3
votes
1answer
72 views

What are the differences in mathematical notation around the world?

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 ...
21
votes
14answers
2k 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 ...
3
votes
3answers
150 views

Mental Math Techniques [on hold]

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 ...
2
votes
0answers
25 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 ...
8
votes
1answer
125 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 ...
0
votes
0answers
31 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
29 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 ...
2
votes
5answers
300 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 ...
5
votes
0answers
98 views

What is the best Mathematical Insight you have had? - PLEASE MOVE TO META [closed]

I've used this site a lot but am new to the actual forum. Basically, I am wondering if we could collect a list of mathematical insights / explanations / neat proofs etc. that people on this forum have ...
24
votes
5answers
2k views

Tell me problems that can trick you

I am looking for problems that can easily lead the solver down the wrong path. For example take a circle and pick $N$, where $N>1$, points along its circumference and draw all the straight lines ...
74
votes
19answers
12k views

Visually deceptive “proofs” which are mathematically wrong

Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually ...
1
vote
2answers
108 views

Recommendation for Number Theory Textbook

. Greetings, every mathematicians! I'm a foreigner (meaning English is not my first language) and an undergraduate student. I'm currently studying linear algebra, set theory and have already studied ...
320
votes
41answers
139k 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 ...
2
votes
2answers
91 views

Simple & Intuitive Statements that are Difficult to Prove

Looking through the webcomic, I came across one of my favorite comics: (from Saturday Morning Breakfast Cereal) It seems that people have an ongoing interest in results in mathematics that are ...
3
votes
3answers
109 views

Most inspirational mathematical books [closed]

I would like to know which books on mathematics (from university texts to divulgative pop-math books) inspired you the most. My choice is Spivak's Calculus, which is, IMHO one of the most ...
8
votes
4answers
766 views

Interesting mathematical problems for 1st year university students

Can you explain some mathematical problems that you find the most interesting (NB: the problem must be accessible to a 1st year university student: that is, a problem for which there is an elegant ...
21
votes
7answers
465 views

“Here's a cool problem”: a collection of short questions with clever solutions

This game will be familiar to many mathematicians, and it is always good fun to play. I am looking to find a list of good questions with short, when-you-see-it solutions. The kind of question one ...
8
votes
2answers
420 views

Surprising applications of cohomology

The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the ...
1
vote
3answers
94 views

University-level books focusing on intuition?

I help some students with difficulties in Mathematics and Physics (especially math, physics, and engineering majors). While in high school they usually don't study, or are not interested, etc., in ...
0
votes
0answers
24 views

List of crucial results deserving more attention for first course in Real Analysis

Can it help to form a list of crucial results for basic courses that are concealed as exercises or neglected? I don't know of other resources for this, as I wrote here. I am happy for this to be ...
5
votes
1answer
82 views

Why Did You Specialize in X?

For those of you who are researchers or graduate students, why did you choose to specialize in the field of mathematics X (as opposed to some other field Y)? Is it because you think X is important, ...
6
votes
3answers
169 views

Definitions which should be propositions/theorems

I am asking for a list of concepts which some sources present as definitions whereas other sources pose them as propositions/theorems. For example, most abstract algebra books will define a group ...
74
votes
29answers
15k views

Examples of mathematical results discovered “late”

What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple ...
1
vote
1answer
116 views

Have any definitions in mathematics been redefined

Based on certain intuitions and motivations we make certain definitions and then proceed to use these concepts in further developing our intuition. For example, we have an intuition that a line has ...
73
votes
18answers
7k views

How do you describe your mathematical research in layman's terms?

"You do research in mathematics! Can you explain your research to me?" If you're a research mathematician, and you have any contact with people outside of the mathematics community, I'm sure ...
6
votes
3answers
205 views

What are some properties that imply that a group must be the trivial group?

In the problem posed in this question of mine we want to show that a particular group is both perfect and solvable, and therefore trivial, and this turns out to be useful in proving the result. What ...
14
votes
3answers
255 views

which exact integration techniques belong in a first year calculus/analysis course?

At our university we are now discussing changes to the course contents and there is some heated discussion regarding integration in the first year calculus courses. Currently, the techniques of exact ...
2
votes
2answers
95 views

When are analytical solutions preferred over numerical solutions in practical problems?

In most engineering or applied math papers that I read, the authors seem to obtain solutions to say, a system of differential equations, using numerical methods, rather than analytical techniques. ...
0
votes
1answer
107 views

Most “beautiful” presentations of the basic proofs for vector spaces?

