chapter 3

CHAPTER 3
REAL & COMPLEX NUMBER SYSTEM

TS. SYAMSIAH HUSSIN

3.1 Describe the concepts of real numbers

3.1.1 Identify type of real numbers
Real numbers is one type of numbers classification. The set of real numbers are:

a. Natural numbers
These are real numbers that have no decimal and are bigger than zero.

b. Whole numbers
These are positive real numbers that have no decimals, and also zero. Natural numbers are
also whole numbers.

c. Integers
The full set of numbers {. . . −3,−2,−1, 0, 1, 2, 3 . . .} is called the set of integers. It consists
of both positive and negative whole numbers, as well as zero.

d. Rational numbers (fractions)

Numbers that can be written in the form:

integer 1 −3
non − zero integer ( . . 2 , 4 )

are called rational numbers or fractions. All measurements of a physical nature (weight,

length, voltage, etc.) can only be expressed in terms of rational numbers.

e. Irrational numbers
Numbers which cannot be expressed as ratios of integers, are called irrational numbers.
Examples are √5 and π.

3.1.2 Define real number lines
A real number line, or simply number line, allows us to visually display real numbers by
associating them with unique points on a line.
To construct a number line, draw a horizontal line with arrows on both ends to indicate that
it continues without bound. Next, choose one point to represent the number zero; this point
is called the origin (normally, origin is at the centre of the line).

0
Mark off consistent lengths on both sides of the origin and label each tick mark to define
the scale. Positive real numbers lie to the right of the origin and negative real numbers lie to
the left. The number zero (0) is neither positive nor negative. Typically, each tick represents
one unit.

- - 3 -2 -1 0 1 2 3 4
4
In number line, numbers on the left are smaller than on the right.

3.1.3 Define graph of inequalities
The real numbers are ordered. That is, we can always say whether one number a is
less than, equal to, or greater than another given number b.
To denote this we use the ‘comparator’ symbols or inequalities:

a. Strict inequalities (< and >)
x > b means x is greater than b; x < b means x is less than b.

 Example: x < −8 “x is less than negative 8”
Therefore, we can define this on the graph of inequalities as follows.

-16 -12 -8 -4 0 4 8
define strict inequalities. As for the example given we put it right
Use an empty circle
above number -8.

b. Inclusive inequalities (≤ and ≥)
a ≥ b means a is greater than or equal to b, and similarly a ≤ b means a is less than or
equal to b.

 Example: x  3 “x is equal or more than 3”
Therefore, on the number line:

-1 0 1 2 3 4 5
Use a solid circle define inclusive inequalities.

 Example: -2 < x ≤ 4

-2 -1 0 12 3 4

c. Interval Notations

We can also represent the number inequality by using interval notations.

Notations Name Example Meaning

( , ) Open interval ( -2, 5) From -2 to 5, excluding -2 and

5.

[ , ] Closed interval [3, 9] From 3 to 9, including 3 and 9.

[ , ) Half-closed interval [0, 6) From 0 to 6, including 0 but 6

excluded.

( , ] Half-open interval ( -10, 7] From -10 to 7, including 7 but

not -10.

EXERCISE 3.1
Show graph of enequalities for x:
i) -5 ≤ x < 5
ii) 3.0 > x  - 0.5
iii) x < 0 , x  4
iv) – ¾ < x < 2 ¼
v) -  < x ≤ 2

3.2 Describe the concepts of complex numbers
First, let us key in this number on our calculator: √−1 . Of course, we will get “Math error” as
the answer. Because, there is no mathematical object that results in a negative number when
squared. In real life we don’t need such quantities – no one ever measured the length of a line to
be, for example √−1 metres. However, it does turn out that such ‘numbers’ are very useful in
electrical alternating current theory. As the solution, we replace √−1 with a symbol i (in some
other references uses j). Therefore:

Because when we square i we get −1 2 = −1

The object i is sometimes called ‘imaginary’, and is an example of a ‘complex number’. Examples

of imaginary numbers: 7 , 1.06 , −2.7 , 1 , (√5) , 1310
5

3.2.1 State that complex numbers are derived by combining the real parts with the
imaginary parts
Any number of this form = + where a and b are real numbers, is called a complex
number. We will usually write z rather than x for such a number since z always signals that
we are talking about complex numbers. A complex number of the form + is called a
Cartesian complex number.

