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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106
votes
44answers
12k views

What's your favorite proof accessible to a general audience? [closed]

What math statement with proof do you find most beautiful and elegant, where such is accessible to a general audience, meaning you could state, prove, and explain it to a general audience in ...
4
votes
2answers
39 views

What are statements about the natural numbers where induction is impossible or unnecessary to prove?

I'm looking for statements like "for all natural numbers, ____" where induction would be impossible or unnecessarily complicated. This is for pedagogical reasons. When students first learn induction, ...
14
votes
4answers
2k views

problem books in functional analysis

There are many excellent problem books in real analysis. I'm looking for a problem book in functional analysis or a book which contains a lot of problems in functional analysis (Easy and hard ...
2
votes
1answer
91 views

Challenging problems in algebra (book recommendation)

Could you suggest me a book/web page where I can find challenging/hard problems in algebra (possibly with solutions) for an undergraduate student (groups, rings, fields, Galois theory)? Thanks in ...
306
votes
26answers
30k views

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

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
75
votes
22answers
2k views

Open mathematical questions for which we really, really have no idea what the answer is

There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both ...
3
votes
1answer
76 views

Suplement books for calculus course?

I'm looking for books that could enhance the learning of calculus. At the moment, I have the following titles: Counterexamples in Calculus; Irresistible integrals; Inside calculus; The Calculus ...
20
votes
10answers
416 views

What are the theorems in mathematics which can be proved using completely different ideas?

I would like to know about theorems which can give different proofs using completely different techniques. For example: When I read from the book Proof from the Book, I saw there were ...
3
votes
1answer
3k views

Large numbers in real world

I am a high school math teacher and I am looking for a comprehensive list of large numbers which occur in real world. For example There are $10^{14}$ cells in the human body $10^{100}$ is called ...
54
votes
21answers
5k views

What is your favorite application of the Pigeonhole Principle?

The pigeonhole principle states that if $n$ items are put into $m$ "pigeonholes" with $n > m$, then at least one pigeonhole must contain more than one item. I'd like to see your favorite ...
26
votes
6answers
1k views

Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?

I understand that naive set theory, whose axioms are extensionality and unrestricted comprehension, is inconsistent, due to paradoxes like Russell, Curry, Cantor, and Burali-Forti. But these all ...
6
votes
1answer
56 views

Books that start with questions? [closed]

Does anyone know of any books that start with a relevant question, study it from different perspectives and then show some mathematics? I would appreciate it if you posted any that you know of, ...
66
votes
38answers
9k views

Unusual mathematical terms [closed]

From time to time, I come across some unusual mathematical terms. I know something about strange attractors. I also know what Witch of Agnesi is. However, what prompted me to write this question is ...
14
votes
10answers
2k views

Math problems that are impossible to solve [closed]

I recently read about the impossibility of trisecting an angle using compass and straight edge and its fascinating to see such a deceptively easy problem that is impossible to solve. I was wondering ...
6
votes
1answer
362 views

Structuralist slogans

I am afraid to make a bad impression by misusing this forum but I am looking for as-many-as-possible mathematically inspired formulations and references to one (sometimes vague) idea. The idea is ...
20
votes
11answers
5k views

How to prove $[a,b]$ is compact?

Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familar to us. Here I want to collect the ways to prove $[a,b]$ ...
0
votes
0answers
58 views

What are some examples of principal, proper ideals that have height at least $2$?

Krull's principal ideal theorem states that in a Noetherian ring $R$, any principal proper ideal $I$ has height at most $1$. Presumably the Noetherian hypothesis is required, so what are some ...
10
votes
3answers
382 views

Which statements are equivalent to the parallel postulate?

I would like to have a long-ish list of statements that are equivalent to the parallel postulate. If a line segment intersects two straight lines forming two interior angles on the same side that ...
93
votes
24answers
22k views

List of Interesting Math Blogs

I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level ...
26
votes
17answers
1k views

Accidents of small $n$

In studying mathematics, I sometimes come across examples of general facts that hold for all $n$ greater than some small number. One that comes to mind is the Abel–Ruffini theorem, which states that ...
4
votes
0answers
49 views

What are some other examples of this phenomenon: if $S$ is a finite set, then all possible total orderings of $S$ are isomorphic (as posets).

Finite sets have the amazing property that if $S$ is a finite set, then all possible total orderings of $S$ are isomorphic (as posets). Said another way: finite totally-ordered sets that are ...
0
votes
4answers
82 views

Easy proof of $\mathcal{P}(\mathbb{Q})$ is uncountable [Big list]

I'm looking for a easy proof of uncountability of $\mathcal P(\mathbb Q)$. I'll contribute with this: Let $\mathcal{P}(A)$ denote the power set of $A$, since ...
311
votes
33answers
19k views

Examples of apparent patterns that eventually fail

Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
16
votes
1answer
723 views

The most active fields of mathematics?

