How to Think About Multiplication: Part 2
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.
Jason Marshall, PhD
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How to Think About Multiplication: Part 2
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!
How to Think About Multiplying Variables
Of course, just like every good arithmetic operator, we’re not stuck doing multiplication on integers alone—variables are quite happy to play the game too. Using the sort of thinking we developed last time in which multiplication is viewed as a kind of “stretching,” we see that when we multiply the variables x • y, we’re really just stretching or scaling y so that it’s x times as large as it was to begin with.
If you think about it, you’ll see that this means that x • y is just the area of the rectangle that has sides that are x and y units long. It shouldn’t come as much of a surprise that we can also multiply variables by themselves. When we multiply a number by itself, we square the number. And when we multiply a variable by itself, we square the variable.
What “shape” does this action create? Again, it should not come as a surprise that in our way of viewing multiplication, we get a square … hence the term “squared!”
How to Think About Multiplying Lots of Variables
Now, if you’re really on top of things, you may have already asked yourself: What happens when we try to multiply more than two variables? The easy answer is that everything works exactly as you’d expect it to and you end up with, say, x • y • z. But how can you picture the meaning of this?
Well, as we’ve seen, we can picture the meaning of multiplying a pair of variables in terms of a rectangle created by stretching a stack of boxes. But does that still work when we throw a third variable into the mix? Sure it does. Multiplying by the third variable, say z, simply “stretches” the entire 2-dimensional rectangle formed from the first multiplication to create a 3-dimensional stack of rectangles.
When we actually plug in numbers for x, y, and z, we end up with a number that’s equal to the total number of squares in our stack. That’s the meaning of x • y • z—it’s the volume of the 3-dimensional stack.
How to Know When to Stop Thinking About Multiplication
Variables are adventurous creatures—not only can we multiply x, y, and z, we can multiply any combination that strikes our fancy. And that includes multiplying x by itself three, four, five, or any other number of times. If we multiply x • x • x we get a cube—hence the origin of the term “cubed.” Just in case you don’t see where the cube comes from, remember that when we multiply x • x • x we are effectively stretching an x • x square up to a height of x. Which makes a cube!
And it’s a nice, shiny, well polished cube, don’t you think? Although if we polish it any more, we might accidently round off its edges and turn it into a cannonball! In fact, I think I hear the wise words of my advisor echoing back into my head letting me know that this is a perfect place to stop polishing and to start enjoying the fact that we now truly understanding the meaning of multiplication.
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!
Chalkboard 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.