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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4
votes
5answers
240 views

Examples of open problems solved through short proof

Are there good examples of reasonable open problems in mathematics that had an 'obvious' solution via application of a theorem already known/not yet found in mathematics but could have been found with ...
8
votes
3answers
649 views

What are some easily-stated recently proven theorems? [on hold]

What are some easily-stated relatively recently proven theorems? I don't mean they were necessarily easy to prove, just easy to state. Here are a few examples: The proof of Fermat's Last Theorem ...
2
votes
0answers
20 views

List of textbooks that take a historical approach

As the title suggests my aim in this topic is to make a big list of textbooks on any mathematical topic that take a historical approach. I will start with the ones I know: Thomas Muir - The theory of ...
6
votes
0answers
58 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
11
votes
1answer
349 views
+50

What is the most cited mathematical paper?

Just out of curiosity: What is the paper with the largest number of citations in all of mathematics? I think it is Shannon's A Mathematical Theory of ...
3
votes
4answers
389 views

What are some uses for Monte Carlo simulations in mathematics?

I've recently been interested in Monte Carlo simulations and their uses, unfortunately most of the examples I find are difficult to understand for a beginner. What are some simple examples of using ...
30
votes
12answers
750 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 ...
333
votes
27answers
34k 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 ...
77
votes
23answers
23k views

Complete course of self-study

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 ...
3
votes
1answer
260 views

I'm searching for some books with guidance into mathematical study. [closed]

Yesterday, I've found this. It's a PDF file with this purpose, from Oxford. Some weeks ago I've also found two books tha seems to fill this purpose: Prelude to Mathematics; I Want to Be a ...
11
votes
3answers
9k views

I want to start mathematics from scratch. What should I begin with?

I've just finished high school but I don't feel my knowledge of mathematics is good enough. I'd like to start again from scratch, possibly adding a bit (a lot) of problem solving to it. What is the ...
12
votes
6answers
7k views

Resources/Books for Discrete Mathematics

I am going to a Computer Science Course in University next year. I heard that Discrete Mathematics is whats required for Comp Sci so, I am looking for resources/books that I can read to get started ...
23
votes
15answers
940 views

New Idea to prove $1+2x+3x^2+\cdots=(1-x)^{-2}$

Given $|x|<1 $ prove that $\\1+2x+3x^2+4x^3+5x^4+...=\frac{1}{(1-x)^2}$. 1st Proof: Let $s$ be defined as $$ s=1+2x+3x^2+4x^3+5x^4+\cdots $$ Then we have $$ \begin{align} ...
19
votes
2answers
688 views

“Bad” Mathematics in Movies

There's a website and a companion book to it about bad physics in movies, called "Insultingly Stupid Movie Physics". Similar issues may exist about mathematics: What are the differences between ...
39
votes
14answers
4k views

Examples of famous problems resolved easily

Have there been examples of seemingly long standing hard problems, answered quite easily possibly with tools existing at the time the problems were made? More modern examples would be nice. An example ...
98
votes
19answers
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 ...
7
votes
5answers
1k views

discrete math book suitable for younger person?

When I took discrete math as an adult I realized that this was a subject I would have enjoyed and done well at much earlier in life, even in my early teens. Does anyone know if there are good books, ...
31
votes
12answers
15k 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?
8
votes
12answers
2k views

Best Quotes by a Mathematician [closed]

If I were to ask this: Which quote by a "Mathematician" do you like, then what would be your answer!
4
votes
4answers
177 views

Do you know any almost identities?

Recently, I've read an article about almost identities and was fascinated. Especially astonishing to me were for example $\frac{5\varphi e}{7\pi}=1.0000097$ and ...
3
votes
1answer
38 views

Applications of algebraic graph theory

What are some subtle, or non-obvious applications of algebraic graph theory? Obviously it can be used to study anything directly involving graphs (for instance, the wikipedia page mentions ...
1
vote
0answers
47 views

What families of transcendental equations do we have solved?

