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1500 questions
74
votes
3 answers
How does a calculator calculate the sine, cosine, tangent using just a number?
Sine $\theta$ = opposite/hypotenuse
Cosine $\theta$ = adjacent/hypotenuse
Tangent $\theta$ = opposite/adjacent
In order to calculate the sine or the cosine or the tangent I need to know $3$ sides of a right triangle. $2$ for each corresponding…
themhz
- 1,233
74
votes
4 answers
Why it is important to write a function as sum of even and odd functions?
For the function $f(x)$ we can write it as sum of even and odd functions:
$$f(x)=\underbrace{\frac{f(x)+f(-x)}{2}}_{\text{Even}}+\underbrace{\frac{f(x)-f(-x)}{2}}_{\text{Odd}}$$
My question is why it is important for us to write a function as sum of…
User
- 8,033
74
votes
2 answers
Are there any valid continuous Sudoku grids?
A standard Sudoku is a $9\times 9$ grid filled with digits such that every row, column, and $3\times 3$ box contains all the integers from $1$ to $9$.
I am thinking about a generalization of Sudoku which I call "continuous Sudoku", which consists of…
ZKG
- 1,357
74
votes
6 answers
$1 + 2 + 4 + 8 + 16 \ldots = -1$ paradox
I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1:
Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 \ldots$
$x = 1 + 2 + 4 + 8 + 16 \ldots$
Multiply…
Christian
- 851
74
votes
13 answers
How to prove every closed bounded interval in $\mathbb{R}$ is compact?
Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familiar to us. Here I want to collect the ways to prove $[a,b]$ is compact.
Thanks for your help and any link.
Paul
- 21,141
74
votes
8 answers
Using Gröbner bases for solving polynomial equations
In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean algorithm and Gaussian elimination". I've tried to…
J. M. ain't a mathematician
- 76,540
74
votes
2 answers
Numerical phenomenon. Who can explain?
I was doing some software engineering and wanted to have a thread do something in the background to basically just waste CPU time for a certain test.
While I could have done something really boring like for(i < 10000000) { j = 2 * i }, I ended up…
Jake Mirra
- 3,273
74
votes
10 answers
List of problem books in undergraduate and graduate mathematics
I would like to know some good problem books in various branches of undergraduate and graduate mathematics like group theory, galois theory, commutative algebra, real analysis, complex analysis, topology etc. The books should contain solution to…
Mohan
- 15,494
74
votes
5 answers
Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$
Show that :
$$
\int_{0}^{\Large\frac\pi2}
{\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right)
\ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right)
\over
\cos\left(x\right)\sin\left(x\right)}\,{\rm d}x
={1 \over…
Ryan
- 4,035
74
votes
3 answers
Why isn't several complex variables as fundamental as multivariable calculus?
One typically studies analysis in $\mathbb{R}^n$ after studying analysis in $\mathbb{R}$. Why can't the same be said of $\mathbb{C}$?
user60042
- 751
74
votes
15 answers
Solving $DEF+FEF=GHH$, $KLM+KLM=NKL$, $ABC+ABC+ABC=BBB$
She visits third class and is $8$ years old (you can imagine how ashamed I felt when I said so to her). I helped her with lots of maths stuff today already but this is very unknowable for me. Sorry it's in German but I have translated it :)
It's…
cnmesr
- 4,910
74
votes
6 answers
Why is $A^TA$ invertible if $A$ has independent columns?
How can I understand that $A^TA$ is invertible if $A$ has independent columns? I found a similar question, phrased the other way around, so I tried to use the theorem
$$
rank(A^TA) \le min(rank(A^T),rank(A))
$$
Given $rank(A) = rank(A^T) = n$ and…
Chewers Jingoist
- 1,193
74
votes
1 answer
What is the logic/rationale behind the vector cross product?
I don't think I ever understood the rationale behind this.
I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from the cosine rule. (Do correct me if I'm…
Danxe
- 1,703
74
votes
2 answers
Continuity and the Axiom of Choice
In my introductory Analysis course, we learned two definitions of continuity.
$(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ satisfies $f(z_n) \to f(a)$.
$(2)$ A function $f:E \to…
John Gowers
- 25,678
74
votes
2 answers
Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$
About a month ago, I got the following :
For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that
$$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{c^\color{red}{3}+d^\color{red}{3}}.$$
For $r=p/q$…
mathlove
- 151,597