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1500 questions
207
votes
34 answers
List of interesting math videos / documentaries
This is an offshoot of the question on Fun math outreach/social activities . I have listed a few videos/documentaries I have seen. I would appreciate if people could add on to this list. $1.$ Story of maths Part1 Part2 Part3 Part4 $2.$ Dangerous…
user17762
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
- 8,632
206
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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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
- 11,840
205
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
- 10,043
204
votes
0 answers
Sorting of prime gaps
Let $g_i$ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we rearrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ , if the gaps are arranged from smallest to largest, we have a new sequence $(\hat{g}_{n,i})_{i=1}^n.$ For example, for…
daniel
- 10,601
204
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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204
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
- 2,331
204
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
- 6,996
203
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
- 186,861
202
votes
12 answers
What is $dx$ in integration?
When I was at school and learning integration in maths class at A Level my teacher wrote things like this on the board. $$\int f(x)\, dx$$ When he came to explain the meaning of the $dx$, he told us "think of it as a full stop". For whatever reason…
Sachin Kainth
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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
- 2,147
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
- 2,213
200
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 form…
Moos Hueting
- 2,197
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
- 2,178