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1500 questions
118
votes
0 answers
On the Constant Rank Theorem and the Frobenius Theorem for differential equations.
Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the Implicit Function Theorem (finite-dimensional vector…
Elias Costa
- 14,658
118
votes
15 answers
The math behind Warren Buffett's famous rule – never lose money
This is a question about a mathematical concept, but I think I will be able to ask the question better with a little bit of background first.
Warren Buffett famously provided 2 rules to investing:
Rule No. 1: Never lose money. Rule No. 2: Never…
118
votes
5 answers
A multiplication algorithm found in a book by Paul Erdős: how does it work?
I am trying to understand the following problem from Erdős and Surányi's Topics in the theory of numbers (Springer), chapter 1 ("Divisibility, the Fundamental Theorem of Number Theory"):
We can multiply two (positive integer) numbers together in…
iadvd
- 8,875
118
votes
6 answers
How Do You Actually Do Your Mathematics?
Better yet, what I'm asking is how do you actually write your mathematics?
I think I need to give brief background: Through most of my childhood, I'd considered myself pretty good at math, up through the high school level. I easily followed…
Uticensis
- 3,311
118
votes
26 answers
If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself?
If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself?
For example the square of $2$ is $2^2=2 \cdot 2=4 $ .
But square root of $2$ is not $\frac{2}{2}=1$ .
bluebellae
- 1,623
117
votes
16 answers
Do mathematicians, in the end, always agree?
I've been trying to study some different sciences in my life, ranging from biology to mathematics, and if I try to explain to people why I like mathematics above the others, I think the most important reason for me is that mathematicians, in the…
Kasper
- 13,528
117
votes
4 answers
What are the Axiom of Choice and Axiom of Determinacy?
Would someone please explain:
What does the Axiom of Choice mean, intuitively?
What does the Axiom of Determinancy mean, intuitively, and how does it contradict the Axiom of Choice?
as simple words as possible?
From what I've gathered from the…
user541686
- 13,772
117
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
- 1,305
117
votes
6 answers
How often does it happen that the oldest person alive dies?
Today, we are brought the sad news that Europe's oldest woman died. A little over a week ago the oldest person in the U.S. unfortunately died. Yesterday, the Netherlands' oldest man died peacefully. The Gerontology Research Group keeps records:…
Řídící
- 3,210
117
votes
13 answers
What are some interpretations of Von Neumann's quote?
John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics.
Did he mean that with experience and practice,…
NebulousReveal
- 13,935
116
votes
14 answers
Can you give an example of a complex math problem that is easy to solve?
I am working on a project presentation and would like to illustrate that it is often difficult or impossible to estimate how long a task would take. I’d like to make the point by presenting three math problems (proofs, probably) that on the surface…
Judy
- 1,271
116
votes
8 answers
Are there real world applications of finite group theory?
I would like to know whether there are examples where finite group theory can be directly applied to solve real world problems outside of mathematics. (Sufficiently applied mathematics such as cryptography, coding theory, or statistics still…
Alexander Gruber
- 26,963
116
votes
1 answer
$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?
If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and $2n-2$. A proof can be found here.
Two weeks and…
Lincoln Blackham
- 1,161
116
votes
1 answer
Convergence of $\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$
Is there a way to assess the convergence of the following series?
$$\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$$
From numerical estimations it seems to be convergent but I don't know how to prove it.
Leonardo Massai
- 1,849
116
votes
11 answers
Prove that every convex function is continuous
A function $f : (a,b) \to \Bbb R$ is said to be convex if
$$f(\lambda x+(1-\lambda)y)\le \lambda f(x)+(1-\lambda)f(y)$$
whenever $a < x, y < b$ and $0 < \lambda <1$. Prove that every convex function is continuous.
Usually it uses the fact:
If $a…
cowik
- 1,203