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1500 questions
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votes
5 answers

Intuition for the Importance of Modular Forms

I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things. I was wondering if the following interpretation of why modular forms are important is correct a)…
Alex Youcis
  • 54,059
69
votes
8 answers

Good books on Math History

I'm trying to find good books on the history of mathematics, dating as far back as possible. There was a similar question here Good books on Philosophy of Mathematics, but mostly pertaining to Philosophy, and there were no good recommendations on…
Philoxopher
  • 431
69
votes
3 answers

Why isn't several complex variables as fundamental as multivariable calculus?

One typically studies analysis in $\mathbb{R}^n$ after studying analysis in $\mathbb{R}$. Why can't the same be said of $\mathbb{C}$?
user60042
  • 701
69
votes
1 answer

Simplicial homology of real projective space by Mayer-Vietoris

Consider the $n$-sphere $S^n$ and the real projective space $\mathbb{RP}^n$. There is a universal covering map $p: S^n \to \mathbb{RP}^n$, and it's clear that it's the coequaliser of $\mathrm{id}: S^n \to S^n$ and the antipodal map $a: S^n \to S^n$.…
Zhen Lin
  • 90,111
69
votes
29 answers

Understandable questions which are hard for non-mathematicians but easy for mathematicians

A friend of mine has set me the challenge of finding an example of the following: Is there a question, that everyone (both mathematicians and non-mathematicians) can understand, that most mathematicians would answer correctly, instantly, but that…
Zestylemonzi
  • 4,103
69
votes
4 answers

A nasty integral of a rational function

I'm having a hard time proving the following $$\int_0^{\infty} \frac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1} \, dx = \frac{\pi}{2}.$$ Mathematica has no problem evaluating it while I haven't the slightest…
user54031
69
votes
7 answers

The relation between trace and determinant of a matrix

Let $M$ be a symmetric $n \times n$ matrix. Is there any equality or inequality that relates the trace and determinant of $M$?
TPArrow
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69
votes
7 answers

Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$?

Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?
Clayton Kershaw
  • 693
69
votes
8 answers

Dominoes and induction, or how does induction work?

I've never really understood why math induction is supposed to work. You have these 3 steps: Prove true for base case (n=0 or 1 or whatever) Assume true for n=k. Call this the induction hypothesis. Prove true for n=k+1, somewhere using the…
bobobobo
  • 9,502
69
votes
12 answers

Is it morally right and pedagogically right to google answers to homework?

This is a soft question that I have been struggling with lately. My professor sets tough questions for homework (around 10 per week). The difficulty is such that if I attempt the questions entirely on my own, I usually get stuck for over 2 hours per…
yoyostein
  • 19,608
69
votes
5 answers

Are there an infinite number of prime numbers where removing any number of digits leaves a prime?

Suppose for the purpose of this question that number $1$ is a prime number. Consider the prime number $311$. If we remove one $1$ from the number we arrive at the number $31$ which is also prime. If we removed $3$ instead of $1$ we would arrived at…
Farewell
  • 5,008
69
votes
7 answers

$100$-th derivative of the function $f(x)=e^{x}\cos(x)$

I've got this task I'm not able to solve. So i need to find the 100-th derivative of $$f(x)=e^{x}\cos(x)$$ where $x=\pi$. I've tried using Leibniz's formula but it got me nowhere, induction doesn't seem to help either, so if you could just give me a…
windircurse
  • 1,894
69
votes
5 answers

What's so special about characteristic 2?

I've often read about things which do not work in a field with a characteristic $2$, mainly things which have to do with factoring, or similar things. I'm not exactly sure why, but the only example of such a field I could think of is…
Juan Sebastian Lozano
  • 2,704
69
votes
4 answers

Topological spaces admitting an averaging function

Let $M$ be a topological space. Define an averaging function as a continuous map $f:M \times M \to M$ which satisfies $f(a,b) = f(b,a)$ for all $a,b \in M$ and $f(a,a) = a$ for all $a \in M$. These seem like reasonable properties for a function…
Steven Gubkin
  • 9,145
69
votes
3 answers

If I know the order of every element in a group, do I know the group?

Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ have the same number of elements for each order,…
Stanley
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