Why Does Multiplying Two Negatives Give a Positive?

“Two debts make a fortune” may be memorable, but it does not prove that two negatives multiply to a positive. The reason is that when multiplication is extended to negative numbers, we want the distributive law we already use to keep working. That leaves no other result.

Consider negative two multiplied by zero. Positive three and negative three add to zero, so negative two times their sum must be zero. Expanding by the distributive law, negative two times positive three is negative six. The other product, negative two times negative three, must therefore be positive six for the terms to add to zero. This does not quietly assume the very rule at issue: negative two times positive three is three negative twos added together, and addition tells us what its opposite must be.

This establishes consistency between mathematical rules, not a fact about two bad things becoming good in nature. Turning around twice is a useful memory aid, but a metaphor is not a proof. When defining multiplication of signed rational numbers so that it agrees with existing integer arithmetic and distributivity, the positive answer follows. Whether a real loss can be reversed depends on the actual circumstances.