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Elementary Mathematics
Chapter 1
CHAPTER 1:
FUNDAMENTAL CONCEPTS OF ALGEBRA
1.1 Real Numbers
Set
Before we describe the set of real numbers, let’s be sure you are
familiar with some basic ideas about sets. For example, the set of
digits consist of the collection of numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and
9. If we use the symbol A to denote the set of digits, then we can
write
A  0,1,2,3,4,5,6,7,8,9
the braces { } are used to enclose the objects, or elements.
In listing the elements of set, we do not list an element more
than once because the elements of a set are distinct. Also, the order in
which the elements are listed is not relevant. Thus, for example {2,3}
and {3,2} both represent the same set.
If every element of a set A is also element of set B, then we
say that A is subset of B or represent as A  B . For example {1,2}
subset of {1,2,3,4}. If two set A and B have the same elements, then
we say that A is equal to B or represent as A = B. For example, {1, 2,
3} is equal to {3, 1, 2}.
1.1.1 Natural numbers
Natural numbers is denoted by N and can be written as N  {1,2,3,...}
which is the number from 1 until infinity. Natural numbers include
prime numbers and non prime numbers.
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Elementary Mathematics
Chapter 1
In mathematics, a prime number (or a prime) is a natural
number that has exactly two distinct natural number divisors: 1 and
itself. The first twenty-five prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97.
For example
2 is a prime number because can only be divided by 2 and 1.
6 is not a prime number because 6 can be divided by 2 and 3.
1.1.2 Whole numbers
The set of whole numbers is a set W = {0}  N = {0, 1, 2, 3,…}
which is the number from 0 until infinity.
1.1.3 Integers
The combination of whole numbers and their negatives are known as
integers. The set of all integers is denoted by Z.
Z= {..., –3, –2, –1, 0, 1, 2, 3,…}
Set of positive integers is denoted by Z+= {1, 2, 3,…}
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, with k  Z }
The set of odd numbers = {2k + 1, with k  Z }
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1.1.4 Rational numbers
The number system can be expended further to set of all rational
numbers denoted by Q. Where
a

Q   ; a, b  Z , b  0
b

For example 5,
3
3
and  . Rational numbers can be expressed as
2
2
terminating or repeating decimals. For example,
2
 0.6666...  0.6 .
3
1.1.5 Irrational numbers
Irrational numbers is denoted by Q . This is the set of all numbers
whose decimal representations are neither terminating nor repeating.
Irrational numbers cannot be expressed as a quotient of integers. For
example,
2 , 3 ,  and

.
2
We can show that N  W  Z  Q  R in figure 1.1 below which is
the Venn diagram of Number Sets.
R
Q
Z
Figure 1.1
W
N
Venn diagram of Number Sets
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Elementary Mathematics
Chapter 1
Q
Q
Figure 1.2
QQ  R
Figure 1.2 shows that Q  Q  R .
Figure 1.3 below shows the relationship of various types of numbers.
Real Numbers
Rational Numbers
{
2 1
, , 2,...}
3 2
Integers
{...,1,0,1,2,...}
Irrational Numbers
{3 5 , 2 ,  ,..}
Fractions
{
2 1
, ,...}
3 2
Whole Numbers
{0,1,2,3,...}
Negative Integers
{..., 2,1}
Natural Numbers
{1,2,3,...}
Zero
{0}
Figure 1.3
The relationship of various types of numbers
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Elementary Mathematics
Chapter 1
Example 1
State 4 natural numbers and determine whether it is prime
number or not a prime number.
Solution
4 = not a prime number;
5 = prime number
17 = prime number;
25 = not a prime number
Example 2
For the set {-5, -3, -1, 0, 3, 8}, identify the set of
(a) Natural numbers
(b) Whole numbers
(c) Prime numbers
(d) Even numbers
(e) Negative integers
(f) Odd numbers
Solution
(a) { 3, 8 }
(b) { 0, 3, 8}
(c) { 3 }
(d) { 0, 8 }
(e) { -5, -3, -1 }
(f) { -5, -3, 3 }
Example 3
1


