What areas of math are related to both $\pi$ and $\varphi$?
There are at least two answers.
Puzzling Stack Exchange is a question and answer site for those who create, solve, and study puzzles. It only takes a minute to sign up.
Sign up to join this communityWhat areas of math are related to both $\pi$ and $\varphi$?
There are at least two answers.
Number theory,
because
in number theory, $\pi(n)$ denotes the prime-counting function (the number of primes $\le n$), and $\varphi(n)$ denotes the Euler totient function (the number of integers between $1$ and $n-1$ that are relatively prime to $n$).
I hope I'm missing the point, but I'll just answer this like it was a normal question:
Geometry and Number Theory
Because
Pi and the Golden Ratio show up in both Geometry and Number Theory. Though I feel like if this is a valid answer, there are dozens of other valid answers
The area of a circle with radius $\phi$ is $\pi \phi^2$, which is very obviously related to both numbers.
In algebra (specfically polynomials) , the $n$-th cyclotomic polynomial is defined as $$\Phi_n(x) = \prod_{\stackrel{1\leq k \leq n}{\gcd(k,n)=1}} \left(x-e^{2i\pi\frac{k}{n}}\right)$$
Also,
In probability theory, the probability density function of the normal distribution. $$\phi(x) = (2\pi)^{-1/2}\cdot e^{-x^2/2}$$
Please see that $\varphi$ is the same as $ϕ$ for both the above mentioned functions.
Are they both
Transcendental
numbers?