nLab real part

Definition

Given a star-algebra $(A ,\ast)$ in characteristic zero, then for $a\in A$ any element

• its real part is $Re(a) \coloneqq \frac{1}{2}(a + a^\ast)$;

• its imaginary part is $Im(a) \coloneqq \frac{1}{2}(a - a^\ast)$,

where $(-)^\ast$ is the given star-anti-involution (e.g. complex conjugation, quaternionic conjugation, …).

Examples

Standard situations where real and imaginary parts are considered include the real normed division algebra

• $A$ the complex numbers;

• $A$ the quaternions;

• $A$ the octonions

Related concepts

• imaginary number