CHAPTER 1

C O M P L E X NU M B E R

CHAPTER OUTLINE1.1 INTRODUCTION AND DEFINITIONS1.2 OPERATIONS OF COMPLEX NUMBERS1.3 THE COMPLEX PLANE1.4 THE MODULUS AND ARGUMENT OF A COMPLEX NUMBER1.5 THE POLAR FORM OF COMPLEX NUMBERS1.6 THE EXPONENTIAL FORM OF COMPLEX NUMBERS1.7 DE MOIVRE`S THEOREM1.8 FINDING ROOTS OF A COMPLEX NUMBER1.9 EXPANSION FOR COS AND SIN IN TERMS OF COSINES AND SINES 1.10 LOCI IN THE COMPLEX NUMBER

1.1 INTRODUCTION AND DEFINITIONS• Complex numbers were discovered in the

sixteenth century.• Purpose:- Solving algebraic equations which

do not have real solutions.• Complex number, as z, in form of • The number a is real part while b is imaginary

part which is combine with j as bj. where and • By combining the real part and imaginary

part, it can solve more quadratic equations.

z a bj

2 1 j 1j

Example 1.1Write down the expression of the square roots ofi. 25 ii. -25

Definition 1.1If z is a complex number then Where a is real part and b is imaginary part.

Example 1.2Express in the form i. ii.

z a bj

9

12

z a bj

2

2

iii. 36 v. 9 0

iv. 28 vi. 2 2 0

x

x x

Exercise 1.1 :Simplify

Exercise 1.2:Express in the form

7 3

5 4

8 5

10

i. v.

ii. vi.

iii. vii. (2 )

iv.

j j

j j

j j

j

z a bj

i. 7 64

ii. 24 45

Definition 1.2

• Two complex numbers are said to be equal if and only if they have the same real and imaginary parts.

Example 1.3Given 5x+2yj = 15 + 4j

Exercise 1.3Given 3x + 7yj = 9 + 28j

1.2 OPERATIONS OF COMPLEX NUMBERS Definition 1.3If

Example 1.4Given and . Findi. ii.iii.iv. Determine the value of

1 2 and z a bj z c dj

1 2

1 2

21 2

. ( ) ( )ii . ( ) ( )

iii. ( )( ) ( ) ( )

i z z a c b d jz z a c b d j

z z a bj c dj ac adj bcj bdj ac bd ad bc j

1 3 5z j 2 1 2z j

1 2z z

1 2z z

1 2*z z3(1 5 ) (4 2 )( 1 8 )z j j j

Definition 1.4The complex conjugate of z = a + bj can defined as

Example 1.5Find the complex conjugate of

i.ii.iii.iv.

3 7z j 5z j2 8z j

1 6z j

z a bj a bj

Exercise 1.4 (complex conjugate):Find the complex conjugate of

1i. 32

ii. 12 5iii. 1iv. 45jv. 101

j

jj

Definition 1.5: Division of Complex NumbersIf then

Example 1.6Find the following quantities.

Exercise 1.5

1 2 and z a bj z c dj

12

2

2 2

, 0,

=

( ) =

z a bj zz c dj

a bj c djc dj c djac bd bc ad j

c d

4 3 1i. ii. 1 2 5

j jj j

9 4 3 2i. ii. 1 5 2 7 4 3

j jj j j j

1.3 THE COMPLEX PLANE

• A useful way to visualizing complex numbers is to plot as points in a plane.

• The complex number, is plotted as coordinate (a,b).

• The x-axis called real axis, y-axis called the imaginary axis.

• The Cartesian plane referred as the complex plane or z-plane or Argand diagram.

z a bj

Example 1.7Plot the following complex numbers on an Argand diagram.

Example 1.8 Given that and that are two complex numbers. Plot in an Argand diagram.

i. 4 ii. 2 2 iii. 3 iv. 2 3j j j

1 2 4z j 2 3 2z j

Additional Exercises :1. Represent the following complex numbers on an Argand

diagram:

2. Let a) Plot the complex numbers on an Argand diagram and label them.b) Plot the complex numbers and on the same Argand diagram.

(a) 3 2 (b) 4 5 (c) 2z i z i z i

1 2 3 45 2 , 1 3 , 2 3 , 4 7z i z i z i z i 1 2 3 4, , ,z z z z

1 2z z 1 2z z

1.4 THE MODULUS AND ARGUMENT OF A COMPLEX NUMBER

Definition 1.6 Modulus of Complex NumbersThe norm or modulus or absolute value of z is defined by

Modulus is the distance of the point (a,b) from the origin.

2 2r z a b

Example 1.9Find the modulus of the following complex numbers.

