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.

learn more… | top users | synonyms

2
votes
1answer
30 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
646 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?)
0
votes
0answers
39 views

Nice statements about the “opposite properties” of $0$ and $1$.

Some of $0$ and $1$'s properties as natural numbers are pleasingly opposite of one another, such as: 1a.) $0$ has infinitely many divisors, but no (non-zero) multiples. 1b.) $1$ has infinitely many ...
6
votes
5answers
106 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
40 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 ...
59
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
89 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
95 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
74 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
49 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
36 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
58 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
322 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 ...
45
votes
13answers
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
114 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
114 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
193 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
120 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
85 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
52 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
68 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
367 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
99 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
248 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
108 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
63 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
26 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
47 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
111 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
46 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
92 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 $A^\perp$ ...
18
votes
6answers
583 views

Honest application of category theory

I believe that category theory is one of the most fundamental theories of mathematics, and is becoming a fundamental theory for other sciences as well. It allows us to understand many concepts on a ...
9
votes
4answers
454 views

Big List of examples of recreational finite unbounded games

What are some examples of mathematical games that can take an unbounded amount of time (a.k.a. there are starting positions such that for any number $n$, there is a line of play taking $>n$ times) ...
3
votes
1answer
94 views

What is the prerequisite knowledge for Navier–Stokes Existence and Smoothness problem?

I am highly interested in the Millennium Problem of Navier–Stokes Existence and Smoothness (also here) and my aim is to reach some level of knowledge to do research on it. The problem seems simple to ...
29
votes
4answers
2k views

Avoiding proof by induction

Proofs that proceed by induction are almost always unsatisfying to me. They do not seem to deepen understanding, I would describe something that is true by induction as being "true by a technicality". ...
44
votes
9answers
1k views

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do ...
0
votes
1answer
40 views

Uniform convergence in series definitions of functions

Are there examples of well-known functions which are defined as the limit of a sequence of functions (for example, power series definitions) and are not uniformly convergent? Thanks!
3
votes
1answer
93 views

Suplement books for calculus course?

I'm looking for books that could enhance the learning of calculus. At the moment, I have the following titles: Counterexamples in Calculus; Irresistible integrals; Inside calculus; The Calculus ...
6
votes
1answer
94 views

Surveys: problems, conjectures, and questions in some areas of nonlinear analysis

I would like to create a "big-list" of resources (e.g., survey papers, webpages, conference proceedings, monographs, etc.) that collect and offer some context and ...
12
votes
6answers
759 views

Induction - Examples where the induction step is correct but the base case is always wrong [duplicate]

I'd like to present to my students some induction examples that always satisfy the inductive step but never the base case. It could be for natural numbers, graphs or anything else.
11
votes
2answers
316 views

Open/publicly available textbooks worth their salt

I've been reading a bit about "open textbooks", i.e. textbooks made available for easy, online access. These can be nice for those without access to a great library, or who might not be willing to ...
5
votes
5answers
592 views

Counterexamples for “every linear map on an infinite dimensional complex vector space has an eigenvalue”

Every linear map on a finite dimensional complex vector space has an eigenvalue. Not so in the infinite case. I'm interested in nice counterexamples anyone might have. Here's one: Consider the ...
-1
votes
1answer
70 views

Does Analysis Appear on Putnam?

I was recently searching past Putnam problems, I believe I saw some sort of Real Analysis, but I am not sure. Is Real Analysis (Mathematical Analysis) a topic, which appears on the Putnam Exam? Which ...
2
votes
0answers
66 views

GRE Mathematics Practice Exams

I will be taking the subject test in the near future. Can you recommend me some sources (online or print) from which I can find realistic practice exams? I would like to get my hands on as many ...
1
vote
2answers
169 views

Doing Michael Spivak's Exercises

I am doing Spivak's Calculus, and I find it EXTREMELY difficult. I usually ask questions here because I cannot do the problems on my own. How long should it take to do a Spivak problem? Is it ...