# 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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