real part in nLab

Contents

Contents

Definition

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

• its real part is Re(a)≔12(a+a *)Re(a) \coloneqq \frac{1}{2}(a + a^\ast);

• its imaginary part is Im(a)≔12(a−a *)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

• AA the complex numbers;

• AA the quaternions;

• AA the octonions

• imaginary number