Calculating with negatives

At night the temperature drops to −5 °C, your bank account slips into the red after shopping, the lift goes down to level −2. Negative numbers appear wherever something can fall below a zero point. You can calculate with them just like with familiar numbers, you just need a solid feel for the signs.

Across zero

On the number line, things continue to the left of zero: −1, −2, −3 and so on. Plus means a jump to the right, minus a jump to the left. For 3 − 5 you start at 3 and jump 5 to the left: across zero down to −2. And −4 + 7 means: start at −4, jump 7 to the right, and you land at 3. If you're unsure, a stopover at zero helps: from −4 it's 4 steps to 0 first, then 3 more.

Brackets and double signs

When two signs meet, we use brackets: 5 − (−2) instead of 5 − −2. The rule: two identical signs in a row act like plus, two different ones like minus. 5 − (−2) = 5 + 2 = 7. That matches everyday life: if the bank cancels 2 euros of your debt (takes away a −2), you're 2 euros better off. And 5 + (−2) is simply 5 − 2 = 3. If a problem contains several pluses and minuses, resolve all the double signs first and then work step by step from left to right.

Minus times minus makes plus

For multiplication: identical signs give plus, different signs give minus. So (−3) · 4 = −12, but (−3) · (−4) = 12. Why? Look at the pattern: 3 · (−4) = −12, then 2 · (−4) = −8, then 1 · (−4) = −4, then 0 · (−4) = 0. With each step the result grows by 4. If the pattern continues, (−1) · (−4) must be 4 and (−2) · (−4) must be 8. The rule isn't arbitrary, it keeps the pattern alive.