NCERT book Class 8 Maths - Rational Numbers

 Page 1


RATIONAL NUMBERS  1
1.1  Introduction
In Mathematics, we frequently come across simple equations to be solved. For example,
the equation x + 2 = 13 (1)
is solved when x = 11, because this value of x satisfies the given equation. The  solution
11 is a natural number. On the other hand, for the equation
x + 5 = 5 (2)
the solution gives the whole number 0 (zero). If we consider only natural numbers,
equation (2) cannot be solved. T o solve equations like (2), we added the number zero to
the collection of natural numbers and obtained the whole numbers. Even whole numbers
will not be sufficient to solve equations of type
x + 18 = 5 (3)
Do you see ‘why’? W e require the number –13 which is not a whole number. This
led us to think of integers, (positive and negative). Note that the positive integers
correspond to natural numbers. One may think that we have enough numbers to solve all
simple equations with the available list of integers. Now consider the equations
2x  = 3 (4)
5x + 7 = 0 (5)
for which we cannot find a solution from the integers. (Check this)
W e need the numbers 
3
2
 to solve equation (4) and 
7
5
-
 to solve
equation (5). This leads us to the collection of rational numbers.
We have already seen basic operations on rational
numbers. W e now try to explore some properties of operations
on the different types of numbers seen so far.
Rational Numbers
CHAPTER
1
Reprint 2024-25
Page 2


RATIONAL NUMBERS  1
1.1  Introduction
In Mathematics, we frequently come across simple equations to be solved. For example,
the equation x + 2 = 13 (1)
is solved when x = 11, because this value of x satisfies the given equation. The  solution
11 is a natural number. On the other hand, for the equation
x + 5 = 5 (2)
the solution gives the whole number 0 (zero). If we consider only natural numbers,
equation (2) cannot be solved. T o solve equations like (2), we added the number zero to
the collection of natural numbers and obtained the whole numbers. Even whole numbers
will not be sufficient to solve equations of type
x + 18 = 5 (3)
Do you see ‘why’? W e require the number –13 which is not a whole number. This
led us to think of integers, (positive and negative). Note that the positive integers
correspond to natural numbers. One may think that we have enough numbers to solve all
simple equations with the available list of integers. Now consider the equations
2x  = 3 (4)
5x + 7 = 0 (5)
for which we cannot find a solution from the integers. (Check this)
W e need the numbers 
3
2
 to solve equation (4) and 
7
5
-
 to solve
equation (5). This leads us to the collection of rational numbers.
We have already seen basic operations on rational
numbers. W e now try to explore some properties of operations
on the different types of numbers seen so far.
Rational Numbers
CHAPTER
1
Reprint 2024-25
2  MATHEMATICS
1.2  Properties of Rational Numbers
1.2.1  Closure
(i) Whole numbers
Let us revisit the closure property for all the operations on whole numbers in brief.
Operation Numbers Remarks
Addition 0 + 5 = 5, a whole number Whole numbers are closed
4 + 7 = ... . Is it a whole number? under addition.
In general, a + b is a whole
number  for any two whole
numbers a and b.
Subtraction 5 – 7 = – 2, which is not a Whole numbers are not closed
whole number . under subtraction.
Multiplication 0 × 3 = 0, a whole number Whole numbers are closed
3 × 7 = ... . Is it a whole number? under multiplication.
In general, if a and b are any two
whole numbers, their product ab
is a whole number.
Division 5 ÷ 8 = 
5
8
, which is  not a
whole number .
Check for closure property under all the four operations for natural numbers.
(ii) Integers
Let us now recall the operations under which integers are closed.
Operation Numbers Remarks
Addition – 6 + 5 = – 1, an integer Integers are closed under
Is – 7 + (–5) an integer? addition.
Is 8 + 5 an integer?
In general, a + b is an integer
for any two integers a and b.
Subtraction 7 – 5 = 2, an integer Integers are closed under
Is 5 – 7 an integer? subtraction.
– 6 – 8 = – 14, an integer
Whole numbers are not closed
under division.
