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1500 questions
207
votes
9 answers

"Advice to young mathematicians"

I have been suggested to read the Advice to a Young Mathematician section of the Princeton Companion to Mathematics, the short paper Ten Lessons I wish I had been Taught by Gian-Carlo Rota, and the Career Advice section of Terence Tao's blog, and I…
Dal
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206
votes
91 answers

Which one result in mathematics has surprised you the most?

A large part of my fascination in mathematics is because of some very surprising results that I have seen there. I remember one I found very hard to swallow when I first encountered it, was what is known as the Banach Tarski Paradox. It states that…
KalEl
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205
votes
14 answers

How many sides does a circle have?

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this: If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have? My first reaction was "0" or "undefined". But my son wrote…
Fixee
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205
votes
7 answers

How to intuitively understand eigenvalue and eigenvector?

I’m learning multivariate analysis and I have learnt linear algebra for two semesters when I was a freshman. Eigenvalues and eigenvectors are easy to calculate and the concept is not difficult to understand. I found that there are many applications…
Jill Clover
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203
votes
6 answers

Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least some of the higher contexts where we would use the…
GPerez
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202
votes
8 answers

Are there any series whose convergence is unknown?

Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will eventually be shown to converge or diverge? EDIT: People…
pseudosudo
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202
votes
8 answers

How do we prove that something is unprovable?

I have read somewhere there are some theorems that are shown to be "unprovable". It was a while ago and I don't remember the details, and I suspect that this question might be the result of a total misunderstanding. By the way, I assume that…
polfosol
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202
votes
9 answers

What Does it Really Mean to Have Different Kinds of Infinities?

Can someone explain to me how there can be different kinds of infinities? I was reading "The man who loved only numbers" by Paul Hoffman and came across the concept of countable and uncountable infinities, but they're only words to me. Any help…
Allain Lalonde
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201
votes
14 answers

Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\\\\ \frac1{\sqrt{-1}} &= \frac1i \\\\ \frac{\sqrt1}{\sqrt{-1}} &= \frac1i \\\\ \sqrt{\frac1{-1}} &= \frac1i \\\\…
Wilhelm
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200
votes
3 answers

Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

The following integral \begin{align*} \int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{dx}{x^{2}+1} = \frac{5\pi^{2}}{96} \tag{1} \end{align*} is called the Ahmed's integral and became famous since its first discovery in 2002.…
Sangchul Lee
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200
votes
6 answers

Using proof by contradiction vs proof of the contrapositive

What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by contradiction, and the other proves the…
Kasper
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200
votes
21 answers

Calculate Rotation Matrix to align Vector $A$ to Vector $B$ in $3D$?

I have one triangle in $3D$ space that I am tracking in a simulation. Between time steps I have the previous normal of the triangle and the current normal of the triangle along with both the current and previous $3D$ vertex positions of the…
user1084113
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199
votes
1 answer

Derivative of Softmax loss function

I am trying to wrap my head around back-propagation in a neural network with a Softmax classifier, which uses the Softmax function: \begin{equation} p_j = \frac{e^{o_j}}{\sum_k e^{o_k}} \end{equation} This is used in a loss function of the…
Moos Hueting
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197
votes
5 answers

A Topology such that the continuous functions are exactly the polynomials

I was wondering which fields $K$ can be equipped with a topology such that a function $f:K \to K$ is continuous if and only if it is a polynomial function $f(x)=a_nx^n+\cdots+a_0$. Obviously, the finite fields with the discrete topology have this…
Dominik
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197
votes
13 answers

Convergence of the series $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ for $p > 1$

I am trying to prove the convergence of the $p$-series $$\sum_{n=1}^{\infty} \frac1{n^p}.$$ for $p > 1$. I am wondering if there is a proof that this series converges, either directly or by applying some test for convergence.
admchrch
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