing-right-side-out | First, a warning. I suspect this response is likely not going to be immediately comprehensible. There is a formal set-up for your question, there are tools available to understand what's going on. They're not particularly light tools, but they exist and they're worthy of being mentioned. Before I write down the main th... | 292 | false | Q: Why can you turn clothing right-side-out?
My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids are easily a... | 2026-01-26T11:29:32.045700 |
562,694 | Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$ | I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The approximate numeric value of the integral: $$I\approx8.372211626601275661625747121...$$ Neither Mathematica nor Maple cou... | 632 | calculus, integration, definite-integrals, contour-integration, closed-form | https://math.stackexchange.com/questions/562694/integral-int-11-frac1x-sqrt-frac1x1-x-ln-left-frac2-x22-x1 | I will transform the integral via a substitution, break it up into two pieces and recombine, perform an integration by parts, and perform another substitution to get an integral to which I know a closed form exists. From there, I use a method I know to attack the integral, but in an unusual way because of the 8th degre... | 1,075 | true | Q: Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$
I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The app... | 2026-01-26T11:29:33.677351 |
11,669 | Mathematical difference between white and black notes in a piano | The division of the chromatic scale in $7$ natural notes (white keys in a piano) and $5$ accidental ones (black) seems a bit arbitrary to me. Apparently, adjacent notes in a piano (including white or black) are always separated by a semitone. Why the distinction, then? Why not just have scales with $12$ notes? (apparen... | 601 | music-theory | https://math.stackexchange.com/questions/11669/mathematical-difference-between-white-and-black-notes-in-a-piano | The first thing you have to understand is that notes are not uniquely defined. Everything depends on what tuning you use. I'll assume we're talking about equal temperament here. In equal temperament, a half-step is the same as a frequency ratio of $\sqrt[12]{2}$; that way, twelve half-steps makes up an octave. Why twel... | 562 | true | Q: Mathematical difference between white and black notes in a piano
The division of the chromatic scale in $7$ natural notes (white keys in a piano) and $5$ accidental ones (black) seems a bit arbitrary to me. Apparently, adjacent notes in a piano (including white or black) are always separated by a semitone. Why the ... | 2026-01-26T11:29:35.209337 |
75,130 | How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? | How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving ... | 567 | calculus, limits, trigonometry, limits-without-lhopital | https://math.stackexchange.com/questions/75130/how-to-prove-that-lim-limits-x-to0-frac-sin-xx-1 | The area of $\triangle ABC$ is $\frac{1}{2}\sin(x)$. The area of the colored wedge is $\frac{1}{2}x$, and the area of $\triangle ABD$ is $\frac{1}{2}\tan(x)$. By inclusion, we get $$ \frac{1}{2}\tan(x)\ge\frac{1}{2}x\ge\frac{1}{2}\sin(x)\tag{1} $$ Dividing $(1)$ by $\frac{1}{2}\sin(x)$ and taking reciprocals, we get $$... | 671 | true | Q: How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found... | 2026-01-26T11:29:36.800032 |
154 | 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... | 546 | soft-question, complex-numbers, education, philosophy | https://math.stackexchange.com/questions/154/do-complex-numbers-really-exist | There are a few good answers to this question, depending on the audience. I've used all of these on occasion. A way to solve polynomials We came up with equations like $x - 5 = 0$, what is $x$?, and the naturals solved them (easily). Then we asked, "wait, what about $x + 5 = 0$?" So we invented negative numbers. Then w... | 378 | true | Q: 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 co... | 2026-01-26T11:29:38.544477 |
302,023 | Best Sets of Lecture Notes and Articles | Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting through books to find [the best source]". It strikes me now that while I love books (I really do), I often find that I lea... | 538 | self-learning, big-list, learning, online-resources | https://math.stackexchange.com/questions/302023/best-sets-of-lecture-notes-and-articles | In no particular order: Algebraic number theory notes by Sharifi: http://math.arizona.edu/~sharifi/algnum.pdf Dalawat's first course in local arithmetic: http://arxiv.org/abs/0903.2615 Intro to top grps: http://www.mat.ucm.es/imi/documents/20062007_Dikran.pdf Representation theory resources: http://www.math.columbia.ed... | 72 | false | Q: Best Sets of Lecture Notes and Articles
Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting through books to find [the best source]". It strikes me now that while I love ... | 2026-01-26T11:29:40.041236 |
206,890 | "The Egg:" Bizarre behavior of the roots of a family of polynomials. | In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ counted the number of subspaces of an $n$-dimensional vector space over $GF(x)$ (which I was using to determine the number ... | 529 | abstract-algebra, complex-analysis, algebraic-geometry, numerical-methods, recreational-mathematics | https://math.stackexchange.com/questions/206890/the-egg-bizarre-behavior-of-the-roots-of-a-family-of-polynomials | First, has anybody ever seen anything at all like this before? Yes, and in fact the interesting patterns that arise here are more than just a mathematical curiosity, they can be interpreted to have a physical context. Statistical Mechanics In a simple spin system, say the Ising model, a discrete set of points are arran... | 157 | false | Q: "The Egg:" Bizarre behavior of the roots of a family of polynomials.
In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ counted the number of subspaces of an $n... | 2026-01-26T11:29:41.614944 |
199,676 | What are imaginary numbers? | At school, I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number that has something to do with the square root of $-1$. When I tried to calculate the square root of $-1$ on my calculator, it gave me an error. To this day I still do not understand imag... | 526 | complex-numbers, definition | https://math.stackexchange.com/questions/199676/what-are-imaginary-numbers | Let's go through some questions in order and see where it takes us. [Or skip to the bit about complex numbers below if you can't be bothered.] What are natural numbers? It took quite some evolution, but humans are blessed by their ability to notice that there is a similarity between the situations of having three apple... | 828 | true | Q: What are imaginary numbers?
At school, I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number that has something to do with the square root of $-1$. When I tried to calculate the square root of $-1$ on my calculator, it gave me an error. To this da... | 2026-01-26T11:29:43.181514 |
3,869 | What is the intuitive relationship between SVD and PCA? | Singular value decomposition (SVD) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional data set into fewer dimensions while retaining important information. Online articles say that these methods are 'related' but never specify the exact relation. What is the intuitive re... | 466 | linear-algebra, matrices, statistics, svd, principal-component-analysis | https://math.stackexchange.com/questions/3869/what-is-the-intuitive-relationship-between-svd-and-pca | (I assume for the purposes of this answer that the data has been preprocessed to have zero mean.) Simply put, the PCA viewpoint requires that one compute the eigenvalues and eigenvectors of the covariance matrix, which is the product $\frac{1}{n-1}\mathbf X\mathbf X^\top$, where $\mathbf X$ is the data matrix. Since th... | 388 | true | Q: What is the intuitive relationship between SVD and PCA?
Singular value decomposition (SVD) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional data set into fewer dimensions while retaining important information. Online articles say that these methods are 'related' bu... | 2026-01-26T11:29:44.598509 |
39,802 | To sum $1+2+3+\cdots$ to $-\frac1{12}$ | $$\sum_{n=1}^\infty\frac1{n^s}$$ only converges to $\zeta(s)$ if $\text{Re}(s)>1$. Why should analytically continuing to $\zeta(-1)$ give the right answer? | 466 | analysis, complex-analysis, riemann-zeta, divergent-series | https://math.stackexchange.com/questions/39802/to-sum-123-cdots-to-frac112 | there are many ways to see that your result is the right one. What does the right one mean? It means that whenever such a sum appears anywhere in physics - I explicitly emphasize that not just in string theory, also in experimentally doable measurements of the Casimir force (between parallel metals resulting from quant... | 248 | false | Q: To sum $1+2+3+\cdots$ to $-\frac1{12}$
$$\sum_{n=1}^\infty\frac1{n^s}$$ only converges to $\zeta(s)$ if $\text{Re}(s)>1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
A: there are many ways to see that your result is the right one. What does the right one mean? It means that whenever su... | 2026-01-26T11:29:46.564273 |
54,506 | Is this Batman equation for real? | HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real? Batman Equation in text form: \begin{align} &\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-3}}+\left(\frac y3\right)^2\sqrt{\frac{\left|y+\frac{3\sqrt{33}}7\right|}{y+\frac{3\sqrt{33}}7}}-1 \right) \\ &\qquad \qquad ... | 466 | geometry, algebra-precalculus, graphing-functions, plane-curves | https://math.stackexchange.com/questions/54506/is-this-batman-equation-for-real | As Willie Wong observed, including an expression of the form $\displaystyle \frac{|\alpha|}{\alpha}$ is a way of ensuring that $\alpha > 0$. (As $\sqrt{|\alpha|/\alpha}$ is $1$ if $\alpha > 0$ and non-real if $\alpha The ellipse $\displaystyle \left( \frac{x}{7} \right)^{2} + \left( \frac{y}{3} \right)^{2} - 1 = 0$ loo... | 1,075 | true | Q: Is this Batman equation for real?
HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real? Batman Equation in text form: \begin{align} &\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-3}}+\left(\frac y3\right)^2\sqrt{\frac{\left|y+\frac{3\sqrt{33}}7\right|}{y+\frac{3\sqr... | 2026-01-26T11:29:48.230889 |
178,940 | Proofs that every mathematician should know. | There are mathematical proofs that have that "wow" factor in being elegant, simplifying one's view of mathematics, lifting one's perception into the light of knowledge, etc. So I'd like to know what mathematical proofs you've come across that you think other mathematicians should know, and why. | 460 | soft-question, big-list | https://math.stackexchange.com/questions/178940/proofs-that-every-mathematician-should-know | Here is my favourite "wow" proof . Theorem There exist two positive irrational numbers $s,t$ such that $s^t$ is rational. Proof If $\sqrt2^\sqrt 2$ is rational, we may take $s=t=\sqrt 2$ . If $\sqrt 2^\sqrt 2$ is irrational , we may take $s=\sqrt 2^\sqrt 2$ and $t=\sqrt 2$ since $(\sqrt 2^\sqrt 2)^\sqrt 2=(\sqrt 2)^ 2=... | 353 | false | Q: Proofs that every mathematician should know.
