We know that $\pi$ is an irrational number, which has infinite number of digits. At the other side we know that the set of Natural numbers($\mathbb{N}$), is an infinite set, I wanted to know if ever numbers of $N$ ever become devisable by $\pi$ number?! Or in the other words, does $sin(n)$ ever meet $zero$?
p.s. This questions came from another question where it wanted to ask is $n \in O(|n^2sin(n)|)$ ?
I will appreciate you if you also provide a proof with your answer.
Let's assume there is a $n \in \mathbb{N}*$ so that $sin(n) = 0$, then $\exists k \in\mathbb{N}*$, so that $n = 0 +k\pi$, then $\pi = \frac{n}{k}$, which would mean $\pi \in \mathbb{Q}$ which is absurd. In other word there is no such $n$. Hope this answers your question.