# 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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