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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3
votes
4answers
2k 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.
25
votes
19answers
474 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 ...
379
votes
29answers
42k 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 ...
5
votes
9answers
502 views

Looking for functions $f$ with $\int_{-\infty}^{\infty}f(x)\,dx = 1$. [on hold]

I am looking for functions and/or constants that when being integrated from minus infinity to infinity produce 1. I think the Dirac delta function is one example but perhaps there are some more? ...
-2
votes
2answers
74 views

Is the area of linear programming dead right now? [on hold]

By dead i mean not much/completely no research there . Is the area of linear programming dead right now? If it is not dead, what are the active area called for example except computer science?
3
votes
1answer
101 views

Do any mathematican still reserach about trigonometry?

Do any mathematican still reserach about applied trigonometry? If so, what are the subject area called in the PhD level except fourier analysis? In many area, you could see a lot of trig and ...
-2
votes
4answers
20 views

Collecting sufficient conditions for Sorli's conjecture on odd perfect numbers

(Note: This question has been cross-posted from MO.) Sorli's conjecture predicts that, for an odd perfect number $N$ given in the Eulerian form $N = {q^k}{n^2}$ (where $q$ is prime with $\gcd(q, n) ...
0
votes
0answers
17 views

What are some easy to prove results on the density of primes?

Bertrand's postulate states that for any integer $n>3$, there's always a prime $p$ between $n$ and $2n-2$. That result sets a reasonable 'lower bound' on how often we can expect primes to show up, ...
0
votes
0answers
42 views

Record-holding mathematical proofs [closed]

Which mathematical theorems admit proofs that are extreme in some sense? Here is what I have in mind: The classification theorem for finite simple groups is the longest proof mathematics has seen. ...
1
vote
1answer
40 views

Examples of holomorphic, complex differentiable, always positive functions

I am looking for classes of functions which are: 1) holomorphic 2) |f(z)|>0 for all z 3) complex differentiable (i.e. f(z)=mod(z) is not valid) ...
595
votes
158answers
37k 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 ...
21
votes
15answers
3k views

Useful examples of pathological functions

What are some particularly well-known functions that exhibit pathological behavior at or near at least one value and are particularly useful as examples? For instance, if $f'(a) = b$, then $f(a)$ ...
2
votes
2answers
35 views

Areas where closed form solutions are of particular interest

Assuming the definition of 'Closed Form' given in the table of: Closed Form Wikipedia entry, what areas tend to have problems that are traditionally expressed in closed form? EDIT: Given the comment ...
36
votes
10answers
958 views

What are Different Approaches to Introduce the Elementary Functions?

Motivation We all get familiar with elementary functions in high-school or college. However, as the system of learning is not that much integrated we have learned them in different ways and the ...
1
vote
2answers
255 views

Mathematicians average in student life but later became significant

What are the examples of mathematicians who were below the average in their student life (say, upto university level but it may be less) but later in life became significant mathematicians. Up until ...
4
votes
0answers
45 views

Open problems in Lie theory

I been studying lie theory for some time. Beside classification related problems what are some examples of open problems in the lie world? Especifically in the topological/differentiable structure of ...
23
votes
5answers
559 views

Famous Problems Where We Only Know the Elementary

Define a graph with vertex set $\mathbb{R}^2$ and connect two vertices if they are unit distance apart. The famous Hadwiger-Nelson problem is to determine the chromatic number $\chi$ of this graph. ...
0
votes
0answers
19 views

Examples of sans serif Greek used in published mathematics

Unicode contains several thousand mathematical symbols, including individual code points for different maths alphabets. For example, U+1D434 is "mathematical italic capital A" (𝐴), U+1D63C is "math ...
27
votes
20answers
2k views

Elementary books by good mathematicians

I'm interested in elementary books written by good mathematicians. For example: Gelfand (Algebra, Trigonometry, Sequences) Lang (A first course in calculus, Geometry) I'm sure there are many other ...
14
votes
1answer
351 views

Most wanted reproducible results in computational algebra

I am interested in suggestions for major computational results obtained with the help of mathematical software but not easily verifiable using computers. "Most wanted" could refer, for example, to ...
1
vote
0answers
26 views

Uniform continuity with respect to parameter.

Let $\mathbb{X},\mathbb{Y}$ and $T$ metric spaces. A family $\{f_t\}_{t\in T}$ of maps $f_t:\mathbb{X}\to\mathbb{Y}$ is uniformly continuous with respect to parameter $t$ if, $$ (\forall ...
653
votes
48answers
408k 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 ...
27
votes
6answers
3k 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 ...
21
votes
3answers
3k views

Research Experience for Undergraduates: Summer Programs (that accept non-American applicants)

There are many summer research programs in the United States, targeted at good motivated undergraduate students majoring in mathematics. The main aspects that characterize such programs are: (a) a ...
58
votes
13answers
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 ...
98
votes
19answers
4k views

Past open problems with sudden and easy-to-understand solutions

What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved ...
32
votes
6answers
2k views

Do We Need the Digits of $\pi$?

I was reading today that someone found $\pi$ to the ten trillionth digit. Whenever I read that $\pi$ has been calculated to more digits, I ask myself whether this is useful. I know that there are ...
41
votes
14answers
4k 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 ...
76
votes
12answers
10k views

Conjectures that have been disproved with extremely large counterexamples?

I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture. I'm sure that everyone here is familiar with it; it describes an operation on a ...
9
votes
8answers
923 views

Examples of fallacies in arithmetic and/or algebra [closed]

I'm currently preparing for a talk to be delivered to a general audience, consisting primarily of undergraduate students from diverse majors. My proposed topic would be Examples of fallacies in ...
3
votes
1answer
39 views

What are the most common handwavy calculus rules used in statistics?

I am two weeks into my first stats course and already I have noticed that, because my class ignores measure theory, the instructors are being sloppy about explaining which kinds of functions are ...
61
votes
17answers
10k views

Interesting “real life” applications of serious theorems

As student in mathematics, one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: ...
6
votes
4answers
452 views

University-level books focusing on intuition?

I help some students with difficulties in Mathematics and Physics (especially math, physics, and engineering majors). While in high school they usually don't study, or are not interested, etc., in ...
2
votes
2answers
75 views

Simple, stable $n$-body orbits in the plane with some fixed bodies allowed

I'm working on a visual simulator for the $n$-body problem in the plane (here). The goal is to show how complex behavior can arise from the simple inverse-square law of gravity. To that end, I want ...
40
votes
7answers
2k views

False beliefs about Lebesgue measure on $\mathbb{R}$

I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is ...
33
votes
7answers
2k views

What are some math concepts which were originally inspired by physics?

There are a number of concepts which were first introduced in the physics literature (usually in an ad-hoc manner) to solve or simplify a particular problem, but later proven rigorously and adopted as ...
14
votes
10answers
7k views

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

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 ...
95
votes
23answers
21k views
0
votes
0answers
40 views

Projects like Stacks?

I am interessted for other projects like Stacks Project which works on algebraic geometry.My questions are : Are there other projects like that? Are there projects like that which are not on research ...
20
votes
8answers
10k views

Best Math books or apps for adults to learn math from the beginning

I lost a possible job because I didn't know how to multiply and subtract negative valued integers. I also don't know how fraction manipulation works. What reference books can I read that can help for ...
351
votes
33answers
21k views

Examples of apparent patterns that eventually fail

Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
64
votes
12answers
8k views

Is there something special about 2015? [closed]

Is there some property which is satisfied only by the number 2015 (among natural numbers, say) or is there a relatively simple question for which the answer is, surprisingly, 2015? This is inspired ...
1
vote
0answers
80 views

Obscure/hard definite integrals evaluating to 2016? [closed]

I wish to send a tricky integral to my math-loving friend which evaluates to 2016 owing to the advent of the upcoming year. What are some interesting integrals which evaluate to 2016? Something ...
6
votes
1answer
348 views

Bounds for $n$-th prime

In this Wikipedia page I have found that the bounds for $n$-th prime is given by, $$n(\ln n+\ln \ln n)>p_n>n(\ln n+\ln \ln n-1)$$ for all $n\ge6$. Are there even stronger bounds for the $n$-th ...
1
vote
1answer
15 views

Index notation of tensors and mnemonics

I've been trying to learn to manipulate tensors but I've got probably too comfortable with all the matrices in my Linear Algebra course, that it gets really difficult beyond rank-3 tensors. So, ...
30
votes
5answers
1k views

Simplicity of $A_n$

I have seen two proofs of the simplicity of $A_n,~ n \geq 5$ (Dummit & Foote, Hungerford). But, neither of them are such that they 'stick' to the head (at least my head). In a sense, I still do ...
76
votes
8answers
16k views

Lesser-known integration tricks

I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now ...
2
votes
4answers
417 views

Abstract Algebra Book Request

I am looking for a good undergraduate level book on Abstract Algebra. By a 'good book' I mean a book which gives equal importance to both, rigor and the historical perspective of the subject. For ...
7
votes
4answers
142 views

A new approach to find value of $x^2+\frac{1}{x^2}$

When I was teaching in college class ,I write this question on board . if we now $x+\frac{1}{x}=4$ show the value of $x^2+\frac{1}{x^2}=14$ Some student ask me for multi idea to show or prove that ...