Complex Numbers: The Phase of a Complex Number

[pamparana]

I just wanted to check something. If I have a complex number of the form

a = C * \exp(i \phi)

where C is some non-complex scalar constant. Then the phase of this complex number is simply \phi. Is that correct?

 

[jbunniii]

I just wanted to check something. If I have a complex number of the form

a = C * \exp(i \phi)

where C is some non-complex scalar constant. Then the phase of this complex number is simply \phi. Is that correct?

If $C > 0$ then this is almost correct. However, the phase is not well-defined under this definition, because $C\exp(i\phi) = C\exp(i(\phi+2\pi n))$ for any integer $n$. You can get around this by defining the phase to be the coset $\phi + 2\pi \mathbb{Z}$ or by constraining it to be in the interval $[0,2\pi)$ or $[-\pi, \pi)$ or some other half-open interval of length $2\pi$.

If $C < 0$, then you need to absorb the sign of $C$ into the phase:
$$a = -|C|\exp(i \phi) = |C|\exp(i(\phi + \pi))$$

If $C = 0$ then the phase is undefined.

 

[pamparana]

Thank you for this detailed answer!