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1500 questions
85
votes
6 answers
Why is a full turn of the circle 360°? Why not any other number?
I was just wondering why we have 90° degrees for a perpendicular angle. Why not 100° or any other number?
What is the significance of 90° for the perpendicular or 360° for a circle?
I didn't ever think about this during my school time.
Can someone…
P K
- 1,121
85
votes
0 answers
Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles?
The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a known regular way to do this? Just playing and not…
2'5 9'2
- 54,717
85
votes
5 answers
Limit of the nested radical $x_{n+1} = \sqrt{c+x_n}$
(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)
For $c \gt 0$, consider the quadratic equation
$x^2 - x - c = 0, x > 0$.
Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been…
cnuulhu
- 995
85
votes
1 answer
Direct proof that $\pi$ is not constructible
Is there a direct proof that $\pi$ is not constructible, that is, that squaring the circle cannot be done by rule and compass?
Of course, $\pi$ is not constructible because it is transcendental and so is not a root of any polynomial with rational…
lhf
- 216,483
84
votes
1 answer
Can one deduce Liouville's theorem (in complex analysis) from the non-emptiness of spectra in complex Banach algebras?
As you probably know, the classical proof of the non-emptiness of the spectrum for an element $x$ in a general Banach algebra over $\mathbb{C}$ can be proven quite easily using Liouville's theorem in complex analysis: every bounded, entire function…
Mark
- 5,824
84
votes
2 answers
Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?
I thought it was easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational, but found a gap in my proof. Simple finite approximations show the denominator cannot be small, though, strongly suggesting irrationality. However, can it be…
user2566092
- 26,142
84
votes
5 answers
Math behind rotation in MS Paint
For those who don't know, MS Paint only has the options to rotate an image by right angles. To carry out an arbitrary rotation ($\theta^\circ$), the following hack is suggested:
Horizontal skew by $\theta$
Vertical Stretch by $\displaystyle…
kuch nahi
- 6,789
84
votes
6 answers
How deep is the liquid in a half-full hemisphere?
I have a baking recipe that calls for $1/2$ tsp of vanilla extract, but I only have a $1$ tsp measuring spoon available, since the dishwasher is running. The measuring spoon is very nearly a perfect hemisphere.
My question is, to what depth (as a…
Holly
- 913
84
votes
1 answer
Conjectured formula for the Fabius function
The Fabius function is the unique function ${\bf F}:\mathbb R\to[-1, 1]$ satisfying the following conditions:
a functional–integral equation$\require{action}
\require{enclose}{^{\texttip{\dagger}{a poet or philosopher could say "it knows and…
Vladimir Reshetnikov
- 47,122
84
votes
3 answers
Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
Consider the sequence defined as
$x_1 = 1$
$x_{n+1} = \sin x_n$
I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't too happy about.
(I think I did this by showing that…
Aryabhata
- 82,206
84
votes
0 answers
Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?
Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. Is it true, that for any group word $w$ there…
Chain Markov
- 15,564
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- 116
84
votes
14 answers
How to effectively and efficiently learn mathematics
How do you effectively study mathematics? How does one read a maths book instead or just staring at it for hours?
(Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd…
user7273
84
votes
6 answers
20 circles in the plane, all passing through the origin
Suppose I draw $20$ circles in the plane, all passing through the origin, but no two tangent at the origin. Also, except for the origin, no three circles pass through a common point. How many regions are created in the plane?
laser01
- 1,180
84
votes
5 answers
Are all infinities equal?
A friend of mine was trying to explain to me how all infinities are equal. For example, they were saying that there are the same amount of numbers between $0$–$1$ as there are between $0$–$2$.
The way they explained it, you could prove that there…
Ephraim
- 1,878
84
votes
8 answers
Equivalent Definitions of the Operator Norm
How do you prove that these four definitions of the operator norm are equivalent?
$$\begin{align*}
\lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\
&=\sup\{ \lVert Av\rVert\;\colon\;…
KiaSure
- 911