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36
votes
6answers
1k views

Need a result of Euler that is simple enough for a child to understand

Talking to my 8 yr old about "the greatest mathematician of all time", I said it was probably Gauss in my opinion, but that Gauss was not very kind to his kids (for example, forbidding them to go into ...
36
votes
7answers
6k views

I can't do math?

So after failing another math test today I have realized that I can't do math on my own. For whatever reason when I get to a test I just can't do math without extra resources. I think this is because ...
36
votes
7answers
3k views

Open math problems which high school students can understand

I request people to list some moderately and/or very famous open problems which high school students,perhaps with enough contest math background, can understand, classified by categories as on ...
36
votes
2answers
3k views

What does “communicated by” mean in math papers?

[This question involves mostly math papers, and may be relevant to graduate students learning to write and cite papers, although this is my only justification for this being a math question.] Usually ...
36
votes
5answers
3k views

What is mathematical research like?

I'm planning on applying for a math research program over the summer, but I'm slightly nervous about it just because the name math research sounds strange to me. What does math research entail ...
36
votes
2answers
712 views

How do I find partners for study?

I want to find partners to study some post-graduate topics together online (because I'm pretty much out of steam as far as self-study goes, and I have problems finding a decent grad school). Are there ...
36
votes
9answers
7k views

List of interesting math podcasts?

mathfactor is one I listen to. Does anyone else have a recommendation?
35
votes
7answers
4k views

How to tell if I'm good enough for graduate school?

I'm concerned about my level of preparation for graduate school. I have decent grades (3.64 GPA, 4.0 in math), great recommendations lined up, and some OK research experience (no publications), but I ...
35
votes
10answers
2k views

Mind-blowing mathematics experiments

We've all heard of some mind-blowing phenomena involving the sciences, such as the double-slit experiment. I was wondering if there are similair experiments or phenomena which seem very ...
35
votes
12answers
12k views

Real life applications of Topology

The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here What ...
35
votes
7answers
6k views

Can someone please explain the Riemann Hypothesis to me… in English?

I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
35
votes
8answers
3k views

Does mathematics require axioms?

I just read this whole article: http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf which is also discussed over here: Infinite sets don't exist!? However, the paragraph which I found most ...
35
votes
5answers
1k views

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and ...
35
votes
3answers
2k views

A path to truly understanding probability and statistics

I'm embarrassed to say that I have a PhD and hold an asst professorship, but get tripped up when reading statistics research. I am in a field of Business that is similar to IO Psychology or Social ...
34
votes
13answers
3k views

Examples of famous problems resolved easily

Have there been examples of seemingly long standing hard problems, answered quite easily possibly with tools existing at the time the problems were made? More modern examples would be nice. An example ...
34
votes
6answers
2k views

A Question about Doctoral Theses in Mathematics [closed]

This is most definitely a soft question, which I'm sure may get some negative attention, and perhaps even be voted closed. However, I genuinely would like to generate answers on this matter as it ...
34
votes
16answers
4k views

What exactly is a number?

We've just been learning about complex numbers in class, and I don't really see why they're called numbers. Originally, a number used to be a means of counting (natural numbers). Then we extend ...
34
votes
6answers
3k views

How hard is the proof of $\pi$ or e being transcendental?

I understand that $\pi$ and e are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
34
votes
10answers
4k views

Paradox: increasing sequence that goes to $0$?

It is $10$ o'clock, and I have a box. Inside the box is a ball marked $1$. At $10$:$30$, I will remove the ball marked $1$, and add two balls, labeled $2$ and $3$. At $10$:$45$, I will remove the ...
34
votes
5answers
2k views

How to write a good mathematical paper?

I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar. A good time always goes too fast! Two years are fled. In the third year of ...
33
votes
17answers
3k views

Non-Scientific questions solved by mathematics

I have a general question about the applications of mathematics. What are some applications of mathematics that are not scientific, perhaps maybe literary or philosophical, or political. I am ...
33
votes
9answers
2k views

Literary statements that are false as mathematics

I recently wanted to use the title of the famous short story "Everything that Rises must Converge" in a poem of mine. However, the mathematician in me insisted on changing it to "Everything that ...
33
votes
16answers
6k views

Pen, pencils and paper to write math [closed]

The question is inspired by this thread on MO. My father is a mathematician, so I remember seeing very nice pens all around. When I started to do math by myself I've realized how important can be a ...
33
votes
6answers
2k views

Should I be worried that I am doing well in Analysis and not well in Algebra?

I attend a mostly liberal arts focused university, in which I was able to test out of an "Introduction to Proofs" class and directly into "Advanced Calculus 1" (Introductory Analysis I) and I loved ...
33
votes
7answers
2k views

What are some good math specific study habits?

