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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6
votes
1answer
86 views

Theorems discovered without observation

Can you name me a few theorems that were discovered without first observing some special cases? In other words, by brute logic: Starting from the known and logically deducing the unknown? EDIT: As an ...
0
votes
0answers
50 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 ...
3
votes
3answers
117 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 ...
1
vote
0answers
168 views

New proofs of the Fundamental Theorem of Calculus

Apart from the standard one, are there any other proofs of the Fundamental Theorem of Calculus which have been published recently?
4
votes
2answers
385 views

An example of a great explanation or freely accessible article on a math concept [closed]

Question: Give an example of a great explanation or freely accessible article on a math concept (suitable at the undergraduate or lower level), and explain why you think it is great. Possible ...
106
votes
44answers
12k views

What's your favorite proof accessible to a general audience? [closed]

What math statement with proof do you find most beautiful and elegant, where such is accessible to a general audience, meaning you could state, prove, and explain it to a general audience in ...
5
votes
2answers
150 views

Toy examples for Kan extensions

Background: If $\mathcal{C}$ is a cocomplete category and $f : I \to J$ is a functor between small categories, then $f^* : \mathrm{Hom}(J,\mathcal{C}) \to \mathrm{Hom}(I,\mathcal{C})$ has a left ...
2
votes
3answers
86 views

Examples of orthogonal/orthonormal functions which are not finite degree polynomials?

I've been reading "Fourier Series & Orthogonal Polynomials" by Dunham Jackson. Great introductory read for anyone interested by the way! My question is, what are other examples of Orthogonal ...
1
vote
0answers
48 views

Methods for evaluating polynomials quickly

I am wondering what methods exist for effectively evaluating polynomials (manually or in the head) in a quick, efficient fashion. For example, one of my favorite methods is the "nested form of a ...
24
votes
8answers
436 views

Big list of “guided discovery” books

K. P. Bogart wrote Combinatorics through Guided Discovery, available freely online. In the preface, he writes (emphasis mine): The point of learning from this book is that you are learning how to ...
66
votes
38answers
9k views

Unusual mathematical terms [closed]

From time to time, I come across some unusual mathematical terms. I know something about strange attractors. I also know what Witch of Agnesi is. However, what prompted me to write this question is ...
8
votes
4answers
619 views

What are some elementary results (number theory) using theorems that went long-unproven?

Firstly, I do not mind if there are examples from fields other than number theory! This was just the first field, and where I think the richest examples, may come from. Now to elaborate on the title, ...
7
votes
1answer
87 views

For finding limits of functions, when are graphs deceiving?

What are some examples of limits which exist of functions $f:A \to B$ where $A$, $B \subseteq \mathbb{R}$ that require by-hand, "analytical" methods and the value of the limit is seemingly ...
1
vote
2answers
140 views

Elementary Applications of Cayley's Theorem in Group Theory

The Cayley's theorem says that every group $G$ is a subgroup of some symmetric group. More precisely, if $G$ is a group of order $n$, then $G$ is a subgroup of $S_n$. In the course on group theory, ...
3
votes
1answer
117 views

Innocent looking open problems in real analysis

Are there any apparently easy problems or conjectures in basic real analysis (that is, calculus) that are still open? By apparently easy, I mean: so much so, that, if it was for the statement alone, ...
3
votes
2answers
138 views

Connectedness arguments in elementary mathematics?

To begin, let me explain a proof strategy (which I'll call the connectedness principle for want of a better, more canonical term): One way to prove that a mathematical object $O_1$ has some property ...
9
votes
0answers
205 views

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 ...
69
votes
9answers
5k views

Besides proving new theorems, how can a person contribute to mathematics?

There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example: Organizing known results into a coherent ...
2
votes
0answers
63 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 ...
0
votes
1answer
82 views

Common conditions on functions to be morphisms. [closed]

When coming in contact with the concept of morphism one may start to wonder what makes different structured objects of the same kind to be similar in a "morphical" way. At least I did. Below ...
1
vote
0answers
49 views

Open problems for which all cases except one have been solved

Keller's conjecture states that in any tiling of Euclidean $n$-space by identical hypercubes there are two cubes that meet face to face. The conjecture has been shown to be true for $n<7$ and ...
3
votes
1answer
54 views

Group structures on Hausdorff space

Could anyone give me some practical (and possibly intuitive) examples of Group structures on Hausdorff spaces? Let us say you had to get freshmen university students interested into fields of maths ...
36
votes
12answers
3k views

What are some theorems that currently only have computer-assisted proofs?

What are some theorems that currently only have computer-assisted proofs? For example, there's the four colour theorem. I am very curious about this and would like to generate a list.
15
votes
1answer
260 views

Learning roadmap request: compiling a “Mathematics Stack Exchange Undergraduate Bibliography” [closed]

[Book recommendation] questions are quite popular on this website, which is, at least for me, one of the best places to get useful and insightful suggestions ...
2
votes
0answers
84 views

Big list of references [divided by categories] that collect commented open problems and conjectures [closed]

The aim of this question is to collect a big list of books or survey papers or websites which collect an up-to-date, comprehensive, well-organized, and possibly commented list of open problems. I ...
2
votes
1answer
110 views

Integration by nonobvious substitutions

The standard technique for evaluating the integral $$\int \sec x \,dx$$ is making the nonobvious substitution $$u = \sec x + \tan x, \qquad du = (\sec x \tan x + \sec^2 x) dx,$$ which transforms the ...
2
votes
4answers
123 views

