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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6
votes
1answer
199 views

What does Hartshorne do wrong?

I'm currently trying to learn algebraic geometry from Hartshorne's Algebraic Geometry. I've often heard it said, both on MathOverflow and in my department, that Hartshorne's treatment of certain ...
1
vote
0answers
15 views

Examples of generalized geometric series.

I'm preparing a short presentation on the convergence of the geometric series of matrices, and I'd love some examples of their uses. I've encountered them when approximating inverses of matrices ...
1
vote
2answers
70 views

Theorems relating to the limitation of mathematics

At one point, mathematicians believed that they may be capable of expressing all of mathematics in one system of ideas, and that their abilities were unlimited. Unfortunately, things like the Godel's ...
6
votes
1answer
149 views

Examples of bi-implications ($\Leftrightarrow$) where the $\Rightarrow$ direction is used in the proof of the $\Leftarrow$ direction.

[I'm asking for examples of proofs with a certain structure. There is quite a lot of text before arriving at the questions. This is because asking for examples of a phenomenon is best carried out by ...
15
votes
6answers
248 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 ...
64
votes
16answers
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 ...
3
votes
0answers
92 views

Exercises on group theory [closed]

What are some difficult, challenging and fair exercises in group theory? I know it is quite general, in particular I am referring to these areas of group theory: theory of automorphism group ...
10
votes
15answers
608 views

$2=1$ Paradoxes repository

I really like to use paradoxes in my math classes, in order to awaken the interest of my students. Concretely, these last weeks I am proposing paradoxes that achieve the conclusion that 2=1. After one ...
2
votes
1answer
215 views

How many six digit numbers start with the same two digits and end with the same three digits?

Say that there is a 6 digit number the first digit is not allowed to be 0 or 1 so How many number combinations start with the same two digits and end with the same three digits ie.119333, 448222, ...
4
votes
6answers
220 views

What are some false proofs for true or false statements where the error in the proof is not obvious? [closed]

I'm looking for examples of subtle errors in reasoning in a mathematical proof. An example of a 'false' proof would be Let $a=b>0$. Then $a^2 - b^2 = ab - b^2$. Factoring, we have $(a-b)(a+b) = ...
46
votes
18answers
6k views

What are some things we can prove they must exist, but have no idea what they are?

What are some things we can prove they must exist, but have no idea what they are? Examples I can think of: Values of the Busy beaver function: It is a well-defined function, but not computable. It ...
15
votes
11answers
519 views

Different ways finding the derivative of $\sin$ and $\cos$.

I am looking for different ways of differentiating $\sin$ and $\cos$, especially when using the geometric definition, but ways that use other defintions are also welcome. Please include the ...
0
votes
1answer
57 views

Books that emphasize physical applications

I made a book recommendation thread recently which got deleted, so I'm hoping this one doesn't have the same fate as the question seems much more defined. I was reading an interesting article by VI ...
0
votes
0answers
30 views

Composition of non-differentiable functions to produce a differentiable one

What are examples of non-differentiable functions which when composed form a differentiable function? It would even better if they are $C^\infty$ I am hoping this will turn into a big list, if it ...
2
votes
1answer
48 views

Useful matrix inner products

What are some interesting/useful examples of matrix inner products on square complex matrices, other than $\langle A, B \rangle = Tr(A^{\dagger}B)$?
5
votes
4answers
534 views

Graph Theory for Dummies Book [duplicate]

Does anyone have a good book on Graph Theory that will introduce me to some of the basic concepts without being so filled with terminology that it's hard to read? I have taken an introductory course (...
3
votes
1answer
67 views

List of textbooks that take a historical approach

As the title suggests my aim in this topic is to make a big list of textbooks on any mathematical topic that take a historical approach. I will start with the ones I know: Thomas Muir - The theory of ...
10
votes
1answer
163 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
4
votes
2answers
234 views

Applications of algebraic graph theory

What are some subtle, or non-obvious applications of algebraic graph theory? Obviously it can be used to study anything directly involving graphs (for instance, the wikipedia page mentions ...
1
vote
0answers
66 views

What families of transcendental equations do we have solved?

