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
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