How to Think About Division: Part 1
What does it really mean to divide two numbers? Or two variables? How can you visualize these processes? And how are they related to the process of multiplication? Keep on reading to find out.
Over the past few months we’ve talked about how to think about three of the big four arithmetic operations: addition, subtraction, and multiplication. Which, perhaps not surprisingly, means that a conversation about division is on tap for today.
A lot of people despise division because they think it’s hard. This perception isn’t wrong—it usually is harder to do than the other arithmetic operations. And the truth is that once you know how to do long division by hand, you may as well use a calculator for the rest of your life (except in those increasingly rare occasions in which you don’t have one in your pocket).
If you, like so many other people, don’t have warm-and-fuzzy super-fond memories of division, rest assured that today we’re not talking about the mechanics of doing long division. Instead, we have the more noble goal of attaining a new vision for division. By which I mean we’re going to talk about the underlying meaning of division.
So, how should you think about division? Let’s think about it!
How to Think About Division with Numbers
To begin with, let’s think about what we’re really doing when we divide one number by another. And, as we’ve done with the other arithmetic processes, let’s think about all of this in terms of manipulating stacks of blocks.
First of all, we know that we can think of a problem like 3 • 5 = 15 as being represented by a grid of 15 blocks arranged into an array that’s 3 blocks high by 5 blocks wide. We’ve talked about how we can think about this as stretching the row of 5 blocks until it’s 3 times its original height. Thus, we can view multiplication as a process that scales or stretches a number. So why are we reviewing multiplication in a discussion about division? Because the two processes are intimately related. In fact, they’re what we call inverse processes.
And that means that in the same way that subtraction “undoes” addition, division “undoes” multiplication. In other words, we can think of division as a process that scales down—or compresses—a number until it is some other number of times smaller.
In this case, we can think of 15 / 3 visually as a process that takes a grid of 3 rows of 5 blocks and compresses it until it’s squished down to 1 row of 5 blocks. Which means that 15 / 3 = 5. No surprise, but it is a great way to think about the meaning of division.
How to Think About Division with Variables
Let’s now extend this idea to variables. The good news is that nothing really changes. For example, let’s think about the problem (x • y) / x:
We can picture the numerator of this expression as a rectangle whose area is x • y. That is, we’ve taken the length y and stretched it until it is x times taller. But now the question is what happens when we divide this by the variable x? Well, it’s basically the same as when we divided (3 • 5) by 3—we compress the rectangle by the amount we divide by.
So, in this case, we’re compressing (x • y) by an amount given by the value of x—which means that we “undo” the stretching that the multiplication originally did. And the answer to the problem (x • y) / x is therefore simply y. Again, I know this is a fairly obvious result, but the point is that we now have a nice way to visualize what we’re doing when we do division.
Division, Multiplication, and Numbers
With that basic idea now firmly tucked away in our mathematical tool belts, it’s time for me to let you in on a big “secret” about division: It’s pretty much the same as multiplication. What do I mean? Well, even though this might sound kind of counterintuitive, I mean that stretching and compressing (i.e., multiplication and division) are actually one in the same process—in the sense that each of these things can be turned into the other.
To see what I mean, let’s take a look again at the problem 15 / 3 = 5 that we talked about earlier. But instead, let’s think about the problem 15 • ⅓ = 5 … because these two problems are really one and the same. In our way of thinking about the world, the problem 15 / 3 is asking you to figure out how many blocks you’ll have left after compressing a 3 by 5 array of blocks into a stack that’s 3 times smaller than it was to begin with.
But what about 15 • ⅓—how is that the same? Well, as we’ve seen, when we multiply one number by some other number we are scaling that first number to be some number of times bigger. And when we multiply some number by a fractional number like ⅓, we are scaling the first number to be some number of times—⅓ in this case—smaller than its original size. So scaling by ⅓ has exactly the same effect as compressing the number until it’s 3 times smaller. Obvious, yes—but powerful too.
Division, Multiplication, and Variables
Of course, not only does this concept help us deal with problems containing numbers, it also helps us with problems containing variables. For example, let’s take another look at the division problem we solved earlier: (x • y) / x. But this time, let’s turn the division problem into the exactly analogous problem of multiplying by a fraction. That means that instead of dividing by x, we need to multiply by 1/x. In other words, we scale the value of x • y until it is 1/x times as large as it was at the outset. The answer is the same.
This connection between division and multiplication is one of the reasons that we often use the “/” symbol to indicate division instead of the classic “÷” symbol we used in elementary school. After all, the “/“ symbol is also used to separate the numerator and denominator of a fraction, so this notation makes the connection between fractions and divisions that we’ve discovered obvious.
And, most importantly, this connection gives us a great way to think about the true underlying meaning of division.
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.
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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.
