Topic 6 Real and Complex Number Systems II 9.1 – 9.5, 12.1 – 12.2

1. Topic 6 Real and Complex Number Systems II9.1 – 9.5, 12.1 – 12.2 Algebraic representation of complex numbers including: • Cartesian, trigonometric (mod-arg) and polar form • definition of complex numbers including standard and trigonometric form • geometric representation of complex numbers including Argand diagrams • powers of complex numbers • operations with complex numbers including addition, subtraction, scalar multiplication, multiplication and conjugation

2. Topic 6 Real and Complex Number Systems II

3. Definition i2 = -1 i = -1 A complex number has the form z = a + bi(standard form) where a and b are real numbers We say that Re(z) = a [the real part of z] and that Im(z) = b [the imaginary part of z]

4. i = i i2 = -1 i3 = -i i4 = 1 i5 = i i6 = -1 i7 = -i i8 = 1 Question : What is the value of i2003 ?

6. Equality If a + bi = c + di then a = c and b = d Addition a+bi + c+di = (a+c) + (b+d)i e.g. 3+4i + 2+6i = 5+10i e.g. 2+6i – (4-5i) = 2+6i-4+5i = -2+11i Scalar Multiplication 3(4+2i) = 12+6i

8. Multiplication (3+4i)(2+5i) = 6+8i+15i+20i2 = 6 + 23i + -20 = -14 + 23i (2+3i)(4-5i) = 8-10i+12i-15i2 = 8 + 2i -15 i2 = 23 + 2i In general (a+bi)(c+di) = (ac-bd) + (ad+bc)i

9. Exercise NewQ P 227, 234 Exercise 9.1, 9.3 Exercise FM P 168 Exercise 12.1

10. Determine the nature of the roots of each of the following quadratics: (a) x2 – 6x + 9 = 0 (b) x2 + 7x + 6 = 0 (c) x2 + 4x + 2 = 0 (d) x2 + 4x + 8 = 0

11. Determine the nature of the roots of each of the following quadratics: (a) x2 – 6x + 9 = 0 (b) x2 + 7x + 6 = 0 (c) x2 + 4x + 2 = 0 (d) x2 + 4x + 8 = 0 (a) x2 – 6x + 9 = 0 = 36 – 4x1x9 = 0 ∴ The roots are real and equal [ x = 3 ]

12. Determine the nature of the roots of each of the following quadratics: (a) x2 – 6x + 9 = 0 (b) x2 + 7x + 6 = 0 (c) x2 + 4x + 2 = 0 (d) x2 + 4x + 8 = 0 (b) x2 + 7x + 6 = 0 = 49 – 4x1x6 = 25 ∴ The roots are real and unequal [ x = -1 or -6 ]

13. Determine the nature of the roots of each of the following quadratics: (a) x2 – 6x + 9 = 0 (b) x2 + 7x + 6 = 0 (c) x2 + 4x + 2 = 0 (d) x2 + 4x + 8 = 0 (c) x2 + 4x + 2 = 0 = 16 – 4x1x2 = 8 ∴ The roots are real, unequal and irrational [ x = -2  2 ]

14. Determine the nature of the roots of each of the following quadratics: (a) x2 – 6x + 9 = 0 (b) x2 + 7x + 6 = 0 (c) x2 + 4x + 2 = 0 (d) x2 + 4x + 8 = 0 (d) x2 + 4x + 8 = 0 = 16 – 4x1x8 = -16 ∴ The roots are complex and unequal [ x = -2  4i ]

15. Exercise FM P 232 Exercise 9.2

16. Division of complex numbers Try this on your GC

17. Exercise NewQ P 239 Exercise 9.4

18. Exercise • Prove that the set of complex numbers under addition forms a group • Prove that the set of complex numbers under multiplication forms a group

19. Model : Show that the set {1,-1,i,-i} forms a group under multiplication • Since every row and column contains every element , it must be a group

20. Exercise NewQ P 245 Exercise 9.5

21. Argand Diagrams Model : Represent the complex number 3+2i on an Argand diagram or

22. Model : Show the addition of 4+i and 1+2i on an Argand diagram

23. Draw the 2 lines representing these numbers

24. Complete the parallelogram and draw in the diagonal.This is the line representing the sum of the two numbers

26. Exercise New Q P300 Ex 12.1

27. Model : Express z=8+2i in mod-arg form (8,2)

28. Model : Express z=8+2i in mod-arg form (8,2)

29. Model : Express z=8+2i in mod-arg form (8,2)

30. r θ

31. Model: Express 3 cis/3 in standard form

32. Exercise New Q P306 Ex 12.2