For two numbers $a,b$, if $a,b\in\mathbb{Q}$, then $ab\in\mathbb{Q}$, but if $a,b\in\mathbb{I}$, then $ab\in\mathbb{I}$ OR $ab\in\mathbb{Q}$, depending on the particular values of $a$ and $b$, where $\mathbb{I}=\mathbb{R}- \mathbb{Q}$. (For instance, $\sqrt2\cdot\sqrt2=2$, but $\sqrt2\cdot\sqrt3=\sqrt6\not\in\mathbb{Q}$.)
Is it possible to pick nontrivial values for $a\in\mathbb{Q}$ and $b\in\mathbb{I}$ (read: not $0$), such that $ab\in\mathbb{Q}$, or will it always be the case that $ab\in\mathbb{I}$? If it can be a rational number, can it be demonstrated whether such examples only exist among non-transcendental numbers or whether they exist even among transcendental ones?
Assuming $a \neq 0,$ if $a \in \Bbb Q, b \in \Bbb I,$ then $ab \in \Bbb I$. Let $ab=c$ and assume $c \in \Bbb Q$. Then $\frac ca=b$ must be in $\Bbb Q$ because the rationals are closed under multiplication and inverses.