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14
votes
4answers
6k views

How to do well on Math Olympiads

I'm a high school student who really likes maths and I'm quite good at school. I want to start training maths by myself but I think I need some guidelines. I want to do well on IMO but I don't know ...
14
votes
7answers
2k views

Why do statements which appear elementary have complicated proofs?

The motivation for this question is : Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$ and some other problems in Mathematics which looks as if they are elementary but their ...
14
votes
8answers
6k views

How would you describe calculus in simple terms?

I keep hearing about this weird type of math called calculus. I only have experience with geometry and algebra. Can you try to explain what it is to me?
13
votes
6answers
2k views

Self studying math, how can I learn the most?

I am currently studying Pre-Calculus on my own. I have a few texts I am working with but feel like I could learning a lot more than I am. When people typically ask these kind of questions the common ...
13
votes
5answers
221 views

Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
13
votes
3answers
1k views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
13
votes
1answer
760 views

Compact = Closed + Bounded + (?)

In $\mathbb{R}^n$ we know (Heine-Borel Theorem) that a set is compact if and only if it is closed and bounded. In $C(X)$ for a compact metric space $X$, we know (corollary of Ascoli-Arzela Theorem) ...
13
votes
5answers
2k views

Software to draw links or knots

I am looking for software that can aid me in drawing knots and links. There are of course (examples) knotplotters all over the web, but they can only draw specific knots. What I am looking for is the ...
12
votes
1answer
2k views

What kind of matrices are non-diagonalizable?

I'm trying to build an intuitive geometric picture about diagonalization. Let me show what I got so far. Eigenvector of some linear operator signifies a direction in which operator just ''works'' ...
11
votes
2answers
688 views

How can I pick up analysis quickly?

I have a 2-3 week recess from university for winter break. In this time, I would like to learn analysis, starting with Walter Rudin's Principles of Mathematical Analysis, and then, if at all possible, ...
9
votes
5answers
3k views

How do you rebuild your Math skills after college? [closed]

I minored in Math in college (B.S. Software Engineering), and went through advanced Calc, Differential Equations, etc. But about 6 months after I graduated I had lost all my formulas. Now, about 5 ...
7
votes
3answers
947 views

Exercise books in analysis

I'm studying Rudin's Principles of mathematical analysis and I was wondering if there are some exercise books (that is, books with solved problems and exercises) that I can use as a companion to ...
41
votes
2answers
4k 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 ...
37
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 ...
22
votes
7answers
2k views

Can you use both sides of an equation to prove equality?

For example: $\color{red}{\text{Show that}}$$$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$ In high school my maths teacher told me To prove ...
19
votes
2answers
496 views

A structural proof that $ax=xa$ forms a monoid

During the discussion on this problem I found the following simple observation: If $M$ is a monoid and $a \in M$ then $\{x: ax = xa\}$ is a submonoid. This is trivial to prove by checking ...
19
votes
5answers
1k views

Results that were widely believed to be false but were later shown to be true

What are some results which were widely believed to be false, but were later to be shown to be true, or vice-versa?
18
votes
11answers
1k views

Why limits work

I'm currently a first year student in electrical engineering and computer science. I know how to compute limits, derivatives, integrals with respect to one variable that is things from one variable ...
18
votes
1answer
2k views

Strategies and Tips: What to do when stuck on math?

Since math students will be stuck on some math at some point, what strategies or tips can help (to assuage this recurrent reality of maths)? Certainly, this wonderful website helps; here: one can ...
17
votes
1answer
1k views

Arxiv - what should I expect

I went through arxiv.org primer and some questions at MO and here about arxiv. There are many of them so I could have missed answers to some of my questions. My apologies in that case. I have never ...
16
votes
1answer
235 views

Is it a good approach to heavily depend on visualization to learn math?

I am a third year undergraduate and I am a beginner on these "real mathematics" (no pun intended). Before contacting the "real math", my math level should be considered to be "good", although I was ...
15
votes
9answers
633 views

Definite integrals with interesting results [closed]

I just stumbled across the fact that $\int_{-\infty}^{+\infty}{e^{-x^2}dx}=\sqrt{\pi}$. This intrigued my already-existing interest in integrals. It made me wonder, are there other integrals with ...
14
votes
4answers
3k views

mathematical maturity

So, I finished my undergrad with a degree in applied mathematics, but when reading some graduate level texts and/or papers, I often find myself struggling. I eventually get there, but I often feel ...
13
votes
6answers
538 views

Just How Strong is Associativity?

A friend of mine is using a lot of algebra that is not associative for an advanced Chemistry project. We were discussing it recently and I found it rather amusing how often she said things like ...
13
votes
3answers
273 views

How to recognize adjointness?

Reading math has gradually expanded my understanding of the word "symmetry", so that now I can recognize symmetries that I would have not noticed before, and without having them pointed out to me. I ...
13
votes
3answers
654 views

Where are the geometric figures in those “advanced geometry” textbooks?

It is me again to bother you. Since my last post I started to look at some seemingly "serious" mathematics for background study. Accidentally I went into a university bookshop and came across some ...
11
votes
3answers
6k views

Baby Rudin vs. Abbott

I am considering Stephen Abbott's Understanding Analysis and Walter Rudin's Principles of Mathematical Analysis. I am looking for a comparison between the two that addresses both of the following ...
11
votes
2answers
1k views

What would be a good way to memorize theorems about algebra?

