Definition 1. The field of rational numbers is the fraction field of $\Z$:
\[\Q = \fract \Z.\]Definition 2. $\Z$ is embedded into $\Q$ by
\[z \mapsto \frac{z}{1}.\]Then we can say
\[\N \subseteq \Z \subseteq \Q.\]Definition 3. The following defines a total order on $\Q$:
\[\frac{a}{b} \leq \frac{x}{y}\]if and only if
\[(by > 0 \,\land\, ay \leq bx) \enspace\lor\enspace (by < 0 \,\land\, ay \geq bx).\]Definition 4. The set of non-zero rational numbers is
\[\Q^* = \Q \setdif \{0\}.\]Definition 5. The set of positive rational numbers is
\[\Q^+ = \{q \in \Q : q > 0\}.\]Definition 6. The set of negative rational numbers is
\[\Q^- = \{q \in \Q : q < 0\}.\]