The Complex Plane The Complex Plane The complex
- Slides: 25
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
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