1. CHAPTER 1 COMPLEX NUMBERS
THE MODULUS AND ARGUMENT
THE POLAR FORM
THE EXPONENTIAL FORM
DE MOIVRE’S THEOREM
FINDING ROOTS

2. The Complex Plane / Argand Diagram
The complex number z = a + bi can plotted as a point with coordinates z(a,b).
Re (z) x – axis
Im (z) y – axis
Im(z)
Re(z)
O(0,0)
z(a,b) a b

3. The Complex Plane / Argand Diagram
Definition 1.6
(Modulus of Complex Numbers)
The modulus of z is defined by
Im(z)
Re(z)
O(0,0)
z(a,b) a b r

4. The Complex Plane / Argand Diagram
Example 1.7 :
Find the modulus of z:

5. The Complex Plane / Argand Diagram
The Properties of Modulus

6. Argument of Complex Numbers
Definition 1.7
The argument of the complex number z = a + bi is defined as
1st QUADRANT
2nd QUADRANT
4th QUADRANT
3rd QUADRANT

7. Argument of Complex Numbers
Example 1.8 :
Find the arguments of z:

8. THE POLAR FORM OF COMPLEX NUMBER
(a,b) r
Re(z)
Im(z)
Based on figure above:

9. The polar form is defined by:
Example 1.9:
Represent the following complex number in polar form:

10. Example 1.10 :
Express the following in standard form of complex number:

11. Theorem 1: If z1 and z2 are 2 complex numbers in polar form where then,

12. Example 1.11 :
If z1 = 2(cos40+isin40) and z2 = 3(cos95+isin95) . Find :
If z1 = 6(cos60+isin60) and z2 = 2(cos270+isin270) . Find :

13. THE EXPONENTIAL FORM DEFINITION 1.8
The exponential form of a complex number can be defined as
Where θ is measured in radians and

14. THE EXPONENTIAL FORM Example 1.15
Express the complex number in exponential form:

15. THE EXPONENTIAL FORM
Theorem 2 If and , then:

16. THE EXPONENTIAL FORM
Example 1.16
If and , find:

17. DE MOIVRE’S THEOREM
Theorem 3 If is a complex number in polar form to any power of n, then De Moivre’s Theorem: Therefore :

18. DE MOIVRE’S THEOREM
Example 1.17
If , calculate :

19. FINDING ROOTS
Theorem 4 If then, the n root of z is: (θ in degrees) OR (θ in radians) Where k = 0,1,2,..n-1

20. FINDING ROOTS
Example 1.18 If then, r =1 and : Let n =3, therefore k =0,1,2 When k =0: When k = 1:

21. FINDING ROOTS When k = 2: Sketch on the complex plane: y1 y x
nth roots of unity:
Roots lie on the circle with radius 1

22. FINDING ROOTS
Example 1.18 If then, and : Let n =4, therefore k =0,1,2,3 When k =0: When k = 1:

23. FINDING ROOTS
Let n =4, therefore k =0,1,2,3 When k =2: When k = 3:

24. FINDING ROOTS
Sketch on complex plane: y x