Can Both Be Rational?

Suppose that it is possible. Let $p$ and $q$ be irrational numbers, and let their sum and difference both be rational. Then,

$\begin{aligned} p+q &= \frac{a}{b} \\ p-q &= \frac{c}{d} \end{aligned}$

where $a,$ $b,$ $c,$ and $d$ are integers. Solving for $p:$

$\begin{aligned} 2p &= \frac{a}{b} + \frac{c}{d} \\ 2p &= \frac{ad+bc}{bd} \\ p &= \frac{ad+bc}{2bd} \end{aligned}$

But, $ad+bc$ and $2bd$ are integers. This implies that $p$ is rational, a contradiction . Therefore, it is impossible.