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1500 questions
58
votes
3 answers
Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$
Do you know an explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? Using the axiom of choice, every vector space admits a norm but have you an explicit formula on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$?
A related question is: Can we proved…
Seirios
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58
votes
6 answers
Is Complex Analysis equivalent Real Analysis with $f:\mathbb R^2 \to \mathbb R^2$?
Am I correct in noticing that Complex Analysis seems to be a synonym for analysis of functions $\mathbb R^2 \to \mathbb R^2$?
If this is the case, surely all the results from complex analysis carry over to the study of these $\mathbb R^2 \to…
providence
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58
votes
10 answers
Is it an abuse of language to say "*the* integers," "*the* rational numbers," or "*the* real numbers," etc.?
I'm finding that the more math I learn, the more concepts I thought were well-defined seem to be intuitive and naive. Here I'm asking about whether it's an abuse of language to refer to "the integers," "the rational numbers," or "the real…
justin
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58
votes
4 answers
Does this Fractal Have a Name?
I was curious whether this fractal(?) is named/famous, or is it just another fractal?
I was playing with the idea of randomness with constraints and the fractal was generated as follows:
Draw a point at the center of a square.
Randomly choose any…
SilverSlash
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58
votes
11 answers
What is an operator in mathematics?
Could someone please explain the mathematical difference between an operator (not in the programming sense) and a function? Is an operator a function?
Nik
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58
votes
27 answers
Big List of Fun Math Books
To be on this list the book must satisfy the following conditions:
It doesn't require an enormous amount of background material to understand.
It must be a fun book, either in recreational math (or something close to) or in philosophy of…
RougeSegwayUser
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58
votes
12 answers
Easy way of memorizing values of sine, cosine, and tangent
My math professor recently told us that she wanted us to be able to answer $\sin\left(\frac{\pi }{2}\right)$ in our head on the snap. I know I can simply memorize the table for the test by this Friday, but I may likely forget them after the test. So…
James Smith
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58
votes
1 answer
Is $1+x+\frac{x^2}2+\dots+\frac{x^n}{n!}$ irreducible?
The polynomial $f(x)=1+x+\frac{x^2}2+\dots+\frac{x^n}{n!}$ often appears in algebra textbooks as an illustration for using derivative to test for multiple roots.
Recently, I stumbled upon Example 2.1.6 in Prasolov's book Polynomials (Springer,…
Martin Sleziak
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58
votes
12 answers
Calculating pi manually
Hypothetically you are put in math jail and the jailer says he will let you out only if you can give him 707 digits of pi. You can have a ream of paper and a couple pens, no computer, books, previous pi memorization or outside help.
What is the best…
Neil
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58
votes
1 answer
Semi-direct v.s. Direct products
What is the difference between a direct product and a semi-direct product in group theory?
Based on what I can find, difference seems only to be the nature of the groups involved, where a direct product can involve any two groups and the…
retro
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58
votes
8 answers
How can a set contain itself?
In Russell's famous paradox ("Does the set of all sets which do not contain themselves contain itself?") he obviously makes the assumption that a set can contain itself. I do not understand how this should be possible and therefore my answer to…
jimmyorpheus
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57
votes
7 answers
Why doesn't induction extend to infinity? (re: Fourier series)
While reading some things about analytic functions earlier tonight it came to my attention that Fourier series are not necessarily analytic. I used to think one could prove that they are analytic using induction
Let $P(n)$ be some statement…
David Z
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57
votes
1 answer
Evaluating $\int_0^\infty \frac{dx}{\sqrt{x}[x^2+(1+2\sqrt{2})x+1][1-x+x^2-x^3+...+x^{50}]}$
My brother's friend gave me the following wicked integral with a beautiful result
\begin{equation}
{\Large\int_0^\infty} \frac{dx}{\sqrt{x} \bigg[x^2+\left(1+2\sqrt{2}\right)x+1\bigg]…
Anastasiya-Romanova 秀
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57
votes
5 answers
Self-learning mathematics - help needed!
First, I apologise for the nebulous nature of my title but I can't adequately explain myself concisely.
I am about to start an MSc in pure maths after a fairly shaky undergraduate degree. I am very passionate about maths but I have several problems…
Elliptic2005
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57
votes
2 answers
Geometry problem involving infinite number of circles
What is the sum of the areas of the grey circles? I have not made any progress so far.
Dan
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