The rational numbers, being the field_of_fractions of the integers, have the following field structure:

Addition is given by

Multiplication is given by $\frac{a}{b} \frac{c}{d} = \frac{a c}{b d}$

The identity under addition is $\frac{0}{1}$

The identity under multiplication is $\frac{1}{1}$

The additive inverse of $\frac{a}{b}$ is $\frac{- a}{b}$

The multiplicative inverse of $\frac{a}{b}$ (where $a \neq 0$ ) is $\frac{b}{a}$ .

It additionally inherits a total ordering which respects the field structure: $0 < \frac{c}{d}$ if and only if $c$ and $d$ are both positive or $c$ and $d$ are both negative. All other information about the ordering can be derived from this fact: $\frac{a}{b} < \frac{c}{d}$ if and only if $0 < \frac{c}{d} - \frac{a}{b}$ .