Why Is a Negative Times a Negative Positive?
The French novelist Stendhal loved mathematics as a boy. In his autobiography, The Life of Henry Brulard, he describes how that love was shaken when none of his teachers could explain why minus times minus equals plus. One answer he got, as he recalls it, amounted to "it's the rule."
"It's the rule" is a terrible explanation. There's a much better one. In fact, there are several, and the strongest shows that (−1) × (−1) = 1 isn't a choice at all. It's forced by rules we already accept.
The Rule Isn't a Convention
It's tempting to think mathematicians decided that negative × negative is positive, and could have decided otherwise. They couldn't.
If you want the distributive law, a(b + c) = ab + ac, to keep working with negative numbers, as it does with positive ones, then (−1) × (−1) must equal +1. Any other answer would break arithmetic.
Explanation 1: Follow the Pattern
Start with a multiplication table you trust, and keep going:
3 × (−2) = −6
2 × (−2) = −4
1 × (−2) = −2
0 × (−2) = 0
Each time the first number goes down by 1, the answer goes up by 2. Continue:
(−1) × (−2) = 2
(−2) × (−2) = 4
(−3) × (−2) = 6
The pattern only continues smoothly if negative × negative is positive. Check the positive rows on the multiplication tables.
Explanation 2: A Short Proof
Let's use just three facts that everyone accepts:
- Any number times 0 is 0
- (−1) + 1 = 0
- The distributive law: a(b + c) = ab + ac
Start with:
(−1) × 0 = 0
Replace 0 with (−1 + 1):
(−1) × (−1 + 1) = 0
Distribute:
(−1) × (−1) + (−1) × 1 = 0
(−1) × (−1) + (−1) = 0
What number plus (−1) gives 0? Only 1:
(−1) × (−1) = 1
That's a complete proof. And for any numbers, (−a) × (−b) = (−1)(−1) × ab = ab. The rule follows from the laws in the algebra formulas.
Explanation 3: Removing Debts
Think of positive numbers as money you have and negative numbers as debts. Multiplying by a positive number means "add this many times." Multiplying by a negative number means "take away this many times."
- 3 × (−$5): add three $5 debts → you're $15 worse off (−15)
- (−3) × $5: take away three $5 credits → you're $15 worse off (−15)
- (−3) × (−$5): take away three $5 debts → you're $15 better off (+15)
Having a debt canceled is good news. Removing a negative is a positive.
Explanation 4: Turning Around on the Number Line
Picture the number line. Multiplying by a positive number stretches a number but keeps its direction. Multiplying by −1 flips it to the other side of zero, a 180° turn.
- 3 × 2 = 6: stretch, stay pointing right
- (−1) × 6 = −6: turn around, now pointing left
- (−1) × (−6) = 6: turn around again, pointing right
Two half-turns make a full turn. You end up facing the way you started. This view leads directly to imaginary numbers, where i is a quarter-turn, and i × i = −1 is two quarter-turns. See The Strange World of Imaginary Numbers.
An Insider Reference: Brahmagupta and the Rhyme
The rule is ancient. In 628 AD, the Indian mathematician Brahmagupta stated in his Brāhmasphuṭasiddhānta:
The product of two debts is a fortune.
Brahmagupta treated positive numbers as fortunes and negative numbers as debts, and set out rules for all combinations. European mathematicians took more than a thousand years to accept negative numbers fully, and the multiplication rule was a large part of their discomfort.
A modern rhyme, often attributed to the poet W. H. Auden, captures how the rule is too often taught:
Minus times minus equals plus, The reason for this we need not discuss.
Stendhal would have hated it. The reason is worth discussing, and it takes only four lines of algebra.
A Quick Check With Real Quantities
The rule appears in physics and everyday life:
- Temperature: if temperature is falling 2°C per hour, then 3 hours ago (−3 hours) it was (−3) × (−2) = 6°C warmer than now. Try the temperature converter.
- Video: a car driving backward (negative velocity), played in reverse (negative time), appears to move forward.
Two Concepts Worth Knowing
Distributive Law
The distributive law says a(b + c) = ab + ac. It links addition and multiplication, and it's the reason negative × negative must be positive.
Additive Inverse
The additive inverse of a is −a, the number that adds with a to give 0. Proving (−1)(−1) = 1 comes down to showing it's the additive inverse of −1.
Quick Answer: Why Is Negative Times Negative Positive?
Because the distributive law requires it. Since (−1) × (−1 + 1) = (−1) × 0 = 0, distributing gives (−1)(−1) + (−1) = 0, so (−1)(−1) must equal 1. Intuitively, multiplying by a negative reverses direction, and reversing twice returns you to the positive direction.
Try Them Yourself
- Multiplication Tables: extend the pattern into negatives
- Multiplication Flash Cards: practice sign rules quickly
- Algebra Formulas: the distributive law
- Temperature Converter: negative numbers in real measurement
- Polar to Rectangular Converter: multiplication by −1 as a 180° turn
- Why Negative Numbers Exist: where negatives came from
Explain the rule to someone using debts, then using the pattern. Whichever one makes them nod is the explanation Stendhal's teachers should have used.