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