Introduction

In this lesson, we're going to learn about the commutative and associative properties. Now, in math, whenever you hear the word property, just think of it as a rule that tells us what we are allowed to do or not do when we're doing math problems. With these two properties, they apply to all real numbers. So it doesn't matter if the numbers are fractions, decimals integers, positive or negative numbers - it'll work for all real numbers. And they only will work for addition and multiplication problems. These properties will not work for subtraction or division. So that's just something to keep in mind.

Commutative Properties

First, let's look at the commutative properties. The commutative properties tell us that when we're adding or multiplying numbers together, we can do it in any order. We’ll, first look at addition. Say if we want to add the numbers two and five together, we could write it as two plus five. Or we can switch the order around and write it as five plus two. Two plus five will give us seven. Or when we switch it around as five plus two, we still get seven. So it gives us the same answer no matter what order we add the numbers in. And of course, this doesn't just work for the numbers two and five, it works for all real numbers. In algebra, whenever we want to show that something works for all real numbers, we use variables. So we can use the variables A and B to show this rule instead of just using the numbers two and five. So we can say that A plus B is equal to or the same as B plus A.

Now let's look at what happens with multiplication. If we want to find the product of three and four, we can write it as three times four or four times three - we can switch the order around. Three times four will give us twelve, and four times three will still give us twelve. So we can multiply in any order. And then of course, we can represent this rule with the variables A and B instead of just the numbers three and four. And it would look like this - A times B is equal to B times A.

Associative Properties

Next, we'll look at the associative properties. The associative properties tell us that when we’re adding or multiplying a bunch of numbers together, we can start anywhere. Our first example will be for additions. If we want to find the total sum of the numbers, five, seven, and ten, we can start by adding the five and seven first, or we can add the seven and ten first. It doesn’t matter where we start, we’ll still get the same answer in the end. Let’s take a look. So here, I wrote that problem over again, but with parentheses around the five plus seven. Remember, in math, parentheses can tell us where we’re starting. So it just kind of tells us what part of the problem we're beginning with. So if we start with the five plus seven, we know that that will give us twelve, and then we can come back and add the ten to that, which will give us 22. Or if we want to start by adding the seven and ten first, we can write parentheses around the seven plus ten just so we can see where we're starting. Seven plus ten will give us 17. Now we have five added to 17, which still gives us the same total of 22. So when we're adding a bunch of numbers together, it doesn't matter which numbers you start with, as long as the end you end up adding everything together, you will get the same answer. And we can represent this rule using the variables A, B and C to show, just like what we did with the numbers five, seven and ten, that we can start with the first two numbers or the last two numbers. We can start anywhere and we'll still get the same total.

This also works with multiplication. If we're multiplying a bunch of numbers together, doesn't matter where we start, we'll get the same answer in the end. So let's see what happens when we get the product of two, three and four. If we start by multiplying the two and three together, two times three will give us six, and then we can come back and multiply the four in and we'll get 24. Or we can start with three times four, which will give us twelve, and then multiply the two times that two times twelve gives us 24. We still get the same answer. And just like before, we can represent that rule with the variables A, B and C showing that we can multiply starting with any of the two numbers, and we'll still get the same answer either way.

Summary

So just to summarize, the commutative and associative properties apply to all real numbers and they only work for addition and multiplication when we're adding numbers together, or if we're multiplying numbers together, these two properties, these two rules will work. So the commutative properties tell us that we can add or multiply in any order. The associative properties tell us that we can start anywhere. When we're multiplying or adding a bunch of numbers together, we'll still get the same answer. So now you know how to apply the commutative and associative properties properties for addition and multiplication.