Exercise 1.6Find the modulus of the following complex numbers.

i. 12 5 ii. 1 10j j

i. 3 4 ii. 5j j

Definition 1.7 Argument of Complex NumbersThe argument of the complex number, is defined as

Example 1.10Find the arguments of the following complex numbers

Exercise 1.7Find the arguments of the following complex numbers

z a bj

1tan ba

i. 2 3 ii. 6 5j j

i. -2 ii. 5j j

Additional Exercises:Find the modulus and argument of complex number below:

(a) 5 4(b) 2 7(c) 5 2(d) 3 7( ) 4 2

z jz jz jz j

e z j

:

) 41, 218.66

) 53, 74.05

) r 29, 158.2

) 58, 293.2

) 20, 153.43

Ans

a r

b r

c

d r

e r

1.5 THE POLAR OF COMPLEX NUMBERS

Example 1.11Represent the following complex numbers in polar form.

Exercise 1.8State the following complex numbers in polar form.

Example 1.12Express the following in form.

Exercise 1.9

i. 2 2 ii. 5 12z j z j

i. 3 ii. 9 3z j z j

z a bj i. 6(cos60 + sin 60 ) ii. 2(cos135 + sin135 )z j z j

i. 8(cos90 + sin90 ) ii. 3(cos75 + sin 75 )z j z j

Example 1.13Given that Find

Exercise 1.10IfFind

1 23(cos30 sin30 ) 6(cos90 sin90 ).z j z j

11 2

2

i. ii. zz zz

1 24(cos60 sin 60 ) and 5(cos135 sin135 ).z j z j

11 2

2

i. ii. zz zz

Additional Exercises:1. Write the following numbers in form:

2. Express the numbers and in the polar form. Find

,r

(i) 7 2(ii) 3(iii) 4 6

(iv) 3

jj

j

j

1 1 3z j 2 3 3z j

1 2z z

:

i. ( 53,15.95 )

ii. ( 10,341.57 )

iii. ( 52,123.69 )

iv. (2,210 )

Ans

z

z

z

z

: 6 2(cos 285 sin 285 )Ans j

1.6 THE EXPONENTIAL FORM OF COMPLEX NUMBERSDefinition 1.8The exponential form of complex number can be defined asWhere is measured in radians and

Example 1.14State the following angles in radians.

Example 1.15 (Exercise 1.12 in Textbook)

jz recos sinje j

i. 150

ii. 45

i. 1 ii. 4 5 z j z j

Theorem 2If and , then

Example 1.16 (Exercise 1.13 in Textbook)If

1 2

1 2

( )1 2 1 2

( )1 1

22 2

i.

zii. , 0z

j

j

z z r r e

re zr

11 1

jz re 22 2

jz r e

3 21 25 and 4 ,find

j jz e z e

1 2

1

2

i.

ii.

z zzz

Additional Exercises:1. Write in exponential form:

2. Given that

Answer:1. 2.i. i. ii. Ii.

i. 1+ ii. 1 3j j

23 6

1 23 and 3j j

z e z e

11 2

2

i. ii. zz zz

42j

z e

232

jz e

29j

z e

56

jz e

1.7 DE MOIVRE’S THEOREMTheorem 3If is a complex number in polar form

to any power n, then with any value n.

Example 1.17(Exercise 1.14 in Textbook)If

(cos sin )z r j

(cos sin )n nz r n j n

2(cos 25 sin 25 ). Calculatez j 3

1 5i. ii. z z

Identity Trigonometrycos( ) cossin( ) sin

Additional Exercises:

1. If

Ans: i. 1.732+j ii. -642.

Ans:3. Calculate the

Ans: 32 ab

4(cos60 sin 60 ). Calculatez j 1

32i. ii. z z101 1Find .

2 2j

8 2Im(( 1) ) for j z z a bj

132

j

1.8 FINDING ROOTS OF A COMPLEX NUMBERTheorem 4If the n root of z is (cos sin )then, z r j

1 1

1 1

360 360cos sin if in degrees

or

2 2cos sin if in radians

for 0,1,2,..., 1

n n

n n

k kz r jn n

k kz r jn n

k n

Example 1.18(Exercise 1.15 in Textbook):Findi. The square roots of ii. The cube roots of

Additional Exercises:1. Find the square roots of

Ans: i. 5.6568 +5.6568j, -5.6568-5.6568j ii. 3.5355 +5.5355j, -3.5355-3.5355j

81z j64z j

i. 64 ii. 25z j z j

1.10 LOCI IN THE COMPLEX NUMBERDefinition 1.9A locus in a complex plane is the set of points that have a

specified property. A locus of a point in a complex plane could be a straight line, circle, ellipse and etc.

Example 1.20If find the equation of the locus defined by:

,z a bj 2i. 11

ii. (2 3 ) 2

z jzz j

2 2

2 2

2 2

straight line

( ) ( ) circle

1 ellipse

y mx c

x h y k

x ya b

Additional Exercises:If , find the equations of the locus defined by:z a bj

1i. 1 2 ii. 2 2 iii. 21

zz j z z z jz

:i. straight line with slope 1

2 4 20ii. circle with centre - , ,3 3 95 4iii. circle with centre ,0 , radius is 3 3

Ans

radius