Why does multiplying two negative numbers equal a positive number?
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The identity property of multiplication: Any number multiplied by 1 is equal to itself. If you have one $20 bill, you have $20. 20 x 1 = 20.
The identity property also applies to negative numbers: -5 x 1 = -5. If you have one negative five, you have one negative five.
Nonzero real numbers are either negative or positive; there’s no other directions you can go on the number line. If a nonzero real number is not positive, it must be negative. Therefore, if a negative times a negative does not equal a positive, it must equal a negative.
But if you let a negative times a negative equal a negative, that would mean -5 x -1 would equal -5 again. You could then substitute that into the original equation we had in the second step and that would give you -5 x 1 = -5 x -1. Divide both sides by -5 and now you have -1 = 1, which cannot be true.
Therefore, by contradiction, a negative times a negative cannot be a negative, and must be a positive.
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