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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421
votes
33answers
48k 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 ...
360
votes
34answers
22k 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 ...
44
votes
13answers
4k views

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
112
votes
19answers
18k views

Striking applications of integration by parts

What are your favorite applications of integration by parts? (The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!) Thanks for your ...
65
votes
12answers
23k 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 ...
9
votes
5answers
745 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.)? ...
86
votes
9answers
19k 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 ...
64
votes
17answers
37k views

What is the most elegant proof of the Pythagorean theorem? [closed]

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite ...
26
votes
9answers
41k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
69
votes
19answers
12k views

Different ways to prove there are infinitely many primes?

This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show ...
83
votes
12answers
12k views

Conjectures that have been disproved with extremely large counterexamples?

I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture. I'm sure that everyone here is familiar with it; it describes an operation on a ...
205
votes
32answers
11k views

Can't argue with success? Looking for “bad math” that “gets away with it”

I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare"). One example would be "cancelling" the 6s in $$\frac{64}{16}.$$ ...
47
votes
5answers
12k views

(Theoretical) Multivariable Calculus Textbooks [duplicate]

(Note that I have used bold text frequently simply to highlight the key points of my question for those who do not have the time to read through it thoroughly (it is not very long, however); I hope ...
721
votes
53answers
424k views

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain, but are ...
222
votes
65answers
51k views

'Obvious' theorems that are actually false

It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true". Now, I know a fair few examples of things that are obviously true and that can be proved ...
113
votes
20answers
14k views

Your favourite application of the Baire Category Theorem

I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications ...
67
votes
8answers
13k views

Are there real world applications of finite group theory?

I would like to know whether there are examples where finite group theory can be directly applied to solve real world problems outside of mathematics. (Sufficiently applied mathematics such as ...
632
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,...
211
votes
92answers
18k views

Surprising identities / equations [closed]

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 ...
156
votes
27answers
27k views

Best Fake Proofs? (A M.SE April Fools Day collection) [closed]

In honor of April Fools Day 2013, I'd like this question to collect the best, most convincing fake proofs of impossibilities you have seen. I've posted one as an answer below. I'm also thinking of a ...
105
votes
23answers
24k views
32
votes
3answers
2k views

Create a Huge Problem

I am wondering if any problems have been designed that test a wide range of mathematical skills. For example, I remember doing the integral $$\int \sqrt{\tan x}\;\mathrm{d}x$$ and being impressed at ...
32
votes
6answers
2k views

Do We Need the Digits of $\pi$?

I was reading today that someone found $\pi$ to the ten trillionth digit. Whenever I read that $\pi$ has been calculated to more digits, I ask myself whether this is useful. I know that there are ...
49
votes
10answers
13k views

Best Algebraic Geometry text book? (other than Hartshorne)

Lifted from Mathoverflow: I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, ...
60
votes
22answers
7k 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 ...
12
votes
6answers
4k views

Examples of non symmetric distances

It is well known that the symmetric property is $d(x,y)=d(y,x)$ is not necessary in the definition of distance if the triangle inequality is carefully stated. On the other hand there are examples of ...
4
votes
1answer
319 views

Seemingly hard integrals which are made easy via differentiation under the integral sign a.k.a Feynman Integration [closed]

I recently discovered Differentiation under the integral sign a.k.a Feynman Integration and I read an article which says it can be substituted for contour integration. Therefore, I am assuming this ...
156
votes
28answers
30k views

List of Interesting Math Videos/ Documentaries

This is an offshoot of the question on Fun math outreach/social activities. I have listed a few videos/documentaries I have seen. I would appreciate if people could add on to this list. Story of ...
236
votes
29answers
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 ...
92
votes
24answers
75k views

Software for drawing geometry diagrams

What software do you use to accurately draw geometry diagrams?
47
votes
10answers
57k views

What is the best book for studying discrete mathematics?

As a programmer, mathematics is important basic knowledge to study some topics, especially Algorithms. Many websites, and my fellows suggest me to study Discrete Mathematics before going to Algorithms,...
45
votes
6answers
42k views

Good books for self-studying algebra?

I have a few weeks off from school soon, and I was hoping to self-study a bit of algebra. I don't think this question has been asked on here before, but does anyone have any suggestions for algebra ...
53
votes
10answers
2k views

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do ...
26
votes
12answers
9k views

How to prove every closed interval in R 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 $[a,b]$ ...
205
votes
65answers
14k views

Funny identities [closed]

Here is a funny exercise $$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$ (If you prove it don't publish it here please). Do you have similar examples?
36
votes
12answers
18k views

Good books on mathematical logic?

I just started to learn mathematical logic. I'm a graduate student. I need a book with relatively more examples. Any recommendation?
73
votes
18answers
30k 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 ...
51
votes
13answers
3k views

Pseudo Proofs that are intuitively reasonable

What are nice "proofs" of true facts that are not really rigorous but give the right answer and still make sense on some level? Personally, I consider them to be guilty pleasures. Here are examples ...
38
votes
19answers
9k views

Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
27
votes
14answers
12k views
17
votes
1answer
16k 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 ...
7
votes
1answer
575 views

Bounds for $n$-th prime

In this Wikipedia 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 ...
92
votes
5answers
4k views

“Advice to young mathematicians”

I have been suggested to read the Advice to a Young Mathematician section of the Princeton Companion to Mathematics, the short paper Ten Lessons I wish I had been Taught by Gian-Carlo Rota, and the ...
78
votes
6answers
17k views

Studying for the Putnam Exam

This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the ...
33
votes
12answers
880 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. Motivation: When I read from the book Proof from the Book, I saw there were many ...
20
votes
2answers
3k views

Good introductory books on homological algebra

Which books would you recommend, for self-studying homological algebra, to a beginning graduate (or advanced undergraduate) student who has background in ring theory, modules, basic commutative ...
15
votes
4answers
6k views

What are some classic fallacious proofs? [duplicate]

If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already: Let $a = b$. Then $a^2 = ab$ $a^2 - b^2 = ...
7
votes
5answers
398 views

Provide different proofs for the following equality.

Let $a$ and $b$ be positive reals. Show that $$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$
7
votes
8answers
1k views

Fundamental Theorem of Trigonometry [closed]

This is a pretty open ended question and I apologize, in advance, if this is not the place for it. But what do you recommend should be given the title of the Fundamental Theorem of Trigonometry and ...
229
votes
7answers
16k 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 ...