NCERT Textbook: Rational Numbers | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


MATHEMATICS 124
Rational
Numbers
8.1  INTRODUCTION
Y ou began your study of numbers by counting objects around you. The
numbers used for this purpose were called counting numbers or natural
numbers. They are 1, 2, 3, 4, ... By including 0 to natural numbers, we
got the whole numbers, i.e., 0, 1, 2, 3, ... The negatives of natural
numbers were then put together with whole numbers to make up
integers. Integers are ..., –3, –2, –1, 0, 1, 2, 3, .... We, thus, extended
the number system, from natural numbers to whole numbers and from
whole numbers to integers.
Y ou were also introduced to fractions. These are numbers of the form 
numerator
denominator
,
where the numerator is either 0 or a positive integer and the denominator, a positive integer.
You compared two fractions, found their equivalent forms and studied all the four basic
operations of addition, subtraction, multiplication and division on them.
In this Chapter , we shall extend  the number system further . W e shall introduce the concept
of rational numbers alongwith their addition, subtraction, multiplication and division operations.
8.2  NEED FOR RATIONAL NUMBERS
Earlier, we have seen how integers could be used to denote opposite situations involving
numbers. For example, if the distance of 3 km to the right of a place was denoted by 3, then
the distance of 5 km to the left of the same place could be denoted by –5. If a profit of ` 150
was represented by 150 then a loss of ` 100 could be written as –100.
There are many situations similar to the above situations that involve fractional numbers.
You can represent a distance of 750m above sea level as 
3
4
 km. Can we represent 750m
below sea level in km? Can we denote the distance of 
3
4
 km below sea level by 
3
4
-
? W e can
see 
3
4
-
 is neither an integer, nor a fractional number. We need to extend our number system
to include such numbers.
Chapter  8
2024-25
Page 2


MATHEMATICS 124
Rational
Numbers
8.1  INTRODUCTION
Y ou began your study of numbers by counting objects around you. The
numbers used for this purpose were called counting numbers or natural
numbers. They are 1, 2, 3, 4, ... By including 0 to natural numbers, we
got the whole numbers, i.e., 0, 1, 2, 3, ... The negatives of natural
numbers were then put together with whole numbers to make up
integers. Integers are ..., –3, –2, –1, 0, 1, 2, 3, .... We, thus, extended
the number system, from natural numbers to whole numbers and from
whole numbers to integers.
Y ou were also introduced to fractions. These are numbers of the form 
numerator
denominator
,
where the numerator is either 0 or a positive integer and the denominator, a positive integer.
You compared two fractions, found their equivalent forms and studied all the four basic
operations of addition, subtraction, multiplication and division on them.
In this Chapter , we shall extend  the number system further . W e shall introduce the concept
of rational numbers alongwith their addition, subtraction, multiplication and division operations.
8.2  NEED FOR RATIONAL NUMBERS
Earlier, we have seen how integers could be used to denote opposite situations involving
numbers. For example, if the distance of 3 km to the right of a place was denoted by 3, then
the distance of 5 km to the left of the same place could be denoted by –5. If a profit of ` 150
was represented by 150 then a loss of ` 100 could be written as –100.
There are many situations similar to the above situations that involve fractional numbers.
You can represent a distance of 750m above sea level as 
3
4
 km. Can we represent 750m
below sea level in km? Can we denote the distance of 
3
4
 km below sea level by 
3
4
-
? W e can
see 
3
4
-
 is neither an integer, nor a fractional number. We need to extend our number system
to include such numbers.
Chapter  8
2024-25
RA TIONAL NUMBERS 125
8.3  WHAT ARE RATIONAL NUMBERS?
The word ‘rational’ arises from the term ‘ratio’. Y ou know that a ratio like 3:2 can also be
written as 
3
2
. Here, 3 and 2 are natural numbers.
Similarly, the ratio of two integers p and q (q ? 0), i.e., p:q can be written in the form
p
q
. This is the form in which rational numbers are expressed.
A rational number is defined as a number that can be expressed in the
form 
p
q
, where p and q are integers and q ?  0.