Reprint 2024-25
Page 3


RATIONAL NUMBERS  1
1.1  Introduction
In Mathematics, we frequently come across simple equations to be solved. For example,
the equation x + 2 = 13 (1)
is solved when x = 11, because this value of x satisfies the given equation. The  solution
11 is a natural number. On the other hand, for the equation
x + 5 = 5 (2)
the solution gives the whole number 0 (zero). If we consider only natural numbers,
equation (2) cannot be solved. T o solve equations like (2), we added the number zero to
the collection of natural numbers and obtained the whole numbers. Even whole numbers
will not be sufficient to solve equations of type
x + 18 = 5 (3)
Do you see ‘why’? W e require the number –13 which is not a whole number. This
led us to think of integers, (positive and negative). Note that the positive integers
correspond to natural numbers. One may think that we have enough numbers to solve all
simple equations with the available list of integers. Now consider the equations
2x  = 3 (4)
5x + 7 = 0 (5)
for which we cannot find a solution from the integers. (Check this)
W e need the numbers 
3
2
 to solve equation (4) and 
7
5
-
 to solve
equation (5). This leads us to the collection of rational numbers.
We have already seen basic operations on rational
numbers. W e now try to explore some properties of operations
on the different types of numbers seen so far.
Rational Numbers
CHAPTER
1
Reprint 2024-25
2  MATHEMATICS
1.2  Properties of Rational Numbers
1.2.1  Closure
(i) Whole numbers
Let us revisit the closure property for all the operations on whole numbers in brief.
Operation Numbers Remarks
Addition 0 + 5 = 5, a whole number Whole numbers are closed
4 + 7 = ... . Is it a whole number? under addition.
In general, a + b is a whole
number  for any two whole
numbers a and b.
Subtraction 5 – 7 = – 2, which is not a Whole numbers are not closed
whole number . under subtraction.
Multiplication 0 × 3 = 0, a whole number Whole numbers are closed
3 × 7 = ... . Is it a whole number? under multiplication.
In general, if a and b are any two
whole numbers, their product ab
is a whole number.
Division 5 ÷ 8 = 
5
8
, which is  not a
whole number .
Check for closure property under all the four operations for natural numbers.
(ii) Integers
Let us now recall the operations under which integers are closed.
Operation Numbers Remarks
Addition – 6 + 5 = – 1, an integer Integers are closed under
Is – 7 + (–5) an integer? addition.
Is 8 + 5 an integer?
In general, a + b is an integer
for any two integers a and b.
Subtraction 7 – 5 = 2, an integer Integers are closed under
Is 5 – 7 an integer? subtraction.
– 6 – 8 = – 14, an integer
Whole numbers are not closed
under division.
Reprint 2024-25
RATIONAL NUMBERS  3
– 6 – (– 8) = 2, an integer
Is 8 – (– 6) an integer?
In general, for any two integers
a and b, a – b is again an integer.
Check if b – a is also an integer.
Multiplication 5 × 8 = 40, an integer Integers are closed under
Is – 5 × 8 an integer? multiplication.
– 5 × (– 8) = 40, an integer
In general, for any two integers
a and b, a × b is also an integer.
Division 5 ÷ 8 = 
5
8
, which is not Integers are not closed
an integer.
under division.
Y ou have seen that whole numbers are closed under addition and multiplication but
not under subtraction and division. However , integers are closed under addition, subtraction
and multiplication but not under division.
(iii) Rational numbers
Recall that a number which can be written in the form 
p
q
, where p and q are integers
and q ? 0 is called a rational number. For example, 
2
3
-
, 
6
7
, 
9
5 -
 are all rational
numbers. Since the numbers 0, –2, 4 can be written in the form 
p
q
, they are also
rational numbers. (Check it!)
(a) Y ou know how to add two rational numbers. Let us add a few pairs.
3 ( 5)
8 7
-
+
=
21 ( 40) 19
56 56
+ - -
=
(a rational number)
3 ( 4)
8 5
- -
+
 =
15 ( 32)
...
40
- + -
=
Is it a rational number?
4 6
7 11
+
 = ... Is it a rational number?
W e find that sum of two rational numbers is again a rational number. Check it
for a few more pairs of rational numbers.
We say that rational numbers are closed under addition. That is, for any
two rational numbers a and b, a + b is also a rational number.
(b) Will the difference of two rational numbers be again a rational number?
