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Proof that Product of Two Negative Numbers is Positive

by Ron Kurtus (revised 28 November 2014)

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.



Goal

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

(−a)(−b) = ab

where a and b can be:

Proof

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)

Thus

x = ab + (−a)[b + (−b)]

Since b + (−b) = 0

x = ab + (−a)(0)

Thus

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)

Thus

x = (−a)(−b)

Result

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

Summary

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

Websites

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

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Top-rated books on Algebra


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