Consider the rule for multiplying complex numbers in polar form
Squaring Complex Numbers
Suppose that the two numbers are equal. i.e. z 1 = z 2
Then z2 = r cis θ. r cis θ = r2cis(θ + θ)
Cubing Complex Numbers
Let z1 = r cis θ and z2 = r cis(2θ)
Then z3 = r cisθ. r cis(2θ) = r cis(θ + 2θ)
De Moivre's Theorem
The above two derivations can be extended into a general rule known as De Moivre's Theorem:
Find (3 cis )6
Equations involving Complex Numbers
The aim of this section is to solve equation of the type zn = r cis θ
Taking the nth root of both sides z = (r cis θ)1/n
By De Moivre's Theorem
This provides one of the roots but because complex numbers in polar form repeat every 360° or 2π radians there will be other roots as well.
The full solution set of the nth roots of r cis θ is given by:
This will mean that when finding the nth root there will be n solutions.
Note that the three roots are equally spaced around a circle of radius 2 (cube root of 8) on the Argand diagram. This is true for all equations of the form zn = r cis θ. Once one root has been found the other roots can be placed, equally spaced, around a circle of radius the n th root of r.