There are mathematical proofs that have that "wow" factor in being elegant, simplifying one's view of mathematics, lifting one's perception into the light of knowledge, etc. So I'd like to know what mathematical proofs you've come across that you think other mathematicia... | 2026-01-26T11:29:49.943020 |
1,002 | Fourier transform for dummies | What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on Kevin Lin's question, which didn't quite fit in MathOverflow. Answers at any level of sophistication are welcome. | 457 | soft-question, fourier-analysis, fourier-transform | https://math.stackexchange.com/questions/1002/fourier-transform-for-dummies | The ancient Greeks had a theory that the sun, the moon, and the planets move around the Earth in circles. This was soon shown to be wrong. The problem was that if you watch the planets carefully, sometimes they move backwards in the sky. So Ptolemy came up with a new idea - the planets move around in one big circle, bu... | 452 | false | Q: Fourier transform for dummies
What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on Kevin Lin's question, which didn't quite fit in MathOverflow. Answers at any level of sophistication are welcome.
A: The ancient Greeks had a theory that... | 2026-01-26T11:29:51.529036 |
30,732 | How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? | How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method. In general, how can ... | 457 | sequences-and-series, convergence-divergence, power-series, geometric-series, faq | https://math.stackexchange.com/questions/30732/how-can-i-evaluate-sum-n-0-inftyn1xn | No need to use Taylor series, this can be derived in a similar way to the formula for geometric series. Let's find a general formula for the following sum: $$S_{m}=\sum_{n=1}^{m}nr^{n}.$$ Notice that \begin{align*} S_{m}-rS_{m} & = -mr^{m+1}+\sum_{n=1}^{m}r^{n}\\ & = -mr^{m+1}+\frac{r-r^{m+1}}{1-r} \\ & =\frac{mr^{m+2}... | 405 | true | Q: How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that ... | 2026-01-26T11:29:53.076805 |
617,625 | On "familiarity" (or How to avoid "going down the Math Rabbit Hole"?) | Anyone trying to learn mathematics on his/her own has had the experience of "going down the Math Rabbit Hole." For example, suppose you come across the novel term vector space, and want to learn more about it. You look up various definitions, and they all refer to something called a field. So now you're off to learn wh... | 453 | soft-question, self-learning | https://math.stackexchange.com/questions/617625/on-familiarity-or-how-to-avoid-going-down-the-math-rabbit-hole | Your example makes me think of graphs. Imagine some nice, helpful fellow came along, and made a big graph of every math concept ever, where each concept is one node and related concepts are connected by edges. Now you can take a copy of this graph, and color every node green based on whether you "know" that concept (un... | 221 | true | Q: On "familiarity" (or How to avoid "going down the Math Rabbit Hole"?)
Anyone trying to learn mathematics on his/her own has had the experience of "going down the Math Rabbit Hole." For example, suppose you come across the novel term vector space, and want to learn more about it. You look up vari... | 2026-01-26T11:29:54.856201 |
406,099 | My son's Sum of Some is beautiful! But what is the proof or explanation? | My youngest son is in $6$th grade. He likes to play with numbers. Today, he showed me his latest finding. I call it his "Sum of Some" because he adds up some selected numbers from a series of numbers, and the sum equals a later number in that same series. I have translated his finding into the following equation: $$(10... | 447 | sequences-and-series | https://math.stackexchange.com/questions/406099/my-sons-sum-of-some-is-beautiful-but-what-is-the-proof-or-explanation | Factor out the $2^n$ and you get: $2^n (100+20+8) = 2^n 128 = 2^{n+7}$ since $2^7 = 128$ | 459 | false | Q: My son's Sum of Some is beautiful! But what is the proof or explanation?
My youngest son is in $6$th grade. He likes to play with numbers. Today, he showed me his latest finding. I call it his "Sum of Some" because he adds up some selected numbers from a series of numbers, and the sum equals a later number in t... | 2026-01-26T11:29:56.395250 |
3,852 | If $AB = I$ then $BA = I$ | If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$. I do not understand anything more than the following. Elementary row operations. Linear dependence. Row reduced forms and their relations with the original matrix. If the entries of the matrix are not from a mat... | 409 | linear-algebra, matrices, inverse | https://math.stackexchange.com/questions/3852/if-ab-i-then-ba-i | Dilawar says in 2. that he knows linear dependence! So I will give a proof, similar to that of TheMachineCharmer, which uses linear independence. Suppose each matrix is $n$ by $n$. We consider our matrices to all be acting on some $n$-dimensional vector space with a chosen basis (hence isomorphism between linear transf... | 326 | true | Q: If $AB = I$ then $BA = I$
If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$. I do not understand anything more than the following. Elementary row operations. Linear dependence. Row reduced forms and their relations with the original matrix. If the entries of... | 2026-01-26T11:29:57.886417 |
630,339 | Pedagogy: How to cure students of the "law of universal linearity"? | One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”: $$ \frac{1}{a+b} \mathrel{\text{“=”}} \frac{1}{a} + \frac{1}{b} $$ $$ 2^{-3} \mathrel{\text{“=”}} -2^3 $$ $$ \sin (5x + 3y) \mathrel{\text{“=”}} \si... | 407 | algebra-precalculus, education | https://math.stackexchange.com/questions/630339/pedagogy-how-to-cure-students-of-the-law-of-universal-linearity | I think this is a symptom of how students are taught basic algebra. Rather than being told explicit axioms like $a(x+y)= ax+ay$ and theorems like $(x+y)/a = x/a+y/a,$ students are bombarded with examples of how these axioms/theorems are used, without ever being explicitly told: hey, here's a new rule you're allowed to ... | 182 | false | Q: Pedagogy: How to cure students of the "law of universal linearity"?
One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”: $$ \frac{1}{a+b} \mathrel{\text{“=”}} \frac{1}{a} + \frac{1}{b} $$... | 2026-01-26T11:29:59.429272 |
820,686 | 'Obvious' theorems that are actually false | It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true". Now, I know a fair few examples of things that are obviously true and that can be proved to be true (like the Jordan curve theorem). But what are some theorems (preferably short ones) which, when put into laym... | 394 | soft-question, big-list | https://math.stackexchange.com/questions/820686/obvious-theorems-that-are-actually-false | Theorem (false): One can arbitrarily rearrange the terms in a convergent series without changing its value. | 285 | true | Q: 'Obvious' theorems that are actually false
It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true". Now, I know a fair few examples of things that are obviously true and that can be proved to be true (like the Jordan curve theorem). But what are some the... | 2026-01-26T11:30:00.885855 |
243,770 | Can every proof by contradiction also be shown without contradiction? | Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantages/disadvantages of proving by contradiction? As an aside, how is proving by contradiction viewed in general by 'advanced' mathematicians. Is it a ... | 389 | logic, proof-writing, propositional-calculus, proof-theory | https://math.stackexchange.com/questions/243770/can-every-proof-by-contradiction-also-be-shown-without-contradiction | To determine what can and cannot be proved by contradiction, we have to formalize a notion of proof. As a piece of notation, we let $\bot$ represent an identically false proposition. Then $\lnot A$, the negation of $A$, is equivalent to $A \to \bot$, and we take the latter to be the definition of the former in terms of... | 309 | true | Q: Can every proof by contradiction also be shown without contradiction?
Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantages/disadvantages of proving by contradiction? As an aside, how is proving... | 2026-01-26T11:30:02.606322 |
290,435 | Find five positive integers whose reciprocals sum to $1$ | Find a positive integer solution $(x,y,z,a,b)$ for which $$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$ Is your answer the only solution? If so, show why. I was surprised that a teacher would assign this kind of problem to a 5th grade child. (I'm a college student tutor) This girl goes to... | 381 | algebra-precalculus, egyptian-fractions | https://math.stackexchange.com/questions/290435/find-five-positive-integers-whose-reciprocals-sum-to-1 | The perfect number $28=1+2+4+7+14$ provides a solution: $$\frac1{28}+\frac1{14}+\frac17+\frac14+\frac12=\frac{1+2+4+7+14}{28}=1\;.$$ If they’ve been doing unit (or ‘Egyptian’) fractions, I’d expect some to see that since $\frac16+\frac13=\frac12$, $$\frac16+\frac16+\frac16+\frac16+\frac13=1$$ is a solution, though not ... | 435 | true | Q: Find five positive integers whose reciprocals sum to $1$
Find a positive integer solution $(x,y,z,a,b)$ for which $$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$ Is your answer the only solution? If so, show why. I was surprised that a teacher would assign this kind of problem to a 5th... | 2026-01-26T11:30:04.086086 |
505,367 | Collection of surprising identities and equations. | What are some surprising equations/identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, etc. I'd request to avoid 'standard' / well-known results like $ e^{i \pi} + 1 = 0$. Please write a single identity (or ... | 380 | soft-question, big-list | https://math.stackexchange.com/questions/505367/collection-of-surprising-identities-and-equations | This one by Ramanujan gives me the goosebumps: $$ \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{ (4k)! (1103+26390k) }{ (k!)^4 396^{4k} } = \frac1{\pi}. $$ P.S. Just to make this more intriguing, define the fundamental unit $U_{29} = \frac{5+\sqrt{29}}{2}$ and fundamental solutions to Pell equations, $$\big(U_{29}\big... | 268 | false | Q: Collection of surprising identities and equations.
