\begin{definition}[Definition:Addition/Rational Numbers]
The addition operation in the domain of rational numbers $\Q$ is written $+$.
Let:
:$a = \dfrac p q, b = \dfrac r s$
where:
:$p, q \in \Z$
:$r, s \in \Z_{\ne 0}$
Then $a + b$ is defined as:
:$\dfrac p q + \dfrac r s = \dfrac {p s + r q} {q s}$
This definition follows from the definition of and proof of existence of the field of quotients of any integral domain, of which the set of integers is an example.
\end{definition}