for a rational number $a$ and an irrational number $b$ , if $ab$ is rational or $\frac{a}{b}$ is rational can we say that $a=0$?

Question

I guess product or division of a rational and an irrational number can be rational and that also only $0$, when the rational number is $0$. In all other cases it will be irrational . Please correct me if I am wrong .

Answer 1

Yes, here's a proof :

Assume $a \neq 0$, then for any $b \neq 0$ we have

• if $c:= a \times b$ is rational, then $b = \frac{c}{a}$ is rational.
• if $d:= \frac{a}{b}$ is rational = , then $b = (\frac{d}{a})^{-1}$ is rational.

Your claim is simply the contraposition of the above.