What are some surprising equations/identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, etc. I'd request to avoid 'standard' / well-known results like $ ... | 2026-01-26T11:30:05.677455 |
11,150 | Zero to the zero power – is $0^0=1$? | Could someone provide me with a good explanation of why $0^0=1$? My train of thought: $$x>0\\ 0^x=0^{x-0}=\frac{0^x}{0^0}$$ so $$0^0=\frac{0^x}{0^x}=\,?$$ Possible answers: $0^0\cdot0^x=1\cdot0^0$, so $0^0=1$ $0^0=\frac{0^x}{0^x}=\frac00$, which is undefined PS. I've read the explanation on mathforum.org, but it isn't ... | 379 | algebra-precalculus, exponentiation, indeterminate-forms, faq | https://math.stackexchange.com/questions/11150/zero-to-the-zero-power-is-00-1 | In general, there is no good answer as to what $0^0$ "should" be, so it is usually left undefined. Basically, if you consider $x^y$ as a function of two variables, then there is no limit as $(x,y)\to(0,0)$ (with $x\geq 0$): if you approach along the line $y=0$, then you get $\lim\limits_{x\to 0^+} x^0 = \lim\limits_{x\... | 364 | true | Q: Zero to the zero power – is $0^0=1$?
Could someone provide me with a good explanation of why $0^0=1$? My train of thought: $$x>0\\ 0^x=0^{x-0}=\frac{0^x}{0^0}$$ so $$0^0=\frac{0^x}{0^x}=\,?$$ Possible answers: $0^0\cdot0^x=1\cdot0^0$, so $0^0=1$ $0^0=\frac{0^x}{0^x}=\frac00$, which is undefined PS. I've read the ex... | 2026-01-26T11:30:07.552890 |
155 | 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... | 373 | real-analysis, calculus, integration, faq, differential-algebra | https://math.stackexchange.com/questions/155/how-can-you-prove-that-a-function-has-no-closed-form-integral | 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 solutio... | 169 | true | Q: 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 trig... | 2026-01-26T11:30:09.361707 |
23,312 | What is the importance of eigenvalues/eigenvectors? | What is the importance of eigenvalues/eigenvectors? | 372 | linear-algebra, matrices, eigenvalues-eigenvectors, soft-question | https://math.stackexchange.com/questions/23312/what-is-the-importance-of-eigenvalues-eigenvectors | Short Answer Eigenvectors make understanding linear transformations easy. They are the "axes" (directions) along which a linear transformation acts simply by "stretching/compressing" and/or "flipping"; eigenvalues give you the factors by which this compression occurs. The more directions you have along which you unders... | 385 | true | Q: What is the importance of eigenvalues/eigenvectors?
What is the importance of eigenvalues/eigenvectors?
A: Short Answer Eigenvectors make understanding linear transformations easy. They are the "axes" (directions) along which a linear transformation acts simply by "stretching/compressing" and/or "flipping"; eigenv... | 2026-01-26T11:30:11.244342 |
11 | 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? | 371 | real-analysis, algebra-precalculus, real-numbers, big-list, decimal-expansion | https://math.stackexchange.com/questions/11/is-it-true-that-0-999999999-ldots-1 | 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... | 445 | true | Q: 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?
A: 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 ... | 2026-01-26T11:30:13.085506 |
1,633,704 | Calculating the length of the paper on a toilet paper roll | Fun with Math time. My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with small error) the total paper length of a toilet roll? Writing down some math, I came to this study, which I share with yo... | 365 | calculus, integration, summation, recreational-mathematics | https://math.stackexchange.com/questions/1633704/calculating-the-length-of-the-paper-on-a-toilet-paper-roll | The assumption that the layers are all cylindrical is a good first approximation. The assumption that the layers form a logarithmic spiral is not a good assumption at all, because it supposes that the thickness of the paper at any point is proportional to its distance from the center. This seems to me to be quite absur... | 275 | true | Q: Calculating the length of the paper on a toilet paper roll
Fun with Math time. My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with small error) the total paper length of a toilet roll? Writ... | 2026-01-26T11:30:14.733982 |
259,584 | Why don't we define "imaginary" numbers for every "impossibility"? | Before, the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had. So we declared $i$ to be a new type of number. How come we don't do the same for other "impossible" equations, such as $x = x + 1$, or $x = 1/0$? Edit: OK, a lot of people have said that a n... | 361 | soft-question | https://math.stackexchange.com/questions/259584/why-dont-we-define-imaginary-numbers-for-every-impossibility | Here's one key difference between the cases. Suppose we add to the reals an element $i$ such that $i^2 = -1$, and then include everything else you can get from $i$ by applying addition and multiplication, while still preserving the usual rules of addition and multiplication. Expanding the reals to the complex numbers i... | 194 | true | Q: Why don't we define "imaginary" numbers for every "impossibility"?
Before, the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had. So we declared $i$ to be a new type of number. How come we don't do the same for other "impossi... | 2026-01-26T11:30:16.330727 |
258,736 | Limit of sequence of growing matrices | Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ $K_1=\left(\begin{array}{c}1 \\ 0\end{array}\right)$ and consider the sequence of matrices defined by $$ K_L = \underset{2^{L}\times 2^{L}}{\underbrace{\left[H\otimes I_{2^{L-2}}\rig... | 350 | linear-algebra, matrices, convergence-divergence, operator-theory | https://math.stackexchange.com/questions/258736/limit-of-sequence-of-growing-matrices | 0 | false | Q: Limit of sequence of growing matrices
Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ $K_1=\left(\begin{array}{c}1 \\ 0\end{array}\right)$ and consider the sequence of matrices defined by $$ K_L = \underset{2^{L}\times 2^{L}}{... | 2026-01-26T11:30:17.999085 | |
250 | A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language | The following is a quote from Surely you're joking, Mr. Feynman. The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently t... | 348 | soft-question, big-list, examples-counterexamples | https://math.stackexchange.com/questions/250/a-challenge-by-r-p-feynman-give-counter-intuitive-theorems-that-can-be-transl | 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 ar... | 323 | false | Q: A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
The following is a quote from Surely you're joking, Mr. Feynman. The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challen... | 2026-01-26T11:30:19.590326 |
362,446 | Nice examples of groups which are not obviously groups | I am searching for some groups, where it is not so obvious that they are groups. In the lecture's script there are only examples like $\mathbb{Z}$ under addition and other things like that. I don't think that these examples are helpful to understand the real properties of a group, when only looking to such trivial exam... | 348 | abstract-algebra, group-theory, examples-counterexamples, big-list | https://math.stackexchange.com/questions/362446/nice-examples-of-groups-which-are-not-obviously-groups | Homological algebra. Let $A,B$ be abelian groups (or more generally objects of an abelian category) and consider the set of isomorphism classes of abelian groups $C$ together with an exact sequence $0 \to B \to C \to A \to 0$ (extensions of $A$ by $B$). It turns out that this set has a canonical group structure (isn't ... | 184 | false | Q: Nice examples of groups which are not obviously groups
I am searching for some groups, where it is not so obvious that they are groups. In the lecture's script there are only examples like $\mathbb{Z}$ under addition and other things like that. I don't think that these examples are helpful to understand the real pr... | 2026-01-26T11:30:21.004596 |
318,299 | Any open subset of $\Bbb R$ is a countable union of disjoint open intervals | Let $U$ be an open set in $\mathbb R$. Then $U$ is a countable union of disjoint intervals. This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as possible. A professor told me tha... | 342 | real-analysis, general-topology, big-list, faq | https://math.stackexchange.com/questions/318299/any-open-subset-of-bbb-r-is-a-countable-union-of-disjoint-open-intervals | Here’s one to get things started. Let $U$ be a non-empty open subset of $\Bbb R$. For $x,y\in U$ define $x\sim y$ iff $\big[\min\{x,y\},\max\{x,y\}\big]\subseteq U$. It’s easily checked that $\sim$ is an equivalence relation on $U$ whose equivalence classes are pairwise disjoint open intervals in $\Bbb R$. (The term in... | 234 | false | Q: Any open subset of $\Bbb R$ is a countable union of disjoint open intervals
Let $U$ be an open set in $\mathbb R$. Then $U$ is a countable union of disjoint intervals. This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many ... | 2026-01-26T11:30:22.676209 |
3,444 | Intuition for the definition of the Gamma function? | In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity $$n! = \int_{0}^{\infty} t^n e^{-t} dt$$ , coming from the Gamma function. I have a mathematical confession to make: I have never "grokked" this identity. Why should I expect the... | 338 | probability-theory, special-functions, intuition, gamma-function | https://math.stackexchange.com/questions/3444/intuition-for-the-definition-of-the-gamma-function | I haven't quite got this straight yet, but I think one way to go is to think about choosing points at random from the positive reals. This answer is going to be rather longer than it really needs to be, because I'm thinking about this in a few (closely related) ways, which probably aren't all necessary, and you can dec... | 117 | true | Q: Intuition for the definition of the Gamma function?
In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity $$n! = \int_{0}^{\infty} t^n e^{-t} dt$$ , coming from the Gamma function. I have a mathematical confession to make: I hav... | 2026-01-26T11:30:24.346163 |
942,263 | Really advanced techniques of integration (definite or indefinite) | Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? Every time I search for "Advanced Techniques of Symbolic Integration" or "Super Advanced Integration Techniques", I get the ... | 333 | integration, soft-question, definite-integrals, indefinite-integrals, big-list | https://math.stackexchange.com/questions/942263/really-advanced-techniques-of-integration-definite-or-indefinite | Here are a few. The first one is included because it's not very well known and is not general, though the ones that follow are very general and very useful. A great but not very well known way to find the primitive of $f^{-1}$ in terms of the primitive of $f$, $F$, is (very easy to prove: just differentiate both sides ... | 254 | true | Q: Really advanced techniques of integration (definite or indefinite)
Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? Every time I search for "Advanced Techniques of Symbo... | 2026-01-26T11:30:25.883317 |
158,219 | Is a matrix multiplied with its transpose something special? | In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together. Is $A A^\mathrm T$ something special for any matrix $A$? | 332 | matrices | https://math.stackexchange.com/questions/158219/is-a-matrix-multiplied-with-its-transpose-something-special | The main thing is presumably that $AA^T$ is symmetric. Indeed $(AA^T)^T=(A^T)^TA^T=AA^T$. For symmetric matrices one has the Spectral Theorem which says that we have a basis of eigenvectors and every eigenvalue is real. Moreover if $A$ is invertible, then $AA^T$ is also positive definite, since $$x^TAA^Tx=(A^Tx)^T(A^Tx... | 219 | true | Q: Is a matrix multiplied with its transpose something special?
In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together. Is $A A^\mathrm T$ something special for any matrix $A$?
A: The main thing is presumably that $AA^T$ is symmetric. Indeed $(AA^T)^T=(A^T... | 2026-01-26T11:30:27.460757 |
21,792 | Norms Induced by Inner Products and the Parallelogram Law | Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$. It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ for some (real) inner product $\langle \cdot, \cdot \rangle$, then the parallelogram equality $$ 2\lVert u\rVert^2 + 2\lVe... | 331 | linear-algebra, functional-analysis, normed-spaces, inner-products | https://math.stackexchange.com/questions/21792/norms-induced-by-inner-products-and-the-parallelogram-law | Since this question is asked often enough, let me add a detailed solution. I'm not quite following Arturo's outline, though. The main difference is that I'm not re-proving the Cauchy-Schwarz inequality (Step 4 in Arturo's outline) but rather use the fact that multiplication by scalars and addition of vectors as well as... | 257 | false | Q: Norms Induced by Inner Products and the Parallelogram Law
Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$. It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ for some (real) inner product $\langle \cdot, \cdot \rangle$... | 2026-01-26T11:30:28.807365 |
1,594,740 | V.I. Arnold says Russian students can't solve this problem, but American students can -- why? | In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is $10$ inches, the altitude dropped onto it is 6 inches. Find the area of the triangle. American school students had been coping successfully with this problem for over a deca... | 325 | geometry, triangles, area, puzzle, word-problem | https://math.stackexchange.com/questions/1594740/v-i-arnold-says-russian-students-cant-solve-this-problem-but-american-student | There is no such right triangle. The maximum possible altitude is half the hypotenuse (inscribe the triangle into a circle to see this), which here is $5$ inches. You would only get $30$ square inches if you tried to compute the area without checking whether the triangle actually exists. | 305 | true | Q: V.I. Arnold says Russian students can't solve this problem, but American students can -- why?
In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is $10$ inches, the altitude dropped onto it is 6 inches. Find the area of... | 2026-01-26T11:30:30.391358 |
76,491 | Multiple-choice question about the probability of a random answer to itself being correct | I found this math "problem" on the internet, and I'm wondering if it has an answer: Question: If you choose an answer to this question at random, what is the probability that you will be correct? a. $25\%$ b. $50\%$ c. $0\%$ d. $25\%$ Does this question have a correct answer? | 318 | probability | https://math.stackexchange.com/questions/76491/multiple-choice-question-about-the-probability-of-a-random-answer-to-itself-bein | No, it is not meaningful. 25% is correct iff 50% is correct, and 50% is correct iff 25% is correct, so it can be neither of those two (because if both are correct, the only correct answer could be 75% which is not even an option). But it cannot be 0% either, because then the correct answer would be 25%. So none of the ... | 291 | true | Q: Multiple-choice question about the probability of a random answer to itself being correct
I found this math "problem" on the internet, and I'm wondering if it has an answer: Question: If you choose an answer to this question at random, what is the probability that you will be correct? a. $25\%$ b. $50\%$ c. $0\%$ d... | 2026-01-26T11:30:31.916850 |
112,067 | How discontinuous can a derivative be? | There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many "highly" discontinuous Darboux functions are known--the only one I am aware of being the Conway base 13 function--and few (none... | 316 | real-analysis, derivatives, examples-counterexamples | https://math.stackexchange.com/questions/112067/how-discontinuous-can-a-derivative-be | What follows is taken (mostly) from more extensive discussions in the following sci.math posts: http://groups.google.com/group/sci.math/msg/814be41b1ea8c024 [23 January 2000] http://groups.google.com/group/sci.math/msg/3ea26975d010711f [6 November 2006] http://groups.google.com/group/sci.math/msg/05dbc0ee4c69898e [20 D... | 308 | true | Q: How discontinuous can a derivative be?
There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many "highly" discontinuous Darboux functions are known--the only one I am aware of being... | 2026-01-26T11:30:33.444844 |
139,503 | In the history of mathematics, has there ever been a mistake? | I was just wondering whether or not there have been mistakes in mathematics. Not a conjecture that ended up being false, but a theorem which had a proof that was accepted for a nontrivial amount of time before someone found a hole in the argument. Does this happen anymore now that we have computers? I imagine not. But ... | 315 | math-history, big-list, fake-proofs | https://math.stackexchange.com/questions/139503/in-the-history-of-mathematics-has-there-ever-been-a-mistake | In 1933, Kurt Gödel showed that the class called $\lbrack\exists^*\forall^2\exists^*, {\mathrm{all}}, (0)\rbrack$ was decidable. These are the formulas that begin with $\exists a\exists b\ldots \exists m\forall n\forall p\exists q\ldots\exists z$, with exactly two $\forall$ quantifiers, with no intervening $\exists$s. ... | 185 | false | Q: In the history of mathematics, has there ever been a mistake?
I was just wondering whether or not there have been mistakes in mathematics. Not a conjecture that ended up being false, but a theorem which had a proof that was accepted for a nontrivial amount of time before someone found a hole in the argument. Does t... | 2026-01-26T11:30:34.870590 |
223,235 | Please explain the intuition behind the dual problem in optimization. | I've studied convex optimization pretty carefully, but don't feel that I have yet "grokked" the dual problem. Here are some questions I would like to understand more deeply/clearly/simply: How would somebody think of the dual problem? What thought process would lead someone to consider the dual problem and to recognize... | 313 | optimization, convex-optimization, intuition, lagrange-multiplier, karush-kuhn-tucker | https://math.stackexchange.com/questions/223235/please-explain-the-intuition-behind-the-dual-problem-in-optimization | Here's what's really going on with the dual problem. (This is my attempt to answer my own question, over a year after originally asking it.) (A very nice presentation of this material is given in Ekeland and Temam. These ideas are also in Rockafellar.) Let $V$ be a finite dimensional normed vector space over $\mathbb R... | 176 | false | Q: Please explain the intuition behind the dual problem in optimization.
I've studied convex optimization pretty carefully, but don't feel that I have yet "grokked" the dual problem. Here are some questions I would like to understand more deeply/clearly/simply: How would somebody think of the dual problem? What though... | 2026-01-26T11:30:36.533073 |
514 | Conjectures that have been disproved with extremely large counterexamples? | I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture. I'm sure that everyone here is familiar with it; it describes an operation on a natural number – $n/2$ if it is even, $3n+1$ if it is odd. The conjecture states that if this operation is repeated, all numb... | 312 | big-list, conjectures, big-numbers | https://math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples | My favorite example, which I'm surprised hasn't been posted yet, is the conjecture: $n^{17}+9 \text{ and } (n+1)^{17}+9 \text{ are relatively prime}$ The first counterexample is $n=8424432925592889329288197322308900672459420460792433$ | 178 | false | Q: Conjectures that have been disproved with extremely large counterexamples?
I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture. I'm sure that everyone here is familiar with it; it describes an operation on a natural number – $n/2$ if it is even, $3n+1$ i... | 2026-01-26T11:30:38.273872 |
96,331 | In Russian roulette, is it best to go first? | Assume that we are playing a game of Russian roulette (6 chambers) and that there is no shuffling after the shot is fired. I was wondering if you have an advantage in going first? If so, how big of an advantage? I was just debating this with friends, and I wouldn't know what probability to use to prove it. I'm thinking... | 308 | probability | https://math.stackexchange.com/questions/96331/in-russian-roulette-is-it-best-to-go-first | For a $2$ Player Game, it's obvious that player one will play, and $\frac16$ chance of losing. Player $2$, has a $\frac16$ chance of winning on turn one, so there is a $\frac56$ chance he will have to take his turn. (I've intentionally left fractions without reducing them as it's clearer where the numbers came from) Pl... | 359 | false | Q: In Russian roulette, is it best to go first?
Assume that we are playing a game of Russian roulette (6 chambers) and that there is no shuffling after the shot is fired. I was wondering if you have an advantage in going first? If so, how big of an advantage? I was just debating this with friends, and I wouldn't know ... | 2026-01-26T11:30:40.004225 |
49,229 | Why can ALL quadratic equations be solved by the quadratic formula? | In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use it. I have tried to figure it out by proving these two equations are equal, but I can't. Why can I use $x = \dfrac{-b\pm... | 307 | algebra-precalculus, polynomials, quadratics | https://math.stackexchange.com/questions/49229/why-can-all-quadratic-equations-be-solved-by-the-quadratic-formula | I would like to prove the Quadratic Formula in a cleaner way. Perhaps if teachers see this approach they will be less reluctant to prove the Quadratic Formula. Added: I have recently learned from the book Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-first Century (Ranj... | 547 | false | Q: Why can ALL quadratic equations be solved by the quadratic formula?
