View source for Complex conjugate

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Two [[complex number]]s of the form $z = x + iy$ and $\bar z = x - iy$: in polar form, $r e^{i\theta}$ and $r e^{-i\theta}$.  In terms of the [[Argand diagram]], they are symmetric in the $x$-axis.  A complex number is its own complex conjugate if and only if it is a [[real number]].

'''Complex conjugation''' is the map $z \mapsto \bar z$.  We have $\overline{z+w} = \bar z + \bar w$, $\overline{z\cdot w} = \bar z \cdot \bar w$, $\overline{z^{-1}} = \bar z^{-1}$, $\overline{\bar z} = z$.  Complex conjugation is an automorphism of the field of complex numbers of order two.  The [[absolute value]] $|z|$ is the positive square root of $z \bar z$.