# Check if this proof about real numbers with an irrational product is correct.

Can anyone confirm if my proof is correct, please?

Claim:- “If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.”

Proof:-

Assume that both $x$ and $y$ are rational. Now, let $x = \dfrac pq$ and $y = \dfrac mn$ since both of them are rational. $xy =\dfrac pq * \dfrac mn = \dfrac{pm}{qn}$ Thus, if the product $xy$ can be written as a fraction, it's not a irrational number. Therefore if one of $x$ and $y$ is not irrational, then the product is not irrational.

By the principle of proof by contraposition, If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.

• Please do not deface your questions. – user642796 Oct 1 '13 at 13:18

I would specify that $p,q,m,n\in\mathbb{Z},q\neq0,n\neq0$, in your first line of the proof.
Looks good to me. ${}{}{}{}{}{}{}{}$