Mathematics Stack Exchange
Mathematics Stack Exchange

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1500 questions
88
votes
10 answers

Is mathematics just a bunch of nested empty sets?

When in high school I used to see mathematical objects as ideal objects whose existence is independent of us. But when I learned set theory, I discovered that all mathematical objects I was studying were sets, for example: $ 0 = \emptyset $ What…
Vinicius L. Deloi
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88
votes
7 answers

Rigorous nature of combinatorics

Context: I'm a high school student, who has only ever had an introductory treatment, if that, on combinatorics. As such, the extent to which I have seen combinatoric applications is limited to situations such as "If you need a group of 2 men and 3…
frog1944
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88
votes
6 answers

Alice and Bob play the determinant game

Alice and Bob play the following game with an $n \times n$ matrix, where $n$ is odd. Alice fills in one of the entries of the matrix with a real number, then Bob, then Alice and so forth until the entire matrix is filled. At the end, the…
pad
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88
votes
4 answers

Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that $\textrm{det}(J(\phi_{\alpha} \circ…
Richard G
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88
votes
3 answers

Why did mathematicians introduce the concept of uniform continuity?

I have solved many problems regarding uniform continuity, but still I can't understand the following: Is there any practical application of this concept, or it is just a theoretical concept? Is there any wide application of this concept in any…
ramanujan
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87
votes
5 answers

How to use the Extended Euclidean Algorithm manually?

I've only found a recursive algorithm of the extended Euclidean algorithm. I'd like to know how to use it by hand. Any idea?
Andrew
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87
votes
4 answers

Is there any easy way to understand the definition of Gaussian Curvature?

I am new to differential geometry and I am trying to understand Gaussian curvature. The definitions found at Wikipedia and Wolfram sites are too mathematical. Is there any intuitive way to understand Gaussian curvature?
Shan
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87
votes
4 answers

Difference between complete and closed set

What is the difference between a complete metric space and a closed set? Can a set be closed but not complete?
ABC
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87
votes
6 answers

Why are differentiable complex functions infinitely differentiable?

When I studied complex analysis, I could never understand how once-differentiable complex functions could be possibly be infinitely differentiable. After all, this doesn't hold for functions from $\mathbb R ^2$ to $\mathbb R ^2$. Can anyone explain…
Casebash
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87
votes
9 answers

What is the proof that the total number of subsets of a set is $2^n$?

What is the proof that given a set of $n$ elements there are $2^n$ possible subsets (including the empty-set and the original set).
Celeritas
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87
votes
9 answers

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 interesting is not really discussed there. I think…
Kasper
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87
votes
3 answers

Is there a definitive guide to speaking mathematics?

Is there a definitive guide to speaking mathematics to avoid ambiguity? I'm writing a program to generate text for a variety of mathematical expressions and would like to code it so that it adheres to some standard. I've found Handbook for Spoken…
Michael Prescott
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87
votes
9 answers

How to find the inverse modulo $m$?

For example: $$7x \equiv 1 \pmod{31} $$ In this example, the modular inverse of $7$ with respect to $31$ is $9$. How can we find out that $9$? What are the steps that I need to do? Update If I have a general modulo equation: $$5x + 1 \equiv 2…
roxrook
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87
votes
5 answers

Compact sets are closed?

I feel really ignorant in asking this question but I am really just don't understand how a compact set can be considered closed. By definition of a compact set it means that given an open cover we can find a finite subcover the covers the…
InsigMath
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