CBSE Class 6 Mathematics – 3. Whole Numbers (Notes)

Revision Notes on Whole Numbers

Learning Goal

1. Predecessor and Successor
2. Number Line
3. Addition on the Number Line
4. Subtraction on the Number Line
5. Multiplication on the Number Line
6. Properties of Whole Numbers

(Closure, Commutative, Associative, Distributivity of Multiplication over Addition)

7. Identity for Addition, Identity for Multiplication

Natural Numbers

Counting numbers are called Natural Numbers.

Ex- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ………….

Whole Numbers

All natural numbers together with zero are called Whole Numbers.

Ex- 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ………….

Note All natural numbers are whole number but zero is a whole number which is not a natural number.

Predecessor and Successor

If we add 1 to any natural number, we get the next number, which is called the Successor of that number.

12 + 1 = 13

So 13 is the successor of 12.

If we subtract 1 from any natural number, we get the predecessor of that number.

12 – 1 = 11

So 11 is the predecessor of 12.

Remark: There is no predecessor of 1 in natural numbers.

Properties of Whole Numbers

1. Closure Property

Two whole numbers are said to be closed if their operation is also the whole number.

Operation	Meaning	Example	Closed or not
Addition	Sum of whole numbers is also a whole number.	2 + 5 = 7	Yes
Subtraction	Difference of whole numbers is not always a whole number.	9 – 2 = 7 2 – 9 = (-7) which is not a whole number.	No
Multiplication	Product of whole number is a whole number.	9 × 5 = 45	Yes
Division	Division is not closer because their result is not always a whole number.	5 ÷ 1 = 5 5 ÷ 2 =, not a whole number.	No

2. Commutative Property

Two whole numbers are said to be commutative if their result remains the same even if we swap the positions of the numbers.

Operation	Meaning	Example	Commutative  or not
Addition	Sum remains the same even if we interchange the position of the numbers.	2 + 5 = 7 5 + 2 = 7	Yes
Subtraction	Difference may be different if we interchange the position of the numbers.	9 – 2 = 7 2 – 9 = (-7) which is not a whole number.	No
Multiplication	Product remains the same even if we interchange the position of the numbers.	9 × 5 = 45 5 × 9 = 45	Yes
Division	The division is not commutative because their result may change.	5 ÷ 1 = 5 1 ÷ 5 =, not a whole number.	No

3. Associative Property

The three whole numbers are said to be associative if the result remains the same even if we change the grouping of the numbers.

 

Operation	Meaning	Example	Associative or not
Addition	Sum remains the same even if we change the grouping of the numbers.	3 + (2 + 5) = (3 + 2) + 5 3 + 7 = 5 + 5 10 = 10	Yes
Subtraction	Difference may change if we change the grouping of the numbers.	8 – (10 – 2) ≠ (8 – 10) – 2 8 – (8) ≠ (-2) – 2 0 ≠ (-4)	No
Multiplication	Product remains the same even if we change the grouping of the numbers.	3 × (5 × 2) = (3 × 5) × 2 3 × (10) = (15) × 2 30 = 30	Yes
Division	The division is not associative because their result may change.	24 ÷ 3 ≠ 4 ÷ 2 8 ≠ 2	No

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4. Distributivity of Multiplication over Addition

This property says that if we have three whole numbers x, y and z, then

x(y + z) = xy + xz

Example

Evaluate 15 × 45

Solution

15 × 45 = 15 × (40 + 5)

= 15 × 40 + 15 × 5

= 600 + 75

= 675

5. Identity for Addition

If we add zero to any whole number the result will the same number only.

So zero is the additive identity of whole numbers.

a + 0 = 0 + a = a

Ex- 2 + 0 = 0 + 2 = 2

6. Identity for Multiplication

If we multiply one to any whole number the result will be the same whole number.

So one is the multiplicative identity of whole numbers.

a × 1 = a Or 1 × a = a

Ex- 5 × 1 = 5 Or 1 × 5 = 5

Patterns

Patterns are used for easy verbal calculations and to understand the numbers better.

We can arrange the numbers using dots in elementary shapes like triangle, square, rectangle and line.

1. We can arrange every number using dots in a line

2. We can arrange some numbers using a rectangle.

3. We can arrange some numbers using a square.

4. We can arrange some numbers using a triangle.

Use of Patterns

Patterns can be used to simplify the process.

1. 123 + 9 = 123 + 10 – 1 = 133 -1 = 132

123 + 99 = 123 + 100 – 1 = 223 – 1 = 222

2. 83 × 9 = 83 × (10-1) = 830 – 83 = 747

83 × 99 = 83 × (100-1) = 8300 – 83 = 8217

Number Line

To draw a number line, follow these steps-

Draw a line and mark a point 0 on it.

Now mark the second point to the right of zero and label it as 1.

The distance between the 0 and 1 is called the unit distance.

Now you can mark other points as 2, 3, 4 and so on with the unit distance.

This is the number line for the whole numbers.

1. The distance between two points

The distance between 3 and 5 is 2 units. Likewise, the distance between 1 and 6 is 5 units.

2. The greater number on the number line

The number on the right is always greater than the number on the left.

As number 5 is on the right of the number 2, Hence 5>2.

3. A smaller number on the number line

The number on the left of any number is always smaller than that number.

As number 3 is on the left of 7, so 3 < 7.

Addition on the Number Line

If we have to add 2 and 5, then start with 2 and make 5 jumps to the right. As our 5^th jump is at 7 so the answer is 7.

The sum of 2 and 5 is 2 + 5 = 7

Subtraction on the Number Line

If we have to subtract 6 from 10, then we have to start from 10 and make 6 jumps to the left. As our 6^th jump is at 4, so the answer is 4.

The subtraction of 6 from 10 is 10 – 6 = 4.

Multiplication on the Number Line

If we have to multiply 4 and 3, then Start from 0, make 4 jumps using 3 units at a time to the right, as you reach to 12. So, we say, 3 × 4 = 12.