Complex Numbers
Polar Form of Complex Numbers
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 z 1 = r 1(cos(θ 1) + ısin(θ 1))andz 2 = r 2(cos(θ 2) + ısin(θ 2)) be complex numbers in polar form.
multiplicationanddivision
| z 1×z 2 = r 1 r 2[cos(θ 1 + θ 2) + ısin(θ 1 + θ 2)] |
 =  [cos(θ
1 - θ
2) + ısin(θ
1 - θ
2)]
|
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 z n = [r(cos(θ) + ısin(θ)]n = r n(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
n
th 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
n
th 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.
