Sample Notes: Types of Number

O Level and IGCSE Mathematics – Chapter 1.1: Types of Number

Definition and Explanation of Types of Numbers

Natural Numbers

• Definition: Natural numbers are the set of positive whole numbers starting from 1.
• Notation: ℕ = {1, 2, 3, 4, 5, …}
• Examples:
  • 7 is a natural number.
  • 103 is a natural number.
• Application: Used for counting (e.g., number of books, students, etc.)

Whole Numbers

• Definition: Whole numbers include all natural numbers and zero.
• Notation: {0, 1, 2, 3, 4, …}
• Example: 0 is a whole number but not a natural number.

Integers

• Definition: Integers include all whole numbers and their negative counterparts.
• Notation: ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}
• Examples:
  • -5 is an integer.
  • 0 and 9 are also integers.

Prime Numbers

• Definition: A prime number is a natural number greater than 1 that has exactly two distinct factors: 1 and itself.
• Examples:
  • 2, 3, 5, 7, 11, 13
  • Note: 2 is the only even prime number.
• Non-examples:
  • 1 is not a prime (it has only one factor).
  • 4 is not a prime (factors are 1, 2, 4).

Composite Numbers

• Definition: A natural number greater than 1 that is not prime; i.e., it has more than two factors.
• Examples: 4, 6, 8, 9, 10

Square Numbers

• Definition: A number which can be expressed as the product of an integer multiplied by itself.
• Examples:
  • 1² = 1
  • 2² = 4
  • 3² = 9
  • 10² = 100
• Square roots of square numbers are integers.

Cube Numbers

• Definition: A number which can be expressed as the product of an integer multiplied by itself twice.
• Examples:
  • 1³ = 1
  • 2³ = 8
  • 3³ = 27
  • 4³ = 64

Rational Numbers

• Definition: A number that can be written as a fraction (a/b) where a and b are integers and b ≠ 0.
• Examples:
  • 1/2, -3/4, 0.25 (since 0.25 = 1/4), 7 (as 7 = 7/1)
• Key Property: Decimal expansions either terminate or repeat.

Irrational Numbers

• Definition: Numbers that cannot be written as a simple fraction.
• Examples:
  • √2, π, √3, e
• Key Property: Decimal expansions are non-terminating and non-repeating.

Real Numbers

• Definition: All rational and irrational numbers combined.
• Examples: -5, 0, 2/3, √7, π

Reciprocals

• Definition: The reciprocal of a number x is 1/x.
• Examples:
  • Reciprocal of 5 is 1/5
  • Reciprocal of 1/3 is 3
  • Reciprocal of -4 is -1/4
• Note: 0 has no reciprocal (division by 0 is undefined).

Prime Factorisation

• Definition: Expressing a number as a product of its prime factors.
• Method:
  • Use a factor tree or repeated division by prime numbers.
• Example:
  • Express 72 as a product of prime factors:
    • 72 ÷ 2 = 36
    • 36 ÷ 2 = 18
    • 18 ÷ 2 = 9
    • 9 ÷ 3 = 3
    • 3 ÷ 3 = 1
    • So, 72 = 2³ × 3²

HCF (Highest Common Factor)

• Definition: The largest factor that two or more numbers have in common.
• Method:
  • Prime factorisation of both numbers.
  • Multiply the common prime factors (with the lowest powers).
• Example:
  • Find the HCF of 60 and 72.
    • 60 = 2² × 3 × 5
    • 72 = 2³ × 3²
    • HCF = 2² × 3 = 12

LCM (Lowest Common Multiple)

• Definition: The smallest number that is a multiple of two or more numbers.
• Method:
  • Prime factorisation of both numbers.
  • Multiply all prime factors (with the highest powers).
• Example:
  • Find the LCM of 60 and 72.
    • 60 = 2² × 3 × 5
    • 72 = 2³ × 3²
    • LCM = 2³ × 3² × 5 = 360

Conversions Between Numbers and Words

• Examples:
  • Six billion = 6,000,000,000
  • 10007 = “Ten thousand and seven”

Key Tips for Exam

• Always express your answer in its simplest form.
• Memorize primes up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29…
• When asked to show working, use factor trees or ladder method clearly.

Practice Questions

1. Express 84 as a product of its prime factors.
2. Find the HCF and LCM of 30 and 45.
3. Write 50003 in words.
4. State whether √5 is rational or irrational.
5. Identify all the types of numbers that 9 belongs to.