CHAPTER 1 COMPLEX NUMBERS THE MODULUS AND ARGUMENT THE POLAR FORM THE EXPONENTIAL FORM DE MOIVRE’S THEOREM FINDING ROOTS
Argument of Complex Numbers Definition 1.7 The argument of the complex number z = a + bj 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+jsin40) and z2 = 3(cos95+jsin95) . Find : If z1 = 6(cos60+jsin60) and z2 = 2(cos270+jsin270) . 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