 What if our divisions don't come out even? For those divisions, we'll need to introduce fractions. The ancient Egyptians used unit fractions, which correspond to the aliquot parts of the divisor. So in case these terms aren't familiar to you, here's a quick review. A unit fraction emerges as follows. Suppose we take a whole object and divide it into equal parts. For example, we can take a cake and divide it into four equal parts. Four equal parts? A unit fraction corresponds to one of those equal parts. So if we divide a cake into three equal pieces, each of these pieces is a third. Or we could divide the cake into six equal pieces, each of which is going to be a sixth. Or we could divide the cake into four equal parts, each of which is a fourth. Aliquot parts are very closely related. In a cake, we can cut the cake into any number of pieces. But if our quantity consists of a number of objects that can't be broken apart, then the possible divisions correspond to the aliquot parts. So for example, let's say we have ten objects, say ten eggs. If we have ten eggs, we can separate them into two equal parts, each of which is a tooth. Well, okay, this division into two equal parts is common enough so that we actually use the term half. Now we could try to separate the eggs into three equal parts, but we can't. And similarly, if we try to separate them into four equal parts, we can't do that either. But we can separate them into five equal parts, each of which is a fifth. Or we could take our eggs and separate them into ten equal parts, each of which is a tenth. So let's find the aliquot parts of twelve. And so the aliquot parts correspond to the fractions of twelve that give us whole numbers. So we could take half of twelve, that's six, one-third of twelve, that's four. We could take one-fourth of twelve, which will be three, and then we can also take the fractional parts one-sixth and one-twelfth. To indicate these unit fractions, the ancient Egyptians did one of two things. In hieroglyphic, they used a row, which represents an open mouth over the hieroglyphic number. So this represents ten, but if I put a row over it, it represents one-tenth. On the other hand, the more commonly used hieratic had to have a dot over the leading figure. So we have our hieratic twelve. If I want to indicate one-twelfth, I'm going to put a dot over this leading figure. And remember, Egyptian is going to be written from right to left. So the leading figure is actually the ten. Now these hieratic conventions, these are often represented in modern typography by a line over the number. So if I want to write the fraction one-third, how I'm going to write that is I'm going to write down a three with a line over it. So let's go back to those aliquote parts. So we've found the aliquote parts of twelve, or one-half, one-third, one-fourth, one-sixth, and one-twelfth, and so I can write them this way. One-half, two with a line over it, of twelve is six, one-third, three with a line over it, is four, and so on, for the fourths, the sixth, and the twelfths. So let's say I wanted to divide thirty-one by twelve, so we'll form our table of multiples of twelve, one-twelve is twelve, two is twenty-four, four would be too much, so I don't have to go any further, but I don't have enough yet. And so to get the remainder, we'll use the aliquote parts. So remember we found the aliquote parts of twelve, they're half, a third, a fourth, a sixth, and a twelfth. So I need thirty-one, so I need this two of twelve, this twenty-four piece here, and I need seven more, and I can pick up seven more by taking the half, and then one more, the twelfth. And so the pieces that I need to make up my thirty-one are the two of twelve, the half of twelve, and the twelfth of twelve. So I need these three pieces, and by quotient, two, and a half, and a twelfth. Well let's take a look at something like sixty-two divided by eighteen. So we'll find our multiples and aliquote parts of eighteen. So one of the eight-teens is eighteen. We can double it. Two of the eight-teens is thirty-six. If I double this again, I'm going to get seventy-two, which is more than I need. So I'm not going to bother doubling it again. If we're fantastically lucky, we'll have everything we need. So I need to make sixty-two. I have thirty-six and eighteen, which is not enough. It's only fifty-four, so I'm going to need eight more. So let's find the aliquote parts of eighteen. So one easy way to figure out what those aliquote parts are is they correspond to the things that will divide eighteen. So eighteen is even, so I can take half of eighteen, which will be nine. Eighteen is also divisible by three, so I can take one-third of eighteen, which will be six. Eighteen is also divisible by three, so I can take a sixth of eighteen, which gives me three. Eighteen is also divisible by nine, so I can take a ninth of eighteen, which is going to be two. And eighteen is also divisible by itself, so I can also take an eighteenth of eighteen, which is just going to give me one. So I need to make sixty-two. So I'll take this package of two, that gives me thirty-six. I'll take this set of eighteen, that brings us up to fifty-four. The nine is too big. I can take the third, that gets me up to sixty, and I need the ninth for the last two. And so our quotient is going to be three, and a third, and a ninth. Now almost all fractions in Egyptian mathematics were unit fractions. The only common exception was the fraction two-thirds, which had a special symbol. And a little bit later on we'll see the significance of this special symbol. We usually designate this in modern typography as three with two lines over it. Now once I have a third, I can actually find two-thirds by doubling the amount. So two-thirds is going to be ten. Fifteen is also divisible by five, so I can also take a fifth of fifteen, that's three. And then finally a fifteenth of fifteen is one. And since I want to make twenty-eight, I need the package of one, that gets me fifteen. The two-thirds gets me ten more for a total of twenty-five, and the one-fifth brings us up to twenty-eight. And we don't need the other parts, so we'll blur them out. And so our quotient is going to rely on these three rows, and it's going to be one, and two-thirds, and a fifth. Since I let's divide twenty-eight by fifteen, and we'll construct our table of aliquot and two-thirds parts, so one of the fifteens is fifteen. Two is going to be thirty, which is already too much, so I'm not going to bother doubling it. But I do want to find those aliquot parts, so I know fifteen is divisible by three. So a third is five.