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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0
votes
6answers
221 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 ...
19
votes
15answers
737 views
+100

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

If $|x|<1 $ prove that $\\1+2x+3x^2+4x^3+5x^4+...=\frac{1}{(1-x)^2}$ 1st proof:suppose ...
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 $[a,b]$ ...
-1
votes
0answers
46 views

A book on functions! [on hold]

I am looking for a good book on algebra which covers functions very comprehensively. It should cover functions of the type: linear: $f(x) = ax + b$ quadratic: $f(x) = ax^2 + bx + c$ cubic: $f(x) = ...
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
786 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 ...
34
votes
6answers
29k 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 ...
1
vote
2answers
50 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 ...
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
90 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
22k views

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

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 ...
1
vote
1answer
35 views

Applications of statistics to pure mathematics [closed]

Are there any "applications" of statistical methods to pure mathematics?
2
votes
0answers
51 views

(Theoretical) Complex Analysis Textbooks

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 ...
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 ...
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 ...
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 $\bullet+\cdots\cdots\cdots+\bullet$ ...
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 ...
19
votes
17answers
15k views

Applications of the Fibonacci sequence

The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any ...
30
votes
12answers
648 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 ...
4
votes
2answers
105 views

What is the most general notion of “Fourier transform?”

I know the definition of a classical Fourier transform that maps a function f(x) on the real line X to a function F(p) on a dual space (here another real line and borrowing some physics notation) P. ...
27
votes
14answers
795 views

Examples where it is easier to prove more than less

Especially (but not only) in the case of induction proofs, it happens that a stronger claim $B$ is easier to prove than the intended claim $A$ (e.g. since the induction hypothesis gives you more ...
20
votes
6answers
1k views

What Mathematics questions can be better solved with concepts from Physics?

Over the years, I've seen several questions in mathematics that can be solved using concepts borrowed from Physics. Having seen these question, I'm interested to find out what other mathematics ...
0
votes
0answers
29 views

Final year dissertation/project ideas for numerical methods

In my final year, I have to submit a project/dissertation on Numerical Methods. I have done a course on it, which included some proofs and programming. Just eager to get ideas that I can look at. PS ...
3
votes
2answers
94 views

Interesting facts and problems to motivate high school combinatorics students

I will give some classes in combinatorics to high school students and I would like to know some facts (and proof) I can show to my students to motivate them to study this beautiful subject. I'm ...
56
votes
25answers
7k views

Easy example why complex numbers are cool

I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary. The best example would be possible to explain rigourously and also be clearly ...
4
votes
3answers
671 views

What are your favorite integration tricks?

I'm learning to integrate and I'd like to hear what are you favorite integration tricks? I can't contribute much to this thread, but I like the fact that: $$\int_{-a}^{a}{f(x)}dx=0 ...
51
votes
15answers
1k views

What is the most unusual proof you know that $\sqrt{2}$ is irrational?

What is the most unusual proof you know that $\sqrt{2}$ is irrational? Here is my favorite: Theorem: $\sqrt{2}$ is irrational. Proof: $3^2-2\cdot 2^2 = 1$. (That's it) That is a ...
3
votes
4answers
68 views

Are there any books with lots of questions of “Fill in The holes” type

Does anyone knows books which have lots of questions ,whose format are like fill in the holes type . . Same goes for theorems and exercises . I am looking on pure math especially Real analysis ...
47
votes
10answers
1k 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 ...
5
votes
2answers
146 views

Characterizations of the cross-ratio

$$ (z_1,z_2;z_3,z_4) = \frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)} $$ What are the most prominent or most interesting theorems of the following form? Theorem: The cross-ratio is the only function ...
0
votes
3answers
80 views

Alternative infinite summations that equal $e$

Everyone (and I mean everyone) knows this sum: $$\sum_{n=0}^\infty \frac{1}{k!} =e$$ Are there any lesser known infinite sums that go to e?
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 ...
18
votes
7answers
1k views

What are some examples of induction where the base case is difficult but the inductive step is trivial?

According to Wikipedia: ...proofs by mathematical induction have two parts: the "base case" that shows that the theorem is true for a particular initial value such as n = 0 or n = 1 and ...
5
votes
1answer
67 views

Elegant applications of advanced techniques to “olympiad” problems

I am interested in applications of somewhat "advanced machinery" (with respect to the usual machinery involved in these cases, which is usually elementary) to olympiad or (high school-level) contest ...
1
vote
2answers
47 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 ...
3
votes
2answers
103 views

List of first-order foundations of mathematics

I would like to get an overview over what different first-order systems suitable for the formalization of "classical" mathematics are currently known. So far I only know the variants of ZFC, ...
0
votes
0answers
23 views

List of divergent series and their summations

On the web one can manage to find a lot of lists of convergent series and their summation btw I didn't find (at least on a quick search) a corrispective list of divergent series, does anyone know one ...
0
votes
0answers
29 views

Looking for problems which can be solved by the similar technique

While browsing on internet for different proofs of Fermat's theorem on sums of two squares, I came across Zagier's "one-sentence proof" which seems to be the most elegant and short proof. It invokes a ...
5
votes
0answers
62 views

“Toys” spaces in algebraic topology

I did follow a course of algebraic topology last semester and I still want to continue to do some computations. But in many books it's all the time the same examples which comes back for computing ...
79
votes
2answers
6k views

Open problems in General Relativity

I would like to know if there are some open mathematical problems in General Relativity, that are important from the point of view of Physics. Is there something that still needs to be justified ...
1
vote
2answers
444 views

Practice Problem Books

The Analysis I/II/III (Differentiation and Continuity/Sequence and Series/Integration) published by AMS. The first one is this. It's a problem-solution book. I found it excellent because of the ...
3
votes
4answers
242 views

Theorems in number theory whose first proofs were long and difficult

What are the examples of important theorems of number theory that has been shown to have surprisingly simple proofs though their first demonstration wasn't at all simple enough. Now simple proof is an ...
11
votes
10answers
685 views

What are the most prominent uses of transfinite induction outside of set theory?

What are the most prominent uses of transfinite induction in fields of mathematics other than set theory? (Was it used in Cantor's investigations of trigonometric series?)
17
votes
8answers
1k views

Properties of the number 50

I will shortly be engaging with my 50th (!) birthday. 50 = 1+49 = 25+25 can perhaps be described as a "sub-Ramanujan" number. I'm trying to put together a quiz including some mathematical content. ...
2
votes
1answer
35 views

open problems regarding functions

I am looking for some open problems regarding functions. Problems like, Whether a function satisfying some properties say, X,Y,Z, exists or not, is unknown. Like there is no function $f(x)$ such ...