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1500 questions
73
votes
1 answer
Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?
Related: Can a sum of square roots be an integer?
Except for the obvious cases $n=0,1$, are there any values of $n$ such that $\sum_{k=1}^n\sqrt k$ is an integer? How does one even approach such a problem? (This is not homework - just a problem I…
Mario Carneiro
- 27,438
73
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
- 1,223
73
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…
Etemon
- 6,437
73
votes
5 answers
Why do engineers use derivatives in discontinuous functions? Is it correct?
I am a Software Engineering student and this year I learned about how CPUs work, it turns out that electronic engineers and I also see it a lot in my field, we do use derivatives with discontinuous functions. For instance in order to calculate the…
Santiago Pardal
- 896
73
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
- 841
73
votes
5 answers
Can someone explain the Yoneda Lemma to an applied mathematician?
I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood everything until we covered the Yoneda Lemma,…
Chris Taylor
- 28,955
73
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
- 3,198
73
votes
16 answers
Why can a real number be defined as a Dedekind cut, that is, as a set of rational numbers?
I don't know if my textbook is written poorly or I'm dumb. But I can't bring myself to understand the following definition.
A real number is a cut, which parts the rational numbers into two classes. Let $\mathbb{R}$ be the set of cuts. A cut is a…
God bless
- 2,049
73
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
- 4,720
- 7
- 44
- 83
73
votes
4 answers
How to know if a point is inside a circle?
Having a circle with the centre $(x_c, y_c)$ with the radius $r$ how to know whether a point $(x_p, y_p)$ is inside the circle?
Ivan
- 939
73
votes
9 answers
What is the most expensive item I could buy with £50?
I was set the following question during the discrete mathematics module of my degree and despite my instructor explaining his working to me I still disagree with the answer he says is correct.
Can someone please help me either understand where my…
Sam
- 788
73
votes
8 answers
Is linear algebra laying the foundation for something important?
I'm majoring in mathematics and currently enrolled in Linear Algebra. It's very different, but I like it (I think). My question is this: What doors does this course open? (I saw a post about Linear Algebra being the foundation for Applied…
Mallory
- 1,187
73
votes
8 answers
Why does this "miracle method" for matrix inversion work?
Recently, I answered this question about matrix invertibility using a solution technique I called a "miracle method." The question and answer are reproduced below:
Problem: Let $A$ be a matrix satisfying $A^3 = 2I$. Show that $B = A^2 - 2A + 2I$ is…
David Zhang
- 8,835
73
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
- 139,939
73
votes
3 answers
Is $ 0.112123123412345123456\dots $ algebraic or transcendental?
Let $$x=0.112123123412345123456\dots $$
Since the decimal expansion of $x$ is non-terminating and non-repeating, clearly $x$ is an irrational number.
Can it be shown whether $x$ is algebraic or transcendental over $\mathbb{Q}$ ? I think $x$ is…
ASB
- 3,999