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1500 questions
84
votes
10 answers
What's a proof that the angles of a triangle add up to 180°?
Back in grade school, I had a solution involving "folding the triangle" into a rectangle half the area, and seeing that all the angles met at a point: However, now that I'm in university, I'm not convinced that this proof is the best one (although…
Joe Z.
- 6,759
84
votes
2 answers
84
votes
6 answers
How to take the gradient of the quadratic form?
It's stated that the gradient of: $$\frac{1}{2}x^TAx - b^Tx +c$$ is $$\frac{1}{2}A^Tx + \frac{1}{2}Ax - b$$ How do you grind out this equation? Or specifically, how do you get from $x^TAx$ to $A^Tx + Ax$?
victor
- 963
84
votes
6 answers
20 circles in the plane, all passing through the origin
Suppose I draw $20$ circles in the plane, all passing through the origin, but no two tangent at the origin. Also, except for the origin, no three circles pass through a common point. How many regions are created in the plane?
laser01
- 1,200
84
votes
10 answers
Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?
I have seen the Fresnel integral $$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$ evaluated by contour integration and other complex analysis methods, and I have found these methods to be the standard way to evaluate this integral. I was…
Argon
- 26,059
84
votes
5 answers
Are all infinities equal?
A friend of mine was trying to explain to me how all infinities are equal. For example, they were saying that there are the same amount of numbers between $0$–$1$ as there are between $0$–$2$. The way they explained it, you could prove that there…
Ephraim
- 1,888
84
votes
8 answers
Finding a point along a line a certain distance away from another point!
Let's say you have two points, $(x_0, y_0)$ and $(x_1, y_1)$. The gradient of the line between them is: $$m = (y_1 - y_0)/(x_1 - x_0)$$ And therefore the equation of the line between them is: $$y = m (x - x_0) + y_0$$ Now, since I want another point…
Kel196
- 1,009
84
votes
3 answers
Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$
Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same degree as $f(x)$, the polynomial $f(x)$ is…
spin
- 12,377
84
votes
8 answers
Intuition behind conjugation in group theory
I am learning group theory, and while learning automorphisms, I came across conjugation as an example in many textbooks. Though the definition itself, (and when considering the case of abelian groups), it seems pretty innocent, I have to admit that…
Nikhil
- 1,097
83
votes
7 answers
Algebraic Intuition for Homological Algebra and Applications to More Elementary Algebra
I am taking a course next term in homological algebra (using Weibel's classic text) and am having a hard time seeing some of the big picture of the idea behind homological algebra. Now, this sort of question has been asked many times on forums such…
Alex Youcis
- 57,478
83
votes
1 answer
Theorem that von Neumann proved in five minutes.
In "How To Solve It", George Pólya writes: "There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann…
Salech Alhasov
- 7,002
- 2
- 31
- 48
83
votes
1 answer
Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius?
It is well-known that the function $$f(x) = \begin{cases} e^{-1/x^2}, \mbox{if } x \ne 0 \\ 0, \mbox{if } x = 0\end{cases}$$ is smooth everywhere, yet not analytic at $x = 0$. In particular, its Taylor series exists there, but it equals $0 + 0x +…
The_Sympathizer
- 20,437
83
votes
6 answers
The Intuition behind l'Hopitals Rule
I understand perfectly well how to apply l'Hopital's rule, and how to prove it, but I've never grokked the theorem. Why is it that we should expect that $$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x \to a}\frac{f'(x)}{g'(x)},$$ but only when specific…
user71641
83
votes
3 answers
How to find this limit: $A=\lim_{n\to \infty}\sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}}$
Question: Show that $$A=\lim_{n\to \infty}\sqrt{1+\sqrt{\dfrac{1}{2}+\sqrt{\dfrac{1}{3}+\cdots+\sqrt{\dfrac{1}{n}}}}}$$ exists, and find the best estimate limit $A$. It is easy to show that…
math110
- 95,328
- 17
- 156
- 531
83
votes
3 answers
Eigenvalues of the rank one matrix $uv^T$
Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in ${\mathbb R}^n$, $n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of $A$ since $$Au = (uv^T)u=u(v^Tu)=(v^Tu)u.$$ Here…
user9464