What does it mean to multiply a positive number by a negative number? What does it mean to multiply two negative numbers? How about two variables? Or lots and lots of variables? Keep on reading to find out.
If you’ve been following along the past few months, you’ve probably noticed that I’m a big fan of thinking about things. Some people might say I like to over-think things, but I think they’re crazy. I mean, can you really over-think something? Actually, yes.
When I was a graduate student, my thesis advisor used to warn me against the dangers of “polishing the cannonball.” After a few weeks of perfecting the paper I was writing or the code I was developing, I would realize that he was right: my "cannonball" was good enough. After all, an ugly cannonball is still a pretty effective cannonball.
It was good advice, and I still think about it often as I’m thinking about things. So I’m generally aware of when I’m steering myself into over-polishing mode—and I can say that we’re definitely not heading down that road. Because when it comes to the basic arithmetic operations—addition, subtraction, multiplication, and division—most people never really finish even forming their cannonball … and very few ever get to the stage of polishing it.
Which is why it's time to continue down our meandering pathway and spend a little more time thinking about how to think about the meaning of multiplication. Because there’s still a healthy bit of healthy polishing to do. So let’s get started.
How to Think About Multiplying Positive and Negative Numbers
Last time as we began thinking about the meaning of multiplication, we conveniently ignored the existence of roughly half the numbers in existence—namely, negative numbers. What happens when you multiply a negative number by a positive number? Or by another negative number? How should you think about the meaning of these things? Let's think about it.
For my money, the easiest way to picture what happens when you multiply a positive number by a negative number is to imagine somehow “flipping” a number from the positive side of the number line over to the negative side of the number line:
Notice that I’ve drawn this “flipping” as sort of a rotation so that you can see what’s happening. But keep in mind that it’s really more of a flip through the origin than a rotation since there’s nowhere to actually rotate through. After all, the number line is a line—there’s no above or below! No matter how you think about it, they key point is that we end up on the opposite side of the number line relative to the location of zero.
How to Think About Multiplying Two Negative Numbers
Imagine that we’ve multiplied 6 by –1 and ended up at –6. What happens if we then multiply this result, –6, by another negative number? The answer is pretty simple:
Multiplying by another negative number once again flips everything across or around or through the origin of the number line. So –6 gets flipped around when we multiply it by, say, –1 so that it ends up right back where it started from at +6. And thus we have found the origin of the saying that is often recited but rarely understood: “A negative times a negative is a positive.” Sure, it’s true … but the question that everybody should ask is: Why is it true? And now you have a way to think about it!