Class 7 - Maths - Properties of Integers (#cbsenotes)(#eduvictors)(#class7maths)

Class 7 - Mathematics - Properties of Integers 

1. Closure Property Under Addition

Integers are closed under addition, i.e. for any two integers,a and b, a+b is an integer. 
eg: 3 + 4 = 7
   -9 + 7 = 2


2. Closure Property under Subtraction: 

Integers are closed under subtraction, i.e. for any two integers,a and b, a-b is an integer.
eg:
-21 -(-9) = -12
8  -3    = 5

3. Closure Property under Multiplication: 

Integers are closed under multiplication, i.e. for any two integers,a and b, ab is an integer.
e.g.
5 × 6 = 30
-3 × -8 = 24
-3 × 7 = -21



4. Closure Property under Division:
Integers are NOT closed under division, i.e. for any two integers, a and b, a/b need not be an integer.

e.g. 2 ÷ 0 = undefined
2 ÷ 3 = 2/3 (a fraction, not an integer)


5. Commutative Property under Addition:  
Addition is commutative for integers.  For any two integers, a and b, a+b = b+a

e.g.
5 + (-3) = -3 + 5 = 2
(-6) + 5 = -6 + 5 =-1
5 + (-6) = (-6) + 5 = -1


6.  Commutative Property under Subtraction:

Subtraction is NOT commutative for integers.  For any two integers a and b, a-b ≠ b-a
e.g. 8 - (-6) = 14 and -6 - 8 = -14


7. Commutative Property under Multiplication: 

Multiplication is commutative for integers.  For any two integers, a and b, a × b = b × a
e.g. 9 × (-6) = (-6) × 9 = -54



8. Commutative Property under Division: 

Division is NOT commutative for integers.  For any two integers a and b,  a/b ≠ b /a

e.g. 3/6=1/2 while 6/3 = 2


9.  Associative Property under Addition
Addition is associative for integers.  For any three integers, a, b and c, a + (b + c) = (a + b) + c

e.g.
5 + (7 + 3) = 5 + 10 = 15
(5 + 7) + 3 = 12 + 3 = 15

4 + (-6 + 3) = 4 -3 = 1
(4 + (-6)) + 3 = (4 - 6) + 3 = -2 + 3 = 1


10.  Associative Property under Subtraction: 
Subtraction is NOT associative for integers.  For any three integers ,  a-(b-c) ≠ (a-b)-c

e.g. 5 — (6-4) = 5-2 = 3
    (5-6) - 4 =-1-4=-5


11. Associative Property under multiplication: 
Multiplication is associative for integers.  For any three integers, a, b and c, (a × b) × c = a × (b × c)

e.g. [(-3) × (-2)] × 5 = 6 × 5 = 30
(-3) × [(-2) × 5] = (-3) × (-10) = 30


12. Associative Property under Division: 
Division is NOT associative for integers.


13.  Distributive property of multiplication over addition: 
For any three integers, a, b and c, a × (b + c) = a × b + a × c
e.g. -2 (4 + 3) =  -2(7)  = -14
     -2 (4 + 3) = (-2 × 4) + (-2 × 3) = (-8) + (-6) = -14


14. Identity under Addition: 
Integer 0 is the identity under addition.  That is, for an integer a, a+0=0+a=a.
e.g. 5 + 0 = 0 + 5 = 5


15. Identity under Multiplication:  
The integer 1 is the identity under multiplication.  That is, for an integer a, 1 × a = a × 1 = a
e.g. (-5) × 1 = 1 × (-5) =-5

Ch 1 - Introduction to Integers
Ch 1 - Integers (NCERT Ex 1.2)
Ch 1 - Integers (NCERT Ex 1.3)
Ch 1 - Integers (Key Points)


Ch 2 - Decimals - Multiplication and Division (Worksheet)

Ch 2 - Fractions and Decimals - NCERT Exercise Answers 2.6