1. B.J.P.S Samiti’s
M V HERWADKAR ENGLISH MEDIUM
HIGH SCHOOL
CLASS : VII SUBJECT : MATHEMATICS
TOPIC : INTEGERS
BY: SHRAMMI CHAUHAN
2. Introduction
Natural Numbers : Natural numbers are the set of positive integers from 1 to infinity. They are also
known as Counting Numbers. They are denoted by ‘N’.
Eg:1,2,3,80,500,1230,etc.
It does not include zero or any fractional or decimal part.
Whole Numbers : Whole numbers are the set of counting numbers and zero. They are denoted by
‘W’.
Eg:0,1,2,5,6,10,100,121,40001etc.
It does not include any fractional or decimal part.
Integers : An integer is a number that includes positive numbers, negative numbers and 0. They are
denoted by ‘Z’.
Eg: -8,-5,-3,-1,0,2,4,5,6,100,etc.
It does not include any fractional or decimal part.
3. Operations On Number Line
Addition on a Number Line: When we add on a number line, we move
towards right.
Subtraction on a Number Line: When we subtract on a number line, we
move towards left.
4. Arithmetic Operations on Integers
1. Addition of Integers :
When two positive integers are added we get a positive integer.
Example: 44+ 71 = 116.
When two negative integers are added we get a negative integer .
Example:(-44)+(-71)=-116.
When one positive and one negative integers is added, we take their difference and place
the sign of the bigger integer.
Example: (-44) + (71) = 27
PROPERTY NAME OPERATION EXAMPLE REMARK
CLOSURE PROPERTY a+b = c 9 + 4 = 13 ; -9 + 4 = -5 YES
COMMUTATIVE PROPERTY a+b = b+a 5 + 8 = 8 +5 YES
ASSOCIATIVE PROPERTY a+(b+c) = (a+b)+c -5 + (4 + 7) = (-5 + 4) + 7 YES
5. Arithmetic Operations on Integers
2. Subtraction of Integers :
Q1. Solve the following:
a) 5 + 3 = ?
b) 5 + (-3) = ?
c) (-5) + (-3) = ?
d) (-5) - (-3) = ?
PROPERTY NAME OPERATION EXAMPLE REMARK
CLOSURE PROPERTY a-b = c 9 - 4 = 5 ; -9 - 4 = -13 YES
COMMUTATIVE PROPERTY a-b ≠ b-a 5 - 8 = 8 - 5 NO
ASSOCIATIVE PROPERTY a-(b-c) ≠ (a-b)-c -5 - (4 - 7) ≠ (-5 - 4) - 7 NO
6. Arithmetic Operations on Integers
3. Multiplication of Integers :
Product of a positive integer and a negative integer is a negative integer.
a × (–b) = – ab, where a and b are integers.
Product of two negative integers is a positive integer.
(–a) × (–b) = ab, where a and b are integers
PROPERTY NAME OPERATION EXAMPLE REMARK
CLOSURE PROPERTY a x b = c 9 x 4 = 36 ; -9 x 4 = -36 YES
COMMUTATIVE PROPERTY a x b = b x a 5 x 8 = 8 x 5 YES
ASSOCIATIVE PROPERTY ax(bxc) = (axb)xc -5 + (4 + 7) = (-5 + 4) + 7 YES
DISTRIBUTIVE PROPERTY ax(b+c) = axb + axc 6x(-4+9) = 6x(-4) + 6x9 YES
7. Arithmetic Operations on Integers
Rules of Integers in Multiplication / Division :
:
Product of even number of negative integers is positive, where as the product of odd
number of negative integers is negative
PRODUCT OF SIGNS RESULT EXAMPLE
(+) x (+) + 9 x 4 = 36
(+) x (-) - 5 x (-8) = -40
(-) x (+) - -5 x 7 = -35
(-) x (-) + (-4) x (-9) = 36
8. Arithmetic Operations on Integers
4. Division of Integers :
For any integer a,
a ÷ 1 = a and
a ÷ 0 is not defined.
PROPERTY NAME OPERATION EXAMPLE REMARK
CLOSURE PROPERTY a ÷ b = c 4 ÷ 7 = 4/7 NO
COMMUTATIVE PROPERTY a ÷ b = b ÷ a 5 ÷ 8 = 8 ÷ 5 NO
ASSOCIATIVE PROPERTY a ÷ (b÷c) = (a÷b) ÷ c -5 ÷ (4 ÷ 7) = (-5 ÷ 4) ÷ 7 NO
9. The Role of Zero (0)
Additive Identity : Zero is called the identity for integers under addition because
adding zero to any integer, the result is same integer.
This is known as Additive identity property
a + 0 = 0 + a = a
example : 5 + 0 = 0 + 5 = 5
The Role of One (1)
Multiplicative Identity : When 1 is multiplied to any number, the is the same
number.
Therefore, 1 is the multiplicative identity for whole numbers, integers and rational
numbers.
a x 1 = 1 x a = a
Example : -6 x 1 = 1 x -6 = -6
10. Additive Inverse or Negative of a Number
Additive Inverse or Negative of a number is defined as the value which on adding
with the original number results in zero.
For integer a,
a + (-a) = a – a = 0
Example : 9 + (-9) = 9 – 9 = 0
For integer – a,
-a + a = -a + a = 0
Example : -9 + 9 = 0
