Question

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

Answer 1

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[cos(phi_1)+i*sin(phi_1)]`
`z_2=r_2[cos(phi_2)+i*sin(phi_2)]`
Then their product is
`z_1*z_2=r_1[cos(phi_1)+i*sin(phi_1)]*r_2[cos(phi_2)+i*sin(phi_2)]`
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*z_2=r_1r_2[cos(phi_1+phi_2)+i*sin(phi_1+phi_2)]`
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:
`{r[cos(phi)+i*sin(phi)]}^t=r^t[cos(t*phi)+i*sin(t*phi)]`

Addition of complex numbers is much more convenient in canonical form `z=a+i*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=[r_1cos(phi_1)+r_2cos(phi_2)]+i*[r_1sin(phi_1)+r_2sin(phi_2)]`

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*b` where
`a=r_1cos(phi_1)+r_2cos(phi_2)` and
`b=r_1sin(phi_1)+r_2sin(phi_2)`

This general rule states that
`z=r[cos(phi)+i*sin(phi)]` where
`r=sqrt(a^2+b^2)` and
angle `phi` (usually, in radians) is defined by its trigonometric functions
`sin(phi)=b/r`,
`cos(phi)=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!=0`, `tan(phi)=b/a`. Or, if `b!=0`, `cot(phi)=a/b`.