# All Questions

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### 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 ...
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### 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 ...
42k 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 ...
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### 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 ...
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### 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 ...
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### 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|-...
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### “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 $f_n(x)$ ...
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### 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 ...
208k 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
15k 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 ...
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+500

### 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, $f(S)\subset Y$ ...
332k 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 in Mathoverflow. ...
190k 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 data set into fewer dimensions while retaining important ...
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### 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 ...