by Jason Marshall
In the previous article we talked about using the number line to help you keep your signs straight when adding positive and negative integers. In this article we’re taking things one step further—not just adding positive and negative integers, but subtracting them too.
Quick Review of Adding Negative Integers
But before we get to that, let’s quickly review what we covered last time. The main thing to take away was the idea that you can use the number line to visualize what’s actually going on when you add positive and negative numbers. Let’s use the first practice problem I gave at the end of the last article to demonstrate what I mean: 3 + (-13) + 14? We can solve this problem by visualizing walking along the number line. Here’s how. Start at zero, and walk three steps in the positive direction to your right. Then, turn around, and walk thirteen steps in the negative direction. Where are you? You should be at “-10.” But we’re not done yet. You next need to walk fourteen steps in the positive direction to your final destination. Which is...? That’s right, it’s “4.” So, 3 + (-13) + 14 = 4. Make sure that makes sense before moving on since things are about to get a little bit tougher.
How to Subtract Negative Integers
Now, this mental meandering along the number line works for more complex problems too. How about something like 2 - (-3). Huh? How do you subtract a negative number?! Well, we’ve already established that adding a positive or negative integer can be thought of as walking that number of steps in the positive or negative direction along the number line. An addition sign is like a green light saying: “Keep on walking in the positive or negative direction.” But subtraction? Well, in contrast to addition, a subtraction sign is like a red light saying: “Stop! Turn around, and head in the other direction.”
An Example of Subtracting Negative Integers
Let’s go back to our example problem, 2 - (-3), and see how we can use the number line to solve it. As always, start out at zero on your number line. The first number in 2 - (-3) is positive two, so you need to walk two steps in the positive direction. Now, the second number in 2 - (-3) is negative three. Starting at your current location of “2,” imagine you turn as if to start walking those three steps in the negative direction—exactly as you’d do if there was an addition sign. But, wait a minute—that’s a subtraction sign! And remember, a subtraction sign is like a bright red light warning you to turn around and walk in the opposite direction. So, instead of walking three steps in the negative direction, you have to do the opposite and walk three steps in the positive direction. Starting from “2,” you do exactly that and find yourself to be at “5.” You’ve calculated that 2 - (-3) = 5.
So Adding a Negative Number is Like Subtracting It — Yes!
Let’s take another look at adding positive and negative numbers—say, 10 + (-5). Walking ten steps in the positive direction, then five in the negative, gives us the answer: 5. Notice anything funny about that problem? How about the fact that the answer to 10 + (-5) is exactly the same as the answer to 10 - 5! And, thinking about the number line a bit, we can see why that has to be true: walking in the negative direction when adding a negative number is exactly the same thing as turning around and walking in the negative direction in reaction to a subtraction sign. So here’s a quick and dirty tip for you: Whenever you see a problem asking you to add a negative number to another number, you can always just think of the problem as asking you to subtract that number instead. In other words, solving a problem like 133 + (-43) is identical to solving the problem 133 - 43. They’re both 90.
So Subtracting a Negative Number is Like Adding It — Yes, Again!
And here’s another thing: take a second look at the problem 2 - (-3) = 5. Notice anything funny there? How about the fact that 2 - (-3) gives the exact same answer as 2 + 3?! Yes, once again, thinking about the number line makes it clear why this has to be true: subtracting a negative by turning to walk in the negative direction, seeing the “red” light subtraction sign, and then walking in the positive direction is the exact same thing as just walking in the positive direction from the outset. So here’s another quick and dirty tip to go along with the first: Whenever you see a problem asking you to subtract a negative number from another number, you can always simplify the problem by adding that number instead. In other words, 133 - (-43) is identical to solving 133 + 43. They both equal 176.
Why Bother Visualizing the Meaning of Math?
Perhaps you’re wondering: “Math Dude, why not forget all that number line business and instead just give us these ‘rules’ to help us solve problems right from the get-go?” Well, in my experience, this sort-of shortcut-to-answers method of teaching and learning math doesn’t really work that well. It’s how many of us first learned math—the “just use these formulas and follow these rules” method—but it doesn’t really help you to understand math. Although the problems we’ve looked at have been fairly simple, they’ve been chosen to help you “see” what the math they describe really means—once you understand that, you have a very powerful tool at your disposal for solving much more complex problems.
Wrap Up
In the next article, we’ll look at a real life example of how all this addition and subtraction of positive and negative numbers is used in the financial world of banks and balance sheets. In the meantime, here’s a problem for you to think about: What’s the sum of all the positive even integers less-than-or-equal-to ten, minus the sum of all the positive odd integers less-than-or-equal-to ten. That’s quite a mouthful, I know. Don’t worry if you aren’t familiar with all those terms just yet...here’s the translation: What is 2 + 4 + 6 + 8 + 10 - 1 - 3 - 5 - 7 - 9? Hint: there’s an easy way and a hard way to do it—try to figure out the easy way if you can.
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