I'm particularly interested in transcendental equations but searching in internet gives me only results about the classical linear-exponential equation (which is solved with Lambert's W) and its ...
9
votes
5answers
145 views

$32$ Goldbach Variations - Papers presenting a single gem in number theory or combinatorics from different point of view

A short time ago I found the nice paper Thirty-two Goldbach Variations written by J.M. Borwein and D.M. Bradley. It presents $32$ different proofs of the Euler sum identity \begin{align*} \zeta(2,1)=\...
8
votes
3answers
365 views

Elementary topology examples

I'm preparing (to teach) my first class of undergraduate topology and I'm looking for some elementary, motivating applications of topology for the first day. We'll be following Munkres, starting with ...
5
votes
1answer
160 views

A beautiful book on arithmetic doesn't treat you like a little baby

The state of arithmetic today is disgusting. The textbooks on it are absolutely repelling, the authors treat it like a subject that will be of concern to only babies. They don't show any love, they ...
1
vote
1answer
175 views

Intuitive functional analysis book

I would like a functional analysis book like Terence Tao's Real Analysis and Measure Theory book, full of intuition. I am ...
3
votes
7answers
308 views

Alternate ways to prove that $4$ divides $5^n-1$

I was working for various method to solve this: For all $n\in \mathbb N$: $4\;\mid\;(5^{n}-1)$. My try was: 1st: $$n=1 \to 4|5^1-1\\n \geq 2 \to 5^n=25,125,625,3125,...\\ n\geq 2 \to 5^n=\...
1
vote
2answers
108 views

What are some unfamiliar and/or special tricks used to evaluate limits?

What are some neat tricks used to evaluate limits that might be otherwise a problem to deal with? I'm not asking for methods akin to L'Hopital's rule.. which is often used. My question is geared ...
0
votes
4answers
192 views

Proof of Pythagorean theorem without using geometry for a high school student?

There are some proofs of Pythagoras theorem which don't even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs of ...
2
votes
1answer
178 views

(Theoretical) Complex Analysis Textbooks [closed]

Most books I've seen on complex analysis do not develop it theoretically, which can be somewhat infuriating for the budding pure mathematician. What I am looking for are some comprehensive, rigorous ...
23
votes
15answers
1k views

New Idea to prove $1+2x+3x^2+\cdots=(1-x)^{-2}$

Given $|x|<1 $ prove that $\\1+2x+3x^2+4x^3+5x^4+...=\frac{1}{(1-x)^2}$. 1st Proof: Let $s$ be defined as $$ s=1+2x+3x^2+4x^3+5x^4+\cdots $$ Then we have $$ \begin{align} xs&=x+2x^2+3x^3+4x^...
0
votes
0answers
246 views

Final year dissertation/project ideas for numerical methods

In my final year, I have to submit a project/dissertation on Numerical Methods. I have done a course on it, which included some proofs and programming. Just eager to get ideas that I can look at. PS :...
4
votes
2answers
740 views

Interesting facts and problems to motivate high school combinatorics students

I will give some classes in combinatorics to high school students and I would like to know some facts (and proof) I can show to my students to motivate them to study this beautiful subject. I'm ...
3
votes
4answers
139 views

Are there any books with lots of questions of “Fill in The holes” type

Does anyone knows books which have lots of questions ,whose format are like fill in the holes type . . Same goes for theorems and exercises . I am looking on pure math especially Real analysis ,...
4
votes
2answers
171 views

What is the most general notion of “Fourier transform?”