Which fields of mathematics are the most active at this time -- by number of papers published, grant money, people working in them or by any other measure? Any trends in this regard?
15
votes
7answers
534 views

Looking for Open Source Math Software with Poor Documentation

I'm a techie who is looking to transition to technical writing as a career. It's been suggested that I produce documentation for an Open Source project to demonstrate my ability to do technical ...
1
vote
0answers
46 views

Mathematical structures with name reffering to a country

I am looking for a list of mathematical structures (not theorems) that refer to a country or nationality. I only know of Polish spaces and Polish groups. Does anyone have other examples? Note: many ...
-1
votes
0answers
7 views

Difference between the (Euclidean) hyperboloid and the (Hyperbolic) hyperboloid model.

I am getting completely confused on the differences and similarities between the (Euclidean) Hyperboloid and the (Hyperbolic) Hyperboloid Model and it looks like some people just mixthem upo ...
0
votes
1answer
214 views

Crazy Set Theory Analogies

I think the following analogies are too interesting to be ignored: Union = Least Common Multiple If $G_1,...,G_n$ denote a number of sets of points (either linear or in any number of dimensions), ...
255
votes
23answers
31k views

Proofs that every mathematician should know?

There are mathematical proofs that have that "wow" factor in being elegant, simplifying one's view of mathematics, lifting one's perception into the light of knowledge etc. So I'd like to know what ...
3
votes
1answer
56 views

Bounds for $n$-th prime

In this page I have found that the bounds for $n$-th prime is given by, $$n(\ln n+\ln \ln n)>p_n>n(\ln n+\ln \ln n-1)$$ for all $n\ge6$. Are there even stronger bounds for the $n$-th prime? If ...
7
votes
1answer
7k views

Best Book For Differential Equations?

I know this is a subjective question, but I need some opinions on a very good book for learning differential equations. Ideally it should have a variety of problems with worked solutions and be ...
11
votes
4answers
335 views

Elementary problems that would've been hard for past mathematicians, but are easy to solve today? [closed]

I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous ...
17
votes
5answers
743 views

Examples of non-obvious isomorphisms following from the first isomorphism theorem

I am learning the first isomorphism theorem, and I am working with some isomorphisms to practice for my upcoming test. I know some of the basic ones like: $\mathbb{R}/\mathbb{Z} \cong \mathcal{C}$, ...
53
votes
17answers
15k views

Anecdotes about famous mathematicians or physicists

I'm not sure whether this question suits this website, however, I don't know where else I could ask it. It is no mathematical problem or something similar, still I hope it won't be closed. A few ...
183
votes
90answers
15k views

Surprising identities / equations

What are some surprising equations / identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic ...
89
votes
28answers
7k views

What are some examples of notation that really improved mathematics? [closed]

I've always felt that the concise, suggestive nature of the written language of mathematics is one of the reasons it can be so powerful. Off the top of my head I can think of a few notational ...
29
votes
10answers
2k 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 ...
10
votes
15answers
12k views

What concepts were most difficult for you to understand in Calculus? [closed]

I'm developing some instructional material for a Calculus 1 class and I wanted to know from experience for yourself, tutoring others, and/or helping people on this site where is the most difficulty in ...
4
votes
0answers
84 views

Request for counter examples in group theory

I am looking for books, papers, or even webpages, that have collected many counter examples in group theory (which, I guess, are just examples in group theory). I am particularly interested in ...
36
votes
10answers
8k views

Examples of finite nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
4
votes
2answers
479 views

A table of homology and cohomology groups

Does anyone know where I can find a table of the homology and cohomology groups, with different coefficients, of standard spaces - $S^1\times S^1$, Klein bottle, projective space, etc.?
6
votes
5answers
209 views

Proof writing: how to write a clear induction proof?

What is an effective way to write induction proofs? Essentially, are there any good examples or templates of induction proofs that may be helpful (for beginners, non-English-native students, etc.)? ...
170
votes
6answers
8k 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 ...
2
votes
1answer
97 views

Amazing integrals and how is solved it [closed]

There a lot of integrals, however many people solved it in different ways, we can find interesting integrals in Table of Integrals, Series, and Products. I wonder What is the most exciting integral ...
29
votes
4answers
2k views

Avoiding proof by induction

Proofs that proceed by induction are almost always unsatisfying to me. They do not seem to deepen understanding, I would describe something that is true by induction as being "true by a technicality". ...
2
votes
0answers
31 views

Perimeters Areas and Volumes

I have to write an article for a school magazine. I thought it is better to choose a simple topic like Perimeter, Area and Volume. I am looking for historical fact and surprising facts about ...
56
votes
16answers
8k views

Interesting “real life” applications of serious theorems

As a student one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: "You have a ...
4
votes
0answers
72 views

Proofs shorter than the statement of the theorem

In Postnikov's first book in his Lectures in Geometry Series, Analytic Geometry, he states and then proves the Desargues theorem. Then he writes (in my English translated copy) "The proof has turned ...
186
votes
28answers
17k 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 ...
32
votes
10answers
12k views

Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs]

This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as ...