Suppose that you have any two real numbers: $x_1$ and $x_2$ (different from $0$).
If we know that the product $x1x2$ is rational and the division $\frac{x_1}{x_2}$ is also rational, is it possible to show that $x_1$ and $x_2$ are rational as well?
Many thanks!
No. Here is a counter example: $$x_1=x_2 = √{2}$$
with
\begin{align*} x_1x_2 &= 2 \\ \frac{x_1}{x_2} &= 1 \end{align*}