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.

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56
votes
10answers
3k views

Big List of Erdős' elementary proofs

Paul Erdős was one of the greatest mathematicians of all time and he was famous for his elegant proofs from The Book. I posted a question about one of his theorem and got a reference, and I have other ...
1
vote
1answer
49 views

In which topologies do open sets maintain open under countable or arbitrary intersection?

We know that in the usual topology, countable or arbitrary intersection of open sets can zoom into a singleton, hence is not in the topology. I am curious if there is well known classes of ...
31
votes
11answers
3k views

Are there any interesting semigroups that aren't monoids?

Are there any interesting and natural examples of semigroups that are not monoids (that is, they don't have an identity element)? To be a bit more precise, I guess I should ask if there any ...
2
votes
4answers
55 views

Proving convergence or divergence of series: Tips and Tricks

I currently write an article where I collect some tips for students for proving the convergence or divergence of series. What tips and tricks do you know or use or teach? Remark: I will add some ...
36
votes
16answers
4k views

Intuitive Understanding of the constant “$e$”

Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate. I know there are many ways of calculating (or should I say "...
239
votes
30answers
18k views

A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language

The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...
6
votes
2answers
169 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 ...
64
votes
16answers
2k views

Unconventional mathematics books

I've recently purchased Oliver Byrne's reproduction of Euclid's Elements. It's a beautiful tome, that's rather unique in its presentation of the material as it represents many of Euclid's proof as ...
2
votes
1answer
37 views

Seeking advice from the more experienced on which trig identities are crucial to memorize and which can be derived quickly

This is a bit of a two part question. I also have read some of the related questions, but I think mine is different as whether they can be derived quickly, rather than whether they can be derived, is ...
425
votes
34answers
49k 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 ...
13
votes
8answers
434 views

The proofs of the fundamental Theorem of Algebra [closed]

There are many proofs of the fundamental theorem of algebra. Which are the most beautiful proofs?
21
votes
1answer
969 views

What Do Mathematicians Do?

The American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: What ...
75
votes
19answers
31k views

Real life applications of Topology

The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here What ...
9
votes
1answer
82 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 ...
98
votes
27answers
9k 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 ...
6
votes
2answers
80 views

Noteworthy examples of finite categories

So far all the finite categories I have encountered fall into one of these c̶a̶t̶e̶g̶o̶r̶i̶e̶s̶ sets: finite monoids finite preorders just formal devices to explain, what a "diagram" in another (...
1
vote
3answers
72 views

Recommend book Taylor expansions

I've taken up self-study of math and i start using the book called : Mathematical Analysis I Authors: Canuto, Claudio, Tabacco, Anita I would like to start from zero to understand taylor ...
2
votes
3answers
225 views

Analytical mechanics book

On my PHD I have to learn the following subject: Analysis on manifolds and Analytical mechanics. But my book is really not good to read; it is too hard. So I need some book that explains to me ...
1
vote
1answer
175 views

Intuitive functional analysis book

I would like a functional analysis book like Terence Tao's Real Analysis and Measure Theory book, full of intuition. I am ...
1
vote
1answer
166 views

Representation Theory Symmetric Group Book?

I'm looking for a nice book that discusses the representation theory of the symmetric group. My background is an introductory class in representation theory.
0
votes
2answers
111 views

Good book on Lebesgue Theory [duplicate]

I am a graduate student and I need a suggestion for a good book in Lebesgue Measure Theory with good exercises and if its possibly with hints or solutions. Thank you.
55
votes
19answers
17k views

Good Book On Combinatorics

What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions ...
5
votes
4answers
533 views

Graph Theory for Dummies Book [duplicate]

Does anyone have a good book on Graph Theory that will introduce me to some of the basic concepts without being so filled with terminology that it's hard to read? I have taken an introductory course (...
0
votes
0answers
113 views

Abstract Algebra book

Has anyone read the book "Abstract Algebra: An Introduction" by Thomas Hungerford? I'm in the class Intro to Abstract Algebra, and my professor is, to say the least, not very great at teaching. Great ...
6
votes
0answers
83 views

Advanced stochastic process book

I am looking for the book about advanced stochastic process . It may cover the following content: Stochastic matrices. Ex: $A(k)$, where $k$ is the time index. Stochastic process in space (...
9
votes
6answers
2k views

Distribution theory book

I'm looking for a good book on distribution theory (in the Schwartz sense), I have the basic knowledge as given in Grafakos' Classical Fourier Analysis, but I want to know more about it. Is the ...
3
votes
4answers
464 views

Abstract Algebra Book Request

I am looking for a good undergraduate level book on Abstract Algebra. By a 'good book' I mean a book which gives equal importance to both, rigor and the historical perspective of the subject. For ...
8
votes
5answers
3k views

