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8
votes
2answers
523 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 ...
8
votes
2answers
567 views

Suggestions for high school?

I am currently a sophomore in high school who is very interested in mathematics and (theoretical) physics, and was wondering if the diverse set of mathematicians at MSE had any suggestions as to any ...
8
votes
3answers
1k views

Research in plane geometry or euclidean geometry

I was doing good at school in plane geometry and trigonometry - especially in geometric proofs like proving the equality of two line segments or two angles - more than I was doing in analytic geometry....
7
votes
4answers
700 views

What is… A Parsimonious History?

Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a ...
7
votes
2answers
897 views

What are Diophantine equations REALLY?

Sometimes when you want to solve an equation you can just use algebra and rearrange it then you are done. But sometimes no amount of algebra can ever prove the equation, and then you need an idea, ...
5
votes
3answers
504 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 ...
3
votes
6answers
3k views

Best practice book for calculus [closed]

I tried with every inch in me to not ask a question such as this but I just couldn't resist asking this. What is the best Calculus practice book? I tried looking around but couldn't find a ...
35
votes
3answers
2k 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 $e$. ...
18
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, ...
17
votes
1answer
1k views

Why are there so few Euclidean geometry problems that remain unsolved?

Stillwell mentions in his book Mathematics and its History that: Most of the really old unsolved problems in mathematics, in fact, are simple questions about the natural numbers... What is it ...
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
2answers
236 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}}} - 20379280\...
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 ...
13
votes
2answers
975 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 ...
13
votes
4answers
5k views

Importance of eigenvalues

I know how to find eigenvalues and eigenvector .But I dont know what to do with that. What is there use? Can anyone explain me that?
13
votes
9answers
921 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 ...
12
votes
0answers
229 views

Why are the fundamental theorems of calculus usually associated to the Riemann Integral?

I am writing a "textbook" on Analysis, and I've reached the time I must talk about integrals. I prefer to approach directly the Lebesgue Integral theory. This question is not about the status of this ...
11
votes
1answer
42k views

Why is a full circle 360° degrees? [duplicate]

What's the reason we agreed to setting the number of degrees of a full circle to 360? Does that make any more sense than 100, 1000 or any other number? Is there any logic involved in that particular ...
11
votes
2answers
749 views

What's the difference between “duality” and “symmetry” in mathematics?

Motivated by the answer to this question--"What kind of “symmetry” is the symmetric group about?", I read the article about dual graph. It is said in this article that "the term 'dual' is used because ...
11
votes
2answers
396 views

Is there a geometric interpretation of $F_p,\ F_{p^n}$ and $\overline{F_p}?$

I have been doing some exercises about finite fields lately and I think I've obtained some understanding of what they are. What seems to be missing though is some kind of picture. Learning to work ...
11
votes
6answers
830 views

Suggestions for topics in a public talk about art and mathematics [closed]

I've been giving a public talk about Art and Mathematics for a few years now as part of my University's outreach program. Audience members are usually well-educated but may not have much knowledge of ...
10
votes
2answers
942 views

Applications of Geometry to Computer Science

How is differential geometry (or any type of theoretical math) being used in computer science? Any research I have done on this topic leads me to some sort of applied math concept. I know that there ...
10
votes
4answers
514 views

What is the difference between analytic combinatorics and the theory of combinatorial species?

Yesterday I asked the question Why should a combinatorialist know category theory?, where Chris Taylor suggested me to have a look at combinatorial species. I had heard the term before but I haven't ...
10
votes
4answers
480 views

Application of computers in higher mathematics

Currently the main application of computers in mathematics seems to be to compute things, i.e. to solve equations, evaluate integrals, etc. It is at all possible to delegate the thinking of a ...
9
votes
3answers
1k views

Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
9
votes
3answers
2k views

The most common theorems taught in Abstract Algebra

I am self learning abstract algebra. I want to know which theorems are a must to understand. Now these are limits I have to deal with (please consider when answering): I have limited ...
9
votes
2answers
2k views

When can one expect a classical solution of a PDE?

When solving a PDE, there may be a classical solution or a weak solution (or distribution solution). But I am wondering that when people talk about "finding a solution" to some PDE, what do they refer ...
8
votes
1answer
749 views

What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
7
votes
1answer
355 views

Recommendations for an “illuminating” (explained in the post) group theory/abstract algebra resource?

I recently asked a question regarding why homomorphisms and isomorphisms are important. The best answer to that question was actually a comment, which referred me to Brian M. Scott's answer here: http:...
7
votes
4answers
475 views

What sequence should I study these topics in? [closed]

I will shortly list a series of topics that amount to what is essentially the first two years of an undergraduate degree. I'd like to know what is considered best order in which to study these ...
7
votes
1answer
3k views

Journals that publish papers quickly [closed]

I have written two papers in Mathematics and want to get them published. Can you suggest some journals that publish quickly? Besides, how can I know if a journal is well-regarded or not? I know there'...
7
votes
2answers
796 views

What are Free Objects?

