Answer
See proof.
Work Step by Step
Suppose by contradiction that the sum of an irrational number and a rational number is rational. That is, $\dfrac{p}{q}+r=\dfrac{s}{t}$ where $p,q,s,t$ are integers and $r$ is irrational. It follows that $r=\dfrac{s}{t}-\dfrac{p}{q}$. Finding a common denominator we get that $r=\dfrac{sq-pt}{tq}$ where $sq-pt$ and $tq$ are integers. Thus, $r$ must be a rational number but this contradicts the fact that $r$ is irrational. Therefore, the sum of a rational number and an irrational number is irrational.
