Imaginary and Complex Numbers
Polar or trigonometric notation of complex numbers
Multiplication and division of complex numbers in the polar form
Exponentiation and root extraction of complex numbers in the polar form

Polar or trigonometric notation of complex numbers
A point (x, y) of the complex plane that represents the complex number z can also be specified by its distance r from the origin and the angle j between the line joining the point to the origin and the positive x-axis.
 Cartesian coordinates expressed by polar coordinates: x = r cosj y = r sinj plugged into  z = x + yi  give z = r (cosj + isinj), where
Thus, obtained is the polar or trigonometric form of a complex number where polar coordinates are r, called the absolute value or modulus, and j, that is called the argument, written j = arg(z).
By using Euler's formula  eij = cosj + isinj,  a complex number can also be written as
 z = r eij which is called the exponential form.
To show the equivalence between the algebraic and the trigonometric form of a complex number,
z = r eij = r (cosj + isinj)
express the sine and the cosine functions in terms of the tangent
and substitute into above expression
Example:  Given the complex number  z = 1 - Ö3i,  express  z = x + yi  in the trigonometric form.
 Solution:  The modulus the argument
 the trigonometric form is
Multiplication and division of complex numbers in the polar form
If given  z1 = r1(cosj1 + isinj1)  and  z2 = r2(cosj2 + isinj2)  then
 z1 ·  z2 = r1 r2 · [cos(j1 + j2) + isin(j1 + j2)]
 and
Exponentiation and root extraction of complex numbers in the polar form - de Moivre's formula
We use the polar form for exponentiation and root extraction of complex numbers that are known as de Moivre's formulas.
 zn = rn · [cos(nj) + isin(nj)]
 and
Exponentiation and root extraction of complex numbers in the polar form examples
 Example: Compute
 Solution:
 or in the polar form,
 and
 since exponentiation with integer exponent
 then
 Example: Compute
Solution:  As square root of a complex number is a complex number, then
and, two complex numbers are equal if their real parts are equal and their imaginary parts are equal, that is
 Example: Calculate using de Moivre's formula.
 Solution:
 since then
These complex numbers satisfy the equation z3 = -8 and by the Fundamental theorem of algebra, since this equation is of degree 3, there must be 3 roots.
Thus, for example to check the root zk=2 we cube this solution,
 then
 Example: Calculate
 Solution:
r = 64    and     j = p
 thus,
These complex numbers satisfy the equation z6 = -64 and by the Fundamental theorem of algebra, since this equation is of degree 6, there must be 6 roots.
 Example: Calculate
 Solution:
 Example: Calculate
 Solution:
Intermediate algebra contents