Most Popular
1500 questions
172
votes
34 answers
Can you provide me historical examples of pure mathematics becoming "useful"?
I am trying to think/know about something, but I don't know if my base premise is plausible. Here we go. Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because it has no practical utility, but I guess that…
Red Banana
- 25,024
- 22
- 104
- 211
172
votes
1 answer
Classification of prime ideals of $\mathbb{Z}[X]$
Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$ . My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this? $(0)$ . $(f(X))$ , where $f(X)$ is an irreducible…
Makoto Kato
- 44,710
171
votes
7 answers
Is non-standard analysis worth learning?
As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides some formalism to this type of calculus. So, do you…
user13255
171
votes
1 answer
Rational roots of polynomials
Can one construct a sequence $(a_k)_{k\geqslant 0}$ of rational numbers such that, for every positive integer $n$ the polynomial $a_nX^n+a_{n-1}X^{n-1}+\cdots +a_0$ has exactly $n$ distinct rational roots ? If we cannot construct it explicitly, can…
user84673
- 2,057
171
votes
1 answer
Is there a homology theory that counts connected components of a space?
It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the corresponding statement would be that $\check{H}_0(X)$ is…
Dejan Govc
- 17,437
171
votes
4 answers
The Hole in One Pizza
In a recent issue of Crux, at the end of the editorial (which is public), it appears the following very nice problem by Peter Liljedahl. I couldn't resist sharing it with the MSE community. Enjoy!
Robert Z
- 147,775
171
votes
19 answers
What actually is a polynomial?
I can perform operations on polynomials. I can add, multiply, and find their roots. Despite this, I cannot define a polynomial. I wasn't in the advanced mathematics class in 8th grade, then in 9th grade I skipped the class and joined the more…
Travis
- 3,614
170
votes
2 answers
Example of infinite field of characteristic $p\neq 0$
Can you give me an example of infinite field of characteristic $p\neq0$? Thanks.
Aspirin
- 5,919
170
votes
7 answers
Is infinity an odd or even number?
My 6 year old wants to know if infinity is an odd or even number. His 38 year old father is keen to know too.
Kevin
- 1,711
169
votes
15 answers
Monty hall problem extended.
I just learned about the Monty Hall problem and found it quite amazing. So I thought about extending the problem a bit to understand more about it. In this modification of the Monty Hall Problem, instead of three doors, we have four (or maybe $n$)…
Shaurya Gupta
- 4,449
169
votes
10 answers
Intuitive interpretation of the Laplacian Operator
Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian Operator (a.k.a. divergence of gradient)?
koletenbert
- 4,170
169
votes
17 answers
Why does factoring eliminate a hole in the limit?
$$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5} = \lim_{x\rightarrow 5} (x+5)$$ I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and that the original limit is equal to the new and…
Emi Matro
- 5,113
169
votes
1 answer
Overview of basic facts about Cauchy functional equation
The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of mathematics ranging from exercises in freshman…
Martin Sleziak
- 56,568
169
votes
1 answer
What properties of busy beaver numbers are computable?
The busy beaver function $\text{BB}(n)$ describes the maximum number of steps that an $n$ -state Turing machine can execute before it halts (assuming it halts at all). It is not a computable function because computing it allows you to solve the…
Qiaochu Yuan
- 478,279
168
votes
26 answers
Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?
Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very slowly? I'd prefer an easily comprehensible explanation…
bryn
- 10,135