**Introduction**

Mathematics is a game of numbers and numbers are everywhere. Numbers, that we use every day follow the rules what are known as properties in mathematics. Properties help you calculate answers in your head quickly and easily. However, not all sets of numbers follow or satisfy these properties in the same manner. For instance, an operation in whole numbers may satisfy a property that is not satisfied by operations in Integers. Two of these important properties are commutative property and associative property. Let us now learn what we mean by commutative and associative property and their behaviour with respect to the addition of different numbers such as integers, whole numbers, natural numbers etc.

**What is Commutative Property ?**

**Commutative Property states that when an operation is performed on two numbers, the order in which the numbers are placed does not matter. **In the case of an addition, this means that when we add two numbers, a and b, the order of the numbers does not matter, i.e.

a + b = b + a

But, whether this property is satisfied or not depends entirely on the type of numbers being used in the operation.

**What is Associative property ?**

**Associative Property states that when an operation is performed on more than two numbers, the order in which the numbers are placed does not matter. **In the case of an addition, this means that when we add three numbers, a, b and c the order of the numbers does not matter, i.e.

a + ( b + c )= ( a + b ) + c

But, whether this property is satisfied or not depends entirely on the type of numbers being used in the operation.

Let us now discuss both these properties with respect to different types of numbers in our number system

**Commutative and Associative Property for Addition of Natural Numbers**

Before we learn about the commutative and associative properties of the addition of natural numbers, it is important to recall what we mean by natural numbers.

Since our early education years, we use numbers such as 1, 2, 3, 4 ….. to count and calculate. This is the natural way of counting objects. Hence, 1, 2, 3, 4,…….. are called natural numbers. However, fractions such as 37, 118 etc. are not natural numbers. Since we start counting from 1, therefore, is the first natural number. Also, there is no last or greatest natural number.

**Commutative Property for Addition of Natural Numbers**

Now that we are familiar with what we mean by natural numbers, let us check whether the commutative property is satisfied for the operation for the addition of natural numbers.

For the commutative property to be true for the addition of natural numbers it means that if one number is added to the other, it does not matter which number is added to whom. Let us verify this statement.

Suppose we have two numbers 45 and 17.

We want to add 45 and 17. The sum will be 45 + 17 = 62

Now, if we interchange the order of numbers and we get the problem as 17 + 45

What would be the answer?

The sum would be 17 + 45 = 62 which is equal to our previous result.

This means that, if we have two natural numbers, a and b, then

**a + b = b + a**

**Hence, addition is commutative for natural numbers. In other words, commutative property is satisfied in the case of the addition of natural numbers. **

**Associative Property for Addition of Natural Numbers**

For the associative property to be true for the addition of natural numbers, the following statement must be true –

If we want to add 3 natural numbers, two of them can be chosen first. The result of this addition would serve as the first number the original third number will serve as the second number to get the final answer. Let us verify this statement.

Let us take 3 numbers, 7, 6 and 9

Let us take 7 and 6 first.

We get, 7 + 6 = 13

Now, we add 13 and 9, we get 13 + 9 = 22

Now let us reverse the order and choose 6 and 9 first.

We get, 6 + 9 = 15

Now, we add this result to 7, we get, 7 + 15 = 22

Both the processes give us the same answer.

Therefore, we can say that addition of natural numbers satisfies the associative property. We get

**(a + b) + c = a + (b + c)**

**Hence, the addition of natural numbers is associative. In other words, the associative property is satisfied in the case of the addition of natural numbers. **

**Commutative and Associative Property for Addition of Whole Numbers**

We are already familiar with the counting of numbers, 1, 2, 3, 4, 5, 6 and so on. These numbers are called natural numbers. But a point to be noticed here is that while the digit 0 appears in the counting series in the form of 10, 20, 30 etc. it is not present in the list of counting numbers as an individual number. So, in which category do we put the number 0 as it is definitely not a natural number?

Whole Numbers are the set of natural numbers along with the number 0. This means that all natural numbers together with 0 form the whole numbers.

Clearly, every natural number is a whole number, while every whole number is not a natural number.

So, the set of whole numbers can be defined as { 0, 1, 2, 3, 4, 5, ……….. }

This set of whole numbers is denoted by the symbol “W”.

Hence,

W = { 0, 1, 2, 3, 4, 5, ……….. }

**Commutative Property for Addition of Whole Numbers**

For the commutative property to be true for the addition of whole numbers it means that if one number is added to the other, it does not matter which number is added to whom. Let us verify this statement.

Suppose we have two numbers 32 and 15.

We want to add 15 and 32. The sum will be 15 +32 = 47

Now, if we interchange the order of numbers and we get the problem as 32 + 15.

What would be the answer?

The sum would be 32 + 15 = 47 which is equal to our previous result.

This means that, if we have two whole numbers, a and b, then

**a + b = b + a**

**Hence, addition is commutative for whole numbers. In other words, commutative property is satisfied in the case of the addition of whole numbers. **

**Associative Property for Addition of Whole Numbers**

For the associative property to be true for the addition of whole numbers, the following statement must be true –

If we want to add 3 numbers, two of them can be chosen first. The result of this addition would serve as the first number the original third number will serve as the second number to get the final answer. Let us verify this statement.

Let us take 3 numbers, 8, 5 and 2.

Let us take 8 and 5 first.

We get, 8 + 5 = 13

Now, we add 13 and 2, we get 13 + 2 = 15

Now let us reverse the order and choose 5 and 2 first.

We get, 5 + 2 = 7

Now, we add this result to 8, we get, 7 + 8 = 15

Both the processes give us the same answer.

