1 Crore+ students have signed up on EduRev. Have you? 
Ethan is going for a picnic with his friends. He wants to carry cupcakes with him, but he has got only 3 cupcakes and there are 4 friends. What is he going to do now?
So, Ethan decided to borrow one cupcake from his sister, which he would return later.
How many cupcakes does he have now?
After borrowing one cupcake from his sister, he has got 4cupcakes, which he would give to his four friends.
He goes for the picnic, where he gave away the 4 cupcakes to his friends.
Now, how many cupcakes are left with him?
Is your answer zero (0)?
We can say that there are no or 0 cupcakes left with him, but we also have to keep in mind that he has borrowed one cupcake from his sister.
So, in actual Ethan has (1) cupcake, which means that 1 cupcake is borrowed and did not belong to him.
If he buys 3 more cupcakes the next day, he will have to return 1 cupcake to his sister and will be left with 2 cupcakes only.
Natural Numbers: Natural Numbers is a set of counting numbers. They are denoted by N.
Whole Numbers: If zero is included in the collection of naturalnumbers, we get a new collection of numbers known as whole numbers.
Integers: Integers are a set of whole numbers and negative of all natural numbers.
Opposite Integers/ Additive Inverse
The opposite of an integer is at the same distance from zero but on the opposite sides of the number line. Therefore, one integer will have a positive sign and the other will have a negative sign.
So, we can say that 5 and 5 are opposite integers.
Example: Write the opposite of the following integers:
25, 16, 7, 100
(a) 25
The given integer is negative.
The opposite integer of 25 = ( 25) = 25
(b) 16
The given integer is positive.
The opposite integer of 16 = ( 16) = 16
(c) 7
The given integer is positive.
The opposite integer of 7 = (7) = 7
(d) 100
The given integer is negative.
The opposite integer of 100 = (100) = 100
The absolute value of an integer is its distance from 0 on a number line or its numerical value without taking its sign into consideration.The symbol for absolute value is .
Consider the integers 7 and 7 on the number line.
How far are 7 and 7 from zero?
Both 7 and 7 are at the same distance from 0.
Absolute value of 7 = 7  = 7
Absolute value of 7 = 7= 7
Example: Write the absolute values of the following integers.
(a) 40
(b) 21
(a) We know that the absolute value of an integer is its distance from 0 on a number line.
Therefore, the absolute of 40 = 40 = 40
(b) 25
Now, the absolute value of an integer is its distance from 0 on a number line.
The absolute value of 25 = 25 = 25
Example: Which is greater?
23 or 15
We know that the absolute value of an integer is its distance from 0 on a number line.
First, we will locate 23 and 25 on the number line and compare their distance from 0.
Distance of 0 from 23 is more than its distance from 15.
Absolute value of 23 = 23 = 23
Absolute value of 15 = 15 = 15
So, 23>15
Example: Find the value of,
(a) 8 – 17
8 – 17 = 9
So, 9 = 9
(b) 6 + 7
We see, 6 = 6 and 7 = 7
6+ 7 = 6 + 7 = 13
Ordering of Absolute Values
Example: Arrange the following integers in descending order.
3, 12, 3 – 12, 312, 3+12
We know that the absolute value of an integer is the distance of the integer from 0.
3 = 3
12 = 12
3 – 12= 9 = 9
3 12 = 3 – (12) = 3 – 12 = 9
3 + 12 = 3 + 12 = 15
Now, arranging the integers in descending order we get,
15 > 12 > 9 > 3 > 9
Absolute values as the distance between Numbers
Let a and b be the two numbers marked on the number line as shown below.
As b lies to the right of a on the number line, so b > a.
Distance between a and b = b − a
Now, we interchange the position of a and b on the number line.
Here, a > b
Distance between a and b = a − b
If we don’t know which of the two numbers is greater, we find the absolute value of the distance between the two numbers.
a − b = b − a
Example: Find the distance between 5 and 6.
(a) Distance between 5 and 6
5 – 6 = 11 = 11
6 – (5) = 6 + 5 = 11
Distance between 5 and 6 = 11
If we add 6 and 4 (both are positive integers), we add their values and the result will also be a positive integer.
