Properties of Subtraction

Properties of Subtraction are explained.


(1) Closure under Subtraction

Numbers under subtraction do not hold closure property.

Condition for closure property:

x – y = z

Here, x, y and z are numbers.


(i) 5-3=2

But, in example given below

(ii) 4 – 6= – 2[integer]

Hence, closure property under subtraction for numbers is not true.

(2)Commutative Property

Numbers cannot be subtracted in any order.

Condition for commutative property:

x – y = y – x

Here, x and y are numbers.


(i) 5-3=2

After changing order 3-5=-2

2 and -2 are not equal.

(ii) 4- 2 = 2

After changing order 2 – 4 = – 2

2 and -2 are not equal.

In above examples, we are not getting same result after subtracting in both sides.

Hence, subtraction is not commutative for numbers.

(3) Associative property

Numbers cannot grouped differently in subtraction.

Condition for associative property:

x – (y – z) = (y – x) – z

Here, x, y and z are numbers.


(i) 5- (6-8) can be grouped as (5-6) -8

Now, check the total

5- (6-8) = 7and

(5-6)-8= -9

Results are not same.

(ii) 3- (2-5) = (3-2)-5

6 = -4

Results are not same.

Hence, subtraction is not associative for numbers.

(4)Identity property

Condition for subtractive identity

Subtractive identity of a number states the

x – identity = x = identity – x

Here, x is a number.


(i) 7- 0 =7


0 – 7 = -7

Results are not same.

(ii)6 – 0 = 6


0 – 6 = – 6

Results are not same.

Hence, subtraction does not hold identity for numbers.

(5)Distributive property

Numbers under subtraction 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:

picture1distributive-propertya is distribute to b and c by multiplying a inside the terms of brackets.

RHS of condition for distributive property for subtraction will be like a – (b + c)

a – (b + c) can not be distribute a inside parenthesis because of (-) sign outside the parenthesis or brackets.  It is clear that we do not have sign of multiplication in property of subtraction. Therefore, it is not possible to distribute subtraction over all the terms inside the parenthesis or brackets.

Hence, distributive property cannot be executed in subtraction.

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