Reference request: self-contained rigorous introductions to “cool” topics

I am looking for some self-contained (i.e., providing all necessary background information) rigorous introductions to topics perceived as "cool" to propose to (really) advanced high school students ...
15
votes
4answers
417 views

Problems from the Kourovka Notebook that undergraduate students can fully appreciate

The Kourovka Notebook is a collection of open problems in Group Theory. My question is: could you point out some (a "big-list" of) problems [by referencing them] presented in this book that ...
0
votes
0answers
44 views

Books on contemporary set theory [duplicate]

I have gone through Halmos' Naive Set Theory. Now, could you recommend me a good follow-up book for a rigorous treatment of contemporary set theory? (For example, I've been suggested to look at ...
2
votes
3answers
207 views

“Methods of Theoretical Physics for Mathematicians”

I read in the Princeton Companion to Mathematics that even pure mathematicians should know some theoretical physics. However, I see that there are many reference books of mathematical methods for ...
12
votes
1answer
198 views

Websites that promote co-operation and social networking among mathematicians

Are there some websites that could be defined as social networks for mathematicians and scientists? What I have in mind is something similar to Academia.edu or ResearchGate, but more specific towards ...
19
votes
2answers
500 views

Collections of undergraduate research projects

I would like to compile a "big list" of undergraduate research projects in the following areas of mathematics: calculus; analysis; abstract algebra; linear algebra; number theory; geometry; ...
2
votes
2answers
187 views

Little, unknown, English or French research journals with good mathematics

In this article by Gian-Carlo Rota, you can read: "I bought a copy of Frederick Riesz' Collected Papers as soon as the big thick heavy oversize volume was published. [...] It was clear that ...
55
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 ...
7
votes
3answers
370 views

On progress in mathematics: some long-open problems and long-standing conjectures

I would like to ask a question here on Math Stack Exchange taking inspiration (and therefore combining) from two well-known threads on MathOverflow: (1) Not especially famous, long-open problems which ...
4
votes
6answers
2k views

What's behind the Banach-Tarski paradox? [closed]

The discovery of the Banach-Tarski paradox was of course a great thing in mathematics but raises the issue of the relation between mathematics and reality. Empirically there are good reasons for faith ...
2
votes
1answer
136 views

Natural progression in a curriculum for self-study of analysis

Would you list what is a natural and effective progression to self-study topics in analysis in order to gain a broad knowledge of the enormous corpus of knowledge that modern analysis involves. As a ...
1
vote
1answer
115 views

Combinatorial techniques, methods, and ideas in (“undergraduate”) real analysis

This question is dual to Probabilistic techniques, methods, and ideas in ("undergraduate") real analysis: I would like to collect some examples of combinatorial arguments to undergraduate ...
3
votes
3answers
179 views

Examples of combinatorial/probabilistic proofs of theorems in linear algebra

Are there any examples of combinatorial/probabilistic proofs of theorems in linear algebra? Motivation: I see here, the inverse is true.
-1
votes
2answers
126 views

What is some pure math news website by a publisher? [closed]

Why aren't there be any pure math website by a publisher? I google a lot and resulting only applied math news or math journal that is difficult and inaccessible even to advanced reader I am looking ...
15
votes
7answers
518 views

What are some mathematically productive ways to waste time? [closed]

What are some productive things that can be done (other than directly studying Mathematics) during leisure time that has a side effect to improve oneself at Mathematics? For example, reading ...
6
votes
4answers
251 views

Probabilistic techniques, methods, and ideas in (“undergraduate”) real analysis

As the book Probabilistic Techniques in Analysis by Richard F. Bass shows, nowadays techniques drawn from probability are used to tackle problems in analysis. The mentioned book presents a survey of ...
4
votes
2answers
103 views

Problems whose first solutions had been using Calculus but later was shown to be done by non-Calculus methods

I was wondering about mathematical problems whose first published solutions was obtained by using methods of Calculus but later was shown (or known) to be solvable by using non-Calculus methods. ...
5
votes
1answer
73 views

What are some remarkable and interesting uses of AM-GM Inequality ? Cite and explain with problems.

There are really lot of problems on AM-GM inequality because of its elementary nature and powerful applications. What I want is a collection of questions/problems which look very complex but get ...
20
votes
19answers
2k views

What are some interesting sole exceptions or counterexamples? [duplicate]

Many theorems assert that a particular property holds for all objects in a class except those in a given list of exceptions. Examples of rules that admit precisely one exception include: All primes ...
1
vote
0answers
61 views

Where can Gaussian Elimination be used?

I have searched for this and came to know about it that it is traditionally used to solve linear equations, finding determinant, rank of matrix, inverse of matrix. There was a problem on codechef: ...
0
votes
2answers
118 views

Why $1$ isn't a prime? [duplicate]

I was wondering the reason behind defining the Prime Numbers in a manner of which $1$ isn't an example. I read in Rotman's A First Course in Abstract Algebra that one reason that $1$ is not called a ...
1
vote
1answer
84 views

subtle/annoying fallacious proofs [duplicate]

I've been invited to a maths themed Xmas after party. I need to prepare a selection of interesting, and relatively simple fallacious proofs which other guests will try and find the flaw in. I'm trying ...
44
votes
18answers
4k views

What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside ...
9
votes
3answers
302 views

Examples of useful, insightful, and interesting hand-waving [closed]

It seems to me that some hand-waving (by which I mean some arguments that aim at giving some form of intuition on the problem even at expenses of complete rigour [and not mnemonics for high-schoolers ...