1.0 Number System

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NUMBER SYSTEM

1.1 REAL NUMBERS

Natural Whole REAL Rational Irrational
Numbers numbers NUMBER (Q) ( ത )

(N) (W) S

Prime Integers
Number (Z)

Positive Negative
( +)⬚ ( −)

Types of Number Definition
Natural Numbers (N) The set of counting numbers or natural numbers and denoted by N.

N = {1, 2, 3 …}
Natural numbers include prime. Prime numbers are the natural numbers that are
greater than 1 and only can be divided by itself and 1.

Prime number = {2, 3, 5, 7 …}

Whole Numbers (W) The natural numbers, together with the number 0 are called the whole numbers.
The set of the whole number is written as follow: W = {0, 1, 2, 3 …}.

Integers (Z) The whole numbers together with the negative of counting numbers form the set
of integers and denoted by Z.

Z = {…, -3, -2, -1, 0, 1, 2, 3 …}
The set of positive integers is denoted by Z  = {1, 2, 3 …} and the set of negative
integers is denoted by Z  = {…, -3, -2, -1}. Hence Z = Z-  {0}  Z+.

The elements in Z can be classified as even and odd numbers where
the set of even numbers = {2k, k  Z}

the set of odd numbers = {2k + 1, k  Z}

Rational NUMBERS A rational number is any number that can be represented as a ratio (quotient) of
(Q) NUMBERS
two integers and can be written as Q=  a ; a, b  Z , b  0 .
irrational  b 
( ̅ ) 

Rational number can be expressed as terminating or repeating decimals for

example 5, 3 and  3 .
22

Irrational number is the set of numbers whose decimal representations are
neither terminating nor repeating. Irrational numbers cannot be expressed as a

quotient for example 3, 5 and  .

Real numbers (R) Consist of rational and irrational numbers

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Relationship of Number Sets

When we put the irrational numbers together with the rational numbers, it is a complete set of real numbers. Any
number such as a weight, a volume, or the distance between two points, will always be a real number. The
following diagram illustrates the relationships of the sets that make up the real numbers.

Example 1: Exercise 1:

For the set of {-5, -3, -1, 0, 3, 8}, identify the set of Express each of this number as a quotient a
b
(a) Natural numbers (b) Whole numbers
(a) 1.555…
= {3,8} = {0,3,8} (b) 5.45959…

(c) Prime numbers (d) Even numbers

. = {3} . = {0,8}

(e) Negative integers (f) Odd numbers
− = {−5, −3, −1} . = {−5, −3, −1,3}

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Algebraic Operation on Real Numbers
For all a, b, c and d  R

Properties of Real Numbers Addition Multiplication
1. Closure
2. Commutative a+b=c, cR ab = d , d  R
3. Associative
4. Distributive 6 + 7 = 13  R 6  7 = 42  R
5. Identity
a+b=b+a ab = ba
6. Inverse 2+5=5+2 25=52

(a + b) + c = a + (b + c) (ab)c = a(bc)
(1 + 3) + 2 = 1 + (3 + 2) (4  3)  2 = 4  (3  2)

a(b + c) = ab + ac

4  (2 + 3) = 4  2 + 4 3

a+0=0+a=a a1=1a=a
5+0=0+5=5 31=13=3

a + (–a) = 0 = (–a) + a a 1 =1= 1 a
7 + (–7) = 0 = (–7) + 7 aa

5 1 =1= 1 5
55

The Number Line
The set of numbers that corresponds to all point on number lines is called the set of real number. The real
numbers on the number line are ordered in increasing magnitude from the left to the right.

For example for –3.5, 2 and  can be shown on a real number line as
3

4 3 2 1 012 34

3.5 2 Description 
3 a equal to b
Symbol a less than b Example
a=b a greater than b 3=3
a<b 4 < 4
a>b 5>0

Note: The symbols ‘<’ or ‘>’ are called inequality sign

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All sets of real numbers between a and b, where a  b can be written in the form of intervals as shown in the
following table.

Type of Interval Notation Inequalities Representation on the number line
Closed interval [a, b] ab
Open interval (a, b) axb ab
Half-open interval (a, b] ab
Half-open interval [a, b) a<x<b ab
Open interval (, b) b
Half-open interval (, b] a<x  b b
Open interval (a, ) a  x<b a
Half- open interval [a, ) a
 < x < b

 < x  b

a<x<

a  x<

Note : The symbol  is not a numerical. When we write [a, ), we are simply referring to the interval starting
at a and continuing indefinitely to the right.

Example 2
List the number described and graph the numbers on a number line.

(a) The whole number less than 4
(b) The integer between 3 and 9
(c) The integer greater than -3

Solution
(a) W = {0 , 1, 2 , 3}

 

-3 -2 -1 0 1 234

Exercise 2
Represent the following interval on the real number line and state the type of the interval.

(a) [-1, 4] (b) {x : 2  x  5}

(c) [2, ) (d) {x : x  0, x  R}

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INTERSECTION AND UNION

Intersection and union operation can be performed on intervals.
For example,
A = [1, 6) and B = (2, 4),
Intersection of set A and set B is a half-open interval [1, 4).
The union of set A and set B is given by A  B = (2, 6) is an opened interval.

