Expert: Josh - 1/3/2006

Question
My son is in 5th grade. He asked me a very basic question why is (-3)*(-3)=+9. I know this is a rule but no logical explanation to convinve my child that the rule works on logic.

Any help will be really appreciated.

Thanks and regards,
Ash


Answer
Hi Ash,

An excellent question from your son.

Negative numbers were in fact not used by some ancient civilizations, because they have limited practical value. For instance, negative numbers cannot represent physical measurements (consider distance) in space.

Today, we can appreciate the need for negative numbers. Just think about how we measure temperature. In fact, the choice of the Celsius temperature scale is also somewhat arbitrary. The significance of zero Celsius is that it happens to coincide with the freezing point of water. Once a reference point is established, it becomes a convention. In colder regions of the globe, it is only natural to contemplate a representation for colder temperatures relative to the established temperature scale. A negative extension of natural (positive) numbers seems like an obvious choice, in keeping with a uniform temperature scale. So, that's just a bit of motivation. The story may be summarized as follows:

* There is no intuitive explanation as to why the product of two negative numbers should yield a positive number.
* From a historical perspective, established practice becomes a standard convention that people simply adhere to.
* Fundamentally, the foundation of mathematics builds on a common set of rules that everyone agrees on -- these are called the twelve axioms. Proofs of mathematical facts rely on these so called "ground truths" (the rules of the game). e.g., (-1)*(-1)=1. Then, -2*-1=2*(-1)*(-1)=2 follows from it. A definition needs not follow from something else. Like, why is an apple called an "apple"?

* We have come too far to question whether our existing number system was a good choice and why we have such definitions (or rules). One could argue that it is a "good choice" if it produces "useful outcomes" and it leads us somewhere. What makes "(-1)*(-1)=1" a "nice" definition is symmetry. We have:
1*1=1 POSITIVE times POSITIVE gives a POSITIVE quantity.
-1*1=1 NEGATIVE times POSITIVE gives a NEGATIVE quantity.
1*(-1)=1 POSITIVE times NEGATIVE gives a NEGATIVE quantity.
(-1)*(-1)=1 NEGATIVE times NEGATIVE gives a POSITIVE quantity.

Josh