We have,
5 2
7 3
-
-
=
5 3 – 2 7 29
21 21
- × × -
=
(a rational number)
Reprint 2024-25
Page 4


RATIONAL NUMBERS  1
1.1  Introduction
In Mathematics, we frequently come across simple equations to be solved. For example,
the equation x + 2 = 13 (1)
is solved when x = 11, because this value of x satisfies the given equation. The  solution
11 is a natural number. On the other hand, for the equation
x + 5 = 5 (2)
the solution gives the whole number 0 (zero). If we consider only natural numbers,
equation (2) cannot be solved. T o solve equations like (2), we added the number zero to
the collection of natural numbers and obtained the whole numbers. Even whole numbers
will not be sufficient to solve equations of type
x + 18 = 5 (3)
Do you see ‘why’? W e require the number –13 which is not a whole number. This
led us to think of integers, (positive and negative). Note that the positive integers
correspond to natural numbers. One may think that we have enough numbers to solve all
simple equations with the available list of integers. Now consider the equations
2x  = 3 (4)
5x + 7 = 0 (5)
for which we cannot find a solution from the integers. (Check this)
W e need the numbers 
3
2
 to solve equation (4) and 
7
5
-
 to solve
equation (5). This leads us to the collection of rational numbers.
We have already seen basic operations on rational
numbers. W e now try to explore some properties of operations
on the different types of numbers seen so far.
Rational Numbers
CHAPTER
1
Reprint 2024-25
2  MATHEMATICS
1.2  Properties of Rational Numbers
1.2.1  Closure
(i) Whole numbers
Let us revisit the closure property for all the operations on whole numbers in brief.
Operation Numbers Remarks
Addition 0 + 5 = 5, a whole number Whole numbers are closed
4 + 7 = ... . Is it a whole number? under addition.
In general, a + b is a whole
number  for any two whole
numbers a and b.
Subtraction 5 – 7 = – 2, which is not a Whole numbers are not closed
whole number . under subtraction.
Multiplication 0 × 3 = 0, a whole number Whole numbers are closed
3 × 7 = ... . Is it a whole number? under multiplication.
In general, if a and b are any two
whole numbers, their product ab
is a whole number.
Division 5 ÷ 8 = 
5
8
, which is  not a
whole number .
Check for closure property under all the four operations for natural numbers.
(ii) Integers
Let us now recall the operations under which integers are closed.
Operation Numbers Remarks
Addition – 6 + 5 = – 1, an integer Integers are closed under
Is – 7 + (–5) an integer? addition.
Is 8 + 5 an integer?
In general, a + b is an integer
for any two integers a and b.
Subtraction 7 – 5 = 2, an integer Integers are closed under
Is 5 – 7 an integer? subtraction.
– 6 – 8 = – 14, an integer
Whole numbers are not closed
under division.
Reprint 2024-25
RATIONAL NUMBERS  3
– 6 – (– 8) = 2, an integer
Is 8 – (– 6) an integer?
In general, for any two integers
a and b, a – b is again an integer.
Check if b – a is also an integer.
Multiplication 5 × 8 = 40, an integer Integers are closed under
Is – 5 × 8 an integer? multiplication.
– 5 × (– 8) = 40, an integer
In general, for any two integers
a and b, a × b is also an integer.
Division 5 ÷ 8 = 
5
8
, which is not Integers are not closed
an integer.
under division.
Y ou have seen that whole numbers are closed under addition and multiplication but
not under subtraction and division. However , integers are closed under addition, subtraction
and multiplication but not under division.
(iii) Rational numbers
Recall that a number which can be written in the form 
p
q
, where p and q are integers
and q ? 0 is called a rational number. For example, 
2
3
-
, 
6
7
, 
9
5 -
 are all rational
numbers. Since the numbers 0, –2, 4 can be written in the form 
p
q
, they are also
rational numbers. (Check it!)
(a) Y ou know how to add two rational numbers. Let us add a few pairs.
3 ( 5)
8 7
-
+
=
21 ( 40) 19
56 56
+ - -
=
(a rational number)
3 ( 4)
8 5
- -
+
 =
15 ( 32)
...
40
- + -
=
Is it a rational number?
4 6
7 11
+
 = ... Is it a rational number?
W e find that sum of two rational numbers is again a rational number. Check it
for a few more pairs of rational numbers.
We say that rational numbers are closed under addition. That is, for any
two rational numbers a and b, a + b is also a rational number.
(b) Will the difference of two rational numbers be again a rational number?
We have,
5 2
7 3
-
-
=
5 3 – 2 7 29
21 21
- × × -
=
(a rational number)
Reprint 2024-25
4  MATHEMATICS
TRY THESE
5 4
8 5
-
 = 
25 32
40
-
 = ... Is it a rational number?
3
7
8
5
-
- ?
?
?
?
?
