Let z = a + bı. Consider the complex number z plotted in the complex plane. The distance between z and the origin is called the modulus of z (symbolized | z|) and is equal to . The angle θ between the positive real axis and the ray containing z whose endpoint is the origin is called the argument of z, and is equal to arctan(). The polar form of a complex number z = a + bı is this: z = r(cos(θ) + ısin(θ)), where r = | z| and θ is the argument of z. Polar form is sometimes called trigonometric form as well.
The polar form of a complex number is especially useful when we're working with powers and roots of a complex number. First, we'll look at the multiplication and division rules for complex numbers in polar form. Let z1 = r1(cos(θ1) + ısin(θ1))andz2 = r2(cos(θ2) + ısin(θ2)) be complex numbers in polar form.
|z1×z2 = r1r2[cos(θ1 + θ2) + ısin(θ1 + θ2)]|
|= [cos(θ1 - θ2) + ısin(θ1 - θ2)]|
These equations arise from the sum and difference formulas for the trigonometric functions sine and cosine.
The power of a complex number is given by an equation known as De Moivre's Theorem: Let z = r(cos(θ) + ısin(θ). Then zn = [r(cos(θ) + ısin(θ)]n = rn(cos(nθ) + ısin(nθ), where n is any positive integer.
The roots of a complex number are also given by a formula. A complex number a + bı is an nth root of a complex number z if z = (a + bı)n, where n is a positive integer. A complex number z = r(cos(θ) + ısin(θ) has exactly nnth roots given by the equation [cos() + ısin()], where n is any positive integer, and k = 0, 1, 2,..., n - 2, n - 1.
When the nth roots of a complex number are graphed in the complex plane, they all lie on the same circle with radius r1n. They are also all evenly spaced around the circle, like spokes on a bike. This is because the arguments of consecutive roots differ by a measure of radians.