Most Popular
1500 questions
81
votes
2 answers
Inscribing square in circle in just seven compass-and-straightedge steps
Problem Here is one of the challenges posed on Euclidea, a mobile app for Euclidean constructions: Given a $\circ O$ centered on point $O$ with a point $A$ on it, inscribe $\square{ABCD}$ within the circle — in just seven elementary steps . Euclidea…
PDE
- 1,487
81
votes
6 answers
What's so special about characteristic 2?
I've often read about things which do not work in a field with a characteristic $2$ , mainly things which have to do with factoring, or similar things. I'm not exactly sure why, but the only example of such a field I could think of is…
Juan Sebastian Lozano
- 2,862
81
votes
20 answers
Proof of the hockey stick/Zhu Shijie identity $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$
After reading this question , the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1},$$ or, what is equivalent, $$\sum_{t=k}^n \binom{t}{k} = \binom{n+1}{k+1}.$$ What's the name of this identity? Is it the identity…
hlapointe
- 1,680
80
votes
6 answers
Matrix is conjugate to its own transpose
Mariano mentioned somewhere that everyone should prove once in their life that every matrix is conjugate to its transpose. I spent quite a bit of time on it now, and still could not prove it. At the risk of devaluing myself, might I ask someone else…
George
- 2,067
80
votes
14 answers
Good books on mathematical logic?
I just started to learn mathematical logic. I'm a graduate student. I need a book with relatively more examples. Any recommendation?
ciciplus
80
votes
10 answers
Examples of finite nonabelian groups.
Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
kalpeshmpopat
- 3,382
80
votes
7 answers
Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis
I am studying for an exam and I have been studying my butt off during the winter break for it. During the course of my study I have written down quite a number of tricks, which in my opinion were 'outrageous' :-). Meaning there was no way I would…
Cousin
- 3,625
80
votes
8 answers
Koch snowflake paradox: finite area, but infinite perimeter
The Koch snowflake has finite area, but infinite perimeter, right? So if we make this snowflake have some thickness (like a cake or something), then it appears that you can fill it with paint like this ($\text{finite area} \times \text{thickness}…
ArtisanRtasin
- 743
80
votes
4 answers
"Closed" form for $\sum \frac{1}{n^n}$
Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating series etc. One series that popped into our…
user17762
80
votes
5 answers
Drunk man with a set of keys.
I found this problem in a contest of years ago, but I'm not very good at probability, so I prefer to see how you do it: A man gets drunk half of the days of a month. To open his house, he has a set of keys with $5$ keys that are all very similar,…
iam_agf
- 5,538
80
votes
7 answers
Is "The empty set is a subset of any set" a convention?
Recently, I learned that for any set $A$ , we have $\varnothing \subset A$ . I found some explanation of why it holds. $\varnothing\subset A$ means "for every object $x$ , if $x$ belongs to the empty set, then $x$ also belongs to the set $A$ ". This…
Searene
- 1,021
- 1
- 8
- 8
80
votes
8 answers
Why do books titled "Abstract Algebra" mostly deal with groups/rings/fields?
As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I've been looking through some books on the topic, and most seem to 'only' cover groups, rings and fields. Why is…
GeorgW
- 809
80
votes
11 answers
Highest power of a prime $p$ dividing $N!$
How does one find the highest power of a prime $p$ that divides $N!$ and other related products? Related question: How many zeros are there at the end of $N!$? This is being done to reduce abstract duplicates. See Coping with *abstract* duplicate…
user17762
80
votes
7 answers
Functions that are their own inverse.
What are the functions that are their own inverse? (thus functions where $ f(f(x)) = x $ for a large domain) I always thought there were only 4: $f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ f(x) = \frac {-1}{x} $ Later I heard about a fifth…
Willemien
- 6,830
80
votes
4 answers
Is the product of symmetric positive semidefinite matrices positive definite?
I see on Wikipedia that the product of two commuting symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices? My proof of the positive definite case falls apart…
nullUser
- 28,933