I've read the wikipedia article, but couldn't grasp the concept. Is there an informal definition? Are there examples of uses of free objects in calculus? Are free objects somehow connected to ...
6
votes
2answers
730 views

Linear algebra and geometric insight: a rigorous approach to vector spaces, matrices, and linear applications

I'm searching for some material (books or lecture notes) that extensively uses a geometric approach to explain the meaning of the concepts realted regarding to vector spaces, matrices, and linear ...
6
votes
2answers
237 views

Are there any theories being developed which study structures with many operations and many distributive laws?

"Algebraic structure" will mean a set with some n-ary operations defined on it. This does not include vector spaces for example. During my study of algebra I have encountered mostly algebraic ...
6
votes
1answer
473 views

A layman's motivation for non-standard analysis and generalised limits

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. Background: I have recently ...
5
votes
2answers
183 views

The two distinct cultures in mathematics

I once read an article about two distinct culture within mathematics, "analysts" and "algebraists" when I was in high school. I am not still a graduate student, but I really want to hear what it feels ...
-1
votes
2answers
662 views

Is it to the students' advantage to learn the language of infinitesimals? [closed]

A colleague of mine asked an interesting question reproduced below with his permission. It is reasonable to ask whether it is to the students' advantage to learn the language of infinitesimals - ...
22
votes
5answers
1k views

What does it mean for a set to exist?

Is there a precise meaning of the word 'exist', what does it mean for a set to exist? And what does it mean for a set to 'not exist' ? And what is a set, what is the precise definition of a set?
20
votes
3answers
1k views

What are the applications of finite calculus

I'm reading through Concrete Mathematics [Graham, Knuth, Patashnik; 2nd edition], and in the section regarding Summation, they have a sub-section entitled "Finite and Infinite Calculus". In this ...
16
votes
4answers
620 views

Banach spaces over fields other than $\mathbb{C}$?

Sorry, this is a rather vague question. I was just wondering if there is any kind of theory about normed (if possible Banach) spaces over fields other than the real or complex numbers. I'm guessing ...
16
votes
3answers
7k views

Importance of Linear Algebra

In one of his online lectures Benedict Gross comments that one can never have too much Linear Algebra. Also, looking around it seems like I can find comments to the effect that Linear Algebra has ...
14
votes
3answers
1k views

Transcendental number

While reading on Wikipedia about transcendental numbers, i asked myself: Why is it so hard and difficult to prove that $e +\pi, \pi - e, \pi e, \frac{\pi}{e}$ etc. are transcendental numbers? ...
12
votes
7answers
7k views

Why don't we use base 6 or 11?

Another question on this site asks why we have chosen our number system to be decimal base 10. There are others asking basically the same thing as well. I'm not really satisfied with any of the ...
12
votes
5answers
2k views

Common misconceptions about math

YARFMO (Yet another reposting from Mathoverflow) ;-) The more you know about math the more you find conceptions previously thought correct to be false: 1.) math is not as exact as many believe - in ...
12
votes
9answers
1k views

Strangest Notation? [closed]

While this may be a fruitless pursuit of anecdotes, I still ask: what is the strangest (or most blatantly wrong (at least in the eyes of common notation)) mathematical notation you have ever seen?
11
votes
4answers
3k views

Angle brackets for tuples

I've recently noticed that use of angle brackets for writing tuples, e.g. $\langle x, y \rangle$ instead of the usual round brackets in a few books I've been reading — Lawvere's Sets for Mathematics, ...
11
votes
2answers
995 views

Abstract algebra book with real life applications

Is there an abstract algebra book that emphasizes the applications to "real world" problems? Update: By real world, I mean mostly related to physics or other sciences. But references to coding theory ...
10
votes
4answers
255 views

Is there a “natural” / “categorical” definition of the “parity” of a permutation?

Given a permutation $\sigma$ on $n$ elements (i.e. $\sigma \in S_n$), there is a notion of "parity" (or "sign" or "signature") of $\sigma$, which can be defined in several equivalent ways (look here). ...
10
votes
7answers
795 views

Advice on self study of category theory

I'm very intrested in start to study category theory with aim to use this in algebraic geometry. I already took courses (besides the basics) on commutative algebra and general topology with a soft ...
10
votes
4answers
1k views

Why are nets not used more in the teaching of point-set topology?

I just finished working through a proof of Tychonoff's Theorem that uses nets (specifically, as a corollary of the fact that a net in a product space converges iff the projected nets in the components ...