Questions asking for a "big list" of examples, illustrations, etc. Please do not ask too many of these. Please do not use this as the only tag for a question.

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6
votes
5answers
209 views

Different ways to prove $\sqrt p$ irrational for $p$ prime.

I know this fact can be proved by contradiction(reductio ad absurdum) but please give proofs by different methods.
0
votes
0answers
12 views

How to suggest the intractability of a problem that is not known to be $\mathcal{NP}$-complete

If a proof of a a decision problem in $\mathcal{NP}$ being $\mathcal{NP}$-complete can be found, it is a strong evidence that the problem is intractable: people have not found efficient algorithms for ...
6
votes
2answers
152 views

List of (pre-graduate level) exercises

I am about to get my undergraduate degree in (pure) mathematics, but I feel like I'm ill prepared to go through a graduate program. This is why I'm looking for texts like this one ...
5
votes
3answers
164 views

Differences between infinite-dimensional and finite-dimensional vector spaces

I've just started a course in Representation Theory, and in solving our first homework I've used a couple of theorems about finite-dimensional vector spaces (for an example, rank-nullity theorem). My ...
1
vote
2answers
87 views

Interesting problems using group/representation theory

I've been going through this representation theory lecture notes, and I've found the ''hungry knights'' problem very interesting. Do you know any interesting problems similar to that one? To define ...
2
votes
1answer
148 views

Best book to learn Affine Geometry?

I'm going to learn Affine plane as well as affine Geometry. Unfortunately, my text book (not in English) is not good at all, so please recommend some book you think it's good for self-learning (and ...
0
votes
1answer
15 views

What other statements of this general form can we prove about the direct image function?

Given a relation $R : X \rightarrow Y,$ write $R_* : \mathcal{P}(X) \rightarrow \mathcal{P}(Y)$ for the direct image function defined by asserting that $$b \in R_*(A) \Leftrightarrow \exists a \in A : ...
1
vote
4answers
476 views

Real world examples of quadratic and/or finding roots of a quadratic?

Anyone ever come across a good situation where a) a situation is modeled by a quadratic equation $y=ax^2+bx+c$ and/or b) you've even needed to find where $y=0$ (roots, $x$-intercept, etc)
15
votes
11answers
1k views

Undergraduate Schools for Mathematics

I am currently a senior in high school, and I have been studying mathematics for about nine or ten years now in my personal time outside of school. I am not familiar with academia or in general higher ...
3
votes
1answer
70 views

Second order linear differential equation

I have to teach the following methods to my juniors at college to solve differential equations: 1) partial fractions 2) reduction of order 3) variation of parameter 4) power series 5) green's ...
16
votes
9answers
448 views

What is the most surprising result that you have personally discovered?

This question is inspired by my answer to this one: Surprising identities / equations In that question, people were asked about the most surprising result that they knew. Almost all of them quoted ...
10
votes
0answers
210 views

Papers with unorthodox writing style

I'm not sure if this is the right forum for this question, in any case probably CW is appropriate? I've been looking around the mathblogosphere for the past few weeks and ran into mathgen. It's ...
30
votes
8answers
2k views

Literary statements that are false as mathematics

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 ...
5
votes
6answers
333 views

“$n$ is even iff $n^2$ is even” and other simple statements to teach proof-writing

I am supposed to teach undergraduate students who do not major in mathematics and I would like to give them a short introduction to mathematical reasoning and to the concept of proof. I am looking for ...
11
votes
2answers
274 views

What was the largest ratio (result size)/(integrand size) you have seen?

Sometimes a definite or indefinite integral of a simple-looking one-liner integrand can give astonishingly huge result. What was the largest ratio of the size of shortest known closed-form result to ...
4
votes
2answers
170 views

Examples of non-obvious isomorphisms following from the first isomorphism theorem

I am learning the first isomorphism theorem, and I am working with some isomorphisms to practice for my upcoming test. I know some of the basic ones like: $\mathbb{R}/\mathbb{Z} \cong \mathcal{C}$, ...
4
votes
3answers
291 views

What are some easily-stated recently proven theorems?

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 ...
3
votes
3answers
152 views

Alternative Creative Proofs that $A_4$ has no subgroups of order 6

Since I've been so immersed in group theory this semester, I have decided to focus on a certain curious fact: $A_4$ has no subgroups of order $6$. While I know how to prove this statement, I am ...
132
votes
72answers
11k 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 ...
22
votes
10answers
1k views

Puzzles or short exercises illustrating mathematical problem solving to freshman students

At high school, the solution method to almost all mathematical exercises is to apply some technique or algorithm you have learned before. At the university, the situation is fundamentally different. ...
10
votes
0answers
264 views

What Do Mathematicians Do?

The American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: What ...
0
votes
1answer
97 views

What are some good subtly incorrect proofs of obviously incorrect results? [duplicate]

I'm interested in compiling a list of proofs that look logically correct at a glance, but "prove" something obviously incorrect. Here are some examples. $e^{i \pi} = -1$ $e^{2i\pi} = 1$ $2i\pi = ...
39
votes
13answers
5k views

Interesting math-facts that are visually attractive

To give a talk to 17-18 years old (who have a knack for mathematics) about how interesting mathematics (and more specifically pure mathematics) can be, I wanted to use nice facts accompanied by nice ...
61
votes
20answers
3k views

What are some examples of mathematics that had unintended useful applications much later?

