Nib asked
Why is a negative times a negative a positive?
A negative times a negative is positive because arithmetic's other rules force it: it is the only answer that keeps the multiplication pattern and the distributive law working.
6 steps, drawn by Nib. Read it here, or watch him draw it. Updated .
1the pattern
The rule 'a negative times a negative is a positive' is not an arbitrary decree. It is the only choice that keeps the rest of arithmetic working. Start with a pattern: 3 × (−2) = −6, 2 × (−2) = −4, 1 × (−2) = −2, 0 × (−2) = 0. Each time the first number drops by one, the answer goes up by 2.
It is not a memorized rule: it is the answer that keeps the pattern going.
a common mix-upA negative times a negative being positive is not an arbitrary rule or a language trick like a double negative: arithmetic breaks without it.
2keep going
Carry on down the same pattern: (−1) × (−2) = 2, (−2) × (−2) = 4 and (−3) × (−2) = 6. A pattern alone is only a hint, though, so the next two steps show why nothing else can work.
The pattern says negative times negative is positive.
3the proof
Suppose the distributive law, a(b + c) = ab + ac, must keep working with negative numbers. Then (−1) × ((−2) + 2) = (−1) × 0 = 0, and also (−1)(−2) + (−1)(2) = 0. Since (−1)(2) = −2, the term (−1)(−2) has to be the number that cancels −2: it must be 2. Any other answer would break the distributive law.
Only a positive answer keeps the distributive law true.
If (−1)(−2) were −2 instead, what would break?
Show the answer
(−1) × ((−2) + 2) would equal −4, not (−1) × 0 = 0, so the distributive law would fail.
4flip twice
On the number line, multiplying by −1 flips a point to the other side of zero. Flip 2 and you land on −2. Flip again and you are back on 2. So (−1) × (−1) × 2 = 2, and since (−1) × 2 = −2, that says (−1) × (−2) = 2.
Multiplying by minus one flips across zero; flipping twice gets you home.
5walking back
Say you walk left along a line at −2 meters per second and are at position 0 now. Your position at time t seconds is (−2) × t. Three seconds ago, t = −3, so your position was (−2) × (−3) = +6 meters: six meters to the right, which is exactly where you must have started to walk left to 0 in 3 seconds.
Walking left, three seconds ago you were six meters to the right.
6the keepsake
So the whole idea fits on one board: the pattern, the distributive law and the flip all demand it. Negative times negative is positive, forced, not a trick.
A negative times a negative is positive because every other rule of arithmetic would break if it were anything else.