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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86
votes
20answers
8k views

Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
-5
votes
2answers
128 views
+400

How many ways are there to prove Cayley-Hamilton Theorem?

I see many proofs for Cayley-Hamilton Theorem in textbooks and net, so I want to know how many proofs are there for this important and applicable theorem.
104
votes
22answers
11k views

Most ambiguous and inconsistent phrases and notations in maths

What are some examples of notations and words in maths which have been overused or abused to the point of them being almost completely ambiguous when presented in new contexts? For instance, a ...
0
votes
1answer
12 views

Calculating the amount of times a binary search could run (worse case) without a calculator/calculating base 2 logs without a calculator.

Ok so I had a question on a test that I had to do without a calculator. And I can not figure out how in the world I am supposed to do it without a calculator. The question asked to find how many ...
26
votes
6answers
837 views

Open source lecture notes and textbooks

This question is inspired by the popular "Best Sets of Lecture Notes and Articles". Indeed, I would like to collect a "big-list" of open source (that is, with $\LaTeX$ code available) high-quality ...
114
votes
10answers
9k views

Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
-2
votes
0answers
68 views

Active, beautiful fields of mathematics [on hold]

Some fields of mathematics are attributed more beauty than others. There are many historical quotes of mathematicians celebrating the beauty of number theory -of which I know little- in comparison to ...
13
votes
9answers
858 views

Elevator pitch for a (sub)field of maths?

When I first saw the title of this question, I forgot for a moment I was on meta, and thought it was asking about quick, catchy, attractive, informative one-or-two-liner summaries of various fields of ...
0
votes
1answer
36 views

What are some pairs of mathematically-important functions that differ only at a few points?

Examples would include things like $$f(x, y) = \begin{cases} x^y & \text{ if } (x, y) \neq 0 \\ 0 & \text{ else} \end{cases}$$ versus $$g(x, y) = \begin{cases} x^y & \text{ if } (x, y) ...
15
votes
3answers
611 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 ...
0
votes
1answer
595 views

How many ways to find the center of an inscribed circle?

I want to find the coordinates of center of the inscribed circle triangle $ABC$, where $A(-274,-253)$, $B(-1,7)$, $C(14,7)$. I tried First way. We have $c = AB=377$, $a = BC=15$, $b = AC=388$. Let ...
697
votes
53answers
416k 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 ...
24
votes
9answers
38k views

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

I've always enjoyed solving problems in the complex numbers 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 ...
-4
votes
0answers
50 views

Big list of mathematical facts [closed]

There is a lot of good books and articles dedicated to school competitive math problems solving. Sometimes they contain some list of methods and facts which can be used to solve problems. Those lists ...
89
votes
5answers
4k views

“Advice to young mathematicians”

I have been suggested to read the Advice to a Young Mathematician section of the Princeton Companion to Mathematics, the short paper Ten Lessons I wish I had been Taught by Gian-Carlo Rota, and the ...
1
vote
1answer
31 views

Results on “subtraction” of measures and outer measures?

Most results I have seen involves addition of measures For example, let $m^*$ and $m$ be Lebesgue outer measure and Lebesgue measure respectively, then given $A = \bigcup\limits_{n = 1}^\infty E_n, ...
90
votes
24answers
71k views

Software for drawing geometry diagrams

What software do you use to accurately draw geometry diagrams?
613
votes
160answers
38k 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 ...
65
votes
30answers
31k views

Best Maths Books for Non-Mathematicians

I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often ...
14
votes
5answers
208 views

Is there a property in $\mathbb{N}$ that we know some number must satisfy but don't know which one?

I have two questions. $(1.)$ Is there a property of the natural numbers such that we know at least one number satisfies it but we don't know which one? Even more, $(2.)$ Is there a property ...
103
votes
34answers
17k 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 ...
59
votes
15answers
2k views

Unconventional mathematics books

I've recently purchased Oliver Byrne's reproduction of Euclid's Elements. It's a beautiful tome, that's rather unique in its presentation of the material as it represents many of Euclid's proof as ...
212
votes
64answers
50k views

'Obvious' theorems that are actually false

It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true". Now, I know a fair few examples of things that are obviously true and that can be proved ...
42
votes
20answers
3k views

Which mathematicians have influenced you the most? [closed]

This question is lifted from Mathoverflow.. I feel it belongs here too. There are mathematicians whose creativity, insight and taste have the power of driving anyone into a world of beautiful ideas, ...
3
votes
1answer
41 views

Foliations vs Laminations

What's the big difference/similarity between foliations and laminations? What kind of theorems hold for both of them? Is there something that makes them essentially the same/different?
13
votes
2answers
129 views

$a^x+b^x=c^x$ in geometry

The Pythagorean theorem. Let $A$, $C$, $B$ be three points on a line in this order, and let $D$ be another point, such that $\angle ADC =\angle CDB = 60^\circ$. Let $a=AD$, $b=BD$, $c=CD$. Then, ...
42
votes
10answers
12k views

Examples of finite nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
18
votes
4answers
588 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 ...
1
vote
1answer
38 views

Desirable properties of statistical estimators?

