Polar Form of a Complex Number
Example 1
Writing a Complex Number in Polar Form
Write the complex number
in polar form.
Solution
The absolute value of z is
and the angle θ is given by
Because
and because
lies in Quadrant III, you
choose θ to be
θ = π + π/3 = 4π/3. Thus, the polar form is
(See the figure below)
The polar form adapts nicely to multiplication and division of complex numbers.
Suppose you are given two complex numbers
z_{1} = r_{1}(cos
θ_{1} + i sin θ_{1} )
and z_{2} = r_{2}(cos
θ_{2} + i sin θ_{2} )
The product of z_{1} and z_{2} is
z_{1}
z_{2} 
= r_{1}
r_{2 }cos
θ_{1} + i sin θ_{1} )(cos
θ_{2} + i sin θ_{2} )
= r_{1}
r_{2} [(cos
θ_{1}cos θ_{2}  i sin
θ_{1}sin θ_{2}) + i(sin θ_{1}cos θ_{2} +_{
}cos θ_{1}sin
θ_{2})] 
Using the sum and difference formulas for cosine and sine, you can rewrite this
equation as
z_{1} z_{2}
= r_{1} r_{2}
[(cos (θ_{1}+
θ_{2}) + i sin (θ_{1}+ θ_{2})]
This establishes the first part of the following rule. Try to establish the second part on
your own.
Product and Quotient of Two Complex Numbers
Let z_{1} = r_{1}(cos
θ_{1} + i sin θ_{1} )
and z_{2} = r_{2}(cos
θ_{2} + i sin θ_{2} ) be complex
numbers.
z_{1} z_{2}
= r_{1} r_{2}
[(cos (θ_{1}+
θ_{2}) + i sin (θ_{1}+ θ_{2})] 
Product 

Quotient 
Note that this rule says that to multiply two complex numbers you multiply
moduli and add arguments, whereas to divide two complex numbers you divide
moduli and subtract arguments.
