Why a negative times a negative is positive

Sooner or later, in Class 7 or Class 8, someone asks: "Why is minus times minus equal to plus?" The usual answer is "it is a rule, just learn it". That works until an exam asks something a little different. A rule you understand is much harder to forget, so here are three ways to see why it must be true.

Way 1: keep the pattern going

Multiply −3 by a list of numbers that goes down by 1 each time, and watch the answers:

(−3) × 3 = −9
(−3) × 2 = −6
(−3) × 1 = −3
(−3) × 0 = 0
(−3) × (−1) = ?
(−3) × (−2) = ?

Look at the first four answers: −9, −6, −3, 0. Each one is 3 more than the one before. If the pattern is to carry on, the next answers have to be 3 and then 6. So (−3) × (−1) = 3 and (−3) × (−2) = 6. A pattern that suddenly changed its mind half way would not be a pattern.

Way 2: the distributive law forces it

In algebra you use a(b + c) = ab + ac all the time, and you want it to work for negative numbers too. Let us see what that means for (−1) × (−1). We know that −1 + 1 = 0. Multiply both sides by −1:

(−1) × (−1 + 1) = (−1) × 0 = 0
Open the bracket: (−1) × (−1) + (−1) × 1 = 0
We know (−1) × 1 = −1, so (−1) × (−1) + (−1) = 0
Add 1 to both sides: (−1) × (−1) = 1

If (−1) × (−1) were anything other than 1, the distributive law, which you rely on in every algebra question, would break. The rule is not a custom. It is what keeps the rest of maths working.

Way 3: going backwards in time

Think of a train moving along a straight track. Right is positive and left is negative. Suppose it moves at 3 km per minute to the left, so its velocity is −3. Now it is at the station. Where was it 2 minutes ago? Two minutes ago is −2 minutes. Its position then was (−3) × (−2) = 6 km, and that is 6 km to the right of the station. It was on the right, and it has moved left since, which is exactly what we would expect. Backwards in time, and backwards in direction, gives forwards.

The rules, in one line

Same signs, multiply or divide: the answer is positive. (−4) × (−5) = 20 and (−12) ÷ (−3) = 4.
Different signs: the answer is negative. (−4) × 5 = −20 and 12 ÷ (−3) = −4.

Two mistakes to avoid

(−3)² is (−3) × (−3) = 9, but −3² means −(3 × 3) = −9. The brackets decide what is squared.
The sign rule is for multiplying and dividing only. For adding, −2 + (−3) = −5. A minus next to a minus does not turn into a plus when you add.

Try it yourself

Work out (−4) × (−5) and (−4) × 5. Which is bigger?
Which is larger, (−6)² or −6²? Why?
Use Way 2 to show that (−2) × (−3) = 6. Start from −3 + 3 = 0 and multiply both sides by −2.

When a rule feels strange, do not just memorise it. Ask what would go wrong if it were different. Integers and rational numbers are part of the Class 8 syllabus, and you can revise them with the free Class 8 notes and tests, or build a fresh test on them with the Practice Generator.