Why Is Negative Times Negative Positive? (2 Proofs)

Step by step

1. Negative × negative = positive. But why?
   Negative times negative is positive. But why?
2. Explained why? Or just memorized?
   Did a teacher ever explain why? Or did you just memorize the rule?
3. Pattern: 3, 2, 1, 0 times (−2) → −6, −4, −2, 0
   Way one: follow the pattern. Three times negative two is negative six. Two times negative two, negative four. One times, negative two. Zero times, zero.
4. Each step down: +2 → (−1)(−2) = 2 · 4 · 6
   Each step down, the answer goes up by two. Keep going: negative one times negative two is two. Then four. Then six!
5. Way 2: (−3)(2) + (−3)(−2)
   Way two, the proof. Take negative three times two, plus negative three times negative two.
6. = (−3)(2 + (−2)) = (−3)(0) = 0
   Pull out the negative three. Two plus negative two is zero, and anything times zero is zero.
7. −6 + (−3)(−2) = 0 → (−3)(−2) = +6
   So negative six, plus our mystery product, makes zero. The only number that does that is positive six!
8. Take away 3 debts of $2 → you're $6 richer
   Picture it with money. You owe three debts of two dollars each. Take all three away, and you're six dollars richer.
9. 628 AD · Brahmagupta: negative × negative = positive
   This rule is old. In the year six twenty-eight, the Indian mathematician Brahmagupta wrote it down: the product of two negatives is positive.
10. Where you'll see it: (x − 3)² → (−3)(−3) = +9
    Where you'll see it: all over algebra. Expand x minus three, squared, and negative three times negative three gives plus nine.
11. On a test: distributive property → 0 → product = 6
    On a test: by the distributive property, negative three times two plus negative three times negative two is negative three times zero: zero. Negative three times two is negative six. So negative six plus the product is zero, and the product is six.