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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0
votes
4answers
80 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 ...
1
vote
1answer
34 views

Applications of statistics to pure mathematics [on hold]

Are there any "applications" of statistical methods to pure mathematics?
2
votes
0answers
48 views

(Theoretical) Complex Analysis Textbooks

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 ...
15
votes
14answers
600 views
+100

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

If $|x|<1 $ prove that $\\1+2x+3x^2+4x^3+5x^4+...=\frac{1}{(1-x)^2}$ 1st proof:suppose ...
0
votes
0answers
29 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 ...
3
votes
2answers
91 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
68 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
105 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. ...
5
votes
1answer
65 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 ...
1
vote
2answers
45 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
79 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
23 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
29 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
146 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
62 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
35 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 ...
11
votes
10answers
684 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?)
6
votes
5answers
109 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
49 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 ...
64
votes
34answers
6k 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
107 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
96 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
2answers
77 views

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

Here are some list to start with $$\begin{array}{c|c|c|} \hline Space(S)& \pi_1(S) \\ \hline \mathbb{R}^2&0 \\ \hline \mathbb{S}^1& \mathbb{Z} \\ \hline 1-Torus& ...
3
votes
1answer
55 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 ...
2
votes
1answer
38 views

Benefit from measure theory

With your help I want to list the benefits from measure theory and the lebesgue integral. (Advantages to the Riemann integral) What I know: With the Lebesgue integral we need less requirements to ...
19
votes
11answers
1k views

What are some results that shook the foundations of one or more fields of mathematics? [closed]

An example would be the proof that $\sqrt{2}$ is not rational, which was a violation of some fundamental assumptions that mathematicians at the time made about numbers. Another would be Russell's ...
1
vote
1answer
59 views

Website for Mathematics enigmas?

I seek some website just for the pleasure of solving mathematics enigmas. I know this website : Brilliant.org I just want to know if you know some others good sites ! Thank you
7
votes
3answers
335 views

Space on which all real-valued continuous functions achieve maximum but not compact?

A friend is writing a book for non-mathematicians; he has asked me some questions... One possible direction I suggested was whether a topological space (metric space can probably be assumed given what ...
51
votes
15answers
1k views

What is the most unusual proof you know that $\sqrt{2}$ is irrational?

What is the most unusual proof you know that $\sqrt{2}$ is irrational? Here is my favorite: Theorem: $\sqrt{2}$ is irrational. Proof: $3^2-2\cdot 2^2 = 1$. (That's it) That is a ...
8
votes
1answer
128 views

What are some interesting blogs about general topology?

We have several question asking about book recommendations for general topology - for example the posts linked to Best book for topology? or the posts mentioned in the relevant section of List of ...
2
votes
0answers
117 views

Comprehensive Linear Algebra Text

Occasionally I come across a fact from linear algebra that I have not seen before. These facts are often obscured in search engines by either introductory texts or unrelated papers, and it is ...
13
votes
7answers
199 views

What are statements about the natural numbers where induction is impossible or unnecessary to prove?

I'm looking for statements like "for all natural numbers, ____" where induction would be impossible or unnecessarily complicated. This is for pedagogical reasons. When students first learn induction, ...
3
votes
1answer
135 views

Challenging problems in algebra (book recommendation) [closed]

Could you suggest me a book/web page where I can find challenging/hard problems in algebra (possibly with solutions) for an undergraduate student (groups, rings, fields, Galois theory)? Thanks in ...
4
votes
0answers
68 views

All possible total orderings of a finite set are isomorphic. What are some other examples of this phenomenon?

All possible total orderings of a finite set are isomorphic. I find these kinds of results remarkable. Here's a few more. Assume that $S$ is a finite set. Then: All possible field structures on $S$ ...
27
votes
5answers
1k views

Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?

I understand that naive set theory, whose axioms are extensionality and unrestricted comprehension, is inconsistent, due to paradoxes like Russell, Curry, Cantor, and Burali-Forti. But these all ...
0
votes
4answers
86 views

Easy proof of $\mathcal{P}(\mathbb{Q})$ is uncountable [Big list]

I'm looking for a easy proof of uncountability of $\mathcal P(\mathbb Q)$. I'll contribute with this: Let $\mathcal{P}(A)$ denote the power set of $A$, since ...
14
votes
10answers
2k views

Math problems that are impossible to solve [closed]

I recently read about the impossibility of trisecting an angle using compass and straight edge and its fascinating to see such a deceptively easy problem that is impossible to solve. I was wondering ...
1
vote
0answers
54 views

Mathematical structures with name reffering to a country

I am looking for a list of mathematical structures (not theorems) that refer to a country or nationality. I only know of Polish spaces and Polish groups. Does anyone have other examples? Note: many ...
3
votes
1answer
78 views

Bounds for $n$-th prime

In this 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 prime? If ...
11
votes
4answers
381 views

Elementary problems that would've been hard for past mathematicians, but are easy to solve today? [closed]

I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous ...
5
votes
0answers
102 views

Request for counter examples in group theory

I am looking for books, papers, or even webpages, that have collected many counter examples in group theory (which, I guess, are just examples in group theory). I am particularly interested in ...
6
votes
5answers
273 views

Proof writing: how to write a clear induction proof?

What is an effective way to write induction proofs? Essentially, are there any good examples or templates of induction proofs that may be helpful (for beginners, non-English-native students, etc.)? ...
2
votes
1answer
114 views

Amazing integrals and how is solved it [closed]

There a lot of integrals, however many people solved it in different ways, we can find interesting integrals in Table of Integrals, Series, and Products. I wonder What is the most exciting integral ...
0
votes
1answer
65 views

What are some examples of principal, proper ideals that have height at least $2$?

Krull's principal ideal theorem states that in a Noetherian ring $R$, any principal proper ideal $I$ has height at most $1$. Presumably the Noetherian hypothesis is required, so what are some ...
2
votes
0answers
27 views

How much regularity is needed, anyway?

When doing real analysis, the difference between functions which are continuous and functions which are not is intuitive. The graph of the later may exhibit shearing, or extreme distortion (in higher ...
1
vote
2answers
50 views

Representative Pedagogical Examples of Groups, Real Functions, Modules, etc.

In the preface of Munkres's Topology, he writes, Fortunately, one does not need too many counterexamples for a first course; there is a fairly short list that will suffice for most purposes. Let ...
3
votes
0answers
114 views

Examples of categories which appear naturally without objects

Regarding the morphisms-only-definition of a category (which is equivalent to the usual one dealing with objects and morphisms), I would like to ask: Which examples of categories in practice appear ...
0
votes
0answers
48 views

Looking for examples on Jordan Form

I am not asking a very specific question, but rather I am looking for any good examples that illustrate the following; $\mathbf{Theorem}: $ Let $T: V \to V$ be a linear operator with characteristic ...
3
votes
0answers
94 views

Funny translations of mathematical words [closed]

As already noticed in this question there are some mathematical words that literally translated from a language to english (or from english to this language) means something totally different. A few ...
3
votes
2answers
53 views

What other classes of commutative rings can be defined by requiring that $\{0\}$ is the only proper ideal satisfying some condition?

A field is just a commutative ring $R$ such that $\{0_R\}$ is the only proper ideal. Interestingly, there's a similar characterization of integral domains. Given a subset $A$ of $R$, let $A^\perp$ ...