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[[File:Complex conjugate picture.svg|right|thumb|upright|An illustration of the [[complex plane]]. The real part of a complex number {{math|1=''z'' = ''x'' + ''iy''}} is {{mvar|x}}, and its imaginary part is {{mvar|y}}.]]
A complex number is a number of the form {{math|1=''a'' + ''bi''}}, where {{mvar|a}} and {{mvar|b}} are [[real numbers]], and {{math|''i''}} is an indeterminate satisfying {{math|1=''i''<sup>2</sup> = −1}}. For example, {{math|2 + 3''i''}} is a complex number.<ref>{{cite book|title=College algebra |url=https://archive.org/details/collegealgebrawi00axle |url-access=limited |last=Axler |first=Sheldon |page=[https://archive.org/details/collegealgebrawi00axle/page/n285 262]|publisher=Wiley|year=2010}}</ref><ref name=":1" />
This way, a complex number is defined as a [[polynomial]] with real coefficients in the single indeterminate {{math|''i''}}, for which the relation {{math|''i''<sup>2</sup> + 1 {{=}} 0}} is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation {{math|''i''<sup>2</sup> + 1 {{=}} 0}} induces the equalities {{math|''i''<sup>4''k''</sup> {{=}} 1, ''i''<sup>4''k''+1</sup> {{=}} ''i'', ''i''<sup>4''k''+2</sup> {{=}} −1,}} and {{math|''i''<sup>4''k''+3</sup> {{=}} −''i'',}} which hold for all integers {{mvar|k}}; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in {{mvar|i}}, again of the form {{math|1=''a'' + ''bi''}} with real coefficients {{mvar|a, b.}}
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