? = ... Is it a rational number?
Try this for some more pairs of  rational numbers. W e find that rational numbers
are closed under subtraction. That is, for any two rational numbers a and
b, a – b is also a rational number.
(c) Let us now see the product of two rational numbers.
2 4
3 5
-
×
 =
8 3 2 6
;
15 7 5 35
-
× =
(both the products are rational numbers)
4 6
5 11
-
- ×
 = ... Is it a rational number?
T ake some more pairs of rational numbers and check that their product is again
a rational number.
We say that rational numbers are closed under multiplication. That
is, for any two rational numbers a and b,  a × b is also a rational
number.
(d) We note that 
5 2 25
3 5 6
- -
÷ = (a rational number)
2 5
...
7 3
÷ =
 . Is it a rational number? 
3 2
...
8 9
- -
÷ =
. Is it a rational number?
Can you say that rational numbers are closed under division?
We find that for any rational number a, a ÷ 0 is not defined.
So rational numbers are not closed under division.
However , if we exclude zero then the collection of, all other rational numbers is
closed under division.
Fill in the blanks in the following table.
Numbers Closed under
addition subtraction multiplication division
Rational numbers Y es Y es ... No
Integers ... Y es ... No
Whole numbers ... ... Y es ...
Natural numbers ... No ... ...
Reprint 2024-25
Page 5


RATIONAL NUMBERS  1
1.1  Introduction
In Mathematics, we frequently come across simple equations to be solved. For example,
the equation x + 2 = 13 (1)
is solved when x = 11, because this value of x satisfies the given equation. The  solution
11 is a natural number. On the other hand, for the equation
x + 5 = 5 (2)
the solution gives the whole number 0 (zero). If we consider only natural numbers,
equation (2) cannot be solved. T o solve equations like (2), we added the number zero to
the collection of natural numbers and obtained the whole numbers. Even whole numbers
will not be sufficient to solve equations of type
x + 18 = 5 (3)
Do you see ‘why’? W e require the number –13 which is not a whole number. This
led us to think of integers, (positive and negative). Note that the positive integers
correspond to natural numbers. One may think that we have enough numbers to solve all
simple equations with the available list of integers. Now consider the equations
2x  = 3 (4)
5x + 7 = 0 (5)
for which we cannot find a solution from the integers. (Check this)
W e need the numbers 
3
2
 to solve equation (4) and 
7
5
-
 to solve
equation (5). This leads us to the collection of rational numbers.
We have already seen basic operations on rational
numbers. W e now try to explore some properties of operations
on the different types of numbers seen so far.
Rational Numbers
CHAPTER
1
Reprint 2024-25
2  MATHEMATICS
1.2  Properties of Rational Numbers
1.2.1  Closure
(i) Whole numbers
Let us revisit the closure property for all the operations on whole numbers in brief.
Operation Numbers Remarks
Addition 0 + 5 = 5, a whole number Whole numbers are closed
4 + 7 = ... . Is it a whole number? under addition.
In general, a + b is a whole
number  for any two whole
numbers a and b.
Subtraction 5 – 7 = – 2, which is not a Whole numbers are not closed
whole number . under subtraction.
Multiplication 0 × 3 = 0, a whole number Whole numbers are closed
3 × 7 = ... . Is it a whole number? under multiplication.
In general, if a and b are any two
whole numbers, their product ab
is a whole number.
Division 5 ÷ 8 = 
5
8
, which is  not a
whole number .
Check for closure property under all the four operations for natural numbers.
(ii) Integers
Let us now recall the operations under which integers are closed.
Operation Numbers Remarks
Addition – 6 + 5 = – 1, an integer Integers are closed under
Is – 7 + (–5) an integer? addition.
Is 8 + 5 an integer?
In general, a + b is an integer
for any two integers a and b.
Subtraction 7 – 5 = 2, an integer Integers are closed under
Is 5 – 7 an integer? subtraction.
– 6 – 8 = – 14, an integer
Whole numbers are not closed
under division.
Reprint 2024-25
RATIONAL NUMBERS  3
– 6 – (– 8) = 2, an integer
Is 8 – (– 6) an integer?
In general, for any two integers
a and b, a – b is again an integer.
Check if b – a is also an integer.
Multiplication 5 × 8 = 40, an integer Integers are closed under
Is – 5 × 8 an integer? multiplication.
– 5 × (– 8) = 40, an integer
In general, for any two integers
a and b, a × b is also an integer.
