1. Standard form Operations The Cartesian Plane Modulus and Arguments
COMPLEX NUMBERS
Standard form
Operations
The Cartesian Plane
Modulus and Arguments

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
In real life, problems usually involve Real Numbers(R). Imaginary number: If we combined Real number and imaginary number:
A number that cannot be solved.

4. Introduction Why do we need to study complex numbers, C ?
Many applications especially in engineering:
Electrical engineering, Quantum Mechanics and so on.
Allow us to solve any polynomial equation, such as:

5. Introduction
To solve algebraic equations that don’t have the real solutions
Since, the imaginary number is then
Real solution
No real solution

6. Introduction Simplifying a complex number: Since we know that
To simplify a higher order of imaginary number:

7. Introduction
Try to simplify
Solution

8. Introduction
Simplify
(a)
(b)
(c)

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

10. Introduction
Example: Express in the standard form, z:

11. 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

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

13. Operations of Complex Numbers
Definition 1.3 If z1 = a + bi and z2 = c + di, then:

14. Operations of Complex Numbers
Example Given and find:

15. Operations of Complex Numbers
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
Im(z)
Re(z) 3 -3 2
z(2,3)

16. Operations of Complex Numbers
Example : Find the conjugate of:

17. The Properties of Conjugate Complex Numbers

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

19. Operations of Complex Numbers
Example: Simplify and write in standard form, z:

20. The Complex Plane/ Cartesian 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
Re(z)
O(0,0) a b
Im(z)
z(a,b)

21. The Complex Plane/ Cartesian Plane/ Argand Diagram
Definition 1.6 (Modulus of Complex Numbers) The modulus of z is defined by *Distance from the origin to z(a,b). r
O(0,0)
Re(z) a b
Im(z)
z(a,b)

22. The Complex Plane/ Cartesian Plane/ Argand Diagram
Definition 1.7 The argument of the complex number z = a + bi is defined as arg(z) is not unique. Therefore it can also be written as:
1st QUADRANT
2nd QUADRANT
4th QUADRANT
3rd QUADRANT

23. The Complex Plane/ Cartesian Plane/ Argand Diagram
Example: Find the modulus and the argument of z:

24. The Complex Plane/ Cartesian Plane/ Argand Diagram
The Properties of Modulus

25. Polar form Exponential Form De Moivre’s Theorem Finding Roots
COMPLEX NUMBERS
Polar form
Exponential Form
De Moivre’s Theorem
Finding Roots

26. The Polar Form of Complex Numbers
(a,b) r
Re(z)
Im(z)
Based on Figure 1:
Applying the Pythagorean trigonometric identity,
Therefore,
(1)

27. The Polar Form of Complex Numbers
The standard form of complex numbers is given by: Definition: Then the polar form is defined by:

28. The Polar Form of Complex Numbers
Example: Represent the following complex numbers in polar form: a) b) c)

29. The Polar Form of Complex Numbers
Example: Express the following in the standard form complex numbers: a) b) c)

30. The Polar Form of Complex Numbers
Theorem 1: If z1 and z2 are complex numbers in polar form where, Then, a) b)

31. The Polar Form of Complex Numbers
Example: Find and , if: a) b)

32. The Exponential Form of Complex Numbers
Euler’s formula state that for any real number Where is the exponential function, i is the imaginary unit, sine and cosines are trigonometric function and arg (z) = is in radians. Definition: The exponential form of a complex numbers can be defined as:

33. The Exponential Form of Complex Numbers
Example: Represent the following complex numbers in exponential form: a) b) c)

34. The Exponential Form of Complex Numbers
Example: Express the following in the standard form complex numbers: a) b)

35. The Exponential Form of Complex Numbers
Theorem 2: If z1 and z2 are complex numbers in exponential form where, Then, a) b)

36. The Exponential Form of Complex Numbers
Example: Find and , if: a) b)

37. De Moivre’s Theorem
Let z1 and z2 be complex numbers where Then: From the properties of polar form:

38. De Moivre’s Theorem
From the properties of modulus: And suppose:

39. De Moivre’s Theorem
Using all these facts; (3),(4) and (5), we can compute the square of a complex number. Suppose so Then

40. 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 :

41. De Moivre’s Theorem
Example: a)Let Find b) Use De Moivre’s theorem to find: (i) (ii)

42. De Moivre’s Theorem : Finding Roots
We know that argument of z is not unique, then we can also defined
Using the fact above and DMT, we can find the roots of a complex
number,

43. De Moivre’s Theorem : 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

44. De Moivre’s Theorem : Finding Roots
Example Find all complex cube roots of

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46. De Moivre’s Theorem : Finding Roots
Sketch on the complex plane: 1 y x
nth roots of unity:
Roots lie on the circle with radius 1

47. De Moivre’s Theorem : Finding Roots
Example: Solve and show the roots on the Argand diagram.

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49. De Moivre’s Theorem : Finding Roots
Sketch on complex plane: y x

50. COMPLEX NUMBERS
Expansions for cosn and sinn in terms of Cosines and Sines of multiple n
Loci in the Complex Numbers

51. Expansion of Sin and Cosine
Theorem 5: If , then: Theorem 6: (Binomial Theorem) If , then:

52. Expansion of Sin and Cosine
Example Expand using binomial theorem, then write in standard form of complex number:

53. Expansion of Sin and Cosine
Example State in terms of cosines.

54. Expansion of Sin and Cosine
Example: By using De Moivre’s theorem and Binomial theorem, prove that:

55. Expansion of Sin and Cosine
Example Using appropriate theorems, state the following in terms of sine and cosine of multiple angles :

56. Loci in the Complex Numbers
Since any complex number, z = x+iy correspond to point (x,y) in complex plane, there are many kinds of regions and geometric figures in this plane can be represented by complex equations or inequations. Definition 1.9 A locus in a complex plane is the set of points that have specified property. A locus in a complex plane could be a straight line, circle, ellipse and etc.

57. Loci in the Complex Numbers
(i) Standard form of circle equation Equation of circle with center at the origin, z0 , O(0,0) and z=x+iy, P(x,y) and radius, r y x
P on circumference:
P outside circle:
P inside circle:

58. Loci in the Complex Numbers
(i) Standard form of circle equation Equation of circle with center, z1= x1+ iy1, A(x1, y1) and z=x+iy, P(x,y) and radius, r

59. Loci in the Complex Numbers
Example: What is the equation of circle in complex plane with radius 2 and center at 1+i Solution: Re Im
Distance from center to any point P must be the same

60. Loci in the Complex Numbers
(ii) Perpendicular bisector where the distance from point z=x+iy, P(x,y) to z1 =x1+iy1 , A(x1,y1 ) and z2 = x2+iy2, B(x2,y2 ) are equal. Locus of z represented by perpendicular bisector of line segment joining the points to z1 and z2

61. Loci in the Complex Numbers
Example Find the equation of locus if:

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63. Loci in the Complex NumbersRe Im
Distance from point (0,-1) and (2,0) to any point P must be the same

64. Loci in the Complex Numbers
(ii) Argument: The locus of z represented by a half line from the fixed point z1 =x1+iy1 , making an angle, with a line from the fixed point z1 which is parallel to x-axis

65. Loci in the Complex Numbers
Example If , determine the equation of loci and describe the locus of z

66. Loci in the Complex Numbers
Example Find the equation of locus if: a) b) c)