Thus, 
4
5
 is a rational number. Here, p = 4 and q = 5.
Is 
3
4
-
 also a rational number? Y es, because p = – 3 and q = 4 are integers.
l Y ou have seen many fractions like 
3
8
4
8
1
2
3
, ,
 etc. All fractions are rational
numbers. Can you say why?
How about the decimal numbers like 0.5, 2.3, etc.? Each of such numbers can be
written as an ordinary fraction and, hence, are rational numbers. For example, 0.5 = 
5
10
,
0.333 = 
333
1000
 etc.
1. Is the number 
2
3 -
 rational? Think about it. 2. List ten rational numbers.
Numerator and Denominator
In 
p
q
, the integer p is the numerator, and the integer q (? 0) is the denominator.
Thus, in 
3
7
-
, the numerator is –3 and the denominator is 7.
Mention five rational numbers each of whose
(a) Numerator is a negative integer and denominator is a positive integer.
(b) Numerator is a positive integer and denominator is a negative integer.
(c) Numerator and denominator both are negative integers.
(d) Numerator and denominator both are positive integers.
l Are integers also rational numbers?
Any integer can be thought of as a rational number. For example, the integer – 5 is a
rational number, because you can write it as 
5
1
-
. The integer 0 can also be written as
0
0
2
0
7
= or
 etc. Hence, it is also a rational number.
Thus, rational numbers include integers and fractions.
TRY THESE
2024-25
Page 3


MATHEMATICS 124
Rational
Numbers
8.1  INTRODUCTION
Y ou began your study of numbers by counting objects around you. The
numbers used for this purpose were called counting numbers or natural
numbers. They are 1, 2, 3, 4, ... By including 0 to natural numbers, we
got the whole numbers, i.e., 0, 1, 2, 3, ... The negatives of natural
numbers were then put together with whole numbers to make up
integers. Integers are ..., –3, –2, –1, 0, 1, 2, 3, .... We, thus, extended
the number system, from natural numbers to whole numbers and from
whole numbers to integers.
Y ou were also introduced to fractions. These are numbers of the form 
numerator
denominator
,
where the numerator is either 0 or a positive integer and the denominator, a positive integer.
You compared two fractions, found their equivalent forms and studied all the four basic
operations of addition, subtraction, multiplication and division on them.
In this Chapter , we shall extend  the number system further . W e shall introduce the concept
of rational numbers alongwith their addition, subtraction, multiplication and division operations.
8.2  NEED FOR RATIONAL NUMBERS
Earlier, we have seen how integers could be used to denote opposite situations involving
numbers. For example, if the distance of 3 km to the right of a place was denoted by 3, then
the distance of 5 km to the left of the same place could be denoted by –5. If a profit of ` 150
was represented by 150 then a loss of ` 100 could be written as –100.
There are many situations similar to the above situations that involve fractional numbers.
You can represent a distance of 750m above sea level as 
3
4
 km. Can we represent 750m
below sea level in km? Can we denote the distance of 
3
4
 km below sea level by 
3
4
-
? W e can
see 
3
4
-
 is neither an integer, nor a fractional number. We need to extend our number system
to include such numbers.
Chapter  8
2024-25
RA TIONAL NUMBERS 125
8.3  WHAT ARE RATIONAL NUMBERS?
The word ‘rational’ arises from the term ‘ratio’. Y ou know that a ratio like 3:2 can also be
written as 
3
2
. Here, 3 and 2 are natural numbers.
Similarly, the ratio of two integers p and q (q ? 0), i.e., p:q can be written in the form
p
q
. This is the form in which rational numbers are expressed.
A rational number is defined as a number that can be expressed in the
form 
p
q
, where p and q are integers and q ?  0.
Thus, 
4
5
 is a rational number. Here, p = 4 and q = 5.
Is 
3
4
-
 also a rational number? Y es, because p = – 3 and q = 4 are integers.
l Y ou have seen many fractions like 
3
8
4
8
1
2
3
, ,
 etc. All fractions are rational
numbers. Can you say why?
