Special Numbers Chapter Notes | Year 4 Mathematics IGCSE (Cambridge) - Class 4 PDF Download

Ordering and comparing numbers

• The objective is to compare positive and negative numbers using the symbols <, >, or =.
• Order positive and negative numbers starting with either the smallest or largest.
• Negative numbers represent values below zero, such as depths below sea level or temperatures below 0°C.
• Ordering examples:
  • Starting with the smallest: −90, −80, −70, −60, ….
  • Starting with the largest: 10, 0, −10, −20, ….
• Comparing numbers:
  • Use a number line to visualize positions, where numbers to the right are larger.
  • Example: −3 > −5 because −3 is to the right of −5 on the number line.
  • Other comparisons: −30 < 0, −50 > −60.

Working with special numbers

• The objective is to use odd and even numbers, factors, multiples, and square numbers.
• Even numbers: Divisible by 2, e.g., 0, 2, 4, 6, 8, 10, ….
• Odd numbers: Not divisible by 2, e.g., 1, 3, 5, 7, 9, 11, …, often seen in nature, such as flowers with 3 or 5 petals.
• Factors:
  • Numbers that divide another number without a remainder.
  • Example: Factors of 12 are 1, 2, 3, 4, 6, 12.
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
  • Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40.
  • Common factors of 30 and 40: 1, 2, 5, 10.
• Multiples:
  • Numbers obtained by multiplying a number by integers.
  • Example: Multiples of 5: 5, 10, 15, 20, 25, ….
• Square numbers:
  • Numbers that are the product of an integer with itself, e.g., 1 = 1², 4 = 2², 9 = 3².
  • Example: 10² = 100, so two square numbers with a sum of 100 could be 6² + 8² = 36 + 64 = 100.
• Venn diagrams can organize numbers by properties, such as factors or multiples, with overlapping sections for numbers meeting multiple criteria.

Tests of divisibility

• The objective is to find numbers divisible by 2, 5, 10, 25, 50, 100.
• Divisibility rules simplify checking if one number divides another without extensive calculation, useful for validating product codes or performing quick checks.
• Divisibility rules:
  • Divisible by 2: Number ends in 0, 2, 4, 6, 8 (even numbers).
  • Divisible by 5: Number ends in 0 or 5, e.g., 2 + 3 = 5 or 3 + 2 = 5 using digits 2, 3, 4, 5.
  • Divisible by 10: Number ends in 0.
  • Divisible by 25: Number ends in 00, 25, 50, 75.
  • Divisible by 50: Number ends in 00 or 50.
  • Divisible by 100: Number ends in 00.
• Relationships between divisibility:
  • If a number is divisible by 100, it is also divisible by its factors: 2, 5, 10.
  • If a number is divisible by 50, it is also divisible by 2, 5.
  • If a number is divisible by 25, it is also divisible by 5.
• Example for product code validation:
  • For code 065407954097:
    • Sum of even-position digits: 7 + 9 + 0 + 5 + 4 + 9 = 34.
    • Sum of odd-position digits: 0 + 6 + 5 + 4 + 0 + 7 = 22, then 22 × 3 = 66.
    • Total: 34 + 66 = 100.
    • Check: 100 ÷ 10 = 10 remainder 0, so the code is valid.