Properties of whole numbers
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We want to figure out how many balloons we have here. And obviously, we could just count these. But now, we have other ways of thinking about it, especially because they're arranged in this nice array, this nice grid pattern here. And the reason why it's useful to not just always have to count it, but to be able to use little multiplication with the number of rows and the number of columns is that you might run into things and you will run into things where it's very hard to count each of the objects individually, but it might be a little bit easier to count the rows and to count the columns. So for example, right over here, we see that we have 1, 2, 3, 4 rows. And we have 1, 2, 3, 4, 5, 6, 7 columns. So you could view this 4 as an array of objects where we have 4 rows. Let me write that down. We have 4 rows and we have 7 columns. And you might already remember that we can calculate the total number of objects by multiplying the rows times the columns-- 4 rows times 7 columns. Now, why does this work? Why will this give us the actual number of objects? Well, we could view this-- we have 4 rows. So we have 4 groups of things. And how many things are in each of those rows? Well, the number of columns. We have 7 things in each of those 4 rows-- so 4 groups of 7. Or you could view it the other way around. You could view that each column is a group. So then you have 7 groups. And how many objects do you have in each? Well, that's what the rows tell you. You have 4 things in each of those columns. And we already know that both of these quantities are going to find the exact same number, the number of things that we have right over here. So these two things are equivalent. 4 times 7 is equal to 7 times 4. And there's a bunch of ways that we can calculate either one of these. We can skip count by 4. We say 4 times 4, 8, 12, 16, 20, 24, 28. And let's see. Let me make sure that that's 7. So 4 times 1, 2, 3, 4, 5, 6, 7-- so we get 28. We could just calculate that there are 28 objects here. And likewise, we could skip count by 7. So we could go 7-- 7 times 2 is 14, times 3 is 21, times 4 is 28, just adding seven every time. And so we could get 28 the other way. Let's do that in that same color. We could get 28 the other way. But if you had a situation where you didn't know-- where you either didn't want to do these techniques or it had been hard to do these techniques or you didn't know what 4 times 7 was off the top of your head, which you should know at some point in the very, very near future. Is there any way to break this down to something that maybe you do know or maybe that's a little bit easier to compute? Well, you could realize that 7 columns is the same thing as 5 columns and then 2 columns. So you could view 7 columns as 5 columns-- so this is 5 columns right over here-- plus 2 columns. So that's just like saying that 4 times 7 is the same thing as 4 times 5 plus 2. And all I do is I replace the seven with a 5 plus 2. 7 has been replaced with a 5 plus 2. Now, why is this interesting? Well, now, I can break this up into two separate arrays. So I could say, well, look, there's the array that has 4 rows and 2 columns right over here. And then there's the array that has 4 rows and 5 columns right over here. So how many objects are in this one, in the yellow one right over here? Well, there's 4 times 5 objects. So there's 4 times 5 objects in the yellow grid or yellow array. And how many in this orange-ish looking thing? Well, there's going to be 4 times 2. And if we take the sum of the 4 times the 5 and the 4 times the 2, what are we going to get? Well, we're going to get the 4 times the 7. We're going to get the 4 times the 5 plus 2. So if we take sum of these things-- and we want to do the multiplication first, so I'll just put a parentheses around that to emphasize that-- this is going to be the same thing as these things up here. And so you might say, oh, well, I know what 4 times 5 is. 4 times 5 is 20. 4 times 2 is 8. 20 plus 8 is 28. And you might say, OK, Sal, I get it. 4 times 7 is 28, which is the same thing as 4 times 5 plus 2. And I see that that's the same thing as 4 times 5 plus 4 times 2. And actually, this is called the distributive property-- that 4 times 5 plus 2 is the same thing as 4 times 5 plus 4 times 2. But I could just do one of these first techniques you talked about. Why is this distributive property that you just showed me useful for computing or doing multiplication problems? Well, let me give you a slightly more difficult one. Let's imagine you wanted to multiply 6 times 36. Actually, I don't need to write that parentheses. So how could you do this? Well, you could decompose 36 into two products or into two numbers where it's easier to find the product of that and 6. So for example 36, is the same thing as 30 plus 6. So this is going to be equal to 6 times 30 plus 6. And what's this going to be? Well, we just saw. 6 times these two things added together first, this is going to be equal to 6 times 30 plus 6 times 6. Notice, we distributed the 6-- 6 times 30 plus 6 times 6. Now, why is this useful? Why was this useful at all? I'm going to put parentheses to emphasize. We're going to do the multiplication first. In general, when you see multiplication and addition in a row like this, or division, you want to do your multiplication and division first, then do your addition and subtraction. So what's 6 times 30? Well, this is easier to calculate. 6 times 3, we know to be 18. So 6 times 30 is going to be 180. And 6 times 6-- well, we know that's going to be 36. So this is going to be 180 plus 36. And what's that going to be? 180 plus 36-- well, 0 plus 6 is 6. 8 plus 3 is 11. 1 plus 1 is 2. So you just figured out that 6 times 36 is equal to 216. And what we just did with the distributive property, this is actually going to be how you're going to multiply all sorts of larger numbers, way larger than what we just saw. So the distributive property, which hopefully you're pretty convinced by, based on how we broke things up, is a super useful thing as you want to compute larger and larger numbers. And you're going to find it even more useful when you go even further in your mathematical career.