How about the decimal numbers like 0.5, 2.3, etc.? Each of such numbers can be
written as an ordinary fraction and, hence, are rational numbers. For example, 0.5 = 
5
10
,
0.333 = 
333
1000
 etc.
1. Is the number 
2
3 -
 rational? Think about it. 2. List ten rational numbers.
Numerator and Denominator
In 
p
q
, the integer p is the numerator, and the integer q (? 0) is the denominator.
Thus, in 
3
7
-
, the numerator is –3 and the denominator is 7.
Mention five rational numbers each of whose
(a) Numerator is a negative integer and denominator is a positive integer.
(b) Numerator is a positive integer and denominator is a negative integer.
(c) Numerator and denominator both are negative integers.
(d) Numerator and denominator both are positive integers.
l Are integers also rational numbers?
Any integer can be thought of as a rational number. For example, the integer – 5 is a
rational number, because you can write it as 
5
1
-
. The integer 0 can also be written as
0
0
2
0
7
= or
 etc. Hence, it is also a rational number.
Thus, rational numbers include integers and fractions.
TRY THESE
2024-25
MATHEMATICS 126
Equivalent rational numbers
A rational number can be written with different numerators and denominators. For example,
consider the rational number 
– 2
3
.
– 2
3
 =
–2 2 – 4
3 2 6
×
=
×
. We see that 
– 2
3
 is the same as 
– 4
6
.
Also,
–2
3
 =
( ) ( )
( )
–2 –5 10
3 –5 –15
×
=
×
. So, 
– 2
3
 is also the same as 
10
15 -
.
Thus, 
– 2
3
 = 
4
6
-
 =
10
15 -
. Such rational numbers that are equal to each other are said to
be equivalent to each other.
Again,
10
15 -
 =
-10
15
 (How?)
By multiplying the numerator and denominator of a rational
number by the same non zero integer, we obtain another rational
number equivalent to the given rational number. This is exactly like
obtaining equivalent fractions.
Just as multiplication, the division of the numerator and denominator
by the same non zero integer, also gives equivalent rational numbers. For
example,
10
–15
 = 
( )
( )
10 –5 –2
–15 –5 3
÷
=
÷
,     
–12
24
 = 
12 12 1
24 12 2
- ÷ -
=
÷
We write 
–2
3
as –
2
3
, 
–10
15
as –
10
15
, etc.
8.4  POSITIVE AND NEGATIVE RATIONAL NUMBERS
Consider the rational number 
2
3
. Both the numerator and denominator of this number are
positive integers. Such a rational number is called a positive rational number. So, 
3
8
5
7
2
9
, ,
etc. are positive rational numbers.
The numerator of 
–3
5
 is a negative integer, whereas the denominator
is a positive integer. Such a rational number is called a negative rational
number. So, 
5 3 9
, ,
7 8 5
- - -
 etc. are negative rational numbers.
TRY THESE
Fill in the boxes:
(i)
5 25 15
4 16
-
= = =
(ii)
3 9 6
7 14
- -
= = =
TRY THESE
1. Is 5 a positive rational
number?
2. List five more positive
rational numbers.
2024-25
Page 4


MATHEMATICS 124
Rational
Numbers
8.1  INTRODUCTION
Y ou began your study of numbers by counting objects around you. The
numbers used for this purpose were called counting numbers or natural
numbers. They are 1, 2, 3, 4, ... By including 0 to natural numbers, we
got the whole numbers, i.e., 0, 1, 2, 3, ... The negatives of natural
numbers were then put together with whole numbers to make up
integers. Integers are ..., –3, –2, –1, 0, 1, 2, 3, .... We, thus, extended
the number system, from natural numbers to whole numbers and from
whole numbers to integers.
Y ou were also introduced to fractions. These are numbers of the form 
numerator
denominator
,
where the numerator is either 0 or a positive integer and the denominator, a positive integer.
You compared two fractions, found their equivalent forms and studied all the four basic
operations of addition, subtraction, multiplication and division on them.
In this Chapter , we shall extend  the number system further . W e shall introduce the concept
of rational numbers alongwith their addition, subtraction, multiplication and division operations.
