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# Proof that Product of Two Negative Numbers is Positive

by Ron Kurtus (revised 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 = (y**,^{2}+ 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

## Resources and references

### Websites

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

### Books

(Notice: The *School for Champions* may earn commissions from book purchases)

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## Proof that Product of Two Negative Numbers is Positive