Learn about Properties of Real Numbers In this lesson, let's remember about the associative, commutative and distributive properties when it comes to addition and multiplication. So we are given a few problems and we have to name the property that is illustrated in each equation. This is the first equation. What property is being used here? As we can see, what we’re seeing is first we had 4+x grouped together and then added to y. In the second side, we have 4+x and y added and grouped together. All we've done, the values in expressions, variables and values are the same, 4xy, 4xy. It’s just that the grouping is changed. This is called the associative property of addition, which basically says a + b + c, it doesn’t matter how you group them. If you're adding, the grouping really doesn’t matter. I can add a + b, same as b + a, it makes no difference. This is the associative property. The same thing applies to multiplication by the way; a x b x c, whether I multiplied b x c first or a x b x c, do it this way. It does not make a difference. That’s called the associative property. If you look here, what we have is -3 x b = b x -3, similar right? But, what we've done is applied the commutative property which basically says that it does not make a difference whether we multiply a x b or b x a. It’s not going to make a difference; a x b = b x a. This is called a commutative property. The reason these are important is because when I'm trying to solve bigger equations, I can use these. Let me give you an example. Let's do the third problem. We need to find the product of 15 and 103 using the distributive property. I'll show you how to use these, associative, commutative and distributive properties to simplify these. If I wanted to multiply these two numbers in my head, it’s pretty difficult. But, here is what I can do. I can use the distributive property which says a x b + c = a x b + a x c. So, instead of writing 15 x 103, instead of 103 I can write 100 + 3. That’s the same as 100+ 3. Distributive property tells me that I can distribute this multiplication here and multiplication here. I can distribute this. This is the same as 15 x 100 + 15 x 3. This is distributive property. 15 x 100 is 1500, 15 x 3 is 45, 1500 + 45 is 1545. Note how we were able to solve this, simplify this without using a calculator. I could do this by using a distributive property. Similarly, we have ways to use the associative and commutative properties, right when we’re solving problems.