I would like to prove the conditional statement "If a real number $x$ is rational, then its square is rational", and was wondering if this is the correct approach.
Let $x$ be a rational, real number, $\frac{a}{b}$, where $a,b\in \mathbb{Z}$ and $b \neq 0$.
Let us square $x$, giving us $x^2 = (\frac{a}{b})^2 = \frac{a^2}{b^2} = \frac{a\cdot a}{b \cdot b}$.
We know that an integer multiplied by another integer results in an integer, so $a^2 = a\cdot a\in \mathbb{Z}$ and $b^2 = b\cdot b\in\mathbb{Z}$
Because $x^2$ can be rewritten as the ratio between two integers, $a^2$ and $b^2$, we have thus shown that given a real, rational number $x$, its square is also rational.
