For any irrational number $r$, $r^2$ is irrational

Question

Is the following statement true of false?

For any irrational number $r$, $r^2$ is irrational.

How can I prove this? I'm confused. Can I use $r=\frac{a}{b}$?

Answer 1

It's false; $\sqrt{2}$ is irrational and $\sqrt{2}^2=2$ is not.

Answer 2

HINT: It may be easier to consider the contrapositive.

The statement you're looking at is true if and only if the following statement is true: $$\mbox{Every rational number has a rational square root.}$$

(i) Do you see why these two statements are equivalent?

(ii) Can you think of a rational number with an irrational square root?