 So division is the last and by far the most complicated of our elementary operations. So we'll start out by introducing a definition, and one of the things that makes division complicated is that there's actually two slightly different definitions. And the first definition is the following. It emerges from our definition of multiplication, and if I have whole numbers a, b, and c, then I'm going to say a divided by b gives us the quotient c, if and only if a is the product c times b, which is the divisor, the dividend, divided by the divisor is the quotient, and that's going to be the dividend is the product of the quotient and the divisor. And one of the things that also further complicates the problem of division is that this actually gives us two very different approaches to how we actually perform a division. Well, let's start off by what division actually is. So here's a proved statement, proved that 12 divided by 3 is equal to 4. And since we're trying to prove this statement, we want to go back to our definition of division. a divided by b is equal to c, if and only if a is c multiplied by b. So comparing what we want to prove with what our definition is tells us that a is 12, b is 3, and c is 4. So I have my a, b, and c value. And since I want to prove this, that means I need to start with this statement a equals c times b. Well, I have a, b, and c, and I'll fill it out. And again, because it's part of the definition of division, I don't really need to comment on whether how we are able to find this. You may want to actually verify that it is actually true. And so you might want to think about it. Is it true that 12 is equal to 4 times 3? Yeah, that sounds right. So I can start there. I'm starting with the true statement. And I can just follow my definition of division. Since I have a product, I also have the corresponding quotients. It's 12 equals 4 times 3. That tells me that 12 divided by 3 is equal to 4. Now, because division is defined in terms of multiplication, we can find an interpretation for any quotient that's based on what a division is. Actually, because multiplication is commutative, there's actually two different interpretations for any given quotient. And my example, 12 divided by 3 equals 4. Well, from this we have our multiplication fact that corresponds to 12 equals 4 times 3. And from the definition of multiplication, that means that 12 is equal to 3 plus 3 plus 3 plus 3. And what we see here is that I'm going to write 12 as the sum of a bunch of 3s, in fact, specifically 4 3s. And so the quotient 4 is the number of 3s I need to make 12. So this gives us what's called the quotitive interpretation of division. If I want to look at A divided by B, what I'm really asking to do is I'm going to break my first number A into bits of size B. And the quotient is the number of bits that I need. So here, 12 broken into parts of size 3. I need 1, 2, 3, 4 such parts. And so 12 broken into parts of size 3 gives me a quotient of 4. Let's see how that works. Show as quotient of division, we want to use a concrete representation 12 divided by 3. And so by our quotitive interpretation, 12 divided by 3 means we're going to break 12 into parts with a size, a quota, in other words, of 3, and the quotient is going to correspond to the number of parts that I have. So let's start with the set of size 12, and I want to use a concrete representation, so I'm actually going to show 12 pieces there. So I have my set of size 12, and I'm going to break this into pieces with 3 of these objects in each piece. So I'll form sets of size 3. So here's a set, here's another set, and another, and another, and the quotient is going to be the number of sets that I have. So here I have 1, 2, 3, 4 sets of size 3. And so that tells me that my quotient, 12 divided by 3, is equal to 4. Now because multiplication is commutative, we can actually get a second interpretation of a division. So again, we have our division 12 divided by 3 equals 4, and again, by definition, I know that this must have come from the state of 12 equals 4 times 3. Because multiplication is commutative, I can reorder the factors 3 times 4, and by the definition of multiplication, 3 times 4 is the sum of 3 4s. And the thing to notice here is I've now written 12 as the sum of 3 equal numbers, 4, 4, and 4. And this leads to a new interpretation of division, known as the partitive interpretation of division. And the idea behind the partitive interpretation is I'm going to treat a division like this as being a way of taking my original number, and I'm going to break it into this many equal pieces, and the quotient is the size of each piece. Now, it's worth keeping in mind that when we describe division, we often begin with this notion of a partitive interpretation of the division. But in practice, as we'll see, we actually abandoned the partitive interpretation because it doesn't give us a good way of doing the division. How do we know that? Well, let's try an example. So let's show a partitive division using a concrete representation. Again, same quotient we've been using, 12 divided by 3. And so as before, my partitive interpretation says that I'm going to take 12, I'm going to break it into three equal parts, and the actual quotient is going to be the size of each of those parts. So again, I'll begin with a set of 12, and I want to break this into three equal parts. The problem we have right from the beginning is I don't know how big those parts are. So I can't just draw lines that's going to break this into three equal parts because I don't know how big the parts are. With a quotitive division, I knew how big each part was. If I interpreted this quotitively, I know that I'm going to be taking 12, I'm going to be breaking it into parts of size 3 and see where I go. But with the partitive division, I don't know how big the parts are. So in some sense, what I have to do is I set up some spaces for them. And so here, I'm going to make sure that what I put into each of these three spaces is going to be equal in amount. And so I can now start to set up my partitive division. And because I have three boxes, three bins, I'm going to be putting these in, and I want to make sure that the number in each of these bins is the same, what I might do is I might start off with a set of three. Note that we have a quotitive aspect to this. And if I take this set of three, I know I could put one of these in each of my places. So this one is going to go here, this one is going to go there, this one is going to go there. And I'm going to have the right amount, an equal amount in each of those three places. Well, I'm not done. I have not finished dividing 12. So I'm going to select another set of three. And again, I'm going to drop one in each of my places. Another set of three, drop one in each place. Another set of three, drop one in each place. And now I have done my division. I have taken my set of 12 and broken it into one, two, three equal sets. And the quotient is going to be the size of each of those sets. So here I notice that every one of these has one, two, three, four. Every one of these has four things in it. So 12 divided by three is equal to four.