 While we know how the Mesopotamians wrote numbers and multiplied and divided, we don't actually know how they added or subtracted, but it's useful to remember the important idea, arithmetic is bookkeeping. How much of what units? So let's say we want to add 1, 25, 40, and 4, 10. Now, it's useful to keep the following idea in mind. When working base 60, it's helpful. To think about times in hours, minutes, and seconds. And again, remember the base is independent of the item counted. A number in base 60 could be a length, an area, a number of people, or an amount of money. So while we might think about these numbers as time, they don't necessarily represent times. And so if we think about this as 1, 2, 40, being really 1 hour, 2 minutes, 40 seconds, and 4, 10, well, that's really 4 minutes, 10 seconds. And so if I want to add 1, 2, 40, and 4, 10, it's like adding 1 hour, 2 minutes, 40 seconds, and 4 minutes, 10 seconds. Arithmetic is bookkeeping. And so here we see here we have 40 seconds and 10 seconds. Well, that's really 50 seconds. We have 2 minutes and 4 minutes. That's 6 minutes. And we have 1 hour and, well, no hours. So that's 1 hour. And so if we add our times together, it's 1 hour, 6 minutes, 50 seconds. Now, while we might think about these as hours, minutes, and seconds, they're just numbers. And so they don't actually have these units there. And so we should record our final answer as a base 60 number, 1, 6, 50. How about a subtraction? Let's subtract 1, 5, 20 from 5, 10, 50. So again, let's think about this. We have 5, 10, 50. That's 5 hours, 10 minutes, 50 seconds. And we want to subtract 1, 5, 20, 1 hour, 5 minutes, 20 seconds. Arithmetic is bookkeeping. So let's see. We have 5 hours, we're subtracting 1 hour, and so that leaves us with 4 hours. Here, we have a 10 and a 5, and so we'll add them to get 15. Wait, no, we want to subtract. So that's 10 minutes minus 5 minutes, and that's 5 minutes. And this 50 seconds and this 20 seconds, we subtract to get 30 seconds. And again, we don't want to write our answer in hours, minutes, and seconds because that's not part of the original problem. We want to write our answer in base 60 by recording how many of which units. So that's 4, 5, 30. So at this point, the following math joke is obligatory. One day, your trip to school took 90 minutes. On another day, it took 1 hour and 30 minutes. Why was it different? And in fact, 90 minutes is the same as 1 hour, 30 minutes, so there was no difference. And we say that we bundled 60 minutes and trade it for 1 hour. So why is that admittedly lame math joke important? Well, let's consider the following. Let's add 30, 40, and 45, 50. So again, thinking about these as times, 30, 40 is really 30 minutes, 40 seconds. 45, 50 is really 45 minutes, 50 seconds, and we're adding. So we'll just add these normally, and that gets us 75 minutes and 90 seconds. The thing to recognize here is that because we're working base 60, any group of 60 can be traded. Here, it's helpful to have those hours, minutes, and seconds written down. And so we note here that we can trade 90 seconds. Well, that's really the same as 1 minute and 30 seconds. Remember, arithmetic is bookkeeping. If I'm trading this 90 seconds for 1 minute and 30 seconds, I have to give up that 90 seconds. It's gone. So let's cross it out. In exchange, I'm getting 1 minute, 30 seconds. Arithmetic is bookkeeping. I have 75 minutes and 1 minute, that's 76 minutes, and 30 seconds. Which we write down as 30 seconds. But wait, there's more. This 76 minutes we can trade, 76 minutes is an hour and 16 minutes. Again, let's trade it. We'll get rid of the 76 minutes and then write in 1 hour, 16 minutes. Arithmetic is bookkeeping, and what I have is 1 hour, 16 minutes, 30 seconds, or writing this in base 16 notation, 1, 16, 30. Well, let's do a subtraction. So let's subtract 4, 0, minus 1, 20. So let's write that down. And we note that we can subtract 1 hour from 4 hours. But this says we're supposed to subtract 20 minutes from 0 minutes, and we can't do that. So what we do next has often been referred to as borrowing. But how you speak influences how you think. Borrowing implies returning. Trading, on the other hand, implies giving something up in exchange for something else. So we're not borrowing, we're actually going to trade. So what shall we trade? Well, we need more minutes. So we can trade 1 hour for 16 minutes. So remember, arithmetic is bookkeeping. How much of what units? We have 4 hours. So if I trade 1 hour, I'll still have 3 hours. And the hour I've traded gives me 60 minutes. So we no longer have 4 hours, 0 minutes. We actually have 3 hours, 60 minutes. Arithmetic is bookkeeping. And now we can do the subtraction. 3 hours, subtract 1 hour, leaves me with 2 hours. 60 minutes, minus 20 minutes, leaves me with 40 minutes. And as a Bay 16 number, that's 2 comma 40.