In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use it. I have tried to figure it out by proving the... | 2026-01-26T11:30:41.614650 |
513,239 | One question to know if the number is 1, 2 or 3 | I've recently heard a riddle, which looks quite simple, but I can't solve it. A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "Yes", "No", or "I don't know," and after the girl answers it, he knows what the number is. What is the que... | 307 | puzzle | https://math.stackexchange.com/questions/513239/one-question-to-know-if-the-number-is-1-2-or-3 | "I am thinking of a number which is either 0 or 1. Is the sum of our numbers greater than 2?" | 280 | true | Q: One question to know if the number is 1, 2 or 3
I've recently heard a riddle, which looks quite simple, but I can't solve it. A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "Yes", "No", or "I don't know," and after the girl answ... | 2026-01-26T11:30:43.075672 |
260,656 | Can't argue with success? Looking for "bad math" that "gets away with it" | I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare"). One example would be "cancelling" the 6s in $$\frac{64}{16}$$ Another one would be something like $$\frac{9}{2} - \frac{25}{10} = \frac{9 - 25}{2 - 10} = \frac{-16}{-8} = 2 \;\;$$ Yet ... | 306 | examples-counterexamples, recreational-mathematics, big-list | https://math.stackexchange.com/questions/260656/cant-argue-with-success-looking-for-bad-math-that-gets-away-with-it | I was quite amused when a student produced the following when cancelling a fraction: $$\frac{x^2-y^2}{x-y}$$ He began by "cancelling" the $x$ and the $y$ on top and bottom, to get: $$\frac{x-y}{-}$$ and then concluded that "two negatives make a positive", so the final answer has to be $x+y$. | 454 | false | Q: Can't argue with success? Looking for "bad math" that "gets away with it"
I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare"). One example would be "cancelling" the 6s in $$\frac{64}{16}$$ Another one would be... | 2026-01-26T11:30:44.726394 |
1,833,459 | Math without pencil and paper | For someone who is physically unable to use a pencil and paper, what would be the best way to do math? In my case, I have only a little movement in my fingers. I can move a computer mouse and press the left button. Currently I do very little math and when I do I use MS Office but this is a very slow way of doing it and... | 305 | math-software | https://math.stackexchange.com/questions/1833459/math-without-pencil-and-paper | (Background: I have a chronic pain condition, and it is extremely painful for me to do any sort of repetitive fine motor activity, including writing and typing. I earned an undergraduate degree in math with quite little handwriting, and I'm currently a graduate student.) Unfortunately, I've never been able to find any ... | 221 | false | Q: Math without pencil and paper
For someone who is physically unable to use a pencil and paper, what would be the best way to do math? In my case, I have only a little movement in my fingers. I can move a computer mouse and press the left button. Currently I do very little math and when I do I use MS Office but this ... | 2026-01-26T11:30:46.405361 |
24,241 | Why do mathematicians use single-letter variables? | I have much more experience programming than I do with advanced mathematics, so perhaps this is just a comfort thing with me, but I often get frustrated when I try to follow mathematical notation. Specifically, I get frustrated trying to keep track of what each variable signifies. As a programmer, this would be complet... | 301 | soft-question, notation, math-history | https://math.stackexchange.com/questions/24241/why-do-mathematicians-use-single-letter-variables | I think one reason is that often one does not want to remember what the variable names really represent. As an example, when we choose to talk about the matrix $(a_{ij})$ instead of the matrix $(\mathrm{TransitionProbability}_{ij})$, this expresses the important fact that once we have formulated our problem in terms of... | 240 | false | Q: Why do mathematicians use single-letter variables?
I have much more experience programming than I do with advanced mathematics, so perhaps this is just a comfort thing with me, but I often get frustrated when I try to follow mathematical notation. Specifically, I get frustrated trying to keep track of what each var... | 2026-01-26T11:30:48.380895 |
2,667,980 | Why does this innovative method of subtraction from a third grader always work? | My daughter is in year $3$ and she is now working on subtraction up to $1000.$ She came up with a way of solving her simple sums that we (her parents) and her teachers can't understand. Here is an example: $61-17$ Instead of borrowing, making it $50+11-17,$ and then doing what she was told in school $11-7=4,$ $50-10=40... | 301 | arithmetic | https://math.stackexchange.com/questions/2667980/why-does-this-innovative-method-of-subtraction-from-a-third-grader-always-work | So she is doing \begin{align*} 61-17=(60+1)-(10+7)&=(60-10)-(7-1)\\ & = 50-6\\ & =44 \end{align*} She manage to have positive results on each power of ten group up to a multiplication by $\pm 1$ and sums at the end the pieces ; this is kind of smart :) Conclusion : If she is comfortable with this system, let her do... | 291 | false | Q: Why does this innovative method of subtraction from a third grader always work?
My daughter is in year $3$ and she is now working on subtraction up to $1000.$ She came up with a way of solving her simple sums that we (her parents) and her teachers can't understand. Here is an example: $61-17$ Instead of borrowing, ... | 2026-01-26T11:30:50.049865 |
8,814 | Funny identities | Here is a funny exercise $$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$ (If you prove it don't publish it here please). Do you have similar examples? | 300 | big-list | https://math.stackexchange.com/questions/8814/funny-identities | $$\int_0^1\frac{\mathrm{d}x}{x^x}=\sum_{k=1}^\infty \frac1{k^k}$$ | 247 | true | Q: Funny identities
Here is a funny exercise $$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$ (If you prove it don't publish it here please). Do you have similar examples?
A: $$\int_0^1\frac{\mathrm{d}x}{x^x}=\sum_{k=1}^\infty \frac1{k^k}$$ | 2026-01-26T11:30:51.895448 |
166,869 | Is '$10$' a magical number or I am missing something? | It's a hilarious witty joke that points out how every base is '$10$' in its base. Like, \begin{align} 2 &= 10\ \text{(base 2)} \\ 8 &= 10\ \text{(base 8)} \end{align} My question is if whoever invented the decimal system had chosen $9$ numbers or $11$, or whatever, would this still be applicable? I am confused - Is $10... | 296 | notation, number-systems | https://math.stackexchange.com/questions/166869/is-10-a-magical-number-or-i-am-missing-something | Short answer: your confusion about whether ten is special may come from reading aloud "Every base is base 10" as "Every base is base ten" — this is wrong; not every base is base ten, only base ten is base ten. It is a joke that works better in writing. If you want to read it aloud, you should read it as "Every base is ... | 256 | true | Q: Is '$10$' a magical number or I am missing something?
It's a hilarious witty joke that points out how every base is '$10$' in its base. Like, \begin{align} 2 &= 10\ \text{(base 2)} \\ 8 &= 10\ \text{(base 8)} \end{align} My question is if whoever invented the decimal system had chosen $9$ numbers or $11$, o... | 2026-01-26T11:30:53.600410 |
384,861 | Is mathematics one big tautology? | Is mathematics one big tautology? Let me put the question in clearer terms: Mathematics is a deductive system; it works by starting with arbitrary axioms, and deriving therefrom "new" properties through the process of deduction. As such, it would seem that we are simply creating a string of equivalences; each property ... | 295 | soft-question, philosophy, foundations | https://math.stackexchange.com/questions/384861/is-mathematics-one-big-tautology | Disclaimer: different people view this differently. I side with Lakatos: Logic is a tool. Proofs are a way to verify one's intuition (and in many cases to improve one's intuition) and it is a tool to check the consistency of theories in a process of refining the axioms. The fact that every proof boils down to a tautolo... | 281 | true | Q: Is mathematics one big tautology?
Is mathematics one big tautology? Let me put the question in clearer terms: Mathematics is a deductive system; it works by starting with arbitrary axioms, and deriving therefrom "new" properties through the process of deduction. As such, it would seem that we are simply creating a ... | 2026-01-26T11:30:55.248736 |
2,072,308 | Help with a prime number spiral which turns 90 degrees at each prime | I awoke with the following puzzle that I would like to investigate, but the answer may require some programming (it may not either). I have asked on the meta site and believe the question to be suitable and hopefully interesting for the community. I will try to explain the puzzle as best I can then detail the questions... | 291 | prime-numbers, visualization, pattern-recognition | https://math.stackexchange.com/questions/2072308/help-with-a-prime-number-spiral-which-turns-90-degrees-at-each-prime | Just for visual amusement, here are more pictures. In all cases, initial point is a large red dot. Primes up to $10^5$: Primes up to $10^6$: Primes up to $10^6$ starting gaps of length $>6$: Primes up to $10^7$ starting gaps of length $>10$: Primes up to $10^8$ starting gaps of length $>60$: For anyone interested, all ... | 213 | false | Q: Help with a prime number spiral which turns 90 degrees at each prime
I awoke with the following puzzle that I would like to investigate, but the answer may require some programming (it may not either). I have asked on the meta site and believe the question to be suitable and hopefully interesting for the community.... | 2026-01-26T11:30:56.910664 |
38,517 | In (relatively) simple words: What is an inverse limit? | I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of the guys in my office, and despite a very shady explanation he ended up muttering that "you usually take an already know... | 290 | category-theory, intuition, universal-algebra, limits-colimits | https://math.stackexchange.com/questions/38517/in-relatively-simple-words-what-is-an-inverse-limit | I like George Bergman's explanation (beginning in section 7.4 of his Invitation to General Algebra and Universal Constructions). We start with a motivating example. Suppose you are interested in solving $x^2=-1$ in $\mathbb{Z}$. Of course, there are no solutions, but let's ignore that annoying reality for a moment. We ... | 287 | true | Q: In (relatively) simple words: What is an inverse limit?
I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of the guys in my office, and despite a very shady explanation ... | 2026-01-26T11:30:58.754849 |
671,076 | Is $7$ the only prime followed by a cube? | I discovered this site which claims that "$7$ is the only prime followed by a cube". I find this statement rather surprising. Is this true? Where might I find a proof that shows this? In my searching, I found this question, which is similar but the answers seem focused on squares next to cubes. Any ideas? | 289 | prime-numbers | https://math.stackexchange.com/questions/671076/is-7-the-only-prime-followed-by-a-cube | This is certainly true. Suppose $n^3 - 1$ is prime, for some $n$. We get that $n^3-1 = (n-1)(n^2 + n + 1)$ and so we have that $n-1$ divides $n^3 - 1$. If $n-1>1$ then we're done, as we have a contradiction to $n^3 - 1$ being prime. | 300 | true | Q: Is $7$ the only prime followed by a cube?
