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

 Page 1


1.1 PROPERTIES OF ADDITION AND SUBTRACTION OF
INTEGERS
W e have learnt about whole numbers and integers in Class VI. W e have also learnt about
addition and subtraction of integers.
1.1.1  Closure under Addition
We have learnt that sum of two whole numbers is again a whole number. For example,
17 + 24 = 41 which is again a whole number. W e know that, this property is known as the
closure property for addition of the whole numbers.
Let us see whether this property is true for integers or not.
Following are some pairs of integers. Observe the following table and complete it.
Statement Observation
(i) 17 + 23 = 40 Result is an integer
(ii) (–10) + 3 = _____ ______________
(iii) (– 75) + 18 = _____ ______________
(iv) 19 + (– 25) = – 6 Result is an integer
(v) 27 + (– 27) = _____ ______________
(vi) (– 20) + 0 = _____ ______________
(vii) (– 35) + (– 10) = _____ ______________
What do you observe? Is the sum of two integers always an integer?
Did you find a pair of integers whose sum is not an integer?
Since addition of integers gives integers, we say integers are closed under addition.
In general, for any two integers a and b, a + b is an integer .
Chapter  1
Integers
2024-25
Page 2


1.1 PROPERTIES OF ADDITION AND SUBTRACTION OF
INTEGERS
W e have learnt about whole numbers and integers in Class VI. W e have also learnt about
addition and subtraction of integers.
1.1.1  Closure under Addition
We have learnt that sum of two whole numbers is again a whole number. For example,
17 + 24 = 41 which is again a whole number. W e know that, this property is known as the
closure property for addition of the whole numbers.
Let us see whether this property is true for integers or not.
Following are some pairs of integers. Observe the following table and complete it.
Statement Observation
(i) 17 + 23 = 40 Result is an integer
(ii) (–10) + 3 = _____ ______________
(iii) (– 75) + 18 = _____ ______________
(iv) 19 + (– 25) = – 6 Result is an integer
(v) 27 + (– 27) = _____ ______________
(vi) (– 20) + 0 = _____ ______________
(vii) (– 35) + (– 10) = _____ ______________
What do you observe? Is the sum of two integers always an integer?
Did you find a pair of integers whose sum is not an integer?
Since addition of integers gives integers, we say integers are closed under addition.
In general, for any two integers a and b, a + b is an integer .
Chapter  1
Integers
2024-25
MATHEMATICS 2
1.1.2  Closure under Subtraction
What happens when we subtract an integer from another integer? Can we say that their
difference is also an integer?
Observe the following table and complete it:
Statement Observation
(i) 7 – 9 = – 2 Result is an integer
(ii) 17 – (– 21) = _______ ______________
(iii) (– 8) – (–14) = 6 Result is an integer
(iv) (– 21) – (– 10) = _______ ______________
(v) 32 – (–17) = _______ ______________
(vi) (– 18) – (– 18) = _______ ______________
(vii) (– 29) – 0 = _______ ______________
What do you observe? Is there any pair of integers whose difference is not an integer?
Can we say integers are closed under subtraction? Yes, we can see that integers are
closed under subtraction.
Thus, if a and b are two integers then a – b is also an intger. Do the whole numbers
satisfy this property?
1.1.3  Commutative Property
We know that 3 + 5 = 5 + 3 = 8, that is, the whole numbers can be added in any order. In
other words, addition is commutative for whole numbers.
Can we say the same for integers also?
We have 5 + (– 6) = –1 and (– 6) + 5 = –1
So, 5 + (– 6) = (– 6) + 5
Are the following equal?
(i) (– 8) + (– 9) and (– 9) + (– 8)
(ii) (– 23) + 32 and 32 + (– 23)
(iii) (– 45) + 0 and 0 + (– 45)
Try this with five other pairs of integers. Do you find any pair of integers for which the
sums are different when the order is changed? Certainly not. We say that addition is
commutative for integers.
2024-25
Page 3


1.1 PROPERTIES OF ADDITION AND SUBTRACTION OF
INTEGERS
W e have learnt about whole numbers and integers in Class VI. W e have also learnt about
addition and subtraction of integers.
1.1.1  Closure under Addition
We have learnt that sum of two whole numbers is again a whole number. For example,
17 + 24 = 41 which is again a whole number. W e know that, this property is known as the
closure property for addition of the whole numbers.
