Let $\mathbb N$ be the set of positive integers, and let $f:\mathbb N\to\mathbb N$ be a strictly increasing function such that $f(2)=2$ and $f(m)f(n)=f(mn)$ for all positive integers $m,n$ such that $\gcd(m,n)=1$. Find all such functions $f$.
Watered down version:
Let $\mathbb N$ be the set of positive integers, and let $f:\mathbb N\to\...
I mainly teach physics, only a little math on the side. If I was teaching this math topic and wanted to spend 3 minutes giving the appropriate physical motivation, I would do something like the following.
There is something called energy. It comes in various forms. Food has energy, which is what we're talking about when we talk about calories. Heat, sound, ...
I would start with the scalar. Work = force times distance. Starting with the scalar makes things easier, more intuitive. (Not a math logic point, just a psychological one.) You can then easily generalize, perhaps without showing a derivation and say "well this is what work is in these trickier situations (vectors, non constant force, more dimensions, ...
The best way to illustrate the sense of $W=|F||AB|\cos(\alpha)$ is to show that it makes sense that each of the factors is proportional to the work done.
If you double the amount of force to an object to move it a certain distance, it doubles the amount of work needed.
If you double the distance to move an object by applying a certain force, it doubles the ...
Clearly there is no historical data that addresses this question
I want to know if there are any numerical bases that are notably
well-suited for humans to learn and use at an elementary or
since we have ten fingers and humans have learned only decimal arithmetic for everyday use.
I just finished four weekly sessions with fifth ...
Here is another one. Prove that the power of $13$ can be writen as a sum of two squares.
I will give two profs of it. First one is more involved and includes lemma $$(a^2+b^2)(x^2+y^2)= (ax+by)^2+(bx-ay)^2$$ yet second takes step 2 and it is much more elegant.
Base: $n=1$, then $13 = 2^2+3^2$ and we are done.
We know that $13^n = a^2+b^2$ ...
A simple consequnce of:
Postage Stamp Problem, which states that for any two relatively prime positive integers $m,n$, the greatest integer that cannot be written in the form $am + bn$ for nonnegative integers $a, b$ is $mn-m-n$,
is that every natural number greater or equal to $mn-m-n+1$ can be writen in a form $am+bn$ for some $a,b$.
And for some ...
I suspect that you might get more informed answers from the History of Science and Mathematics SE. You're sure to find more people who are more learned about the relationship between Hardy and Ramanujan than I am.
That said, I suspect that Hardy is talking about Ramanujan's relative inability to contribute to the mathematical progress of his day because ...