Sometimes we need numbers below zero — a temperature of −5°C on a frosty morning, a bank balance overdrawn by −£20, or pressing "−3" in a lift to go down three floors. These are negative numbers, and once you can picture them, the rules for adding, subtracting, multiplying and dividing all fall into place.
Why this matters: negatives are everywhere in real life — money you owe, depths below sea level, golf scores under par, temperatures in winter. They also underpin nearly all the algebra to come: solving equations, working out gradients and rearranging formulae all rely on being confident with signs. Get comfortable here and the rest of maths gets a lot smoother.
← smaller · negatives on the left, positives on the right · bigger →
The number line — your map of the negatives
Picture a line with 0 in the middle. Positive numbers march off to the right, negative numbers stretch out to the left. Think of it like a thermometer laid on its side, or the floors of a building with the ground floor at 0 and basements below.
The further left you go, the smaller the number — so −8 is actually less than −3 (colder, or deeper in debt). To put numbers in order, just read them off left to right: −7, −2, 0, 3, 5.
Common mistake: −8 is not "bigger" than −3. With negatives, the bigger-looking digit is the smaller number. A £8 overdraft is worse than a £3 one.
Adding and subtracting — walk the line
Treat the number line like a pavement you walk along: + moves you right, − moves you left. Start on the first number, then take the steps.
−6 + 10: start at −6, move 10 places to the right → you land on 4.
Start at −6, step 10 to the right, land on 4.
3 − 10: start at 3, move 10 places left → you pass through 0 and land on −7.
Two signs next to each other
Sometimes a plus or minus bumps up against a negative number, giving you two signs in a row. There's a neat rule for tidying them up:
Same signs → plus, different signs → minus. So − (−3) becomes + 3, and + (−3) becomes − 3.
Matching signs join to +, mismatched signs give −.
5 − (−3): the two minuses make a plus → 5 + 3 = 8. Taking away a debt of £3 leaves you £3 better off.
Multiplying and dividing — just check the signs
For × and ÷ you don't need the number line at all. Work out the calculation as if both numbers were positive, then decide the sign with one rule:
Same signs → positive. Different signs → negative.
Calculation
Signs
Answer
−4 × −3
same (− and −)
12
−4 × 3
different
−12
−20 ÷ −4
same
5
−20 ÷ 4
different
−5
Same signs land on the purple diagonal (positive); mixed signs are negative.
Why two negatives make a positive: taking away a debt leaves you better off. Removing −3 from a total has the same effect as adding 3 — so "the opposite of the opposite" brings you back to positive.
In a nutshell: picture the number line for + and − (right for plus, left for minus); for × and ÷ just check the signs — matching signs give a positive, mismatched give a negative.