1. CHAPTER 1 COMPLEX NUMBERS
STANDARD FORM
OPERATIONS
THE COMPLEX PLANE
THE MODULUS AND ARGUMENT
THE POLAR FORM

2. Classification of Numbers
INTEGERS
(Z)
COMPLEX NUMBERS
(C)
REAL NUMBERS
(R)
RATIONAL NUMBERS
(Q)
IRRATIONAL NUMBERS
WHOLE NUMBERS
(W)
NATURAL NUMBERS
(N)

3. Introduction
To solve algebraic equations that don’t have the real solutions
To solve complex numbers :
Since :
Real solution
No real solution

4. Introduction
Example 1
Simplify:

5. Introduction Definition 1.1
If z is a complex number, then the standard equation of Complex number denoted by:
where
a – Real part of z (Re z)
b – Imaginary part of z (Im z)

6. Introduction
Example 1.2 :
Express in the standard form, z:

7. Introduction Definition 1.2
2 complex numbers, z1 and z2 are said to be equal if and only if they have the same real and imaginary parts:
If and only if a = c and b = d

8. Introduction
Example 1.3 :
Find x and y if z1 = z2:

9. Operations
Definition 1.3
If z1 = a + bi and z2 = c + di, then:

10. Operations
Example 1.4 :
Given z1 = 3-2i and z2= 4-2i

11. Operations Definition 1.4
The conjugate of z = a + bi can be defined as:
**the conjugate of a complex number changes the sign of the imaginary part only!!!
**obtained geometrically by reflecting point z on the real axis

12. Operations
Example 1.5 :
Find the conjugate of

13. The Properties of Conjugate Complex Numbers

14. Operations Definition 1.5 (Division of Complex Numbers)
If z1 = a + bi and z2 = c + di then:
Multiply with the conjugate of denominator

15. Operations
Example 1.6 :
Simplify and write in standard form, z:

16. The Complex Plane / Argand Diagram
The complex number z = a + bi is 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

17. 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

18. 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

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

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

21. 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

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