The document is about integers and their properties. Some key points: - Integers include whole numbers and their negatives, but not fractions or imaginary numbers. - The modulus or absolute value of a number gives its numerical value regardless of sign. - Every integer has an additive inverse, such that when added the result is 0. - Addition and subtraction follow rules based on sign: unlike signs subtract, like signs add. - Multiplication and division of integers with unlike signs results in a negative product or quotient. - Integers have properties for addition, subtraction, multiplication and division like commutativity, associativity and distribution.

1. Integers
Class 7th
Nisha Mahajan

2. Introduction
 Whole Number – Number from 0 till infinity
 Negatives of whole number + whole numbers = Integers
 Integers does not include fraction or imaginary numbers.
 When we move from L to R , value of integers increase.

3. Additive Inverse and Modulus
 Modulus is absolute value if integer a is numeral value of a regardless of its
sign. It is denoted by |a|.
Eg. |9| = 9 |18| = 18
|-9| = 9 |-18| = 18
 Additive inverse – for every integer A there exist an additive inverse –A , such
that A + (-A) = (-A) + A = 0
Eg. Additive inverse of 8 = - 8
Additive inverse of – 9 = 9

4. Addition Subtraction
(Same Sign / Type number)
+ + +
- - +
Eg.
a) +2 + 2 = +4
(+2) + (+2) = +4
b) - 2 – 2 = -4
(-2) + (-2) = -4
(Different Sign / Type number)
- + -
+ - -
Eg.
a) +3 - 2 = +1
(+3) + (-2) = +1
b) -3 + 2 = -1
(-3) + (+2) = -1
*Sign would be of the Greater number would be placed in Answer

5. Predecessor
 It is one more than the integer
Successor of A = A + 1
Eg.
 7
= 7 + 1 = 8
 -7
= -7 + 1 = - 6
 It is one less the integer
Predecessor of A = A -1
Eg.
 7
= 7 – 1 = 6
 -7
= -7 - 1 = - 8
Successor

6. Practice
1. 9 – 3
2. 10 – (-5)
3. (-11) – (-6)
4. (-12) – 8
1. 20 - |-11|
2. |-23| - |-16|
3. |137| + |-13|
4. 6 - |-4|
Q 2 . Find the Successor of : a) -70 b) 30
Q 3 . Find the Predecessor of : a) 60 b) -35
Q1 Evaluate

7. Properties of addition and subtraction
 1) Commutative Property – can add integer in any order
A+ B = B + A
Eg. 6 + 4 = 4 + 6
10 = 10
 2) Associative Property – Grouping does not matter while adding or subtracting
A , B , C
(A + B) + c = A + (B + C) = (A + B ) + C
Eg. 2 , 3, 5
(2+3) + 5 = 2 + (3 + 5) = (2 + 5) + 3 = 10
 3) Repeated addition –
A + A + A = 3 x A
Eg . 2 + 2 + 2 + 2 = 4 x 2

8. Practice
a) Fill in the blanks ( using Additive property of integers)
a) 6 + ___ = - 5 + _____
b) 6 + (8 + ____) = ____ + ____ (-2)
b) Subtract the sum of -189 and -67 from the sum of -789 and 291
c) Verify
a) (-7) + [(-6) + 11] = [(-7) + (-6) ] + 11
b) 17 + (-13) = -13 + 17
d) Evaluate :
a) (-13) – 6
b) -4 – ( - 6)
c) - 24 – 16
d) 4 - 64

9. Integer Sign
 Addition or Subtraction – Sign of greater number is placed in answer.
Eg. -3 + 2 = -1 3 – 2 = + 1 3 + 2 = +5
 Multiplication or Division – Product of two integers with unlike signs is always
negative
Eg. -2 X -3 = + 6 -2 X 3 = - 6
- 8
= - 2
-10
= + 2
4 -5

10. Multiplicative properties of integers
 Commutative Property
A x B = B x A
Eg. 3 x (-2) = -5 (-2) x 3 = -5
 Associative Property
A x ( B x C) = (A x B ) x C = ( A x C) x B
Eg. 3 x ( 2 x 5) = 30 (3 x 2 ) x 5 = 30 (3 x 5) x 2 = 30
 Distributive Property
A, B , C
A x ( B + C ) = (A x B) + (A x C)
Eg. -3 , 2 ,5
-3 x ( 2 + 5 ) (-3 x 2) + (-3 x 5)
= -3 x ( 7) = -6 + (-15)
= -21 = -21
• Multiplicative inverse:
A =
1 1
= A
A A

11. Zero and Integer
 Addition
0 + A = A
A + 0 = A
 Subtraction
A – 0 = A
0 – A = -A
0 – ( -A ) = + A
-A – 0 = -A
 Multiplication
0 X A = 0
A X 0 = 0
-A X 0 = 0
0 X –A = 0
 Division
A
= ∞
0
= 0
0 A

12. Practice
Q1. Evaluate
1. 18 x 4
2. (-6) X (-5) X (-11)
3. (-25) x 6
4. (-12) x (-15)
5. (-9) x 0
6. { (-9) x 8 } x (-5)
7. { (-2) x (-5) } x (-3)
 Verify
A) (-16) x { (-5) + (-6) } = { (-16) x (-5) } + { (-16) x (-6) }
B) (-4) x (-8) = (-8) x (-4)

13. Division of Integers
 In case of two integers having like or same sign are divided ( ++ or --) ,
resulting answer would be positive (+)
 In case two integers having different or unlike signs are divided ( +- or - +)
then result would be in negative.
 0 ÷ A = 0
 a ÷ b ≠ b ÷ a
 (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)
 Even Integers – Integers Exactly Divisible by 2. Eg -2 , -6 , 8 , 20 , -8
 Odd Integers – Integers Not Exactly Divisible by 2. Eg. -7 , 9 , 11 , -2
- 8
= - 2
-10
= + 2
4 - 5

14. Practice
Q1. Evaluate ( Find the quotient ) :
1. (-36) ÷ 9
2. (54) ÷ (-6)
3. (-72) ÷ (-18)
4. (-20) ÷ (-1)
Q2. Write all even integers between -9 and 6
Q3. Write all odd integers between -10 and 8
Q4. Write four consecutive even integers succeeding -64
Q5. Write four consecutive odd integers preceding – 37
Q6. The product of two integers is -18 , if one of them is 6 . What is the other?

16. Evaluate :