= +

Real part Imaginary part

Real numbers are special cases of complex numbers with zero imaginary part.

Complicated?
Complex does not mean complicated.
It means the two types of numbers, real and imaginary, together form
a complex, just like a shopping complex (shops joined together in

one building).

 Example: Change the following expressions in term of imaginary number.
i) √−7
= √7 . √−1
= √7 @ 2.65 i

ii) √−49
= √49 . √−1
= 7

iii) √4√−6
= √4 . √6 . √−1
= 2 √6 @ 4.899 i

 Example: Express the following in terms of + .
i) 5 + √−9
= 5 + √9. √−1
= +

ii) −3 − √−16
= −3 − √16. √−1
= − −

3.2.2 Identify that product of two imaginary number is real number
We define the arithmetic operations with i such that the ordinary rules of arithmetic continue
to hold. Let's consider some consequences of this fact. We obtain in particular:
0 = 1
1 =
2 = −1
 Example:
3 = 2 × = (−1) × = −
4 = 2 × 2 = (−1) × (−1) = 1
5 = 4 × = 1 × =

3.2.3 State the conjugate of complex numbers
A conjugate is where we change the sign in the middle (if the number written in correct
order, + ). i.e. we change the sign of the imaginary number.
Complex Conjugate, ̅ :

A conjugate is often written with a bar over it. Example: ̅5̅̅−̅̅̅3̅̅ = ̅ = 5 + 3

 Example: Complex numbers and its conjugate.

i) = −1 + 3 ̅ = −1 − 3

ii) = 2 − 5 ̅ = 2 + 5

iii) = 7 + 4 ̅ = 7 − 4

iv) = −6 − 9 ̅ = −6 + 9

3.2.4 Describe the operations such as addition, subtraction, multiplication, division and
equivalent complex numbers
We work with complex numbers as we would with ordinary algebraic expressions
containing the variable i, except that we replace i2 with -1 wherever it occurs.

Let's look at the four basic operations. For the sake of illustration we will combine the
complex numbers = 4 − 3 and = −2 + 5 .

a. Addition & Subtraction
To add or subtract two complex numbers, just add or subtract the corresponding real and
imaginary parts.
 Example:
+ = (4 − 3 ) + (−2 + 5 )
= 4 − 2 − 3 + 5
= 2 + 2

− = (4 − 3 ) − (−2 + 5 )
= 4 + 2 − 3 − 5
= 6 − 8

b. Multiplication Write the answer in
We can apply basic method of expansion. Cartesian Form (a + bi)
 Example:
. = (4 − 3 )(−2 + 5 )
= −8 + 20 + 6 − 15 2
Remember, we have to replace i2 with -1.
. = −8 + 20 + 6 − 15(−1)
= −8 + 26 + 15
= 7 + 26

*Quicker way to solve complex number multiplication, use this rule:
( + )( + ) = ( − ) + ( + )

c. Division
The conjugate is used to help complex division.

 Example:

= −2+5
4−3

The trick is to multiply both top and bottom by the conjugate of the bottom.

