Negative numbers and order of operations
Maths · Key Stage 3 · The School
Negatives on a number line
Think of a number line running both ways from zero. ADDING moves right, SUBTRACTING moves left, whatever sign you start from. So 3 − 7 goes seven left from three and lands on −4. Subtracting a negative moves right — 5 − (−3) is 8 — because removing a debt leaves you better off, which is the version most people find sticks.
Signs in multiplication
Two numbers with the SAME sign multiply to a positive; two with DIFFERENT signs give a negative. So −4 × −3 = 12 and −4 × 3 = −12. Division follows exactly the same rule. This is one of the few places in school maths where memorising the rule is genuinely the efficient route, provided you can also check it against a pattern.
Order of operations
Without a shared order, 2 + 3 × 4 could be 20 or 14. Mathematicians agreed one: brackets first, then indices, then multiplication and division left to right, then addition and subtraction left to right. It is a CONVENTION, not a discovery — but a universal one, which is why a calculator and a person following it always agree.
First: 5 − (−3). Subtracting a negative moves you RIGHT on the number line, so 5 − (−3) = 8. Second: −5 + (−3). Adding a negative moves you LEFT, so the answer is −8, not 8 — the 'two minuses make a plus' rule does not apply to addition. Now one with order of operations: 2 + 3 × 4² = 2 + 3 × 16 = 2 + 48 = 50. Indices before multiplication, multiplication before addition; doing it left to right gives 400.