Editing Field (mathematics) (section)

=== Rational numbers ===
{{Main|Rational number}}
Rational numbers have been widely used a long time before the elaboration of the concept of field.
They are numbers that can be written as [[fraction (mathematics)|fractions]]
{{math|''a''/''b''}}, where {{math|''a''}} and {{math|''b''}} are [[integer]]s, and {{math|''b'' ≠ 0}}. The additive inverse of such a fraction is {{math|−''a''/''b''}}, and the multiplicative inverse (provided that {{math|''a'' ≠ 0}}) is {{math|''b''/''a''}}, which can be seen as follows:
: <math> \frac b a \cdot \frac a b = \frac{ba}{ab} = 1.</math>

The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows:<ref>{{harvp|Beachy|Blair|2006|loc=p. 120, Ch. 3}}</ref>
: <math>
\begin{align}
& \frac a b \cdot \left(\frac c d + \frac e f \right) \\[6pt]
= {} & \frac a b  \cdot \left(\frac c d  \cdot \frac f f + \frac e f \cdot \frac d d \right) \\[6pt]
= {} & \frac{a}{b} \cdot \left(\frac{cf}{df} + \frac{ed}{fd}\right) = \frac{a}{b} \cdot \frac{cf + ed}{df} \\[6pt]
= {} & \frac{a(cf + ed)}{bdf} = \frac{acf}{bdf} +  \frac{aed}{bdf} = \frac{ac}{bd} +  \frac{ae}{bf} \\[6pt]
= {} & \frac a b \cdot \frac c d + \frac a b \cdot \frac e f.
\end{align}
</math>