Multiplying & Dividing Integers

The rule is short. Two negatives multiply to a positive. One negative gives a negative. Division follows exactly the same pattern.

For longer products, just count the negative signs. (−2)(−3)(−4) has three negatives, an odd number, so the answer is negative: −24.

Why does negative times negative come out positive? Look at a pattern: −3 × 3 = −9, −3 × 2 = −6, −3 × 1 = −3, −3 × 0 = 0. Each time the second factor drops by 1, the answer rises by 3. Keep going and −3 × −1 has to be 3.

The positive result is forced by the distributive property, not chosen for convenience. Consider:

0 = (−3)·0 = (−3)(2 + (−2)) = (−3)(2) + (−3)(−2) = −6 + (−3)(−2)

The only value of (−3)(−2) that makes that sum zero is 6. Any other convention would break distributivity, and distributivity is what makes the whole arithmetic system consistent.

In general, for a product of n nonzero factors the sign is (−1)^k where k is the number of negative factors, so parity of k decides the sign. Division inherits the rule because a ÷ b = a · b⁻¹ and the reciprocal of a negative is negative.