The Commutative and Associative

The Commutative and Associative Properties and Algebraic Expressions

The order of operations is important and useful, but a few mathematical properties highlight cases where order doesn't matter. In this lesson, we'll learn about the commutative and associative properties, which may save you time and effort.

Math Rules

Math is full of rules. You have to divide before you subtract. 2 + 2 has to equal 4. You can only eat pie after you finish your vegetables. Working with pi will always make you think of pie.

Fortunately, there are a few rules that actually make math simpler. These are like the Casual Fridays of mathematics. They're rules, yes, but they define how you can loosen up a bit and lose that tie.

Commutative Property

Let's say you want to know what 3 + 8 is. Do you have to add 8 to 3? Or, could you add 3 to 8? It doesn't matter, right? And, the same thing is true if you want to know 2 * 5. That's the same as 5 * 2.

These examples illustrate the commutative property, which states that the order of the numbers when you add or multiply doesn't affect the sum or product. In other words, a + b = b + a and ab = ba.

The name comes from the word 'commute.' When you commute, what are you doing? You're moving from one place to another. When you get to the end of your commute, you're still the same person. Well, unless you're commuting using a malfunctioning teleporter.

Note that the commutative property doesn't work for subtraction or division. 3 - 8 does not equal 8 - 3. And, 10/5 does not equal 5/10. But, it does apply with addition and multiplication.

Commutative Practice

Let's practice. Let's say you have 2 * 5 * 3. That's 30. What if we rearrange it to 3 * 5 * 2? Still 30. 5 * 2 * 3? Still 30. 2 * 3 * 5? Yep, still 30.

Here's one with addition: 10 + 7 + 9. That's 26. If we rearrange it to 9 + 10 + 7? Still 26.

You could use the commutative property to justify eating your dessert first at dinner. After all, no matter which order you eat the food, it all ends up in your stomach. So, why not polish off that ice cream before getting to the broccoli? Well, that's sort of the same thing, but not exactly.

When you add 10 and 7 and 9, you're always dealing with constant numbers. If you had all the parts of your meal laid out, and you were sure to have room for all of them in your stomach, then order really doesn't matter. Granted, parents everywhere may still not approve.

Associative Property

There's another law that's similar to the commutative property. To understand this one, let's imagine the world's saddest yard sale. You're selling three things: a broken hair dryer for $1, a three-legged chair for $4 and a box of old VHS tapes for $2.

Let's say your neighbor Mrs. Lake buys the hair dryer. Then your other neighbor, Mr. Rivers, buys the chair and tapes. You just made $1 from Mrs. Lake and $4 + $2 from Mr. Rivers - that's $7. While that won't buy you nicer stuff, it will buy you a burrito with guacamole.

But what if Mrs. Lake bought the hair dryer and the chair? And then, Mr. Rivers bought just the tapes? You'd then make $1 + $4 from Mrs. Lake and $2 from Mr. Rivers. You'd still get $7. And, you'd still get that burrito.

This sad yard sale illustrates the associative property, which states that the way you group numbers when you add or multiply doesn't affect the sum or product. In other words (a + b) + c = a + (b + c) and a(bc) = (ab)c.

Whether Mrs. Lake buys two items and Mr. Rivers buys one or Mrs. Lake buys one and Mr. Rivers buys two, you still get $7.

That was an addition example, but it works the same with multiplication. Let's say you have this: (5 * 2) * 3. If you remember the order of operations, you need to handle the stuff inside the parentheses first. That gets you 10 * 3, which is 30. But, the associative property says that (5 * 2) * 3 is the same as 5 * (2 * 3). That latter format gets you 5 * 6, which is, yep, also 30.

Note that I said the property works for addition and multiplication. The associative property doesn't work for subtraction and division.

(7 - 4) - 2 does not equal 7 - (4 - 2). With (7 - 4) - 2, you first subtract 7 - 4 to get 3. Then you do 3 - 2, to get 1. In 7 - (4 - 2), you start with 4 - 2, which is 2. You then do 7 - 2, which is 5.

Associative Practice

Let's try a few of these. Here's one: (5 + 10) + 7. Again, the order of operations says we need to do that 5 + 10 first. But, since everything here is addition, the grouping doesn't matter. So, you could add the 10 and 7 first. In other words (5 + 10) + 7 = 5 + (10 + 7). No matter how you group it, you get 22.

How about this one: 6 * (2 * 5)? If you do 2 * 5 first, you get 6 * 10, which is 60. But, the associative property tells us that we could go (6 * 2), which is 12, then multiply that by 5, which still gets us 60.

Lesson Summary

In summary, the commutative property states that order doesn't matter when you are performing only addition or only multiplication. a + b = b + a and ab = ba. Your terms can commute around without changing the result.

The associative property states that the way you group numbers when you add or multiply doesn't matter. (a + b) + c = a + (b + c) and a(bc) = (ab)c. Your terms can associate with whomever they'd like.