Imaginary and Complex Numbers Multiplication and division of complex numbers Polar or trigonometric notation of complex numbers
Multiplication and division of complex numbers in the polar form
Multiplication and division of complex numbers
 Multiplication: z1· z2 = (a + bi) · (c + di) = ac + bci + adi + bdi2 = (ac - bd) + (ad + bc)i
 Division: Examples:  Given are complex numbers,  z1 = -3 + 2i  and  z2 = 4 + 3i, find  z1 ·  z2  and  z1 / z2.
Solutions:      z1 ·  z2 = (-3 + 2i) · (4 + 3i) = -3 · 4 + 2 · 4i + (-3) · 3i +  2 · 3 i2 = -18 - i
 and Example:  For what real number a the real part of the complex number equals 1.
 Solution:  Example:  Evaluate the expression where  z = 1 - i.
 Solution: 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    Functions contents A 