Tagged Questions

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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17
votes
4answers
735 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 ...
1
vote
0answers
25 views

What [precisely] is special, algebraically, about fundamental Pell solutions?

I'm asking for a list of algebraic identities which "uniquely identify" (or nearly so) the fundamental solution (or those immediately around it) of a Pell equation. As a concrete [numerical] example, ...
3
votes
2answers
83 views

Examples of misleading notation that gives correct results

A lot of the time, in maths, especially when I'm trying to remember a formula, I'm taught to remember it in a way that is not notationally correct, but produces the right result. e.g. when learning ...
3
votes
2answers
151 views

What jobs in Mathematics are always in demand, and are deeply Mathematically specialised or greatly general?

I am wondering what jobs in the field of Mathematics are (seemingly) always in demand. I am also wondering what jobs there are that are (once again seemingly) greatly Mathematically demanding in ...
154
votes
62answers
33k views

'Obvious' theorems that are actually false

It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true". Now, I know a fair few examples of things that are obviously true and that can be proved ...
2
votes
1answer
52 views

Good resource for learning braid theory?

I recently heard about braid theory and read the Wikipedia article on it, and it seems really beautiful. What is a good resource for learning more about it? I have a background in mathematics at the ...
9
votes
0answers
199 views

Paradoxical models of $\sf ZF$ without choice [closed]

There are some models of $\sf ZF$ without the Axiom of choice, where some paradoxical statements hold that are not possible in $\sf ZFC$ (we do not require that all those statements necessarily hold ...
16
votes
6answers
2k 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 ...
0
votes
1answer
51 views

Theorems for Pascal's triangle

The mathematician Donald Knuth (b. 1938) once indicated that there are so many relations and patterns in Pascal's triangle that when someone finds a new identity, there aren't many people who get ...
9
votes
2answers
247 views

Examples of theorems that haven't been proven without AC in practice but can be proven without it in principle

It is possible to prove theorems of the form "if $\phi$ is provable in ZFC, then $\phi$ is provable in ZF". For example, let $\phi$ be a statement that is absolute between $V$ and $L$. If $\phi$ were ...
0
votes
1answer
53 views

Exhaustive list of recreational mathematical concepts

There are many simple yet elegant, addictive and entertaining mathematical concepts. For example, drinker paradox, pigeon hole principle, Monty Hall problem, Hilbert's paradox of the Grand Hotel, etc. ...
1
vote
2answers
55 views

When trees are the answer: what is the question?

For which optimization problems are (abstract) trees the best solution? E.g. binary search trees are somehow optimal data structures for quick search. But why for example do botanic trees grow as ...
0
votes
0answers
27 views

A compendium of proof-techniques per objective

Please consider this as an on-going list of techniques preferably per objective or subject. Many mathematical books (at least lately) are focusing on "design patterns" if you like of proof-techniques ...
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}{ ...
0
votes
1answer
17 views

List of Bounds of $n$-th composite

I am looking for a list of all the bounds on $c_n$, the $n$-th composite. There is a trivial bound $2n \geq c_n >n$ $\forall n \geq 5$. But I am looking for bounds stronger than this. I have ...
1
vote
1answer
47 views

Interesting Identities in First-Order Logic

Are there more identities of this sort (http://en.wikipedia.org/wiki/First-order_logic#Provable_identities) that are interesting/non-trivial? It seems that most further work in first-order logic is ...
14
votes
3answers
343 views

What are the uses of “squeezing”?

Off hand, the uses of "squeezing" that I can think of are: showing that $\lim_{x\to0}\dfrac{\sin x}x = 1$, which is then used in finding derivatives (PS: I've just remembered this item showing ...
6
votes
4answers
158 views

The Power of Dimensional Analysis

What are some illustrative examples of dimensional analysis at work? Especially, where can it be used to significantly reduce the difficultly of a computation?
1
vote
0answers
44 views

Are there such prime giving functions?

Here let us define a function $f : \mathbb{N} \rightarrow \mathbb{N}$ , such that for every $n$ , The sequence $\{f(n) ,f(n)+1 ,f(n)+2 , f(n)+3, \dots , f(n)+n\}$ contains atleast $1$ prime . Let us ...
37
votes
18answers
4k views

Nuking the Mosquito — ridiculously complicated ways to achieve very simple results [closed]

Here is a toned down example of what I'm looking for: Integration by solving for the unknown integral of $f(x)=x$: $$\int x \, dx=x^2-\int x \, dx$$ $$2\int x \, dx=x^2$$ $$\int x \, ...
2
votes
1answer
109 views

Reference Request - Early Calculus Papers

Question: I am looking for good references on the early calculus papers. Optimally, I want emphasis on the mathematics itself and I want that mathematics to be translated into modern terminology and ...
48
votes
16answers
6k views

Interesting “real life” applications of serious theorems

As a student 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: "You have a ...
0
votes
0answers
39 views

List of prime graphs

Prime graph under action $\circ$ (where $\circ$ can be Cartesian, strong, direct, lexicographic, etc. product of graphs) is a graph $G$ for which if $G=H\circ J$ then $H=E\vee J=E$, where $E$ is ...
1
vote
0answers
45 views

Repository of functions which do not have elementary integrals [duplicate]

If there is some function and I suspect that the primitive function cannot be expressed using elementary functions, I would like to have some argument that there indeed is no such expression. One ...
2
votes
2answers
69 views

Introductions to posets on algerbaic structures (Everything I need to know about them)

I need a good and complete introduction to Tree-like orders and partial orders on algebraic structures with one operations. I accept basic texts too. I'm looking for free online texts mostly because ...
3
votes
2answers
106 views

Applications of powerful theorems in Bruns -Herzog's book “Cohen-Macaulay Rings”

It seems that theorem 1.4.13 and it's corollary of Bruns and Herzog's book Cohen-Macaulay Rings, are powerful tools but I don't see any example that shows the power of it. My original question was an ...
0
votes
3answers
294 views

Integers with interesting properties. [closed]

A few weeks ago I found the book "Lure of the Integers" by Joe Roberts, in my schools library, and promptly ordered it from Amazon. It is a wonderful book for those of us who are interested in number ...
5
votes
1answer
124 views

Examples of interesting/unconventional solutions to rather “standard” problems?