All these can be shown on a number line given below:

AB A B
AB 4 6

2 1

Example 3

Solve the following using the number line

(a) [0, 5)  (4, 7) (b) (, 5)  (1, 9)
(c) (, 0]  [0, ) (d) (4, 2)  (0, 4]  [2, 2)

Solution

(a)

0 45 7

 [0, 5)  (4, 7) = [0, 7)

Example 4
Given A = {x : -2  x  5} and B = {x : 0  x  7}. Show that A  B = (0, 5].

Solution

2 0 5 7
 (2, 5]  (0, 7] = (0, 5]

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1.2 INDICES NUMBER SYSTEM

Rules of Indices

1. × = + 6. √ = ( )

=

2. = − 7. √

=

3. ( ) = 8. ( ) =

4. = 9. ( ) =



5. − =


Example 5

Simplify:

(a) 35×36 = 37 (b) 18 2 5 = 6 −2 4 (c) (3 5)2 = 9 10
34 3 4
3
Exercise 3
(c) (81)4
Without using calculator, evaluate:

(a) 3 1 (b) (− 1)−31

9 2= 3 8

92

= 1

 9 1  3
2

=1
27

Exercise 4
Simplify the following expression

a) ( −2 1 − 32)3 ( −23 −1 1 )

2 3

b) 22 +4−24.22( −1)
10(2 )2

Answer: a) 9⁄2 b) 1
10 19⁄3

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Exercise 5 : Solve the following equations

a) 49 2 = 1 b) 27 +1 = 9 −1 c) 52 +1 = 6(5 ) − 1
73 −2

d) 4 +1 − 5(2 ) + 1 = 0 e) 2 − 3 + 2= 0 f) 9 +1 − 3 +3 − 3 + 3= 0

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1.2 SURD

 A number expressed in terms of root sign is radical or a surd
 For example, √4 = 2 and 3√27 = 3 are called radical expressiom
 However, radical expression in the form of √7 or 3√71 which equal to irrational number are known

as surds.

RULES OF SURDS: ADDITIONAL RULES:
√ × √ =
Property 1: √ = √ × √ ; , ≥ 0
√ + √ = 2√
Property 2: √ = √ √ ÷ √ = √
√
2
Property 3: √ + √ = ( + )√
(√ + √ ) = + + 2√
Property 4: √ − √ = ( − )√ (√ + √ )(√ − √ ) = −

Example 6: Remark: √ + ≠ √ + √
a) √45 = 3√5
b) √24 = MULTIPLYING RADICALS:
√ × √ = √
c) 6√7 + 2√7 =
√ × √ = √
d) 5√3 − √27 =

Example 7: Multiply
1. 3√6 × 5√7 = 15√42
2. 7√8 × 10√6 = 280√3
3. √3(4√7 − √3) =

4. 8√2(5√6 + √2) =

5. (2√3 + 4√2)(6√3 + 2√2) =

Exercise 6: Expand and Simplify (√8 − √3)(√8 + √3)

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RATIONALISING THE DENOMINATOR

 If the denominator of a fraction is a surd; we have to rationalize the denominator
 How??

~ Multiply the numerator and the denominator of the fraction with conjugate of the denominator
that will result the denominator to become a rational number.

State the conjugate and rationalize the denominator:

If Denominator To Obtain Denominator

Contains the Factor Multiply by conjugate Free from surds

√ √ √ × √ =
√ + √ − (√ ) − =

√ − √ +

√ − √ √ + √

In rationalizing the denominator of a quotient, be sure to multiply both the numerator and the denominator by the
same expression.

Exercise 7 : Rationalise

a) 5 b) 3 c) 1 d) 2√3
√3 2√3 7−√2 5−√3

SURD EQUATION
Exercise 8 : Solve the following equation:

a) √2 − 1 − 5 = 0
b) √3 + 1 + 1 =
c) √ + √ + 2 = 2
d) √ + 13 − √7 − = 2

Ans: a) = 13, b) = 5, c) = 1 d) = 3
4

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1.4 LOGARITHMS

Natural Logarithms
The natural logarithmic function and the natural exponential function are inverse functions of each other. The
symbol ln x is an abbreviation for logex, and we refer to it as the natural logarithms of x.

Definition of Common Logarithms Definition of Natural Logarithms

= log10 for every > 0 ln = log for every > 0

In general, = ↔ =

The Laws of Logarithms

1. Product rule : log = log + log

2. The Quotient rule : log = log − log


3. The power rule : log = log

4. Change of Base : log ≡ log
log

The following table lists the general properties for natural logarithmic form.

Logarithms with base a ( base : 10) ( base : e )
1) log 1 = 0 Common logarithms Natural logarithms
2) log = 1
3) log = log 1 = 0 ln 1 = 0
4) log =
log 10 = 1 ln = 1
log 10 = ln =

10log = ln =

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Exercise 9:
1. Write as a single logarithm.

