How to Think About Division: Part 2
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?
Jason Marshall, PhD
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How to Think About Division: Part 2
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.
How to Picture Division with Negative Numbers
We can get a little better feeling for what this means by thinking about the problems 6 / –1 and –6 / –1. If we think about these division problems as compressing stacks of blocks, we see that we’ve actually posed a rather simple example since the 1 in the denominator means that the number of boxes in each stack doesn’t actually change. But what does change? Well, that’s where those negative signs come into the picture.
Just as we found for multiplication, the negative sign has the effect of flipping the result from one side of the number line through the origin over to the other side. And, as we found for multiplication, the two negative signs in the case of –6 / –1 end up flipping the answer from the positive side of the number line to the negative side and then right back once again to the positive side.
Make sense? Hopefully so. And hopefully you now have a good feeling for what it really means to divide negative numbers!
Division and the Commutative and Associative Properties
Last up today, let’s talk about the commutative and associative properties. Just in case you can’t quite remember what those are, here’s a quick reminder of how they work with addition and multiplication. The commutative property simply tells us that you can add or multiply a pair of numbers in whatever order you want and the answer will be the same. So 3 + 2 = 2 + 3 and 3 • 2 = 2 • 3. Simple. But does this idea hold for division, too?
Well, does 20 / 4 give the same result as 4 / 20? No! So the commutative property does not work for division. How about the associative property? Does (20 / 4) / 2 = 20 / (4 / 2)? As you can check, the answer is no! So it appears that these properties don’t hold for division. And that’s sort of true … but not entirely. It’s true that they don’t work when you treat division like division, but we know that we can also treat division like multiplication. And that changes things.
Once you know how to think about multiplication, you know how to think about division, too.
To see how, let’s again look at the problem 20 / 4, but let’s write the expression as 20 • ¼ instead. Does the commutative property hold now? In other words, does 20 • ¼ = ¼ • 20? You bet it does! How about the associative property and the problem (20 / 4) / 2? Well, let’s instead write the expression (20 / 4) / 2 as (20 • ¼) • ½. Then all we have to check is whether or not (20 • ¼) • ½ = 20 • (¼ • ½). As you can check, both of these do indeed equal the same value. Which means that the associative property does indeed hold if we treat division in terms of multiplying equivalent fractions.
In case you haven’t noticed, there’s theme going on here. Simply put, so long as you think about division in terms of the equivalent process of multiplying fractions, everything behaves exactly as you expect it to. Which means that once you know how to think about multiplication, you know how to think about division, too.
Wrap Up
Okay, that’s all the math we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. Also, remember to become a fan of The Math Dude on Facebook and to follow me on Twitter.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Division image from Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.