Dividing Complex Numbers

Division is the same as multiplying by the multiplicative inverse. For a complex number a+bi, the inverse is 1/(a+bi), but it is helpful to transform this ito rectangular form:

 1         a-bi          a-bi         a           b
---- = ------------ = --------- = --------- - --------- i
a+bi   (a+bi)(a-bi)   a^2 + b^2   a^2 + b^2   a^2 + b^2

So, if we have problem (a+bi)/(c+di), we actually have a+bi multiplied by the inverse of c+di. We can transform this to rectangular form also:

             c           d
(a+bi) ( --------- - --------- i )
         c^2 + d^2   c^2 + d^2

     ac          ad            bc            bd
= --------- - --------- i + --------- i + ---------
  c^2 + d^2   c^2 + d^2     c^2 + d^2     c^2 + d^2

   ac + bd     bc - ad
= --------- + --------- i
  c^2 + d^2   c^2 + d^2

So,

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

If you happen to be working in polar form instead, the calculation is much simpler, as you are simply dividing with the magnitudes and subtracting with the angles.

ae^(ic)   a  i(c-d)
------- = - e
be^(id)   b