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The Complex Plane

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 ). z( a,b ). b. Re( z ). O(0,0). a. The Complex Plane. Definition 1.6 ( Modulus of Complex Numbers)

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The Complex Plane

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  1. The Complex Plane

  2. The Complex Plane • The complex number z = a + bican plotted as a point with coordinates z(a,b). • Re (z) x – axis • Im (z) y – axis Im(z) z(a,b) b Re(z) O(0,0) a

  3. The Complex Plane • Definition 1.6 (Modulus of Complex Numbers) The modulus of z is defined by Im(z) z(a,b) b r Re(z) O(0,0) a

  4. The Complex Plane • Example 1.7 : • Find the modulus of z:

  5. The Complex Plane • The Properties of Modulus

  6. Argument of Complex Numbers • Definition 1.7 The argument of the complex number z = a + bi is defined as 2nd QUADRANT 1st QUADRANT 4th QUADRANT 3rdQUADRANT

  7. Argument of Complex Numbers • Example 1.8 : • Find the arguments of z:

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

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

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

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

  12. 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 :

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

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

  15. THE EXPONENTIAL FORM Theorem 2 If and , then:

  16. THE EXPONENTIAL FORM Example 1.16 If and , find:

  17. 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 :

  18. DE MOIVRE’S THEOREM Example 1.17 If , calculate : If , calculate :

  19. 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

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

  21. FINDING ROOTS y nth roots of unity: Roots lie on the circle with radius 1 x 0 1 When k = 2: Sketch on the complex plane:

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

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

  24. FINDING ROOTS y x 0 Sketch on complex plane:

  25. Thank You Prepared By Smt. Sasmitakumari swain Lecturer in Mathematics Khemundi College

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