Argument of Quotient equals Difference of Arguments

Theorem
Let $z_1$ and $z_2$ be complex numbers.

Then:
 * $\arg \left({\dfrac {z_1} {z_2} }\right) = \arg \left({z_1}\right) + \arg \left({z_1}\right)$

where $\arg$ denotes the argument of a complex number.

As the argument is a multifunction it is understood that the equation holds for some values of $\arg$ and may not hold for the principal argument.

Proof
Let $z_1$ and $z_2$ be expressed in polar form.
 * $z_1 = \left\langle{r_1, \theta_1}\right\rangle$
 * $z_2 = \left\langle{r_2, \theta_2}\right\rangle$

From Division of Complex Numbers in Polar Form:
 * $\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} \left({\cos \left({\theta_1 - \theta_2}\right) + i \sin \left({\theta_1 - \theta_2}\right)}\right)$

By the definition of argument:
 * $\arg \left({z_1}\right) = \theta_1$
 * $\arg \left({z_2}\right) = \theta_2$
 * $\arg \left({\dfrac {z_1} {z_2} }\right) = \theta_1 - \theta_2$