Negative times a negative

I often have students ask me the following question: “Mr. Peters, why is a negative times a negative a positive?” Until recently, I didn’t have a good answer for them.

I have tried to use analogies, like “two wrongs make a right”, or “taking away a penalty in football moves the team forward.” However, these analogies were never very good.

A few weeks ago, I stumbled upon a research article that articulated my dilemma and proposed a beautiful response. I want to submit a slightly modified version of that answer here.

First, we need to answer a simpler question, what is 1*(-1)=??

Consider the equation:

1*(1-1) = 0

If we distribute the one in front, we get the following:

(1*1) +1*(-1) = 0

Since 1*1 = 1 , we can simplify this slightly:

1 +1*(-1) = 0

Now subtract 1 from both sides:

1*(-1) = -1

Tada! 1*(-1)=-1 . But you already knew that. Now we will play the same game, but start with a slightly different equation.

Consider the equation:

-1*(1-1) = 0

If we distribute the negative one in front, we get the following:

(-1*1)+(-1)*(-1) = 0

Since (-1)*(1) = -1 , we can simplify this slightly:

-1+(-1)*(-1) = 0

Subtract -1 from both sides (add plus one):

(-1)*(-1) = 1

I want to dwell on this argument for a moment. First, we did not assume anything about the value of (-1)*(-1). This argument is not circular, which is a good thing. Second, the main power of the argument comes from using the distributive property. In fact, you could argue that the reason a negative times a negative is a positive stems directly from the distributive property. Third, this argument does not provide intuitive understanding. While the argument is airtight, something is lost in the rigour of it. Personally, there is something human about “two wrongs make a right” and something very impersonal about the argument presented above.

Which begs the question, should our answers to students be logically airtight or personally convincing? Should we value rigour or intuition? What is the point of a proof?

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