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1500 questions
76
votes
8 answers
A bestiary about adjunctions
What is your favourite adjoint? Following Mac Lane philosophy adjoints are everywhere , so I would like to draw a (possibly but unprobably) exhaustive list of adjunctions one faces in studying Mathematics. For the sake of clarity I would like you to…
fosco
- 12,739
76
votes
18 answers
How to convince a layperson that the $\pi = 4$ proof is wrong?
The infamous "$\pi = 4$" proof was already discussed here: Is value of $\pi = 4$? And I have read all the answers, yet I think that they will not be of much help to me if I try to explain this thing to a non mathematician. The main missing point, in…
Gadi A
- 19,969
76
votes
6 answers
Why do engineers use derivatives in discontinuous functions? Is it correct?
I am a Software Engineering student and this year I learned about how CPUs work, it turns out that electronic engineers and I also see it a lot in my field, we do use derivatives with discontinuous functions. For instance in order to calculate the…
Santiago Pardal
- 936
76
votes
14 answers
"Naturally occurring" non-Hausdorff spaces?
It is not difficult for a beginning point-set topology student to cook up an example of a non-Hausdorff space; perhaps the simplest example is the line with two origins. It is impossible to separate the two origins with disjoint open sets. It is…
Eric
- 1,720
76
votes
2 answers
Order of general- and special linear groups over finite fields.
Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have? $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$ Here GL is the general linear group , the group of invertible n × n matrices,…
user9656
76
votes
3 answers
How did Euler prove the Mersenne number $2^{31}-1$ is a prime so early in history?
I read that Euler proved $2^{31} -1$ is prime. What techniques did he use to prove this so early on in history? Isn't very large number stuff done with computers? Do you know if Euler had a team of people to follow algorithms for him, dubbed…
SwimBikeRun
- 1,077
- 1
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- 18
76
votes
2 answers
A comprehensive list of binomial identities?
Is there a comprehensive resource listing binomial identities? I am more interested in combinatorial proofs of such identities, but even a list without proofs will do.
user813
76
votes
8 answers
How come $32.5 = 31.5$? (The "Missing Square" puzzle.)
Below is a visual proof (!) that $32.5 = 31.5$ . How could that be? (As noted in a comment and answer, this is known as the "Missing Square" puzzle .)
Mehper C. Palavuzlar
- 2,186
76
votes
5 answers
Is the rank of a matrix the same of its transpose? If yes, how can I prove it?
I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view: "The rank of a matrix A is the number of non-zero rows in the reduced row-echelon form of A". The…
Vivi
- 1,519
76
votes
2 answers
When can we interchange the derivative with an expectation?
Let $ (X_t) $ be a stochastic process, and define a new stochastic process by $ Y_t = \int_0^t f(X_s) ds $. Is it true in general that $ \frac{d} {dt} \mathbb{E}(Y_t) = \mathbb{E}(f(X_t)) $? If not, under what conditions would we be allowed to…
Jonas
- 2,613
76
votes
25 answers
Prove $0! = 1$ from first principles
How can I prove from first principles that $0!$ is equal to $1$?
Ssegawa Victor
- 1,059
76
votes
10 answers
Why can't the second fundamental theorem of calculus be proved in just two lines?
The second fundamental theorem of calculus states that if $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$ on the same interval, then: $$\int_a^b f(x) dx= F(b)-F(a).$$ The proof of this theorem in both my textbook and Wikipedia is…
Newton
- 1,260
76
votes
3 answers
Is it possible to find an infinite set of points in the plane where the distance between any pair is rational?
The question is written like this: Is it possible to find an infinite set of points in the plane, not all on the same straight line, such that the distance between EVERY pair of points is rational? This would be so easy if these points could be on…
Ahmed Amir
- 1,043
76
votes
9 answers
Why does the symbol for the multiplication operation change shape?
Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing…
user64742
- 2,226
76
votes
8 answers
Finding $\int x^xdx$
I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this: Let $u=x^x$. $$\begin{align} \int x^x \, dx&=\int u \, du\\[6pt] &=\frac{u^2}{2}\\[6pt] &=\dfrac{\left(x^x\right)^2}{2}\\[6pt] &=\frac{x^{2x}}{2} \end{align}$$ But…
Garmen1778
- 2,426