Learn to take advantage of the commutative property of addition to simplify problem solving. Make math easy!
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 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 to receive more than one befuddled 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 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, 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.