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
By using general rather than specific complex numbers we can, in the same way,find the outcome of a general division of complex numbers in terms of their real and imaginary parts as we have done above for general complex multiplication. However, as long as the technique is understood, there is no pressing need to produce and to memorize the resulting formula.