GCSE Maths Number revision

One number line orders everything. Negative numbers sit left of zero and get smaller as they move left, so −7 is less than −3.5 even though 7 is bigger than 3.5 — a favourite trap. Decimals compare place by place: 0.68 beats 0.7? No — seven tenths beats six tenths, so 0.7 > 0.68; the extra digit in 0.68 does not make it bigger. Fractions compare most safely with a common denominator: ⅔ = 8⁄12 beats 7⁄12, so ⅔ > 7⁄12.

The six symbols carry precise meanings: = (equal), ≠ (not equal), < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to). The open small end of < and > faces the bigger value. For mixed lists, convert every entry to the same type first — decimals to three places or a common denominator — then read off the order.

GCSE papers expect formal written methods: column addition and subtraction, short and long multiplication, and short division — even where a calculator would be allowed, non-calculator Paper 1 does not. Keep place value aligned in columns, and track the decimal point: 3.86 × 7 means 386 × 7 = 2702, then restore two decimal places → 27.02. Multiplying by 100 shifts every digit two places right of the point (3.08 × 100 = 308); dividing by 1000 shifts three places left.

With fractions and mixed numbers, convert mixed numbers to improper fractions first. Add and subtract with a common denominator: 2⅓ + 1½ = 7⁄3 + 3⁄2 = 14⁄6 + 9⁄6 = 23⁄6 = 3⅚. Multiply numerators and denominators across (simplify first where you can); divide by flipping the second fraction and multiplying. When subtraction would push a fraction negative — 4⅖ − 1¾ = 4 8⁄20 − 1 15⁄20 — borrow one whole as 20⁄20: 3 28⁄20 − 1 15⁄20 = 2 13⁄20.

Negative numbers obey sign rules worth memorising: adding a negative subtracts; two negatives multiplied or divided give a positive; a negative times a positive stays negative. So −4 + 9 = 5, then 5 − 6 = −1; and (−6)² = 36 but −6² = −36, because in −6² the power acts on 6 only.

The priority of operations runs: brackets → indices (powers and roots) → multiplication and division (left to right) → addition and subtraction (left to right). So 3 + 4 × 2² = 3 + 4 × 16 = 3 + 64? No — 2² = 4 first, then 4 × 4 = 16, then 3 + 16 = 19. The classic error is adding before multiplying: (3 + 4) × 2² would instead give 7 × 4 = 28, which is exactly why the bracket position matters.

A reciprocal multiplies a number to 1: the reciprocal of ⅖ is 5⁄2, of 4 is ¼, of 1¼ = 5⁄4 is ⅘. Reciprocals turn division into multiplication and reappear in algebraic fractions later in the course. Inverse operations undo each other — subtraction undoes addition, division undoes multiplication, square roots undo squares (for positive numbers) — and they are the tool for checking: 27.02 ÷ 7 = 3.86 confirms the multiplication above.

A factor divides a number exactly; a multiple is the times table; a prime has exactly two factors. Every whole number greater than 1 is either prime or a product of primes, and that product is unique apart from the order — this is why prime factorisation is the reliable route to HCF and LCM. To test whether a number such as 53 is prime, try dividing by the primes up to its square root (2, 3, 5, 7 for √53 ≈ 7.3): if none divide, it is prime.

Product notation writes repeated factors as powers: 360 = 2 × 2 × 2 × 3² × 5 = 2³ × 3² × 5. Given two numbers in this form, the HCF multiplies the shared primes to their lowest powers, and the LCM multiplies all primes to their highest powers. For 36 = 2² × 3² and 90 = 2 × 3² × 5: HCF = 2 × 3² = 18, LCM = 2² × 3² × 5 = 180. A sanity check always works: HCF × LCM = 18 × 180 = 3240 = 36 × 90.

Systematic listing means organising so nothing is missed — fix the first item and pair it with each second item in order. The product rule for counting is the shortcut: if one decision has *m* outcomes and an independent decision has *n*, the pair has *m × n* outcomes. Three soups with five sandwiches give 15 lunches, not 8. For a four-digit PIN from digits 0–9 with repetition allowed there are 10 × 10 × 10 × 10 = 10 000 codes; if digits may not repeat, the choices shrink: 10 × 9 × 8 × 7 = 5040.

Powers count repeated multiplication: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Learn the small power tables: powers of 2 up to 2¹⁰, cubes to 10³ = 1000, 3⁴ = 81, 4⁴ = 256, 5⁴ = 625. Roots reverse them: √225 = 15, ∛64 = 4. Roots of non-square numbers are estimated by bracketing between known squares or cubes — 72 sits between 8² = 64 and 9² = 81, so √72 is between 8 and 9, closer to 9 because 72 is more than halfway.