Quick Summary
Integers are whole numbers (positive, negative, zero); natural numbers are the positive integers including 0.
Multiples come from multiplying; factors are values that divide exactly with no remainder.
A prime number has exactly two distinct factors — itself and 1. Note: 1 is NOT prime.
Square numbers = n×n. Cube numbers = n×n×n. Their inverses are square roots and cube roots.
A reciprocal is "1 divided by the number" — multiply a number by its reciprocal and you always get 1.
Rational numbers can be written as a fraction a/b; irrational numbers cannot (e.g. π, √2, √3, √5).
Negative number rules: same signs → positive result; different signs → negative result (for × and ÷).
BIDMAS/BODMAS tells you the order: Brackets → Indices → Division/Multiplication → Addition/Subtraction.
Column methods (with carrying/borrowing) handle addition, subtraction, and multiplication of large numbers.
For decimals: line up decimal points for +/−; for ×/÷, convert to integers, calculate, then undo the conversion.
1. Types of Number
Before you can operate on numbers, you need to know what "family" each number belongs to. Think of this section as learning the vocabulary of maths — examiners use these words constantly, and if you don't know what they mean, you can't answer the question even if you know the maths.
Integers and Natural Numbers
Integers are whole numbers — no fractions, no decimals. They can be positive, negative, or zero: ..., −3, −2, −1, 0, 1, 2, 3, ...
Natural numbers are the positive integers, thought of as "counting numbers": 0, 1, 2, 3, 4, ... Notice that 0 is included in this syllabus's definition — don't forget it!
Memory Trick
Integers = "In-teger" sounds like "whole" thing intact — no cutting into pieces (no fractions/decimals). Natural numbers are what a toddler counts on their fingers: 0, 1, 2, 3...
Multiples
A multiple of a number is what you get when you multiply it by an integer. 12 is a multiple of 3 because 12 ÷ 3 = 4 exactly (no remainder). A common multiple is shared between two or more numbers — e.g. 12 is a common multiple of both 4 and 6.
- Even numbers (2, 4, 6, 8, 10, ...) are multiples of 2.
- Odd numbers (1, 3, 5, 7, 9, ...) are not multiples of 2.
- Multiples can be algebraic too — the multiples of k are k, 2k, 3k, 4k, 5k, ...
Factors
A factor of a number divides it exactly, leaving no remainder. For example, 6 is a factor of 18 because 18 ÷ 6 = 3 exactly. Every integer greater than 1 has at least two factors: itself, and 1. A common factor is shared between numbers — 3 is a common factor of both 21 and 18.
Method
Finding factors using factor pairs:
Worked Example — Finding all factors of 18
Answer: Factors of 18 = {1, 2, 3, 6, 9, 18}