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1500 questions
75
votes
4 answers
Does every set have a group structure?
I know that there is no vector space having precisely $6$ elements. Does every set have a group structure?
user23505
74
votes
7 answers
What’s the difference between analytical and numerical approaches to problems?
I don't have much (good) math education beyond some basic university-level calculus.
What do "analytical" and "numerical" mean? How are they different?
jbrennan
- 859
74
votes
5 answers
A new imaginary number? $x^c = -x$
Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would allow solving for logs with negative bases or with…
Warren L.
- 853
74
votes
3 answers
Path-connected and locally connected space that is not locally path-connected
I'm trying to classify the various topological concepts about connectedness. According to 3 assertions ((Locally) path-connectedness implies (locally) connectedness. Connectedness together with locally path-connectedness implies…
Qian
- 1,167
74
votes
2 answers
Integration with respect to counting measure.
I am having trouble computing integration w.r.t. counting measure. Let $(\mathbb{N},\scr{P}(\mathbb{N}),\mu)$ be a measure space where $\mu$ is counting measure. Let $f:\mathbb{N}\rightarrow{\mathbb{R}}$ be a non-negative bounded measurable…
user54992
74
votes
3 answers
How likely is it not to be anyone's best friend?
A teenage acquaintance of mine lamented:
Every one of my friends is better friends with somebody else.
Thanks to my knowledge of mathematics I could inform her that she's not alone and $e^{-1}\approx 37\%$ of all people could be expected to be in…
hmakholm left over Monica
- 291,611
74
votes
11 answers
Is Euclid's proof on the infinitude of primes flawed because it yields some composites?
I've expressed Euclid's proof on the infinitude of primes on Mathematica:
f[x_] := Product[Prime[n], {n, 1, x}] + 1
TableForm[Table[{f[x], PrimeQ[f[x]]}, {x, 1, 20}]]
Which results in:
$\begin{array}{ll}
3 & \text{True} \\
7 & \text{True} \\
31…
Red Banana
- 24,885
- 21
- 101
- 207
74
votes
5 answers
Integers $n$ such that $i(i+1)(i+2) \cdots (i+n)$ is real or pure imaginary
A couple of days ago I happened to come across [1], where the curious fact that $i(i-1)(i-2)(i-3)=-10$ appears ($i$ is the imaginary unit). This led me to the following question:
Problem 1: Is $3$ the only positive integer value of $n$ such that…
Dave L. Renfro
- 38,688
74
votes
6 answers
Probability for the length of the longest run in $n$ Bernoulli trials
Suppose a biased coin (probability of head being $p$) was flipped $n$ times. I would like to find the probability that the length of the longest run of heads, say $\ell_n$, exceeds a given number $m$, i.e. $\mathbb{P}(\ell_n > m)$.
It suffices to…
Sasha
- 71,686
74
votes
2 answers
coincidental (?) patterns in logs of repeating decimals, e.g. $\ln(2/3)$ vs. $\ln(0.6666666)$
I was playing with logarithms under an arbitrary-precision calculator, and got some odd results. I happened to have the precision set to 20 places, and these are the initial results I got:
ln 0.6666666 = -0.40546520810816938198
ln 2/3 =…
Steve Summit
- 759
74
votes
3 answers
Geometry question about a six-pack of beer
On a hot summer day like today, I like to put a six-pack of beer in my cooler and enjoy some cold ones outdoors.
My cooler is in the shape of a cylinder. When I place the six-pack in the cooler against the wall, with three beer cans touching the…
Dan
- 35,053
74
votes
8 answers
How can one prove that $e<\pi$?
This question is inspired by another one, asking to prove that something approximately equal to $1.2$ is bigger than something approximately equal to $0.9$. The numerical answer to this question was (expectedly) downvoted, though in my opinion it is…
Start wearing purple
- 54,260
74
votes
9 answers
Introductory texts on manifolds
I was studying some hyperbolic geometry previously and realised that I needed to understand things in a more general setting in terms of a "manifold" which I don't yet know of.
I was wondering if someone can recommend to me some introductory texts…
user38268
74
votes
13 answers
What would have been our number system if humans had more than 10 fingers? Try to solve this puzzle.
Try to solve this puzzle:
The first expedition to Mars found only the ruins of a civilization.
From the artifacts and pictures, the explorers deduced that the
creatures who produced this civilization were four-legged beings with
a tentatcle…
Vishnu Vivek
- 1,441
74
votes
4 answers
How to cut a cube out of a tree stump, such that a pair of opposing vertices are in the center?
I saw this picture of a cube cut out of a tree stump.
I've been trying to craft the same thing out of a tree stump, but I found it hard to figure out how to do it.
One of the opposing vertices pair is on the center of the tree stump:
I've been…
Aaron Cheuk
- 711