• INTRODUCTION• OPERATIONS OF COMPLEX NUMBER

• THE COMPLEX PLANE• THE MODULUS & ARGUMENT

• THE POLAR FORM

CHAPTER 1:COMPLEX NUMBER

CLASSIFICATION

COMPLEX NUMBERS

(C)

REAL NUMBERS

(R)

INTEGERS(Z)

RATIONAL NUMBERS

(Q)

IRRATIONAL

NUMBERS(Q)

WHOLE NUMBERS

(W)

NATURAL NUMBERS

(N)

To solve algebraic equations that do not have real solutions.

To solve Complex number:

Since,

INTRODUCTION TO COMPLEX NUMBERS

4

04

2

04

2

2

x

x

x

xReal solution

No real solution

ix 24

112 ii

Example 1 :Solve

8

5

)

)

ib

ia

Example :Solution

1)1()()

)1()()2428

2225

iib

iiiiia

Definition 1.1If z is a complex number, then the

standard equation of Complex number denoted by:

where a, b Ra – Real part of z (Re z) b – Imaginary part of z (Im z)

biaz

Example 1.2 :Express in the standard form, z:

493)

42)

b

a

Re(z) = 2, Im (z) = -2

Example 1.2 :Solution:

izzb

izza

1023403)

2242)

Re(z) = 3, Im (z) = 2√10

Definition 1.22 complex numbers are said equal if

and only if they have the same real and imaginary parts:

Iff a = c and b = d

dicbia

Example 1.3 :Find x and y if z1 = z2:

iyixb

iyixa

20105)

9432)

Definition 1.3If z1 = a + bi and z2 = c + di, then:

OPERATIONS OF COMPLEX NUMBERS

ibcadbdaczziii

idbcazzii

idbcazzi

)()()

)()()

)()()

21

21

21

Example 1.4 :Given z1 = 2+4i and z2= 1-2i

21

21

21

)

)

)

zzc

zzb

zza

Definition 1.4The conjugate of z = a + bi can be defined as:

***the conjugate of a complex number changes the sign of the imaginary part only!!!

biabiaz

Example 1.5 :Find the conjugate of

izd

zc

izb

iza

10)

10)

23)

2)

The Properties of Conjugate Complex Numbers

)Im(2

)

)Re(2

)

;)

11)

..)

)

)

)

2121

2121

2121

zzz

viii

zzz

vii

nzzvi

zzv

zzzziv

zzzziii

zzzzii

zzi

nn

Definition 1.5 (Division of Complex Numbers)

If z1 = a + bi and z2 = c + di then:

22

2

1

)(

dc

iadbcbdacdic

dic

dic

bia

dic

bia

z

z

Multiply with the conjugate of denominator

Example 1.6 :Simplify and write in standard form, z:

i

ib

i

ia

31

43)

1

2)

The complex number z = a + bi is plotted as a point with coordinates (a,b).Re (z) x – axis Im (z) y – axis

THE COMPLEX PLANEOR

ARGAND DIAGRAM

Im(z)

Re(z)O(0,0)

z(a,b)

a

b

Definition 1.6 (Modulus of Complex Numbers)

The modulus of z is defined by

THE MODULUS & ARGUMENT OF A COMPLEX NUMBER

22 bazr Im(z)

Re(z)O(0,0)

z(a,b)

a

b

r

Example 1.7 :Find the modulus of z:

izb

iza

53)

2)

The Properties of Modulus

2121

22

1

2

1

2121

2

)

)

0,)

)

)

)

zzzzvi

zzv

zz

z

zz

iv

zzzziii

zzzii

zzi

nn

Definition 1.7 (Argument of Complex Numbers)

The argument of the complex number z = a + bi is defined as

a

b1tan

1st QUADRANT

2nd QUADRANT

4th QUADRANT

3rd QUADRANT

900 18090

360270 270180

Example 1.8 :Find the arguments of z:

izd

izc

izb

iza

2)

1)

53)

2)

Based on figure above:

THE POLAR FORM OF COMPLEX NUMBER

b

a

(a,b)

r

Re(z)

Im(z)

a

b

rb

ra

1tan

sin

cos

The polar form is defined by:

Example 1.9:Represent the following complex number in

polar form:

izc

izb

iza

2)

53)

2)

,

@

sincos

rz

irz

Answer 1.9 :Polar form of z:

90sin90cos2)

96.120sin96.120cos34)

43.333sin43.333cos5)

izc

izb

iza

Example 1.10 :Express the following in standard form of

complex number:

270sin270cos2)

)180sin180(cos3)

)45sin45(cos2)

izc

izb

iza

Answer 1.10 :Standard form:

izc

zb

iza

2)

3)

22)

Theorem 1:If z1 and z2 are 2 complex numbers in

polar form where

then,

1111 sincos irz 2222 sincos irz

21212

1

2

1

21212121

sincos)

sincos)

ir

r

z

zii

irrzzi

Example 1.11 :a) If z1 = 4(cos30+isin30) and z2 =

2(cos90+isin90) . Find :

b) If z1 = cos45+isin45 and z2 = 3(cos135+isin135) . Find :

2

1

21

)

)

z

zii

zzi

2

1

21

)

)

z

zii

zzi

Answer 1.11 :

iz

zii

zzi

b

iz

zii

izzi

a

3

1)

3)

)

31)

344)

)

2

1

21

2

1

21

Think of Adam and Eve like an imaginary number, like the square root of minus one: you can never see any concrete proof that it exists, but if you include it in your equations, you can calculate all manner of things that couldn't be

imagined without it.

Philip PullmanIn The Golden Compass (1995, 2001),

372-373.

QUOTES