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1500 questions
61
votes
1 answer
Is there a "good" reason why $\left\lfloor \frac{n!}{11e}\right\rfloor$ is always even?
(A follow-up of sorts to this question.)
The quantity $\left\lfloor \frac{n!}{11e}\right\rfloor$ is always even, which can be proved as follows.
Using the sum for $\frac{1}{e}$, we split the fraction up into three parts:
$A_n=\sum_{k=0}^{n-11}…
Micah
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61
votes
7 answers
Is Complex Analysis equivalent Real Analysis with $f:\mathbb R^2 \to \mathbb R^2$?
Am I correct in noticing that Complex Analysis seems to be a synonym for analysis of functions $\mathbb R^2 \to \mathbb R^2$?
If this is the case, surely all the results from complex analysis carry over to the study of these $\mathbb R^2 \to…
providence
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61
votes
1 answer
Baby/Papa/Mama/Big Rudin
Recently, I was looking for the reviews of some Analysis books while encountered terms such as Baby/Papa/Mama/Big Rudin. Firstly, I thought that these are the names of a book! But it turned out that these are some nick names used for the books of…
Hosein Rahnama
- 14,882
61
votes
5 answers
Golden Number Theory
The Gaussian $\mathbb{Z}[i]$ and Eisenstein $\mathbb{Z}[\omega]$ integers have been used to solve some diophantine equations. I have never seen any examples of the golden integers $\mathbb{Z}[\varphi]$ used in number theory though. If anyone happens…
quanta
- 12,425
61
votes
9 answers
How many 7-note musical scales are possible within the 12-note system?
This combinatorial question has a musical motivation, which I provide below using as little musical jargon as I can. But first, I'll present a purely mathematical formulation for those not interested in the motivation:
Define a signature as a…
MGA
- 9,636
61
votes
8 answers
How to debug math?
May seem strange as I'm good in programming, but I just started diving into math. ATM I'm learning combinatorics at Khan Academy, and here's an example of a question that I struggled with (that's not the question):
The answer to «ways of getting…
61
votes
14 answers
Are there theoretical applications of trigonometry?
I am a high school student currently taking pre-calculus. We have just finished a unit on analytic trigonometry.
Are any purely theoretical uses for trigonometry? More specifically, can trigonometric concepts (or even functions) be used to…
Conan G.
- 1,102
61
votes
6 answers
How to avoid perceived circularity when defining a formal language?
Suppose we want to define a first-order language to do set theory (so we can formalize mathematics).
One such construction can be found here.
What makes me uneasy about this definition is that words such as "set", "countable", "function", and…
IssaRice
- 1,155
61
votes
7 answers
Is there an easy way to see associativity or non-associativity from an operation's table?
Most properties of a single binary operation can be easily read of from the operation's table. For example, given
$$\begin{array}{c|ccccc}
\cdot & a & b & c & d & e\\\hline
a & e & d & b & a & c\\
b & d & c & e & b & a\\
c & b & e & a &…
celtschk
- 43,384
61
votes
1 answer
Generating correlated random numbers: Why does Cholesky decomposition work?
Let's say I want to generate correlated random variables. I understand that I can use Cholesky decomposition of the correlation matrix to obtain the correlated values. If $C$ is the correlation matrix, then we can do the cholesky…
Flux Capacitor
- 773
61
votes
12 answers
What are the Laws of Rational Exponents?
On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number seems to be shown as equal to its opposite…
Daniel R. Collins
- 8,380
61
votes
2 answers
What are exact sequences, metaphysically speaking?
Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a very useful and efficient framework to work…
Joshua Seaton
- 2,218
61
votes
12 answers
$\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite
How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
Breton
- 1,638
61
votes
11 answers
How to calculate the area of a 3D triangle?
I have coordinates of 3d triangle and I need to calculate its area. I know how to do it in 2D, but don't know how to calculate area in 3d. I have developed data as follows.
(119.91227722167969, 122.7717056274414, 39.3568115234375),…
iamgopal
- 643
61
votes
3 answers
"Every linear mapping on a finite dimensional space is continuous"
From Wiki
Every linear function on a finite-dimensional space is continuous.
I was wondering what the domain and codomain of such linear function are?
Are they any two topological vector spaces (not necessarily the same), as along as the domain is…
Tim
- 47,382