So, 6 + 4 = 10
Let’s do this addition on a number line also.
When we add two or more integers on the number line then we move towards the right of any one of the given numbers.
So we start from 0 and jump 6 units towards the right and then again jump 4 units towards right from 6.
Example: Add 25 and 46
25 and 46 are positive integers. So, we add their values.
25 + 46 = 71
The result will be a positive integer. (a common sign of both the integers)
Add: 6 + (9)
Here, one integer is positive and the other integer is negative.
So, we find the difference of the integers, (9 – 6 = 3)
Out of the two integers, which one is greater (do not consider the sign of the integers here)?
9 is the greater integer, but it is a negative integer.
Therefore, result 3 will have a negative sign.
So, 9 + 6 = 3
When we add 9 and 6 on a number line, we first start from 0 and jump 6 places to the right of zero.
We reach 6 on the number line, and then we jump 9 places to the left of 6. (When we add a negative integer on a number line, we move towards left)
We reach 3 on the number line.
So, 9 + 6 = 3
Example: Add 67 and 32
67 + 32
Here, one integer is positive and the other is negative so we find the difference of their numerical values.
67 – 32 = 35
Now, 67 is the greater integer (signs of the integers are not considered).
As 67 is a negative integer, the result will take the negative sign.
67 + 32 = 35
Example: Write down a pair of integers whose:
(i) sum is 7
Consider the pair of integers (10, 3).
As one of the integers is negative, we find the difference between their numerical values and put
the sign of the greater integer.
Sum of 10 and 3 = 10 + 3 = 7
(ii) sum is 0
Consider the pair of integers (7, 7)
One of the integers is negative, we find the difference between the numerical values.
Sum of 7 and 7 = 7 + 7 = 0
Add: 5 and 3
5 + (3) = 5 – 3 = 8
If we do the addition of 5 and 3 on the number line, then we start from 0 and jump 5 places to the left of 0.
We reach 5 on the number line, and then we jump 3 places to the left of 5. (As we are adding a negative integer on a number line, we move towards left)
We reach 8 on the number line.
5 + (3) = 8
Example: Add 78 and 36
Both the integers are negative, so we add their values and the result will take the negative sign.
78 + (36) = 78 – 36 = 114
When we subtract one integer from the other, we convert the integer to be subtracted to its negative and then add the two integers.Subtract 3 from 7.
7 – 3
Additive inverse of 3 is 3
So we add 7 to the additive inverse of 3 that is 3.
7 3 = 7 + (3) = 4
On a number line, we jump 7 places to the right of zero.
We reach 7 on the number line, and then we jump 3 places to the leftof 7. As we are adding a negative integer we move towards left.
We reach 4 on the number line.
So, 7 + (3) = 4
Example: Subtract 18 from 76
76 – 18
Additive inverse of 18 is 18. So, we add 76 to the additive inverse of 18, which is 18.
= 76 + (18) = 58
Example: Subtract 45 from 34
34 – 45
Now, we add 34 to the additive inverse of 45, which is 45.
= 34 + (45) = 11
When we subtract a negative integer from another integer, it is the same as adding the two integers.
Subtract 3 from 9.
9 – (3) = 9 + 3 = 12
On a number line we jump 9 places to the right of 0.
We reach 9 on the number line. When we subtract a negative integer from any integer we move towards the right. So we jump 3 places to the right of 9.
We reach 12 on the number line.
So, 9 – (3) = 9 + 3 = 12
Example: Subtract 26 from 48
Here, we are subtracting the negative integer, 26 from 48. So we simply add the two integers.
48 – (26)
48 – (26) = 48 + 26 = 74
Example: 89 from 67
Now, we have to subtract a negative integer, 89 from another negative integer, 67.
67 – (89) = 67 + 89
We know that when we add a positive and a negative integer, then we find their difference and put the sign of the greater integer.
67 + 89 = 22 (sign will be positive as the greater integer, 89 has a positive sign)
Example: Shyam deposits Rs 2,000 in his bank account and withdraws Rs 1,542 from it, the next day. If the withdrawal of the amount from the account is represented by a negative integer, then how will you represent the amount deposited? Find the balance in Shyam’s account after the withdrawal.
