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1500 questions
72
votes
7 answers
How to find the factorial of a fraction?
From what I know, the factorial function is defined as follows:
$$n! = n(n-1)(n-2) \cdots(3)(2)(1)$$
And $0! = 1$. However, this page seems to be saying that you can take the factorial of a fraction, like, for instance, $\frac{1}{2}!$, which they…
Cisplatin
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72
votes
7 answers
Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers?
I self-realized an interesting property today that all numbers $(a,b)$ belonging to the infinite set $$\{(a,b): a=(2l+1)^2, b=(2k+1)^2;\ l,k \in N;\ l,k\geq1\}$$ have their AM and GM both integers.
Now I wonder if there exist distinct real numbers…
Gaurang Tandon
- 6,665
72
votes
2 answers
Is there an intuitive reason for a certain operation to be associative?
When I read Pinter's A Book of Abstract Algebra, Exercise 7 on page 25 asks whether the operation
$$x*y=\frac{xy}{x+y+1}$$
(defined on the positive real numbers) is associative. At first I considered this to be false, because the expression is so…
Zirui Wang
- 1,547
72
votes
12 answers
What is the largest eigenvalue of the following matrix?
Find the largest eigenvalue of the following matrix
$$\begin{bmatrix}
1 & 4 & 16\\
4 & 16 & 1\\
16 & 1 & 4
\end{bmatrix}$$
This matrix is symmetric and, thus, the eigenvalues are real. I solved for the possible eigenvalues and,…
Cloud JR K
- 2,536
72
votes
11 answers
How can I find the points at which two circles intersect?
Given the radius and $x,y$ coordinates of the center point of two circles how can I calculate their points of intersection if they have any?
Joe Elder
- 729
72
votes
2 answers
Let, $A\subset\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \setminus \{p\}$.
A challenge problem from Sally's Fundamentals of Mathematical Analysis.
Problem reads: Suppose $A$ is a subset of $\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \setminus \{p\}$ with the usual…
David Bowman
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72
votes
18 answers
Why is radian so common in maths?
I have learned about the correspondence of radians and degrees so 360° degrees equals $2\pi$ radians. Now we mostly use radians (integrals and so on)
My question: Is it just mathematical convention that radians are much more used in higher maths…
Slater
- 795
72
votes
3 answers
Why is the Daniell integral not so popular?
The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some serious flaws.
The most natural way to fix all the…
gifty
- 2,311
72
votes
6 answers
Fourier transform of function composition
Given two functions $f$ and $g$, is there a formula for the Fourier transform of $f \circ g$ in terms of the Fourier transforms of $f$ and $g$ individually?
I know you can do this for the sum, the product and the convolution of two functions. But I…
MathematicalOrchid
- 6,365
72
votes
14 answers
Are proofs by contradiction really logical?
Let's say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction.
My question is this: How does one go from "so there's a contradiction if we don't have $A$" to concluding that "we have $A$"?
That, to me, seems the…
Simp
- 735
72
votes
11 answers
Have there been efforts to introduce non Greek or Latin alphabets into mathematics?
As a physics student, often I find when doing blackboard problems, the lecturer will struggle to find a good variable name for a variable e.g. "Oh, I cannot use B for this matrix, that's the magnetic field".
Even ignoring the many letters used for…
Mark Allen
- 833
72
votes
4 answers
Difference between supremum and maximum
Referring to this lecture , I want to know what is the difference between supremum and maximum. It looks same as far as the lecture is concerned when it explains pointwise supremum and pointwise maximum
user31820
- 1,033
72
votes
7 answers
Why $9$ & $11$ are special in divisibility tests using decimal digit sums? (casting out nines & elevens)
I don't know if this is a well-known fact, but I have observed that every number, no matter how large, that is equally divided by $9$, will equal $9$ if you add all the numbers it is made from until there is $1$ digit.
A quick example of what I…
JD Isaacks
- 883
72
votes
2 answers
Continuous functions do not necessarily map closed sets to closed sets
I found this comment in my lecture notes, and it struck me because up until now I simply assumed that continuous functions map closed sets to closed sets.
What are some insightful examples of continuous functions that map closed sets to non-closed…
Aaa
- 862
72
votes
3 answers
How and why does Grothendieck's work provide tools to attack problems in number theory?
This is probably a horrible question to experts, but I think it is reasonable from someone who knows nothing.
I have always been fascinated with Grothendieck and the way he did mathematics.
I've heard Mochizuki's work on the abc conjecture heavily…
Arrow
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