 A key concept, arithmetic, is bookkeeping. You're tracking how many of what unit? And as long as we have symbols for the how many, and some way to represent the unit, we can record any amount we want. If we have a large number of objects, we can bundle them in sets, then trade for a larger unit. To avoid the problem of naming the larger units, we'll just refer to them as small, medium, large, extra large. And if we need additional names, we'll use them, but we probably won't be dealing with amounts that large. So let's express the amount shown in base five. In base five, any set of five can be traded for one of the next larger unit. So we'll identify a couple of sets of five. So here's a set of five, and we'll bundle. Here's another set of five, we'll bundle. And well, we don't have quite enough left here to make another set. Now we can trade our sets of five for one of the next larger unit. So let's make those trades. And so we see that all together, we have two medium and three small. Now as long as we explicitly indicate the units and the base, order doesn't matter. Here, we have two medium and three small, and that's writing it in base five. We could also describe this as three small and two medium, base five. Or we could be creative and say something like base five, three small, two medium. But that's a lot of writing, and you might wonder if there's an easier way of doing this. And in fact, we can write fewer words at a price. And that price is we have to be a little bit organized. We must choose an order to list the units and we must always use that order. For example, if we wanted to start with the largest units first, we have two medium and three small. We can just record two and three. Now to avoid confusion, we should also record what base we're working with. And so we'll spell that out and subscript it. Also, to avoid confusion, we should really read this as two, three, base five. So let's express the amount shown in base three, four, and five. In base three, we begin by bundling sets of three. And so we have, well, actually we have more than three of these sets of three. And remember, in base n, n of any unit form one of the next larger unit. And since we have more than threes of the sets of three, we can bundle to form a larger unit. So arithmetic is bookkeeping, and what we have is one large, one medium, and two small. And so we can record our amount as one, one, two, base three. Now if we want to express our amount in base four, we'll break our collections apart. And in base four, we begin by bundling sets of four. And again, arithmetic is bookkeeping, and so we have three medium, two small, and the amount is three, two, base four. Again, if we want to express the amount in base five, we'll break our collections apart. And in base five, we begin by bundling sets of five. So arithmetic is bookkeeping, and we have two medium, four small, and so the amount shown is two, four, base five. Now because we know, so to speak, how to write numbers, sometimes we'll write down incorrect amounts. So let's see if we can fix something. Suppose we write down one, five, base four. Let's write the amount correctly. So arithmetic is bookkeeping how many of which units? And what we've written appears to say that we have one medium and five small. Let's go ahead and draw that amount. But since this is base four, we're supposed to bundle sets of four. So here's a set of four we can bundle. And so we actually have two medium and one small. And the number should be written as two one, base four. One final idea, as long as we explicitly indicate our units, we have no problem recording amounts. Four large, two medium, or four medium, two small, or four large, two small. And these are all obviously different from each other. But what if we omit the units? The problem is that if we do that, all of these become four, two, and then whatever our base is. And at that point, how could we tell them apart? So it helps to be organized. In a place value chart, we sort the amounts into each category, then record how many of which units, including when we have none of a unit. So here in this first row, we have four of these things, two of these things, and zero of these. So we'll record those amounts four, two, zero. In the second row, we have four of these and two of these. So we'll write those amounts down. And we might indicate that we have zero of these, but we usually omit leading zeros. And at the last row, we have four of these and two of these. So we record those amounts and zero of these amounts in the middle. So let's say we want to express the amount shown in base four. So bundling and trading sets of four. And we have another set of four we can trade. We can organize these on our place value chart. And so bundling and trading, we see that we eventually have one large, zero, medium and three small. And so we write this as one, zero, three, base four.