Whole Numbers | Speed Notes

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Whole Numbers The numbers 1,2, 3, ……which we use for counting are known as natural numbers. If you add 1 to a natural number, we get its successor. If you subtract 1 from a natural number, you get its predecessor. (Scroll down to continue …)

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Every natural number has a successor. Every natural number except 1 has a predecessor.

Whole Numbers

Whole numbers are formed by adding zero to the collection of natural numbers. Hence, the set of whole numbers includes 0, 1, 2, 3, and so on.

Key Properties of Whole Numbers:

1. Successors and Predecessors:
   • Every whole number has a successor. For example:
     • The successor of 0 is 1.
     • The successor of 1 is 2.
     • The successor of 2 is 3.
   • Every whole number except zero has a predecessor. For example:
     • The predecessor of 1 is 0.
     • The predecessor of 2 is 1.
     • The predecessor of 3 is 2.
2. Relationship with Natural Numbers:
   • All natural numbers (1, 2, 3, …) are whole numbers, but not all whole numbers are natural numbers since whole numbers include 0.
3. Number Line Representation:

1. To visualize whole numbers, we can draw a number line starting from 0:
   • Mark points at equal intervals to the right: 0, 1, 2, 3, …
   • This number line allows us to carry out operations:
     • Addition: Moving to the right (e.g., 1 + 2 = 3).
     • Subtraction: Moving to the left (e.g., 3 – 1 = 2).
     • Multiplication: Making equal jumps (e.g., 2 × 3 means jumping twice the distance of 2, reaching 6).
     • Division: Although division can be tricky, it involves partitioning. For example, 6 ÷ 2 means splitting 6 into 2 equal parts, resulting in 3.

Closure Properties:

• Adding two whole numbers always results in a whole number:
  • Examples:
    • 2 + 3 = 5
    • 0 + 4 = 4
    • 1 + 1 = 2
• Multiplying two whole numbers also results in a whole number:
  • Examples:
    • 2 × 3 = 6
    • 0 × 5 = 0
    • 1 × 4 = 4
• Whole numbers are closed under subtraction only if the result is non-negative:
  • Examples:
    • 2 – 1 = 1
    • 5 – 3 = 2
    • 3 – 3 = 0
  • Yet, if the result is negative, they are not closed under subtraction:
    • Example: 2 – 3 = -1 (not a whole number).
  • So, the whole numbers are not not closed under subtraction.
• Division by whole numbers is defined only when the divisor is not zero, and the result is a whole number:
  • Examples:
    • 6 ÷ 2 = 3
    • 8 ÷ 4 = 2
    • 0 ÷ 5 = 0
  • Division by zero is undefined (e.g., 5 ÷ 0).
  • So, the whole numbers are not not closed under division.

Identity Elements:

• Zero acts as the identity for addition:
  • Example: 5 + 0 = 5.
• The whole number 1 acts as the identity for multiplication:
  • Example: 3 × 1 = 3.

Commutative and Associative Properties:

• Addition is commutative:
  • Examples:
    • 2 + 3 = 3 + 2
    • 1 + 4 = 4 + 1
    • 0 + 5 = 5 + 0
• Multiplication is also commutative:
  • Examples:
    • 2 × 3 = 3 × 2
    • 1 × 4 = 4 × 1
    • 0 × 5 = 5 × 0
• Both addition and multiplication are associative:
  • Examples for addition:
    • (1 + 2) + 3 = 1 + (2 + 3)
    • (0 + 4) + 1 = 0 + (4 + 1)
    • (2 + 2) + 2 = 2 + (2 + 2)
  • Examples for multiplication:
    • (1 × 2) × 3 = 1 × (2 × 3)
    • (0 × 4) × 1 = 0 × (4 × 1)
    • (2 × 2) × 2 = 2 × (2 × 2)

Distributive Property:

• Multiplication distributes over addition:
  • Example: 2 × (3 + 4) = 2 × 3 + 2 × 4.

Understanding these properties helps simplify calculations. It enhances our grasp of numerical patterns. These patterns are not only interesting but also practical for mental math.