Most Popular
1500 questions
253
votes
11 answers
"Integral milking": working backward to construct nontrivial integrals
I begin this post with a plea: please don't be too harsh with this post for being off topic or vague. It's a question about something I find myself doing as a mathematician, and wonder whether others do it as well. It is a soft question about…
Franklin Pezzuti Dyer
- 39,754
- 9
- 73
- 166
252
votes
5 answers
What is the difference between linear and affine function?
I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated.
user34790
- 4,192
252
votes
3 answers
Why study Algebraic Geometry?
I'm going to start self-stydying algebraic geometry very soon. So, my question is why do mathematicians study algebraic geometry? What are the types of problems in which algebraic geometers are interested in? And what are some of the most beautiful…
Mohan
- 14,856
251
votes
7 answers
Can you answer my son's fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?
My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question:
I am thinking of a number...
It is prime.
The digits add up to $10.$
It has a $3$ in the…
JDH
- 44,236
251
votes
4 answers
The Integral that Stumped Feynman?
In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods that he could not do with real…
Argon
- 25,303
249
votes
27 answers
Best Fake Proofs? (A M.SE April Fools Day collection)
In honor of April Fools Day $2013$, I'd like this question to collect the best, most convincing fake proofs of impossibilities you have seen.
I've posted one as an answer below. I'm also thinking of a geometric one where the "trick" is that it's…
Potato
- 40,171
249
votes
9 answers
What is the importance of the Collatz conjecture?
I have been fascinated by the Collatz problem since I first heard about it in high school.
Take any natural number $n$. If $n$ is even, divide it by $2$ to get $n / 2$, if $n$ is odd multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the…
Dan Brumleve
- 17,796
248
votes
6 answers
What books must every math undergraduate read?
I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden).
I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on if I want to study beyond Masters. I'm not there…
Adam
- 1,978
246
votes
26 answers
What are some examples of when Mathematics 'accidentally' discovered something about the world?
I do not remember precisely what the equations or who the relevant mathematicians and physicists were, but I recall being told the following story. I apologise in advance if I have misunderstood anything, or just have it plain wrong. The story is as…
Trogdor
- 10,331
243
votes
1 answer
Is this really a categorical approach to integration?
Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical Approach to Integration".
The Abstract:
"We…
Shaun
- 44,997
242
votes
11 answers
Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?
If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer.
If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ appears in the sum, but $\frac{1}{2p}$ does not. Aside from…
Anton Geraschenko
- 4,580
241
votes
11 answers
What is the result of $\infty - \infty$?
I would say $\infty - \infty=0$ because even though $\infty$ is an undetermined number, $\infty = \infty$. So $\infty-\infty=0$.
Pacerier
- 3,369
240
votes
8 answers
What are the Differences Between a Matrix and a Tensor?
What is the difference between a matrix and a tensor? Or, what makes a tensor, a tensor? I know that a matrix is a table of values, right? But, a tensor?
Aurelius
- 2,801
238
votes
35 answers
What are some counter-intuitive results in mathematics that involve only finite objects?
There are many counter-intuitive results in mathematics, some of which are listed here. However, most of these theorems involve infinite objects and one can argue that the reason these results seem counter-intuitive is our intuition not working…
Burak
- 3,696
236
votes
27 answers
Good books and lecture notes about category theory.
What are the best books and lecture notes on category theory?
