Commutative Property – Definition | Commutative Law and Examples

Commutative Property states that the result of an operation between two numbers remains the same irrespective of the position of the numbers. It is an important property of mathematics. Commutative property is satisfied by the addition(+) and multiplication(×) operations.

Commutative Property:

• X + Y = Y + X (Commutative Property of Addition)
• X × Y = Y × X (Commutative Property of Multiplication)

In this article, we will explore the commutative property, commutative property definition, commutative property formula, commutative property examples, and others in detail.

What is Commutative Property?

The commutative property is the property in which the order of the two operands does not affect the result of the expression. The commutative word is derived from the word commute which means switching. The arithmetic operator addition and multiplication satisfies the commutative property.

Commutative Property Definition

Examples,

• 4 + 7 = 7 + 4 = 11
• 5 + 4 = 4 + 5 = 9
• 4 × 7 = 7 × 4 = 28
• 5 × 4 = 4 × 5 = 20

Commutative Property Definition

Commutative Property is the property in which the sequence of two operands does not change the result. In commutative property the switching of the operands does not reflect the result of the given expression.

Commutative Property Formula

Commutative property is satisfied by the addition and multiplication arithmetic operators only. For two numbers P and Q, the commutative property formula is given by:

• P + Q = Q + P
• P × Q = Q × P

Commutative Property Meaning

Commutative Property is the basic property of mathematics that tells us that changing order if number in adding or multiplying two numbers does not impact the result of the number. For example, adding 3 and 4 is equal to adding 4 and 3, i.e.

• 3 + 4 = 4 + 3 = 7

Commutative Property Examples

Following are some examples of commutative property:

• 3 + 5 = 5 + 3
• 3 × 5 = 5 × 3
• 6 + 8 = 8 + 6
• 6 × 8 = 8 × 6

Commutative Property of Addition

Commutative property for addition is the property in which the order of two operands related with addition operator has no affect in the result of addition.

Commutative Property for Addition Formula

For addition of two numbers X and Y, the commutative property for addition formula is given below:

X + Y = Y + X

Commutative Property of Addition Examples

Following are some examples for the commutative property for addition:

• 4 + 9 = 9 + 4
• (-12) + 5 = 5 + (-12)
• 8/9 + 1/3 = 1/3 + 8 / 9

Commutative Property of Multiplication

The commutative property for multiplication is the property in which the order of two operands related with multiplication operator has no affect in the result of multiplication.

Commutative Property for Multiplication Formula

For multiplication of two numbers X and Y, the commutative property for addition formula is given below:

X × Y = Y × X

Commutative Property of Multiplication Examples

Following are some examples for the commutative property for addition:

• 4 × 9 = 9 × 4
• (-12) × 5 = 5 × (-12)
• (8 / 9) × (1 /3) = (1 / 3) × (8 / 9)

Commutative Property for Subtraction and Division

The arithmetic operators subtraction and division does not satisfy the commutative property as changing the order of the operands changes the result of the expression.

For two operands R and S,

• R – S ≠ S – R
• R / S ≠ S / R

Commutative Property vs Associative Property

The differences between commutative property and associative property is explained in the table below,

Commutative Property	Associative Property
Commutative property states that the order of the number in some operations(such as multiplication and addition) dose not effects the results of the operation.	Associative property states that grouping various number in different operations (such as multiplication and addition) does not changes the result of the operation.
Commutative Formula: A + B = B + A A × B = B × A	Associative Formula: A + (B + C) = (A + B) + C = (A + C) + B A × (B × C) = (A × B) × C = (A × C) × B
Example: 4 + 5 = 5 + 4 = 9	Example: (3 + 4) + 5 = 3 + (4 + 5) = 12

Non-Commutative Property

Non-Commutative property means that changing the order of the numbers in the operations changes the seult of the operation. This can be explained by taking the example of Division Operation (÷). Suppose we take two numbers 12 and 3 then dividing 12 by 3 and dividing 3 by 12 gives the separate results, i.e.

• 12/3 = 4
• 3/12 = 1/4

Read More,

• Properties of Rational Numbers
• Properties of Real Numbers
• Properties of Integers

Examples on Commutative Property

Example 1: Which of the following satisfies the commutative property:

(i) 4 × 5

(ii) 7 + 8

(iii) 9 / 2

(iv) 15 – 6

Solution:

• 4 × 5

4 × 5 = 20

5 × 4 = 20

Commutative Property Satisfies

• 7 + 8

7 + 8 = 15

8 + 7 = 15

Commutative Property Satisfies

• 8 / 2

8 / 2 = 4

2 / 8 = 1 / 4

4 ≠ 1/4

Commutative Property Does Not Satisfies

• 15 – 6

15 – 6 = 9

6 – 15 = -9

9 ≠ -9

Commutative Property Does Not Satisfies

Example 2: Prove that p + q = q + p if p = 9 and q = 7.

Solution:

p + q = 9 + 7 = 16

q + p = 7 + 9 = 16

p + q = q + p

Hence proved

Example 3: Prove that p × q = q × p if p = 10 and q = 3.

Solution:

p × q = 10 × 3 = 30

q × p = 3 × 10 = 30

p × q = q × p

Hence Proved

Commutative Property – Practice Problems

P1. Which of Following Satisfies Commutative Property

• 42 × 15
• 12 + 7
• 23 / 3
• 25 – 8

P2. Prove x + y = y + x if x = 10 and y = 17

P3. Prove x × y = y × x if x = 2 and y = 23

Commutative Property-FAQs

1. What is Commutative Property in Math?

Commutative property in Math is the property in which the switching of the two operands does not change the result of the expression.

2. What is Commutative Law for Multiplication?

Commutative property of multiplication states that the change in position of the two operands for multiplication does not change the result.

A × B = B × A

3. What is Commutative Law for Addition?

Commutative property of addition states that the change in position of the two operands for addition does not change the result.

A + B = B + A

4. Can Commutative Property be used for Subtraction and Division?

No, the commutative property cannot be used for subtraction and division as these operators does not satisfy the commutative property.

5. What do you Mean by Commutative Laws?

Commutative laws include the commutative property for addition and commutative property of multiplication.

• A + B = B + A
• A × B = B × A

6. What is Commutative Property for Addition?

Commutative Property for Addition states that, adding two numbers the order of the numbers in addition is not important, i.e. a + b = b + a.