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1500 questions
103
votes
5 answers
If $x$, $y$, $x+y$, and $x-y$ are prime numbers, what is their sum?
Suppose that $x$, $y$, $x−y$, and $x+y$ are all positive prime numbers. What is the sum of the four numbers?
Well, I just guessed some values and I got the answer.
$x=5$, $y=2$, $x-y=3$, $x+y=7$. All the numbers are prime and the answer is…
chndn
- 2,863
103
votes
4 answers
Importance of matrix rank
What is the importance of the rank of a matrix? I know that the rank of a matrix is the number of linearly independent rows or columns (whichever is smaller).
Why is it a problem if a matrix is rank deficient?
Also, why is the smaller value between…
0x0
- 2,641
103
votes
4 answers
A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
As the title says, I would like to launch a community project for
proving that the series $$\sum_{n\geq 1}\frac{\sin(2^n)}{n}$$ is
convergent.
An extensive list of considerations follows. The first fact is that the inequality
$$…
Jack D'Aurizio
- 353,855
103
votes
18 answers
Is arrow notation for vectors "not mathematically mature"?
Assuming that we can't bold our variables (say, we're writing math as opposed to typing it), is it "not mathematically mature" to put an arrow over a vector?
I ask this because in my linear algebra class, my professor never used arrow notation, so…
hlin117
- 1,171
102
votes
5 answers
True or false? $x^2\ne x\implies x\ne 1$
Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following:
$$x^2\ne x\implies x\ne 1$$
I immediately answered true, but for some reason, everyone (including…
Chris
- 1,041
102
votes
2 answers
Help me put these enormous numbers in order: googol, googol-plex-bang, googol-stack and so on
Popular mathematics folklore provides some simple tools
enabling us compactly to describe some truly enormous
numbers. For example, the number $10^{100}$ is commonly
known as a googol,
and a googol
plex is
$10^{10^{100}}$. For any number $x$, we…
JDH
- 44,236
102
votes
1 answer
In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?
I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their double duals; they are isomorphic to their duals as…
Ben Blum-Smith
- 20,766
102
votes
14 answers
Why can't you square both sides of an equation?
Why can't you square both sides of an equation?
I've been asked this many times and can never quite give a good, clear, concise answer (for beginning algebra students) in plain language. I just searched the web and still couldn't find a…
Jeff
- 3,405
102
votes
2 answers
What's the difference between simple induction and strong induction?
I just started to learn induction in my first year course. I'm having a difficult time grasping the concept. I believe I understand the basics but could someone summarize simple induction and strong induction and explain what the differences are?…
Jake Park
- 1,277
102
votes
7 answers
What kind of "symmetry" is the symmetric group about?
There are two concepts which are very similar literally in abstract algebra: symmetric group and symmetry group. By definition, the symmetric group on a set is the group consisting of all bijections of the set (all one-to-one and onto functions)…
user9464
102
votes
19 answers
Good Physical Demonstrations of Abstract Mathematics
I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to use.
I'm looking for nontrivial ideas in abstract…
Jamie Banks
- 12,942
102
votes
2 answers
What is the solution to Nash's problem presented in "A Beautiful Mind"?
I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve (obviously, an exaggeration). Nonetheless, one…
Mlagma
- 1,677
102
votes
7 answers
Mathematicians' Tensors vs. Physicists' Tensors
It seems, at times, that physicists and mathematicians mean different things when they say the word "tensor." From my perspective, when I say tensor, I mean "an element of a tensor product of vector spaces."
For instance, here is a segment about…
msm
- 2,734
102
votes
4 answers
How is a group made up of simple groups?
I've read more than once the analogy between simple groups and prime numbers, stating that any group is built up from simple groups, like any number is built from prime numbers.
I've recently started self-studying subgroup series, which is supposed…
Bruno Stonek
- 12,527
102
votes
17 answers
How do you find the center of a circle with a pencil and a book?
Given a circle on a paper, and a pencil and a book. Can you find the center of the circle with the pencil and the book?
zdd
- 1,053