The Complex Plane The Complex Plane The complex

The Complex Plane • The complex number z = a + bi can plotted

The Complex Plane • Definition 1. 6 (Modulus of Complex Numbers) The modulus of

The Complex Plane • Example 1. 7 : • Find the modulus of z:

The Complex Plane • The Properties of Modulus

Argument of Complex Numbers • Definition 1. 7 The argument of the complex number

Argument of Complex Numbers • Example 1. 8 : • Find the arguments of

THE POLAR FORM OF COMPLEX NUMBER Im(z) (a, b) r b Re(z) a •

• The polar form is defined by: • Example 1. 9: • Represent

• Example 1. 10 : • Express the following in standard form of

Theorem 1: If z 1 and z 2 are 2 complex numbers in polar

• Example 1. 11 : a) If z 1 = 2(cos 40+isin 40)

THE EXPONENTIAL FORM • DEFINITION 1. 8 The exponential form of a complex number

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 any power of n, then De Moivre’s Theorem:

DE MOIVRE’S THEOREM • Example 1. 17 If , calculate :

FINDING ROOTS Theorem 4 If (θ in degrees) OR (θ in radians) Where k

FINDING ROOTS Example 1. 18 If then, r =1 and Let n =3, therefore

FINDING ROOTS When k = 2: Sketch on the complex y plane: 0 1

FINDING ROOTS Example 1. 18 If then, Let n =4, therefore k =0, 1,

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

FINDING ROOTS Sketch on complex plane: y 0 x

Thank You Prepared By Smt. Sasmita kumari swain Lecturer in Mathematics Khemundi College

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The Complex Plane

The Complex Plane • 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) b O(0, 0) z(a, b) a Re(z)

The Complex Plane • Definition 1. 6 (Modulus of Complex Numbers) The modulus of z is defined by Im(z) z(a, b) b r O(0, 0) a Re(z)

The Complex Plane • Example 1. 7 : • Find the modulus of z:

The Complex Plane • The Properties of Modulus

Argument of Complex Numbers • Definition 1. 7 The argument of the complex number z = a + bi is defined as 2 nd QUADRANT 1 st QUADRANT 3 rd QUADRANT 4 th QUADRANT

Argument of Complex Numbers • Example 1. 8 : • Find the arguments of z:

THE POLAR FORM OF COMPLEX NUMBER Im(z) (a, b) r b Re(z) a • 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 z 1 and z 2 are 2 complex numbers in polar form where then,

• Example 1. 11 : a) If z 1 = 2(cos 40+isin 40) and z 2 = 3(cos 95+isin 95). Find : b) If z 1 = 6(cos 60+isin 60) and z 2 = 2(cos 270+isin 270). 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 any power of n, then De Moivre’s Theorem: Therefore : is a complex number in polar form to

DE MOIVRE’S THEOREM • Example 1. 17 If , calculate :

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

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 y plane: 0 1 x nth roots of unity: Roots lie on the circle with radius 1

FINDING ROOTS Example 1. 18 If then, Let n =4, therefore k =0, 1, 2, 3 When k =0: When k = 1: and :

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

FINDING ROOTS Sketch on complex plane: y 0 x

Thank You Prepared By Smt. Sasmita kumari swain Lecturer in Mathematics Khemundi College