Amount deposited = Rs 2,000
Amount withdrawn = Rs 1,542
If the withdrawal of the amount from the account is represented by a negative integer then the amount deposited will be represented by a positive integer.
Balance amount = Rs 2,000 – Rs 1,542 = Rs 458
Closure property of Addition of Integers:
If a and b are two integers, then a + b is also an integer.
When we add any two integers, the result will always be an integer.
This is true for all integers.
If we add 4 to 5 on the number line, we start from 4 and jump 5 places to the right of 4. We reach 1 on the number line.
So, 4 + 5 = 1, an integer
Closure Property of Subtraction of Integers
If a and b are two integers, then a − b is also an integer.
When we subtract an integer from another, the result will always be an integer. This is true for all integers.
If we subtract4 from5,(5 4) on the number line, we start from 4 and jump 5 places to the left of 4. We reach9 on the number line.
So, 5  4 = 9, which is an integer
Commutative property of addition of integers
If a and b are two integers, then a + b = b + a
Hence, we can add two integers in any order.
Commutative property of subtraction of integers
If a and b are two integers, then a − b ≠ b − a
Example: Verify the following and state the property used.
(5) + (8) = (8) + (5)
LHS
5 + (8) = 5 – 8 = 13
RHS
8 + (5) = 8 – 5 = 13
LHS = RHS
Here, we have used the commutative property of addition of integers which states that, if a and b are two integers, then a + b = b + a
Associative property of Addition of Integers
If a, b & c are any three integers, then
(a + b) + c = a + (b + c)
When we are adding integers, they can be grouped in any order and the result remains the same.
Consider the three integers, 2, 4 and 6
Case 1: [2 + (4)] + (6)
In the first case, we group 2 and 4.
[2 + (4)] + (6) = 6 + (6)
On a number line, we start from 6 and jump 6 places to the left of 6.
We reach 12 on the number line.
Case 2: (2) + [4 + (6)]
In the second case we group together 4 and 6.
(2) + [4 + (6)] = 2 + (10)
On a number line, we start from 2 and jump 10 places to the left of 2.
We reach 12 on the number line.
We see that the result is the same in both cases.
Associative property of Subtraction of Integers
For any three integers a, b and c,
(a – b) – c ≠ a – (b – c)
Consider the integers, 3, 5 and 6
Case 1: [3  (5)]  (6)
In the first case, we group together 3 and 5.
[3  (5)]  (6) = 2 + 6 = 8
Case 2: (3) – [5 – (6)]
In the second case, we group together 5 and 6.
(3) – [5 – (6)] = 3 – 1 = 4
[3  (5)]  (6) ≠ (3) – [5 – (6)]
Example: Fill in the blanks to make the following statements true.
(i) [13 + (12)] + (___) = 13 + [(12) + (7)]
We have used the associative property of addition of integers which states that, if a, b & c are any three integers, then
(a + b) + c = a + (b + c)
If a = 13, b = 12 and c = 7 then,
[13 + (12)] + (7) = 13 + [(12) + (7)]
(ii) (4) + [15 + (3)] = [4 + 15] + (__)
We use the associative property of addition of integers which states that, if a, b & c are any three integers, then
(a + b) + c = a + (b + c)
If a = 4, b = 15 and c = 3 then,
(4) + [15 + (3)] = [4 + 15] + (3)
Additive Identity Property:
If a is any integer, then a + 0 = a = 0 + a
The number 'zero' has a special role in addition. When we add zero to any integer the result is the same integer again. Zero is the additive identity for integers.
Example: Fill in the blanks
(i) (23) + 0 = ____
If we add zero to any integer the result is the same integer again. This property is known as additive identity property.
So, (23) + 0 = 23
(ii) 0 + ___ = 43
We again use the additive identity property.
So, 0 + (43) = 43
(iii) 8 + ___ = 8
Using the additive identity property, we get, 8 + 0 = 8
Rule 1: To find the product of two integers with the same sign, we find the product of their values and put the positive sign before the product.