I discovered this site which claims that "$7$ is the only prime followed by a cube". I find this statement rather surprising. Is this true? Where might I find a proof that shows this? In my searching, I found this question, which is similar but the answers seem focused on s... | 2026-01-26T11:31:00.392767 |
976,462 | A 1400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I | The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I wondered how much this could be improved using our computers and so I tried (very immodestly) to see if we could do better using... | 287 | trigonometry, approximation, math-history | https://math.stackexchange.com/questions/976462/a-1400-years-old-approximation-to-the-sine-function-by-mahabhaskariya-of-bhaskar | One simple way to derive this is to come up with a parabola approximation. Just getting the roots correct we have $$f(x)=x(\pi-x)$$ Then, we need to scale it (to get the heights correct). And we are gonna do that by dividing by another parabola $p(x)$ $$f(x)=\frac{x(\pi-x)}{p(x)}$$ Let's fix this at three points (thus ... | 156 | false | Q: A 1400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I
The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I wondered how much this could be improved... | 2026-01-26T11:31:02.132135 |
5,248 | Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$? | A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int\limits_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$ Well, can anyone prove this without using Residue theory? I actually thought of using the series representation of $\sin x$: $$\in... | 286 | calculus, complex-analysis, analysis, definite-integrals, improper-integrals | https://math.stackexchange.com/questions/5248/evaluating-the-integral-int-0-infty-frac-sin-x-x-mathrm-dx-frac-pi | I believe this can also be solved using double integrals. It is possible (if I remember correctly) to justify switching the order of integration to give the equality: $$\int_{0}^{\infty} \Bigg(\int_{0}^{\infty} e^{-xy} \sin x \,dy \Bigg)\, dx = \int_{0}^{\infty} \Bigg(\int_{0}^{\infty} e^{-xy} \sin x \,dx \Bigg)\,dy$$ ... | 277 | false | Q: Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int\limits_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$ Well, can anyone prove this without using ... | 2026-01-26T11:31:03.751545 |
860,294 | How does one prove the determinant inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$? | Reposted on MathOverflow Let $\,A,B,C\in M_{n}(\mathbb C)\,$ be Hermitian and positive definite matrices such that $A+B+C=I_{n}$, where $I_{n}$ is the identity matrix. Show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n \det \left(A^2+B^2+C^2\right)$$ This problem is a test question from China (xixi). It is said o... | 286 | linear-algebra, matrices, inequality, determinant, hermitian-matrices | https://math.stackexchange.com/questions/860294/how-does-one-prove-the-determinant-inequality-det-left6a3b3c3i-n-ri | Here is a partial and positive result, valid around the "triple point" $A=B=C= \frac13\mathbb 1$. Let $A,B,C\in M_n(\mathbb C)$ be Hermitian satisfying $A+B+C=\mathbb 1$, and additionally assume that $$\|A-\tfrac13\mathbb 1\|\,,\,\|B-\tfrac13\mathbb 1\|\,,\, \|C-\tfrac13\mathbb 1\|\:\leqslant\:\tfrac16\tag{1}$$ in the ... | 12 | false | Q: How does one prove the determinant inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?
Reposted on MathOverflow Let $\,A,B,C\in M_{n}(\mathbb C)\,$ be Hermitian and positive definite matrices such that $A+B+C=I_{n}$, where $I_{n}$ is the identity matrix. Show that $$\det\left(6(A^3+B^3+C^3)+... | 2026-01-26T11:31:05.511769 |
80,246 | The Mathematics of Tetris | I am a big fan of the old-school games and I once noticed that there is a sort of parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no other piece in the game. Background: The Tetris playing field has width $10$. Rotation is allowed, so there are then ex... | 284 | recreational-mathematics, problem-solving, packing-problem, parity, invariance | https://math.stackexchange.com/questions/80246/the-mathematics-of-tetris | My colleague, Ido Segev, pointed out that there is a problem with most of the elegant proofs here - Tetris is not just a problem of tiling a rectangle. Below is his proof that the conjecture is, in fact, false. | 206 | true | Q: The Mathematics of Tetris
I am a big fan of the old-school games and I once noticed that there is a sort of parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no other piece in the game. Background: The Tetris playing field has width $10$. Rotation is... | 2026-01-26T11:31:07.154301 |
364,452 | Evaluate $\int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\,\Bbb dx$ | Evaluate the following integral $$ \tag1\int_{0}^{\frac{\pi}{2}}\frac1{(1+x^2)(1+\tan x)}\,\Bbb dx $$ My Attempt: Letting $x=\frac{\pi}{2}-x$ and using the property that $$ \int_{0}^{a}f(x)\,\Bbb dx = \int_{0}^{a}f(a-x)\,\Bbb dx $$ we obtain $$ \tag2\int_{0}^{\frac{\pi}{2}}\frac{\tan x}{\left(1+\left(\frac{\pi}{2}-x\ri... | 284 | calculus, real-analysis, integration, definite-integrals, improper-integrals | https://math.stackexchange.com/questions/364452/evaluate-int-0-frac-pi2-frac11x21-tan-x-bbb-dx | Here is an approach. We give some preliminary results. The poly-Hurwitz zeta function The poly-Hurwitz zeta function may initially be defined by the series $$ \begin{align} \displaystyle \zeta(s\mid a,b) := \sum_{n=1}^{+\infty} \frac{1}{(n+a)^{s}(n+b)}, \quad \Re a >-1, \, \Re b >-1, \, \Re s>0. \tag1 \end{align} $$ Th... | 253 | false | Q: Evaluate $\int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\,\Bbb dx$
Evaluate the following integral $$ \tag1\int_{0}^{\frac{\pi}{2}}\frac1{(1+x^2)(1+\tan x)}\,\Bbb dx $$ My Attempt: Letting $x=\frac{\pi}{2}-x$ and using the property that $$ \int_{0}^{a}f(x)\,\Bbb dx = \int_{0}^{a}f(a-x)\,\Bbb dx $$ we obtain $$ \ta... | 2026-01-26T11:31:08.894647 |
23,902 | What is the practical difference between a differential and a derivative? | I ask because, as a first-year calculus student, I am running into the fact that I didn't quite get this down when understanding the derivative: So, a derivative is the rate of change of a function with respect to changes in its variable, this much I get. Thing is, definitions of 'differential' tend to be in the form o... | 280 | calculus | https://math.stackexchange.com/questions/23902/what-is-the-practical-difference-between-a-differential-and-a-derivative | Originally, "differentials" and "derivatives" were intimately connected, with derivative being defined as the ratio of the differential of the function by the differential of the variable (see my previous discussion on the Leibnitz notation for the derivative). Differentials were simply "infinitesimal changes" in whate... | 215 | false | Q: What is the practical difference between a differential and a derivative?
I ask because, as a first-year calculus student, I am running into the fact that I didn't quite get this down when understanding the derivative: So, a derivative is the rate of change of a function with respect to changes in its variable, thi... | 2026-01-26T11:31:10.530341 |
885,879 | Meaning of Rays in Polar Plot of Prime Numbers | I recently began experimenting with gnuplot and I quickly made an interesting discovery. I plotted all of the prime numbers beneath 1 million in polar coordinates such that for every prime $p$, $(r,\theta) = (p,p)$. I was not expecting anything in particular, I was simply trying it out. The results are fascinating. Whe... | 279 | prime-numbers, polar-coordinates | https://math.stackexchange.com/questions/885879/meaning-of-rays-in-polar-plot-of-prime-numbers | What we're seeing is arithmetic progressions (not prime-producing polynomials) of primes, combined with a classical phenomenon about rational approximations. When the integers (or any subset of them) are represented by the polar points $(n,n)$, of course integers that are close to a multiple of $2\pi$ apart from each o... | 200 | true | Q: Meaning of Rays in Polar Plot of Prime Numbers
I recently began experimenting with gnuplot and I quickly made an interesting discovery. I plotted all of the prime numbers beneath 1 million in polar coordinates such that for every prime $p$, $(r,\theta) = (p,p)$. I was not expecting anything in particular, I was sim... | 2026-01-26T11:31:12.135441 |
279,079 | How to read a book in mathematics? | How is it that you read a mathematics book? Do you keep a notebook of definitions? What about theorems? Do you do all the exercises? Focus on or ignore the proofs? I have been reading Munkres, Artin, Halmos, etc. but I get a bit lost usually around the middle. Also, about how fast should you be reading it? Any advice i... | 278 | soft-question | https://math.stackexchange.com/questions/279079/how-to-read-a-book-in-mathematics | This method has worked well for me (but what works well for one person won't necessarily work well for everyone). I take it in several passes: Read 0: Don't read the book, read the Wikipedia article or ask a friend what the subject is about. Learn about the big questions asked in the subject, and the basics of the theo... | 241 | true | Q: How to read a book in mathematics?
How is it that you read a mathematics book? Do you keep a notebook of definitions? What about theorems? Do you do all the exercises? Focus on or ignore the proofs? I have been reading Munkres, Artin, Halmos, etc. but I get a bit lost usually around the middle. Also, about how fast... | 2026-01-26T11:31:13.693919 |
17,152 | Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet? | Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet." I have a (not very mathematically intelligent) friend who presented it as if it were a mathematical fact, which got me thinking... Is this really true? Of course, I've learned that deali... | 277 | probability, infinity | https://math.stackexchange.com/questions/17152/given-an-infinite-number-of-monkeys-and-an-infinite-amount-of-time-would-one-of | I found online the claim (which we may as well accept for this purpose) that there are $32241$ words in Hamlet. Figuring $5$ characters and one space per word, this is $193446$ characters. If the character set is $60$ including capitals and punctuation, a random string of $193446$ characters has a chance of $1$ in $60^... | 336 | false | Q: Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet?
Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet." I have a (not very mathematically intelligent) friend who presented it as if it we... | 2026-01-26T11:31:15.440578 |
160,248 | Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ | I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$, which is a pretty interesting and nice result. I wonder in which ways we may approach it. | 277 | calculus, real-analysis, sequences-and-series, limits | https://math.stackexchange.com/questions/160248/evaluating-lim-limits-n-to-infty-e-n-sum-limits-k-0n-fracnkk | The probabilistic way: This is $P[N_n\leqslant n]$ where $N_n$ is a random variable with Poisson distribution of parameter $n$. Hence each $N_n$ is distributed like $X_1+\cdots+X_n$ where the random variables $(X_k)$ are independent and identically distributed with Poisson distribution of parameter $1$. By the central ... | 200 | false | Q: Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$
I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$, which is a pretty interesting and nice result. I wonder in which ways we may a... | 2026-01-26T11:31:17.000320 |
78,575 | Derivative of sigmoid function $\sigma (x) = \frac{1}{1+e^{-x}}$ | In my AI textbook there is this paragraph, without any explanation. The sigmoid function is defined as follows $$\sigma (x) = \frac{1}{1+e^{-x}}.$$ This function is easy to differentiate because $$\frac{d\sigma (x)}{d(x)} = \sigma (x)\cdot (1-\sigma(x)).$$ It has been a long time since I've taken differential equations... | 276 | calculus, derivatives | https://math.stackexchange.com/questions/78575/derivative-of-sigmoid-function-sigma-x-frac11e-x | Let's denote the sigmoid function as $\sigma(x) = \dfrac{1}{1 + e^{-x}}$. The derivative of the sigmoid is $\dfrac{d}{dx}\sigma(x) = \sigma(x)(1 - \sigma(x))$. Here's a detailed derivation: $$ \begin{align} \dfrac{d}{dx} \sigma(x) &= \dfrac{d}{dx} \left[ \dfrac{1}{1 + e^{-x}} \right] \\ &= \dfrac{d}{dx} \left( 1 + \mat... | 460 | true | Q: Derivative of sigmoid function $\sigma (x) = \frac{1}{1+e^{-x}}$
In my AI textbook there is this paragraph, without any explanation. The sigmoid function is defined as follows $$\sigma (x) = \frac{1}{1+e^{-x}}.$$ This function is easy to differentiate because $$\frac{d\sigma (x)}{d(x)} = \sigma (x)\cdot (1-\sigma(x... | 2026-01-26T11:31:18.686987 |
165,696 | Your favourite application of the Baire Category Theorem | I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications that confirm its neatness and/or power. Here's the theorem (with proof) and two applications: (Baire) A non-empty compl... | 275 | general-topology, functional-analysis, big-list, baire-category | https://math.stackexchange.com/questions/165696/your-favourite-application-of-the-baire-category-theorem | If $P$ is an infinitely differentiable function such that for each $x$, there is an $n$ with $P^{(n)}(x)=0$, then $P$ is a polynomial. (Note $n$ depends on $x$.) See the discussion in Math Overflow. | 79 | false | Q: Your favourite application of the Baire Category Theorem
I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications that confirm its neatness and/or power. Here's the theorem... | 2026-01-26T11:31:20.774269 |
264,371 | Fun but serious mathematics books to gift advanced undergraduates. | I am looking for fun, interesting mathematics textbooks which would make good studious holiday gifts for advanced mathematics undergraduates or beginning graduate students. They should be serious but also readable. In particular, I am looking for readable books on more obscure topics not covered in a standard undergrad... | 275 | reference-request, soft-question, big-list | https://math.stackexchange.com/questions/264371/fun-but-serious-mathematics-books-to-gift-advanced-undergraduates | Check into Generatingfunctionology by Herbert Wilf. From the linked (author's) site, the second edition is available for downloading as a pdf. There is also a link to the third edition, available for purchase. It's a very helpful, useful, readable, fun, (and short!) book that a student could conceivably cover over wint... | 79 | false | Q: Fun but serious mathematics books to gift advanced undergraduates.
I am looking for fun, interesting mathematics textbooks which would make good studious holiday gifts for advanced mathematics undergraduates or beginning graduate students. They should be serious but also readable. In particular, I am looking for re... | 2026-01-26T11:31:22.577252 |
82,467 | Eigenvectors of real symmetric matrices are orthogonal | Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = Q\Lambda Q^{-1} = Q\Lambda Q^{T}$ where $\Lambda$ is diagonal. | 272 | linear-algebra, matrices, reference-request, eigenvalues-eigenvectors, symmetric-matrices | https://math.stackexchange.com/questions/82467/eigenvectors-of-real-symmetric-matrices-are-orthogonal | For any real matrix $A$ and any vectors $\mathbf{x}$ and $\mathbf{y}$, we have $$\langle A\mathbf{x},\mathbf{y}\rangle = \langle\mathbf{x},A^T\mathbf{y}\rangle.$$ Now assume that $A$ is symmetric, and $\mathbf{x}$ and $\mathbf{y}$ are eigenvectors of $A$ corresponding to distinct eigenvalues $\lambda$ and $\mu$. Then $... | 371 | true | Q: Eigenvectors of real symmetric matrices are orthogonal
Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = Q\Lambda Q^{-1} = Q\Lambda Q^{T}$ where $\Lambda$ is diagona... | 2026-01-26T11:31:24.172790 |
3,749 | Why do we care about dual spaces? | When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are, but I don't really understand why we want to study them within linear algebra. I was wondering if anyone knew a nice intuitive motivation for the study of dual spaces and whether or n... | 271 | linear-algebra, soft-question, intuition, dual-spaces | https://math.stackexchange.com/questions/3749/why-do-we-care-about-dual-spaces | Let $V$ be a vector space (over any field, but we can take it to be $\mathbb R$ if you like, and for concreteness I will take the field to be $\mathbb R$ from now on; everything is just as interesting in that case). Certainly one of the interesting concepts in linear algebra is that of a hyperplane in $V$. For example,... | 273 | true | Q: Why do we care about dual spaces?
When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are, but I don't really understand why we want to study them within linear algebra. I was wondering if anyone knew a nice intuitive motivation for the... | 2026-01-26T11:31:25.688221 |
275,310 | What is the difference between linear and affine function? | I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated. | 271 | linear-algebra, affine-geometry | https://math.stackexchange.com/questions/275310/what-is-the-difference-between-linear-and-affine-function | A linear function fixes the origin, whereas an affine function need not do so. An affine function is the composition of a linear function with a translation, so while the linear part fixes the origin, the translation can map it somewhere else. Linear functions between vector spaces preserve the vector space structure (... | 245 | false | Q: What is the difference between linear and affine function?
I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated.
A: A linear function fixes the origin, whereas an affine function need not do so. An affine function is the composition of a linear funct... | 2026-01-26T11:31:27.404183 |
237,002 | Too old to start math | I'm sorry if this question goes against the meta for posting questions - I attached all the "beware, this is a soft-question" tags I could. This is a question I've been asking myself now for some time. In most areas, there's a "cut off age" to be good at something. For example, you're not going to make the NHL if you s... | 270 | soft-question, self-learning, advice, learning | https://math.stackexchange.com/questions/237002/too-old-to-start-math | Karl Weierstrass was in his 40's when he got his PHD. There are a dozen other counterexamples, a number fairly recent. A good set of examples can be found in the thread on MO here.This myth of "science is a game for the young" is one of the falsest and most destructive canards in modern society. Don't listen to it. You... | 252 | true | Q: Too old to start math
I'm sorry if this question goes against the meta for posting questions - I attached all the "beware, this is a soft-question" tags I could. This is a question I've been asking myself now for some time. In most areas, there's a "cut off age" to be good at something. For example, you're not goin... | 2026-01-26T11:31:28.965350 |
2,855,975 | What is the maximum volume that can be contained by a sheet of paper? | I was writing some exercises about the AM-GM inequality and I got carried away by the following (pretty nontrivial, I believe) question: Q: By properly folding a common $210mm\times 297mm$ sheet of paper, what is the maximum amount of water such a sheet is able to contain? The volume of the optimal box (on the right) i... | 270 | geometry, optimization, recreational-mathematics, volume | https://math.stackexchange.com/questions/2855975/what-is-the-maximum-volume-that-can-be-contained-by-a-sheet-of-paper | This problem reminds me of tension field theory and related problems in studying the shape of inflated inextensible membranes (like helium balloons). What follows is far from a solution, but some initial thoughts about the problem. First, since you're allowing creasing and folding, by Nash-Kuiper it's enough to conside... | 142 | true | Q: What is the maximum volume that can be contained by a sheet of paper?
I was writing some exercises about the AM-GM inequality and I got carried away by the following (pretty nontrivial, I believe) question: Q: By properly folding a common $210mm\times 297mm$ sheet of paper, what is the maximum amount of water such ... | 2026-01-26T11:31:30.700501 |
61,497 | Why are rings called rings? | I've done some search in Internet and other sources about this question. Why the name ring to this particular object? Just curiosity. Thanks. | 269 | abstract-algebra, ring-theory, terminology, math-history | https://math.stackexchange.com/questions/61497/why-are-rings-called-rings | The name "ring" is derived from Hilbert's term "Zahlring" (number ring), introduced in his Zahlbericht for certain rings of algebraic integers. As for why Hilbert chose the name "ring", I recall reading speculations that it may have to do with cyclical (ring-shaped) behavior of powers of algebraic integers. Namely, if ... | 256 | true | Q: Why are rings called rings?
I've done some search in Internet and other sources about this question. Why the name ring to this particular object? Just curiosity. Thanks.
A: The name "ring" is derived from Hilbert's term "Zahlring" (number ring), introduced in his Zahlbericht for certain rings of algebraic integers... | 2026-01-26T11:31:32.450708 |
2,821,112 | "Integral milking": working backward to construct nontrivial integrals | I begin this post with a plea: please don't be too harsh with this post for being off topic or vague. It's a question about something I find myself doing as a mathematician, and wonder whether others do it as well. It is a soft question about recreational mathematics - in reality, I'm shooting for more of a conversatio... | 269 | integration, definite-integrals, soft-question, recreational-mathematics, big-list | https://math.stackexchange.com/questions/2821112/integral-milking-working-backward-to-construct-nontrivial-integrals | Yes, definitely. For example, I found that $$ m\int_0^{\infty} y^{\alpha} e^{-y}(1-e^{-y})^{m-1} \, dy = \Gamma(\alpha+1) \sum_{k \geq 1} (-1)^{k-1} \binom{m}{k} \frac{1}{k^{\alpha}} $$ (and related results for particular values of $\alpha$) while mucking about with some integrals. Months later, I was reading a paper a... | 170 | false | Q: "Integral milking": working backward to construct nontrivial integrals
I begin this post with a plea: please don't be too harsh with this post for being off topic or vague. It's a question about something I find myself doing as a mathematician, and wonder whether others do it as well. It is a soft questio... | 2026-01-26T11:31:34.676562 |
2,217,248 | "Which answer in this list is the correct answer to this question?" | I received this question from my mathematics professor as a leisure-time logic quiz, and although I thought I answered it right, he denied. Can someone explain the reasoning behind the correct solution? Which answer in this list is the correct answer to this question? All of the below. None of the below. All of the abo... | 268 | logic, soft-question, puzzle | https://math.stackexchange.com/questions/2217248/which-answer-in-this-list-is-the-correct-answer-to-this-question | // gcc ImpredictivePropositionalLogic1.c -o ImpredictivePropositionalLogic1.exe -std=c99 -Wall -O3 /* Which answer in this list is the correct answer to this question? (a) All of the below. (b) None of the below. (c) All of the above. (d) One of the above. (e) None of the above. (f) None of the above. */ #include #defi... | 230 | false | Q: "Which answer in this list is the correct answer to this question?"
