This is not as simple as

complex number multiplication, and it
works on

complex numbers in the representation

*a + bi*.
See below for a (shorter) solution to polar form complex numbers.

a + bi
? = ------
c + di
a + bi c - di
= ------ x ------
c + di c - di
(a + bi)(c - di)
= ----------------
(c + di)(c - di)
(a + bi)(c - di)
= ----------------
c^2 - d^2i^2
(a + bi)(c - di)
= ---------------- since i^2 = -1
c^2 + d^2

(

*c - di* is called the

conjugate of

*c + di*)

The point of all that was to get a real denominator, which
we can easily divide into the numerator. As stupot says,
"We multiply the numerator and
denominator by the complex conjugate of the denominator in order to obtain a real denominator".
The *(a + bi)(c - di)* term
may be expanded by complex number multiplication,
to yield the final answer:

**
a + bi (ac + bd) + (bc - ad)i
------ = ----------------------
c + di c^2 + d^2**

A simpler

notation can be developed
as follows:

**
_
z zw
- = -----
w |w|^2**
where
**_**
w = conjugate of w
|w| = absolute value of w = sqrt(*real*^2 + *imaginary*^2)

**Polar form**

ariels and jpfed have informed
me of a simple solution for polar form complex numbers.
In this representation, any complex number *z = r*exp(θ*i)*,
due to certain features of *exp(θ*i)*. I will seperate *z* and *w*
as follows: *z = r1*exp(θ1*i)* and *w = r2*exp(θ2*i)*.
Then,

**z** r1*exp(θ1*i)
**-** = ------------ = **(r1/r2) * exp((θ1-θ2)i)**
**w** r2*exp(θ2*i)

So, in English, you "divide radii and subtract thetas".