Types of Number
O Level and IGCSE Mathematics Cheat Sheet – 1.1 Types of Number
| Type of Number | Definition | Examples | Key Points |
|---|---|---|---|
| Natural Numbers (ℕ) | Counting numbers starting from 1 | 1, 2, 3, … | No 0, fractions, or negatives |
| Integers (ℤ) | Whole numbers (positive, negative, zero) | -3, 0, 15 | No fractions or decimals |
| Prime Numbers | >1, only divisible by 1 and itself | 2, 3, 5, 7 | 2 is the only even prime |
| Square Numbers | Integer × same integer | 1, 4, 9, 16 | Root is an integer |
| Cube Numbers | Integer × itself × itself | 1, 8, 27, 64 | Root is an integer |
| Common Factors | Numbers dividing exactly into both | For 12 & 18: 1, 2, 3, 6 | Use factor listing or prime factors |
| Common Multiples | Multiples of both numbers | For 4 & 6: 12, 24, 36 | LCM is the smallest one |
| Rational Numbers (ℚ) | Can be written as p/q, q≠0 | 1/2, -3, 0.75 | Includes terminating/recurring decimals |
| Irrational Numbers | Cannot be written as p/q | π, √2, √3 | Non-terminating, non-repeating |
| Reciprocal | Number that gives 1 when multiplied | Reciprocal of 4 = 1/4 | Reciprocal of 2/3 = 3/2 |
Quick Conversions: Numbers ↔ Words
| Words | Numbers |
|---|---|
| Six billion | 6,000,000,000 |
| Ten thousand and seven | 10,007 |
Prime Factorisation
Example: 72
72 ÷ 2 = 36 → ÷ 2 = 18 → ÷ 2 = 9 → ÷ 3 = 3 → ÷ 3 = 1
Answer: 72 = 2³ × 3²
HCF (Highest Common Factor)
Example: 24 and 36
24 = 2³ × 3, 36 = 2² × 3²
HCF = 2² × 3 = 12
LCM (Lowest Common Multiple)
Example: 8 and 12
8 = 2³, 12 = 2² × 3
LCM = 2³ × 3 = 24
Exam Special Tips
- Always check if a number is prime before starting prime factorisation.
- For HCF, take lowest powers of common primes.
- For LCM, take highest powers of all primes present.
- If a decimal terminates or recurs, it’s rational.
Common Mistakes & Confusions
- Thinking 1 is prime — It is NOT prime.
- Forgetting that 0 is an integer.
- Confusing LCM with HCF — LCM is larger, HCF is smaller.
- Writing irrational numbers with rounded decimals in exams — write as π, √2, etc.
- Forgetting that 2 is the only even prime.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course