What does it mean to divide an integer by a fraction? Or a positive number by a negative number? Or even a negative number by another negative number?
As I mentioned last time, a lot of people have a strong distaste for division. This makes me feel a bit bad for division because division itself isn’t a terrible fellow. Sure, it is a little more difficult in practice to divide numbers than to multiply them, but that’s not division’s fault—it’s just the lot it was given in the numerical universe.
So, at least for today, I ask you to forgive division’s difficulties and spend a bit of time thinking about the meaning of this sometimes frustrating arithmetic operation. We got off to a good start talking about division last time, but we’ve still got a few loose ends to tie up. Things like what happens when you divide by fractions and negative numbers, whether or not division is a stickler for the order in which you perform its operations, and some other fun stuff like that.
And of course, as with all of our discussions in this series, our goal isn’t to develop lightning quick division skills, it’s to develop your intuition for the true meaning of division.
How to Think About Division with Fractions
When it comes to dividing fractions, most people fall back on the old trusty slogan: “invert and multiply.” While that is ultimately a good thing to do (since it works), it doesn’t shed much light on what’s really going on. And that’s a shame because last time when we talked about the connection between fractions and division we actually figured it out.
So, what is going on? Let’s start by recapping how division by an integer works. As a simple example, we’ve seen that a problem like 10 / 2 = 5 is really just representing the idea of taking a 10 block tall stack and compressing it down by a factor of 2. In other words, it’s identical to the problem 10 • ½ = 5 since this problem is simply telling us to take that same original stack of 10 blocks and stretch it down to be half its original size—it’s the same thing.
Invert and multiply works because of the relationship between fractions and division.
Just to be sure this makes sense, take another look at what we did. The bottom part of the division problem 10 / 2 is the number 2, which we can write as the fraction 2/1. What do you get when you invert 2/1? The fraction ½. When you multiply this by 10 you get 10 • ½. If you follow the logic, you’ll see that we’ve just discovered that we’ve actually been using “invert and multiply” all along—it’s nothing new.
So, yes—invert and multiply works. But remember that it works because of the relationship between fractions and division.
How to Think About Division with Negative Numbers
Next up on our list of loose ends to tidy up: What happens when we divide positive and negative numbers? As luck (or rather arithmetic) would have it, things work in the exact same way as they do for multiplication. Whenever you divide a positive number by a negative number (or vice versa), the result is a negative number. And whenever you divide one negative number by another negative number, the result is a positive number.
But why is that? Well, you’ve heard this before—it all goes back to the connection between fractions and division. Take the problem 100 / 4. As we’ve described, you can write this as the equivalent problem 100 • ¼. If we instead have the problem –100 / 4 = 100 / –4, we can use what we’ve discovered to rewrite these as –100 • ¼ = 100 • –¼ = –25. In other words, once you understand what it means to multiply by a negative number, you also understand what it means to divide by a negative number.