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.

learn more… | top users | synonyms

34
votes
6answers
27k 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 ...
10
votes
9answers
561 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?)
60
votes
34answers
6k views

Easy math proofs or visual examples to make high school students enthusiastic about math [on hold]

I'm a teacher in mathematics at a high school. Math has fascinated me for almost my entire life, so I would like to bring that enthusiasm to my students with beautiful yet easy to understand proofs or ...
76
votes
23answers
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 ...
180
votes
7answers
9k 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 ...
0
votes
0answers
33 views

Nice statements about the “opposite properties” of $0$ and $1$.

Some of $0$ and $1$'s properties as natural numbers are pleasingly opposite of one another, such as: 1a.) $0$ has infinitely many divisors, but no (non-zero) multiples. 1b.) $1$ has infinitely many ...
35
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 ...
315
votes
25answers
32k 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 ...
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 ...
0
votes
0answers
81 views

How do i remove my Guilt? [closed]

When i see a theorem ,which i cannot prove in real analysis ,i think about it but still i couldn't figure it out .Then i look for its solution ,after understanding the proof i feel very guilt that i ...
2
votes
1answer
39 views

Real analysis : Preliminary topics for - Measure Theory, Integration Theory, Differentiation and Integration [closed]

I have following syllabus to study in Real Analysis Subject. I want to know, What are necessary topics that I have to cover as a prerequisite for below syllabus. Actually I am unable to get direction ...
39
votes
19answers
2k views

Literary statements that are false as mathematics [closed]

I recently wanted to use the title of the famous short story "Everything that Rises must Converge" in a poem of mine. However, the mathematician in me insisted on changing it to "Everything that ...
6
votes
5answers
106 views

$e^{i\theta}$ versus $\cos\theta+i\sin\theta$

I am teaching an basic university maths course, and have been thinking about the complex numbers part. Specifically, I was wondering why I should include Euler's formula in my course. This led me to ...
17
votes
4answers
635 views

Pathologies in module theory

Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar ...
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 ...
2
votes
1answer
82 views

Coordinate Geometry and Trigonometry book recommendation for GRE Math Subject Test

I am currently a math major at university and I plan to take GRE Math Subject Test in future (most probably next year). Can you please suggest any good book for revising and brushing up Coordinate ...
6
votes
2answers
225 views

Which mathematical game or puzzle did you invent?

A couple of weeks ago, a friend of mine showed me a extension of a game we are all familiar with that he was working on. The game we know is called Tic-Tac-Toe, and he was working on his own version ...
45
votes
13answers
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 ...
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 $[a,b]$ ...
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 ...
1
vote
2answers
94 views

Counting the number of solutions of equation $x^2 + y^2 = 1$ over $\Bbb Z/p$

List proofs of the fact that the number of solutions to $x^2 + y^2 = 1$ over $\Bbb Z/p$, where $p$ is a prime $\neq 2$, is $p-(-1)^{\frac{p-1}2}$. I thought of two. I write one below.
16
votes
5answers
998 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 ...
19
votes
3answers
972 views

The most active fields of mathematics? [closed]

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?
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?
58
votes
17answers
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 ...
13
votes
11answers
1k views

Favourite open problem?

Do you have any favorite open problem? Let me mention one of my favorites. Let $A(\mathbb{T})$ be the Wiener algebra, that is, the linear space of absolutely convergent Fourier series on the unit ...
37
votes
10answers
3k views

Mathematical literature to lose yourself in

H.M. Edwards in the preface to his book on the Riemann Zeta Function, summarises his philosophy on learning Mathematics: ...I have tried to say to students of mathematics that they should read the ...
18
votes
16answers
14k 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 ...
0
votes
2answers
73 views

Lists of the first fundamental group of spaces. [closed]

Here are some list to start with $$\begin{array}{c|c|c|} \hline Space(S)& \pi_1(S) \\ \hline \mathbb{R}^2&0 \\ \hline \mathbb{S}^1& \mathbb{Z} \\ \hline 1-Torus& ...
20
votes
10answers
459 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
48 views

Are there unsolved problems known to be not independent of the axiomatic system it is proposed in?

Are there unsolved problems known to be not independent of the axiomatic system it is proposed in? For example, is Goldbach's conjecture known to be provable using the axioms of PA? I believe I ...
19
votes
14answers
8k views
2
votes
1answer
35 views

Benefit from measure theory

With your help I want to list the benefits from measure theory and the lebesgue integral. (Advantages to the Riemann integral) What I know: With the Lebesgue integral we need less requirements to ...
538
votes
152answers
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 ...
19
votes
11answers
1k views

What are some results that shook the foundations of one or more fields of mathematics? [closed]

An example would be the proof that $\sqrt{2}$ is not rational, which was a violation of some fundamental assumptions that mathematicians at the time made about numbers. Another would be Russell's ...
8
votes
4answers
299 views

Mathematical results that were generally accepted but later proven wrong?

I am giving a presentation on mathematical results that were widely accepted for a period of time and then later proven wrong, or vice versa. This talk is geared towards undergraduates who are likely ...
1
vote
1answer
57 views

Website for Mathematics enigmas?

I seek some website just for the pleasure of solving mathematics enigmas. I know this website : Brilliant.org I just want to know if you know some others good sites ! Thank you
36
votes
18answers
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 $\bullet+\cdots\cdots\cdots+\bullet$ ...
57
votes
18answers
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 ...
7
votes
3answers
321 views

Space on which all real-valued continuous functions achieve maximum but not compact?

A friend is writing a book for non-mathematicians; he has asked me some questions... One possible direction I suggested was whether a topological space (metric space can probably be assumed given what ...
4
votes
0answers
67 views

All possible total orderings of a finite set are isomorphic. What are some other examples of this phenomenon?

All possible total orderings of a finite set are isomorphic. I find these kinds of results remarkable. Here's a few more. Assume that $S$ is a finite set. Then: All possible field structures on $S$ ...
12
votes
6answers
6k views

Prerequisites/Books for A First Course in Linear Algebra

What mathematical knowledge do I need to begin studying linear algebra? In particular, how much calculus do I need to know? Also, do you have a favorite linear algebra book you can recommend?
8
votes
1answer
110 views

What are some interesting blogs about general topology?

We have several question asking about book recommendations for general topology - for example the posts linked to Best book for topology? or the posts mentioned in the relevant section of List of ...
3
votes
1answer
75 views

Generalizations of de l'Hospital rule

Are there any useful generalizations of de l'Hospital rule? Could you point out some references? Edit: Something in the spirit of http://arxiv.org/abs/1403.3006, but not necessarily for functions of ...
79
votes
4answers
10k views

Books that every student “needs” to go through

I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden). I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on ...
10
votes
3answers
441 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 ...
36
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 ...
1
vote
1answer
421 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 ...
12
votes
8answers
552 views

Why is the 3D case so rich?

The Banach--Tarski theorem applies only in the case of three or more dimensions. In 3D, there are five regular solids, two of them being not at all obvious, and the 4D case is also interesting; but ...
179
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 ...