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1500 questions
83
votes
3 answers
What is the "standard basis" for fields of complex numbers?
What is the "standard basis" for fields of complex numbers?
For example, what is the standard basis for $\Bbb C^2$ (two-tuples of the form: $(a + bi, c + di)$)? I know the standard for $\Bbb R^2$ is $((1, 0), (0, 1))$. Is the standard basis…
Casey Patton
- 1,453
83
votes
11 answers
Fake induction proofs
Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do not answer with the very common all horses are the…
Daniel W. Farlow
- 22,531
83
votes
8 answers
Choose a random number between $0$ and $1$ and record its value. Keep doing it until the sum of the numbers exceeds $1$. How many tries do we need?
Choose a random number between $0$ and $1$ and record its value. Do this again and add the second number to the first number. Keep doing this until the sum of the numbers exceeds $1$. What's the expected value of the number of random numbers needed…
user25329
- 1,037
- 1
- 9
- 7
83
votes
1 answer
Numbers $n$ such that the digit sum of $n^2$ is a square
Let $S(n)$ be the digit sum of $n\in\mathbb N$ in the decimal system. About a month ago, a friend of mine taught me the…
mathlove
- 139,939
82
votes
4 answers
Gradient of 2-norm squared
Could someone please provide a proof for why the gradient of the squared $2$-norm of $x$ is equal to $2x$?
$$\nabla\|x\|_2^2 = 2x$$
user167133
- 943
82
votes
6 answers
What's so "natural" about the base of natural logarithms?
There are so many available bases. Why is the strange number $e$ preferred over all else?
Of course one could integrate $\frac{1}x$ and see this. But is there more to the story?
user218
82
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
82
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…
math110
- 93,304
82
votes
1 answer
How to solve fifth-degree equations by elliptic functions?
From F. Klein's books, It seems that one can find the roots of a quintic equation
$$z^5+az^4+bz^3+cz^2+dz+e=0$$
(where $a,b,c,d,e\in\Bbb C$) by elliptic functions. How to get that?
Update: How to transform a general higher degree five or higher…
ziang chen
- 7,771
82
votes
15 answers
Simplest proof of Taylor's theorem
I have for some time been trawling through the Internet looking for an aesthetic proof of Taylor's theorem.
By which I mean this: there are plenty of proofs that introduce some arbitrary construct: no mention is given of from whence this beast came.…
P i
- 2,136
82
votes
1 answer
In $n>5$, topology = algebra
During the study of the surgery theory I faced following sentence:
Surgery theory works best for $n > 5$, when
"topology = algebra".
I don't know what is the meaning of topology=algebra. Can someone clear the sentence to me?
Sepideh Bakhoda
- 5,816
82
votes
3 answers
Escaping infinitely many pursuers
The fugitive is at the origin. They move at a speed of 1. There's a guard at every integer coordinate except the origin. A guard's speed is 1/100. The fugitive and the guards move simultaneously and continuously. At any moment, the guards only move…
Eric
- 1,909
82
votes
6 answers
If a two variable smooth function has two global minima, will it necessarily have a third critical point?
Assume that $f:\mathbb{R}^2\to\mathbb{R}$ a $C^{\infty}$ function that has exactly two minimum global points. Is it true that $f$ has always another critical point?
A standard visualization trick is to imagine a terrain of height $f(x,y)$ at the…
Jyrki Lahtonen
- 133,153
82
votes
1 answer
Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$
Consider the following integral:
$$\mathcal{I}(\mu,\nu)=\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx,$$
where $J_\mu(x)$ is the Bessel function of the first kind:
$$J_\mu(x)=\sum…
Vladimir Reshetnikov
- 47,122
82
votes
13 answers
What does "surface area of a sphere" actually mean (in terms of elementary school mathematics)?
I know what "surface area" means for:
a 2d shape
a cylinder or cone
but I don't know what it actually means for a sphere.
For a 2d shape
Suppose I'm given a 2d shape, such as a rectangle, or a triangle, or a drawing of a puddle. I can cut out a…
silph
- 775