Associative property of multiplication

• [Instructor] So, what we're gonna do is get a little bit of practicing multiple numbers together, and we're gonna discover some things.

So, first I want you to figure out what four times five times two is. Pause the video and try to figure it out on your own.

Alright, so whatever your answer is, some of you might have done it this way, some of you might have said, "Hey, what is four times five?" and then you multiplied it by two. So, what you would really have done is you would have done four times five first, so that's why I put parentheses around that, and then you would have multiplied by two.

And what would you have gotten? Well, the four times five part, that is of course 20, and then you multiply that times two, and you would get 40, which of course would be correct. Four times five times two is indeed equal to 40.

Now, what I want you to do now is as quickly as possible try to figure out what five times two times four is. Really quick, pause the video, try to figure that out.

Well, some of you might have tried, and you might have done it in a similar way where you tried to figure out five times two first, and you said, "Okay, five times two is equal to 10," and then I'd multiply that times four, and then you would say, "Well, gee, this is the same thing as I got last time. Is there something interesting going on?"

And the interesting thing that you might realize is in both cases, we're multiplying the same three numbers. We are just doing it in a different order. Here we multiplied four times, we wrote it out in a different order, four times five times two. Here we wrote five times two times four.

Here we did the four times five first, here we did the five times two first, but notice we got the same result. Now, I'd encourage you, pause this video. Try to multiply these numbers in any order. Maybe you do two times four first.

In fact, let's just do that. Let's do two times four, two times four, and then multiply that by five. What is this going to be equal to? Well, you might notice, again, this is two times four is eight; you multiply that times five. Well, once again, we got 40, so you might see a pattern here.

It doesn't matter which order we multiply these things in. In fact, you could write four times five times two. You could do the four times five first, four times five times two, or you could do four times five times two, so you could do four times five times two.

So, it doesn't matter which order you multiply these things in. In every case, you are going to get 40. Now, there's a very fancy term for this, the associative property of multiplication, but the main realization is—and it's not just true with the three numbers—in fact, you've seen something similar with two numbers where it doesn't matter what order you multiply them in.

But what you see with three numbers, and even if you tried it with four or five or really 1,000 numbers being multiplied together, as long as you're just multiplying them all, it doesn't matter what order you're doing it with. It doesn't matter in what order you associate them with.

Here we did four times five first, four times five first; here we did five times two first, but in either case, we got the same result. And I'd encourage you, after this video, try to draw it out. Try to think about why that actually makes intuitive sense, why this is true in the world, and it's nice because it simplifies our life when we're doing mathematics and not only now but in our future mathematical career.