+ −

2. Find the value of x if
a) 52 = 8
b) 3 +1 = 4 −1

3. Solve the following equation:
a)log3 − 4 log 3 + 3 = 0
b) 3 ln 2 − 4 = 2 ln 2
c) log4 + log 4 = 2.5
(Note: checking the last answer)

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1.5 COMPLEX NUMBERS

The set of all numbers in the Cartesian form, = + with real numbers a and b, and i the imaginary unit, is
called the set of complex numbers.

Algebraic Operations on Complex Number

Equality of Complex Numbers.

For two complex numbers:

1 = + and 2 = +
1 = 2 if and only a = c and b = d

Example 7 : Find a and b for the following equations

(a) + + ( − ) = 6 + 4 (b) + 2 + ( − ) = 9

+ = + =
− = − =

Solve simultaneously: ∴ = =
= , =

Conjugate of Complex Numbers (Z* @ ̅ )

The conjugate of a complex number 1 = + is ̅ 1 = −
The conjugate of a complex number 2 = − is ̅ 2 = +

Thus, 1 ̅ 1 = ( + )( − ) = 2 + 2
2 ̅ 2 = ( − )( + ) = 2 + 2

This fact is used in simplifying expressions where the denominator of a quotient is complex and 2 = −1

Example 8 : Simplify the expressions

a) 1 b) 4+7 c) 2+
2+5 3− √2

Solution: 4 + 7 2 − 5
1 − = 2 + 5 × 2 − 5

= × −
−

= − 2
= −

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Addition and Subtraction complex numbers

( + ) + ( + ) = ( + ) + ( + )
( + ) − ( + ) = ( − ) + ( − )

Multiplication of complex number
( + )( + ) = + + +
= + ( + ) −
= ( − ) + ( + )

Example 9

Given that z = 2 + 3i and w = 7- 6i , find

a) + = 2 + 3 + (7 − 6 ) b) − = 7 − 6 − (2 + 3 ) c) = (2 + 3 )(7 − 6 )
= 9 − 3 = 5 − 9 = 14 − 12 + 21 − 18 2

= 32 + 9

Example 10: Given that c) = 4 − 2 and = −7 + 5 , find
(a) = 4 + 3 and = 7 + 5 . Find

Example 11: Find the square roots of the complex number 6 + 8

Exercise 10:

1. Solve the complex number given that (1 + 2 ) = 2 + 5

2. Solve ( + )(3 − ) = 1 + 2 where x and y are real.

3. If = 12+− , find ̅ in the Cartesian form + .
1
4. If = 1 − 2 , express + in the form of +

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Argand Diagram

The complex number x  yi is representing as the vector  x  . For this reason it is known as the Cartesian Form
y

of Complex Number.

For example, the complex number z = 3+4i is represented as a point in the xy plane with coordinates (3,4) as shown
in Figure 1. That is, the real part, 3, is plotted on the x axis, and the imaginary part, 4, is plotted on the y axis.

y
5
4 P(3,4)

3

2

1

0 123 45 x

Figure 1. Argand diagram which represents the complex number 3+4i by the point P (3,4).
The complex number = + is plotted as a point with coordinates (a,b) as shown in Figure 2.

imaginary axis P(a,b)
b

0 a real axis

Figure 2. Argand diagram which represents the complex number a + bi by the point P(a,b).
The real part of z is plotted on the horizontal axis (the real axis). The imaginary part of z is plotted on the vertical
axis (imaginary axis). Such a diagram is called an Argand diagram. Engineers often refer to this diagram as the
complex plane.

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Example 22

Plot the complex numbers z1 = 2 + 3i, z2 = −3 + 2i, z3 = −3 − 2i, z4 = 2 − 5i, z5 = 6, z6 = i on an Argand diagram.

The Argand diagram is shown in Figure 3.

5

4
3

2
1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6
-1

-3-2 -2
-3

-4
-5 2-5

Figure 3. Argand diagram showing several complex numbers
Note that purely real numbers lie on the real axis. Purely imaginary numbers lie on the imaginary axis.

Another observation is that complex conjugate pairs (such as −3+2i and −3−2i) lie symmetrically about the x axis.

Finally, because every real number, a say, can be written as a complex number, a + 0i, that is as a complex number
with a zero imaginary part, it follows that all real numbers are also complex numbers. As such we see that complex
numbers form an extension of the sets of numbers with which we were already familiar.

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Modulus & Argument

The length OP is called the modulus of the complex number = + and is written | + |so that
| + | = | | = √ 2 + 2

The angle  is called the argument of + and is written ( + )

Thus ( + ) = = −1 ( ) ; − ≤ ≤



The Polar Form & Trigonometric Form of a Complex Number
If = + is any complex number such that | | = and ( ) = , then

Polar Form : = (cos + sin )
Trigonometric Form : = ∠
Cartesian Form : = +

Note: is a common abbreviation of +

Example 12 b) polar form.
Express = −1 − 2 in
a) Trigonometric Form

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