Whole Numbers

• Basic Concepts

  • Natural Numbers
    • Counting numbers ($1, 2, 3, \dots$)
    • Smallest natural number: 1
  • Whole Numbers
    • Natural numbers including zero ($0, 1, 2, 3, \dots$)
    • Smallest whole number: 0
    • Symbolized by set W
  • Predecessor
    • Given number minus 1 ($n - 1$)
    • Number 0 has no predecessor in whole numbers
  • Successor
    • Given number plus 1 ($n + 1$)
    • Every whole number has a successor

• The Number Line

  • Representation
    • Straight line starting at 0 and extending right indefinitely
  • Addition
    • Move to the right on the number line
  • Subtraction
    • Move to the left on the number line
  • Multiplication
    • Make equal jumps starting from 0 towards the right

• Properties of Whole Numbers

  • Closure Property
    • Addition: $a + b$ is always a whole number
    • Multiplication: $a \times b$ is always a whole number
    • Subtraction & Division: Not closed
  • Commutative Property
    • Addition: $a + b = b + a$
    • Multiplication: $a \times b = b \times a$
    • Subtraction & Division: Not commutative
  • Associative Property
    • Addition: $(a + b) + c = a + (b + c)$
    • Multiplication: $(a \times b) \times c = a \times (b \times c)$
    • Makes calculations easier by regrouping
  • Distributive Property
    • Multiplication over Addition: $a \times (b + c) = (a \times b) + (a \times c)$
    • Multiplication over Subtraction: $a \times (b - c) = (a \times b) - (a \times c)$
  • Identity Properties
    • Additive Identity: 0 ($a + 0 = a$)
    • Multiplicative Identity: 1 ($a \times 1 = a$)
  • Division by Zero
    • Division of any whole number by zero is undefined ($a \div 0 = \text{not defined}$)

• Patterns in Whole Numbers

  • Shapes formed by numbers
    • Line: Every number can be arranged as a line
    • Rectangle: Numbers like 6, 8, 10
    • Square: Numbers like 4, 9, 16 ($n \times n$)
    • Triangle: Numbers like 3, 6, 10
  • Applications of Patterns
    • Simplifies mental arithmetic
    • Helps in quick multiplication and addition shortcuts