8.2  NEED FOR RATIONAL NUMBERS
Earlier, we have seen how integers could be used to denote opposite situations involving
numbers. For example, if the distance of 3 km to the right of a place was denoted by 3, then
the distance of 5 km to the left of the same place could be denoted by –5. If a profit of ` 150
was represented by 150 then a loss of ` 100 could be written as –100.
There are many situations similar to the above situations that involve fractional numbers.
You can represent a distance of 750m above sea level as 
3
4
 km. Can we represent 750m
below sea level in km? Can we denote the distance of 
3
4
 km below sea level by 
3
4
-
? W e can
see 
3
4
-
 is neither an integer, nor a fractional number. We need to extend our number system
to include such numbers.
Chapter  8
2024-25
RA TIONAL NUMBERS 125
8.3  WHAT ARE RATIONAL NUMBERS?
The word ‘rational’ arises from the term ‘ratio’. Y ou know that a ratio like 3:2 can also be
written as 
3
2
. Here, 3 and 2 are natural numbers.
Similarly, the ratio of two integers p and q (q ? 0), i.e., p:q can be written in the form
p
q
. This is the form in which rational numbers are expressed.
A rational number is defined as a number that can be expressed in the
form 
p
q
, where p and q are integers and q ?  0.
Thus, 
4
5
 is a rational number. Here, p = 4 and q = 5.
Is 
3
4
-
 also a rational number? Y es, because p = – 3 and q = 4 are integers.
l Y ou have seen many fractions like 
3
8
4
8
1
2
3
, ,
 etc. All fractions are rational
numbers. Can you say why?
How about the decimal numbers like 0.5, 2.3, etc.? Each of such numbers can be
written as an ordinary fraction and, hence, are rational numbers. For example, 0.5 = 
5
10
,
0.333 = 
333
1000
 etc.
1. Is the number 
2
3 -
 rational? Think about it. 2. List ten rational numbers.
Numerator and Denominator
In 
p
q
, the integer p is the numerator, and the integer q (? 0) is the denominator.
Thus, in 
3
7
-
, the numerator is –3 and the denominator is 7.
Mention five rational numbers each of whose
(a) Numerator is a negative integer and denominator is a positive integer.
(b) Numerator is a positive integer and denominator is a negative integer.
(c) Numerator and denominator both are negative integers.
(d) Numerator and denominator both are positive integers.
l Are integers also rational numbers?
Any integer can be thought of as a rational number. For example, the integer – 5 is a
rational number, because you can write it as 
5
1
-
. The integer 0 can also be written as
0
0
2
0
7
= or
 etc. Hence, it is also a rational number.
Thus, rational numbers include integers and fractions.
TRY THESE
2024-25
MATHEMATICS 126
Equivalent rational numbers
A rational number can be written with different numerators and denominators. For example,
consider the rational number 
– 2
3
.
– 2
3
 =
–2 2 – 4
3 2 6
×
=
×
. We see that 
– 2
3
 is the same as 
– 4
6
.
Also,
–2
3
 =
( ) ( )
( )
–2 –5 10
3 –5 –15
×
=
×
. So, 
– 2
3
 is also the same as 
10
15 -
.
Thus, 
– 2
3
 = 
4
6
-
 =
10
15 -
. Such rational numbers that are equal to each other are said to
be equivalent to each other.
Again,
10
15 -
 =
-10
15
 (How?)
By multiplying the numerator and denominator of a rational
number by the same non zero integer, we obtain another rational
number equivalent to the given rational number. This is exactly like
obtaining equivalent fractions.
Just as multiplication, the division of the numerator and denominator
by the same non zero integer, also gives equivalent rational numbers. For
example,
10
–15
 = 
( )
( )
10 –5 –2
–15 –5 3
÷
=
÷
,     
–12
24
 = 
12 12 1
24 12 2
- ÷ -
=
÷
We write 
–2
3
as –
2
3
, 
–10
15
as –
10
15
, etc.
