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O Level · Mathematics 0580

Number Toolkit

Revise Number Toolkit for Mathematics 0580 (O Level) — revision notes and instant AI marking.

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1. Types of Numbers

Numbers can be classified into different types based on their properties. Understanding these classifications is fundamental to all the maths you'll do in this course.

📌 Integers & Natural Numbers

Integer: A whole number that can be positive, negative, or zero.

Examples: -3, -2, -1, 0, 1, 2, 3

Natural Number: The positive integers (also called counting numbers). Zero is NOT included.

Examples: 1, 2, 3, 4, 5, ...

📌 Multiples & Factors

Multiple: A number formed by multiplying a given integer by another positive integer.

12 is a multiple of 3 because 12 = 3 × 4. Even numbers (2, 4, 6, ...) are multiples of 2.

Factor: A value that divides into a number exactly with no remainder.

6 is a factor of 18 because 18 ÷ 6 = 3

💡 Tip: To find all factors of a number, use factor pairs. For 18: (1,18), (2,9), (3,6). So factors are: 1, 2, 3, 6, 9, 18

📌 Divisibility Tests (No Calculator Needed!)

Divisible by	Test	Example
2	Last digit is 0, 2, 4, 6, or 8	134 ✓ (ends in 4)
3	Sum of digits is divisible by 3	123 ✓ (1+2+3=6)
5	Last digit is 0 or 5	3025 ✓ (ends in 5)
10	Last digit is 0	2710 ✓ (ends in 0)

📌 Prime Numbers

Prime Number: A number with exactly two distinct factors: itself and 1.

First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Remember: 1 is NOT prime (only 1 factor). 2 is the ONLY even prime.

📌 Square & Cube Numbers

Square Number: The result of multiplying a number by itself.

1² = 1, 2² = 4, 3² = 9, ... 15² = 225

Memorise the first 15: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

Cube Number: The result of multiplying a number by itself twice.

1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125

Memorise the first 5: 1, 8, 27, 64, 125. Also remember: 10³ = 1000

📌 Square & Cube Roots

Square Root: The number that when multiplied by itself equals the value. √49 = 7 because 7 × 7 = 49

⚠️ Important: √25 = 5, but -√25 = -5. Sometimes both ±√25 = ±5 are needed.

Surd: The square root of a non-perfect square. √3 is a surd because 3 is not a perfect square.

Surds are IRRATIONAL numbers (we'll cover this next).

Cube Root: The number that when multiplied by itself twice equals the value. ³√64 = 4 because 4 × 4 × 4 = 64

📌 Reciprocals

Reciprocal:

Q1: What are the factors of 24?

Q2: Explain why 51 is not a prime number.

2. Irrational Numbers

📌 Rational Numbers

Rational Number:

📌 Irrational Numbers

Irrational Number: π (pi) √2, √3, √5

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Also in the full note

3. Negative Numbers
4. Mathematical Symbols
5. Order of Operations (BIDMAS/BODMAS)
6. Addition & Subtraction Methods
7. Multiplication Methods
8. Division Methods
9. Operations with Decimals
📋 Chapter Summary

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