数学英语 24 What is the Commutative Property of Addition?

by Jason Marshall

We’ve spent the last few articles building our numerical vocabulary from the ground up. Today, we’re going to talk about something that sounds hard, but is actually pretty easy...and very useful too: the commutative property of addition.

Review of Adding and Subtracting Integers

But before we head down that path, let’s review the world of adding and subtracting positive and negative integers by taking a look at the practice problem I mentioned at the end of the last article: What is 2 + 4 + 6 + 8 + 10 - 1 - 3 - 5 - 7 - 9? I hinted there’s an easy way and a hard way to solve it. Let’s take a look at the hard way first.

Using our imaginary number line walking technique to solve this problem, start at zero, walk two steps in the positive direction, then four more steps in that direction, then six, and so on until after 2 + 4 + 6 + 8 + 10, you arrive at the number 30. Next, to subtract, start from where you are at “30,” turn around, and walk one step in the negative direction, then three more, then five, and so on until after marching out the problem 30 - 1 - 3 - 5 - 7 - 9, you arrive at 5.

Easy enough, but that’s a lot of steps. And that’s where the commutative property of addition is going to come in and save the day. Let’s take a few minutes to fully 1 understand what it means, and then we’ll come back to our practice problem and see how to solve it in a much simpler way.

The Commutative Property in Everyday Life

Does it matter if you put your right shoe on before your left? How about putting on your socks before your shoes, or vise versa? Does the final outcome depend upon the order in which you do these things? Hopefully the answers are obvious. Otherwise, you’re probably destined 2 to receive more than one befuddled 3 look when venturing outside.

But the point is an important one. No, it does not matter if you put your left shoe on and then your right, or your right shoe on and then your left. The result is exactly the same. Namely, you go from a state of not wearing shoes, to a state of wearing shoes. However, the same thing cannot be said about socks and shoes. The outcome is very different when putting your socks on before your shoes versus 4 putting your shoes on before your socks.

So what’s the relationship to math? Well, the process of putting on your left and right shoes satisfies the commutative property. When two processes commute 5, or yield the same result regardless of order, the order in which you do them doesn’t matter. So, putting on your left and rights shoes is a commutative process—the end result doesn’t change whether you put the left one on before or after the right one. Putting on your socks and shoes is not a commutative process.

What is the Commutative Property of Addition?

How about walking two steps forward then one back? Does that yield the same result as walking one step backward then two forward? Sure does. In both cases, the net result is you’ve taken one step forward. But hold on a minute! Does taking steps forward and backward remind you of anything? Isn’t it just like walking along the number line in the positive and negative direction? Why yes, it absolutely is. Here’s how it all comes together.

Addition is a commutative process. You can add a list of numbers in any order you like: 23 + 44 gives the same answer as 44 + 23. That is perfectly 6 reasonable if you think about the number line. You can walk 23 steps then 44 more, or alternatively, you can walk 44 steps then 23 more—you’ll end up at the same place.

Okay, that’s addition; what about subtraction 7? Does the problem 23 - 44 have the same answer as 44 - 23? Absolutely not! Subtraction is NOT a commutative process. Take a few moments to think about walking along the number line to see why. In the first case, 23 - 44, you end up with -21, but in the second, 44 - 23, you get +21. Definitely not the same.

How to Use the Commutative Property to Solve Problems

Let’s return to our practice problem: 2 + 4 + 6 + 8 + 10 - 1 - 3 - 5 - 7 - 9. We already went over the “hard” way to solve it, but that was before we knew about the commutative property! How can that help us? Well, let’s look at a shorter version of the problem to see: How about 2 + 4 - 1 - 3? First, instead of 2 + 4 - 1 - 3, let’s think of the problem as 2 + 4 + (-1) + (-3). Now, since addition is commutative, let’s rearrange the order of these numbers. Let’s try 2 + (-1) + 4 + (-3). Since adding a negative number is the same as subtracting it, let’s instead write this as 2 - 1 + 4 - 3. See a pattern here? Well, 2 - 1 = 1 and 4 - 3 = 1, so the problem simplifies to 1 + 1 = 2.

How about the full problem? Let’s rearrange 2 + 4 + 6 + 8 + 10 - 1 - 3 - 5 - 7 - 9 into (2 – 1) + (4 – 3) + (6 – 5) + (8 – 7) + (10 – 9). The solution is almost immediately obvious now. The answer to each of the sub-problems in parenthesis—or terms—is one, and there are five of those terms in total. So the final answer must be five. Much simpler!

Wrap Up

At the end of the last article I promised to talk about the real world application of the mathematics of money. But before doing that, I want to give you a chance to think about how things like credit card balances, loans, paychecks, gifts, and forgiven debts are represented with positive and negative numbers. So, give it a bit of thought and be sure to check out the next article to read all about it.

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