 Natural Numbers and Whole Numbers
 Properties of Integers
 Properties of Addition and Subtraction of Integers
 Properties of Multiplication of Integers
 Addition and Subtraction of Integers
 Multiplication of Integers
 Division of Integers
 Rules of Addition of Integers
 Rules for subtraction of integers
 Rules for Multiplication and Division of Integers
 How to Tackle Problem Solving Strategy
 Exercise 1
 We have learnt about whole numbers and integers in previous class.
 Everyone know that integers form a bigger collection of numbers which contains whole numbers and negative numbers.
 In this chapter, we will study more about integers, their properties and operations.
Representation of Integers on Number Line:
Natural Numbers and Whole Numbers
Natural numbers are counting numbers but these set of numbers do not include zero.
Example: 1,2,3,4,5,6……etc are all natural numbers.
All natural numbers along with zero are called Whole numbers.
For example: 0, 1, 2, 3, 4, 5, 6………etc are all whole numbers.
Properties of Integers
Integers are closed under:
 Addition
 Subtraction
 Multiplication
 Division
It means that operation on integers will also give integers.
Properties of Addition and Subtraction of Integers
Closure under Addition and subtraction:

Integers
are closed under addition.
 In general, for any two integers a and b, a + b is an integer.

Integers
are closed under subtraction.
 Thus, if a and b are two integers then a – b is also an integer.
Commutativity Property for addition:
In general, for any two integers a and b, we can say a + b = b + a
Associativity Property for addition:
In general for any integers a, b and c, we can say a + (b + c) = (a + b) + c
Additive Identity:

Zero is
an additive identity for integers.
 In general, for any integer a: a + 0 = a = 0 + a
Additive Inverse:
If
we add numbers like (7) and 7 then we get the result as zero.
So these are called the Additive inverse of each other.
If we add (2) + (2), then
first we move 2 steps to the left of zero then we move two steps to the right
of (2).
So, finally we reached to zero.
Hence, if we add the positive and negative of the same number then we get the zero.
Properties of Multiplication of Integers
Closure under Multiplication:
 Integers are closed under multiplication.
 So, a × b is an integer, for all integers a and b.
Commutative Property of Multiplication:

Multiplication
is commutative for integers.
 In general, for any two integers a and b, a × b = b × a
Multiplication by Zero:
 The product of a negative integer and zero is zero.
 So, a × 0 = 0 × a=0
Multiplicative Identity:
 1 is the multiplicative identity for integers.
 a × 1 = 1 × a = a
Associative property of Multiplication:
 Multiplication is associative for integers.
 (a × b) × c = a × (b × c)
Distributive Property of Integers:
 The distributivity of multiplication over addition is true for integers.
 a × (b + c) = a × b + a × c
 The distributivity of multiplication over subtraction is true for integers.
 a × (b – c) = a × b – a × c
Addition and Subtraction of Integers
Addition of Two Positive Integers
 If you have to add two positive integers then simply add them as natural numbers.
 (+6) + (+7) = 6 + 7 = 13
Addition of Two Negative Integers
 If we have to add two negative integers then simply add them as natural numbers and then put a negative sign on the answer.
 (6) + (7) =  (6+7) = 13
Addition of One Negative and One Positive Integer
 If we have to add one negative and one positive integer then simply subtract the numbers and put the sign of the bigger integer.
 We will decide according to the bigger integer and ignoring the sign of the smaller integer.
 (6) + (7) = 1 (bigger integer 7 is positive integer)
 (6) + (7) = 1(bigger integer 7 is negative integer)
Multiplication of Integers
Product of two positive integers is a positive integer.
Example: (+4) x (+2) = +8
Product of two negative integers is a positive integer.
Example: (−5) × (−2) = +10
Product of a positive and a negative integer is a negative integer.
Example: (+6) × (−3) = −18
Example: (−2) × (+5) = −10
Product of even number of negative integers is positive.
Example: (4) × (−3) = 12
Product of odd number of negative integers is negative.
Example: (4) × (−3) × (−2) = −24
Division of Integers
When we divide a positive integer by a negative integer:
 We first divide them as whole numbers.
