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 \not = 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 $~$0 < \frac{c}{d} - \frac{a}{b}$~$.