What are/ were some of your good mathematician's study habits that you found really worked for you? I'm a CS major at a respected school and have a solid GPA... However, I definitely lack when it ...
33
votes
1answer
1k views

Are difficult exercises good for beginners?

I'm self-studying Rudin's Real and Complex Analysis. I've finished the first two chapters so far, and I don't have any major problems understanding the definitions and theorems. I can prove the ...
33
votes
4answers
2k views

A Young Math Enthusiast's Fear

Having been an avid lover of Mathematics, it is my dream to become a mathematician one day. It is purely passion that drives me and I just can't stand the thought of not doing math when I grow older. ...
33
votes
3answers
1k views

What is your method?

Many young, and not so young, mathematicians struggle with how to spend their time. Perhaps this is due to the 90%-10% rule for mathematical insight: 90 pages of work yield only 10 pages of useful ...
33
votes
2answers
1k views

How much rigour is necessary?

I am taking a course in Algebraic Topology. We are using Hatcher as a textbook. One of the main problems I am facing with the textbook is its level of rigour. Example: On Pg 10, Hatcher mentions in ...
32
votes
10answers
3k views

Becoming Better at Math

How can I become excellent at math? It really interests me but when I fail I become demotivated and begin to give up. EDIT: Could anyone suggest books for someone with a math education that just ...
32
votes
6answers
2k views

How to learn from proofs?

Recently I finished my 4-year undergraduate studies in mathematics. During the four years, I've met all kinds of proofs. Some of them are friendly: they either show you a basic skill in one field or ...
32
votes
3answers
1k views

$e$ to 50 billion decimal places

Sorry if this is a really naive question, but in my reading of a lot of textbooks and articles, there is a lot of mention of how many decimals we know of a certain number today, such as $\pi$ or $e$. ...
32
votes
6answers
1k views

Attitude towards exercises in mathematics

I'm doing self study on a couple of topics in mathematics, such as real analysis, abstract algebra, and linear algebra. From time to time, there are always a couple of exercises which I found too ...
32
votes
3answers
3k views

Motivation and methods for self-study

First, a little background: Beginning with calculus in my first semester of college, I fell in love with mathematics. That was the point at which the concepts became interesting to me, and I started ...
31
votes
22answers
7k views

Poems related to mathematics [closed]

I am supposed to be presenting a poem in my undergraduate math class, that relates to mathematics. The point of the presentation is to show the beauty of math , and the fun of it ,and also it will ...
31
votes
7answers
3k views

Are the exercises necessary to understand the subject of a mathematical textbook?

When I am reading a mathematical textbook, I tend to skip most of the exercises. Generally I don't like exercises, particularly artificial ones. Instead, I concentrate on understanding proofs of ...
31
votes
7answers
1k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
31
votes
5answers
3k views

Soft Question - Intuition of the meaning of homology groups

I'm studying homology groups and I'm looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible ...
31
votes
4answers
5k views

The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
30
votes
12answers
1k views

How To Present Algebraic Topology To Non-Mathematicians?

I am writing my master thesis in algebraic topology (fundamental groups) and as a system in my school students must write about one page about their theses explaining for non mathematicians the ...
30
votes
7answers
984 views

Why are topological spaces interesting to study?

In introductory real analysis, I dealt only with $\mathbb{R}^n$. Then I saw that limits can be defined in more abstract spaces than $\mathbb{R}^n$, namely the metric spaces. This abstraction seemed ...
30
votes
13answers
3k views

List of interesting integrals for early calculus students

I am teaching Calc 1 right now and I want to give my students more interesting examples of integrals. By interesting, I mean ones that are challenging, not as straightforward (though not extremely ...
30
votes
5answers
6k views

How hard should a mathematician work?

Of course I would never compare myself to, and don't expect to be, one of the great mathematicians. However, I am curious as to how my work habits, dedication and passion differs from theirs. How ...
30
votes
6answers
1k views

Why are integrals called integrals?

What is the historical background for this term? I cannot quite see what is integral about an integral, even if we go back to the viewing it as the area under a curve. It seems a strange choice of ...
30
votes
14answers
776 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 ...
30
votes
14answers
14k views

Good book for self study of a First Course in Real Analysis

Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction ...
30
votes
4answers
742 views

Why do we look at morphisms?

I am reading some lecture notes and in one paragraph there is the following motivation: "The best way to study spaces with a structure is usually to look at the maps between them preserving structure ...
30
votes
7answers
927 views

Appearance of Formal Derivative in Algebra

When studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are ...
29
votes
15answers
3k views

Best way to express 2014 [closed]

In my class, I always write the date using mathematical formulas, or cool little equations. I want to show my students that even the most mundane seeming number often has fascinating features, and its ...
29
votes
17answers
5k views

Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...