 And then, put a minus sign () before the quotient.
 General Rule: a ÷ (b) = (a) ÷ b where b ≠ 0.
When we divide a negative integer by a negative integer:
 We first divide them as whole numbers.
 And then, put a positive sign (+).
 General Rule: (a) ÷ (b) = a ÷ b where b ≠ 0
Any integer divided by 1 gives the same number.
 a ÷ 1 = a
For any integer a, we have a ÷ 0 is not defined.
Rules of Addition of Integers
Rules for subtraction of integers
Rules for Multiplication and Division of Integers
How to Tackle Problem Solving Strategy
Exercise 1
1. Following number line shows the temperature in degree Celsius (C°) at different places on a particular day.
(a) Observe this number line and write the temperature of the places marked on it.
(b) What is the temperature difference between the hottest and the coldest places among the above?
(c) What is the temperature difference between Lahulspriti and Srinagar?
(d) Can we say temperature of Srinagar and Shimla taken together is less than the temperature at Shimla?
Is it also less than the temperature at Srinagar?
Solution:
(a) From the given number line, we observe the following temperatures.
Cities 
Temperature 
Lahulspriti 
8°C 
Srinagar 
2°C 
Shimla 
5°C 
Ooty 
14°C 
Bengaluru 
22°C 
(b) The temperature of the hottest place = 22°C
The temperature of the coldest place = 8°C
Difference = 22°C – (8°C)
= 22°C + 8°C = 30°C
(c) Temperature of Lahulspriti = 8°C
Temperature of Srinagar = 2°C
Difference = 2°C – (8°C)
= 2°C + 8°C = 6°C
(d) Temperature of Srinagar = 2°C
Temperature of Shimla = 5°C
Temperature of the above cities taken together
= 2°C + 5°C = 3°C
Temperature of Shimla = 5°C
Hence, the temperature of Srinagar and Shimla taken together is less than that of Shimla by 2°C.
i.e., (5°C – 3°C) = 2°C
2. In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Jack’s scores in five successive rounds were 25, –5, –10, 15 and 10, what was his total at the end?
Solution:
Given scores are 25, 5, 10, 15, 10
Marks given for correct answers
= 25 + 15 + 10 = 50
Marks given for incorrect answers
= (5) + (10) = 15
=> Total marks given at the end
= 50 + (15) = 50 – 15 = 35
3. At Srinagar temperature was –5°C on Monday and then it dropped by 2°C on Tuesday. What was the temperature of Srinagar on Tuesday? On Wednesday, it rose by 4°C. What was the temperature on this day?
Solution:
Initial temperature of Srinagar on Monday = 5°C
Temperature on Tuesday = 5°C – 2°C = 7°C
Temperature was increased by 4°C on Wednesday.
=> Temperature on Wednesday
= 7°C + 4°C = 3°C
Hence, the required temperature on Tuesday = 7°C
and the temperature on Wednesday = 3°C
4. A plane is flying at the height of 5000 m above the sea level. At a particular point, it is exactly above a submarine floating 1200 m below the sea level. What is the vertical distance between them?
Solution:
Height of the flying plane = 5000 m
Depth of the submarine = 1200 m
=> Distance between them
= + 5000 m – (1200 m)
= 5000 m + 1200 m = 6200 m
Hence, the vertical distance = 6200 m
5. Mohan deposits Rs. 2,000 in his bank account and withdraws Rs. 1,642 from it, the next day. If withdrawal of amount from the account is represented by a negative integer, then how will you represent the amount deposited? Find the balance in Mohan’s account after the withdrawal.
Solution:
The deposited amount will be represented by a positive integer i.e., Rs. 2000.
Amount withdrawn = Rs. 1,642
=> Balance in the account
= Rs. 2,000 – Rs. 1,642 = Rs. 358
Hence, the balance in Mohan’s account after the withdrawal = Rs. 358.
6. Rita goes 20 km towards east from a point A to the point B. From B, she moves 30 km towards west along the same road. If the distance towards east is represented by a positive integer then, how will you represent the distance traveled towards west? By which integer will you represent her final position from A?