Given S =  9, 7 , ,  2 ,0,4,5.125125, identify the set of
3


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Elementary Mathematics
Chapter 1
(a) natural numbers
(b) whole numbers
(c) integers
(d) rational numbers
(e) irrational numbers
(f) real numbers
Solution
(a) { 4 }
(b) { 0, 4 }
(c) { -9, 0, 4}
(d) { -9, 0, 4,
 7 , 
1
, 5.125125 }
3
2
(e)
1


(f)  9, 7 , ,  2 ,0,4,5.125125  S
3


Example 4
Express each of the following numbers as a quotient
a
b
(a) 1.5555…..
(b) 5.45959..
Solution
(a)
Let
(1)  10
x  1.555....  1.5
(1)
10x  15.555...
(2)
therefore, (2) – (1), 9x  14
x
14
9
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Elementary Mathematics
Chapter 1
x  5.45959...  5.459
(1)
(1)  10
10x  54.59...
(2)
(2)  100
1000x  5459.59....
(3)
(b)
Let
therefore, (3)-(2), 990x  5405
x
5405 1081

990
198
1.1.6 Algebraic Operation on Real Numbers
For all a, b  R
Properties Of
Description
Example
Real
Numbers
Closure
Commutative
Associative
Distributive
Identity
Inverse
a  b  c; c  R
6  7  13
(a )b  d ; d  R
(6)(7)  42
ab  ba
24  42
ab  ba
5(10)  (10)5
(a  b)  c  a  (b  c)
(1  3)  2  1  (3  2)
(ab)c  a (bc)
(4  2)  3  4  (2  3)
a(b  c)  ab  ac
2(6  8)  (2  6)  (2  8)
a(b  c)  ab  ac
3(4  5)  (3  4)  (3  5)
a0  0a  a
90  09  9
a(1)  (1)a  a
 5(1)  1(5)  5
a  (a)  0  (a)  a
10  (10)  0  (10)  10
1
1
a   1    a, a  0
a
a
1
1
5   1   5
5
5
Example 5
Given a, b  R, ab  1. Prove that a  b 1 .
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Elementary Mathematics
Chapter 1
Solution
Given ab  1
ab  1   1 1 
b
b
1 1

a b     
b b

(Associative and identity)
1
a 1   
b
(inverse)
1
a 
b
(identity)
a  b 1
1.1.7 The Number Line
For example –3.5,
2
and  can be shown on real number line as
3
|
|
|
|
|
|
|
|
|
-4
-3
-2
-1
0
1
2
3
4
2
3
- 3.5