Therefore, we can say that addition satisfies the associative property. We get

**(a + b) + c = a + (b + c)**

**Hence, the addition of whole numbers is associative. In other words, the associative property is satisfied in the case of the addition of whole numbers. **

**Commutative and Associative property for Addition of Integers**

Corresponding to natural numbers 1, 2, 3, 4, …….. etc. we create new numbers, -1, – 2, – 3, – 4, ….. etc called minus one, minus two, minus three, minus four etc. respectively such that

1 + ( – 1 ) = 0

2 + ( – 2 ) = 0 and so on.

Combining these numbers we get a new set of numbers which is written as

…….. -3, -2, -1, 0, 1, 2, 3, ………

These numbers are called integers.

Let us now see whether the commutative and the associative property are satisfied in the operation of the addition of integers.

**Commutative Property for Addition of Integers**

For the commutative property to be true for the addition of integers, it means that if one number is added to the other, it does not matter which number is added to whom. Let us verify this statement.

Suppose we have two integers 32 and – 15.

We want to add – 15 and 32. The sum will be – 15 +32 = 17

Now, if we interchange the order of numbers and we get the problem as 32 + (- 15 )

What would be the answer?

The sum would be 32 + ( – 15 ) = 17 which is equal to our previous result.

This means that, if we have two integers, a and b, then

**a + b = b + a**

**Hence, addition is commutative for integers. In other words, the commutative property is satisfied in the case of the addition of integers. **

**Associative Property for Addition of Integers**

For the associative property to be true for the addition of integers, the following statement must be true –

If we want to add 3 integers, two of them can be chosen first. The result of this addition would serve as the first number the original third number will serve as the second number to get the final answer. Let us verify this statement.

Let us take 3 numbers, – 8, 5 and -2

Let us take – 8 and 5 first.

We get, – 8 + 5 = – 3

Now, we add – 3 and – 2, we get – 3 + ( – 2 ) = – 5

Now let us reverse the order and choose 5 and – 2 first.

We get, 5 + ( – 2 ) = 3

Now, we add this result to – 8, we get, – 8 + 3 = – 5

Both the processes give us the same answer.

Therefore, we can say that addition of integers satisfies the associative property. We get

**(a + b) + c = a + (b + c)**

**Hence, the addition of integers is associative. In other words, the associative property is satisfied in the case of the addition of integers. **

**Solved Examples**

**Example 1** Add the integers – 523 and 937 and check whether the addition satisfies the commutative property.

Solution We have been given the numbers, – 523 and 937.

Let us first find the value of – 523 + 937

Therefore, – 523 + 937 = 414 ……………………… ( 1 )

Now, let us reverse the order and add 937 and – 523

We get,

937 + ( – 523 ) = 414 ……………………. ( 2 )

From ( 1 ) and ( 2 ) we get that addition of the integers – 523 and 937 gives us the same result no matter in which we place them. Hence it is proved that this addition satisfies the commutative property.

**Example 2** Find the missing number in this equation :

3 + ( _____ + 5 ) = ( 3 + 7 ) + 5

**Solution** We have been given the equation

3 + ( _____ + 5 ) = ( 3 + 7 ) + 5

The numbers given in the equation are 3, 5 and 7 all of which are natural numbers. We also know that the addition of natural numbers satisfies the associative property. Therefore, if we observe the above equation, we can see that it is a case of demonstration of the associative property of 3 numbers. The number missing on the left-hand side is 7.

Hence, the missing number in the equation is 7.

Let us now summarize the comparison of the commutative and the associative property that we learned for the addition of different numbers.

**Difference between Commutative and Associative property**

Commutative Property | Associative Property |

The commutative property comes from the term “commute” which means ‘move around’ and it refers to being able to switch numbers that you’re adding or multiplying regardless of the order of the numbers. | The associative property comes from the word “associate” or “group” and it refers to the grouping of three or more numbers using parentheses, regardless of how you group them. The result will be the same, no matter how you re-group the numbers or variables. |

The commutative rule of addition states, a + b = b + a, which means adding a and b gives the same result as adding b and a. The orders can be changed without changing the result. This rule of addition is called the commutative property of addition. | The associative property, on the other hand, is the rule that refers to the grouping of numbers. The associative rule of addition states, a + (b + c) is the same as (a + b) + c. |

Example of Commutative Property of addition = 2 + 3 = 3 + 2 = 5 | Example of Associative property of Addition = 2 + ( 3 + 5 ) = ( 2 + 3 ) + 5 = 10 |

**Key Facts and Summary**

- Commutative Property states that when an operation is performed on two numbers, the order in which the numbers are placed does not matter. In the case of an addition, this means that when we add two numbers, a and b, the order of the numbers does not matter, i.e. a + b = b + a
- Associative Property states that when an operation is performed on more than two numbers, the order in which the numbers are placed does not matter. In the case of an addition, this means that when we add three numbers, a, b and c the order of the numbers does not matter, i.e. a + ( b + c )= ( a + b ) + c
- The commutative property comes from the term “commute” which means ‘move around’ and it refers to being able to switch numbers that you’re adding or multiplying regardless of the order of the numbers.
- The associative property comes from the word “associate” or “group” and it refers to the grouping of three or more numbers using parentheses, regardless of how you group them. The result will be the same, no matter how you re-group the numbers or variables
- Counting numbers are called natural numbers. 1 is the first natural number and there is no last natural number.
- Commutative property is satisfied in the case of the addition of natural numbers.
- The associative property is satisfied in the case of the addition of natural numbers.
- Commutative property is satisfied in the case of the addition of whole numbers.
- The associative property is satisfied in the case of the addition of whole numbers.
- Commutative property is satisfied in the case of the addition of integers
- The associative property is satisfied in the case of the addition of integers.

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