Let *z* = *a* + *b*ı. 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* + *b*ı 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* + *b*ı is an *n*th root of a complex number *z* if *z* = (*a* + *b*ı)^{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*^{}1*n*. 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.

Figure %: The roots of a complex number graphed in the coordinate plane

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