8.4  POSITIVE AND NEGATIVE RATIONAL NUMBERS
Consider the rational number 
2
3
. Both the numerator and denominator of this number are
positive integers. Such a rational number is called a positive rational number. So, 
3
8
5
7
2
9
, ,
etc. are positive rational numbers.
The numerator of 
–3
5
 is a negative integer, whereas the denominator
is a positive integer. Such a rational number is called a negative rational
number. So, 
5 3 9
, ,
7 8 5
- - -
 etc. are negative rational numbers.
TRY THESE
Fill in the boxes:
(i)
5 25 15
4 16
-
= = =
(ii)
3 9 6
7 14
- -
= = =
TRY THESE
1. Is 5 a positive rational
number?
2. List five more positive
rational numbers.
2024-25
RA TIONAL NUMBERS 127
l Is 
8
3 -
 a negative rational number? W e know that 
8
3 -
 = 
8× 1
3× 1
-
- -
= 
8
3
-
,  and
8
3
-
 is a negative rational number. So, 
8
3 -
 is a negative rational number.
Similarly, 
5 6 2
, ,
7 5 9 - - -
 etc. are all negative rational numbers. Note that
their numerators are positive and their denominators negative.
l The number 0 is neither a positive nor a negative rational number.
l What about 
3
5
-
-
?
Y ou will see that 
( )
( )
3 1 3 3
5 5 1 5
- × - -
= =
- - × -
. So, 
3
5
-
-
 is a positive rational number.
Thus, 
2 5
,
5 3
- -
- -
 etc. are positive rational numbers.
Which of these are negative rational numbers?
(i)
2
3
-
(ii)
5
7
(iii)
3
5 -
(iv) 0 (v)
6
11
(vi)
2
9
-
-
8.5  RATIONAL NUMBERS ON A NUMBER LINE
Y ou know how to represent integers on a number line. Let us draw one such number line.
The points to the right of 0 are denoted by + sign and are positive integers. The points
to the left of 0 are denoted by – sign and are negative integers.
Representation of fractions on a number line is also known to you.
Let us see how the rational numbers can be represented on a number line.
Let us represent the number 
-
1
2
 on the number line.
As done in the case of positive integers, the positive rational numbers would be marked
on the right of 0 and the negative rational numbers would be marked on the left of 0.
T o which side of 0 will you mark 
-
1
2
? Being a negative rational number, it would be
marked to the left of 0.
Y ou know that while marking integers on the number line, successive integers are
marked at equal intervels. Also, from 0, the pair 1 and –1 is equidistant. So are the pairs 2
and – 2, 3 and –3.
TRY THESE
1. Is – 8 a negative
rational number?
2. List five more
negative rational
numbers.
TRY THESE
2024-25
Page 5


MATHEMATICS 124
Rational
Numbers
8.1  INTRODUCTION
Y ou began your study of numbers by counting objects around you. The
numbers used for this purpose were called counting numbers or natural
numbers. They are 1, 2, 3, 4, ... By including 0 to natural numbers, we
got the whole numbers, i.e., 0, 1, 2, 3, ... The negatives of natural
numbers were then put together with whole numbers to make up
integers. Integers are ..., –3, –2, –1, 0, 1, 2, 3, .... We, thus, extended
the number system, from natural numbers to whole numbers and from
whole numbers to integers.
Y ou were also introduced to fractions. These are numbers of the form 
numerator
denominator
,
where the numerator is either 0 or a positive integer and the denominator, a positive integer.
You compared two fractions, found their equivalent forms and studied all the four basic
operations of addition, subtraction, multiplication and division on them.
In this Chapter , we shall extend  the number system further . W e shall introduce the concept
of rational numbers alongwith their addition, subtraction, multiplication and division operations.
8.2  NEED FOR RATIONAL NUMBERS
Earlier, we have seen how integers could be used to denote opposite situations involving
numbers. For example, if the distance of 3 km to the right of a place was denoted by 3, then
the distance of 5 km to the left of the same place could be denoted by –5. If a profit of ` 150
was represented by 150 then a loss of ` 100 could be written as –100.
