Does $\pi \ | \ 2 \pi$

Question

Does $\pi$ divide $2 \pi?$

Clearly $\frac{2 \pi}{\pi}=2$ and 2 is an integer, so it would seem to make sense to say that $\pi \ | \ 2 \pi$.

Does it make sense to write, for example, $$\pi \ | \ x \implies \sin(x)=0?$$

Answer 1

This isn't completely formal, as usually divisibility is only defined on the integers. However, if we define divisibility on the reals, or say a ring extension of $\mathbb{Z}$, we can use the generalized definition $$ a\mid b \Longleftrightarrow \exists c\in \mathbb{Z}\text{ such that } b=ac $$ In which case your assertion would be correct, taking $c=2$.

Answer 2

Yes, it makes sense. It is not widely used, but it is clear and useful.