I would like to know some examples of interesting (to the layman or young student), easy-to-describe examples of mathematics that has had profound unanticipated useful applications in the real world. ...
3
votes
5answers
388 views

Riddles with a mathematical twist

I am looking for riddles that are understandable for everyone(so especially non-mathematicians) but require mathematical knowledge or deep abstract ideas to be solved. The best answer will be the ...
1
vote
1answer
39 views

How can we transform matrices into scalars?

If we have the three matrices: $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} , \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 ...
14
votes
4answers
736 views

Can we get un-obvious results by defining sophisticated topologies?

What I originally found so interesting about general topology was how general a type of thing a topology is, and how the terminology open, closed, compact, continuous, convergence et cetera means ...
2
votes
6answers
211 views

Neat expressions that equal 1

I would like to see beautiful and elegant expressions involving elementary and non-elementary functions, transcendental numbers, etc. that equal 1. Be creative!
1
vote
3answers
140 views

Uniquely geodesic spaces

The purpose of this list issue is to better understand the class of uniquely geodesic spaces. I'm looking for two different things : Overclass : for example geodesic space or contractible space. ...
1
vote
3answers
117 views

Paradoxes without self-reference?

Most paradoxes involves self-reference, the only exception known to me is Yablo's paradox, however it is still debated if it is really without self-reference. So, I was wondering, are there other ...
7
votes
2answers
143 views

Soft question: Unconventional proofs

I'm not sure if I understood it correctly, but one of my professors told us that one theorem was proved this way: A mathematician assumed the truth of the Riemann hypothesis and was able to prove a ...
3
votes
8answers
279 views

simple theorems that non-mathematicians can understand and appreciate.

For example, I stated this fact/theorem at a dinner to some friends and they were pretty impressed. Given any sequence of n integers, positive or negative, not necessarily all different, some ...
3
votes
3answers
92 views

''Reading'' polynomials at the first glance

I'm reading Proofs from the Book, and I ran into following theorem: Suppose all roots of polynomial $x^n + a_{n-1}x^{n-1} + \dots + a_0$ are real. Then the roots are contained in the interval: $$ - ...
7
votes
3answers
288 views

What are some easy-to-remember prime numbers?

This is a question without much mathematical value, but since I don't immediately see an answer on Google I thought I'd ask anyway ... I'm looking for some largeish (> 10,000) easy-to-remember primes, ...
1
vote
0answers
88 views

What is a good source of problem-solving type problems?

I am not looking for contest problems where there is a clever trick or a standard approach, I am looking for more creative and open-ended problems such as this , and I am not looking for questions ...
1
vote
0answers
368 views

What are real life applications of Diophantine equations?

Are there any real life applications of linear Diophantine equations? I am looking for examples which will motivate students.
3
votes
4answers
159 views

Enlightening Mathematical Models

What is your favourite Mathematical Model? What features make it intuitive or elegant? This question is largely inspired by an example and a desire to find other's like it. Suppose we have two ...
7
votes
3answers
175 views

Examples of $\mathcal{O}_X$-modules that are not quasi-coherent sheaves

Let $X = \operatorname{Spec} k[x]_{(x)}$ which consists of two elements, the generic point $\zeta$ corresponding to the zero ideal and the closed point $(x)$. Define an $\mathcal{O}_X$-module ...
40
votes
14answers
2k views

Notations that are mnemonic outside of English

Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the ...
45
votes
17answers
2k views

Rigour in mathematics

Mathematics is very rigorous and everything must be proven properly even things that may seem true and obvious. Can you give me examples of conjectures/theories that seemed true but through rigorous ...
4
votes
1answer
179 views

List of proofs of non-trivial theorems which were unnoticed to be wrong for at least a few years

For example, the Weber's proof of Kronecker–Weber theorem. I would like to know such proofs. It seems to be important for me to remember that a widely accepted proof might be wrong.
7
votes
7answers
161 views

Mathematical Games suitable for undergraduates

I am looking for mathematical games for an undergraduate `maths club' for interested students. I am thinking of things like topological tic-tac-toe and singularity chess. I have some funding for this ...
18
votes
3answers
564 views

How to Garner Mathematical Intuition

Motivated by Why Is Intuition so Important to Mathematicians but Missing from Mathematics Education? $^{1}$ by Leona Burton, I would like to learn about specific ideas or strategies to attain ...
19
votes
4answers
545 views

What newer mathematics fields helped to solve or solved problems from older fields of mathematics?

I usually have a more or less formed template of conversation to talk with people about mathematics, It's importance, methods, history, etc. I've been for some time interested in newer fields of ...
2
votes
3answers
174 views

Pop Math Book You Would Love to See Written

I don't usually like these types of big list questions, but I think this is actually fairly important as far as education, informing the public about what we do, and getting people excited about math ...
4
votes
4answers
123 views

Resources to help an 8yo struggling with math

Friends of mine asked me for suggestion for one of their children (age 8) who had bad scores at the local Star test (the family is based in California). Both parents work, so they have also limited ...
10
votes
3answers
289 views

categorical generalizations of familiar objects

A couple of days ago I've learned that you can define trace in a very abstract setting. Namely, suppose $F\colon A\to B$ is a functor between two categories. Suppose $E,G\colon B\to A$ are two ...
7
votes
1answer
121 views

Applications of Serre duality

I am reading about Serre duality theory in algebraic geometry from Hartshorne, and am wondering what kinds of applications it has. It seems that most applications go through some version of the ...
5
votes
0answers
82 views

Classes of groups known to be realizable (IGP)

A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of ...
-2
votes
1answer
102 views

Article for a maths magazine [closed]

Any idea for an unique proof or theorem or any kind of mathematical philosophy to be presented in a maths magazine?