Question

Suppose a is rational and b is irrational. Show that a+b is irrational.

Answer 1

Assume that $a+b=r$ is a rational number. Since $r$ and $a$ are rational numbers, we obtain that $r-a$ is a rational number. We obtain a contradiction, because $b=r-a$ is not a rational number.

So our assumption that $a+b\in\mathbb Q$ is false, that is $a+b$.is a irrational number.