Solution:
Distances traveled towards east from point A will be represented by positive integer i.e. +20 km.
Distance traveled towards the west from point B will be represented by negative integer, i.e., –30 km.
Final position of Rita from A
= 20 km – 30 km = – 10 km
Hence, the required position of Rita will be presented by a negative number, i.e., 10.
7. In a magic square each row, column and diagonal have the same sum. Check which of the following is a magic square.
5 
1 
4 
5 
2 
7 
0 
3 
3 
(i)
1 
10 
0 
4 
3 
2 
6 
4 
7 
(ii)
Solution:
(i) Row one R_{1} = 5 + (1) + (4)
= 5 – 1 – 4 = 5 – 5 = 0
Row two R_{2} = (5) + (2 ) + 7
= 5 – 2 + 7 = 7 + 7 = 0
Row three R_{3} = 0 + 3 + (3)
= 0 + 3  3 = 0
Column one C_{1}t = 5 + (5) + 0
= 5 – 5 + 0 = 0
Column two C_{2} = (1) + (2) + (3)
= 1 – 2 + 3 = 3 + 3 = 0
Column three C_{3} = (4) + 7 + (3)
= 4 + 7 – 3 = 7 – 7 = 0
Diagonal d_{12} = 5 + (2) + (3)
= 5 – 23 = 5 – 5 = 0
Diagonal d_{2} = (4) + (2) + 0
= 4 –2 + 0 = 6 + 0 = 6
Here, the sum of the integers of diagonal d2 is different from the others.
Hence, it is not a magic square.
(ii) Row one R_{1} = 1 + (10) + 0
= 1 – 10 + 0 = 9
Row two R_{2} = (4) + (3) + (2)
= 4 – 3 – 2 = 9
Row three R_{3} = (6) + (4) + (7)
= 6 + 4 –7 = 9
Column one C_{3} = 1 + (4) + (6)
= 1 – 4 – 6 = 9
Column two C_{2} = (10) + (3) + 4
= 10 – 3 + 4 = 9
Column C_{3} = 0 + (2) + (7)
= 0 – 2 – 7 = 9
Diagonal d_{1} = 1 + (3) + (7)
= 1 – 3 – 7 = 1 – 10 = 9
Diagonal d_{2} = 0 + (3) + (6)
= 0 – 36 = 9
Here, sum of the integers column wise, row wise and diagonally is same i.e. 9.
Hence, (ii) is a magic square.
8. Verify a – (b) = a + b for the following values of a and b.
(i) a = 21, b = 18
(ii) a = 118, b = 125
(iii) a = 75, b = 84
(iv) a = 28, b = 11
Solution:
(i) a – (b) = a + b
LHS = 21 – (18) = 21 + 18 = 39
RHS = 21 + 18 = 39
LHS = RHS Hence, verified.
(ii) a – (b) = a + b
LHS = 118 – (125) = 118 + 125 = 243
RHS = 118 + 125 = 243
LHS = RHS Hence, verified.
(iii) a – (b) = a + b
LHS = 75 – (84) = 75 + 84 = 159
RHS = 75 + 84 = 159
LHS = RHS Hence, verified.
(iv) a – (b) = a + b
LHS = 28 – (11) = 28 + 11 = 39
RHS = 28 + 11 = 28 + 11 = 39
LHS= RHS Hence, verified.