For any two positive integers a and b,
a × b = ab
For any two negative integers (−a) and (−b)
(−a) × (−b) = ab
Consider two positive integers 6 and 8
6 × 8 = +48
Now, consider the two negative integers 6 and 8
(6) × (8) = +48
We see that the product is positive in both cases.
Example: Find
(i) (11) × (100)
The two integers have the same sign (negative), so we find the product of their values and put the positive sign before the product.
(11) × (100) = +1100
(ii) 25 × 250
As the two integers are positive (same sign) we find the product of their values and give a positive sign to the product.
25 × 250 = 6250
(iii) (60) × (21)
Now, both the integers are negative (same sign) so we find the product of their values and put the positive sign before the product.
(60) × (21) = 1260
Rule 2: To find the product of two integers with unlike signs, we find the product of their values and put the negative sign before the product.
For any two integers a and b,
a × (−b) = (−a) × b = −(a × b)
Example: Find
(i) (31) × 30
The two integers have different signs, one is positive and the other is negative. So we find the product of their values and give the product a negative sign.
(31) × 30 = 930
(ii) 26 × (13)
Here, one integer is positive and the other integer is negative. So, we find the product of their values and put a negative sign before the product.
26 × (13) = 338
(iii) (60) × 14
As the two integers have unlike signs, we find the product of their values and put a negative sign before the product.
(60) × 14 = 840
Product of three or more Negative Integers
We know that the product of two negative integers is a positive integer. What happens if we have to find the product of more than two negative integers?
We see that when the number of negative integers in a product is even, then the product is an even integer and if the number of negative integers in the product is odd, then the product is a negative integer.
Example: The product of (9) × (5) × (6) × (3) is positive whereas the product of (9) × (5) × 6 × (3) is negative. Why?
The product of (9) × (5) × (6) × (3) is positive because the number of negative integers in the product is 4, which is an even number.
(9) × (5) × (6) × (3) = [(9) × (5)]×[ (6) × (3)]
= 45×18 = 810
The product of (9) × (5) × 6 × (3) is negative because the number of negative integers in the product is 3, which is an odd number.
(9) × (5) × 6 × (3) = [(9) × (5)]×[ 6 × (3)]
= 45 × (18) = 810
Example: What will be the sign of the product if we multiply together,
(i) 8 negative integers and 3 positive integers?
When we consider the sign of the product, we count the number of negative integers. Here, the number of negative integers in the product is 8, which is an even number.
So the product of 8 negative integers and 3 positive integers is an even integer.
(ii) 5 negative integers and 4 positive integers?
We know that if the number of negative integers in the product is odd, then the product is a negative integer. Here, the number of negative integers in the product is 5, an odd number. So, the product of 5 negative integers and 4 positive integers is odd.
Closure Property
If a and b are two integers, then a × b is an integer.
(2)× 3 = ( 6)
We start from 0 and jump 2 places to the left of 0. We make 3 such jumps. We reach 6 on the number line.
If we multiply two integers, the product is also an integer.
Commutative Property:
If a and b are two integers, then a × b = b × a
The value of the product does not change even when the order of multiplication is changed.
Associative Property
If a, b&c are any three integers, then
(a × b) × c = a × (b × c)
When we multiply three or more integers, the value of the product remains the same even if they are grouped in any manner and this is called the associative property for multiplication of integers.
Multiplication of Integers by Zero (0)
If a is any integer, then a × 0 = 0 × a = 0
15 × 0 = 0
(100)× 0 = 0
0 ×(25) = 25
The product of a negative integer and zero is always zero.
Multiplicative Identity
If a is any integer, then a × 1 = a = 1 × a
If we multiply any integer by 1, the product is the integer itself.
So, 1 is the multiplicative identity of integers.
Distributive Property of Multiplication over Addition:
If a, b & c are any three integers, then
a × (b + c) = a × b + a × c
Example: Find the product using suitable property.
