Multiplication and division of a pair of real numbers

Question

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!

What if you specify $x\ne y$ and both product and quotient are rational? — Raffaele, Jul 25 '17 at 16:17
@Raffaele No, $x=√12$, $y=√3$. Then $xy= √36 = 6$ and $x/y = √4=2$. — P. Siehr, Jul 25 '17 at 17:34

score 2 · Answer 1 · answered Jul 25 '17 at 09:24

2

No. You could take $x_1=x_2=\sqrt2$.

answered Jul 25 '17 at 09:24

tiefi

395

1

Beat you by 11 seconds ;) – P. Siehr Jul 25 '17 at 09:25
Thank you for the fast response :-D – Marcos Jul 25 '17 at 10:20

score 1 · Accepted Answer · answered Jul 25 '17 at 09:23

1

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*}

answered Jul 25 '17 at 09:23

P. Siehr

3,672

Thank you too for this :-) – Marcos Jul 25 '17 at 10:21

Multiplication and division of a pair of real numbers

2 Answers2