Whole Numbers Class 6 Notes - Chapter 2

We have learned about the natural numbers from 1 to 10. Whole numbers are the set natural numbers, including zero. 0 is the smallest whole number. Whole numbers are 0, 1, 2, 3, ……… All-natural numbers are whole numbers, but all whole numbers are not natural numbers

Properties of Whole Numbers

1. Addition and multiplication of any 2 whole numbers give a whole number.
2. Subtraction and division of any 2 whole numbers may or may not give a whole number.

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What Is a Number line?

A number line is a picture of a graduated straight and horizontal line in which numbers are written. A number written on the left-hand side of the number line is lesser, and a number written on the right-hand side of the number line is greater. Lets us look into some solved example problems.

Find 12 × 35 using distributivity.

12 × 35 = 12 × (30 + 5)

= 12 × 30 +12 × 5

= 360 + 60 = 420.

Calculate – (2 + 3) + 4 = ? = 5+ 4 = 9.
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Evolution of Numbers

Natural Numbers

• Numbers that are used for counting and ordering are called natural numbers.
• 1,2,3,4,5,6… are natural numbers

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Whole Numbers

• Natural numbers, along with zero, form the collection of whole numbers.
• 0,1,2,3,4,5… are called whole numbers.

For more information on Number Systems, watch the below video

Predecessors and Successors

Predecessor and Successor

• A successor of any number is the next number to it, which is obtained by adding 1.
• A predecessor of any number is the previous number to it, which is obtained by subtracting 1.
• For example, the predecessor and successor of the number 12 is 12 – 1 and 12 + 1, which is 11 and 13

For more information on Predecessors and Successors, watch the below video

Where Do Whole Numbers Live?

Number Line

• It is the infinitely long line containing all the whole numbers.
• The line starts at zero, and any two consecutive whole numbers have the same distance between them.

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Describing Number Line

Operations on a Number Line

⇒ Addition on a number line. For example, the addition of 1 and 5 (1 + 5 = 6). First, locate 1 on the number line. Then, moving 5 places to the right will give 6.

⇒ Subtraction on a number line. For example, subtraction of 3 from 7 (7 – 3 = 4). First, locate 7 on the number line. Then move 3 places to the left will give 4.

⇒ Multiplication on a number line. For example, the product of 3 and 4 (3 × 4 = 12). Start from 0 and skip 3 places to the right 4 times.

⇒ Division on a number line. For example, 6 ÷ 3 = 2. Start from 6 and subtract 3 for a number of times till 0 is reached. The number of times 3 is subtracted from gives the quotient.

Properties of Operators: Commutative Associative and Distributive

Division by Zero

Division of any whole number by 0 is not defined.

Mathematical operations are simplified due to certain properties that every number follows. They are:

• Commutative Property

Addition and multiplication are commutative for whole numbers, i.e., whole numbers can be added or multiplied in any order.

For e.g: 2 + 3 = 5 = 3 + 3 × 4 = 12 = 4 × 3

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  Associative Property

Associativity of addition and multiplication

For eg: (5 +6) + 4 = 15 = 5 + (6 + 4)

(2 × 3) × 4 = 24 =2 × (3 × 4)

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  Distributive Property

With distributivity property, 4 × (5 + 3) can be written as (4 × 5) + (4 × 3)

Here, 4 × (5 + 3) = 4 × 13 = 52

Also, (4 × 5) + (4 × 3) = 20 + 32 = 52

There exist certain numbers; when included in mathematical operations like addition and multiplication, the value of the operation remains unchanged. Such numbers are called identities.
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Additive Identity

Additive identity gives the same whole number when added to another whole number.

Zero is the additive identity as a + 0 = a, (a is any whole number).

Multiplicative Identity

Multiplicative identity gives the same whole number when multiplied by another whole number.

1 is the Multiplicative identity as a × 1 = a, (a is any whole number)
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Let’s Play With Whole Numbers

Patterns

• Every number can be arranged as a line.
  • E.g : 5 = •••••
• Some whole numbers can be expressed as squares.
  • E.g : 
• Some whole numbers can be expressed as rectangles.
  • E.g : 6 can be shown as 3 × 2

• Some numbers can also be arranged as triangles.
• 

Numbers between Square Numbers

• Between 2 successive square numbers there exists 2n non-square numbers. Between n² and (n + 1)², there are non-square numbers. Here is a whole number.
• For example, between 9 (3)² and 16 (4)², there are 10 , 11, 12, 13, 14, 15 which is 6 = 2 × 3 numbers.

Adding Odd Numbers

• Sum of the first n natural odd numbers gives n², which is a perfect square.
• For example : Sum of first 5 natural odd numbers ⇒ 1 + 3 + 5 + 7 + 9 = 25 = 5²

Properties of Operators: Closure Properties

Closure Property

Whole numbers are closed under addition and also under multiplication.

Whole numbers are not closed under subtraction and division.

Learn more about the whole numbers from the topics given below: