Products of Complex Numbers and De Moivre's Theorem

Any nonzero complex number z  =  x  +  i y  can be described in polar form in terms of its modulus  and  argument.   If you plot z in the complex plane (where the x axis is the real part and the y axis is the imaginary part) at P(x,  y) , then the modulus of z is the distance, r,  from the origin to P.  The argument of z is the angle, θ, that the ray OP makes with the positive real axis.

Using basic trigonometry, we can write z as
			z  =  r cos θ  +  
            i r sin θ = r(
            cos θ  +  i sin θ) 
This last expression is often abbreviated r CiS(θ).

You might notice that zero, which is  0  +  i 0, has a modulus zero, but has no unique argument.  Since our main reason for considering this form of complex numbers is for multiplying them, this deficiency of zero won't really be an issue.

Now if you multiply (Subscript[z, 1])= (Subscript[r, 1]) CiS((Subscript[θ, 1])) and (Subscript[z, 2])= (Subscript[r, 2]) CiS((Subscript[θ, 2])) and expand this messy expression, you will see some patterns that might be familiar.  In the last step we apply the angle sum formulae for sine and cosine.  

	(Subscript[z, 1])·
            (Subscript[z, 2]) =  (Subscript[r, 1])(
              cos (Subscript[θ, 1])  +  i sin (Subscript[θ, 1])) (Subscript[r, 2])(
              cos (Subscript[θ, 2])  +  i sin (Subscript[θ, 2]))
	⋱
	=\(\(r\_1\) \(\(r\_2\)(\((cos\ \(θ\_1\) 
                            cos\ θ\_2 - 
                          sin\ \(θ\_1\) sin\ θ\_2)\) + 
                      i\ \((sin\ \(θ\_1\) cos\ θ\_2 + \ 
                            sin\ \(θ\_2\) 
                              cos\ θ\_1)\))\)
	 = \(r\_1\) \(\(r\_2\)(
                    cos(θ\_1 + θ\_2) + \ 
                      i\ \(sin(θ\_1 + θ\_2)\))\)\)

The Rule to Remember:  When you multiply complex numbers, you multiply moduli and add arguments.

This is actually the way many mathematicians remember the trig identities.  Once you know this general rule for multiplying complex numbers, you don't need to memorize the details of the two identities we used — you can always reconstruct them by knowing that the last step above is true!

De Moivre's formula, named after Abraham de Moivre (1667 - 1754), follows from this more general rule.  Given any nonzero complex number z and any integer n, the (n^th) power of  z =r CiS(θ)= r (cos  θ  +  
                  i sin  θ)  is 

		(z^n) = (((r(cos  θ  +  i sin  θ)))^n)= (r^n)((cos n θ  +  i sin n θ)= (r^n) CiS(n θ)

Related Links:
http://forum.swarthmore.edu/johnandbetty/