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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3
votes
1answer
254 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
242 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
429 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
70 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
793 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
348 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) ...
28
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 ...
71
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
148 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
2k 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
4k 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 ...
30
votes
12answers
1k 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 ...
-2
votes
4answers
484 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
223 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 ...
106
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
81 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
100 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
224 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
181 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
250 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 ...
0
votes
2answers
102 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
193 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
527 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
2k 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
74 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
8answers
528 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
246 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 ...
33
votes
9answers
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
375 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 ...
12
votes
2answers
309 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
223 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
370 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 ...
4
votes
3answers
157 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 ...
137
votes
77answers
12k 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
303 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
104 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 = ...
41
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 ...
64
votes
21answers
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. ...