# Properties of Addition and Multiplication: Commutative, Associative and Identity

## Commutative Property

Addition and multiplication are commutative in arithmetic operations. Commutative property means that the order of the terms can be changed so that the result does not change. This property is called **Commutative property over addition and mutiplication**.

Let's take an example of commutative over addition:

If This means that

Whether we add 9 to 2 or add 2 to 9 we get the same result

### Note:

Let and represent whole numbers. Then,

Let's take an example of commutative over multiplication:

If This means that

Whether we multiply 3 by 4 or multiply 4 by 3 we get the same result

### Note:

Let and represent whole numbers. Then,

But what about subtraction and division? Do you think they are commutative too? Let's try.

It is clear that , but

So the commutative property over subtraction has not been achieved.

It is also clear that , but .

So the commutative property was not achieved for the division as well.

### Note:

Let and represent whole numbers. Then,

We say that subtraction and division are not commutative.

## Associative Property

Addition and multiplication are associative in arithmetic operations. Associative means in the case of an arithmetic operation in which there are 3 or more terms, we can add or multiply two different terms without paying attention to their order in the operation. This property is called the **associative property over addition and mutiplication**.

Let's take an example of associative over addition.

Obviously means that

Applying BODMAS rule, whether we add 5 to the sum of terms 4 and 3 or we add the sum of terms 5 and 3 to 4 we get the same result.

### Note:

Let and represent whole numbers. Then,

Let's take an example of associative over multiplication.

Obviously means that

Applying BODMAS rule, whether we multiply the product of 2 and 6 by 5 or we multiply 2 by the product of 6 and 5 we get the same result.

### Note:

Let and represent whole numbers. Then,

But what about subtraction and division? Do you think they are associative too? Let's try.

It is clear that , but

So the associative property over subtraction has not been achieved.

It is also clear that , but

So the associative property was not achieved for the division as well.

### Note:

Let and represent whole numbers. Then,

We say that subtraction and division are not associative.

## Identity Property (Neutral Element)

The identity element is every number that does not affect an arithmetic operation. That is, it does not affect the value of the algebraic expression, but not for all operations.

**For addition**:

In addition, the identity element is 0, because whether or not we add 0 to the expression, nothing changes.

For example:

### Note:

Let represent whole number. Then,

**For multiplication**:

In multiplication, the identity element is 1, because whether or not we multiply the expression by 1, nothing changes.

For example:

### Note:

Let represent whole number. Then,