I know the definition of a classical Fourier transform that maps a function f(x) on the real line X to a function F(p) on a dual space (here another real line and borrowing some physics notation) P. ...
7
votes
1answer
223 views

Elegant applications of advanced techniques to “olympiad” problems

I am interested in applications of somewhat "advanced machinery" (with respect to the usual machinery involved in these cases, which is usually elementary) to olympiad or (high school-level) contest ...
2
votes
3answers
106 views

Application of Euler's theorem apart from finding last digits of huge numbers

I am looking for clever applications of Euler's Theorem. On browsing the internet, I see that nearly all the applications of the theorem asks for finding last few digits of a huge number. The only ...
0
votes
3answers
95 views

Alternative infinite summations that equal $e$

Everyone (and I mean everyone) knows this sum: $$\sum_{n=0}^\infty \frac{1}{k!} =e$$ Are there any lesser known infinite sums that go to e?
0
votes
0answers
48 views

List of divergent series and their summations

On the web one can manage to find a lot of lists of convergent series and their summation btw I didn't find (at least on a quick search) a corrispective list of divergent series, does anyone know one ?...
0
votes
0answers
35 views

Looking for problems which can be solved by the similar technique

While browsing on internet for different proofs of Fermat's theorem on sums of two squares, I came across Zagier's "one-sentence proof" which seems to be the most elegant and short proof. It invokes a ...
5
votes
2answers
163 views

Characterizations of the cross-ratio

$$ (z_1,z_2;z_3,z_4) = \frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)} $$ What are the most prominent or most interesting theorems of the following form? Theorem: The cross-ratio is the only function ...
5
votes
0answers
89 views

“Toys” spaces in algebraic topology

I did follow a course of algebraic topology last semester and I still want to continue to do some computations. But in many books it's all the time the same examples which comes back for computing ...
2
votes
1answer
57 views

open problems regarding functions

I am looking for some open problems regarding functions. Problems like, Whether a function satisfying some properties say, X,Y,Z, exists or not, is unknown. Like there is no function $f(x)$ such ...
13
votes
10answers
949 views

What are the most prominent uses of transfinite induction outside of set theory?

What are the most prominent uses of transfinite induction in fields of mathematics other than set theory? (Was it used in Cantor's investigations of trigonometric series?)
7
votes
5answers
126 views

$e^{i\theta}$ versus $\cos\theta+i\sin\theta$

I am teaching an basic university maths course, and have been thinking about the complex numbers part. Specifically, I was wondering why I should include Euler's formula in my course. This led me to ...
2
votes
1answer
157 views

Real analysis : Preliminary topics for - Measure Theory, Integration Theory, Differentiation and Integration [closed]

I have following syllabus to study in Real Analysis Subject. I want to know, What are necessary topics that I have to cover as a prerequisite for below syllabus. Actually I am unable to get direction ...
70
votes
34answers
8k views

Easy math proofs or visual examples to make high school students enthusiastic about math [closed]

I'm a teacher in mathematics at a high school. Math has fascinated me for almost my entire life, so I would like to bring that enthusiasm to my students with beautiful yet easy to understand proofs or ...
2
votes
1answer
275 views

Coordinate Geometry and Trigonometry book recommendation for GRE Math Subject Test

I am currently a math major at university and I plan to take GRE Math Subject Test in future (most probably next year). Can you please suggest any good book for revising and brushing up Coordinate ...
1
vote
2answers
106 views

Counting the number of solutions of equation $x^2 + y^2 = 1$ over $\Bbb Z/p$

List proofs of the fact that the number of solutions to $x^2 + y^2 = 1$ over $\Bbb Z/p$, where $p$ is a prime $\neq 2$, is $p-(-1)^{\frac{p-1}2}$. I thought of two. I write one below.
0
votes
0answers
113 views

Abstract Algebra book

Has anyone read the book "Abstract Algebra: An Introduction" by Thomas Hungerford? I'm in the class Intro to Abstract Algebra, and my professor is, to say the least, not very great at teaching. Great ...
1
vote
2answers
113 views

Lists of the first fundamental group of spaces. [closed]

Here are some list to start with $$\begin{array}{|c|c|c|} \hline \mbox{Space}(S)& \pi_1(S) \\ \hline \mathbb{R}^2&0 \\ \hline \mathbb{S}^1& \mathbb{Z} \\ \hline 1-Torus& \mathbb{...
2
votes
1answer
68 views

Are there unsolved problems known to be not independent of the axiomatic system it is proposed in?

Are there unsolved problems known to be not independent of the axiomatic system it is proposed in? For example, is Goldbach's conjecture known to be provable using the axioms of PA? I believe I ...