How to Think About Multiplication: Part 1
What happens when you multiply one number by another? Is there a right answer to this question? What does it tell us about how you should think about multiplication? Keep on reading to find out.
Jason Marshall, PhD
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How to Think About Multiplication: Part 1
Quick question: How do you prefer to think about multiplication? In other words, how do you like to picture the meaning of multiplication so that you too get that awesome deep-down-in-the-gut feel for it?
In truth, I’m pretty sure multiplication isn’t a topic that most people particularly like to think about at all. If you are like most people, you memorized the multiplication table way back in the day and haven’t given it much thought since. If that sounds familiar, rest assured that I don’t think any less of you. In fact, I’d say you’re perfectly normal.
But I’d also say that it’s good for all of us to spend a few minutes thinking about what we’re actually doing when we’re doing stuff. It’s a healthy habit to get into and it’s actually kind of fun too. And besides, memorization can only take us so far in math … and in life—thinking is a much better option in the long run.
So today let’s think about multiplication.
Is Multiplication (Just) a Handy Way to Add Faster?
People often like to think about multiplication as being nothing more than a sort of shortcut for adding some number to itself some other number of times. For example, you can think of the expression 3 • 5 as a request for you to add 5 to itself 3 times—that is, as 3 • 5 = 5 + 5 + 5 = 15. If you want a picture to have in your head for this view of multiplication, think of the process as stacking up three 5-box-long rows and then counting up all the boxes:
Clearly this picture gives the correct answer—that is, 3 • 5 really does equal 15. Of course, we can extend this idea to the problems 4 • 5 = 5 + 5 + 5 + 5 = 20, 5 • 5 = 5 + 5 + 5 + 5 + 5 = 25, and so on by stacking more and more 5-box-long rows onto the pile. If you think about it, you’ll see that we’re just using multiplication as a convenient way to save ourselves from having to write a bunch of extra terms in an addition problem.
So that’s how we should think about multiplication? It’s just a handy way to do addition more efficiently? No, not exactly—multiplication is certainly this, but it’s a lot more too.
How Should You Think About Multiplication?
The picture we’ve just come up with for multiplication is, as we’ve noted, all well and good … so long as we’re talking about multiplying positive integers. But it kind of fails when it comes to multiplying other things like negative numbers, fractions, and irrational numbers. In truth, it’s not so much that it “fails,” but rather that it starts to get very strange.
It doesn’t make much sense to add 5 to itself 3½ times.
For example, while it makes perfect sense to add 5 to itself 3 times, it doesn’t make much sense to add 5 to itself 3½ times. What would that even look like? 3½ • 5 = 5 + 5 + 5 + what?!? … I can’t write half of a “5” there! Okay, I suppose I could if we just write 5 + 5 + 5 + (½ • 5). But as I said, it starts getting weird to think about multiplication in this way. It’s not wrong, it’s just weird. And it’s even more bizarre if we’re talking about multiplying two fractions—say a problem like ⅓ • ⅕. How do we add the fraction ⅕ to itself ⅓ of a time? I don’t know!
Multiplication Can “Stretch” Integers
So where does that leave us? Well, you’ll be happy to know that all is not lost and that there is a viable alternative. To see what it is, notice that we can view the problem ⅓ • ⅕ as a way of saying that we need to “shrink” the fraction ⅕ to be ⅓ its original size (think “⅓ of ⅕”). And if we extend this idea, we see that this problem is telling us that we can think of any multiplication problem as a process of scaling the magnitude of some number until it is some other number of times its original size.
If you think about it (which, after all, is our job for today), you’ll see that this picture makes sense whether we’re multiplying whole numbers, fractions, or even irrational numbers. So to solve our simple example of 3 • 5, we need to take a 5 box-long-row and stretch it until its three times its original size. Of course, this gives us the same answer for 3 • 5 as before (it had better)—so no surprise there. But the real power of this stretching analogy is that it can help us think about other kinds of multiplication problems too.
Multiplication Can “Stretch” Any Kind of Number
As an example, the problem 3½ • 5 has a very natural interpretation when we think about it as a problem of scaling—or stretching—one number by another:
The picture looks very similar to the previous one. But the new idea here is that we can stretch the 5-box-long row to be any length that we’d like. In other words, we don’t need to end up with a whole number of boxes. Which means we can picture multiplying an integer by a fraction like 3½ • 5, or a fraction by a fraction like ⅓ • ⅕, or even something like 5 times an irrational number like √2. All of these things make perfect sense when viewed this way!
So, going back to the question we started with: How do I think about multiplication? Well, I imagine that multiplication is the process of “stretching” the magnitude of a number.
Multiplication is the process of “stretching” the magnitude of a number.
And it’s a pretty interesting way of thinking about things. Because, as we’ve already discussed, it’s valid for any and all of the number types you run into on a regular basis. On top of that, if you think about it, you’ll see that this view of multiplication is closely related to the idea of finding the area of a rectangle. Each of the 1-by-1 boxes that we stretch out has an area of 1 • 1 = 1 (we call that “unit” area). And the total area of the stretched out boxes is exactly the answer to the multiplication problem.
It’s not just a great way to think about and visualize multiplication, it’s a great way to get a solid deep-down-in-the-gut feel for multiplication 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!
Cherry multiplication 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.