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1176
votes
67answers
495k views

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 ...
952
votes
31answers
133k views

How long will it take Marie to saw another board 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 ...
952
votes
20answers
93k views

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 ...
926
votes
29answers
83k views

Can I use my powers for good? [closed]

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 ...
731
votes
27answers
172k views

How to study math to really understand it and have a healthy lifestyle with free time? [closed]

Here's my problem. I'm studying math and when I really work hard, I think I understand things very good, but that comes at a big cost: in the last few years, I've had practically zero physical ...
704
votes
21answers
78k views

The staircase paradox, or why $\pi\ne4$

What is wrong with this? Is $\pi=4?$
678
votes
43answers
98k views

Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (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 ...
670
votes
0answers
36k views

A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
668
votes
164answers
49k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) [closed]

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,...
634
votes
25answers
65k views

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 ...
609
votes
13answers
114k views

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 ...
597
votes
13answers
159k views

Does Pi contain all possible number combinations?

I came across the following image: Which states: $\pi$ Pi Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that every possible number combination exists somewhere in pi. ...
550
votes
6answers
79k views

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 ...
535
votes
7answers
37k views

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 $...
517
votes
21answers
80k views

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 ...
497
votes
38answers
39k views

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 ...
478
votes
21answers
78k views

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 which has something to do with the square root of $-1$. When I ...
467
votes
10answers
472k views

Is this Batman equation for real? [closed]

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|-...
460
votes
36answers
58k views

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 ...
433
votes
7answers
17k views

“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 $f_n(x)$ ...
416
votes
24answers
121k views

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 ...
416
votes
10answers
27k views

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, ...
404
votes
8answers
189k views

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 ...
400
votes
23answers
68k views

Proofs that every mathematician should know. [closed]

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 ...
382
votes
19answers
33k views

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 ...
379
votes
15answers
46k views

Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
372
votes
21answers
21k views

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 ...
366
votes
8answers
43k views

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 ...
349
votes
14answers
317k views

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 the question of Kevin Lin, which didn't quite fit at Mathoverflow. ...
348
votes
33answers
42k views

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{\...
346
votes
0answers
13k views

The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on MathOverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
336
votes
18answers
37k views

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 ...
335
votes
4answers
167k views

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 dataset into fewer dimensions while retaining important ...
333
votes
0answers
10k views

Does there exist 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, $f(S)\subset Y$ ...
316
votes
14answers
19k views

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 advantage/disadvantages of proving ...
312
votes
24answers
33k views

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: $x>0$ $0^x=0^{x-0}=0^x/0^0$, so $0^0=0^x/0^x=\,?$ Possible answers: $0^0\cdot0^x=1\cdot0^0$, so $0^0=1$ $...
306
votes
37answers
26k views

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 ...
305
votes
8answers
31k views

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 ...
302
votes
23answers
11k views

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 said $i$ to be a new type of number. How come we don't do the ...
299
votes
68answers
66k views

'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 ...
291
votes
6answers
85k views

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? ...
285
votes
100answers
26k views

Surprising identities / 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, ...
285
votes
5answers
65k views

In Russian roulette, is it best to go first?

Assume that we are playing a game of Russian roulette (6 chambers). Assume 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 ...
283
votes
18answers
34k views

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 ...
277
votes
7answers
30k views

How can you prove that a function has no closed form integral?

I've come across statements in the past 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 ...
272
votes
27answers
41k views

Is it true that $0.999999999\dots=1$?

I'm told by smart people that $$0.999999999\dots=1$$ and I believe them, but is there a proof that explains why this is?
268
votes
39answers
47k views

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 ...
266
votes
31answers
28k views

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 lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I don't ...
265
votes
13answers
10k views

Help with a prime number spiral which turns 90 degrees at each prime

I awoke with the following puzzle and 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 ...
263
votes
26answers
59k views

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 ...

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