Let us see whether this property is true for integers or not.
Following are some pairs of integers. Observe the following table and complete it.
Statement Observation
(i) 17 + 23 = 40 Result is an integer
(ii) (–10) + 3 = _____ ______________
(iii) (– 75) + 18 = _____ ______________
(iv) 19 + (– 25) = – 6 Result is an integer
(v) 27 + (– 27) = _____ ______________
(vi) (– 20) + 0 = _____ ______________
(vii) (– 35) + (– 10) = _____ ______________
What do you observe? Is the sum of two integers always an integer?
Did you find a pair of integers whose sum is not an integer?
Since addition of integers gives integers, we say integers are closed under addition.
In general, for any two integers a and b, a + b is an integer .
Chapter  1
Integers
2024-25
MATHEMATICS 2
1.1.2  Closure under Subtraction
What happens when we subtract an integer from another integer? Can we say that their
difference is also an integer?
Observe the following table and complete it:
Statement Observation
(i) 7 – 9 = – 2 Result is an integer
(ii) 17 – (– 21) = _______ ______________
(iii) (– 8) – (–14) = 6 Result is an integer
(iv) (– 21) – (– 10) = _______ ______________
(v) 32 – (–17) = _______ ______________
(vi) (– 18) – (– 18) = _______ ______________
(vii) (– 29) – 0 = _______ ______________
What do you observe? Is there any pair of integers whose difference is not an integer?
Can we say integers are closed under subtraction? Yes, we can see that integers are
closed under subtraction.
Thus, if a and b are two integers then a – b is also an intger. Do the whole numbers
satisfy this property?
1.1.3  Commutative Property
We know that 3 + 5 = 5 + 3 = 8, that is, the whole numbers can be added in any order. In
other words, addition is commutative for whole numbers.
Can we say the same for integers also?
We have 5 + (– 6) = –1 and (– 6) + 5 = –1
So, 5 + (– 6) = (– 6) + 5
Are the following equal?
(i) (– 8) + (– 9) and (– 9) + (– 8)
(ii) (– 23) + 32 and 32 + (– 23)
(iii) (– 45) + 0 and 0 + (– 45)
Try this with five other pairs of integers. Do you find any pair of integers for which the
sums are different when the order is changed? Certainly not. We say that addition is
commutative for integers.
2024-25
INTEGERS 3
In general, for any two integers a and b, we can say
a + b = b + a
l W e know that subtraction is not commutative for whole numbers. Is it commutative
for  integers?
Consider the integers 5 and (–3).
Is 5 – (–3) the same as (–3) –5? No, because  5 – ( –3) = 5 + 3 = 8, and (–3) – 5
= – 3 – 5 = – 8.
T ake atleast five different pairs of integers and check this.
We conclude that subtraction is not commutative for integers.
1.1.4  Associative Property
Observe the following examples:
Consider the integers –3, –2 and –5.
Look at (–5) + [(–3) + (–2)] and [(–5) + (–3)] + (–2).
In the first sum (–3) and (–2) are grouped together and in the second (–5) and (–3)
are grouped together. W e will check whether we get different results.
(–5) + [(–3) + (–2)] [(–5) + (–3)] + (–2)
In both the cases, we get –10.
i.e., (–5) + [(–3) + (–2)] = [(–5) + (–2)] + (–3)
Similarly consider  –3 , 1 and  –7.
( –3) + [1 + (–7)] = –3 + __________ = __________
[(–3) + 1] + (–7) = –2 + __________ = __________
Is (–3) + [1 + (–7)] same as [(–3) + 1] + (–7)?
T ake five more such examples. Y ou will not find any example for which the sums are
different. Addition is associative for integers.
In general for any integers a, b and c, we can say
a + (b + c) = (a + b) + c
2024-25
Page 4


1.1 PROPERTIES OF ADDITION AND SUBTRACTION OF
INTEGERS
W e have learnt about whole numbers and integers in Class VI. W e have also learnt about
addition and subtraction of integers.
1.1.1  Closure under Addition
We have learnt that sum of two whole numbers is again a whole number. For example,
17 + 24 = 41 which is again a whole number. W e know that, this property is known as the
closure property for addition of the whole numbers.
