Mathematics Stack Exchange
Mathematics Stack Exchange

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1500 questions
156
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
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155
votes
1 answer

Overview of basic facts about Cauchy functional equation

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of mathematics ranging from exercises in freshman…
Martin Sleziak
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155
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
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155
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
  • 19,608
155
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
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155
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
  • 10,331
154
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
154
votes
14 answers

Why does an argument similiar to 0.999...=1 show 999...=-1?

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the below argument that shows $999\ldots = -1$ is…
CommonToad
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153
votes
11 answers

Is the inverse of a symmetric matrix also symmetric?

Let $A$ be a symmetric invertible matrix, $A^T=A$, $A^{-1}A = A A^{-1} = I$ Can it be shown that $A^{-1}$ is also symmetric? I seem to remember a proof similar to this from my linear algebra class, but it has been a long time, and I can't find it in…
gregmacfarlane
  • 1,729
153
votes
8 answers

Do most mathematicians know most topics in mathematics?

How many topics outside of his or her specialization is an average mathematician familiar with? For example, does an average group theorist know enough of partial differential equations to pass a test in a graduate-level PDE course? Also, what are…
Sid Caroline
  • 3,729
153
votes
7 answers

Studying Euclidean geometry using hyperbolic criteria

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise. But recently a good friend named Euclid has…
Zach Conn
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152
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
  • 1,873
151
votes
21 answers

Best book of topology for beginner?

I am a graduate student of math right now but I was not able to get a topology subject in my undergrad... I just would like to know if you guys know the best one..
gg1
  • 111
151
votes
10 answers

Are mathematical articles on Wikipedia reliable?

I know that Wikipedia gets a bad rap, and it seems like some teachers of mine have nothing better to do in class than harp on about the Great Academic Pastime of calling Wikipedia untrustworthy, but let's face it - Wikipedia is probably the single…
user140943
  • 2,061
151
votes
4 answers

Does $R[x] \cong S[x]$ imply $R \cong S$?

This is a very simple question but I believe it's nontrivial. I would like to know if the following is true: If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and $S$ are isomorphic. Thanks! If there isn't a proof…
Richard G
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