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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7
votes
1answer
234 views

“All math is useful eventually”

We have all heard the argument : a lot of mathematics that was thought to be useless, abstract constructions with no links to the real world ended up being of use, like some arithmetic is useful in ...
2
votes
2answers
65 views

when is a ring a free module over a subring?

Let $S \subset R$ be rings, $S$ not necessarily an ideal of $R$, and $S \neq R$. Is there anything that can be said about when $R$ is free as an $S$-module?
2
votes
1answer
142 views

Statements equivalent to the Axiom of Choice

The Axiom of Choice reads: The product of a collection of non-empty sets is non-empty. As you know well, this axiom is equivalent to many other statements. A few examples (probably the most ...
10
votes
1answer
332 views

What are some examples of hard theorems in category theory?

I'm currently learning some category theory, but so far I've used it only as a handy way to talk about various related concepts in algebra and topology with some nice, easy-to-prove lemmas like "left ...
18
votes
1answer
526 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 ...
2
votes
3answers
64 views

“Asymmetric” results in maths analogous to “Parity violation” of the weak force?

Disclaimer: I'm not a physicist and I don't claim to be one so if I have any mistakes I’ll be glad to be corrected. One feature of the standard model of particle physics is that the weak force is not ...
30
votes
14answers
799 views

Interesting but short math papers?

Is it ok to start a list of interesting, but short mathematical papers, e.g. papers that are in the neighborhood of 1-3 pages? I like to read them here and there throughout the day to learn a new ...
3
votes
3answers
163 views

What are some examples of “exotic” algebraic structures? [closed]

I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, ...
1
vote
0answers
34 views

limit formalisms

Let $f:\mathbb R \to \mathbb R$ be a function and $a\in \mathbb R$ a point. The Cauchy definition of the limit $\lim _{x\to a}f(x)=L$ is well-known. For pedagogical reasons I'm interesting in a ...
2
votes
1answer
129 views

Book about elementary geometry , triangles, circles … [duplicate]

Currently, I'm studying a little about geometry and I was trying to find out some good book about it on internet, however I didn't find anything that I thought nice to me or what I really expected to ...
2
votes
3answers
468 views

Book with lots of geometry theorems

I want to study geometry and was looking for some book that has lots of theorems and covers almost all Euclidean geometry that is needed for High School and Maths Olympiads. Thanks.
18
votes
1answer
514 views

The most active fields of mathematics?

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?
1
vote
0answers
61 views

How calculators compute. [duplicate]

I would like to teach a class on the "magic" behind the calculator, so I would like to generate a list of "algorithms" for how a calculator computes the things we want it to. I will get the ball ...
3
votes
3answers
275 views

Examples where derivatives are used (outside of math classes)

I want to know what is the use of derivatives in our daily life. I have searched it on google but i haven't find any accurate answer. I think it is mostly used in Maths but I want to know its use in ...
26
votes
5answers
444 views

Other interesting consequences of $d=163$?

Question: Any other interesting consequences of $d=163$ having class number $h(-d)=1$ aside from the list below? Let $\tau = \tfrac{1+\sqrt{-163}}{2}$. We have (see notes at end of list), ...
3
votes
1answer
221 views

Best book ever on Galois theory (and differential galois theory) [closed]

Which is the single best book for Galois theory (that includes differential Galois theory) that everyone who loves pure Mathematics should read?
2
votes
1answer
58 views

Find multiple integrals $I_{\max}(k,n)$ and $I_{\min}(k,n)$ in various ways

$I_{\max}(k,n)=\underbrace{\int\limits_0^1\int\limits_0^1\dots\int\limits_0^1}_k\left(\max\limits_{1\le i\le k}x_i\right)^n\,dx_1dx_2\dots dx_k$ ...
9
votes
4answers
233 views

Math games for car journeys

On long car journeys with kids we are all familiar with "I spy" or "Twenty questions". What math related games can one play on a car journey instead that are fun and offer similar variety?
5
votes
1answer
85 views

Nice examples of finite things which are not obviously finite

This question is in the spirit of the question "Nice examples of groups which are not obviously groups". There are many impressive finiteness results in mathematics. For example: The finiteness of ...
0
votes
1answer
66 views

Real examples of non-positive definite integrals

We need to get some examples of non-positive definite integrals in real practice (mathematical physics, engineering, mathematical statistics and etc).
3
votes
1answer
269 views

Good book for logic self-study

I know a similar question has already been asked, but can anyone suggest a good book on mathematical logic that includes answers to exercises? I am looking for something that is conducive to ...
8
votes
3answers
249 views

Bedtime maths books?

Most of math books require you to copy proofs and do excersices to extract the content from them. Are there any good serious math books which require only reading and no writing? ADDED: One ...
2
votes
0answers
58 views

List of most useful coverings and their applications?

