Commutative Property - Definition | Commutative Law Examples, FormulaCommutative Property

The commutative property applies to the arithmetic operations of addition and multiplication. It means that changing the order or position of two numbers while adding or multiplying them does not change the end result. For example, 4 + 5 gives 9, and 5 + 4 also gives 9. The order of two numbers being added does not affect the sum. The same concept applies to multiplication too. The commutative property does not hold for subtraction and division, as the end results are completely different after changing the order of numbers.

The word 'commutative' originates from the word 'commute', which means to move around. Hence, the commutative property deals with moving the numbers around. So mathematically, if changing the order of the operands does not change the result of the arithmetic operation then that particular arithmetic operation is commutative. Let us discuss the commutative property of addition and multiplication.

Commutative Property Formula

For any two numbers, A and B, the formula of the commutative property of numbers is expressed as follows.

• A + B = B + A
• A × B = B × A
• A - B ≠ B - A
• A ÷ B ≠ B ÷ A

The commutative property formula states that the change in the order of two numbers while adding and multiplying them does not affect the result. However, while subtracting and dividing any two real numbers, the order of numbers are important and hence it can't be changed.

The commutative property of addition says that changing the order of the addends does not change the value of the sum. If 'A' and 'B' are two numbers, then the commutative property of addition of numbers can be represented as shown in the figure given below.

Let us take an example of the commutative property of addition and understand the application of the above formula.

Example: Let us check the Commutative property by adding 10 and 13.

Let us add the given numbers 10 and 13. So, 10 + 13 = 23 and 13 + 10 = 23. Therefore, 10 + 13 = 13 + 10 which proves the commutative property of addition.

The commutative property of multiplication says that the order in which we multiply two numbers does not change the final product. The figure given below represents the commutative property of the multiplication of two numbers.

If 4 and 6 are the numbers, then 4 × 6 = 24, and 6 × 4 is also equal to 24. Thus 4 × 6 = 6 × 4. Therefore, the commutative property holds true for the multiplication of numbers.

Note: The commutative property does not hold for subtraction and division operations. Let us take the example of numbers 6 and 2.

• 6 - 2 = 4, but 2 - 6 = -4. Thus, 6 - 2 ≠ 2 - 6.
• 6 ÷ 2 = 3, but 2 ÷ 6 = 1/3. Thus, 6 ÷ 2 ≠ 2 ÷ 6

Commutative Property of Subtraction

The commutative property is not applicable to subtraction. The commutative law only applies to addition and multiplication. Let us see why it does not apply on subtraction. For example, 8 - 5 = 3, but 5 - 8 = -3. Thus, 8 - 5 ≠ 5 - 8.

Let us learn the difference between the associative and commutative property. Both associative property and commutative property state that the order of numbers does not affect the result of addition and multiplication. So, what is the difference between the two? Let us find out.

Observe the table given below to see the comparison of commutative property vs associative property.

Important Notes:

Some key points to remember about the commutative property are given below.

• The commutative property states that 'changing the order of the operands does not change the result'.
• The commutative property for addition is A + B = B + A.
• The commutative property for multiplication is A × B = B × A.

☛ Related Topics

Check out some interesting articles related to the commutative property in math.

• Properties of Natural Numbers
• Properties of Whole Numbers
• Properties of Rational Numbers
• Properties of Integers

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What is Commutative Property in Math?

The commutative property states that if the order of numbers is interchanged while performing addition or multiplication, the sum or the product obtained does not change. It is to be noted that the commutative property holds true only for addition and multiplication and not for subtraction and division. For example, 6 + 7 is equal to 13 and 7 + 6 is also equal to 13. Similarly, 6 × 7 = 42, and 7 × 6 = 42.

What is the Commutative Property of Addition?

According to the commutative property of addition, when two numbers are added in any order the sum remains the same. For example, 3 + 9 = 9 + 3 = 12.

What is the Commutative Property of Multiplication?

According to the commutative property of multiplication, the order of multiplication of numbers does not change the product. For example, 4 × 5 is equal to 20 and 5 × 4 is also equal to 20. Though the order of numbers is changed, the product is 20.

Can Commutative Property be Used for Subtraction and Division?

The commutative property cannot be applied for subtraction and division, because the changes in the order of the numbers while doing subtraction and division do not produce the same result. For example, 5 - 2 is equal to 3, whereas 2 - 5 is not equal to 3. In the same way, 10 ÷ 2 = 5, whereas, 2 ÷ 10 ≠ 5. Therefore, the commutative property is not applicable for subtraction and division.

What is the Difference Between Commutative Property and Associative Property?

The commutative property states that the change in the order of two numbers in an addition or multiplication operation does not change the sum or the product. The commutative property of addition is expressed as A + B = B + A. The commutative property of multiplication is expressed as A × B = B × A. The associative property states that the grouping or combination of three or more numbers that are being added or multiplied does not change the sum or the product. The associative property of addition is written as: (A + B) + C = A + (B + C) = (A + C) + B. The associative property of multiplication is written as (A × B) × C = A × (B × C) = (A × C) × B.

What is the Difference Between Commutative Property and Distributive Property?

The commutative property states that the change in the order of numbers for the addition or multiplication operation does not change the result. The commutative property of addition for two numbers 'A' and 'B' is A + B = B + A. The distributive property means multiplying a number with every number inside the parentheses. The numbers inside the parentheses are separated by an addition or a subtraction symbol. The distributive property of addition for two numbers 'A', 'B' is: A(B + C) = AB + AC.

How are the Commutative Property of Addition and Multiplication Alike?

In the commutative property of addition and multiplication, the order of numbers does not affect the sum or product. For example, in the commutative property of addition, 7 + 8 = 8 + 7 = 15. Similarly, in the commutative property of multiplication, 6 × 5 = 5 × 6 = 30. So, the commutative property holds true with addition and multiplication operations.

How to Teach the Commutative Property of Addition?

The best way to teach the commutative property of addition is by using real-life objects such as pebbles, dice, seeds, etc. Give 3 marbles to student A and then give 5 marbles to student B. Ask them to count the total number of marbles which will be 8. Then repeat the same process by giving 5 marbles to student A and 3 marbles to student B. Now, ask them to tell the total number of marbles again which will result in 8. Use the commutative property of addition worksheets to examine their understanding.

What are Commutative Laws?

Commutative law is another word for the commutative property that applies to addition and multiplication. The commutative law of addition states that the order of adding two numbers does not change the sum (A + B = B + A). The commutative property of multiplication states that the order of multiplying two numbers does not change the product (A × B = B × A).

What is an Example of Commutative Law?

The Commutative law of addition states that the result of the addition of any two numbers remains the same even when the positions of these numbers are interchanged. For example, 4 + 7 = 11 and 7 + 4 = 11. The same rule applies to multiplication as well. This means that as per the commutative law of multiplication, the result of the multiplication of any two numbers remains the same even when the positions of these numbers are interchanged. For example, 3 × 7 = 21 and 7 × 3 = 21.