I received this question from my mathematics professor as a leisure-time logic quiz, and although I thought I answered it right, he denied. Can someone explain the reasoning behind the correct solution? Which answer in this list is the co... | 2026-01-26T11:31:36.502082 |
94,827 | What books must every math undergraduate read? | I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden). I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on if I want to study beyond Masters. I'm not there yet as I'm on a four year course and had a gap year between Years 3 and... | 264 | reference-request, soft-question, big-list, book-recommendation | https://math.stackexchange.com/questions/94827/what-books-must-every-math-undergraduate-read | EDIT: I now think that this list is long enough that I shall be maintaining it over time--updating it whenever I use a new book/learn a new subject. While every suggestion below should be taken with a grain of salt--I will say that I spend a huge amount of time sifting through books to find the ones that conform best t... | 233 | true | Q: What books must every math undergraduate read?
I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden). I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on if I want to study beyond Masters. I'm not there yet as I'm on a four... | 2026-01-26T11:31:38.179132 |
2,746 | Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer? | If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ appears in the sum, but $\frac{1}{2p}$ does not. Aside from $\frac 1p$, every other term $\frac 1k$ has $k$ divisible only by prim... | 261 | sequences-and-series, number-theory, harmonic-numbers | https://math.stackexchange.com/questions/2746/is-there-an-elementary-proof-that-sum-limits-k-1n-frac1k-is-never-an-int | Hint $ $ There is a $\rm\color{darkorange}{unique}$ denominator $\rm\,\color{#0a0} {2^K}$ having maximal power of $\:\!2,\,$ so scaling by $\rm\,\color{#c00}{2^{K-1}}$ we deduce a contradiction $\large \rm\, \frac{1}2 = \frac{c}d \,$ with odd $\rm\,d \:$ (vs. $\,\rm d = 2c),\,$ e.g. $$\begin{eqnarray} & &\rm\ \ \ \ \co... | 339 | true | Q: Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?
If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ appears in the sum, but $\frac{1}{2p}$ d... | 2026-01-26T11:31:39.820446 |
255,063 | Why study Algebraic Geometry? | I'm going to start self-stydying algebraic geometry very soon. So, my question is why do mathematicians study algebraic geometry? What are the types of problems in which algebraic geometers are interested in? And what are some of the most beautiful theorems in algebraic geometry? | 259 | algebraic-geometry, soft-question | https://math.stackexchange.com/questions/255063/why-study-algebraic-geometry | NEW ADDITION: a big list of freely available online courses on algebraic geometry, from introduction to advanced topics, has been compiled in this other answer. And a digression on motivation for studying the subject along with a self-learning guide of books is in this new answer. There are other similar questions, abo... | 364 | true | Q: Why study Algebraic Geometry?
I'm going to start self-stydying algebraic geometry very soon. So, my question is why do mathematicians study algebraic geometry? What are the types of problems in which algebraic geometers are interested in? And what are some of the most beautiful theorems in algebraic geometry?
A: N... | 2026-01-26T11:31:41.456402 |
348,198 | Best Fake Proofs? (A M.SE April Fools Day collection) | In honor of April Fools Day $2013$, I'd like this question to collect the best, most convincing fake proofs of impossibilities you have seen. I've posted one as an answer below. I'm also thinking of a geometric one where the "trick" is that it's very easy to draw the diagram wrong and have two lines intersect in the wr... | 258 | soft-question, big-list, fake-proofs | https://math.stackexchange.com/questions/348198/best-fake-proofs-a-m-se-april-fools-day-collection | $$x^2=\underbrace{x+x+\cdots+x}_{(x\text{ times})}$$ $$\frac{d}{dx}x^2=\frac{d}{dx}[\underbrace{x+x+\cdots+x}_{(x\text{ times})}]$$ $$2x=1+1+\cdots+1=x$$ $$2=1$$ | 208 | false | Q: Best Fake Proofs? (A M.SE April Fools Day collection)
In honor of April Fools Day $2013$, I'd like this question to collect the best, most convincing fake proofs of impossibilities you have seen. I've posted one as an answer below. I'm also thinking of a geometric one where the "trick" is that it's very easy to dra... | 2026-01-26T11:31:43.037539 |
2,694 | What is the importance of the Collatz conjecture? | I have been fascinated by the Collatz problem since I first heard about it in high school. Take any natural number $n$. If $n$ is even, divide it by $2$ to get $n / 2$, if $n$ is odd multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process indefinitely. The conjecture is that no matter what number you star... | 257 | number-theory, elementary-number-theory, prime-numbers, soft-question, collatz-conjecture | https://math.stackexchange.com/questions/2694/what-is-the-importance-of-the-collatz-conjecture | Most of the answers so far have been along the general lines of 'Why hard problems are important', rather than 'Why the Collatz conjecture is important'; I will try to address the latter. I think the basic question being touched on is: In what ways does the prime factorization of $a$ affect the prime factorization of $... | 383 | true | Q: What is the importance of the Collatz conjecture?
I have been fascinated by the Collatz problem since I first heard about it in high school. Take any natural number $n$. If $n$ is even, divide it by $2$ to get $n / 2$, if $n$ is odd multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process indefinitely.... | 2026-01-26T11:31:44.799405 |
253,910 | The Integral that Stumped Feynman? | In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods that he could not do with real methods: One time I boasted, "I can do by other methods any integral anybody else needs ... | 257 | real-analysis, complex-analysis, reference-request, integration, contour-integration | https://math.stackexchange.com/questions/253910/the-integral-that-stumped-feynman | I doubt that we will ever know the exact integral that vexed Feynman. Here is something similar to what he describes. Suppose $f(z)$ is an analytic function on the unit disk. Then, by Cauchy's integral formula, $$\oint_\gamma \frac{f(z)}{z}dz = 2\pi i f(0),$$ where $\gamma$ traces out the unit circle in a counterclockw... | 83 | false | Q: The Integral that Stumped Feynman?
In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods that he could not do with real methods: One time I boasted, "I can do by other m... | 2026-01-26T11:31:46.779113 |
209,241 | Exterior Derivative vs. Covariant Derivative vs. Lie Derivative | In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them in order of appearance in my education/in descending order of my understanding of them. Note, there may be others that I am... | 256 | reference-request, differential-geometry, differential-forms, exterior-algebra, big-picture | https://math.stackexchange.com/questions/209241/exterior-derivative-vs-covariant-derivative-vs-lie-derivative | Short answer: the exterior derivative acts on differential forms; the Lie derivative acts on any tensors and some other geometric objects (they have to be natural, e.g. a connection, see the paper of P. Petersen below); both the exterior and the Lie derivatives don't require any additional geometric structure: they rel... | 98 | true | Q: Exterior Derivative vs. Covariant Derivative vs. Lie Derivative
In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them in order of appearance in my education/in descending or... | 2026-01-26T11:31:48.418105 |
1,188,845 | Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers | An exam for high school students had the following problem: Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a circle be determined by the points $E$, $B$ and $C$ as shown on the diagram. Which of the geometric figures has the greater perimeter, the square or the circle? Of cour... | 256 | geometry, circles, quadrilateral | https://math.stackexchange.com/questions/1188845/does-the-square-or-the-circle-have-the-greater-perimeter-a-surprisingly-hard-pr | Perhaps the examiner intended the students to notice the square is determined by a $(3, 4, 5)$ triangle, because $3 + 5 = 4 + 4$ (!): Consequently, as several others have noted, $$ \frac{\text{perimeter of the circle}}{\text{perimeter of the square}} = \frac{5 \cdot 2\pi}{4 \cdot 8} = \frac{\pi}{3.2} For an approach le... | 224 | true | Q: Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers
An exam for high school students had the following problem: Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a circle be determined by the points $E$, $B$ and $C$ as show... | 2026-01-26T11:31:50.058507 |
546,155 | Proof that the trace of a matrix is the sum of its eigenvalues | I have looked extensively for a proof on the internet but all of them were too obscure. I would appreciate if someone could lay out a simple proof for this important result. Thank you. | 255 | linear-algebra, matrices, eigenvalues-eigenvectors, trace | https://math.stackexchange.com/questions/546155/proof-that-the-trace-of-a-matrix-is-the-sum-of-its-eigenvalues | These answers require way too much machinery. By definition, the characteristic polynomial of an $n\times n$ matrix $A$ is given by $$p(t) = \det(A-tI) = (-1)^n \big(t^n - (\text{tr} A) \,t^{n-1} + \dots + (-1)^n \det A\big)\,.$$ On the other hand, $p(t) = (-1)^n(t-\lambda_1)\dots (t-\lambda_n)$, where the $\lambda_j$ ... | 339 | true | Q: Proof that the trace of a matrix is the sum of its eigenvalues
I have looked extensively for a proof on the internet but all of them were too obscure. I would appreciate if someone could lay out a simple proof for this important result. Thank you.
A: These answers require way too much machinery. By definition, the... | 2026-01-26T11:31:51.797928 |
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