I've heard that many problems may be simplified when looking at covering spaces, but I haven't been able to find a good list. What are the most common covering spaces one should understand by heart? ...
3
votes
2answers
99 views

Numerically Misleading Results

Are there any calculations or results that have similar answers and when compared numerically look the same, but in actual fact after so much precision, the answers diverge from each other? An ...
3
votes
1answer
111 views

A question about the hierarchy of topologies on a given set

It's easier to understand with examples: Every finer topology than a Hausdorff topology is hausdorff. Every coarser topology than a compact topology is compact. What are the full set of properties ...
8
votes
1answer
436 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ...
1
vote
1answer
74 views

examples of toy non-Euclidean geometries

Today I was working on a problem in euclidean geometry, and I found it immensely useful to compare with Fano geometry, for contrast. Are there any other toy geometries like Fano geometry? I think it ...
3
votes
0answers
67 views

Math for kids with Cuisenaire rods

I work with kids and i am searching some cool stuff to do with Cuisenaire rods. Thinking about an application i thought that i can show to my students what will be the sum of first $N\in\mathbb{N}$ ...
27
votes
3answers
806 views

Common Math Mistakes Made by Scientists

I am a mathematician by training working with a physicist. I have been invited to give an hour-long tutorial/presentation to incoming graduate students. These students are all coming in with physical ...
5
votes
6answers
378 views

Fundamental Theorem of Trigonometry

This is a pretty open ended question and I apologize, in advance, if this is not the place for it. But what do you recommend should be given the title of the Fundamental Theorem of Trigonometry and ...
6
votes
3answers
200 views

What is the fastest way to $\pi$?

There are many known sequences convergent to $\pi$ with different convergence accelerations. For example both of the following sequences are convergent to $\pi$ when $n$ goes to $\infty$: (a) ...
29
votes
20answers
2k views

Good math bed-time stories for children?

What are some good references/books/articles from which to derive some good bed-time math stories to pique a child's interest in math? I am fascinated by math (used to hate it as a kid) and want my ...
76
votes
24answers
6k views

What are some examples of notation that really improved mathematics?

I've always felt that the concise, suggestive nature of the written language of mathematics is one of the reasons it can be so powerful. Off the top of my head I can think of a few notational ...
72
votes
31answers
9k views

What are the most overpowered theorems in mathematics?

What are the most overpowered theorems in mathematics? By "overpowered," I mean theorems that allow disproportionately strong conclusions to be drawn from minimal / relatively simple assumptions. ...
2
votes
1answer
150 views

Proofs that there is no $f(z)$ such that $\exp f(z) = z$ for all $z \in \Bbb{C}\setminus\{0\}$

When I first learned about this result I was completely stunned that there is no holomorphic function $f(z)$ on $\Bbb{C}\setminus\{0\}$ such that $\exp f(z) = z$. What are some interesting proofs of ...
1
vote
3answers
3k views

Rules for Product and Summation Notation

When we deal with summation notation, there are some useful computational shortcuts, e.g.: $$\sum\limits_{i=1}^{n} 2 + 3i = \sum\limits_{i=1}^{n} 2 + \sum\limits_{i=1}^{n} 3i = 2n + ...
20
votes
9answers
5k views

What is the simplest proof of the pythagorean theorem you know? [duplicate]

Maybe enough so to explain it to children.
18
votes
6answers
1k views

What are the most ugly equations in mathematics? [closed]

I feel like most people focus on only the most simple, elegant equations in math, but are there useful ugly ones? When I say ugly, I mean extremely long and generally useless? I can't exactly find an ...
32
votes
13answers
2k views

How To Present Algebraic Topology To Non-Mathematicians?

I am writing my master thesis in algebraic topology (fundamental groups) and as a system in my school students must write about one page about their theses explaining for non mathematicians the ...
-3
votes
4answers
519 views

Examples of $ \sqrt 2$ and $\sqrt[3]{3}$ in nature? [closed]

The question has been written up. By nature i mean in physics, ascetics, etc. @all. I expected some advanced things, the things all of mentioned are already known to the asker.
3
votes
1answer
231 views

Open Problems in Real Analysis [closed]

What are some open problems in Real Analysis? I have found some on the Open Problem Garden, but would like to see some more.
1
vote
1answer
33 views

More values of $a$ and $D $ on conditions set by me and a way to obtain more values.

Here I define -: $\alpha=a+ \sqrt D$ and $\beta=a-\sqrt D$ Then find out values of $\alpha$ and $\beta$ satisfying- $$\alpha>1 \quad and \quad -1< \beta <1 $$ and both the variables are ...
107
votes
19answers
7k 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 ...
1
vote
2answers
82 views

What is the one thing I am going to forget on the math subject GRE tomorrow?

As I finish up my final study session for the subject test tomorrow, I was wondering if anyone had some general advice/ often forgotten facts that might come in handy.
34
votes
13answers
3k 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 ...
2
votes
1answer
101 views

Judging a book by its cover

One of the main things I am using Stack Exchange for lately is finding math texts. As a community, I think we are generally very helpful at suggesting texts for all kinds of topics, but are we too ...
7
votes
5answers
226 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
13 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
182 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 ...
6
votes
3answers
273 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 ...