I'm particularly interested in transcendental equations but searching in internet gives me only results about the classical linear-exponential equation (which is solved with Lambert's W) and its ...
1
vote
1answer
191 views

A question on mean value inequality

It is known that mean value inequality is very useful. It is: For any $0 \le a_i (i=1,2,\dots,n)$, $$ a_1 a_2\dots a_n\le (\frac{a_1+a_2+\dots + a_n}{n})^n \tag1 $$ My question is: how many ...
8
votes
5answers
115 views

$32$ Goldbach Variations - Papers presenting a single gem in number theory or combinatorics from different point of view

A short time ago I found the nice paper Thirty-two Goldbach Variations written by J.M. Borwein and D.M. Bradley. It presents $32$ different proofs of the Euler sum identity \begin{align*} ...
2
votes
3answers
63 views

Application of Euler's theorem apart from finding last digits of huge numbers

I am looking for clever applications of Euler's Theorem. On browsing the internet, I see that nearly all the applications of the theorem asks for finding last few digits of a huge number. The only ...
18
votes
5answers
669 views

Big list of serious but fun “unusual” books

I would like to have some suggestions about serious (that is, with good mathematical content) but fun books that cover topics (or propose problems) in "recreational mathematics"; in any other field ...
8
votes
3answers
299 views

Elementary topology examples

I'm preparing (to teach) my first class of undergraduate topology and I'm looking for some elementary, motivating applications of topology for the first day. We'll be following Munkres, starting with ...
40
votes
11answers
14k 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 ...
5
votes
1answer
106 views

A beautiful book on arithmetic doesn't treat you like a little baby

The state of arithmetic today is disgusting. The textbooks on it are absolutely repelling, the authors treat it like a subject that will be of concern to only babies. They don't show any love, they ...
30
votes
11answers
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 ...
1
vote
2answers
69 views

What are some unfamiliar and/or special tricks used to evaluate limits?

What are some neat tricks used to evaluate limits that might be otherwise a problem to deal with? I'm not asking for methods akin to L'Hopital's rule.. which is often used. My question is geared ...
58
votes
18answers
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 ...
2
votes
1answer
124 views

(Theoretical) Complex Analysis Textbooks [closed]

Most books I've seen on complex analysis do not develop it theoretically, which can be somewhat infuriating for the budding pure mathematician. What I am looking for are some comprehensive, rigorous ...
14
votes
1answer
549 views

successful absurd formalities

Has anyone published in print or on a web site or elsewhere a compilation of successful illogical formal arguments? By those I mean arguments that follow a form in disregard of the legality of its ...
0
votes
6answers
263 views

Alternate ways to prove that $4$ divides $5^n-1$

I was working for various method to solve this: For all $n\in \mathbb N$: $4\;\mid\;(5^{n}-1)$. My try was: 1st: $$n=1 \to 4|5^1-1\\n \geq 2 \to 5^n=25,125,625,3125,...\\ n\geq 2 \to ...
23
votes
11answers
6k 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
78 views

Different ways to prove Fundamental Theorem of Algebra

This is just a curosity .I know some proofs of the fact that Every non constant polynomial with complex coefficient has a complex root via using Liouville's theorem in Complex Analysis.Proof goes as ...
36
votes
2answers
798 views

How do I find partners for study?

I want to find partners to study some post-graduate topics together online (because I'm pretty much out of steam as far as self-study goes, and I have problems finding a decent grad school). Are there ...
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 ...
17
votes
2answers
2k 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 ...
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 ...
35
votes
6answers
30k 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 ...
13
votes
4answers
1k views

Guidelines for learning about Ramanujan's work?

It is well known that one of the first books Ramanujan studied was "Synopsis of Pure and Applied Mathematics" and that it shaped the way Ramanujan thought and wrote about mathematics. Being interested ...
0
votes
4answers
97 views

Proof of Pythagorean theorem without using geometry for a high school student?

There are some proofs of Pythagoras theorem which don't even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs of ...
24
votes
9answers
1k views

Free online mathematical software

What are the best free user-friendly alternatives to Mathematica and Maple available online? I used Magma online calculator a few times for computational algebra issues, and was very much satisfied, ...
38
votes
8answers
4k views

Open math problems which high school students can understand

I request people to list some moderately and/or very famous open problems which high school students,perhaps with enough contest math background, can understand, classified by categories as on ...
54
votes
17answers
24k 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 ...
555
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 ...
56
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 ...
37
votes
19answers
8k 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 $\bullet+\cdots\cdots\cdots+\bullet$ ...