9. Use the sign of >, < or = in the box to make the statements true.
(a) (8) + (4) [] (8) – (4)
(b) (3) + 7 – (19) [] 15 – 8 + (9)
(c) 23 – 41 + 11 [] 23 – 41 – 11
(d) 39 + (24) – (15) [] 36 + (52) – (36)
(e) 231 + 79 + 51 [] 399 + 159 + 81
Solution:
(a) (8) + (4) [] (8) – (4)
LHS = (8) + (4) = 8 – 4 = 12
RHS = (8) – (4) = 8 + 4 = 4
Here – 12 < 4
Hence, (8) + (4) [<] (8) – (4)
(b) (3) + 7 – (19) [] 15 – 8 + (9)
LHS = (3) + 7 – (19) = 3 + 719
= 3 – 19 + 7
= 22 + 1 = 15
RHS = 15 – 8 + (9)
= 1589
= 15 – 17 = 2
Here – 15 < 2
Hence, (3) + 7 – (19) [<] 15 – 8 +(9)
(c) 23 – 41 + 11 [] 23 – 41 – 11
LHS = 23 – 41 + 11 = 23 + 11 – 41 = 34 – 41 = 7
RHS = 23 – 41 – 11 = 23 – 52 = 29 Here, 7 > 29
Hence, 23 – 41 + 11 [>] 23 – 41 – 11
(d) 39 + (24) – (15) [] 36 + (52) – (36)
LHS = 39 + (24) – (15)
= 39 – 24 – 15
= 39 – 39 = 0
RHS = 36 + (52) – (36) = 36 – 52 + 36
= 36 + 36 – 52
= 72 – 52 = 20
Here 0 < 20
Hence, 39 + (24) – (15) [<] 36 + (52) – (36)
(e) 231 + 79 + 51 [] 399 + 159 + 81
LHS = 231 + 79 + 51 = 231 + 130 = 101
RHS = 399 + 159 + 81 = 399 + 240 = 159
Here, 101 > 159
Hence, 231 + 79 + 51 [>] 399 + 159 + 81
10. A water tank has steps inside it. A monkey is sitting on the topmost step (i.e., the first step). The water level is at the ninth step.
(i) He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will he reach the water level?
(ii) After drinking water, he wants to go back. For this, he jumps 4 steps up and then jumps back 2 steps down in every move. In how many jumps will he reach back the top step?
(iii) If the number of steps moved down is represented by negative integers and the number of steps moved up by positive integers, represent his move in part (t) and (ii) by completing the following:
(a) – 3 + 2 – … = 8
(b) 4 – 2 + … = 8. In (a) the sum (8) represents going down by eight steps. So, what will the sum 8 in (b) represent?
Solution:
(i) The position of monkey after the
1^{st} jump J_{1} is at 4^{th} step ¯
2^{nd} jump J_{2} is at 2^{nd} step
3^{rd} jump J_{3} is at 5^{th} step ¯
4^{th} jump J_{4} is at 3^{rd} step
5^{th} jump J_{5} is at 6^{th} step ¯
6^{th} jump J_{6} is at 4^{th} step
7^{th} jump J_{7} is at 7^{th} step ¯
8^{th} jump J_{8} is at 5^{th} step
9^{th} jump J_{9} is at 8^{th} step ¯
10^{th} jump J_{10} is at 6^{th} step
11^{th} jump J_{11} is at 9^{th} step ¯ (Water level)
Hence the required number of jumps = 11.
(ii) Monkey’s position after the
1^{st} jump J_{1} is at 5^{th} step
2^{nd} jump J_{2} is at 7^{nd} step
3^{rd} jump J_{3} is at 3^{rd} step
4^{th} jump J_{4} is at 5^{th} step ¯
5^{th} jump J_{5} is at 1^{st} step
Hence, the required number of jumps = 5.
(iii) According to the given conditions we have the following tables
Jumps 
J_{1} 
J_{2} 
J_{3} 
J_{4} 
J_{5} 
J_{6} 
J_{7} 
J_{8} 
J_{9} 
J_{10} 
J_{11} 
Number of steps 
3 
+2 
3 
+2 
3 
+2 
3 
+2 
3 
+2 
3 
Therefore (a) Total number of steps
= 3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 – 3
= 8 which represents the monkey goes down by 8 steps.
In case (ii), we get
Jumps 
J_{1} 
J_{2} 
J_{3} 
J_{4} 
J_{5} 
Number of steps 
+4 
2 
+4 
2 
+4 
Therefore (b) Total number of steps.
= +4 – 2 + 4 – 2 + 4 = 8
Here, the monkey is going up by 8 steps.
Did You Know
 Negative numbers were accepted into the number system in the 19^{th} century.
 The Indian mathematician Brahmagupta is known to have used negative numbers starting around 630 AD.
 The Chinese are credited with being the first known culture to recognize and use negative numbers.