Foundations

Types of number

Before anything can be calculated, it has to be classified. The families of number are nested, not separate: every natural number is an integer, every integer is rational, and the rationals together with the irrationals make up the real numbers — nothing else does.

The families of number

Every integer, every terminating decimal and every recurring decimal is rational. An irrational number cannot be written as a fraction: √2 and π run on for ever without repeating. Within the reals sit the square numbers (1, 4, 9, …), the cube numbers (1, 8, 27, …) and the primes. The reciprocal of a number is 1 divided by it: the reciprocal of 1¼, or 5/4, is 4/5.

Prime factorisation

Every integer greater than 1 is a product of primes in exactly one way. Divide repeatedly by the smallest prime that goes in: this gives 72 = 2³ × 3², and no other primes multiply to 72.

Highest common factor and lowest common multiple

In prime-factor form the HCF and LCM read straight off. The HCF takes the lowest power of each shared prime; the LCM takes the highest power of every prime present. At Extended the same rule reaches algebraic terms, treating each letter as another prime.

Worked example: 70 = 2 × 5 × 7 and 112 = 2⁴ × 7. The HCF takes the lowest power of each shared prime — both share one 2 and one 7 — so HCF = 2 × 7 = 14. The LCM of 70x⁴y² and 112x³y⁵ takes the highest power of everything present: 2⁴ × 5 × 7 × x⁴y⁵ = 560x⁴y⁵. The HCF keeps only what both have; the LCM keeps everything either of them needs.

Sets and Venn diagrams

Set language is a way of being exact about which numbers you mean. The notation is short, and the marks are given for using it precisely rather than describing it in words. The number types of the previous section are the elements these sets are usually built from.

Counting elements

For any two sets, n(A∪B) = n(A) + n(B) − n(A∩B): the overlap is subtracted because it is otherwise counted twice.

ExtendedWorked example — reading a three-set Venn

Take ℰ = {integers 1 to 18}, A = {factors of 12} = {1, 2, 3, 4, 6, 12}, B = {even numbers} = {2, 4, 6, 8, 10, 12, 14, 16, 18}, C = {multiples of 3}. Then A∩B = {2, 4, 6, 12}, so n(A∪B) = 6 + 9 − 4 = 11. The centre region A∩B∩C is the numbers that are factors of 12, even and multiples of 3: {6, 12}. C is not a subset of B, because 3 ∈ C but 3 ∉ B — one counter-example is enough to disprove a subset claim.

Powers and roots

A power is repeated multiplication; a root undoes it. Most of the marks here are lost not to the method but to the recall — the values below are assumed knowledge.

Squares, cubes and their roots

Squaring an integer gives a square number, and the square root reverses it: √49 = 7. Cubing gives a cube number, undone by the cube root: ∛64 = 4. Every positive number has two square roots, one positive and one negative, but √ means the positive one unless told otherwise.

Known cold: squares 1–15 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225; cubes are 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 10³ = 1000.

Higher powers and roots

The same idea extends: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32, and the fifth root of 32 is 2. The small raised number is the index, and it counts how many times the base is used as a factor — not what the base is multiplied by. Writing 2⁴ as 8 is the single most common slip on this objective.

Worked example (no calculator): √196 + ∛64 − 3³ = 14 + 4 − 27 = −9. The roots and powers are all evaluated before the additions and subtractions.

Fractions, decimals and recurring decimals

A fraction, a decimal and a percentage are three ways of writing one value. Fluent conversion between them is assumed by almost every question in the rest of the syllabus.

Equivalent forms

To convert a fraction to a decimal, divide the numerator by the denominator; to convert a decimal to a percentage, multiply by 100. So 3/8 = 0.375 = 37.5%. An improper fraction has a numerator at least as large as its denominator, and converts to a mixed number by division: 22/5 = 4⅖, since 5 goes into 22 four times with 2 left over.

ExtendedRecurring decimals

Every recurring decimal can be turned back into an exact fraction. Call the decimal x, multiply by whichever power of ten shifts one whole repeating block to the left of the point, then subtract to destroy the repeating tail. For 0.4545…: let x = 0.4545…, so 100x = 45.4545…; subtracting, 99x = 45, so x = 45/99 = 5/11.

The power of ten is chosen by the length of the repeating block, not the number of digits shown. For 0.1777…, one digit repeats but one digit does not, so 100x − 10x is needed, giving 90x = 16 and x = 8/45.

Ordering

Ordering is rarely hard mathematics. It is a question about whether you converted carefully — and whether you noticed which symbol the boundary needs.

The symbols

Five symbols carry the whole objective: = equal to, ≠ not equal to, > greater than, < less than, and the inclusive pair ⩾ (greater than or equal to) and ⩽ (less than or equal to). The strict symbols exclude the endpoint; the inclusive ones allow it. On a number line the distinction is drawn, not written.

Comparing mixed forms

A comparison can only be made once the quantities are written the same way. Convert every value to a decimal, compare, then write the symbol the original question asked for. Worked example: 3/8 vs 0.38 — 3 ÷ 8 = 0.375 < 0.38, so 3/8 < 0.38. And 7/20 vs 0.35 — 7 ÷ 20 = 0.35 exactly, so they are equal.

The four operations

This is the objective the non-calculator paper is built on. The mathematics is elementary; the marks are decided by order, by signs, and by whether the working is written down.

Order of operations

Work through brackets, then indices, then multiplication and division left to right, then addition and subtraction left to right. So 3 + 2 × (−4)² is not 80: the bracket squares to 16, the multiplication gives 32, and only then does the 3 go on, for 35.

Negative numbers

Subtracting a negative adds: 5 − (−3) = 8. Two negatives multiplied or divided give a positive; one negative gives a negative. The commonest slip is applying this to a fraction — −2/3 + 7/6 is 3/6 = 1/2, not a negative answer, because the positive term is the larger of the two.

Fractions without a calculator

Convert mixed numbers to improper fractions first, then put both over a common denominator, then combine. Convert back to a mixed number only if the question asks. Worked example: 4⅖ − 1¾ = 22/5 − 7/4 = 88/20 − 35/20 = 53/20 = 2 13/20, already in simplest form.