What are the Number Properties? (Commutative, Distributive, Associative & Identity) – BYJUS

# Numbers Properties

We have learned to perform basic math operations like addition, subtraction, multiplication, and division. We can simplify these operations by keeping in mind some special properties of numbers. Learn the properties of numbers that will help you evaluate expressions quickly with the help of some examples....Read MoreRead Less

## What are Number Properties?

Number properties lay down some rules that we can follow while performing mathematical operations.

There are four number properties: commutative property, associative property, distributive property and identity property. Number properties are only associated with algebraic operations that are addition, subtraction, multiplication and division. However, some of these properties are not applicable to subtraction and division operations.

## Commutative Property

The word commute means “to travel back and forth”. If a number is commutative, that means it is movable. The commutative property states that changing the order of addends or factors does not change the sum or the product.

Let’s see how this is applicable to the numbers in an expression.

Consider the expression 3 + 5.

We know that  3 + 5 = 8. But 5 + 3 is also equal to 8.

So, 3 + 5 = 5 + 3

When two numbers are added together, the sum remains the same even if we change the order in which the addition operation is performed. That means the expression gives us the same result even if the position of the numbers change. This is known as the commutative property of addition.

Just like we saw in addition, the commutative property is also applicable to multiplication.

For example, $$3 \times 5 = 15$$

And $$5 \times 3 = 15$$

So, when two numbers are multiplied together, the product of the two numbers remain the same irrespective of the order in which they are multiplied. This is known as the commutative property of multiplication.

## Associative Property

Some math expressions with more than two terms can be solved easily by grouping the terms in the expression. To “associate” numbers means to group numbers. The associative property states that changing the grouping of addends or factors does not change the sum or the product.

Let’s see how associative property can be used in addition. Consider the following equation:

5 + 7 + 6 = 18

Whenever we perform this addition in our mind, we usually add two numbers first and then add the third number to the sum of the first two numbers. We can perform this addition in two ways.

5 + (7 + 6) = 5 + 13 = 18

And (5 + 7) + 6 = 12 + 6 = 18

In both cases, the answer remains the same.

So, when three numbers are added, the sum remains the same irrespective of the way in which they were grouped. This is known as the associative property of addition.

Let’s try out associative property in the case of multiplication.

$$1 \times 2 \times 3 = 6$$

We can perform this multiplication in two ways.

$$1 \times (2 \times 3) = 6$$

And $$(1 \times 2) \times 3 = 6$$

When three or more numbers are multiplied, the product remains the same irrespective of the way in which the numbers were grouped. This is known as the associative property of multiplication.

## Distributive Property

The distributive property states that multiplying the sum of two or more addends by a number is the same as multiplying each addend individually by the number and then adding the products together. Interestingly, the distributive property is also applicable to subtraction. Let’s take a look at an example.

$$5 \times (2 + 3) = (5 \times 2) + (5 \times 3)$$

Similarly, $$5 \times (2 – 3) = (5 \times 2) – (5 \times 3)$$

## Identity Property

Identity property states that when a number is added, subtracted, multiplied or divided by a specific number, the result will be the same as the original number. Let’s find out more about the identity property of addition and subtraction and the identity property of multiplication and division.

### Identity property of addition and subtraction

0 is considered as the additive identity in the case of addition and subtraction. When we add or subtract 0 to any number, we get the same number.

For example, 7 + 0 = 0, 0 + 2 = 2, and 5 – 0 = 5

### Identity property of multiplication and division

1 is considered as the multiplicative identity in the case of multiplication. If we multiply any number by 1, we get the same number.

For example,

$$5 \times 1 = 5, 1 \times 7 = 7$$

This holds true for division as well. Any number divided by 1 gives the same number.

For example, $$5 \div 1 = 5$$.

## Solved Number Properties Examples

Example 1: Use the distributive property to solve $$8\times24$$.

Solution:

$$8\times24=8\times(20+4)$$    Write 24 as a sum of two numbers

$$=(8\times20)+(8\times4)$$      Distributive property

$$=160+32$$                      Multiply

$$=192$$                              Add

Example 2: Complete the following equations and identify the property used in each case.

a. $$12\times 5=\_\_\_\_\times 12$$
b. $$15+12+9=\_\_\_\_+15+9$$

Solution:

a. We have to find the missing number in the equation.

According to the commutative property of multiplication,

$$12\times5=5\times12$$

So, the missing number is 5.

b. Both sides of the equation are equal. We have 15 and 9 on both sides of the equation. We have 12 only on the left-hand side of the equation. So, the missing value is 12.

15 + 12  + 9 = 12 + 15 + 9

We used the commutative property of addition to find the missing term in this equation.

Example 3: Complete the equations and identify the property being used.

a. $$1\times\_\_\_\_=115$$
b. $$213+\_\_\_\_=213$$

Solution:

a. In this example, we are multiplying an unknown number with 1 to get 115. But we know that any number multiplied by 1 gives us the same number. Hence, the number that is missing in the equation is also 115.

We used the identity property of multiplication to complete this equation.

b. In this case, we are adding an unknown number to 213 to get 213. We must be adding 0 to 213 on the left-hand side. According to the identity property of addition, when we add 0 to a number, we get the same number as the result.

Example 4: Solve the following equation using a number property.
113 + 4 + 27 = ?

Solution:

We can group 113 and 27 because it adds up to give 140. This makes the addition much easier.

113 + 4 + 27 = 113 + 27 + 4              Using the commutative property to swap position of numbers.

(113 + 27) + 4 = 140 + 4 = 144          Using the associative property of addition to group numbers conveniently.

Frequently Asked Questions on Number Properties

Number properties help us solve equations containing math operations easily. Number properties reduce the number of steps involved in the solution and make it easy to understand.

All number properties are not applicable to subtraction and division. For example, commutative and associative properties are not applicable to subtraction and division. Only the distributive property and identity property are applicable to subtraction and division.