There are many situations similar to the above situations that involve fractional numbers.
You can represent a distance of 750m above sea level as 
3
4
 km. Can we represent 750m
below sea level in km? Can we denote the distance of 
3
4
 km below sea level by 
3
4
-
? W e can
see 
3
4
-
 is neither an integer, nor a fractional number. We need to extend our number system
to include such numbers.
Chapter  8
2024-25
RA TIONAL NUMBERS 125
8.3  WHAT ARE RATIONAL NUMBERS?
The word ‘rational’ arises from the term ‘ratio’. Y ou know that a ratio like 3:2 can also be
written as 
3
2
. Here, 3 and 2 are natural numbers.
Similarly, the ratio of two integers p and q (q ? 0), i.e., p:q can be written in the form
p
q
. This is the form in which rational numbers are expressed.
A rational number is defined as a number that can be expressed in the
form 
p
q
, where p and q are integers and q ?  0.
Thus, 
4
5
 is a rational number. Here, p = 4 and q = 5.
Is 
3
4
-
 also a rational number? Y es, because p = – 3 and q = 4 are integers.
l Y ou have seen many fractions like 
3
8
4
8
1
2
3
, ,
 etc. All fractions are rational
numbers. Can you say why?
How about the decimal numbers like 0.5, 2.3, etc.? Each of such numbers can be
written as an ordinary fraction and, hence, are rational numbers. For example, 0.5 = 
5
10
,
0.333 = 
333
1000
 etc.
1. Is the number 
2
3 -
 rational? Think about it. 2. List ten rational numbers.
Numerator and Denominator
In 
p
q
, the integer p is the numerator, and the integer q (? 0) is the denominator.
Thus, in 
3
7
-
, the numerator is –3 and the denominator is 7.
Mention five rational numbers each of whose
(a) Numerator is a negative integer and denominator is a positive integer.
(b) Numerator is a positive integer and denominator is a negative integer.
(c) Numerator and denominator both are negative integers.
(d) Numerator and denominator both are positive integers.
l Are integers also rational numbers?
Any integer can be thought of as a rational number. For example, the integer – 5 is a
rational number, because you can write it as 
5
1
-
. The integer 0 can also be written as
0
0
2
0
7
= or
 etc. Hence, it is also a rational number.
Thus, rational numbers include integers and fractions.
TRY THESE
2024-25
MATHEMATICS 126
Equivalent rational numbers
A rational number can be written with different numerators and denominators. For example,
consider the rational number 
– 2
3
.
– 2
3
 =
–2 2 – 4
3 2 6
×
=
×
. We see that 
– 2
3
 is the same as 
– 4
6
.
Also,
–2
3
 =
( ) ( )
( )
–2 –5 10
3 –5 –15
×
=
×
. So, 
– 2
3
 is also the same as 
10
15 -
.
Thus, 
– 2
3
 = 
4
6
-
 =
10
15 -
. Such rational numbers that are equal to each other are said to
be equivalent to each other.
Again,
10
15 -
 =
-10
15
 (How?)
By multiplying the numerator and denominator of a rational
number by the same non zero integer, we obtain another rational
number equivalent to the given rational number. This is exactly like
obtaining equivalent fractions.
Just as multiplication, the division of the numerator and denominator
by the same non zero integer, also gives equivalent rational numbers. For
example,
10
–15
 = 
( )
( )
10 –5 –2
–15 –5 3
÷
=
÷
,     
–12
24
 = 
12 12 1
24 12 2
- ÷ -
=
÷
We write 
–2
3
as –
2
3
, 
–10
15
as –
10
15
, etc.
8.4  POSITIVE AND NEGATIVE RATIONAL NUMBERS
Consider the rational number 
2
3
. Both the numerator and denominator of this number are
positive integers. Such a rational number is called a positive rational number. So, 
3
8
5
7
2
9
, ,
etc. are positive rational numbers.
The numerator of 
–3
5
 is a negative integer, whereas the denominator
is a positive integer. Such a rational number is called a negative rational
number. So, 
5 3 9
, ,
7 8 5
- - -
 etc. are negative rational numbers.
