Topic 6 Topic 6 Real and Complex Number Systems II 9.1 – 9.5, 12.1 – 12.2 Algebraic representation of complex numbers including: Cartesian, trigonometric (mod-arg) and polar form Cartesian, trigonometric (mod-arg) and polar form definition of complex numbers including standard and trigonometric form definition of complex numbers including standard and trigonometric form geometric representation of complex numbers including Argand diagrams geometric representation of complex numbers including Argand diagrams powers of complex numbers powers of complex numbers operations with complex numbers including addition, subtraction, scalar multiplication, multiplication and conjugation operations with complex numbers including addition, subtraction, scalar multiplication, multiplication and conjugation

Topic 6 Real and Complex Number Systems II

Definition i 2 = -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]

i = i i 2 = -1 i 3 = -i i 4 = 1 i 5 = i i 6 = -1 i 7 = -i i 8 = 1 Question : What is the value of i 2003 ?

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

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

Exercise FM P 168 Exercise 12.1 Exercise 12.1 ExerciseExerciseExerciseExercise NewQ P 227, 234 Exercise 9.1, 9.3 Exercise 9.1, 9.3

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

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

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

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

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

Exercise FM P 232 Exercise 9.2 Exercise 9.2

Division of complex numbers Try this on your GC

ExerciseExerciseExerciseExercise NewQ P 239 Exercise 9.4 Exercise 9.4

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

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 Since every row and column contains every element, it must be a group x1i-i 11i-i1-ii ii-i1 -i-ii1

ExerciseExerciseExerciseExercise NewQ P 245 Exercise 9.5 Exercise 9.5

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

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

Draw the 2 lines representing these numbers

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

Exercise New Q P300 Ex 12.1

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

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

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

r θ

Model: Express 3 cis /3 in standard form

Exercise New Q P306 Ex 12.2