(i) 26× (48) + (48)× (36)
= (48)×26 + (48)×(36)
(by commutative property,a × b = b × a)
= (48)×[26 + (36)]
= (48)×[26 – 36]
= (48)×(10) = 480
(ii) 8×53×(125)
= 53×[8×(125)]
(by associative property of multiplication (a × b) × c = a × (b × c))
= 53×(1000) = 53000
(iii) (41)×101
= (41)×(100 + 1)
= (41)×100 + (41)× 1
(By the Distributive Property of Multiplication over Addition,a × (b + c) = a × b + a × c)
= 4100 + (41)
= 4100 – 41 = 4141
Example: A certain freezing process requires that room temperature be lowered from 50°C at the rate of 6°C every hour. What will be the room temperature 12 hours after the process begins?
Initial room temperature = 50°C
Decrease in temperature in 1 hour = 6°C
Decrease in temperature in 12 hours = 12 × (6) =  72°C
Final temperature = 50°C + ( 72°C) = 22°C
Rule 1: If two integers of different signs are divided, then we divide them as whole numbers and give a negative sign to the quotient.
Consider the two integers, 12 and 3.
If we divide 12 by 3, we get,
Example: Evaluate each of the following:
(i) (−30) ÷ 10
(−a) ÷ b = − (a/b)
(−30) ÷ (10) = − (30/10) = 3
(ii) 49 ÷ (−49)
a÷ (b) = − (a/b)
49 ÷ (−49) = − (49/49) = 1
(iii) 13 ÷ [(−2) + 1]
= 13 ÷ (−1)
a ÷ (−b) = − (a/b)
= 13 ÷ (−1) = − (13/1)=  13
Rule 2: If two integers of the same signs are divided, then we divide them as whole numbers and give a positive sign to the quotient.
(−a) ÷ (−b) = a/b
a ÷ b = a/b
On dividing 25 by 5 we get,
25 ÷ 5 = 25/5 = 5
If we divide (25) by (5) we get,
(−25) ÷ (−5) = (25/5) = 5
We see that the result is the same in both cases.
Example: Evaluate each of the following:
(i) (−36) ÷ (−4)
(−a) ÷ (−b) = a/b
= (−36) ÷ (−4) = 36/4 = 9
(ii) (−31) ÷ [(−30) + (−1)]
= (−31) ÷ [(−30) + (−1)]
= (−31) ÷ (−31)
(−a) ÷ (−b) = a/b
(−31) ÷ (−31) = 31/31 = 1
(iii) [(−6) + 5] ÷ [(−3) + 2]
= (−1) ÷ (−1)
(−a) ÷ (−b) = a/b
(−1) ÷ (−1) = 1/1 = 1
Example: Write five pairs of integers (a, b) such that a ÷ b = – 4.
Five pairs of integers are,
(i) (8, −2)
a ÷ (−b) = − (a/b)
8 ÷ (−2) = − (8/2) = 4
(ii) (−4 , 1)
(−a) ÷ b =  (a/b)
= (−4 ) ÷ 1 = − 4/1 = 4
(iii) (−16, 4)
(−a) ÷ b = − (a/b)
(−16) ÷ 4 =  16/4) = 4
(iv)(−24, 6)
(−a) ÷ b = − (a/b)
(−24) ÷ 6 = − (24/6) = 4
(v) (36, −9)
a ÷ (−b) =  (a/b)
36 ÷ (−9) = − (36/9) = 4
(i) Closure Property
If a and b are two integers, then a ÷ b is not always an integer.
(ii) Commutative Property
If a and b are two integers, a ÷ b ≠ b ÷ a
d5
(iii) Associative Property
For any 3 integers a, b and c, (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)
(iv) Division of 0 by any integer
If a is any integer other than zero, then a ÷ 0 is not defined but 0 ÷ a = 0,
(v) Division by 1
If a is an integer, then a ÷ 1 = a
Example: Fill in the blanks
(i) ____ ÷ 25 = 0
If we divide 0 by any integer, the result is always zero.
So, 0 ÷25 = 0
(ii) (206) ÷ _____ = 1
If we divide any 206 by 206, the result is one.
(206) ÷ (206) = 1
(iii) _____ ÷ 1 = 87
If any integer is divided by 1 the result is the same integer.
(87) ÷ 1 = 87
Addition of Integers
Subtraction of Integers
When we subtract one integer from the other, we convert the integer to be subtracted to its negative and then add the two integers.
168 videos276 docs45 tests