What are some of the properties that people will consider when designing a statistical estimators? For example, unbiasedness and sufficiency are some of the factors considered. Please give some ...
46
votes
10answers
54k views

What is the best book for studying discrete mathematics?

As a programmer, mathematics is important basic knowledge to study some topics, especially Algorithms. Many websites, and my fellows suggest me to study Discrete Mathematics before going to ...
31
votes
20answers
4k 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 ...
10
votes
14answers
635 views

Mathematics and literature [closed]

Are there novels (or other kinds of books) that include substantial references to topics and ideas closely related to mathematics (even if there are no explicit references to theorems, proofs, ...)?
1
vote
4answers
45 views

What are the principal (different) mechanisms of infinite descent proof?

I’m interested in building a list (including, where possible, links to proofs/papers/examples) which presents all known mechanisms of infinite descent (ID). I think this list would best be presented ...
27
votes
20answers
941 views

What are some surprising appearances of $e$?

I recently came across the following beautiful and seemingly out-of-the-blue appearance of $e$: $E[\xi]=e$, where $\xi$ is a random variable that is defined as follows. It's the minimum number of ...
22
votes
10answers
24k views

Real world uses of hyperbolic trigonometric functions

I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful. Is there any good examples of their uses ...
9
votes
4answers
2k views

What is the deepest / most interesting known connection between Trigonometry and Statistics?

I'm teaching both at the same time to different classes in high school, so I just wondered about this. Added by OP on 16.May.2011 (Beijing time) I mean Statistics only, without Probability. In ...
2
votes
0answers
26 views

On the alternative stamentes of the famous Sperner's Lemma.

The Sperner's lemma can be stated as follows. Lemma of Sperner. Let $\Omega$ an fintie set with $n$ elements. If a family $\{ A_i \}_{1\leq i \leq N}\subset \Omega$ of subsets satisfies the ...
5
votes
2answers
180 views

Ugly solutions to easily stated problems [closed]

I recently saw a very hideous closed form for a quartic equation here: Does a closed form solution exist for $x$? For fun, I'm wondering about surprisingly ugly solutions/ complicated machinery ...
0
votes
0answers
30 views

How to write the mathematic formulation of a pairwise sum?

In the normal summation, we can do this: And that would sum a list X as such in Python: ...
6
votes
1answer
313 views

Examples of interesting/unconventional solutions to rather “standard” problems?

For example, I recently came across the following way to evaluate the integral of $\cos^2 x - \sin^2 x$ without using double angle formulas: $$\int dx (\cos^2 x - \sin^2 x) = \int dx(\cos x + \sin ...
8
votes
0answers
59 views

Categorical formulations of basic results and ideas from functional analysis?

I'm taking a first (undergrad) course on functional analysis. Though the material is nice, the approach seems very ad hoc and in a sense, near-sighted (?). I was wondering whether the/a big picture ...
213
votes
29answers
20k 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 ...
0
votes
0answers
42 views

What are some interesting results that could be derived if some conjectures were true/false?

I recently came up on Djikstra's idea of a fictional company called Mathematics.Inc which produced proofs as trade secrets which could then be used by customers.. For example, it produced the proof of ...
11
votes
2answers
156 views

Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
6
votes
1answer
142 views

When adding zero really counts …

Note: Although adding zero has usually no effect, there is sometimes a situation where it is the essence of a calculation which drives the development into a surprisingly fruitful direction. Here is ...
394
votes
30answers
46k views

Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
0
votes
1answer
30 views

Exceptions in infinite-dimensional spaces

What are the properties that are true in finite-dimensional spaces but fails in the infinite-dimensional space? For example, the closed unit ball is compact only in finite-dimensional normed space.
21
votes
7answers
3k views

What are some common proof strategies in mathematics?

I want to start out by saying that I am new at proof based mathematics. I am used to seeing patterns and using them to solve similar problems. However, I have found this is not a very good way to ...
2
votes
1answer
44 views

Books with SAGE portions

I recently finished working through Adventures in Group Theory and really appreciated the use of SageMath it employs. I considered myself moderately proficient with Sage, but I found working through ...
5
votes
0answers
47 views

Curvature and topology

I am studying Riemannian Geometry and I came across various Theorems which give conditions on the topology of a manifold given conditions on curvature, and vice-versa. Just to mention a few of them: ...