Let us see whether this property is true for integers or not.
Following are some pairs of integers. Observe the following table and complete it.
Statement Observation
(i) 17 + 23 = 40 Result is an integer
(ii) (–10) + 3 = _____ ______________
(iii) (– 75) + 18 = _____ ______________
(iv) 19 + (– 25) = – 6 Result is an integer
(v) 27 + (– 27) = _____ ______________
(vi) (– 20) + 0 = _____ ______________
(vii) (– 35) + (– 10) = _____ ______________
What do you observe? Is the sum of two integers always an integer?
Did you find a pair of integers whose sum is not an integer?
Since addition of integers gives integers, we say integers are closed under addition.
In general, for any two integers a and b, a + b is an integer .
Chapter  1
Integers
2024-25
MATHEMATICS 2
1.1.2  Closure under Subtraction
What happens when we subtract an integer from another integer? Can we say that their
difference is also an integer?
Observe the following table and complete it:
Statement Observation
(i) 7 – 9 = – 2 Result is an integer
(ii) 17 – (– 21) = _______ ______________
(iii) (– 8) – (–14) = 6 Result is an integer
(iv) (– 21) – (– 10) = _______ ______________
(v) 32 – (–17) = _______ ______________
(vi) (– 18) – (– 18) = _______ ______________
(vii) (– 29) – 0 = _______ ______________
What do you observe? Is there any pair of integers whose difference is not an integer?
Can we say integers are closed under subtraction? Yes, we can see that integers are
closed under subtraction.
Thus, if a and b are two integers then a – b is also an intger. Do the whole numbers
satisfy this property?
1.1.3  Commutative Property
We know that 3 + 5 = 5 + 3 = 8, that is, the whole numbers can be added in any order. In
other words, addition is commutative for whole numbers.
Can we say the same for integers also?
We have 5 + (– 6) = –1 and (– 6) + 5 = –1
So, 5 + (– 6) = (– 6) + 5
Are the following equal?
(i) (– 8) + (– 9) and (– 9) + (– 8)
(ii) (– 23) + 32 and 32 + (– 23)
(iii) (– 45) + 0 and 0 + (– 45)
Try this with five other pairs of integers. Do you find any pair of integers for which the
sums are different when the order is changed? Certainly not. We say that addition is
commutative for integers.
2024-25
INTEGERS 3
In general, for any two integers a and b, we can say
a + b = b + a
l W e know that subtraction is not commutative for whole numbers. Is it commutative
for  integers?
Consider the integers 5 and (–3).
Is 5 – (–3) the same as (–3) –5? No, because  5 – ( –3) = 5 + 3 = 8, and (–3) – 5
= – 3 – 5 = – 8.
T ake atleast five different pairs of integers and check this.
We conclude that subtraction is not commutative for integers.
1.1.4  Associative Property
Observe the following examples:
Consider the integers –3, –2 and –5.
Look at (–5) + [(–3) + (–2)] and [(–5) + (–3)] + (–2).
In the first sum (–3) and (–2) are grouped together and in the second (–5) and (–3)
are grouped together. W e will check whether we get different results.
(–5) + [(–3) + (–2)] [(–5) + (–3)] + (–2)
In both the cases, we get –10.
i.e., (–5) + [(–3) + (–2)] = [(–5) + (–2)] + (–3)
Similarly consider  –3 , 1 and  –7.
( –3) + [1 + (–7)] = –3 + __________ = __________
[(–3) + 1] + (–7) = –2 + __________ = __________
Is (–3) + [1 + (–7)] same as [(–3) + 1] + (–7)?
T ake five more such examples. Y ou will not find any example for which the sums are
different. Addition is associative for integers.
In general for any integers a, b and c, we can say
a + (b + c) = (a + b) + c
2024-25
MATHEMATICS 4
TRY THESE
EXAMPLE 1 Write down a pair of integers whose
(a) sum is –3 (b) difference is  –5
(c) difference is 2 (d) sum is 0
SOLUTION (a) (–1) + (–2) = –3 or (–5) + 2 = –3
(b) (–9) – (– 4) = –5 or (–2) – 3 = –5
(c) (–7) – (–9) = 2 or 1 – (–1) = 2
(d) (–10) + 10 = 0 or 5 + (–5) = 0
              Can you write more pairs in these examples?
