 # Commutative Law

In Mathematics, commutative law is applicable only for addition and multiplication operations. But, it is not applied to other arithmetic operations, such as subtraction and division. As per commutative law or commutative property, if a and b are any two integers or variables, then the addition and multiplication of a and b result in the same answer even if we change the position of a and b. Symbolically it may be represented as:

a+b=b+a

a×b=b×a

For example, if 2 and 5 are the two numbers, then;

2 + 5 = 5 + 2 = 7

2 × 5 = 5 × 2 = 10

## Definition

The definition of commutative law states that when we add or multiply two numbers then the resultant value remains the same, even if we change the position of the two numbers. Or we can say, the order in which we add or multiply any two real numbers does not change the result.

Hence, if A and B are two real numbers, then, as per this law;

 A+B = B+A A.B = B.A

Apart from commutative law, there are other two major laws, which are commonly used in Maths, they are:

• Associative law
• Distributive law

Associative Law: As per this law, if A, B and C are three real numbers, then;

• A+(B+C) = (A+B)+C
• A.(B.C) = (A.B).C

Just like commutative rule, this law is also applicable to addition and multiplication.

For example: If 2,3 and 5 are three numbers then;

2+(3+5) = (2+3)+5

⇒2+8 = 5 + 5

⇒10 = 10

&

2.(3.5) = (2.3).5

⇒ 2.(15) = (6).5

⇒ 30 = 30

Hence, proved.

Distributive Law: This law is completely different from commutative and associative law. According to this law, if A, B and C are three real numbers, then;

A.(B+C) = A.B + A.C

For example: If 2,3 and 5 are three numbers then;

2.(3+5) = 2.3+2.5

2.(8) = 6+10

16 = 16

The commutative law of addition states that if two numbers are added, then the result is equal to the addition of their interchanged position.

A+B = B+A

Examples:

• 1+2 = 2+1 = 3
• 4+5 = 5+4 = 9
• -3+6 = 6+(-3) = 6-3 = 3

This law is not applicable for subtraction because if the first number is negative and if we change the position, then the sign of the first number will get changed to positive, such that;

(-A)-B = -A – B ……(1)

After changing the position of the first and second number, we get;

B – (-A) = B + A ….(2)

Hence, from equation 1 and 2, we can see that;

(-A)-B ≠ B-(-A)

For example: (-9)-2 = -9-2 = -11

& 2-(-9) = 2+9 = 11

Hence, -11 ≠ 11.

### Commutative Law of Multiplication

As per this law, the result of multiplication of two numbers stays the same, even if the positions of the numbers are interchanged.

Hence, A.B = B.A

Examples:

• 1×2 = 2×1 = 2
• 4×5 = 5×4 = 20
• -3×6 = 6×(-3) = -18