Imaginary and complex numbers
Addition and subtraction of complex numbers
Multiplication and division of complex numbers
Polar or trigonometric notation of complex numbers
Addition and subtraction of complex numbers
To add or subtract two complex numbers z1 = a + bi and z2 = c + di, we add or subtract the real parts and the imaginary parts.
 Addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i
 Subtraction: z1 - z2 = (a + bi) - (c + di) = (a - c) + (b - d)i
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 + 3)i = 1 + 5i
and             z1 - z2 = (4 + 3i) - (1 + 5i) = (4 - 1) + (3 - 5)i = 3 - 2i
Given addition and subtraction are shown in the complex plane in the figures below.
 z1 + z2 = (-3 + 2i) + (4 + 3i) = 1 + 5i z1 - z2 = (4 + 3i) - (1 + 5i) = 3 - 2i
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
College algebra contents