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1500 questions
127
votes
7 answers
Is infinity a number?
Is infinity a number? Why or why not?
Some commentary:
I've found that this is an incredibly simple question to ask — where I grew up, it was a popular argument starter in elementary school — but a difficult one to answer in an intelligent manner.…
Pops
- 1,405
127
votes
7 answers
What is the difference between a class and a set?
I know what a set is. I have no idea what a class is.
As best as I can make out, every set is also a class, but a class can be "larger" than any set (a so-called "proper class").
This obviously makes no sense whatsoever, since sets are of unlimited…
MathematicalOrchid
- 6,365
127
votes
19 answers
What parts of a pure mathematics undergraduate curriculum have been discovered since $1964?$
What parts of an undergraduate curriculum in pure mathematics have been discovered since, say, $1964?$ (I'm choosing this because it's $50$ years ago). Pure mathematics textbooks from before $1964$ seem to contain everything in pure maths that is…
Suzu Hirose
- 11,949
126
votes
2 answers
What makes a theorem "fundamental"?
I've studied three so-called "fundamental" theorems so far (FT of Algebra, Arithmetic and Calculus) and I'm still unsure about what precisely makes them fundamental (or moreso than other theorems).
Wikipedia claims:
The fundamental theorem of a…
beep-boop
- 11,825
126
votes
9 answers
Why is $\cos (90)=-0.4$ in WebGL?
I'm a graphical artist who is completely out of my depth on this site.
However, I'm dabbling in WebGL (3D software for internet browsers) and trying to animate a bouncing ball.
Apparently we can use trigonometry to create nice smooth curves.…
Starkers
- 1,247
126
votes
0 answers
Ring structure on the absolute Galois group of a finite field
Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the absolute Galois group carries the structure of a topological ring isomorphic to…
Martin Brandenburg
- 181,922
126
votes
5 answers
What concept does an open set axiomatise?
In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ converges to a point also in $F$. This naturally…
Zhen Lin
- 97,105
126
votes
11 answers
Find the average of $\sin^{100} (x)$ in 5 minutes?
I read this quote attributed to VI Arnold.
"Who can't calculate the average value of the one hundredth power of the sine function within five minutes, doesn't understand mathematics - even if he studied supermanifolds, non-standard calculus or…
Please Delete Account
126
votes
3 answers
More than 99% of groups of order less than 2000 are of order 1024?
In Algebra: Chapter 0, the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024.
Is this for real? How can one deduce this result? Is there a nice way or do we just check all…
Hui Yu
- 15,469
126
votes
4 answers
Motivation for Ramanujan's mysterious $\pi$ formula
The following formula for $\pi$ was discovered by Ramanujan:
$$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$
Does anyone know how it works, or what the motivation for it is?
Nick Alger
- 19,977
126
votes
3 answers
Does convergence in $L^p$ imply convergence almost everywhere?
If I know $\| f_n - f \|_{L^p(\mathbb{R})} \to 0$ as $n \to \infty$, do I know that $\lim_{n \to \infty}f_n(x) = f(x)$ for almost every $x$?
187239
- 1,335
126
votes
16 answers
What is the smallest unknown natural number?
There are several unknown numbers in mathematics, such as optimal constants in some inequalities.
Often it is enough to some estimates for these numbers from above and below, but finding the exact values is also interesting.
There are situations…
Joonas Ilmavirta
- 26,345
126
votes
3 answers
When is the closure of an open ball equal to the closed ball?
It is not necessarily true that the closure of an open ball $B_{r}(x)$ is equal
to the closed ball of the same radius $r$ centered at the same point $x$. For a quick example, take $X$ to be any set and define a…
Alex Lapanowski
- 3,136
- 2
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126
votes
1 answer
Solving Special Function Equations Using Lie Symmetries
The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's equation.
Kaufman's article describes algebraic methods…
bolbteppa
- 4,589
125
votes
16 answers
Calculating the integral $\int_0^\infty \frac{\cos x}{1+x^2}\, \mathrm{d}x$ without using complex analysis
Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form?
$$\int_0^\infty\frac{\cos x}{1+x^2}\,\mathrm{d}x$$
Martin Gales
- 7,927