Proofs of Sylow theorems

It seems that there are many ways to prove the Sylow theorems. I'd like to see a collection of them. Please write down or share links to any you know.
86
votes
23answers
34k views

Complete course of self-study [closed]

I am about $16$ years old and I have just started studying some college mathematics. I may never manage to get into a proper or good university (I do not trust fate) but I want to really study ...
105
votes
33answers
13k views

What are the most overpowered theorems in mathematics? [closed]

What are the most overpowered theorems in mathematics? By "overpowered," I mean theorems that allow disproportionately strong conclusions to be drawn from minimal / relatively simple assumptions. I'...
0
votes
1answer
666 views

How many ways to find the center of an inscribed circle?

I want to find the coordinates of center of the inscribed circle triangle $ABC$, where $A(-274,-253)$, $B(-1,7)$, $C(14,7)$. I tried First way. We have $c = AB=377$, $a = BC=15$, $b = AC=388$. Let $...
3
votes
3answers
188 views

What are the most obscure or advanced mathematics with practical application

Throughout my engineering studies there were jokes made by my professors (mostly mathematics professors) that referenced the fact that pure mathematicians strive to create mathematics with no ...
233
votes
7answers
17k views

Best Sets of Lecture Notes and Articles

Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting ...
88
votes
9answers
20k views

Lesser-known integration tricks

I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now ...
4
votes
2answers
88 views

What properties of the real numbers are almost always true and there are no (or very few) known examples of?

What properties of the positive real numbers are almost always true and there are no (or very few) known examples of? Two that come to mind are numbers that are normal in every base and numbers ...
96
votes
20answers
20k views

Visually deceptive “proofs” which are mathematically wrong

Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually ...
3
votes
1answer
22 views

Seeking Additional Solutions for the Number of Network Links

The Problem Show that the number of possible links in a computer network of $n$ computers ($n \in Z \land n \geq 1$) is $\frac{n(n-1)}{2}$ in as many ways as you can. My Work Solution 1 Given $n$ ...
42
votes
10answers
13k views

Examples of finite nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
46
votes
13answers
7k views

List of interesting integrals for early calculus students

I am teaching Calc 1 right now and I want to give my students more interesting examples of integrals. By interesting, I mean ones that are challenging, not as straightforward (though not extremely ...
1
vote
1answer
152 views

Indefinite Integral challenge problems [closed]

This year I am going to participate in an olympiad of indefinite integrals. The level is very high, I would like to know some (hard, olympiad) Indefinite integrals challenge problems Note: Here is ...
2
votes
1answer
1k views

List of Common or Useful Limits of Sequences and Series

There are many sequences or series which come up frequently, and it's good to have a directory of the most commonly used or useful ones. I'll start out with some. Proofs are not required. $$\begin{...
633
votes
162answers
39k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament,...
12
votes
5answers
456 views

Applications of Character Theory

Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
2
votes
0answers
61 views

Problem sets on Abstract Algebra

Many times we ask about what books should we read to learn or know more about a math topic (Abstract Algebra, in this case). However, I would like to get a list of the exercises what should we solve ...
46
votes
10answers
1k views

What are Different Approaches to Introduce the Elementary Functions?

Motivation We all get familiar with elementary functions in high-school or college. However, as the system of learning is not that much integrated we have learned them in different ways and the ...
11
votes
1answer
170 views

Lonely theorems [closed]

What are some instances of theorems which are especially unique in mathematics, i.e. for which there are not many other theorems of a similar character? An example I have in mind is Pick's theorem, ...
17
votes
4answers
651 views

Tough integrals that can be easily beaten by using simple techniques

This question is just idle curiosity. Today I find that an integral problem can be easily evaluated by using simple techniques like my answer to evaluate \begin{equation} \int_0^{\pi/2}\frac{\cos{x}}{...
2
votes
2answers
63 views

Proving/Disproving $M$ has the structure of an $R$-module

Given an abelian group $M$ and a ring $R$, how can one prove or disprove that $M$ has the structure of an $R$-module? When proving $M$ is an $R$-module, if it is not obvious how to define an action $R\...
18
votes
2answers
482 views

Most wanted reproducible results in computational algebra

I am interested in suggestions for major computational results obtained with the help of mathematical software but not easily verifiable using computers. "Most wanted" could refer, for example, to ...
0
votes
0answers
33 views

Conflicting conjectures [duplicate]

I feel like when two conjectures are inconsistent with one another, it's a clear sign of our misunderstanding of deeper mathematics. I was wondering if anyone knew of a comprehensive list of ...