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1500 questions
198
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
- 4,787
198
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
- 9,245
197
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
- 2,167
197
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
- 2,173
197
votes
7 answers
Do we have negative prime numbers?
Do we have negative prime numbers?
$..., -7, -5, -3, -2, ...$
user103028
197
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
- 2,097
196
votes
15 answers
What's the intuition behind Pythagoras' theorem?
Today we learned about Pythagoras' theorem. Sadly, I can't understand the logic behind it.
$A^{2} + B^{2} = C^{2}$
$C^{2} = (5 \text{ cm})^2 + (7 \text{ cm})^2$
$C^{2} = 25 \text{ cm}^2 + 49 \text{ cm}^2$
$C^{2} = 74 \text{ cm}^2$
${x} =…
user123399
- 289
196
votes
2 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
- 167,468
196
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
- 8,214
195
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
- 14,396
195
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
- 2,241
194
votes
22 answers
List of Interesting Math Blogs
I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills.
I'll start with my entries:
Division…
194
votes
8 answers
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:
$$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl(…
Quixotic
- 22,431
193
votes
21 answers
Taking Seats on a Plane
This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless
Imagine there are a 100 people in line to board a plane that seats 100. The first person in…
crasic
- 4,849
193
votes
12 answers
What is $dx$ in integration?
When I was at school and learning integration in maths class at A Level my teacher wrote things like this on the board.
$$\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
- 6,573