Multiplication is another matter. Formally it is easy to do:we multiply two complex numbers together by multiplying out the brackets, remembering that i2=-1. Assuming the Distributive Law continues to hold, which is the algebraic rule that allows us to expand the brackets in the usual way, then multiplication proceeds as follows:
(a+bi)(c+di)=a(c+di)+bi(c+di)=
ac+adi+bci+bdi2=(ac-bd)+(ad+bc)i