Topic 6 Topic 6 Real and Complex Number Systems II9.1 – 9.5, 12.1 – 12.2

Algebraic representation of complex numbers Algebraic representation of complex numbers including:including:

• Cartesian, trigonometric (mod-arg) and polar formCartesian, trigonometric (mod-arg) and polar form

• definition of complex numbers including standard definition of complex numbers including standard and trigonometric formand trigonometric form

• geometric representation of complex numbers geometric representation of complex numbers including Argand diagramsincluding Argand diagrams

• powers of complex numberspowers of complex numbers

• operations with complex numbers including addition, operations with complex numbers including addition, subtraction, scalar multiplication, multiplication and subtraction, scalar multiplication, multiplication and conjugationconjugation

Topic 6Topic 6

Real and Complex Number Systems II

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]

i = i i2 = -1i3 = -i i4 = 1

i5 = i i6 = -1i7 = -i i8 = 1

Question : What is the value of i2003 ?

i

i

a

acbbx

xxSolveModel

12

222

42

2

12422

2

4

022:

2

2

2

Equality If a + bEquality If a + bii = c + d = c + dii

then a = c and b = dthen a = c and b = d

Addition a+bAddition a+bi i + c+d+ c+dii = (a+c) + = (a+c) + (b+d)(b+d)ii

e.g. 3+4e.g. 3+4ii + 2+6 + 2+6ii = 5+10 = 5+10ii

e.g. 2+6e.g. 2+6ii – (4-5 – (4-5ii) = 2+6) = 2+6ii-4+5-4+5ii

= -2+11= -2+11ii

Scalar Multiplication 3(4+2Scalar Multiplication 3(4+2ii) = 12+6) = 12+6ii

Multiplication (3+4Multiplication (3+4ii)(2+5)(2+5ii) ) = 6+8= 6+8ii+15+15ii+20+20ii22

= 6 + 23= 6 + 23ii + -20 + -20 = -14 + 23= -14 + 23ii

(2+3(2+3ii)(4-5)(4-5ii)) = 8-10= 8-10ii+12i-15+12i-15ii22

= 8 + 2= 8 + 2ii -15 -15 ii22

= 23 + 2= 23 + 2ii

In general (a+bIn general (a+bii)(c+d)(c+dii) = (ac-bd) + ) = (ac-bd) + (ad+bc)(ad+bc)ii

EExxeerrcciissee

FM P 168FM P 168

Exercise 12.1Exercise 12.1

EExxeerrcciissee

NewQ P 227, 234NewQ P 227, 234

Exercise 9.1, 9.3Exercise 9.1, 9.3

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

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 ]

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 ]

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 ]

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 ]

EExxeerrcciissee

FM P 232FM P 232

Exercise 9.2Exercise 9.2

Division of complex numbersDivision of complex numbers

i

i

iii

i

i

i

ii

iModel

25

17

25

625

176916

68912

34

34

34

2334

23

2

Try this on your GC

EExxeerrcciissee

NewQ P 239NewQ P 239

Exercise 9.4Exercise 9.4

ExerciseExercise

• Prove that the set of complex Prove that the set of complex numbers under addition forms a numbers under addition forms a groupgroup

• Prove that the set of complex Prove that the set of complex numbers under multiplication forms a numbers under multiplication forms a groupgroup

Model : Show that the set {1,-1,Model : Show that the set {1,-1,ii,-,-ii} } forms a group under multiplicationforms a group under multiplication

• Since every row and column contains every element , it Since every row and column contains every element , it must be a groupmust be a group

xx 11 -1-1 ii -i-i

11 11 -1-1 ii -i-i

-1-1 -1-1 11 -i-i ii

ii ii -i-i -1-1 11

-i-i -i-i ii 11 -1-1

EExxeerrcciissee

NewQ P 245NewQ P 245

Exercise 9.5Exercise 9.5

Argand DiagramsArgand Diagrams

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

or

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

x

y

-6 -4 -2 0 2 4 6

-4

-2

0

2

4

Draw the 2 lines representing these numbersDraw the 2 lines representing these numbers

x

y

-6 -4 -2 0 2 4 6

-4

-2

0

2

4

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

x

y

-6 -4 -2 0 2 4 6

-4

-2

0

2

4

EExxeerrcciissee

New Q P300New Q P300

Ex 12.1Ex 12.1

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

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-4

-2

0

2

4

(8,2)

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

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-4

-2

0

2

4

(8,2)

cisr

ir

irr

iyxi

ryrxr

y

r

x

sincos

sincos

28

sincos

sincos

r

x

y

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

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-4

-2

0

2

4

(8,2)

o

o

cisiz

r

146828

14

tan

68

28

82

22

r

i

i

i

i

ii

formini

ExpressModel

3

13

)3(4

3

3

3

4

3

4

argmod3

4:

6

6

31

22

2

tan

2

1)3(

3

ciscisr

r

cisri

x

y

-1 0 1 2 30

1

2

3

r

θ

Model: Express 3 cis /3 in standard

form

i

i

i

cis

233

23

23

21

33

3

)(3

)sin(cos3

3

EExxeerrcciissee

New Q P306New Q P306

Ex 12.2Ex 12.2