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.

learn more… | top users | synonyms

3
votes
3answers
138 views

Mental Math Techniques [on hold]

What are some interesting mental math techniques that you know? Here's one that I got from my Grandmother who got it from a book: To square a two-digit number (from $26$ to $49$), take the number ...
20
votes
14answers
2k views

If there are obvious things, why should we prove them?

Obviously, there are obvious things in mathematics. Why we should prove them? Prove that $\lim\limits_{n\to\infty}\dfrac{1}{n}=0$? Prove that $f(x)=x$ is continuous on $\mathbb{R}$? $\dotsc$ Just ...
3
votes
1answer
71 views

What are the differences in mathematical notation around the world?

I just learned that $\text{sen}\,x$ is the Portuguese notation for $\sin x$, and I was inspired to ask: What differences are there in how mathematics is written around the world? Note 1: I am likely ...
5
votes
2answers
146 views

Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?

Question: Which inequalities similar to the famous isoperimetric inequality is known? conjectured? I recently learned about some inequalities which are all similar to the famous isoperimetric ...
6
votes
1answer
342 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) ...
8
votes
1answer
124 views

Create a Huge Problem

I am wondering if any problems have been designed that test a wide range of mathematical skills. For example, I remember doing the integral $$\int \sqrt{\tan x}\;\mathrm{d}x$$ and being impressed at ...
0
votes
0answers
4 views

What are some good introductory textbooks on Sieve Theory?

I fail to find a duplicate. If it exists, please give me the link and close the question accordingly. As the title suggests, I am looking for recommendations on introductory books to Sieve Theory. ...
2
votes
0answers
25 views

Revise high school material

Can you suggest me a comprehensive book to revise high school mathematics (up to besic calculus)? It should be extremely clear and complete and "scientific" (not like most high school books). Thank ...
319
votes
41answers
139k views

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain, but are ...
63
votes
18answers
6k views

Your favourite application of the Baire Category Theorem

I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications ...
8
votes
6answers
9k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
21
votes
7answers
465 views

“Here's a cool problem”: a collection of short questions with clever solutions

This game will be familiar to many mathematicians, and it is always good fun to play. I am looking to find a list of good questions with short, when-you-see-it solutions. The kind of question one ...
74
votes
19answers
12k views

Visually deceptive “proofs” which are mathematically wrong

Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually ...
0
votes
0answers
35 views

What are the branches of mathematics? [duplicate]

What are all the branches of mathematics? I want to know this so that I can compare between them, so if you can give me with them a brief look of what they are about that would be awesome.
14
votes
9answers
3k views

What are some good iPhone/iPod Touch/iPad Apps for mathematicians?

There are lots of good apps for teaching mathematics to children but I would like to learn about apps for undergraduate/graduate/research levels. Helper questions Any algebra system (like ...
24
votes
5answers
2k views

Tell me problems that can trick you

I am looking for problems that can easily lead the solver down the wrong path. For example take a circle and pick $N$, where $N>1$, points along its circumference and draw all the straight lines ...
31
votes
4answers
2k views

How to write a good mathematical paper?

I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar. A good time always goes too fast! Two years are fled. In the third year of ...
5
votes
1answer
29 views

Surprising constructions in algebraic topology that facilitate one's understanding of underlying theory

I am recently come into the world of algebraic topology and find it a fascinating place with lots of beautiful constructions that challenge one's intuition. Also, understanding these constructions are ...
0
votes
0answers
30 views

The term $rank$ in methematics

Reading wikipedia's disambiguation page about the "rank" word I see many concept of rank of many different matematical object. I only know about the rank of a graded poset and the rank of a set that ...
-1
votes
1answer
64 views

A big list of examples that a power of a prime ideal is not primary in an algebra of finite type over a field

Let $k$ be a field. Let $A$ be an integral domain which is a $k$-algebra of finite type. I would like to know examples that a power of prime ideal of $A$ is not primary. The more example, the better. ...
138
votes
21answers
12k views

Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
2
votes
1answer
84 views

What are interesting examples of existential proofs based on cardinality arguments?

Probably the most famous example of a proof, where consideration of cardinalities is used to show existence of some object, it the Cantor's proof that there exist transcendental numbers. What are ...
1
vote
0answers
33 views

Matrices of the form $A^p=(a_{ij}^p)$

I am wondering if there is a name for these kind of matrices and if they are interesting or not? Do they even exist? Let $A$ be a $n\times n$ matrix with elements $a_{ij}$. $A= (a_{ij})_{i,j\in\{1, ...
385
votes
131answers
23k 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 ...
2
votes
5answers
298 views

What area of Abstract Algebra do you find most interesting? [closed]

For my Abstract Algebra class, we will be doing small presentations (2 class periods) covering some topic in Abstract Algebra. Thus far, I have studied groups, rings, fields, modules, tensor ...
5
votes
0answers
98 views

What is the best Mathematical Insight you have had? - PLEASE MOVE TO META [closed]

I've used this site a lot but am new to the actual forum. Basically, I am wondering if we could collect a list of mathematical insights / explanations / neat proofs etc. that people on this forum have ...
10
votes
2answers
183 views

Theorems in algebraic geometry which have been proved only by using cohomology

There are many theorems in algebraic geometry which were proved using cohomology. I would like to know examples of such theorems which have been proved only by using cohomology. In other words, those ...
0
votes
1answer
23 views

In how many different ways can this problem be solved?

