Editing Field (mathematics) (section)

=== Real and complex numbers ===
[[File:Complex multi.svg|thumb|255px|The multiplication of complex numbers can be visualized geometrically by rotations and scalings.]]
{{main|Real number| Complex number}}

The [[real number]]s {{math|'''R'''}}, with the usual operations of addition and multiplication, also form a field. The [[complex number]]s {{math|'''C'''}} consist of expressions
: {{math|''a'' + ''bi'',}} with {{math|''a'', ''b''}} real,
where {{math|''i''}} is the [[imaginary unit]], i.e., a (non-real) number satisfying {{math|1=''i''<sup>2</sup> = −1}}.
Addition and multiplication of real numbers are defined in such a way that expressions of this type satisfy all field axioms and thus hold for {{math|'''C'''}}. For example, the distributive law enforces
: {{math|1=(''a'' + ''bi'')(''c'' + ''di'') = ''ac'' + ''bci'' + ''adi'' + ''bdi''<sup>2</sup> = (''ac'' − ''bd'') + (''bc'' + ''ad'')''i''.}}
It is immediate that this is again an expression of the above type, and so the complex numbers form a field. Complex numbers can be geometrically represented as points in the [[Plane (geometry)|plane]], with [[Cartesian coordinates]] given by the real numbers of their describing expression, or as the arrows from the origin to these points, specified by their length and an angle enclosed with some distinct direction. Addition then corresponds to combining the arrows to the intuitive parallelogram (adding the Cartesian coordinates), and the multiplication is – less intuitively – combining rotating and scaling of the arrows (adding the angles and multiplying the lengths). The fields of real and complex numbers are used throughout mathematics, physics, engineering, statistics, and many other scientific disciplines.