Whole Numbers

• Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on)
• There is no fractional or decimal part. And no negatives.
• Example: 5, 49 and 980 are all whole numbers.

Successor

• The number which comes immediately after a particular number
• The successor of n is ‘n+1'
• If n = 20, then successor of 20 is 20 + 1 = 21
  If n = 50, then successor of 50 is 50 + 1 = 51

Predecessor

• The number which comes immediately before a particular number
• The predecessor of x will be x-1.
• If x = 50, then predecessor of 50 is 50-1 = 49
• If x = 100, then predecessor of 100 is 100-1 = 99
• Q. What is the successor of the largest three-digit number?
  • Ans. The largest three-digit number is 999. Therefore, the successor of 999 is 1000.
• Q. What is the predecessor of the smallest three-digit number?
  • Ans. The smallest 3-digit number is 100. Therefore, the predecessor of 100 is 99.

• Smallest natural number is 1 and not zero.

• The smallest whole number is 0.
• All Natural Numbers are Whole Numbers.

• The Natural number 1 has no predecessor. But The Whole Number 1 has a Predecessor 0.
• Addition on the Number LineIf 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.

  

  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	Whole numbers are closed under addition as their sum is also a whole number.	2 + 5 = 7	Yes
  Subtraction	Whole numbers are not closed under subtraction as their difference is not always a whole number.	9 – 2 = 7 2 – 9 = (-7) which is not a whole number.	No
  Multiplication	Whole numbers are closed under multiplication as their product is also a whole number.	9 × 5 = 45	Yes
  Division	Whole numbers are not closed under division as 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	The addition is commutative for whole numbers as their sum remains the same even if we interchange the position of the numbers.	2 + 5 = 7 5 + 2 = 7	Yes
  Subtraction	Subtraction is not commutative for whole numbers as their 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	Multiplication is commutative for whole numbers as their 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 for whole numbers as their result may be different if we interchange the position of the numbers.	5 ÷ 1 = 5 1 ÷ 5 =, not a whole number.	No

  3. Associative Property

  The two 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	The addition is associative for whole numbers as their 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	Subtraction is not associative for whole numbers as their 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	Multiplication is associative for whole numbers as their 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 for whole numbers as their result may change if we change the grouping of the numbers.	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

  

  This clearly shows that if we add zero apples to 2 apples we get the two apples only.

  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.

  

  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