= −2+5 × 4+3

4−3 4+3

= (−2+5 )(4+3 )

(4−3 )(4+3 )

= (−8−15)+(−6+20) Use quick formula
16−9 2 ( − ) + ( + )

= −23+14

16−9(−1)

= −23+14 Write the answer in
Cartesian Form (a + bi)
25

= −23 + 14

25 25

d. Equivalent complex number (complex equations).
If two complex numbers are equal, then their real parts are equal and their imaginary parts
are equal. Hence if + = + , then = and = .
 Example:
Solve the following complex equations.
i) 2( + ) = 6 − 3
ii) (1 + 2 )(−2 − 3 ) = +

Solution:

i) 2( + ) = 6 − 3 hence,

2 + 2 = 6 − 3

Equating the real parts gives:

2 = 6 therefore, = 3

Equating the imaginary parts gives:

2 = −3 therefore, = −3/2

ii) + = (1 + 2 )(−2 − 3 )
+ = −2 − 3 − 4 − 6 2
= −2 − 7 − 6(−1)
Hence + = 4 − 7
Equating real and imaginary terms gives:
= 4 and = −7

EXERCISE 3.2

1. In problems (a) to (f) evaluate in + form. Given 1 = 1 + 2 , 2 = 4 − 3 , 3 =

−2 + 3 and 4 = −5 − .

a) 1 + 2 − 3

b) 2 − 1 + 4

c) 3 4

d) 1 3 + 4

e) 1
2

f) 1+ 3
2− 4

2. Solve the following complex equations.

a) (2 + )(3 − 2 ) = +

2+
b) 1− = ( + )

c) (2 − 3 ) = √ +

d) ( − 2 ) − ( − ) = 2 +

ANSWERS:

1. a) 7 − 4 b) −2 − 6 c) 13 − 13 d) −13 − 2

e) −2 + 11 f) −19 + 43
85
25 25 85

2. a) = 8, = −1 b) = 3/2, = −1/2
c) = −5, = −12 d) = 3, = 1

3.3 Solve the complex numbers using Argand Diagram

3.3.1 Explain graphical representation of complex number through Argand’s Diagrams

An Argand Diagram is a plot of complex numbers as points.
The complex number = + is plotted as the point (a, b), where the real part is
plotted in the horizontal axis and the imaginary part is plotted in the vertical axis.

Thus, we call y-axis as imaginary axis (Im) and x-axis as real axis (Re).

Im

Re

3.3.2 Draw a straight line in an Argand’s Diagram to represent a complex number
For a complex number z = a + bi, draw a straight line from origin to the point (a, b).

Im
b (a , b)
0 a Re

3.3.3 Use Argand’s Diagrams to find the modulus and argument.

From the Argand’s Diagram, we can find the modulus of z (|z| or R) and the argument of
z ().
In order to find the argument, each quadrant have a unique rule.

Im First quadrant: = √ 2 + 2
b (a , b) Modulus of z,

Argument of z, = −1( )



 However in other quadrants, the formula of  is different.
0a
Re We treat the angle between the real axis and the |z| line as

a reference angle, .

Im Second quadrant:
Modulus of z,
= √(− )2 + 2

(-a , b) b

 Argument of z, = 180° −
-a 0 = 180° − −1( )
Re


Im Third quadrant:
Modulus of z,
 = √(− )2 + (− )2
Argument of z,
-a 0 Re = 180° +
 = 180° + −1( )

(-a , - -b
b)

Im Fourth quadrant:
Modulus of z,
a = √ 2 + (− )2
Argument of z,
 Re = 360° −
= 360° − −1( )
-b (a , -b)


Keypoint: The argument,  must be measured from
the positive real axis.  is only the reference angle.

 Example:

Calculate the modulus and argument for = −5 + 3 and sketch the Argand Diagram.

Solution:

Modulus, = √(−5)2 + 32

= √34 = 5.83 Eliminate minus sign is
recommended. Because,
Argument, = 180° −
= 180° − tan−1 3 tan−1 ( 3 ) = −30.96.

5 −5

= 180° − 30.96° Someone would likely make a
mistake by getting
= 149.04° ≈ 149°

180° − (−30.96)
Im

= 180 + 30.96

(-5 , 3) 3 = 210.96°

 Re This is wrong, because the
-5 0 complex number (in this case)

is in the 2nd quadrant.

EXERCISE 3.3
1. Sketch an Argand Diagram and plot the following complex numbers:

a) 1 = 3 + 2
b) 2 = −2 + 4
c) 3 = −3 − 5
d) 4 = 1 − 3
2. Find the modulus and argument for Z1, Z2, Z3 and Z4 as stated in question 1.
3. Given 1 = 2 + 4 and 2 = 3 − . Determine
(a) Z1+Z2,
(b) Z1−Z2,
(c) Z2−Z1
and show the results on an Argand diagram.

ANSWERS: Z2
1. Argand Diagram: Z1

Z4

Z3

2. Modulus and argument:
Z1: R= 3.61,  = 33.69
Z2: R= 4.47,  = 116.56
Z3: R= 5.83,  = 239.04
Z4: R= 3.16,  = 288.43

3. Result of
(a) Z1 + Z2 = 5 + 3
(b) Z1 − Z2 = −1 + 5
(c) Z2 − Z1 = 1 − 5

-1+
5i

5 + 3i

1 – 5i

3.4 Apply the concepts of complex numbers in other forms.

3.4.1 Describe complex numbers in the form of polar, trigonometric and exponential
Complex numbers can be written in four (4) forms.

Cartesian: Trigonometric:
= + = ( + )

* stated in the unit of degree

Polar: Exponential:
= ∠ =

* stated in the unit of degree * stated in the unit of radian

In trigonometric, polar and exponential form we must first find the modulus (R) and
argument (). From Cartesian to other forms, we must use the formula in 3.3.3 to get the
modulus and argument.

 Examples:

a) Express the complex number, = 2 − 4 in the forms of polar, trigonometric and
exponential.

Solution:

Modulus, = √22 + (−4)2 = √20 = 4.47 Im

−1(4) 2

Argument, = 360° − 2  Re

= 360° − 63.43° = 296.57° - 4 (2 , -4)

Polar form: = . ∠ . °
≈ √ ∠ °

Trigonometric: = √ ( ° + °)

Exponential:  in radian = 297° × = 5.2
180°

 = √ .

b) Express the complex number, = 7(cos 130° + sin 130°) in the forms of polar,
Cartesian and exponential.

Solution:

Modulus, = 7
Argument, = 130°

Polar form: = ∠ °

Cartesian: = (7 × cos 130°) + (7 × sin 130°)
= − . + .

Exponential:  in radian = 130° × = 2.3

180°

 = .

c) Express the complex number, = . in the forms of polar, trigonometric and
Cartesian.

Solution:

Modulus, = 6
Argument, = 3.5 × 180° = 200.5° ≈ 200°



Polar form: = ∠ °

Trigonometric: = ( ° + °)

Cartesian: = (6 × cos 200°) + (6 × sin 200°)
= − . − .

EXERCISE 3.4:

1. Plot the following complex numbers on the Argand plane and put them into polar form.
a) i
b) −3
c) 1 −
d) 2 +

2. Convert the following into Cartesian Form:
a) 20
b) 3 
c) 3 135
d) 1∠(− ⁄3)

3. Write = 3(cos 60° + sin 60°) in exponential form.

4. Express = 5 0.79 to trigonometric form.

ANSWERS: Im
1. Argand Diagram plot:

i 2+ i
0
Re
- 3i 1–
i

a) 1 90 b) 3∠270° c) √2∠315° d) √5∠26.6°
c) – 2.12 + 2.12 d) 0.5 – 1.22
2. a) 2 b) – 3
3. = 3 1.05

4. = 5(cos 45 + sin 45)

Reference:

https://math.libretexts.org/Bookshelves/Algebra/Book%3A_Beginning_Algebra_(Redden)/01%3A_
Real_Numbers_and_Their_Operations/1.01%3A_Real_numbers_and_the_Number_Line

Bill Cox, 2001. Understanding Engineering Mathematics. Reed Educational and Professional
Publishing Ltd, Britain

https://www.mathsisfun.com/numbers/complex-numbers.html

Mohamad, Afifa H. & Husin M. Afkar, 2020. Mathematical Computing 2nd Edition. Politeknik
METrO Kuala Lumpur, Malaysia.