I have a math problem. In many different ways can this problem be solved? Here is the problem: $$y''-y'-2y=0, \\ y(0)=1 \\ y'(0)=0$$ I have already found $5$ ways: $(1):$ Characteristic equation ...
5
votes
2answers
122 views

Why the $\log$ is so special?

When I first learn about the logarithm function $\log$ or $\ln$. My professor said that $\log x$ is a function that when we derive we get the inverse function $1/x$. This $\log$ becomes very popular ...
0
votes
0answers
29 views

What are $\Gamma$-semigroups?

I have some problems with $\Gamma$-Semigroups, the definition that I've found is A $\Gamma$-Semigroup is a pair $(M,\Gamma)$ defined as follow If $x,y$ and $z$ are in $M$ and $\alpha$ and ...
129
votes
91answers
36k 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 ...
15
votes
1answer
417 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
2answers
108 views

Recommendation for Number Theory Textbook

. Greetings, every mathematicians! I'm a foreigner (meaning English is not my first language) and an undergraduate student. I'm currently studying linear algebra, set theory and have already studied ...
0
votes
1answer
27 views

Overview of game theory

I have a good high school math background, and I am interested in game theory, so I wanted to know something more about it, but I found very technical things or wikipedia. I am looking for something ...
9
votes
3answers
4k views

I want to start mathematics from scratch. What should I begin with?

I've just finished high school but I don't feel my knowledge of mathematics is good enough. I'd like to start again from scratch, possibly adding a bit (a lot) of problem solving to it. What is the ...
40
votes
14answers
3k 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 ...
14
votes
2answers
1k views

Good introductory books on homological algebra

Which books would you recommend, for self-studying homological algebra, to a beginning graduate (or advanced undergraduate) student who has background in ring theory, modules, basic commutative ...
25
votes
3answers
716 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 ...
115
votes
27answers
14k views

Best Fake Proofs? (A M.SE April Fools Day collection) [closed]

In honor of April Fools Day 2013, I'd like this question to collect the best, most convincing fake proofs of impossibilities you have seen. I've posted one as an answer below. I'm also thinking of a ...
4
votes
1answer
40 views

Inequalities that show if a distribution decays slowly

Often, one is often interested in theorems/inequalities of the following kind: Let $X$ be a random variable then the probability that $X$ is close to typically $\mu$ (or larger than some constant) is ...
16
votes
11answers
3k 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 $[a,b]$ ...
74
votes
29answers
15k views

Examples of mathematical results discovered “late”

What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple ...
0
votes
1answer
105 views

Examples of non-trivial closed subschemes of a complete non-projective non-singular variety

Let $k$ be an algebraically closed field. A variety over $k$ is a separated integral scheme of finite type over $k$. Let $V$ be a complete non-projective non-singular variety over $k$. Let $Z$ be a ...
1
vote
1answer
321 views

Properties of generalized limits aka nets

I want to find some article or a book which contains all general properties of nets. Of course some of them similar to properties of sequences with almost the same proofs, but I don't fill the edge, ...
2
votes
2answers
91 views

Simple & Intuitive Statements that are Difficult to Prove

Looking through the webcomic, I came across one of my favorite comics: (from Saturday Morning Breakfast Cereal) It seems that people have an ongoing interest in results in mathematics that are ...
5
votes
6answers
252 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 ...
38
votes
15answers
8k views

What is the most elegant proof of the Pythagorean theorem?

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite ...
8
votes
4answers
766 views

Interesting mathematical problems for 1st year university students

Can you explain some mathematical problems that you find the most interesting (NB: the problem must be accessible to a 1st year university student: that is, a problem for which there is an elegant ...
3
votes
3answers
108 views

Most inspirational mathematical books [closed]

I would like to know which books on mathematics (from university texts to divulgative pop-math books) inspired you the most. My choice is Spivak's Calculus, which is, IMHO one of the most ...
1
vote
1answer
191 views

Example of non-noetherian algebras which are tensor products of noetherian algebras

We suppose all rings are commutative with unity. I am looking for examples of a tensor product $B\otimes_A C$ which is not noetherian, where $A$ is a noetherian ring and $B, C$ are noetherian ...