nLab
complex phase

Idea

The complex phase of a complex number cc is the real number ϕ\phi in the decomposition c=|c|e iϕc = {\vert c \vert} e^{i \phi} – well defined modulo 2π2 \pi \mathbb{Z} if the modulus/absolute value |c|{|c|} is positive, and modulo π\pi \mathbb{Z} if the modulus is merely real (and nonzero). If cc (equivalently |c|{|c|}) is zero, then the phase is entirely unspecified.