The rational numbers $\mathbb{Q}$ are the fraction field of the ring $\mathbb{Z}$ of integers. In more elementary terms, a rational number is a quotient $a/b$ of two integers $a$ and $b$ where $b$ is nonzero. Two fractions $a/b$ and $c/d$ are equivalent if the product of the cross terms is equal: $$ \frac{a}{b} = \frac{c}{d} \iff ad = bc $$ Addition and multiplication of fractions are given by the formulae \begin{eqnarray*} \frac{a}{b} + \frac{c}{d} & = & \frac{ad + bc}{bd} \\ \frac{a}{b} \cdot \frac{c}{d} & = & \frac{ac}{bd} \end{eqnarray*} The field of rational numbers is an ordered field, under the ordering relation $\leq$ defined as follows: $a/b \leq c/d$ if

1. the inequality $a\cdot d \leq b \cdot c$ holds in the integers, and $b$ has the same sign as $d$ or
2. the inequality $a\cdot d \geq b \cdot c$ holds in the integers, and $b$ has the opposite sign as $d$