# Tagged Questions

Questions asking for a "big list" of examples, illustrations, etc. Ask only when the topic is compelling, and please do not use this as the only tag for a question.

20answers
8k views

### Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
2answers
128 views
+400

### How many ways are there to prove Cayley-Hamilton Theorem?

I see many proofs for Cayley-Hamilton Theorem in textbooks and net, so I want to know how many proofs are there for this important and applicable theorem.
22answers
11k views

### Most ambiguous and inconsistent phrases and notations in maths

What are some examples of notations and words in maths which have been overused or abused to the point of them being almost completely ambiguous when presented in new contexts? For instance, a ...
1answer
12 views

### Calculating the amount of times a binary search could run (worse case) without a calculator/calculating base 2 logs without a calculator.

Ok so I had a question on a test that I had to do without a calculator. And I can not figure out how in the world I am supposed to do it without a calculator. The question asked to find how many ...
6answers
837 views

### Open source lecture notes and textbooks

This question is inspired by the popular "Best Sets of Lecture Notes and Articles". Indeed, I would like to collect a "big-list" of open source (that is, with $\LaTeX$ code available) high-quality ...
10answers
9k views

### Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
0answers
68 views

### Active, beautiful fields of mathematics [on hold]

Some fields of mathematics are attributed more beauty than others. There are many historical quotes of mathematicians celebrating the beauty of number theory -of which I know little- in comparison to ...
9answers
858 views

### Elevator pitch for a (sub)field of maths?

When I first saw the title of this question, I forgot for a moment I was on meta, and thought it was asking about quick, catchy, attractive, informative one-or-two-liner summaries of various fields of ...
1answer
36 views

0answers
59 views

### Categorical formulations of basic results and ideas from functional analysis?

I'm taking a first (undergrad) course on functional analysis. Though the material is nice, the approach seems very ad hoc and in a sense, near-sighted (?). I was wondering whether the/a big picture ...
29answers
20k views

### Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
0answers
42 views

### What are some interesting results that could be derived if some conjectures were true/false?

I recently came up on Djikstra's idea of a fictional company called Mathematics.Inc which produced proofs as trade secrets which could then be used by customers.. For example, it produced the proof of ...
2answers
156 views

### Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
1answer
142 views

### When adding zero really counts …

Note: Although adding zero has usually no effect, there is sometimes a situation where it is the essence of a calculation which drives the development into a surprisingly fruitful direction. Here is ...
30answers
46k views

### Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
1answer
30 views

### Exceptions in infinite-dimensional spaces

What are the properties that are true in finite-dimensional spaces but fails in the infinite-dimensional space? For example, the closed unit ball is compact only in finite-dimensional normed space.
7answers
3k views

### What are some common proof strategies in mathematics?

I want to start out by saying that I am new at proof based mathematics. I am used to seeing patterns and using them to solve similar problems. However, I have found this is not a very good way to ...
1answer
44 views

### Books with SAGE portions

I recently finished working through Adventures in Group Theory and really appreciated the use of SageMath it employs. I considered myself moderately proficient with Sage, but I found working through ...
0answers
47 views

### Curvature and topology

I am studying Riemannian Geometry and I came across various Theorems which give conditions on the topology of a manifold given conditions on curvature, and vice-versa. Just to mention a few of them: ...