Commutative Property
The commutative property deals with 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.
1.  What is Commutative Property? 
2.  Commutative Property of Addition 
3.  Commutative Property of Multiplication 
4.  Commutative Property vs Associative Property 
5.  FAQs on Commutative Property 
What is Commutative Property?
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. Apart from this, there are other properties of numbers: the associative property, the distributive property, and the identity property. They are different from the commutative property of numbers. Let us discuss the commutative property of addition and multiplication briefly.
Commutative Property Formula
If two numbers A and B are given, then the formula of commutative property of numbers is given as,
 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. But while subtracting and dividing any two real numbers, the order of numbers are important and hence it can't be changed.
Commutative Property of Addition
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 below.
Let us take an example of commutative property of addition and understand the application of the above formula. If two numbers are given 10 and 13, then 10 + 13 = 23 and 13 + 10 = 23. Therefore, 10 + 13 = 13 + 10.
Commutative Property of Multiplication
The commutative property of multiplication says that the order in which we multiply two numbers does not change the final product. The image 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, commutative property holds true for multiplication of numbers.
Note: The commutative property does not hold for subtraction and division operations. Let us take 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 vs Associative Property
There are four common properties of numbers: closure, commutative, associative, and distributive property. In this section, we will learn the difference between 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's the difference between the two? Let's find out.
Look at the table giving below showing commutative property vs associative property.
Commutative Property  Associative Property 

Commutative property comes from the word "commute" which means move around, switch or swap the numbers.  Associative property comes from the word "associate" which deals with the grouping of numbers. 
Order of numbers can be changed in the case of addition and multiplication of two numbers without changing the final result.  Grouping of numbers can be changed in the case of addition and multiplication of three numbers without changing the final result. 
Formula: A + B = B + A A × B = B × A 
Formula: A + (B + C) = (A + B) + C = (A + C) + B A × (B × C) = (A × B) × C = (A × C) × B 
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.
Commutative Property Examples

Example 1: Jacky's mother asked him whether the addition of two natural numbers is an example of the commutative property. Can you help Jacky find out whether it is commutative or not?
Solution:
We know that the commutative property of addition states that changing the order of the addends does not change the value of the sum. If we take any two natural numbers, say 2 and 5, then 2 + 5 = 7 = 5 + 2. Therefore, the addition of two natural numbers is an example of commutative property.

Example 2: Find the missing value: 132 × 121 = ___ × 132.
Solution:
The commutative property of multiplication states that if there are two numbers x and y, then x × y = y × x. If you observe the given equation carefully, you will find that the commutative property can be applied here. If x = 132, and y = 121, then we know that 132 × 121 = 121 × 132.
∴ The missing number is 121.

Example 3: State whether the given statement is true or false.
"Division of 12 by 4 satisfies the commutative property."
Solution:
The commutative property does not hold true for division operation. So, the given statement is false. Let's verify it.
12 ÷ 4 = 3
4 ÷ 12 = 1/3 = 0.33
⇒ 12 ÷ 4 ≠ 4 ÷ 12∴ The given statement is false.
FAQs on Commutative Property
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 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?
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 divided by 2, gives 5, whereas, 2 divided by 10, does not give 5. Therefore, commutative property is not true 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 written as A + B = B + A. The commutative property of multiplication is written 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.
Can Commutative Property have 3 Numbers?
When it comes to the grouping of three numbers, then it is called associative property, and not commutative property. Commutative property is applicable with two numbers and states that we can switch the places of those two numbers while adding or multiplying them without altering the result.
How are the Commutative Property of Addition and Multiplication Alike?
In both cases, addition and multiplication, the order of numbers does not affect the sum or product. So, the commutative property holds true with addition and multiplication operations.
How to Teach Commutative Property of Addition?
The best way to teach commutative property of addition is by using reallife objects such as pebbles, dice, seeds, etc. Give 3 marbles to your learner and then give 5 more marbles to her/him. Ask her/him to count the total number of marbles. Then repeat the same process with 5 marbles first and then 3 marbles. In this way, learners will observe this property by themselves. Use 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).
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