# Most Popular

1500 questions

**575**

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 black) are always separated by a semitone. Why…

egarcia

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**529**

votes

**0**answers

### 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, $f(S)\subset Y$ is connected.
Question: Assume…

Willie Wong

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**524**

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 and intuitive meaning. What's the best way to…

Neil Mayhew

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**522**

votes

**27**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 class, we are about to prove that $\sin$ is…

FUZxxl

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**513**

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 tried to calculate the square root of $-1$ on my…

Sachin Kainth

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**507**

votes

**10**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 approximate numeric value of the…

Laila Podlesny

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**500**

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 $f_n(x)$ counted the number of subspaces of an…

Alexander Gruber

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**486**

votes

**10**answers

### 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…

Alex Youcis

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**465**

votes

**10**answers

### 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…

a_hardin

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**445**

votes

**15**answers

### 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?

perplexed

**443**

votes

**23**answers

### 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…

user10389

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**442**

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, and the sum equals a later number in that same…

haugsire

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**439**

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 information. Online articles say that these methods are…

wickedchicken

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**428**

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. Answers at any level of sophistication are…

user218

**427**

votes

**23**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 convergent, but my class has never learned these…

backus

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