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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320
votes
25answers
33k 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 ...
323
votes
34answers
20k 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 ...
91
votes
18answers
16k 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 ...
53
votes
16answers
22k views

What is the most elegant proof of the Pythagorean theorem?

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 ...
110
votes
29answers
38k views

Best book ever on Number Theory

Which is the single best book for Number Theory that everyone who loves Mathematics should read?
38
votes
10answers
13k 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 ...
30
votes
10answers
1k 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 ...
58
votes
17answers
9k 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 ...
18
votes
8answers
24k views

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

I've always enjoyed solving problems in the complex world 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 ...
174
votes
32answers
9k 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 $$\frac{64}{16}.$$ ...
75
votes
22answers
18k views
80
votes
21answers
53k views

Software for drawing geometry diagrams

What software do you use to accurately draw geometry diagrams?
586
votes
48answers
383k views

Visually stunning math concepts which are easy to explain [closed]

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 ...
547
votes
153answers
34k 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 ...
40
votes
5answers
9k 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 ...
6
votes
5answers
271 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.)? ...
30
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 ...
183
votes
64answers
41k 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 ...
135
votes
27answers
21k 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 ...
60
votes
7answers
12k 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 ...
88
votes
20answers
10k 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 ...
69
votes
12answers
8k 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 ...
186
votes
90answers
16k 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 ...
40
votes
8answers
41k 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 ...
29
votes
10answers
4k views

Seeking a layman's guide to Measure Theory

I would like to teach myself measure theory. Unfortunately most of the books that I've come across are very difficult and are quick to get into Lemmas and proofs. Can someone please recommend a ...
29
votes
3answers
1k 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 ...
46
votes
10answers
10k 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, ...
34
votes
6answers
28k 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 ...
2
votes
1answer
227 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 ...
187
votes
66answers
13k 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?
208
votes
29answers
15k 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 ...
55
votes
22answers
6k 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 ...
48
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 ...
139
votes
28answers
24k 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 ...
194
votes
28answers
18k 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 ...
60
votes
8answers
8k 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 ...
37
votes
19answers
7k 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 $\bullet+\cdots\cdots\cdots+\bullet$ ...
10
votes
1answer
8k 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 ...
144
votes
91answers
39k views

Which one result in mathematics has surprised you the most? [closed]

A large part of my fascination in mathematics is because of some very surprising results that I have seen there. I remember one I found very hard to swallow when I first encountered it, was what is ...
67
votes
6answers
11k 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 ...
35
votes
7answers
8k views

Good books on Math History

I'm trying to find good books on the history of mathematics, dating as far back as possible. There was a similar question here Good books on Philosophy of Mathematics, but mostly pertaining to ...
19
votes
14answers
8k views
18
votes
4answers
783 views

Advice for writing good mathematics?

It's been a (far-fetched, possibly) goal of mine to some day write a math Textbook. I've been thinking about writing this question for a while, but reading an exceedingly mediocre text on Mathematical ...
14
votes
4answers
5k 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 = ...
11
votes
2answers
2k views

Categories of mathematics

I am interested in understanding how mathematics is divided into many categories, such as what categories are particular cases of what, what categories do not or have little overlap with what. This is ...
6
votes
7answers
879 views

Fundamental Theorem of Trigonometry

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 ...
6
votes
5answers
376 views

Provide different proofs for the following equality.

Let $a$ and $b$ be positive reals. Show that $$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$
161
votes
37answers
15k views

Fun but serious mathematics books to gift advanced undergraduates.

I am looking for fun, interesting mathematics textbooks which would make good studious holiday gifts for advanced mathematics undergraduates or beginning graduate students. They should be serious but ...
120
votes
20answers
8k views

Are there any open mathematical puzzles?

Are there any (mathematical) puzzles that are still unresolved? I only mean questions that are accessible to and understandable by the complete layman and which have not been solved, despite serious ...
31
votes
11answers
14k 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?