Let $x$ be a rational number. Prove that if the product $xy$ is irrational, then $y$ is irrational.

Question

Let $x$ be a rational number. Prove that if the product $xy$ is irrational, then $y$ is irrational.

Suppose $x$, $y$ are both rational numbers. The product of two rational numbers will always yield another rational number, thus it cant be the case that $xy$ is irrational. If $y$ is rational then $xy$ is rational. Therefore if $xy$ is irrational then $y$ is irrational.

Is it insufficient to just say the product of two rational numbers is also rational? If it is sufficient how can I argue that it is?

Answer 1

Since $x, y$ are rational, let $x=\frac{a}{b}, y=\frac{c}{d}$. Then, $xy=\frac{ac}{bd}$. By the definition of rational numbers, $xy$ is rational. And hence, we are done. Usually, this result is trivial.