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1500 questions
160
votes
7 answers
Is Apple ipad / tablet good for mathematics students?
I am a math student. I'd like to find out if tablets (iPads, Galaxy Note 10.1) are worth the cost.
How good are tablets for the purposes of reading textbooks as PDF and writing mathematics with a stylus?
For writing math in TeX, it looked like the…
T. Webster
- 3,122
159
votes
5 answers
What is the Riemann-Zeta function?
In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?
159
votes
7 answers
Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$
It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant.
Along similar lines, you can show that $\mathbb{R^2}$ isn't…
user7530
- 50,625
158
votes
7 answers
Does a "cubic" matrix exist?
Well, I've heard that a "cubic" matrix would exist and I thought: would it be like a magic cube? And more: does it even have a determinant - and other properties? I'm a young student, so... please don't get mad at me if I'm talking something…
Ian Mateus
- 7,561
158
votes
2 answers
Is there a known well ordering of the reals?
So, from what I understand, the axiom of choice is equivalent to the claim that every set can be well ordered. A set is well ordered by a relation, $R$ , if every subset has a least element. My question is: Has anyone constructed a well ordering on…
Seamus
- 4,205
158
votes
16 answers
Where to start learning Linear Algebra?
I'm starting a very long quest to learn about math, so that I can program games. I'm mostly a corporate developer, and it's somewhat boring and non exciting. When I began my career, I chose it because I wanted to create games.
I'm told that Linear…
Sergio Tapia
158
votes
17 answers
Why do both sine and cosine exist?
Cosine is just a change in the argument of sine, and vice versa.
$$\sin(x+\pi/2)=\cos(x)$$
$$\cos(x-\pi/2)=\sin(x)$$
So why do we have both of them? Do they both exist simply for convenience in defining the other trig functions?
Tdonut
- 4,046
157
votes
33 answers
Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$
I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:
$$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$
I really have no idea why this statement is true. Can someone…
Nathan Osman
- 1,933
157
votes
10 answers
Math and mental fatigue
Just a soft-question that has been bugging me for a long time:
How does one deal with mental fatigue when studying math?
I am interested in Mathematics, but when studying say Galois Theory and Analysis intensely after around one and a half hours, my…
yoyostein
- 20,428
157
votes
15 answers
Has lack of mathematical rigour killed anybody before?
One of my friends was asking me about tertiary level mathematics as opposed to high school mathematics, and naturally the topic of rigour came up.
To provide him with a brief glimpse as to the difference, I said the following.
In high school, you…
Trogdor
- 10,541
156
votes
1 answer
Is the following matrix invertible?
$$\begin{bmatrix} 1235 &2344 &1234 &1990\\
2124 & 4123& 1990& 3026 \\
1230 &1234 &9095 &1230\\
1262 &2312& 2324 &3907
\end{bmatrix}$$
Clearly, its determinant is not zero and, hence, the matrix is invertible.
Is there a more elegant way to do…
Yongkai
- 1,809
155
votes
41 answers
Why is negative times negative = positive?
Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc.
I went ahead and gave them a proof by contradiction like this:
Assume $(-x) \cdot (-y) = -xy$
Then divide both sides…
Sev
- 2,151
155
votes
13 answers
What does the dot product of two vectors represent?
I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent.
The product of two numbers, $2$ and $3$, we say that it is $2$ added to itself $3$ times or something like…
Saturn
- 7,351
155
votes
8 answers
Can you explain the "Axiom of choice" in simple terms?
As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get.
I went to Wikipedia to see what the Axiom of Choice is, but as often happens with things like this,…
Tyler
- 3,147
154
votes
0 answers
Pullback and Pushforward Isomorphism of Sheaves
Suppose we have two schemes $X, Y$ and a map $f\colon X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\mathcal{F}$ is an $\mathcal{O}_X$-module and…
Matt
- 7,608