SM015 - C 01 Notes

Chapter 1 - Number System

Friday, July 10, 2020 11:31 AM

1.1 Real Numbers

• Classification of Real Numbers
• Interval notation & set notation
• Number line.
• Union & Intersection of intervals

1.2 Complex Numbers

• Imaginary Number,

• Complex Number in Cartesian form,

• Basic operation on complex numbers.

• Conjugate of a complex number.

• Equality of complex numbers.

• Rationalize complex number

• Modulus of a complex Number,

• Argument of a complex Number,

• Complex Number in Polar form,

Chapter 1 - SM015 Page 1

1.1 Real Numbers

Friday, July 10, 2020 11:31 AM

Learning Outcomes:
At the end of the lesson, students should be able to
(a) Define natural numbers, whole number, integers, prime numbers,

rational numbers and irrational numbers.
(b) Represent rational and irrational numbers in decimal form.
(c) Represent the relationship of number sets in a real number

system diagrammatically.
(d) Represent open, closed and half-open intervals and their

representations on the number line.
(e) Find union and intersection of two or more intervals with the

aids of number line.

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Classification of Real Numbers - I

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Types Descriptions Number
System
Natural Numbers Positive Numbers used for counting.

Whole Numbers Zero and Natural numbers

Integers Zero, Natural numbers & their negatives. Real
Number C...

Rational Numbers Numbers that can be represented in the

form of where and .

The decimal representation can be either
(a) terminating decimals

E.g.

(b) non-terminating & repeating decimals
E.g.

Irrational Numbers Numbers that are NOT rational numbers.

The decimal representation is
non-terminating & non-repeating
decimals.

E.g.

Real Numbers The union of rational numbers and
irrational numbers

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Classification of Real Numbers - II

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Types Descriptions

Prime Numbers Natural numbers that have exactly 2
distinct natural number divisors

Note:
1 is NOT a prime number.
2 is the only even prime number.

Even Numbers Integers that are divisible by 2.

Odd Numbers Integers that are NOT even numbers.

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Example 1 , identify the set of

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For the set of

(a) Natural numbers
Solutions:
Natural numbers =

(b) Whole numbers
Solutions:
Whole numbers =

(c) Prime numbers
Solutions:
Prime numbers =

(d) Even numbers
Solutions:
Even numbers =

(e) Negative numbers
Solutions:
Negative numbers =

(f) Odd numbers
Solutions:
Odd numbers =

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Example 2 .

Tuesday, June 23, 2020 8:54 AM

Given

Find the set of
(a)

Solutions:

(b)
Solutions:

(c)
Solutions:

(d)
Solutions:

(e)
Solutions:

(f)
Solutions:

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Real Number Line

Tuesday, June 16, 2020 9:19 PM

 Real numbers can be represented geometrically by points
on a straight line, called as the real number line.

 The real numbers on the number line are ordered in
increasing magnitude from the left to the right.

 All sets of real numbers can be written in the form of
either finite intervals or infinite intervals:
(a) Finite Interval
• Open Interval
• Closed Interval
• Half Open Interval
(b) Infinite Interval

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Type of Intervals

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Closed Interval

Inequality Interval Set Notation Number Line
Notation

Open Interval

Inequality Interval Set Notation Number Line
Notation

Half-open Interval

Inequality Interval Set Notation Number Line
Notation

Infinite Interval

Inequality Interval Set Notation Number Line
Notation

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Intersection & Union of Intervals , is the

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Intersection
The intersection of two sets and denoted by
set of all elements which belongs to both and .

E.g. and .
Given

Union , is the set of
The union of two sets and denoted by
all elements which belongs to either or .

E.g. and .
Given

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Example 3

Tuesday, June 23, 2020 8:54 AM

By using number lines, find each of the following:
(a)

Solutions:

(b)
Solutions:

(c)
Solutions:

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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.



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.

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Example 1

Friday, June 26, 2020 9:13 AM

Simplify:
(a)

Solutions:
(b)

Solutions:
(c)

Solutions:

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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.

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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:

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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:

Chapter 1 - SM015 Page 15

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 )( ) ( )( )

(

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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:

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Example 4 .

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If , find in the form
Solutions:

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Example 5

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

(b)
Solutions:

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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

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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

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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]

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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,

Chapter 1 - SM015 Page 23

Exercise

Friday, June 26, 2020 9:13 AM

Given and .
(a)
and

(b) and

Hence, express and in polar form.

Solutions (a):

Solutions (b):

Chapter 1 - SM015 Page 24