Field struc­ture of ra­tio­nal numbers

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

• Addition is given by $\frac{a}{b}+\frac{p}{q}=\frac{aq+bp}{bq}$

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

• 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 $a\ne 0$) 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 $0<\frac{c}{d}-\frac{a}{b}$.