This post is not constructive, so maybe this rather should be posted on CW, but since there is a 'soft-question' tag, i'm posting it here. ===================== I believe the best way to memorize ...
11
votes
3answers
601 views

What is combinatorial homotopy theory?

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I've heard people ...
11
votes
2answers
638 views

Category Theory with and without Objects

Slight Motivation: In Mac Lane and Freyd's books (the latter being a reprint of an older book called "Abelian Categories") they note that instead of defining any Objects in a category we may define an ...
10
votes
0answers
191 views

Is Category Theory geometric?

In "From a Geometrical Point of View" (http://www.amazon.com/gp/aw/d/1402093837?pc_redir=1407132421&robot_redir=1) Marquis states that category theory is thoroughly geometric. Could someone ...
10
votes
5answers
2k views

Advice for benefits to directly use analysis textbook to replace calculus

Main purpose: For self-learning performance, neither for exam nor degree courses. Calculus textbook using now[1]: Calculus I, Weinstein&Marsden, UTM, Springer Question Description: I've been ...
9
votes
5answers
615 views

Examples of open problems solved through short proof

Are there good examples of reasonable open problems in mathematics that had an 'obvious' solution via application of a theorem already known/not yet found in mathematics but could have been found with ...
9
votes
1answer
344 views

“Visualizing” Mathematical Objects - Tips & Tricks

It has been a while since I am kind of stuck with my skills concerning the visualization of mathematical objects. Here there is the problem. First of all, let me point out that I am completely ...
9
votes
2answers
877 views

Local vs. global in the definition of a sheaf

Apologies in advance that this question is inescapably soft. What I am stuck on is squishy; I have the feeling that if I could even make it precise, I'd already be satisfied. To what extent is a ...
9
votes
1answer
451 views

Good references for stacks

I have seen stacks come up in various settings recently. I understand that, at least heuristically, they are some sort of generalization of a scheme, but I don't actually know anything about them. ...
9
votes
4answers
453 views

Beside transcendental or uncomputable numbers what other types of numbers are there?

What other types/categories of numbers are there that we know of today (i.e. some one has done some work on them, like Chaitin's uncomputable $\Omega$ number)? Of course there are uncountably many ...
8
votes
1answer
2k views

Very interesting graph!

I found a VERY interesting graph on http://www.xamuel.com/graphs-of-implicit-equations/. It looked very, very cool, but the equation of the graph is so simple! Here is the image of the graph: My ...
8
votes
2answers
514 views

What is the best approach when things seem hopeless?

I'm finding that I am getting to the point of being hopelessly behind in one of my courses. What is the best thing to do when it feels impossible to get caught up in the literal sense. Being "caught ...
5
votes
3answers
494 views

Are vectors and covectors the same thing?

In Euclidean space, we usually don't distinguish between vectors and covectors (or dual vectors or 1-forms or whatever you want to call them) -- because the spaces overlap. However, a physicist ...
5
votes
4answers
432 views

Intuitive Explanation of Morphism Theorem

Is there an intuitive explanation for the morphism theorem from introductory abstract algebra? First Morphism Theorem: Let $K$ be the kernel of the group morphism $f: G \to H$. Then $G/K$ is ...
5
votes
6answers
2k views

A book for self-study of matrix decompositions

I am a third year math student and I noticed that there are many uses for decomposing a matrix (I mean decompositions like SVD, LU etc'). Is there a good book for self-study of the subject ? Note ...
34
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 $e$. ...
17
votes
7answers
3k views

What is combinatorics?

I've tried to search the web and in books, but I haven't found a good definition or definitive explanation of what combinatorics is. Could anyone give me a definition/explanation of combinatorics, ...
16
votes
3answers
5k views

Are one-by-one matrices equivalent to scalars?

I am a programmer, so to me $[x] \neq x$—a scalar in some sort of container is not equal to the scalar. However, I just read in a math book that for $1 \times 1$ matrices, the brackets are often ...
14
votes
5answers
3k views

Opposite of Fermat's Last Theorem?

So Wiles' proof showed that no three positive integers $a$, $b$, and $c$ can solve the equation $a^n+b^n=c^n$ for any integer value of n greater than $2$. Now what about the opposite? What does this ...
14
votes
2answers
212 views

Bellard's exotic formula for $\pi$

In his webpage, Fabrice Bellard mentions an exotic formula for $\pi$ as follows $$\pi = \frac{1}{740025}\left(\sum_{n = 1}^{\infty}\dfrac{3P(n)}{{\displaystyle \binom{7n}{2n}2^{n - 1}}} - ...
14
votes
5answers
2k views

What does “formal” mean?

I know the definition of formal power series, power series and polynomials. But what does the adjective "formal" mean? In google English dictionary, does it mean "9. Of or relating to linguistic or ...
13
votes
9answers
862 views

Is it possible to alternate the law of mathematics?

I am freelance writer. Recently I have been planning a science fiction - just planning, nothing solid yet - and I was wondering would it be possible for some other universes that have different set of ...
13
votes
2answers
960 views

What is the smallest constant that has explicitly appeared in a published paper? [closed]

You can read a lot about what large number in mathematics are, like Skewes' number, Moser's number or even Graham's number. So just for the sake of non-discrimination, I ask, what is the smallest ...