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 .

score 1 · Answer 1 · answered Jan 18 '20 at 07:57

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.

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

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