Field structure of rational numbers

by Patrick Stevens Jul 3 2016 updated Jul 29 2016

In which we describe the field structure on the rationals.

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

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 $~$0 < \frac{c}{d} - \frac{a}{b}$~$.