# Mathematics - Class 6 / Grade 6

- Properties of Whole Numbers
### Properties of Whole Numbers

Math - Class 6 (CBSE/NCERT) - Properties of Whole Numbers – Practice Questions and Solutions/Notes/Worksheets Tags: Closure for addition and multiplication, Commutative property for addition and multiplication, Associative property for addition and multiplication, Distributive property of multiplication over addition, Identity for addition and multiplication, Whole Numbers And Its Properties for VI Grade, Properties of whole numbers practice pages for Class 6, Properties of whole numbers free worksheets PDF with answers for 6th class, Properties of whole numbers under addition and multiplication Exercises for Grade 6, problem sum on properties of whole number, Find the value of 2100×102 - 2100×2, Find the product using suitable rearrangement. Write the property for each of the following. Solve the following using distributive property.### 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****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,

__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****Download to practice offline.**