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 \(0 < \frac{c}{d} - \frac{a}{b}\).