CBSE Class 6 Maths Notes Chapter 2 Whole Numbers

Whole numbers are the set natural numbers including with 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

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The different topics covered in CBSE Class 6 Mathematics Chapter 2 are tabulated below:

Ex : 2.1 –  Introduction

• Natural numbers: The counting numbers 1,2, 3,4, are called natural numbers.
• Predecessor: If we subtract 1 from a natural number, what we get is its predecessor. For example, the predecessor of 10 is 10 – 1 = 9.
• Successor: If we add 1 to a natural number, what we get is its successor. For example, the successor of 9 is 9 + 1 = 10.
• The natural number 1 has no predecessor in natural numbers.
• There is no largest natural number
• If we add the number 0 to the collection of whole numbers. Thus, the numbers 0, 1, 2, 3,… form the collection of whole numbers, natural numbers, what we get is the collection
• We regard 0 as the predecessor of 1 in the collection of whole numbers.
• Every whole number has a successor.
• Every whole number except zero has a predecessor.
• All the natural numbers are whole numbers but all the whole numbers are not a natural number. [0 is a whole number but not a natural number]

Ex : 2.2 –  Whole Numbers

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

Ex : 2.3 –  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.

Operations on a number line

⇒ Addition on a number line. For example, 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 moving 3 places to the left will give 4.

⇒ Multiplication on a number line. For example 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 gives the quotient.

Ex : 2.4 –  Properties of Whole Numbers

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

• Associative property

Associativity of addition and multiplication

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

(2 × 3) × 4 = 24 =2 × (3 × 4)

• 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 exists certain numbers, when included in mathematical operations like addition and multiplication, the value of the operation remains unchanged. Such numbers are called as identities.

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)

Ex : 2.5 –  Patterns in Whole Numbers

• Some numbers can be arranged in elementary shapes: a line, a rectangle, a square and a triangle only made up of dots.
• Every number can be arranged as a line.
• Some numbers like 6 can be arranged as a rectangle. Note that the number of rows should be smaller than the number of columns. Also, the rectangle should have more than 1 row.
• Some numbers like 4, 9 can be arranged as a square.
• Note that every square number is also a rectangular number.
• Some numbers like 3, 6 can be arranged as a triangle.
• Note that the triangle should be right-angled and its two sides must be equal.
• The number of dots in the rows starting from the bottom row should be like 4, 3,2, 1. The top row should always have 1 dot.
• The patterns with numbers are not only interesting but also useful especially for mental calculations and help us understand the properties of numbers better.

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.

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