Why a Negative Times a Negative Is Positive

3 × (−2) is three jumps of −2, landing on −6. A positive times a negative is negative.

Now run a pattern downward

3 × (−2) = −6, 2 × (−2) = −4, 1 × (−2) = −2, 0 × (−2) = 0. Each step adds 2.

Keep going below zero

The next line is (−1) × (−2). Adding 2 again gives 2. So it must be positive.

Why it has to be this way

Any other answer would break the pattern, and arithmetic has to stay consistent.

The pattern demands it

Look at 3 × 2 = 6, 2 × 2 = 4, 1 × 2 = 2, 0 × 2 = 0, then −1 × 2 = −2 and −2 × 2 = −4. Each step down the first factor drops the product by 2. Continuing the pattern past zero forces the negative results.

Now run the pattern again

Take 3 × (−2) = −6, 2 × (−2) = −4, 1 × (−2) = −2, 0 × (−2) = 0. Each step up is +2. Carry on: −1 × (−2) = 2 and −2 × (−2) = 4. Negative times negative must be positive, or the pattern breaks.

It is forced, not chosen

Mathematicians did not decide this to be awkward. Any other answer would break the distributive property, and everything built on it would collapse. The rule is the only one consistent with the arithmetic that already existed.

The four cases

Positive times positive is positive. Positive times negative is negative. Negative times positive is negative. Negative times negative is positive. Division follows the identical pattern, since dividing is multiplying by a reciprocal.