Argument of Product is Sum of Arguments

Theorem

Let $z_1, z_2 \in \C$ be complex numbers and $\arg$ the argument operator.

Then:

$\arg \left({z_1 z_2}\right) = \arg \left({z_1}\right) + \arg \left({z_2}\right)$

Proof

Let $\theta_1 = \arg \left({z_1}\right), \theta_2 = \arg \left({z_2}\right)$.

Then the polar forms of $z_1, z_2$ are:

$z_1 = \left|{z_1}\right| \left({\cos \theta_1 + i \sin \theta_1}\right)$

$z_2 = \left|{z_2}\right| \left({\cos \theta_2 + i \sin \theta_2}\right)$

By the definition of complex multiplication, factoring $\left|{z_1}\right|\left|{z_2}\right|$ from all terms, we have:

$z_1 z_2 = \left|{z_1}\right| \left|{z_2}\right| \left({\left({\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2}\right) + i \left({\cos \theta_1 \sin \theta_2 + \sin \theta_1 \cos \theta_2}\right)}\right)$

Using the formulas for addition of sines and cosines, we have:

$z_1 z_2 = \left|{z_1}\right| \left|{z_2}\right| \left({\cos \left({\theta_1 + \theta_2}\right) + i \sin \left({\theta_1 + \theta_2}\right)}\right)$

The theorem follows from the definition of $\arg \left({z}\right)$, which says that $\arg \left({z_1 z_2}\right)$ satisfies the equations:

$(1): \quad \dfrac{\left|{z_1}\right| \left|{z_2}\right| \cos \left({\theta_1 + \theta_2}\right)} {\left|{z_1}\right| \left|{z_2}\right|} = \cos \left({\arg \left({z_1 z_2}\right)}\right)$

$(2): \quad \dfrac{\left|{z_1}\right| \left|{z_2}\right| \sin \left({\theta_1 + \theta_2}\right)} {\left|{z_1}\right| \left|{z_2}\right|} = \sin \left({\arg \left({z_1 z_2}\right)}\right)$

which in turn means that:

$\cos \left({\theta_1 + \theta_2}\right) = \cos \left({\arg \left({z_1 z_2}\right)}\right)$

$\sin \left({\theta_1 + \theta_2}\right) = \sin \left({\arg \left({z_1 z_2}\right)}\right)$

therefore:

$\arg \left({z_1 z_2}\right) = \theta_1 + \theta_2 = \arg \left({z_1}\right) + \arg \left({z_2}\right)$

$\blacksquare$