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1629 votes
88 answers

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 ...
1302 votes
27 answers

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 ...
BBSysDyn's user avatar
  • 16.3k
1090 votes
32 answers

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 ...
yuritsuki's user avatar
  • 10.4k
920 votes
29 answers

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 ...
902 votes
24 answers

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

What is wrong with this proof? Is $\pi=4?$
Pratik Deoghare's user avatar
860 votes
54 answers

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 ...
854 votes
27 answers

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

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 ...
842 votes
0 answers

A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals?

Please read this first before answering. This question is only concerned with a proof of the dimension formula using the Coquand-Lombardi characterization below. If you post something that doesn't ...
Martin Brandenburg's user avatar
825 votes
18 answers

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 ...
Jamie Banks's user avatar
  • 13.1k
796 votes
12 answers

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 ...
Chani's user avatar
  • 7,961
695 votes
25 answers

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 ...
VividD's user avatar
  • 16k
660 votes
164 answers

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,...
641 votes
8 answers

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 ...
Martin Ender's user avatar
  • 6,010
630 votes
6 answers

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 ...
Christopher's user avatar
  • 6,303
626 votes
45 answers

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 ...
617 votes
0 answers

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, $f(S)\subset Y$ ...
Willie Wong's user avatar
  • 73.8k
595 votes
14 answers

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 ...
Laila Podlesny's user avatar
591 votes
21 answers

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 ...
egarcia's user avatar
  • 5,797
555 votes
30 answers

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 ...
FUZxxl's user avatar
  • 9,337
541 votes
37 answers

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 ...
523 votes
10 answers

Best Sets of Lecture Notes and Articles

Let me start by apologizing if there is another thread on 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 ...
522 votes
22 answers

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 ...
Sachin Kainth's user avatar
519 votes
7 answers

"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 $f_n(x)$ ...
Alexander Gruber's user avatar
  • 27.2k
465 votes
10 answers

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|-...
a_hardin's user avatar
  • 5,491
457 votes
18 answers

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?
user avatar
456 votes
4 answers

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 ...
wickedchicken's user avatar
455 votes
23 answers

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 ...
452 votes
24 answers

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 ...
backus's user avatar
  • 4,735
446 votes
10 answers

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, ...
haugsire's user avatar
  • 3,441
446 votes
14 answers

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 in Mathoverflow. ...
446 votes
22 answers

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 ...
kjo's user avatar
  • 14.5k
401 votes
33 answers

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{\...
394 votes
34 answers

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 ...
Dilawar's user avatar
  • 6,175
381 votes
73 answers

'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 ...
380 votes
20 answers

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 ...
Low Scores's user avatar
  • 4,575
380 votes
15 answers

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 ...
sonicboom's user avatar
  • 10k
374 votes
24 answers

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: $0^0\cdot0^x=1\cdot0^0$,...
Stas's user avatar
  • 3,999
367 votes
118 answers

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, ...
362 votes
31 answers

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?
362 votes
11 answers

What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?
Ryan's user avatar
  • 5,519
360 votes
23 answers

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 ...
lily's user avatar
  • 3,769
360 votes
7 answers

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/...
Simon Nickerson's user avatar
357 votes
8 answers

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 ...
Enrico M.'s user avatar
  • 26.3k
351 votes
0 answers

Limit of sequence of growing matrices [closed]

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(\...
Eckhard's user avatar
  • 7,735
346 votes
36 answers

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 ...
332 votes
31 answers

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 ...
324 votes
8 answers

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 ...
Qiaochu Yuan's user avatar
321 votes
9 answers

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 $A$?
Martin Ueding's user avatar
315 votes
6 answers

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? ...
user avatar
313 votes
21 answers

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? ...
user3002473's user avatar
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