If $x$ is rational, $x\neq 0$, and y is irrational, prove that $x/y$ is irrational.

Question

For $y\neq 0$, it can be easily proved by contradiction. But what if $y=0$ ?

P.S. This is Exercise 7., p. 28 of Apostol's Calculus book, (vol. 1, second edition).

Answer 1

$0$ is not irrational since $0=\frac{0}{1}$ and $0,1 \in \mathbb{Z}$.

Hence $y \neq 0$.

Answer 2

I strongly believe that $y$ was implied to be non zero. Division by zero makes no sense here. Best you can do is saying $\frac {x}{0}=\infty$ but that is no real number and the irrational numbers are the real numbers that are not rational, hence it would be not irrational but also not rational. Also doesn't work in complex numbers, hyperreal numbers and probably not in surreal numbers either.