# 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:

• What is the goal of the proof?
• What is the starting equation?
• How do you get the final result?

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:

• Numbers (i.e. a = 5, b = 1/2)
• Constants
• Variables
• Expressions [i.e. a = (y2 + 6), b = (h − w + z)]

## 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

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