The order of real numbers is important in presenting the interval on a
number line.
The Order of Real Number
Symbol
Description
Example
a=b
a is equal to b
3=3
a<b
a is less than b
-4 < 4
a>b
a is greater than b
5>0
Note: The symbols ‘<’ or ‘>’ are called inequality sign.
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Elementary Mathematics
Chapter 1
All real numbers between a and b can be written in the form of
intervals as shown in the following table.
Type of
Notation
Inequalities
[ a, b]
a xb
Representation on the number line
interval
Closed
interval
Opened
( a, b)
( a, b]
[ a, b)
a
b
a
b
a xb
Or
Half-opened
b
a xb
interval
Half-closed
a
a xb
interval
Opened
(  , b )
a
b
a
b
a
b
  x  b
interval
( a,  )
Half-closed
(, b]
ax
  x  b
Or
Half-opened
interval
a
[ a,  )
b
ax
a
b
Example 6
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Elementary Mathematics
Chapter 1
Represent the following interval on real number line and state
their type.
(a) [1,4]
(b) ( 2,5)
(c) [2, )
(d) {x : x  0, x  R}
Solution
(a)
-1
4
[-1,4] is a closed interval
(b)
2
5
(2,5) is an opened interval
(c)
2
[2, ) is a half-open interval
(d)
0
{x : x  0, x  R} = (,0] is a half-close interval
Intersection and union operations can be perform on intervals.
If A = [1, 6) and B = (-2, 4). Intersection of set A and set B is a halfopened interval A  B = [1, 4).
Union of set A and set B is a A  B = (-2, 6).
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Elementary Mathematics
Chapter 1
All this can be shown on a number line given below:
B
A
-2
1
4
6
A B
A B
Example 7
Simplify the following using the number line.
(a) [0,5)  (4,7)
(b) (,5)  (1,9)
Solution
(a)
0
4
5
7
 [0,7)
(b)
-1
5
9
 (1,5)
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1.2 Exponents and Surds
1.2.1 Exponents
Exponents such as 2, 3, 4 and so on are used to indicate repeated
multiplication. For example,
25  2  2  2  2  2  32 .
The 2 that is repeated when multiplying is called the base. The small
number above and to the right of the base that is 5 are called
exponent or power. The exponent tells the number of times the base
is to be used when multiplying. So, a x is an expression in which a is
the base and x is the power.
Multiplication and division
Rule 1
a p  aq  a( p  q)
Example 8
33  33  (3  3  3)  (3  3  3)  36
Rule 2
(a p )q  a pq  a pq
Example 9
(a)
(33 )2  33  33
 (3  3  3)  (3  3  3)
 36
(b)
( 42 )5  42  42  42  42  42
 (4  4)  (4  4)  (4  4)  (4  4)  (4  4)
 410
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Elementary Mathematics
Chapter 1
a p  aq 
Rule 3
ap
 a pq
q
a
Example 10
6 5  63 
66666
666
 6  6  62
a 0  1 provided a  0
Rule 4
Example 11
63  63  63  3
 60  1
division when power are equal
a p 
Rule 5
1
ap
Example 12
73  75  73  5
777
 72
77777
1
 72
77
1
 72
2
7
Division when the power of the denominator is greater than the
power of numerator
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Elementary Mathematics
Chapter 1
1
ap  a
p
Rule 6
Example 13
1
a3  3 a
p
1
q
a  a  (a ) p
q
q
Rule 7
p
Example 14
2
3
1
3 2
a  a  (a )
3
2
Example 15
Simplify:
(a)
35  36
34
(b)
18 x 2 y 5
3x 4 y
(c) (3x 5 ) 2
Solution
(a)
35  36 35  6
 4
34
3

311
34
= 311 4  37
(b)
18 x 2 y 5 6 x 2 y 5
 4
3x 4 y
x y
 6 x ( 2  4) y (5 1)
 6 x 2 y 4

(c)
6 y4
x2
(3x5 )2  32  ( x5 )2  9 x10
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Elementary Mathematics
Chapter 1
1.2.2 Surds
is surd or radical sign. The number expressed in term of this sign
cannot be written in decimal form. These numbers have a decimal
expression that does not terminate but to go on without repeating and
known as irrational numbers. However not all irrational are surds
such as  .
Properties of Surd
ab  a  b
1.
2.
3. a b  c b  (a  c) b
a

b
a
b
4. a b  c b  (a  c) b
Example 16
Simplify
(a)
45
(b) 6 7  2 7
(c) 5 3  27
Solution
(a)
45  9  5
 9 5
3 5
(b)
6 7  2 7  (6  2) 7
8 7
(c)
5 3  27  5 3  9  3
 5 3  ( 9  3)
 5 3 3 3  2 3
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Elementary Mathematics
Chapter 1
Example 17
Expand and simplify ( 8  3 )( 8  3 )
Solution
( 8  3 )( 8  3 )  8 ( 8  3 )  3 ( 8  3 )
 ( 8 )2  8 3  3 8  ( 3)2
 ( 8 )2  ( 3)2
 83  5
The above example demonstrates the algebraic result for the
difference of two squares.
(a  b)(a  b)  a 2  b2
Rationalizing the denominator
When square roots occur in quotient, it is customary to rewrite the
quotient so that the denominator contains no square roots. This
process is referred to as rationalizing the denominator.
For example:
(a)
(b)
3  3  ( 3)2  3
( 3  1)  ( 3  1)  ( 3 ) 2  12  3  1  2
In the rationalizing the denominator of a quotient, be sure to multiply
both the numerator and the denominator by the same expression.
Example 18
Rationalize:
(a)
5
3
(b)
1
7 2
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Elementary Mathematics
Chapter 1
Solution
(a)
5
5
3


