nLab complex conjugation

Idea

Complex conjugation is the operation on complex numbers which reverses the sign of the imaginary part, hence the function

$$\array{ \mathbb{C} & \overset{ \;\;\; (-)^\ast \;\;\; }{ \longrightarrow } & \mathbb{C} \\ a + \mathrm{i} b &\mapsto& a - \mathrm{i} b } \phantom{AAAAAA} for\;\; a,b \in \mathbb{R} \,.$$

More generally, the anti-involution on any star-algebra may be referred to as conjugation. For instance one speaks of quaternionic conjugation for the analogous operation on quaternions:

$$\array{ \mathbb{H} & \overset{ \;\;\; (-)^\ast \;\;\; }{ \longrightarrow } & \mathbb{H} \\ a + \mathrm{i} b + \mathrm{j} c + \mathrm{k} d &\mapsto& a - \mathrm{i} b - \mathrm{j} c - \mathrm{k} d } \phantom{AAAAAA} for\;\; a, b, c, d \in \mathbb{R} \,.$$

Related concepts

• real part, imaginary part

• anti-involution

• anti-linear map, anti-dual linear space

• automorphism of the complex numbers

For an unrelated (or vaguely related) notion with a similar name see at conjugacy class and adjoint action.