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1. Cartesian/Polar Form
[A] A complex number can be written in Cartesian Form or Polar form. You must be able to change from one form into the other with confidence.
(Cartesian Form)
(Polar Form)
(i) Find
(ii) Draw a little picture to locate the complex number in a quadrant.
(iii) Find
(iv) Find q in radians.
(v) Change to Polar form.
Example 1
Find z = -1-1i in polar form.Solution
Example 2
Write
in polar form.Solution
[C] Changing from Polars to Cartesians
To change from Polars to Cartesians simply look up the cosine and sine of the angle after changing it from radians into degrees. Remember ASTC.
Example 3
Put
into Cartesian form.Solution
2. Tricks with the Polar Object
[A] When a complex number is in Polar form the first thing students do is to change it back into Cartesians because they are uncomfortable working in Polars. This often makes the problem more difficult.We shall call an object of the form
a polar object.[B] Trick 1: When you multiply polar objects you add the angles.
Example 4
If
and 
evaluate zw in the form x + iy.Solution
[C] Trick 2: When you invert a polar object you change the sign between the real and imaginary components.
This result is also true for
Example 5
If
find z in the form x + iy.Solution
[D] Trick 3: When you divide polar objects you subtract the angles.
Example 6
Simplify
giving your answer in the form x + iy.Solution
Example 7
Simplify
giving your answer in the form 
Solution
[E] Plotting complex numbers in polar form in the Argand diagram.
When a complex number is in polar form
remember that:r = the distance to the origin (draw a circle of radius r)
q = the angle made with the +Re-axis (measure out this angle)
Example 8
Sketch the complex number
in the Argand diagram.Solution
