Negative times a Negative is a Positive

Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

— Paul Halmos on how to study mathematics

Figure 1: My Favorite Advertizing Campaign.

Figure 1: A Math Poster from My
Favorite Advertising Campaign

On my team, I work hard to ensure that we have a non-threatening environment for questions – any questions. In fact, I often ask very basic questions in meetings so that I can make sure that I understand all the nuances of a situation. You would be amazed how often I learn things from asking questions so basic that you would think asking them would not be necessary.

Because of this approach, I frequently have non-technical people come to me and ask questions about many things, mainly remodeling and basic mathematics. I will cover the remodeling questions in other posts.

The other day I had a person come to me with the following basic mathematics question:

Why does a -1 times -1 equal a 1?

This is actually an interesting question and I last dealt with it when my kids were in school. From a mathematical standpoint, the best answer is that -1*-1 = 1 to ensure that the distributive property works, which I illustrate in Equation 1.

Eq. 1 \displaystyle 0=\text{-}1\cdot 0=\text{-}1\cdot \left( {1+\text{-}1} \right)=\text{-}1\cdot 1+\text{-}1\cdot \text{-}1=\text{-}1+1

However, this answer is not appropriate for most folks, especially schoolchildren, because they hate logical arguments and want a concrete example.

The best way to describe basic math operations is to work with analogies from their daily lives. There are numerous analogies for -1*-1=1, but for this individual the following analogy seemed to work best because they have purchased a new car.

When you buy a car, the car usually has a base price that includes a standard set of features. If you add a feature (e.g. high-end audio system), the price of the car increases. If you remove a feature (e.g. leather seats), the price of the car drops. Suppose you have selected a car with a set of features that includes price increases and decreases. What should happen to the price of a car when you remove a price reduction? It should increase. This is the same as saying that a negative times a negative is a positive.

With my kids, I used time analogies.

You hate driving, but you need to go on a long road trip to numerous locations. You assume that each destination  will add an hour of miserable driving, which you view as a negative value. If you have 10 destinations to visit, you have 10 miserable hours of driving (10*-1=-10). Suppose you remove a destination (-1*-1), now you just have 9 miserable hours of driving. Again, a negative times a negative is a positive.

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