Mathematics

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1500 questions
114
votes
4 answers

Why is $e^{\pi \sqrt{163}}$ almost an integer?

The fact that Ramanujan's Constant $e^{\pi \sqrt{163}}$ is almost an integer $$262 537 412 640 768 743.99999999999925...$$ doesn't seem to be a coincidence, but has to do with the $163$ appearing in it. Can you explain why it's almost-but-not-quite…
stevenvh
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114
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13 answers

References for multivariable calculus

Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it…
user9464
114
votes
6 answers

How do you go about learning mathematics?

I really like mathematics, but I am not good at learning it. I find it takes me a long time to absorb new material by reading on my own and I haven't found a formula that works for me. I am hoping a few people out there will tell me how they go…
Henry B.
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114
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6 answers

Self-studying real analysis — Tao or Rudin?

The reference requests for analysis books have become so numerous as to blot out any usefulness they could conceivably have had. So here comes another one. Recently I've began to learn real analysis via Rudin. I would do all the exercises, and if I…
Lee Wang
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114
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11 answers

Finding the limit of $\frac {n}{\sqrt[n]{n!}}$

I'm trying to find $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$ I tried couple of methods: Stolz, Squeeze, D'Alambert Thanks! Edit: I can't use Stirling.
user6163
114
votes
3 answers

Topology: The Board Game

Edit: I've drawn up some different rules, a map and some cards for playing an actual version of the game. They're available at my personal website with a Creative Commons Attribution 4.0 license. This is a math education question that I've been…
Brian Rushton
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114
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4 answers

Projection map being a closed map

Let $\pi: X \times Y \to X$ be the projection map where $Y$ is compact. Prove that $\pi$ is a closed map. First I would like to see a proof of this claim. I want to know that here why compactness is necessary or do we have any other weaker condition…
M.Subramani
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114
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1 answer

Is this continuous analogue to the AM–GM inequality true?

First let us remind ourselves of the statement of the AM–GM inequality: Theorem: ( AM–GM Inequality ) For any sequence $(x_n)$ of $N\geqslant 1$ non-negative real numbers, we have $$\frac1N\sum_k x_k \geqslant \left(\prod_k x_k\right)^{\frac1N}$$ It…
user1892304
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114
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6 answers

How to learn from proofs?

Recently I finished my 4-year undergraduate studies in mathematics. During the four years, I met all kinds of proofs. Some of them are friendly: they either show you a basic skill in one field or give you a better understanding of concepts and…
Roun
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114
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7 answers

What made you choose your research field?

I was wondering how come modern mathematicians do not seem to discover as many theorems as the older mathematicians seem to have done. Have we reached some kind of saturation limit where all commonly needed mathematics has already been discovered or…
Comic Book Guy
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113
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1 answer

Generalization of Liouville's theorem

As proposed in this answer , I wonder if the answer to following question is known. Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure of the set of functions whose derivative lies in…
RghtHndSd
  • 7,955
113
votes
2 answers

Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square

Question: For any $a,b\in \mathbb{N}^{+}$, if $a+b$ is a square number, then $f(a)+f(b)$ is also a square number. Find all such functions. My try: It is clear that the function $$f(x)=x$$ satisfies the given conditions, since: $$f(a)+f(b)=a+b.$$ But…
math110
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113
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18 answers

Mathematical equivalent of Feynman's Lectures on Physics?

I'm slowly reading through Feynman's Lectures on Physics and I find myself wondering, is there an analogous book (or books) for math? By this, I mean a good approach to mathematics given through sweeping motions, appeals to intuition and an emphasis…
Cotton Seed
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113
votes
6 answers

Defining a manifold without reference to the reals

The standard definition I've seen for a manifold is basically that it's something that's locally the same as $\mathbb{R}^n$, without the metric structure normally associated with $\mathbb{R}^n$. Aesthetically, this seems kind of ugly to me. The real…
user13618
113
votes
5 answers

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif Laczkovich gave a solution with many hundreds of triangles,…
Ed Pegg
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