Field structure of rational numbers

https://arbital.com/p/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:

• 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 $0 < \frac{c}{d} - \frac{a}{b}$.