Closure of the rationals
Theorem 5.1 (Closure of the rationals). If \(p, q \in \Q \), then \(p + q\), \(p - q\) and \(pq\) are rational, and so is \(p / q\) when \(q \neq 0\).
Proof. Write \(p = a/b\) and \(q = c/d\) with \(a, b, c, d \in \Z \) and \(b, d \neq 0\). Then
\[ p \pm q = \frac {ad \pm bc}{bd}, \qquad pq = \frac {ac}{bd}, \qquad \frac pq = \frac {ad}{bc} \quad (c \neq 0), \]
and in each case the numerator and denominator are integers, with a nonzero denominator.