I’ve seen several recent posts wondering about an intuitive explanation for the fact that “a minus times a minus equal to a plus”. I understand why people would wonder about this—the way it’s often taught, it seems to be just an arbitrary rule with no “why”! But in fact, it’s quite logical.
Paul at crossedstreams.com gives an explanation involving two quantities with real-world interpretations associated with negative values: net worth (negative means a decrease in net worth), and time (the past is negative). It’s fairly intuitive, but in some sense it only shows why it is nice that a minus times a minus is a plus, since it allows us to model this real-world situation; it doesn’t really show why it is true in a deep sense. Mike at Walking Randomly gives a proof essentially from the field axioms, but (as he himself admits) it is not particularly intuitive.
Here’s how I would explain it. It isn’t rigorous at all, and I’m not even completely satisfied with it, but I hope it is helpful for building some intuition.
Think about the familiar “number line”: positive numbers go off to the right, and negative numbers are to the left. Adding a positive number corresponds to moving right along the number line. Adding a negative number (that is, subtracting a positive one) corresponds to moving left along the number line. So with addition we already see this idea of negative corresponding to doing something “in the opposite direction”.
So, what does multiplication correspond to on the number line? Of course, multiplication corresponds to scaling, or stretching: for example, if we start at a point on the number line and multiply by 3, we will end up at a point three times as far from zero as we started. And what about multiplying by a negative number? It corresponds to a scale in the other direction: for example, if we start at a point on the number line and multiply by -3, we end up at a point on the other side of zero, and three times as far. That is, multiplying by a negative number means that we flip from one side of zero to the other. So, of course if we start from the left of zero (a negative number) and multiply by a negative, we end up on the right of zero (a positive number)!
[As an afterthought, I think trying to explain it any more deeply than this really does require bringing in the distributive property in some way—it is the only thing that formally connects addition (remember, negative numbers are defined as additive inverses) and multiplication. I really like the idea posted by Eric as a comment on Mike’s post regarding an illustration of the distributive property.
Also, I meant to put nice pictures in this post, but for now if I wait to make nice pictures it will never happen. So I’ll just post it for now, and maybe I’ll come back and add some pictures later.]