Properties of Whole Numbers

The property of whole number includes:

1) Closure for addition and multiplication.

2) Commutative property for addition and multiplication.

3) Associative property for addition and multiplication.

4) Distributive property of multiplication over addition.

5) Identity for addition and multiplication.

Closure property

Addition

Take any two whole numbers and add them. Observe the sum carefully.

4 + 5 = 9 (whole number)

8 + 4 = 12 (whole number)

90 + 0 = 90 (whole number)

It is clear from the above examples that sum of any two whole numbers results in whole number. Therefore, we can say that sum of any two whole numbers is a whole number or the collection of whole numbers is closed under addition. This property is known as the closure property for addition of whole numbers.

Multiplication

Multiply any two whole numbers and observe the product.

7 x 8 = 56 (whole number)

5 x 6 = 30 (whole number)

0 x 15 = 0 (whole number)

From the above example we can conclude that multiplication of two whole numbers is also found to be a whole number. Therefore, it is clear that the system of whole numbers is closed under multiplication.

Subtraction

Now subtract any two whole numbers and observe the difference.

5 – 3 = 2 (whole number)

6 – 9 = -3 (not a whole number)

7 – 2 = 5 (whole number)

Here, we see that difference in all the case is not a whole number.

Therefore, we can say that that whole numbers are not closed under subtraction.

Division

Divide any two whole numbers and observe the quotient.

8 ÷ 2 = 4 (whole number)

2 ÷ 4 = ½ (not a whole number)

9 ÷ 3 = 3 (whole number)

In the above examples, quotient is not a whole number in all the 3 case.

Therefore, whole numbers are not closed under division.

Important Note: Division by zero is not defined.

Let us see how.

Division means repeated subtraction.

Example: 9 ÷ 0

    9 – 0 = 9

    9 – 0 = 9

    9 – 0 = 9

In every step we get same result i.e. 9. This will never stop. So we can say that division by zero is not defined.

Commutative property 

Addition

Example: 5 + 4 = 9

        And 4 + 5 = 9

Therefore, 5 + 4 = 4 + 5

This shows that we can add whole numbers in any order.

Therefore, according commutative property for addition the sum of two whole numbers is the same, no matter in which order they are added.

Multiplication

Example: 5 x 4 = 20

        And 4 x 5 = 20

Therefore, 5 x 4 = 4 x 5

This shows that we can multiply whole numbers in any order.

Commutative property for multiplication states that the product of two whole numbers is the same, no matter in which order they are multiplied.

Subtraction

Example: 7 – 5 = 2

        And 5 – 7 = -2

Therefore, 7 – 5 ≠ 5 - 7

Hence, Subtraction is not commutative.

Division

Example: 6 ÷ 2 = 3

        And 2 ÷ 6 = 1/3

Therefore, 6 ÷ 2 ≠ 2 ÷ 6

Hence, Division is not commutative.

Associative property

Addition

Example: (1 + 2) + 3 = 3 + 3 = 6

   And      1 + (2 + 3) = 1 + 5 = 6

Therefore, (1 + 2) + 3 = 1 + (2 + 3)

This shows that result are same even if we change the grouping of numbers. So, while adding whole numbers, we can group them in any order. This is called the associative property of addition.

Multiplication

Example: (2 x 3) x 4 = 6 x 4 = 24

         And 2 x (3 x 4) = 2 x 12 = 24

Therefore, (2 x 3) x 4 = 2 x (3 x 4)

This shows that result are same even if we change the grouping of numbers. So, while multiplying whole numbers, we can group them in any order. This is called the associative property of multiplication.

Subtraction

Example: (5 – 3) – 2 = 2 – 2 = 0

     And    5 – (3 – 2) = 5 – 1 = 4

Therefore, (5 – 3) – 2 ≠ 5 – (3 – 2)

This shows that result are not same if we regroup the numbers except in certain cases.

Example: (3 – 2) – 0 = 1 – 0 = 1

         And 3 – (2 – 0) = 3 - 2 = 1

Therefore, (3 – 2) – 0 = 3 – (2 – 0)

Hence, subtraction doesn’t follow the associative property except in few cases.

Division

Example: (12 ÷ 3) ÷ 2 = 4 ÷ 2 = 2

     And    12 ÷ (3 ÷ 2) = 12 ÷ 1.5 = 8

Therefore, (12 ÷ 3) ÷ 2 ≠ (12 ÷ 3) ÷ 2

This shows that result are not same if we regroup the numbers except in certain cases.

Example: (6 ÷ 3) ÷ 1 = 2 ÷ 1 = 2

     And    6 ÷ (3 ÷ 1) = 6 ÷ 3 = 2

Hence, division doesn’t follow the associative property except in few cases.

Distributive property

Distributive of multiplication over addition

Example 1: 15 (8 + 2)

              = 15 x 10 = 150

Example 2: 290 x 105

To make this multiplication easy, we break 105 into 100 + 5 and then we will use distributive property.

              = 290 (100 + 5)

              = (290 x 100) + (290 x 5)

              = 29000 + 1450

              = 30450

Distributive of multiplication over subtraction

Example 1: 20 (12 - 2)

              = 20 x 10 = 200

Example 2: 200 x 98

To make this multiplication easy, we write 98 as 100 - 2 and then we will use distributive property.

              = 200 (100 - 2)

              = (200 x 100) - (200 x 2)

              = 20000 - 400

              = 19600

Additive Identity

When we add ’0’ to any whole number, we get the same whole number again. Thus, Zero is called an identity for addition of whole numbers or additive identity for whole numbers.

Example 1: 5 + 0 = 5

Example 2: 0 + 5 = 5

Multiplication Property of Zero

Zero plays a special role in multiplication too i.e. any number when multiplied by zero becomes zero.

Example: 450 x 0 = 0

Multiplicative Identity

The multiplicative identity property states that any time we multiply a number by 1, product, is the original number.

Example: 9 x 1 = 9

               7 x 1 = 7