Why Two Negatives Make a Positive

Look at 3 × (−2) = −6, 2 × (−2) = −4, 1 × (−2) = −2, 0 × (−2) = 0.

Each line goes up by 2

The left number falls by one each time, and the answer rises by two each time.

So what comes next

The next line is (−1) × (−2). Continuing the pattern, the answer must be 2.

It is forced, not chosen

Any other answer would break a pattern that held for every line above it.

The distributive argument

Consider (−2) × (3 + (−3)). The bracket is zero, so the whole thing is zero. Distributing gives (−2)(3) + (−2)(−3) = −6 + (−2)(−3). For the total to be zero, (−2)(−3) must be +6. Nothing else works.

Reading it as "the opposite of"

Multiplying by −1 gives the opposite. So (−1) × (−3) is the opposite of −3, which is 3. Taking the opposite twice returns you where you started, which is exactly what two negatives do.

A story version

If you lose £3 a day, then three days ago — that is, −3 days from now — you had 3 × 3 = £9 more. Removing a loss, or looking backwards at one, is a gain. The stories are less airtight than the algebra but they help it stick.

This does not apply to addition

(−3) + (−4) = −7, not 7. The two-negatives-make-a-positive rule is about multiplication and division only. Carrying it into addition is a very common error, and it is worth saying plainly that the rule has a boundary.