# Is it possible to perform basic operations on complex numbers in polar form?

Jan 6, 2015

Yes, of course.

Polar form is very convenient to multiply complex numbers.
Assume we have two complex numbers in polar form:
${z}_{1} = {r}_{1} \left[\cos \left({\phi}_{1}\right) + i \cdot \sin \left({\phi}_{1}\right)\right]$
${z}_{2} = {r}_{2} \left[\cos \left({\phi}_{2}\right) + i \cdot \sin \left({\phi}_{2}\right)\right]$
Then their product is
${z}_{1} \cdot {z}_{2} = {r}_{1} \left[\cos \left({\phi}_{1}\right) + i \cdot \sin \left({\phi}_{1}\right)\right] \cdot {r}_{2} \left[\cos \left({\phi}_{2}\right) + i \cdot \sin \left({\phi}_{2}\right)\right]$
Performing multiplication on the right, replacing ${i}^{2}$ with $- 1$ and using trigonometric formulas for cosine and sine of a sum of two angles, we obtain
${z}_{1} \cdot {z}_{2} = {r}_{1} {r}_{2} \left[\cos \left({\phi}_{1} + {\phi}_{2}\right) + i \cdot \sin \left({\phi}_{1} + {\phi}_{2}\right)\right]$
The above is a polar representation of a product of two complex numbers represented in polar form.

Raising to any real power is also very convenient in polar form as this operation is an extension of multiplication:
${\left\{r \left[\cos \left(\phi\right) + i \cdot \sin \left(\phi\right)\right]\right\}}^{t} = {r}^{t} \left[\cos \left(t \cdot \phi\right) + i \cdot \sin \left(t \cdot \phi\right)\right]$

Addition of complex numbers is much more convenient in canonical form $z = a + i \cdot b$. That's why, to add two complex numbers in polar form, we can convert polar to canonical, add and then convert the result back to polar form.
The first step (getting a sum in canonical form) results is
${z}_{1} + {z}_{2} = \left[{r}_{1} \cos \left({\phi}_{1}\right) + {r}_{2} \cos \left({\phi}_{2}\right)\right] + i \cdot \left[{r}_{1} \sin \left({\phi}_{1}\right) + {r}_{2} \sin \left({\phi}_{2}\right)\right]$

Converting this to a polar form can be performed according to general rule of obtaining modulus (absolute value) and argument (phase) of a complex number represented as $z = a + i \cdot b$ where
$a = {r}_{1} \cos \left({\phi}_{1}\right) + {r}_{2} \cos \left({\phi}_{2}\right)$ and
$b = {r}_{1} \sin \left({\phi}_{1}\right) + {r}_{2} \sin \left({\phi}_{2}\right)$

This general rule states that
$z = r \left[\cos \left(\phi\right) + i \cdot \sin \left(\phi\right)\right]$ where
$r = \sqrt{{a}^{2} + {b}^{2}}$ and
angle $\phi$ (usually, in radians) is defined by its trigonometric functions
$\sin \left(\phi\right) = \frac{b}{r}$,
$\cos \left(\phi\right) = \frac{a}{r}$
(it's not defined only if both $a = 0$ and $b = 0$).
Alternatively, we can use these equations to define angle $\phi$:
If $a \ne 0$, $\tan \left(\phi\right) = \frac{b}{a}$. Or, if $b \ne 0$, $\cot \left(\phi\right) = \frac{a}{b}$.