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

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

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

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

The Complex Plane / Argand Diagram The Properties of Modulus

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

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

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

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

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

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

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 :

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

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

THE EXPONENTIAL FORM Theorem 2 If and , then:

THE EXPONENTIAL FORM Example 1.16 If and , find:

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 :

DE MOIVRE’S THEOREM Example 1.17 If , calculate :

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

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

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

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

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

FINDING ROOTS Sketch on complex plane: y x