TRY THESE
Fill in the boxes:
(i)
5 25 15
4 16
-
= = =
(ii)
3 9 6
7 14
- -
= = =
TRY THESE
1. Is 5 a positive rational
number?
2. List five more positive
rational numbers.
2024-25
RA TIONAL NUMBERS 127
l Is 
8
3 -
 a negative rational number? W e know that 
8
3 -
 = 
8× 1
3× 1
-
- -
= 
8
3
-
,  and
8
3
-
 is a negative rational number. So, 
8
3 -
 is a negative rational number.
Similarly, 
5 6 2
, ,
7 5 9 - - -
 etc. are all negative rational numbers. Note that
their numerators are positive and their denominators negative.
l The number 0 is neither a positive nor a negative rational number.
l What about 
3
5
-
-
?
Y ou will see that 
( )
( )
3 1 3 3
5 5 1 5
- × - -
= =
- - × -
. So, 
3
5
-
-
 is a positive rational number.
Thus, 
2 5
,
5 3
- -
- -
 etc. are positive rational numbers.
Which of these are negative rational numbers?
(i)
2
3
-
(ii)
5
7
(iii)
3
5 -
(iv) 0 (v)
6
11
(vi)
2
9
-
-
8.5  RATIONAL NUMBERS ON A NUMBER LINE
Y ou know how to represent integers on a number line. Let us draw one such number line.
The points to the right of 0 are denoted by + sign and are positive integers. The points
to the left of 0 are denoted by – sign and are negative integers.
Representation of fractions on a number line is also known to you.
Let us see how the rational numbers can be represented on a number line.
Let us represent the number 
-
1
2
 on the number line.
As done in the case of positive integers, the positive rational numbers would be marked
on the right of 0 and the negative rational numbers would be marked on the left of 0.
T o which side of 0 will you mark 
-
1
2
? Being a negative rational number, it would be
marked to the left of 0.
Y ou know that while marking integers on the number line, successive integers are
marked at equal intervels. Also, from 0, the pair 1 and –1 is equidistant. So are the pairs 2
and – 2, 3 and –3.
TRY THESE
1. Is – 8 a negative
rational number?
2. List five more
negative rational
numbers.
TRY THESE
2024-25
MATHEMATICS 128
In the same way, the rational numbers 
1
2
 and 
-
1
2
 would be at equal distance from 0.
We know how to mark the rational number 
1
2
. It is marked at a point which is half the
distance between 0 and 1. So, 
-
1
2
 would be marked at a point half the distance between
0 and –1.
We know how to mark 
3
2
 on the number line. It is marked on the right of 0 and lies
halfway between 1 and 2. Let us now mark 
3
2
-
 on the number line. It lies on the left of 0
and is at the same distance as 
3
2
 from 0.
In decreasing order, we have, 
1 2
, ( 1)
2 2
- -
= -
, 
3 4
, ( 2)
2 2
- -
= -
. This shows that
3
2
-
 lies between – 1 and – 2. Thus, 
3
2
-
 lies halfway between – 1 and – 2.
Mark 
-5
2
 and 
-7
2
 in a similar way.
Similarly, 
-
1
3
 is to the left of zero and at the same distance from zero as 
1
3
 is to the
right. So as done above, 
-
1
3
 can be represented on the number line. Once we know how
to represent 
-
1
3
 on the number line, we can go on representing 
2 4 5
, ,
3 3 3
- - -
 and so on.
All other rational numbers with different denominators can be represented in a similar way .
8.6  RATIONAL NUMBERS IN STANDARD FORM
Observe the rational numbers 
3
5
5
8
2
7
7
11
, , ,
- -
.
The denominators of these rational numbers are positive integers and 1 is
the only common factor between the numerators and denominators. Further,
the negative sign occurs only in the numerator.
Such rational numbers are said to be in standard form.
-3
2
-1
2
0
2
0 = ( )
1
2
2
2
1 = ( )
3
2
4
2
2 = ( ) ( )
2
1
2
-
= - ( )
4
2
2
-
= -
2024-25