I am familiar with the standard proofs presented in textbooks for stuff like linear independence/dependence, the dimensions of common vector spaces, any basis for a vector space V must be linearly ...
4
votes
2answers
98 views

Theorems that have proofs from the outside of the original field of math

I would like to know more examples of theorems, which "belong to one field of math", but their proofs are from the "outside of the field". I am mostly interested in proofs that are not too long ...
6
votes
2answers
110 views

Sources of Elementary Number Theory Problems

I am looking for sources of interesting and challenging problems that would suitably accompany an honors level introductory number theory course. What are some good sources for interesting elementary ...
16
votes
2answers
261 views

Results in graph theory proved using other areas of math, and vice versa

I'm curious about learning graph theory, as it seems to pop up in some unexpected places. In order to get a partial feel for the subject, I was wondering if anyone could point me to some survey ...
7
votes
1answer
188 views

“All math is useful eventually”

We have all heard the argument : a lot of mathematics that was thought to be useless, abstract constructions with no links to the real world ended up being of use, like some arithmetic is useful in ...
3
votes
1answer
119 views

Statements equivalent to the Axiom of Choice

The Axiom of Choice reads: The product of a collection of non-empty sets is non-empty. As you know well, this axiom is equivalent to many other statements. A few examples (probably the most ...
10
votes
1answer
240 views

What are some examples of hard theorems in category theory?

I'm currently learning some category theory, but so far I've used it only as a handy way to talk about various related concepts in algebra and topology with some nice, easy-to-prove lemmas like "left ...
16
votes
1answer
462 views

“Bad” Mathematics in Movies

There's a website and a companion book to it about bad physics in movies, called "Insultingly Stupid Movie Physics". Similar issues may exist about mathematics: What are the differences between ...
4
votes
3answers
55 views

“Asymmetric” results in maths analogous to “Parity violation” of the weak force?

Disclaimer: I'm not a physicist and I don't claim to be one so if I have any mistakes I’ll be glad to be corrected. One feature of the standard model of particle physics is that the weak force is not ...
8
votes
0answers
169 views

Interesting but short math papers?

Is it ok to start a list of interesting, but short mathematical papers, e.g. papers that are in the neighborhood of 1-3 pages? I like to read them here and there throughout the day to learn a new ...
15
votes
1answer
418 views

The most active fields of mathematics?

Which fields of mathematics are the most active at this time -- by number of papers published, grant money, people working in them or by any other measure? Any trends in this regard?
3
votes
3answers
179 views

Examples where derivatives are used (outside of math classes)

I want to know what is the use of derivatives in our daily life. I have searched it on google but i haven't find any accurate answer. I think it is mostly used in Maths but I want to know its use in ...
3
votes
1answer
175 views

Best book ever on Galois theory (and differential galois theory) [closed]

Which is the single best book for Galois theory (that includes differential Galois theory) that everyone who loves pure Mathematics should read?
9
votes
4answers
207 views

Math games for car journeys

On long car journeys with kids we are all familiar with "I spy" or "Twenty questions". What math related games can one play on a car journey instead that are fun and offer similar variety?
2
votes
1answer
192 views

Good book for logic self-study

I know a similar question has already been asked, but can anyone suggest a good book on mathematical logic that includes answers to exercises? I am looking for something that is conducive to ...
8
votes
3answers
210 views

Bedtime maths books?

Most of math books require you to copy proofs and do excersices to extract the content from them. Are there any good serious math books which require only reading and no writing? ADDED: One ...
3
votes
2answers
86 views

Numerically Misleading Results

Are there any calculations or results that have similar answers and when compared numerically look the same, but in actual fact after so much precision, the answers diverge from each other? An ...
25
votes
3answers
717 views

Common Math Mistakes Made by Scientists

I am a mathematician by training working with a physicist. I have been invited to give an hour-long tutorial/presentation to incoming graduate students. These students are all coming in with physical ...
28
votes
20answers
2k views

Good math bed-time stories for children?

What are some good references/books/articles from which to derive some good bed-time math stories to pique a child's interest in math? I am fascinated by math (used to hate it as a kid) and want my ...
73
votes
24answers
6k views

What are some examples of notation that really improved mathematics?

I've always felt that the concise, suggestive nature of the written language of mathematics is one of the reasons it can be so powerful. Off the top of my head I can think of a few notational ...
59
votes
26answers
6k views

What are the most overpowered theorems in mathematics?

What are the most overpowered theorems in mathematics? By "overpowered," I mean theorems that allow disproportionately strong conclusions to be drawn from minimal / relatively simple assumptions. ...
1
vote
3answers
1k views

Rules for Product and Summation Notation

When we deal with summation notation, there are some useful computational shortcuts, e.g.: $$\sum\limits_{i=1}^{n} 2 + 3i = \sum\limits_{i=1}^{n} 2 + \sum\limits_{i=1}^{n} 3i = 2n + ...