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Proof that Product of Two Negative Numbers is Positive

by Ron Kurtus (updated 18 January 2022)

When you multiply a negative number by another negative number, the result is a positive number. This rule is not obvious and proving it is not straightforward.

However, here is a clever way to prove the rule by starting with an equation and factoring out terms.

Questions you may have include:

This lesson will answer those questions.


Prove that the product of two negative numbers or terms is positive:

(−a)(−b) = ab

where a and b can be:


A clever way to prove that (−a)(−b) = ab is to consider the equation:

x = ab + (−a)(b) + (−a)(−b)

You want use this equation to show that x = ab and x = (−a)(−b).

Factor out −a

First, factor out −a from the expression (−a)(b) + (−a)(−b):

x = ab +(−a)(b) + (−a)(−b)


x = ab + (−a)[b + (−b)]

Since b + (−b) = 0

x = ab + (−a)(0)


x = ab

Factor out b

Now, with the original equation, factor out b from the expression ab + (−a)(b):

x = ab + (−a)(b) + (−a)(−b)

x = b[a + (−a)] + (−a)(−b)

x = b(0) + (−a)(−b)


x = (−a)(−b)


Since x = ab and x = (−a)(−b):

(−a)(−b) = ab

This can be extended to any even amount of negative numbers by factoring out in steps:

(−a)(−b)(−c)(−d) = ab(−c)(−d) = abcd


This clever method proves that (−a)(−b) = ab.

The fact that the product of two negative numbers, terms, or expressions is positive can be extended to any even number of negative items.

Dream of the impossible

Resources and references

Ron Kurtus' Credentials


Why is a negative times a negative a positive? - Ask Dr. Math - FAQ

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