 The Rhine papyrus begins with a lot of division problems, and they're not very interesting because they're all of the form 2 divided by something. So we have 2 divided by 3 is 2 thirds. Remember this 3 with the double overline is how we're representing the Egyptian symbol for 2 thirds. 2 divided by 5 is a third and a fifteenth. And again the line over the number is our representation for how the Egyptians wrote this unit fraction. 2 divided by 7 is a fourth and a twenty-eighth and so on. Now it's possible these are exercises given by a particularly uninspiring arithmetic teacher, but maybe there's a deeper reason. So let's think about why the Rhine would begin with these division problems. This first part is known as the recto and it can be understood using two ideas. First, whenever we look at the quotient like A divided by B, that's the same as the fraction A beats. In other words, every division corresponds to a fraction. And the second important idea is the fraction A beats is shorthand for saying A of the unit fraction, one beat. So for example, if I want to talk about 2 divided by 5 in modern terms, that's really the same as saying two-fifths and the fraction two-fifths in modern terms, well that's really the same as saying two of one-fifth. And so since the recto tells us things like 2 divided by 5 is a third and a fifteenth, this means that 2 of the fifths is a third and a fifteenth. And what this means is that the recto of the Rhine gives us the doubles of unit fractions. So why is this useful? Let's consider the division 10 divided by 7. So we want to know how many 7s we need to make up 10. So 1 7 is 7. I don't need to double this because that'll get me too much. But since we don't have 10, we'll begin by finding the aliquot parts of 7. And so we find 1 7th of 7, but there's no other aliquot parts. Well remember that in our table of multiplication or division, we form additional entries by doubling both numbers. So it's easy to double 1, we know what that is, but we need to know what the double of 1 7th is. And here the important thing to remember is that the only fractions the Egyptians wrote were unit fractions, 1 nth and 2 3rds, and they literally could not write the double of 1 7th even though they knew it was 2 7th. There was no way to write this quantity. So what did they do? So remember from the Rhined computations we know that 2 divided by 7 is a 4th and a 28th. But modern terms, we know that 2 divided by 7 is the same thing as 2 7ths, which is also the same as 2 of a 7th. And so that means when we double the 7th, when we get 2 of the 7ths, we actually get a 4th and a 28th. And since I want to make 10, I need this piece, which gets me 7, this piece, which gets me 1 more, and this last piece, which gets me the final 2. And putting these pieces all together, I have 1 and a 7th, and a 4th, and a 28th. Now in a kind and gentle universe, you'd always be given the relevant doubles. OK, I'll go with it. Let's say we want to find 18 divided by 11. And let's say we're also given the fact that 2 divided by 11 is a 6th and a 66th. So 11 is 11. The only adequate part of 11 is the 11th itself. So an 11th of 11 is 1. And now I'll double that. So we need to know what the double of an 11th is. So from the recto, or the given information, we'd know that 2 of the 11ths is a 6th and a 66th. Well, let's say we have 11, 12, 13, 14. We don't have enough yet, but we can double again. Now, that does require us to know what the double of a 6th and a 66th is. And we're not given that information. But remember, a 6th is 1 6th. So two of them is 2 6th, which is the same as 1 3rd. And we can write that as a 3 with a line over it. Similarly, this 66, well, if I take 2 of the 66, well, not in terms, that's really 2 66, which reduces down to 1 33rd, a 33 with a line over it. And so if I want to make 18, I need this piece, this piece, and this piece, and this piece. I need all the pieces we have present. And now I can just list what we have. At the order we list, it doesn't really matter. We could call this 1 and an 11th and a 6th and a 66th and a 3rd and a 33rd. But let's try to put them in some sort of order. So let's see, here's 1 and a 3rd and a 6th and an 11th and a 33rd and a 66th.