Division 5 ÷ 8 = 
5
8
, which is not Integers are not closed
an integer.
under division.
Y ou have seen that whole numbers are closed under addition and multiplication but
not under subtraction and division. However , integers are closed under addition, subtraction
and multiplication but not under division.
(iii) Rational numbers
Recall that a number which can be written in the form 
p
q
, where p and q are integers
and q ? 0 is called a rational number. For example, 
2
3
-
, 
6
7
, 
9
5 -
 are all rational
numbers. Since the numbers 0, –2, 4 can be written in the form 
p
q
, they are also
rational numbers. (Check it!)
(a) Y ou know how to add two rational numbers. Let us add a few pairs.
3 ( 5)
8 7
-
+
=
21 ( 40) 19
56 56
+ - -
=
(a rational number)
3 ( 4)
8 5
- -
+
 =
15 ( 32)
...
40
- + -
=
Is it a rational number?
4 6
7 11
+
 = ... Is it a rational number?
W e find that sum of two rational numbers is again a rational number. Check it
for a few more pairs of rational numbers.
We say that rational numbers are closed under addition. That is, for any
two rational numbers a and b, a + b is also a rational number.
(b) Will the difference of two rational numbers be again a rational number?
We have,
5 2
7 3
-
-
=
5 3 – 2 7 29
21 21
- × × -
=
(a rational number)
Reprint 2024-25
4  MATHEMATICS
TRY THESE
5 4
8 5
-
 = 
25 32
40
-
 = ... Is it a rational number?
3
7
8
5
-
- ?
?
?
?
?
? = ... Is it a rational number?
Try this for some more pairs of  rational numbers. W e find that rational numbers
are closed under subtraction. That is, for any two rational numbers a and
b, a – b is also a rational number.
(c) Let us now see the product of two rational numbers.
2 4
3 5
-
×
 =
8 3 2 6
;
15 7 5 35
-
× =
(both the products are rational numbers)
4 6
5 11
-
- ×
 = ... Is it a rational number?
T ake some more pairs of rational numbers and check that their product is again
a rational number.
We say that rational numbers are closed under multiplication. That
is, for any two rational numbers a and b,  a × b is also a rational
number.
(d) We note that 
5 2 25
3 5 6
- -
÷ = (a rational number)
2 5
...
7 3
÷ =
 . Is it a rational number? 
3 2
...
8 9
- -
÷ =
. Is it a rational number?
Can you say that rational numbers are closed under division?
We find that for any rational number a, a ÷ 0 is not defined.
So rational numbers are not closed under division.
However , if we exclude zero then the collection of, all other rational numbers is
closed under division.
Fill in the blanks in the following table.
Numbers Closed under
addition subtraction multiplication division
Rational numbers Y es Y es ... No
Integers ... Y es ... No
Whole numbers ... ... Y es ...
Natural numbers ... No ... ...
Reprint 2024-25
RATIONAL NUMBERS  5
1.2.2  Commutativity
(i) Whole numbers
Recall the commutativity of different operations for whole numbers by filling the
following table.
Operation Numbers Remarks
Addition 0 + 7 = 7 + 0 = 7 Addition is commutative.
2 + 3 = ... + ... = ....
For any two whole
numbers a and b,
a + b = b + a
Subtraction          ......... Subtraction is not commutative.
Multiplication          ......... Multiplication is commutative.
Division          ......... Division is not commutative.
Check whether the commutativity of the operations hold for natural numbers also.
(ii) Integers
Fill in the following table and check the commutativity of different operations for
integers:
Operation Numbers Remarks
Addition          ......... Addition is commutative.
Subtraction Is 5 – (–3) = – 3 – 5? Subtraction is not commutative.
Multiplication          ......... Multiplication is commutative.
Division          ......... Division is not commutative.
(iii) Rational numbers
(a) Addition
Y ou know how to add two rational numbers. Let us add a few pairs here.
2 5 1 5 2 1
and
3 7 21 7 3 21
- - ? ?
+ = + =
? ?
? ?
So,
2 5 5 2
3 7 7 3
- - ? ?
+ = +
? ?
? ?
Also,
-
+
- ?
?
?
?
?
?
6
5
8
3
 = ... and 
Is
-
+
- ?
?
?
?
?
?
=
- ?
?
?
?
?
?
+
- ?
?
?
?
?
?
6
5
8
3
6
5
8
3
?
Reprint 2024-25