Editing Complex conjugate (section)

{{Short description|Fundamental operation on complex numbers}}
[[File:Complex conjugate picture.svg|thumb|Geometric representation ([[Argand diagram]]) of <math>z</math> and its conjugate <math>\overline{z}</math> in the complex plane. The complex conjugate is found by [[Reflection symmetry|reflecting]] <math>z</math> across the real axis.]]

In [[mathematics]], the '''complex conjugate''' of a [[complex number]] is the number with an equal [[Real number|real]] part and an [[Imaginary number|imaginary]] part equal in [[Magnitude of a complex number|magnitude]] but opposite in [[Sign (mathematics)|sign]]. That is, if <math>a</math> and <math>b</math> are real numbers, then the complex conjugate of <math> a + bi</math> is <math>a - bi.</math>   The complex conjugate of <math>z</math> is often denoted as <math>\overline{z}</math> or <math>z^*</math>.

In [[Polar coordinate system#Complex numbers|polar form]], if <math>r</math> and <math>\varphi</math> are real numbers then the conjugate of <math>r e^{i \varphi}</math> is <math>r e^{-i \varphi}.</math> This can be shown using [[Euler's formula]].

The product of a complex number and its conjugate is a real number: <math>a^2 + b^2</math>&nbsp;(or&nbsp;<math>r^2</math> in [[Polar coordinate system|polar coordinates]]).

If a root of a [[univariate]] polynomial with real coefficients is complex, then its [[Complex conjugate root theorem|complex conjugate is also a root]].