Rational and irrational
Theorem 6.3 (Rational and irrational). Let \(q \in \Q \) and let \(x \in \R \setminus \Q \). Then \(q + x\) is irrational, and so is \(qx\) if \(q \neq 0\).
Proof. If \(q + x = p\) were rational, then \(x = p - q\) would be rational by Theorem 5.1. If \(qx = p\) were rational with \(q \neq 0\), then \(x = p / q\) would be rational. Both contradict \(x \notin \Q \).