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.
multiplicationanddivision
| 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
n nth 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
r
1n. 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.