For example, I recently came across the following way to evaluate the integral of $\cos^2 x - \sin^2 x$ without using double angle formulas: $$\int dx (\cos^2 x - \sin^2 x) = \int dx(\cos x + \sin ...
1
vote
1answer
56 views

Some examples of local and nonlocal properties

Today I learned that continuity at a point is a local property. Concretely, if $f: \mathbb R \to \mathbb R$ is continuous on $[-K,K]$ for all $K \in \mathbb R$ then $f$ is continuous on $ \mathbb ...
4
votes
3answers
65 views

Examples of “eventually reaches y under iteration” other than the Collatz problem

The Collatz conjecture states that iteratively applying the map $$f(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ 3n+1 & \text{if } n\equiv 1 \pmod{2} .\end{cases}$$ to any ...
0
votes
0answers
27 views

Trigonometric identities for angles that sum up to $2\pi$

Given $A + B + C + D = 2\pi$, are there special trigonometric identities that concern these four angles? If possible, I would like to know whether the above relation has any implications on any of the ...
0
votes
0answers
14 views

Discrete and Continuous analogues

What are some famous theorems who have both discrete and continuous versions but only one of them is well-known?
63
votes
23answers
10k views

Mathematicians ahead of their time?

In every field there's always that person who's just years ahead of their time. For instance, Paul Morphy (born 1837) is said to have retired from chess because he found no one to match his technique ...
1
vote
6answers
2k views

Real life applications for logarithms [duplicate]

Can someone please tell me what purposes logarithms have in the everyday world? What non-theoretical applications are they in and when would one use them?
7
votes
7answers
222 views

Examples of properties not preserved under homomorphism

An isomorphism indicates that two structures are the same, using different names for the elements. Therefore it's obvious that every (algebraic) property of the first structure must be present in the ...
9
votes
7answers
1k views

Beautiful Theorems and what constitutes as beautiful [closed]

I often hear the phrase "mathematical beauty". That a proof or formula or theorem is beautiful. and I do agree I was awestruck when I first saw Euler's formula, connecting 3 seemingly unrelated ...
2
votes
0answers
46 views

Good topologies on $\mathcal{P}(X)$

Let $X$ be a topological space, and let $\mathcal{P}(X)$ (resp. $\mathcal{P}_0(X)$) be the set of all subsets of $X$ (resp. the set of all non empty subsets of $X$). Finally, let ...
3
votes
1answer
99 views

What are the differences in mathematical notation around the world? [duplicate]

I just learned that $\text{sen}\,x$ is the Portuguese notation for $\sin x$, and I was inspired to ask: What differences are there in how mathematics is written around the world? Note 1: I am likely ...
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 ...
4
votes
3answers
255 views

Mental Math Techniques [closed]

What are some interesting mental math techniques that you know? Here's one that I got from my Grandmother who got it from a book: To square a two-digit number (from $26$ to $49$), take the number ...
0
votes
1answer
74 views

What are some good introductory textbooks on Sieve Theory?

I fail to find a duplicate. If it exists, please give me the link and close the question accordingly. As the title suggests, I am looking for recommendations on introductory books to Sieve Theory. ...
2
votes
0answers
40 views

Revise high school material

Can you suggest me a comprehensive book to revise high school mathematics (up to besic calculus)? It should be extremely clear and complete and "scientific" (not like most high school books). Thank ...
28
votes
3answers
779 views

Create a Huge Problem

I am wondering if any problems have been designed that test a wide range of mathematical skills. For example, I remember doing the integral $$\int \sqrt{\tan x}\;\mathrm{d}x$$ and being impressed at ...
1
vote
0answers
35 views

What are the branches of mathematics? [duplicate]

What are all the branches of mathematics? I want to know this so that I can compare between them, so if you can give me with them a brief look of what they are about that would be awesome.
0
votes
0answers
37 views

The term $rank$ in methematics

Reading wikipedia's disambiguation page about the "rank" word I see many concept of rank of many different matematical object. I only know about the rank of a graded poset and the rank of a set that ...
5
votes
1answer
35 views

Surprising constructions in algebraic topology that facilitate one's understanding of underlying theory

I am recently come into the world of algebraic topology and find it a fascinating place with lots of beautiful constructions that challenge one's intuition. Also, understanding these constructions are ...
7
votes
3answers
265 views

Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?

Question: Which inequalities similar to the famous isoperimetric inequality is known? conjectured? I recently learned about some inequalities which are all similar to the famous isoperimetric ...
2
votes
1answer
122 views

What are interesting examples of existential proofs based on cardinality arguments?

Probably the most famous example of a proof, where consideration of cardinalities is used to show existence of some object, it the Cantor's proof that there exist transcendental numbers. What are ...
1
vote
1answer
45 views

Matrices of the form $A^p=(a_{ij}^p)$

I am wondering if there is a name for these kind of matrices and if they are interesting or not? Do they even exist? Let $A$ be a $n\times n$ matrix with elements $a_{ij}$. $A= (a_{ij})_{i,j\in\{1, ...
2
votes
5answers
353 views

What area of Abstract Algebra do you find most interesting? [closed]

For my Abstract Algebra class, we will be doing small presentations (2 class periods) covering some topic in Abstract Algebra. Thus far, I have studied groups, rings, fields, modules, tensor ...