Most Popular
1500 questions
640
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 bottom or collar of the shirt. He thought it was…
Christopher
- 6,403
632
votes
19 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 integral:…
601
votes
22 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
- 5,897
569
votes
31 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
- 9,629
547
votes
35 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
- 1,683
540
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
- 57,798
529
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 $n$…
Alexander Gruber
- 28,405
526
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
- 6,723
466
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
- 5,493
466
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?
perplexed
466
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…
wickedchicken
- 4,771
460
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
- 1,275
457
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 convergent, but my class has never learned these…
backus
- 4,803
456
votes
15 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 Kevin Lin's question , which didn't quite fit in MathOverflow. Answers at any level of sophistication are welcome.
user218
452
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 about it. You look up various definitions, and…
kjo
- 15,004