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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148
votes
80answers
13k 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 ...
5
votes
6answers
129 views

Calculate $\pi$ By Hand?

All over the internet the only hand equation i found was $$\frac\pi4 = 1 - \frac13 + \frac15 - \frac17+\cdots.$$ But this takes something like a thousand iterations to get to four digits, is there a ...
0
votes
3answers
72 views

A big list of non-trivial examples of functions from outside mathematics

I will be teaching my students about functions, and want to stress that functions are not only the usual mathematical ones (linear, logs, exponential, ...), but that function is fundamentally a ...
32
votes
13answers
3k views

Research done by high-school students

I'm giving a talk soon to a group of high-school students about open problems in mathematics that high-school students could understand. To inspire them, I would like to give them examples of ...
148
votes
31answers
8k views

Can't argue with success? Looking for “bad math” that “gets away with it”

I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare"). One example would be "cancelling" the 6s in $$\frac{64}{16}.$$ ...
32
votes
5answers
2k views

Examples of “Non-Logical Theorems” Proven by Logic

I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. Question: What are some examples of mathematical logic solving open problem outside of ...
6
votes
2answers
864 views

Mathematics celebrities that every mathematician should know? [closed]

As a mathematician, sometimes I meet across very embarrassing questions which were posted by who does not learn of mathematics, for example, my wife and so on. She or he always posted such questions: ...
1
vote
0answers
52 views

Mathematicians with hyphenated names [on hold]

I just found out that the Levi-Civita symbol is named after only a single person. Which other mathematical theorems or objects are named after mathematicians with hyphens in their names?
7
votes
0answers
79 views

Known exact values of the $\operatorname{Li}_3$ function

We know some exact values of the trilogarithm $\operatorname{Li}_3$ function. Known real analytic values for $\operatorname{Li}_3$: $\operatorname{Li}_3(-1)=-\frac{3}{4} \zeta(3)$ ...
17
votes
4answers
1k views

An equation that generates a beautiful or unique shape for motivating students in mathematics

Could anyone here provide us an equation that generates a beautiful or unique shape when we plot? For example, this is old but gold, I found this equation on internet: $$ \large\color{blue}{ ...
10
votes
5answers
5k views

Resources/Books for Discrete Mathematics

I am going to a Computer Science Course in University next year. I heard that Discrete Mathematics is whats required for Comp Sci so, I am looking for resources/books that I can read to get started ...
6
votes
3answers
362 views

Undergraduate mathematical magazines to improve mathematical knowledge

I'm sorry my ignorance, I don't know very much about mathematical magazines. I'm finishing my master degree in pure mathematics and I'm looking for mathematical magazines which could improve my ...
16
votes
1answer
468 views

Examples of falsified (or currently open) longstanding conjectures leading to large bodies of incorrect results.

In general, the way that modern mathematical research is conducted isn't the way that many would assume is the ideal method of research. That is, mathematics is not the linear progression of ...
60
votes
19answers
12k views
70
votes
21answers
35k views

Software for drawing geometry diagrams

What software do you use to accurately draw geometry diagrams?
446
votes
138answers
28k 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 ...
4
votes
4answers
281 views

Reference for Quantum groups

I would like to know if there are any general references that you would suggest to learn about quantum groups? I have looked at some of the "standard" books, but I am wondering if someone is ...
46
votes
13answers
3k views

Pseudo Proofs that are intuitively reasonable

What are nice "proofs" of true facts that are not really rigorous but give the right answer and still make sense on some level? Personally, I consider them to be guilty pleasures. Here are examples ...
65
votes
29answers
9k views

What is the single most influential book every mathematician should read?

If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
414
votes
45answers
284k 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 ...
13
votes
1answer
519 views

Big-Daddy-Conjectures and Hierarchy of Mathematical Conjectures

I am interested in the Hierarchy and Connections between various different open problems in Mathematics, and the most general conjectures in various fields of Mathematics. Examples of Hierachy ...
4
votes
7answers
105 views

Finite sequences of prime numbers

There is a lot of prime sequences: prime numbers in a special form. For example Mersenne primes are primes of the the form $2^n-1$, or Pythagorean prime are primes of the form $4n+1$. Even primes are ...
3
votes
0answers
35 views

Books with “project”-like questions

I'm looking for a big list of resources for advanced undergraduate - beginning graduate (and even beyond, really) with a particular feature. Namely, I really like solving "project"-like problems that ...
9
votes
2answers
189 views

Which statements are equivalent to the parallel postulate?

I would like to have a long-ish list of statements that are equivalent to the parallel postulate. If a line segment intersects two straight lines forming two interior angles on the same side that ...
8
votes
3answers
253 views

Resources for Integrals?

I want to learn to solve integrals of some type, probably definite integrals with results involving various constants such as Catalan's, Euler-Mascheroni,Golden-ratio etc. and involving various ...
4
votes
0answers
121 views

What do group automorphisms fix? [closed]

I have often found it useful to sit and contemplate what kinds of elements, subsets, or structures do the automorphisms of an object fix or permute. Sometimes the observations do not have immediate ...
25
votes
11answers
4k views

Good books on Philosophy of Mathematics

Where can I learn more about the implications, meta discussions, history and the foundations of mathematics? Is Russell's Introduction to Mathematical Philosophy a good start?
59
votes
12answers
5k 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 ...
7
votes
1answer
114 views

Handwaving gone wrong

My motivation for this question is twofold: On one hand, I'm studying algebraic topology, where - at least in the book written by Hatcher - there is quite a lot of handwaving (e.g. maps are continous ...
7
votes
2answers
158 views

Theorems with one-line proofs [closed]

Inspired by this very concise answer, which proves that $$\sin^2(\theta)+\cos^2(\theta) \equiv 1 $$ as follows: $f(\theta)=\cos^2\theta+\sin^2\theta \quad;$ then it's simple to see that ...
2
votes
3answers
126 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 ...
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 ...
12
votes
5answers
321 views

I need help finding a rigorous Pre-calculus textbook

I dislike modern textbooks; their cookie-cutter approach and appearance, over reliance on breaking things down into little boxes, the general spoon-feeding they engender and most of all the poor ...
51
votes
18answers
6k 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 ...
11
votes
2answers
1k views

Preparation for Putnam?

If all the training I have right now is Calc 1-3, Linear Algebra and some introduction to set theory/discrete math, what would you recommend focusing on over summer in preparation to Putnam? Real ...
15
votes
16answers
6k views

Applications of the Fibonacci sequence

The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any ...
267
votes
31answers
16k 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 ...
0
votes
0answers
25 views

What are some [mostly trivial] Pell transformations?

Euler looked at some transformations which turned one Pell[-type] equation into another. Example 1: $$u^2-av^2=-1 \quad\iff\quad (2u^2+1)^2-a(2uv)^2=1.$$ Example 2: $$u^2-av^2=-2 \quad\iff\quad ...
17
votes
4answers
714 views

Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”

I couldn't find Charles Radin's Miles of Tiles at the local university library or the public library, and cannot afford its Amazon price right now. Thus, while sorely disappointed for the moment, I ...
0
votes
3answers
133 views

What are other unexpected results of integration?

I have integral of $\dfrac{1}{t^2 + 1}$ and integral of $\dfrac{t}{t^2 + 1}$ whose output is $\arctan(t)$ and $\dfrac12\ln(t^2 + 1)$ respectively. Are there any similar unexpected results when we ...
62
votes
32answers
5k views

Theorems with an extraordinary exception or a small number of sporadic exceptions

The Whitney graph isomorphism theorem gives an example of an extraordinary exception: a very general statement holds except for one very specific case. Another example is the classification theorem ...
26
votes
7answers
2k views

Mathematical literature to lose yourself in

H.M. Edwards in the preface to his book on the Riemann Zeta Function, summarises his philosophy on learning Mathematics: ...I have tried to say to students of mathematics that they should read the ...
0
votes
0answers
34 views

What advanced methods in contour integration are there?

It is well known how to evaluate a definite integral like $$ \int_{0}^\infty dx\, R(x), $$ where $R$ is a rational function, using contour integration around a semicircle or a keyhole. Most complex ...
2
votes
2answers
75 views

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $?

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $ for positive integer $n$ and real $a, b$? You can use any techniques you want. My proof just uses algebra, ...
5
votes
5answers
272 views

Examples of advancement in mathematics due to war

It's not a lie that, in most sciences, some of their advancement comes from war. A couple examples would be the Haber process in chemistry and none other than the Manhattan Project in both physics and ...
3
votes
0answers
54 views

Undergraduate Schools for the Mathematically Inclined

I'm a rising senior and working on generating a list of colleges to apply to, but it seems to me that (with few notable exceptions) my two main criteria are mutually exclusive. Are there any schools ...
3
votes
0answers
65 views

In what order should I study?

I would like to study the basic fundamentals of mathematics from the beginning and move on from there as my understanding of the subject is lacking. In what order should I study? Arithmetic ...
3
votes
2answers
107 views

What are some good questions for this trick, if $\frac{a}{b}=\frac{c}{d}=\frac{e}{f}=\dots=\alpha$ then $\alpha=\frac{a+c+e+…}{b+d+f+…}$?

I need some good algebra questions that are applications of this trick, often in a non obvious and elegant way: $$\text{If } \frac{a}{b}=\frac{c}{d}=\frac{e}{f}=\dots=\alpha \text{ then } ...
1
vote
1answer
69 views

Video/audio lectures on differential topology?

Do there exist decent online video lectures, or even audio lectures, covering differential topology? I'm aware of Milnor's talk, but it is more like exposition and doesn't go very far.
0
votes
0answers
74 views

Has this property for algebraic structures got a name?

Given a binary operation $*:M\times M\rightarrow M$. I define the extension of $*$ to the subset of $M$ in the usual way (with $A,B \subseteq M$) $a*A:=\{a\}*A$and $A*a:=A*\{a\}$, ...