Classifying numbers
Numbers can be grouped into natural numbers (1, 2, 3, …), integers (…, −2, −1, 0, 1, 2, …), rational numbers (any number that can be written as a fraction a/b) and irrational numbers such as √2 or π, whose decimals never terminate or repeat.
A prime number has exactly two factors, 1 and itself; 2 is the only even prime. A square number is the result of multiplying an integer by itself, e.g. 49 = 7². The factors of a number divide into it with no remainder, while its multiples are obtained by multiplying it by 1, 2, 3 and so on.
Key idea
HCF = product of the common prime factors; LCM = product of the highest power of every prime that appears.
HCF and LCM
Write each number as a product of primes, then compare. This is the reliable exam method.
Worked example
Find the HCF and LCM of 24 and 36.
24 = 2³ × 3, 36 = 2² × 3².
HCF = 2² × 3 = 12.
LCM = 2³ × 3² = 8 × 9 = 72.
Sets
A set is a collection of elements. Use ∪ for union (all elements in either set), ∩ for intersection (elements in both) and ∈ for "is an element of". If A = {1, 2, 3, 4} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5} and A ∩ B = {3, 4}. A set A is a subset of B, written A ⊂ B, if every element of A is also in B.
Remember
- 1 is not prime.
- A set with n elements has 2ⁿ subsets.
- The empty set is written ∅ or { }.