Complex Number Division
Complex Number division is not as obvious as addition/subtraction or even multiplication.
Suppose one wanted to divide $ (2+3i)/(4+5i) $. The first step is to multiply the top and the bottom by the lower numbers complex conjugate, $ (4-5i) $. The result of the denominator should be a real number now and one can split the numerator with a common denominator.
$ ((2+3i)(4-5i))/((4+5i)(4-5i)) = (8-10i+12i+15)/(16-20i+20i+25) = (23+2i)/(41) = (23/41)+(2i/41) $
Another example:(3+4i)/(5-i)
$ ((3+4i)(5+i))/((5-i)(5+i)) = (15+3i+20i-4)/(25-5i+5i+1) = (11+23i)/(26) = (11/26)+(23i/26) $
General Formula
A General formula can then be determined as $ (a+ib)/(c+id)=(ac+bd+i(bc-ad))/(c^2+d^2) $
