Commutative Property

There are four basic operations in Mathematics namely addition, subtraction, multiplication, and division. These four operations are the most fundamental constituents of today’s world of Mathematics. These operations obey five fundamental properties namely closure property, commutative property, associative property, distributive property and identity property. All the four basic operations may not obey all the properties. However, each Mathematical operation obeys a few properties based on several conditions. The commutative property is also known popularly as commutative law states that the result of a mathematical operation between the operands remains unaffected by changing the order in which the operation is performed. 

Commutative Property Definition:

Commutative law states that the result obtained by performing a Mathematical operation on any number of operands is the same even when the order of the operands is reversed. Mathematically, commutative property definition can be explained as follows. If ‘c’ and ‘d’ are two numbers, then for any operation performed between ‘c’ and ‘d’ is said to be commutative if and only if the result obtained when ‘c’ is the first operand and ‘d’ is the second operand is equal to the result obtained when ‘d’ is the first operand and ‘c’ is the second.

Commutative Property of Addition:

Commutative property of addition states that the sum of any two numbers remains the same even when the position of the numbers are interchanged. For example, if ‘E’ and ‘F’ are any two real numbers, then according to commutative property of addition, it is true that

E + F = F + E

Let us take an example of any two numbers, say 9 and 23. It is true that the sum of 9 and 23 is equal to 32 irrespective of the direction of addition. i.e. 9 + 23 = 23 + 9 = 32

So addition operation satisfies commutative law. 

Commutative Property of Subtraction:

Commutative property does not hold good for subtraction because the value of the difference between the numbers depends on the direction in which the numbers are subtracted. For example, let us consider the subtraction of 9 and 23. 

The difference between 9 and 23 when 9 is subtracted from 23 is 14 and the difference between 9 and 23 when 23 is subtracted from 9 is - 14.

So, 23 - 9 ≠ 9 - 23

Commutative Property of Multiplication:

The multiplication operation obeys commutative law. Commutative property of multiplication states that the product of any two numbers remains the same even if we change the direction in which the multiplication is performed. If ‘G’ and “H’ are the two numbers, then the commutative property of multiplication states that:

G x H = H x G

For example, if a = 8 and b = 4 are the two numbers, then the value of 4 times 8 and 8 times 4 is the same and is equal to 32.

Commutative Property of Division:

Commutative law is not true for division. The quotient obtained is dependent on the values in place of numerator and denominator. So, the answer does not remain the same when numerator and denominator are interchanged. Hence, the commutative law of division does not exist.

Commutative Property Example Problems:

1. State whether the following statements are true or false. Give appropriate reasons to substantiate your answer.  (Hint: Recall what is commutative property)

a. 7 + 8 = 8 + 7

b. 8 - 7 = 7 - 8

Solution:

1. 7 + 8 = 8 + 7 is true

Because the sum obtained is equal to 15 in both the cases according to the commutative property of addition.

2. 8 - 7 = 7 - 8 is not true 

Because the commutative property of subtraction does not exist. 8 - 7 = 1 and 7 - 8 = - 1.

2. Check whether the given statements are correct or not. Give appropriate reasons to support your answers.

1. 7*5=5*7

2. 7/5= 5/7

(Hint: Recall the commutative property)

Solutions:

1. 7*5=5*7 is true

Because the product obtained 35 is equal in both the cases satisfying the commutative property of multiplication.

2. 7/5=5/7 is false

Because the product obtained in both cases is different. In the first case, we get the product as 1.4 whereas, in the second case, the product obtained is 0.71. Hence, it doesn’t satisfy the conditions for commutative properties. 

Comparison Between The Commutative Property And The Associative Property

Below are some differences and similarities between the commutative property and the associative property:

Differences

• In commutative law, the elements are rearranged or reordered whereas, in associative law, the elements are regrouped. 

• In commutative property, only two elements are used whereas, in associative property, three elements are used.

• Commutative Property: W + V = V + W and W x V = V x W

• Associative Property: L + (M + N) = (L + M) + N 

                                       L x (M x N) = (L x M) x N 

Similarities

• In both the properties, the result remains the same when the elements are reordered or regrouped. 

• Both the properties are applicable for algebraic operations of addition and subtraction.

Examples: Commutativity

60 + 40 = 40 + 60 = 100

60 x 40 = 40 x 60 = 2400

Examples: Associativity

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

1 x (2 x 3) = (1 x 2) x 3 = 6

Comparison Between The Commutative Property And The Distributive Property

Underneath are the similarities and differences between the commutative property and the distributive property:

Differences

• According to the commutative law, the elements are moved from their original position whereas, according to the distributive property, if a number is multiplied by a parenthesis, then the number is multiplied by both the elements in the brackets.

• Commutative Property: J + K = K + J and J x K = K x J

• Distributive Property: D x (E + F) = D x E + D x F

Similarities

• According to both laws, the result remains the same when the numbers are reordered (in the case of commutativity) and when the number is multiplied individually with other numbers in the parenthesis (in the case of distributivity).

Examples: Commutativity

70 + 30 = 30 + 70 = 100

70 x 30 = 30 x 70 = 2100

Examples: Distributivity

8 x (3 + 5) = 8 x 3 + 8 x 5 = 64

Commutative Property- Common Uses

The commutative law or commutative property is associated with binary operations and functions. If a pair of elements under a binary operation holds commutative property, then it is said to commute under that operation.

A set S is commutative if it satisfies the given binary operation

X*Y = Y*X, where X and Y belongs to set S

A binary operation that does not satisfy the above property is called non-commutative.

One can say that X commutes with Y or Y commute with X only if

X*Y= Y*X.

In simple words, an operation is commutative when each pair of elements commute.

A binary function f: B*B= A is called commutative if f (X, Y)= f(Y, X), where X and Y belong to set B. This type of function is commonly known as a symmetric function. 

Fun Quiz:

1. Fill in the blanks with appropriate answers.

4 + 7 = ___ + 4

1.5 x ___ = 3.4 x 1.5

2. Match the expressions in column A with the appropriate statements in column B.