Complex Numbers

1. Complex Numbers If we wish to work with , we need to extend the set of real numbers Definitions i is a number such that i2 = -1 C is the set of numbers Z, of the form where a and b are real numbers. a is called the real part of Z and we write a = R(z) of a = Re(z) b is called the imaginary part of Z and we write b = i(z) or b = Im(z)

2. and Given Addition is defined by: Multiplication is defined by: We may write a + bi or a + ib, whichever we find more convenient

3. a) Given find (i) (ii) (iii) (i) (ii) (iii)

4. b) Solve the equation Using the quadratic formula

5. Page 90 Exercise 1 Questions 1, 2, 3, 6, 7, 8

6. Complex Conjugate When , then its complex conjugate is denoted by Note: This is useful when we wish to carry out a division.

7. a) Calculate

8. b) Calculate where Let Then also Equating parts we get:

9. Since

10. Page 91 Ex 2 Questions 1(a), (b), (c), 2(c), (e) 3(a), (b), (f), 5(a), (b) TJ Exercise 2. TJ Exercise 1 - if needed.

11. The complex number is represented on the plane by the point P(x,y). The plane is referred to as “The Complex Plane”, and diagrams of this sort are called Argand Diagrams. p r y x Argand Diagrams Any point on the x-axis represents a purely Real Number Any point on the y-axis represents a purely imaginary number

12. The size of the rotation is called the amplitude or argument of z. It is often denoted Arg z. This angle could be p r y x We refer to the value of Arg z which lies in the range -< as the principal argument. It is denoted arg z, lower case ‘a’. By simple trigonometry: This is referred to as the Polar form of z.

13. a) Find the modulus and argument of the complex number b) Find the modulus and argument of the complex number Since (3,4) lies in the first quadrant, n = 0 Since (-3,-4) lies in the third quadrant, n = -1

14. (2,2) is in Q1

15. Page 94 Exercise 3 Questions 3a, b, d, e, i 6a, b, f 7a, b, c

16. y y 4 -4 4 -4 x x -4 4 -4 4 Loci-Set of points on the complex plane This is a circle, centre the origin radius 4 (i) (ii)

17. y -1 5 x This is a circle centre (2, 0) radius 3 units.

18. This is a straight line through the origin gradient

19. Page 96 Exercise 4 Questions 1a, b, d, f, j 3a, b, 4a, c TJ Exercise 7

20. Polar Form and Multiplication and Note arg(z1z2) lies in the range (-, ) and adjustments have to be made by adding or subtracting 2  as appropriate if Arg(z1z2) goes outside that range during the calculation.

21. Note:

22. Now turn to page 96 Exercise 5 Questions 1 and 2.

23. Let us now look at question 3 on page 99. This leads to the pattern:

24. De Moivre’s Theorem

25. a) Given find

26. Round your answer to the nearest integer b) Given find

27. Page 101 Exercise 6 questions 1 to 3, 4g, h, i, j.

28. Roots of a complex number then It would appear that if

29. The solutions are  radians apart,

30. By De Moivre’s theorem, when finding the nth root of a complex number we are effectively dividing the argument by n. We should therefore study arguments in the range (-n, n) so that we have all the solutions in the range (-, ) after division. The position vectors of the solution will divide the circle of radius r, centre the origin, into n equal sectors.

31. For k = 0 For k = 1

32. For k = 2

34. Page 106 Exercise 7: Question 2 plus a selection from 1

35. Polynomials In 1799 Gauss proved that every polynomial equation with complex coefficients, f(z) = 0, where z  C, has at least one root in the set of complex numbers. He later called this theorem the fundamental theorem of algebra. In this course we restrict ourselves to real coefficients but the fundamental theorem still applies since real numbers are also complex.

36. We need to find z2, z3 and z4 And substitute them into the Original equation.

38. Using Division Hence the complimentary real factor is Hence all four roots are:

39. Page 108 Exercise 8 Questions 2, 3, 4, 5 and 6 Review on Page 110