Complex conjugation

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In mathematics, complex conjugation is an operation on complex numbers which reverses the sign of the imaginary part, that is, it sends  z = x + i y {\displaystyle z=x+iy} to the complex conjugate z ¯ = x − i y {\displaystyle {\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:  z + w ¯ = z ¯ + w ¯ {\displaystyle {\overline {z+w}}={\bar {z}}+{\bar {w}}} and  z w ¯ = z ¯ w ¯ {\displaystyle {\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. One can say it is impossible to tell which is i and which is -i.