3
3
3

(b)
5 3
3
1
1
(7  2 )
(
)
7 2
7  2 (7  2 )

(7  2 )
(7) 2  ( 2 ) 2

7 2
49  2

7 2
47
This choice of multiplier is by no means accidental. When more
complicated expression needs rationalizing, the multiplier is simply
the conjugate of original denominator.
1.3 Logarithms
The logarithm of a number y ( y  0) for any base a ( a  0) written as
log a y .
If log a y  x , then y  a x , x  R.
For example,
23  8

3 is the power to which the base 2 must be raised to obtain 8
or
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Elementary Mathematics

Chapter 1
3 is the logarithm which with a base 2, gives 8. This is written
simply as 3  log 2 8
We see that the base of a logarithm may be any positive number.
Common logarithms have base 10. Using calculator it is found that
the common logarithm of 5 is 0.6990
5
 0.6990 or
i.e 100.6990  5 or log 10
lg 5  0.6990 or
log 5  0.6990
In general,
log a b  c  b  a c
Example 19
For each of the following, write down an expression for a
logarithm in a suitable base:
(a)
81  34
(b)
1
1
 5
32 2
Solution
(a)
81  34 , so 4  log 3 81
(b)
1
1
1
 5  2 5 , so  5  log 2 ( )
32 2
32
1.3.1 Natural Logarithms
The logarithmic function with base e is called the natural logarithmic
function. The function f ( x)  log e x is usually expressed as
f ( x)  ln x .
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Elementary Mathematics
Chapter 1
Properties of Logarithms
General Properties
Natural Logarithms
Properties
log b 1  0
ln 1  0
log b b  1
ln e  1
log b b x  x
ln e x  x
blogb x  x
e ln x  x
The law of logarithms
1. log a b  log a c  log a bc
2. log a b  log a c  log a
b
c
3. n log a b  log a b n
Example 20
Given log 2  0.301 and log 6  0.778 find log 12.
Solution
log 12  log( 2  6)
 log 2  log 6
 0.301  0.778
 1.079
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Elementary Mathematics
Chapter 1
Example 21
Write the following as single logarithm:
log 8  log 6  log 9
Solution
8
log 8  log 6  log 9  log( )  log 9
6
 89 
 log 

 6 
 log 12
Change of Base
log a c 
log b c
log b a
For example if
log 7 2  x
then
7x  2
so
x log 7  log 2
or
x
Therefore
log 7 2 
log 2
log 7
log 2 0.3010

 0.3562
log 7 0.8451
In special case when c = b this identity becomes
or
log a b 
log b b
log b a
log a b 
1
log b a
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Elementary Mathematics
Chapter 1
For example: To solve an equation like 5 x  10 .
 log 5x  log 10
x log 5  log 10
log 10
log 5
1
x
 1.43
0.6990
x
Example 22
Solve the equation
log 3 x  4 log x 3  3  0
Solution
log 3 x 
4
3  0
log 3 x
(log 3 x) 2  4  3 log 3 x  0
y  3y  4  0
; substitute y  log 3 x
( y  4)( y  1)  0
y  4or1
Therefore
log 3 x  4
or
log 3 x  1
x  34
or
x = 31
x
1
81
=3
21