How to Think About Subtraction: Part 2
Do you really know what subtraction is? By which I mean do you really feel it in your gut? Here’s part two of the series that will help you do so.
Jason Marshall, PhD
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How to Think About Subtraction: Part 2
A while back, we talked about how to best think about adding numbers and variables, and that discussion naturally led us to the question of how to best think about subtracting numbers, variables, and whatever else comes your way. Which is exactly what we did when we last got together. But we didn’t get through all of the myriad subtractable things in the world, which is why we’re returning to this topic today.
Keep in mind that our goal in doing this isn’t to learn how to add and subtract—I’m pretty sure everybody here is already good at doing that. Our goal is to figure out how best to think about these processes. In other words, our goal is to develop a gut feeling for what these most fundamental of operations really do to things like numbers and variables.
This kind of intuition for math is something I’ve found that a lot of people unfortunately never developed when they were learning how to go about the mechanical motions of performing arithmetic. So today let’s continue to remedy that situation.
How to Think About Subtracting Negative Numbers: Option 1
When we last talked about subtraction, we went into a lot of detail about how to best think about subtracting positive numbers. But we completely ignored the entire other half of the infinity of numbers—namely, all of those negative numbers. So today we’re going to think about what we’re actually doing when we’re subtracting negative numbers.
I’m willing to wager that you learned a rule at some point in your life saying something like “minus a negative is a positive.” And while that saying is true, regurgitating rules without understanding what they mean is not a great way to develop intuition—so let’s talk a little about what this rule really means. To do so, let’s think about the super simple problem
3 + (–2) – (–2)
Look at that expression—in particular, look at the second and third terms. Hopefully it’s pretty obvious that those two terms cancel each other out. After all, (–2) – (–2) must be equal to 0 since we’re subtracting something from itself. It’s no different than saying 1 – 1 = 0, other than the fact that all of those negative signs makes it seem like something more complicated must be going on. But it really is just that simple.
So how can we picture a problem like this to help develop our intuition for it? As we talked about last time, let’s picture the number 3 as a 3-block-high stack, and let’s picture the number –2 as a 2-block-deep empty container. If we simply add 3 to –2, we end up with a single block sticking out above the top of the container to give us our answer of 1. But in our problem we also must include the term –(–2). How can we do that?
The key thing to see in our minds is that this third term negates the second. In other words, it makes it vanish from the problem, like this:
The second and third terms cancel each other out, which means that the total value of the expression 3 + (–2) – (–2) is just 3. The first term—the 3-block-high stack—now sails right over the newly non-existent empty container to leave us with the same 3-block-high stack.
How to Think About Subtracting Negative Numbers: Option 2
So that’s one way to intuit this meaning of subtracting a negative number, but let’s think a bit more about this and see if we can come up with something even better. First, as we noted before if we add the first two terms of our addition problem we get 3 + (–2) = 1. But the full problem, 3 + (–2) – (–2), is equal to 3—a larger result … which means that we can subtract a number from some number and actually increase its value!
That might not be shocking, but it is interesting … very interesting. Why? Well, usually we imagine that subtracting numbers makes expressions smaller. But, obviously, not always. If we look at the third term of our expression, we see that –(–2) must have the same value as 2 (that’s how we get the entire expression to be 2 greater). And if you think about that for a second, you’ll see that it absolutely makes sense. As an everyday example, if you owe a bank $10, but through some act of kindness the bank later forgives that $10 debt, the net result is a positive gain of $10 to your balance. In other words, it’s the same as if you added positive $10 to your account. Make sense? You can put this idea into pictures like this:
As before, we have a 3-block-high stack and a 2-block-deep empty container. But we now also have a 2-block-high stack. Where did that come from? It came from the fact that subtracting a negative value is the same as adding that value—yes, that good old memorized mantra: “minus a negative is a positive”—telling us that –(–2) = +2. In this view of things, the new 2-block-high stack fills in the 2-block-deep empty container, leaving us with a 3-block-high final answer. Same answer (as it must be), but a slightly different interpretation.
The Rule for Subtracting Negative Numbers
…subtracting a negative number has the same effect as adding the absolute value of that number.
The bottom line is that subtracting a negative number has the same effect as adding the absolute value of that number. As we’ve seen, you can think of this as being a result of the fact that subtracting a negative number subtracts some portion of an empty container. Or, if you prefer, you can equivalently think of it as being a result of the fact that subtracting an empty container that’s some number of blocks deep is the same as adding a stack that’s that same number of blocks high.
Hopefully now that we’ve attached a bit of meaning and a few pictures to the rule for subtracting negative numbers, you’ll no longer think of it as something to memorize and recite. Instead, you’ll be able to intuit where it comes from and even feel it in your gut.
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!
Subtraction 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.