1.1.5  Additive Identity
When we add zero to any whole number, we get the same whole number. Zero is an
additive identity for whole numbers. Is it an additive identity again for integers also?
Observe the following and fill in the blanks:
(i) (– 8) + 0  = – 8 (ii) 0 + (– 8) = – 8
(iii) (–23) + 0 = _____ (iv) 0 + (–37) = –37
(v) 0 + (–59) = _____ (vi) 0 + _____ = – 43
(vii) – 61 + _____ = – 61 (viii) _____ + 0 = _____
The above examples show that zero is an additive identity for integers.
Y ou can verify it by adding zero to any other five integers.
In general, for any integer a
a + 0 = a = 0 + a
1. Write a pair of integers whose sum gives
(a) a negative integer (b) zero
(c) an integer smaller than both the integers. (d) an integer smaller than only one of the integers.
(e) an integer greater than both the integers.
2. Write a pair of integers whose difference gives
(a) a negative integer . (b) zero.
(c) an integer smaller than both the integers. (d) an integer greater than only one of the integers.
(e) an integer greater than both the integers.
2024-25
Page 5


1.1 PROPERTIES OF ADDITION AND SUBTRACTION OF
INTEGERS
W e have learnt about whole numbers and integers in Class VI. W e have also learnt about
addition and subtraction of integers.
1.1.1  Closure under Addition
We have learnt that sum of two whole numbers is again a whole number. For example,
17 + 24 = 41 which is again a whole number. W e know that, this property is known as the
closure property for addition of the whole numbers.
Let us see whether this property is true for integers or not.
Following are some pairs of integers. Observe the following table and complete it.
Statement Observation
(i) 17 + 23 = 40 Result is an integer
(ii) (–10) + 3 = _____ ______________
(iii) (– 75) + 18 = _____ ______________
(iv) 19 + (– 25) = – 6 Result is an integer
(v) 27 + (– 27) = _____ ______________
(vi) (– 20) + 0 = _____ ______________
(vii) (– 35) + (– 10) = _____ ______________
What do you observe? Is the sum of two integers always an integer?
Did you find a pair of integers whose sum is not an integer?
Since addition of integers gives integers, we say integers are closed under addition.
In general, for any two integers a and b, a + b is an integer .
Chapter  1
Integers
2024-25
MATHEMATICS 2
1.1.2  Closure under Subtraction
What happens when we subtract an integer from another integer? Can we say that their
difference is also an integer?
Observe the following table and complete it:
Statement Observation
(i) 7 – 9 = – 2 Result is an integer
(ii) 17 – (– 21) = _______ ______________
(iii) (– 8) – (–14) = 6 Result is an integer
(iv) (– 21) – (– 10) = _______ ______________
(v) 32 – (–17) = _______ ______________
(vi) (– 18) – (– 18) = _______ ______________
(vii) (– 29) – 0 = _______ ______________
What do you observe? Is there any pair of integers whose difference is not an integer?
Can we say integers are closed under subtraction? Yes, we can see that integers are
closed under subtraction.
Thus, if a and b are two integers then a – b is also an intger. Do the whole numbers
satisfy this property?
1.1.3  Commutative Property
We know that 3 + 5 = 5 + 3 = 8, that is, the whole numbers can be added in any order. In
other words, addition is commutative for whole numbers.
Can we say the same for integers also?
We have 5 + (– 6) = –1 and (– 6) + 5 = –1
So, 5 + (– 6) = (– 6) + 5
Are the following equal?
(i) (– 8) + (– 9) and (– 9) + (– 8)
(ii) (– 23) + 32 and 32 + (– 23)
(iii) (– 45) + 0 and 0 + (– 45)
Try this with five other pairs of integers. Do you find any pair of integers for which the
sums are different when the order is changed? Certainly not. We say that addition is
commutative for integers.
2024-25
INTEGERS 3
In general, for any two integers a and b, we can say
a + b = b + a
l W e know that subtraction is not commutative for whole numbers. Is it commutative
for  integers?
Consider the integers 5 and (–3).
Is 5 – (–3) the same as (–3) –5? No, because  5 – ( –3) = 5 + 3 = 8, and (–3) – 5
= – 3 – 5 = – 8.
