**Proof** Suppose $\frac{p}{q} \in \mathbb{Q}$ and $x \not \in \mathbb{Q}$ BWOC if $x+\frac{p}{q}=\frac{p'}{q'} \text{ then } x= \frac{p'q-pq'}{qq'} \in \mathbb{Q}$contradicting the fact that $x$ is irrational. Similarly if $\frac{xp}{q}-\frac{p'}{q'} \implies x=\frac{p'q}{pq'} \in \mathbb{Q}.$