Complex conjugation

In mathematics, complex conjugation is an operation on complex numbers which reverses the sign of the imaginary part, that is, it sends $$z = x + iy$$ to the complex conjugate $$\bar z = x-iy$$.

In the geometrical interpretation in terms of the Argand diagram, complex conjugation is represented by reflection in the x-axis. The complex numbers left fixed by conjugation are precisely the real numbers.

Conjugation respects the algebraic operations of the complex numbers: $$\overline{z+w} = \bar z + \bar w$$ and $$\overline{zw} = \bar z \bar w$$. Hence conjugation represents an automorphism of the field of complex numbers over the field of real numbers, and is the only non-trivial automorphism.