 We have very few examples of Mesopotamian multiplication. Most of them are multiply and get the answer. So here a scribe might have multiplied 4, 10, 50 by 3 and gotten the answer 12, 32, 30. They probably did something like this. So again, it's helpful to think about 4, 10, 50 as a time like 4 hours, 10 minutes, 50 seconds, and we're multiplying this by 3. So 4 hours times 3 is 12 hours, 10 minutes times 3 is 30 minutes, and 50 seconds times 3 is 150 seconds. And now let's trade. 150 seconds, well our trade rate is 60 seconds will give us a minute. So instead of 150 seconds, well that's 60, 60, and 30 seconds. But the 60 seconds, they're both 1 minutes. And arithmetic is bookkeeping. Where we had 150 seconds, we now have 2 minutes, 30 seconds. So we'll trade out that 150 seconds for 2 minutes, 30 seconds. Arithmetic is bookkeeping. So we have 12 hours, 30 minutes and 2 minutes, or 32 minutes, and 30 seconds. Which gives us our final answer, 12, 32, 30. Now on division, we're on slightly more solid ground in that we actually know how the Mesopotamians did division. And to approach Mesopotamian division, it's useful to remember the following idea. A divided by B, well that's equivalent to the fraction A over B, which is equivalent to the product A times 1 over B. And this means we can divide by multiplying by the reciprocal of the divisor. Now to see how this works, let's consider how that plays out in base 10. Suppose we want to divide by 25. So dividing by 25 is the same as multiplying by 1, 25th. But 1, 25th is 0.04. So instead of dividing by 25, we can multiply by 0.04. And what makes this useful is to multiply by 0.04, we can just multiply by 4, and then move the decimal point to places to the left. So let's say I want to divide by multiplying by the reciprocal, 853 divided by 25. So 853 divided by 25 is the same thing as 853 times 0.04. And that's really 853 times 4 is 3, 4, 1, 2. And now we have to figure out where that decimal point is. Since we have two decimals in our factor, we'll move the decimal point to places to the left. And it ends up here. We have to guess the answer, 34.12. Now to find the reciprocal in base 10, we need to rewrite 1 over b as a fraction with a denominator of 10 to the n. So for example, 1 half is 5 tenths. 1 fourth is 25 one hundredths. 1 25th is 4 one hundredths. And well, the problem is, unfortunately, very few divisors can be written this way. And here's the value of base 60. To divide in base 60 this way, we want to rewrite 1 over b as a fraction with a denominator of 60 to the n. And this could be done with many divisors. It's helpful to think of time. 1 half of an hour is 30 minutes. And so 1 half is 30 sixtieth. 1 third of an hour is 20 minutes. And so 1 third is 20 sixtieth. And similarly, 1 fourth, 1 fifth, 1 sixth, these are all whole number of minutes, which means 1 fourth, 1 fifth, and 1 sixth can be written as fractions with a denominator of 60. So we do need to introduce one more idea. In base 10, we distinguish between units and fractions of a unit using a decimal point. So when we write one two, well, that's one 10 and two ones. But when we write 1.2, that's one one and two tenths. Now in modern base 60, we use the sexogesimal semicolon. So one comma 10, well, that's one 60 and 10 ones. But if we write one semicolon 10, we read that as one one and 10 sixtieths. And so we'd write our reciprocals, one sixtieth, well that's zero semicolon, zero one. 15 sixtieths, well we have 15 sixtieths. It's going to be after the semicolon and so on. So for example, what would we multiply by in base 60 to divide by 12? So again, it's helpful to think of time. One twelfth of an hour is five minutes. And so one twelfth is five sixtieths. So to divide by 12, we'd multiply by zero semicolon, zero five. So let's say we want to divide 12 20 by 12. So to divide by 12, we multiply by zero zero five. So first, we'll multiply by five. So that's 12 20 times five. Well that's five times 12 is 60. Five times 20 is 100. We trade 100 seconds is one minute 40 seconds. So the 100 is gone and we now have 140. Arithmetic is bookkeeping. So now we have 61 40. We trade again 61 minutes. Well that's really one hour, one minute. And so all together we have one, one, 40. And since we're actually multiplying by zero semicolon, oh five, we move the sex decimal point one place over. So our final answer is one, one, semicolon, 40. Or let's divide 11 27 semicolon 20 by 15. So we note that one fifteenth of an hour is four minutes. So the reciprocal of 15 is zero semicolon zero four. So we divide by 15 by multiplying by zero semicolon zero four. And so we'll ignore the semicolon for a moment. We'll multiply by four. Four times 11 is 44. Four times 27 is 108. And four times 20 is 80. Now this is our provisional answer, but we have some bundling and trading we can do. So let's take a look at this rightmost value that's 80. Well it's helpful to think about that as 80 seconds, which is the same as one minute 20 seconds. So we bundle and trade that 80 seconds is gone and we've replaced it with one minute 20 seconds. Arithmetic is bookkeeping. We now have 44, 109, 20. And we can bundle and trade again, 109 minutes. Well that's really one hour, 49 minutes. So that 109 is gone. We've replaced it with 149. Arithmetic is bookkeeping. We now have 45, 49, 20. Now since we're actually multiplying by zero semicolon zero four, we need to move these hexagasmal point one place further. So our final answer, 45, semicolon 49, 20.