 The Mesopotamian division relied on multiplying by the reciprocal, and the Mesopotamian scribes had tables of reciprocals at their disposal. But how were these reciprocals found? There were two common methods. We can describe these as either using the reciprocal relationship or by using the trailing heart algorithm. So let's start off with the reciprocal relationship. If A times B is equal to 1, then A and B are reciprocals. And this means that if we can find two numbers that multiply to 1, they are reciprocals. But in base 60, 1 is also equal to 60 sixtieth, and we can move the sex decimal point around. So how does this help us? Well, let's find two numbers that multiply to 60 and then use our values to find the reciprocals in base 60. So let's pick two numbers that multiply to 60, about 5 and 12. Now, remember that 60 is equal to 1, 0 in base 60, and we want the product to be 1, so we need to move the sex decimal place in 1 or both of the factors. And there are two ways of doing this. We can move the sex decimal place 1 space over in the 5, or we can move the sex decimal place 1 place over in the 12. And so the reciprocal of 12 is 5 sixtieth, while the reciprocal of 5 is 12 sixtieth. And it's worth remembering that in mesopotamian numeration, the sex decimal point is never indicated. And what this means is that if you were a mesopotamian scribe, the reciprocal was just 5 or 12. We can extend this a little bit further. Suppose the product of two numbers is 60. Then if we multiply a by n and divide b by n, the product will still be 60. And since we can shift the sex decimal point, we can focus on the actual digits of the numbers. So we might begin with the fact that 4 times 15 is equal to 60, and then we can find other reciprocal pairs. So 4 times 15 equals 60. Now this is going to require us to divide by something. And so what can we divide by without having a reciprocal table? Well, we could probably divide by 2. So if we divide the first factor by 2, then multiply the second factor by 2, the product is still 60. And so from the fact that 4 and 15 are reciprocals, we also find that 2 and 30 are also reciprocals. We can use this approach to find specific reciprocals. So let's say we want to find the reciprocal of 8. Since we'll either be having or doubling a number, we want to find two reciprocals where we can have or double one of the factors to get 8. Well, let's start with 4 times 15. So if we double our first factor, we get 8. Now we have to take half of 15. And so to find half of 15, we could think about 15 as 14 and 60, 60ths. If you want to think about this as time, 15 minutes is 14 minutes and 60 seconds. So that means half of 15 will be half of 14, 60, which is just going to be 7, 30. And if we are worried about where the sex decimal place is located, we can shift the sex decimal place around. The product is 60, which is the same as 1, 0. If we want that to be 1, we have to shift the sex decimal place 1 space. And since we are trying to find the reciprocal of 8, we can't shift the sex decimal place on 8. We have to shift it on 7, 30. And so that gets us the reciprocal of 8. Another method of finding reciprocals is known as the trailing part algorithm, and it works as follows. To find the reciprocal of a number n, find a sequence of multipliers that make the last digits 0 a multiple of 60. If the product of all of these factors is 1, or 1, 0, 0, and so on, the reciprocal is the product of the multipliers. So, for example, let's try to find the reciprocal of 50. We'll begin by choosing any multiple of 50 that gives a multiple of 60. So we'll find 50 times 2, 50 times 3, and so on. And we find that 50 times 6 gives us 300, which is a multiple of 60. So let's multiply 50 by 6, and we'll trade and convert this into a number in base 60, 50. Now, note that the last non-zero digit is 5, so we want to find a multiple of 5 that gives a multiple of 60. And we note 5 times 12 is 60, and so multiplying by 12 gives us, and since our product is 1, 0, 0, that means the reciprocal of 50 is going to be 6 times 12, 72, or as a number in base 60, 1, 12. And there's a sexy decimal place located somewhere in here, but remember in Mesopotamia Numeration, the sexy decimal point is never indicated. Or let's try to find the reciprocal of 9, so we want to find a multiple of 9 that gives us a multiple of 60, and so we find 9 times 20 gives us 180, which we convert into 30. Now, our last non-zero digit is 3, so we want to find a multiple of 3 that's a multiple of 60. 20 works, and so we multiply and trade. And since the product is 1, 0, 0, our reciprocal is the product of our two multipliers, 20 times 20, 400, which we can convert into 640.