Most Popular
1500 questions
106
votes
5 answers
Why are gauge integrals not more popular?
A recent answer reminded me of the gauge integral , which you can read about here . It seems like the gauge integral is more general than the Lebesgue integral, e.g. if a function is Lebesgue integrable, it is gauge integrable. (EDIT - as Qiaochu…
Chris Brooks
- 7,841
106
votes
7 answers
Mathematicians' Tensors vs. Physicists' Tensors
It seems, at times, that physicists and mathematicians mean different things when they say the word "tensor." From my perspective, when I say tensor, I mean "an element of a tensor product of vector spaces." For instance, here is a segment about…
msm
- 2,926
106
votes
3 answers
Can a limit of an integral be moved inside the integral?
After coming across this question: How to verify this limit , I have the following question: When taking the limit of an integral, is it valid to move the limit inside the integral, providing the limit does not affect the limits of integration? For…
Thomas Russell
- 10,585
106
votes
3 answers
How to evaluate $\int_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x$ using complex analysis?
We were told today by our teacher (I suppose to scare us) that in certain schools for physics in Soviet Russia there was as an entry examination the following integral given $$\int\limits_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1}…
user505183
106
votes
9 answers
Why is the product of two rotation matrices not commutative?
Is there any intuition why rotational matrices are not commutative? I assume the final rotation is the combination of all rotations. Then how does it matter in which order the rotations are applied?
Navin Prashath
- 1,195
106
votes
10 answers
How is a system of axioms different from a system of beliefs?
Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
Gabriel
- 1,385
106
votes
18 answers
Is arrow notation for vectors "not mathematically mature"?
Assuming that we can't bold our variables (say, we're writing math as opposed to typing it), is it "not mathematically mature" to put an arrow over a vector? I ask this because in my linear algebra class, my professor never used arrow notation, so…
hlin117
- 1,201
105
votes
14 answers
Why can't you square both sides of an equation?
Why can't you square both sides of an equation? I've been asked this many times and can never quite give a good, clear, concise answer (for beginning algebra students) in plain language. I just searched the web and still couldn't find a…
Jeff
- 3,495
105
votes
5 answers
Prove that the set of all algebraic numbers is countable
A complex number $z$ is said to be algebraic if there are integers $a_0, ..., a_n$, not all zero, such that $a_0z^n+a_1z^{n-1}+...+a_{n-1}z+a_n=0$. Prove that the set of all algebraic numbers is countable. The Hint is: For every positive integer $N$…
PandaMan
- 3,377
105
votes
8 answers
Why is $\frac{987654321}{123456789} = 8.0000000729?!$
Many years ago, I noticed that $987654321/123456789 = 8.0000000729\ldots$. I sent it in to Martin Gardner at Scientific American and he published it in his column!!! My life has gone downhill since then:) My questions are: Why is this so? What…
marty cohen
- 110,916
105
votes
4 answers
Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$
It's not difficult to show that $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$ On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing terms gives us $$\sum_{k=0}^n…
Skatche
- 1,590
105
votes
2 answers
Predicting Real Numbers
Here is an astounding riddle that at first seems impossible to solve. I'm certain the axiom of choice is required in any solution, and I have an outline of one possible solution, but would like to see how others might think about it. $100$ rooms…
Jared
- 32,247
105
votes
12 answers
Is there a bijective map from $(0,1)$ to $\mathbb{R}$?
I couldn't find a bijective map from $(0,1)$ to $\mathbb{R}$. Is there any example?
ieb
- 1,067
105
votes
4 answers
A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
As the title says, I would like to launch a community project for proving that the series $$\sum_{n\geq 1}\frac{\sin(2^n)}{n}$$ is convergent. An extensive list of considerations follows. The first fact is that the inequality $$…
Jack D'Aurizio
- 372,937
- 42
- 421
- 888
105
votes
8 answers
Does mathematics become circular at the bottom? What is at the bottom of mathematics?
I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign), functions and relations. Now is the "=" taken…
user119615
- 10,652
- 6
- 49
- 125