Properties of Addition of Integers

Properties of Addition of Integers: Closure Property, Commutative Property, Associative Property, Identity Property and Distributive Property are explained with examples.

 

Properties of Addition of Integers

(1) Closure under Addition-

Addition of any two integers gives integer. It is known as integers are closed under addition.

Condition for closure property-

x  +  y = z

Here, x, y and z are integers.

Example

(a) 5+3=8

(b) -5+3=-2

In above example 5, 3,-5, 3, 8,-2 all are integers.

Hence, addition is closure for integers.

(2)Commutative Property-

Integers can be added in any order.

Condition for commutative property:

x + y = y + x

Here, x and y are integers.

Example

(a)

5+7=7+5

11=11

(b)

6+ (-9) =-9+6

6-9 = -9+6

-3=-3

In both examples, we are getting same result after addition in both sides.

Here, addition is commutative for Integers.

(3)Associative Property-

Integers can be grouped differently.

Condition for associative property-

x +  (y + x) = (y + x) + z

Here, x, y and z are integers.

Example

(a) 5+ (6+8) can be grouped as (5+6) +8

Now, check the total

5+ (6+8) = (5+6)+8

19=19

Results are same.

(b) -3+ (-2+5) = [-3+(-2)]+5

-3-2+5=-3-2+5

0=0

Results are same.

Hence, addition is associative for integers.

(4) Additive Identity-

Additive identity is an integer when we add it to any integer it gives us same number.

Condition for additive identity-

Additive identity of a whole number states that

x + identity = x = identity +  x

Example

(a) 7+0=7

(b) -6+0=-6

In both examples when we add 0 we get same number.

Hence, zero is an additive identity for integers.

(5)Distributive property-

Integers under addition do not have distributive property.

Because, distributive property can only exists when multiplication and addition both involve in an expression. This property is also known as ‘distributive property of multiplication over addition`. It tells us that we distribute the multiplication over all the terms inside the parenthesis or brackets by multiply terms with terms of brackets.

Condition for distributive property-

a × (b + c) = ab + ac

a is distribute to b and c by multiplying a inside the terms of brackets.

a + (b + c) will be RHS condition for distributive property of division. We do not have multiplication in this condition. Therefore, it is not possible to distribute division over all the terms inside the parenthesis or brackets.

Hence, distributive property cannot be executed in addition.

 

Helping Topics

Integers

Properties of subtraction of integers

Properties of multiplication of integers

Properties of division of integers

NCERT solutions class 7

Practice sheet

 

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