Most Popular
1500 questions
78
votes
1 answer
Overview of basic results on cardinal arithmetic
Are there some good overviews of basic formulas about addition, multiplication and exponentiation of cardinals (preferably available online)?
Martin Sleziak
- 53,687
78
votes
23 answers
An example of a problem which is difficult but is made easier when a diagram is drawn
I am writing a blog post related to problem solving and one of the main techniques used in problem solving is drawing a diagram. Essentially, I want to illustrate that some hard problems (for example, word problems) can be done fairly easily when…
Jeel Shah
- 9,306
77
votes
12 answers
The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$
What is the sum of the 'second half' of the harmonic series?
$$\lim_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$
More precisely, what is the limit of the above sequence of partial sums?
Daniel Pietrobon
- 4,761
77
votes
6 answers
Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$
I've just started to learn about the tensor product and I want to show: $$(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}.$$
Can you tell me if my proof is right:
$\mathbb{Z}/m\mathbb{Z}$…
Rudy the Reindeer
- 46,359
77
votes
4 answers
Why does the google calculator give $\tan 90^{\circ} = 1.6331779e^{+16}$?
I typed in $\tan 90^{\circ}$ in Google and it gave $1.6331779\mathrm{E}16$. How did it come to this answer? Limits? Some magic?
Gizmo
- 939
77
votes
5 answers
Theorems that disappointed mathematicians
I was looking for a list of theorems that disappointed mathematicians, in the sense that some disgruntlement regarding the fact that the theorem in consideration being true/false was notable. An example could be a theorem whose negation was notably…
QED
- 867
77
votes
6 answers
Reference request: introduction to commutative algebra
My goal is to pick up some commutative algebra, ultimately in order to be able to understand algebraic geometry texts like Hartshorne's. Three popular texts are Atiyah-Macdonald, Matsumura (Commutative Ring Theory), and Eisenbud. There are also…
77
votes
13 answers
Why are addition and multiplication commutative, but not exponentiation?
We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not. My question is why.
As background there are plenty of mathematical schemes that can be used to define these operators. One of these…
rob levin
- 771
77
votes
10 answers
Is learning (theoretical) physics useful/important for a mathematician?
I'm starting to read The Princeton Companion to Mathematics, at the beginning it says:
A proper appreciation of
pure mathematics requires some knowledge of applied
mathematics and theoretical physics.
Some of my professors have told me that…
Vicfred
- 2,787
77
votes
8 answers
How do I prove that a function is well defined?
How do you in general prove that a function is well-defined?
$$f:X\to Y:x\mapsto f(x)$$
I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the following two things:
Every element in the domain…
Kasper
- 13,528
77
votes
1 answer
What does the mysterious constant marked by C on a slide rule indicate?
Years ago, before everyone (or anyone) had electronic calculators, I had a pocket slide rule which I used in secondary school until the first TI-30 cane out.
Recently I dug it out. Here's a photo of one end of it.
As you can see, there's a number…
timtfj
- 2,932
77
votes
5 answers
How are mathematicians taught to write with such an expository style?
I wasn't sure if this question was appropriate for MSE.
One of the major complaints we see in industry is a person's ability to communicate which includes writing.
We see the same thing on questions that are posted on MSE by younger students.…
Amzoti
- 56,093
77
votes
11 answers
Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$?
It seems as if no one has asked this here before, unless I don't know how to search.
The Gamma function is
$$
\Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx.
$$
Why is
$$
\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\text{ ?}
$$
(I'll post my own…
77
votes
4 answers
Gaussian distribution is isotropic?
I was in a seminar today and the lecturer said that the gaussian distribution is isotropic. What does it mean for a distribution to be isotropic? It seems like he is using this property for the pseudo-independence of vectors where each entry is…
Astaboom
- 957
77
votes
7 answers
Is "The empty set is a subset of any set" a convention?
Recently I learned that for any set A, we have $\varnothing\subset A$.
I found some explanation of why it holds.
$\varnothing\subset A$ means "for every object $x$, if $x$ belongs to the empty set, then $x$ also belongs to the set A". This is a…
Searene
- 981