nLab complex phase

Idea

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