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".