T ake atleast five different pairs of integers and check this.
We conclude that subtraction is not commutative for integers.
1.1.4  Associative Property
Observe the following examples:
Consider the integers –3, –2 and –5.
Look at (–5) + [(–3) + (–2)] and [(–5) + (–3)] + (–2).
In the first sum (–3) and (–2) are grouped together and in the second (–5) and (–3)
are grouped together. W e will check whether we get different results.
(–5) + [(–3) + (–2)] [(–5) + (–3)] + (–2)
In both the cases, we get –10.
i.e., (–5) + [(–3) + (–2)] = [(–5) + (–2)] + (–3)
Similarly consider  –3 , 1 and  –7.
( –3) + [1 + (–7)] = –3 + __________ = __________
[(–3) + 1] + (–7) = –2 + __________ = __________
Is (–3) + [1 + (–7)] same as [(–3) + 1] + (–7)?
T ake five more such examples. Y ou will not find any example for which the sums are
different. Addition is associative for integers.
In general for any integers a, b and c, we can say
a + (b + c) = (a + b) + c
2024-25
MATHEMATICS 4
TRY THESE
EXAMPLE 1 Write down a pair of integers whose
(a) sum is –3 (b) difference is  –5
(c) difference is 2 (d) sum is 0
SOLUTION (a) (–1) + (–2) = –3 or (–5) + 2 = –3
(b) (–9) – (– 4) = –5 or (–2) – 3 = –5
(c) (–7) – (–9) = 2 or 1 – (–1) = 2
(d) (–10) + 10 = 0 or 5 + (–5) = 0
              Can you write more pairs in these examples?
1.1.5  Additive Identity
When we add zero to any whole number, we get the same whole number. Zero is an
additive identity for whole numbers. Is it an additive identity again for integers also?
Observe the following and fill in the blanks:
(i) (– 8) + 0  = – 8 (ii) 0 + (– 8) = – 8
(iii) (–23) + 0 = _____ (iv) 0 + (–37) = –37
(v) 0 + (–59) = _____ (vi) 0 + _____ = – 43
(vii) – 61 + _____ = – 61 (viii) _____ + 0 = _____
The above examples show that zero is an additive identity for integers.
Y ou can verify it by adding zero to any other five integers.
In general, for any integer a
a + 0 = a = 0 + a
1. Write a pair of integers whose sum gives
(a) a negative integer (b) zero
(c) an integer smaller than both the integers. (d) an integer smaller than only one of the integers.
(e) an integer greater than both the integers.
2. Write a pair of integers whose difference gives
(a) a negative integer . (b) zero.
(c) an integer smaller than both the integers. (d) an integer greater than only one of the integers.
(e) an integer greater than both the integers.
2024-25
INTEGERS 5
EXERCISE 1.1
1. Write down a pair of integers whose:
(a) sum is  –7 (b) difference is  –10 (c) sum is 0
2. (a) Write a pair of negative integers whose difference gives 8.
(b) Write a negative integer and a positive integer whose sum is –5.
(c) Write a negative integer and a positive integer whose difference is –3.
3. In a quiz, team A scored – 40, 10, 0 and team B scored 10, 0, – 40 in three successive
rounds. Which team scored more? Can we say that we can add integers in
any order?
4. Fill in the blanks to make the following statements true:
(i) (–5) + (– 8) = (– 8) + (............)
(ii) –53 + ............ = –53
(iii) 17 + ............ = 0
(iv) [13 + (– 12)] + (............) = 13 + [(–12) + (–7)]
(v) (– 4) + [15 + (–3)] = [– 4 + 15] + ............
1.2  MULTIPLICATION OF INTEGERS
W e can add and subtract integers. Let us now learn how to multiply integers.
1.2.1  Multiplication of a Positive and a Negative Integer
W e know that multiplication of whole numbers is repeated addition. For example,
5 + 5 + 5 = 3 × 5 = 15
Can you represent addition of integers in the same way?
We have from the following number line, (–5) + (–5) + (–5) = –15
But we can also write
(–5) + (–5) + (–5) = 3 × (–5)
Therefore, 3 × (–5) = –15
–20 –15 –10 –5 0
TRY THESE
Find:
4 × (– 8),
8 × (–2),
3 × (–7),
10 × (–1)
using number line.
2024-25