Knowlpedia

Complex conjugate

The map a+bi ↦ a-bi on complex numbers.

For z=a+bi∈Cz=a+bi\in\mathbb{C} with a,b∈Ra,b\in\mathbb{R}, the complex conjugate of zz is

z‾:=a−bi.\overline{z}:=a-bi.
Remarks

Complex conjugation is an algebraic involution: it reverses the sign of the imaginary part and satisfies z‾‾=z\overline{\overline{z}}=z. It is used to express the modulus and to compute inverses: if z≠0z\ne 0 then z−1=z‾/(zz‾)z^{-1}=\overline{z}/(z\overline{z}).

Examples
  • If z=3−2iz=3-2i, then z‾=3+2i\overline{z}=3+2i.
  • (1+i)2‾=2i‾=−2i\overline{(1+i)^2}=\overline{2i}=-2i.
  • If z=iyz=iy with y∈Ry\in\mathbb{R}, then z‾=−iy\overline{z}=-iy.