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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
- 3,132
160
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
- 4,086
160
votes
25 answers
Most ambiguous and inconsistent phrases and notations in maths
What are some examples of notations and words in maths which have been overused or abused to the point of them being almost completely ambiguous when presented in new contexts? For instance, a function $f$: $f^{-1}(x)$ can be an inverse and a…
Frank Vel
- 5,537
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
6 answers
Can someone clearly explain about the lim sup and lim inf?
Can some explain the lim sup and lim inf ? In my text book the definition of these two is this. Let $(s_n)$ be a sequence in $\mathbb{R}$. We define $$\lim \sup\ s_n = \lim_{N \rightarrow \infty} \sup\{s_n:n>N\}$$ and $$\lim\inf\ s_n =…
eChung00
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159
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33 answers
Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$
I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals: $$\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
- 1,953
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 stupid.…
Ian Mateus
- 7,601
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
- 4,215
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
4 answers
What is category theory useful for?
Okay, so I understand what calculus, linear algebra, combinatorics and even topology try to answer (update: this is not the case in hindsight), but why invent category theory? In Wikipedia it says it is to formalize. As far as I can tell, it sort of…
Asinomás
- 107,927
158
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
- 10,619
157
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
- 7,411
157
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
- 3,167
157
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
- 7,698
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
- 20,598