SM015 - 1.2 (Notes Solutions)

1.2 Complex Numbers

Friday, July 10, 2020 11:31 AM

Learning Outcomes:
At the end of the lesson, students should be able to
(a) Represent a complex number in Cartesian form.
(b) Define the equality of two Complex Numbers.
(c) Express the conjugate of a Complex Number.
(d) Determine a Complex Number in polar form.

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Imaginary Numbers

• The imaginary number is denoted by , where

Remarks:



• Also note that

where whenever is the multiple of 4.

Thus, the value of where can be determined based on
the remainder obtained when

E.g.

1.2 Complex Numbers Page 1

Example 1

Friday, June 26, 2020 9:13 AM

Simplify:
(a)

Solutions:
(b)

Solutions:
(c)

Solutions:

1.2 Complex Numbers Page 2

Complex Numbers

Tuesday, June 23, 2020 2:45 PM

 A complex number consists of 2 parts:
(i) Real part
(ii) Imaginary part

 If is complex number, then can be written as
,

where
Real part :

Imaginary part :

E.g. ,
For , For

••
••

Note:
• Complex numbers are NOT ORDERED and cannot be represent on a
number line.
• Real Numbers are the subset of Complex Numbers,

Equality of Complex Numbers

 Given where .
If (i.e. ) , then

and .
.

E.g.

Remarks:

• Before comparison, 2 complex numbers must each simplified to the

form of , .

• Compare real parts on LHS with real parts in RHS.

The same comparison for imaginary parts.

1.2 Complex Numbers Page 3

Basic Operation in Complex Numbers , then

Tuesday, June 23, 2020 3:27 PM

Addition of Complex Numbers

 If and where

E.g. and .
Given

Example:
Evaluate

Solutions:

Subtraction of Complex Numbers , then

 If and where

E.g. and .
Given

Example:
Evaluate

Solutions:

1.2 Complex Numbers Page 4

Multiplication of Complex Numbers , then

Tuesday, June 23, 2020 3:27 PM

 If and where

E.g. and .
Given

Examples:
Evaluate
(i)

Solutions:

(ii)
Solutions:

Note:

1.2 Complex Numbers Page 5

Conjugate of a Complex Number

Tuesday, June 23, 2020 3:27 PM

Conjugate of Complex Numbers

 If where , then its complex conjugate is

denoted by or and it is written as

.

Remarks:
The conjugate of a complex number is obtained by changing the sign of
the imaginary parts.

E.g.

Example 2
State complex conjugate for the following:

(i)
(ii)
(iii)

Multiplication of Complex Conjugate Pair

E.g. . Given .
Given )( ) ( )( )

(

1.2 Complex Numbers Page 6

Division of Complex Numbers

Tuesday, June 23, 2020 3:27 PM

 The division of complex numbers are usually written in fraction
form. The division in such form can be complete by rationalizing
the fraction [get rid of at the denominator] .

(I) For , [Denominator with only imaginary part] Example:
Rationalize .
 Multiply to both the numerator
Solution:
and denominator, i.e

(II) For , [Denominator with both real & imaginary parts] Example:

 Multiply the conjugate of the Rationalize .

denominator [ to both the Solution:

numerator & denominator, i.e.

Example 3 (b)
Simplify the expression: Solution:
(a)

Solution:

1.2 Complex Numbers Page 7

Example 4 .

Friday, June 26, 2020 9:13 AM

If , find in the form
Solutions:

1.2 Complex Numbers Page 8

Example 5

Friday, June 26, 2020 9:13 AM

Solve the following equations for the complex number
(a)
Solutions:

(b)
Solutions:

1.2 Complex Numbers Page 9

Argand Diagram

Tuesday, June 23, 2020 3:27 PM

 A complex number can be plotted as a point with

coordinate in an Argand diagram where as the

horizontal axis and as the vertical axis.

2nd quadrant 1st quadrant

3rd quadrant 4th quadrant

E.g.
For

1.2 Complex Numbers Page 10

Polar Form

Tuesday, June 23, 2020 3:27 PM

 For a complex number ,
• Modulus of :

• Argument of :

[can be obtained based on ]
1st quadrant
where in the

1st quadrant :
2nd quadrant :
3rd quadrant :
4th quadrant :

2nd quadrant

3rd quadrant 4th quadrant

• Polar Form of :

where

1.2 Complex Numbers Page 11

Example 6

Friday, June 26, 2020 9:13 AM

Represent these complex numbers in Argand diagram.
Calculate the modulus and the argument of .
(a)

Solutions: Modulus of z ,

Argument of z ,

[ in 1st quadrant]

(b) Modulus of z ,
Solutions:

Argument of z , [to 3 d.p.]

(c) [ in 2nd quadrant]
Solutions:
Modulus of z ,

Argument of z ,

[ in 3rd quadrant]

(d) Modulus of z ,
Solutions:

Argument of z , [to 3 d.p.]

[ in 4th quadrant]

1.2 Complex Numbers Page 12

Example 7

Friday, June 26, 2020 9:13 AM

Express each of the following complex numbers in polar form:

(a)

Solutions: &

Based on Example 6,

(b) &
Solutions:

Based on Example 6,

(c) &
Solutions:

Based on Example 6,

(d) &
Solutions:

Based on Example 6,

1.2 Complex Numbers Page 13

Exercise 1.0

Friday, June 26, 2020 9:13 AM

Given and .
(a)
and

(b) and

Hence, express and in polar form.

Solutions (a):

Solutions (b):

1.2 Complex Numbers Page 14