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

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