Class VI - Mathematics

Chapter - 1 Whole Numbers

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Natural numbers are the numbers that we use for counting i.e. 1,2,3,4 ….
Adding 1 to a natural number gives its successor and subtracting 1 gives its predecessor. For example,
1. 6 + 1 = 7, therefore, 7 is the successor of 6
2. 6 – 1 = 5, so 5 is the predecessor of 6
All natural numbers have a successor, and all natural numbers except 1 have a predecessor.
If we add 0 to the collection of all natural numbers, we get a collection of whole numbers i.e. 0, 1, 2, 3, 4 …
If we consider whole numbers, 1 has a predecessor because. 1 – 1 = 0, which is a whole number. However, 0 does not have a whole number as a predecessor.
If we draw a line, mark the starting point as 0, and mark subsequent points at equal distance as 1, 2, 3, etc., we get a number line of whole numbers. We can perform addition on this number line by moving towards the right from any point and subtraction by moving towards the left. Multiplication corresponds to jumping equal distances from 0.

Commutative Property
1. Addition and multiplication of whole numbers are commutative. This means that you can add or multiply two whole numbers in any order, but the result will be the same. For example,
  1. 1 + 9 = 10; 9 + 1 = 10
  2. 3 × 4 = 12; 4 × 3 = 12
2. Subtraction and division of whole numbers is not commutative
  1. 6 – 3 = 3, which is not the same as 3 – 6
  2. 10 ÷ 2 = 5, which is not the same as 2 ÷ 10

Associative Property
1. Addition and multiplication of whole numbers are associative. This means that when we need to add or multiply a few whole numbers, we can choose to group them in any way we like while performing the operation.
  1. 32 + 25 + 18 , can be calculated in multiple ways i.e. (32 + 25) + 18 = 57 + 18, or 32 + (25 + 18) = 32 + 43, or (32 + 18) + 25 = 50 + 25. The solution in all cases is 75
  2. 3 × 4 × 5 can be multiple ways i.e. (3 × 4) × 5 = 12 × 5, or 3 × (4 × 5) = 3 × 20, or (3 × 5) × 4 = 15 × 4. The solution in all cases is 60
2. Subtraction and division of whole numbers are not associative

Distributivity of multiplication over addition: This means that multiplying one number with a sum of two numbers is the same as multiplying it with each of the two numbers, and then adding them. For example:
1. 32 × 5 can be written as (30 + 2) × 5 = 160, which is the same as (30 × 5) + (2 × 5) = 150 + 10 = 160
2. 632 x 8 = (600 + 30 + 2) x 8 = (600 x 8) + (30 x 8) + (2 x 8) = 4800 + 240 + 16 = 5056

Additive and Multiplicative Identity of whole numbers:
1. Since 0 plus any whole number results in the same whole number, zero is referred to as the additive identity of whole numbers
2. Similarly, 1 is referred to as the multiplicative identity of whole numbers because any whole number multiplied by 1 results in the same whole number

Patterns in whole numbers
1. Whole numbers except 0 and 1 can be represented in dots pattern to form either of the following shapes –
  1. a line: All natural numbers can form a line.
    - 2 and 3 can be represented as
      
      respectively.
  2. a rectangle:
    - 6 can be represented as
  3. a square: The numbers that can form a square are 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25 and so on.
    - 4 can be represented as
  4. a triangle: The numbers that can form a triangle are 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15 and so on.
    - 9 can be represented as
2. Pattern observation - Observing the patterns can guide us in simplifying mathematical calculations.
  1. Addition:
    (a) 119 + 8 = 119 + 10 - 2 = 129 - 2 = 127
    (b) 212 + 7 = 212 + 10 - 3 = 222 - 3 = 219
  2. Subtraction:
    (a) 117 - 8 = 117 – 10 + 2 = 107 + 2 = 109
    (b) 118 - 99 = 118 - 100 + 1 = 18 + 1 = 19
  3. Multiplication: