Show that the product of two irrational numbers may be irrational. You may use any facts you know about the real numbers.
All we know is that $\sqrt{2}$ is irrational and that $\sqrt{2}\cdot \sqrt{2} = 2$; but this is a rational product of irrational numbers.
Best Answer
Well, if all you know is that $\sqrt{2}$ is irrational, try the pair of $\sqrt{2}$ and $\sqrt{2}+1$ - both of which are clearly irrational, and their product is $2+\sqrt{2}$, which is also clearly irrational. Then we don't have to know anything other than that $\sqrt{2}$ is irrational and an irrational plus a rational is still irrational.