Most Popular
1500 questions
229
votes
10 answers
What does $2^x$ really mean when $x$ is not an integer?
We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so please let me know what you think.
David G
- 4,337
228
votes
4 answers
How many fours are needed to represent numbers up to $N$?
The goal of the four fours puzzle is to represent each natural number using four copies of the digit $4$ and common mathematical symbols.
For example, $165=\left(\sqrt{4} + \sqrt{\sqrt{{\sqrt{4^{4!}}}}}\right) \div .4$.
If we remove the restriction…
David Bevan
- 5,891
228
votes
4 answers
Why can a Venn diagram for $4+$ sets not be constructed using circles?
This page gives a few examples of Venn diagrams for $4$ sets. Some examples:
Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complete $4$-set Venn diagram using only circles as we could…
Larry Wang
- 9,833
227
votes
2 answers
Proving you *can't* make $2011$ out of $1,2,3,4$: nice twist on the usual
An undergraduate was telling me about a puzzle he'd found: the idea was to make $2011$ out of the numbers $1, 2, 3, 4, \ldots, n$ with the following rules/constraints: the numbers must stay in order, and you can only use $+$, $-$, $\times$, $/$, ^…
Kevin Buzzard
- 4,908
226
votes
21 answers
Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$
How to prove
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
Jichao
- 8,182
224
votes
5 answers
What do modern-day analysts actually do?
In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about the classification of finite simple groups, and…
Jesse Madnick
- 32,819
224
votes
6 answers
What is the geometric interpretation of the transpose?
I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the transpose, or even symmetric matrices.
For example,…
Zed
- 2,241
224
votes
5 answers
The sum of an uncountable number of positive numbers
Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$
such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many $\alpha\in A$ ($A$ need not be countable).
Proof: Let…
Benji
- 6,120
223
votes
13 answers
Why is compactness so important?
I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel theorem, and a proof continuous functions on R from…
FireGarden
- 6,031
222
votes
4 answers
Overview of basic results about images and preimages
Are there some good overviews of basic facts about images and inverse images of sets under functions?
Martin Sleziak
- 56,060
222
votes
16 answers
Optimizing response times of an ambulance corp: short-term versus average
Background: I work for an Ambulance service. We are one of the largest ambulance services in the world. We have a dispatch system that will always send the closest ambulance to any emergency call. There is a belief that this results in the fastest…
Laurence
- 1,493
221
votes
7 answers
Transpose of inverse vs inverse of transpose
Given a square matrix,
is the transpose of the inverse equal to the inverse of the transpose?
$$
(A^{-1})^T = (A^T)^{-1}
$$
Void Star
- 2,665
221
votes
18 answers
Do men or women have more brothers?
Do men or women have more brothers?
I think women have more as no man can be his own brother. But how one can prove it rigorously?
I am going to suggest some reasonable background assumptions:
There are a large number of individuals, of whom half…
layman
- 1,887
219
votes
4 answers
Some users are mind bogglingly skilled at integration. How did they get there?
Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this.
Especially in a world where most scientists compute their integrals…
JessicaK
- 8,007
218
votes
13 answers
Is computer science a branch of mathematics?
I have been wondering, is computer science a branch of mathematics? No one has ever adequately described it to me. It all seems very math-like to me. My second question is, are there any books about computer science/programming that are very…
user107952
- 23,768