 So with dispositional notation, we can write down any numerical amount that we want to in any base we want to work with. So by bundling, we found that the number of cubes shown can be expressed in base 4 as one big thing, two medium, three small, or because this is in dispositional notation, because we're explicitly showing what those units are, then it doesn't really matter what order we list them. We could say there's two medium, one large, three small, or three small, two medium, one large. And this has a number of advantages. One, we don't have to worry about the order that we write things down. All three of these say we have one large, two medium, three small, and it's very clear what this means. The disadvantage is that we have to retain these unit symbols. We have to write them down every single time. And so we might ask, is there a more compact way we could write this? And so this leads us to positional notation. So the number of boxes we had in base 4 was, well, let's pick one of these. We'll say it's three small, two medium, one large. And the thing to remember arithmetic is bookkeeping. And we want to keep track of how many of which type of units. So how can we do this in positional form? Well, this is going to take a little bit of organization. And so, first of all, we want to put down the units in order of decreasing size, largest down to smallest. Now, if we were writing our number in dispositional notation, we would then just use our abstract symbols to indicate how many of each type of unit. So of this large thing, we have one. Of this medium, we have two. Of this small thing, we have three. And so at this point, our positional form is really nothing more than our dispositional form just set down in an organized fashion. What takes us to the positional form is two steps. First of all, we're going to drop these unit designations and leave only the abstract symbols. So those unit designations, large, medium, small, they're gone. And now I just have the symbols one, two, three. Now, the problem is that, again, arithmetic is bookkeeping. We want to keep track of how many of which type of unit. When those units were present, we had an idea of how big each of the units were. When they're gone, we need to remind ourselves of how big those units were. And the key here is that we were working in base four. And so that four of each unit made one of the next larger unit. Since arithmetic is bookkeeping, we want to make note of the fact that we were working in base four. And to do that, we'll add in at the very end and subscripted a spelled out number four. And so this tells us that we have one, two, three, base four. And we can read this one large, two medium, three small at a base of four. Now, one final note, it is very, very, very important to remember how you speak influences how you think. You should read this as one, two, three, base four. Because this reminds you we have one of something, two of something, three of something. And we can fill it in later. We have three small, two medium, one large. If you read this as one, two, three, base four, you'll be reminded of that fact and you'll think about this number correctly. Avoid the temptation to read this as something like 123 base four because that means something completely different that has nothing to do with what we have here. You should read this as one, two, three, base four, one of something, two of something, three of something. All right, so let's take a look. So here we have a number of objects and let's see if we can express the number of objects in base three. And so since we're working in base three, we can begin by bundling sets of three. So here's a set, we'll bundle it. Here's another set. And we don't have any more things that we can bundle together as a set of three so we can write down our positional form. Now as a transition we might want to start with our dispositional form so we'll write down our units in decreasing size order. So we have the medium size unit. We have the small unit. We want to record the number of each type. I have two of these mediums and one of these smalls. Now to get to the positional form we'll drop those unit designations and then we'll remind ourselves that we're working in base three. So we'll write down three there. And there's our number in base three. Now let's see if we can do something with a larger set. So here's a larger set and let's go ahead and express the amount in base three again. So as before, since we're working in base three, we'll begin by bundling sets of three objects. So here's a set of three. Another set of three. Another set of three. Another set of three. Another set of three. Another set of three. And I now have these three things. I can bundle them into a single object. Again, still working base three so I can take these three things and bundle them into a single object. And now I've done I can't bundle anything else. So I'll write down our units in decreasing size, I have this big thing, I have this small thing. I'll write down the number of each type, I have two big, one small. I'll drop the unit symbols, and then I'll put a reminder that we're working in base three. And remember arithmetic is bookkeeping, but there's a problem. It is very bad bookkeeping to write two different numbers in the same way. We wrote this set also as two, one, base three. And yet these two sets have very different amounts of them. So there's something going on that we have to fix. And this comes down to the importance of nothing. To avoid the problem of expressing different amounts using the same written form, we have to write down all the units, even if none of that unit is actually present. So for this amount, which we wrote as two large, one small, there is an intermediate unit that we didn't have any of. But we should, if we're going to write our number in positional form, include that intermediate unit. So we'll include the intermediate unit. We'll write down the amounts. We have two large, zero medium, and one small. We'll drop the unit designations and write the number spelled out. And so there's our number here. There's two, zero, one, base three things there. Well, let's try to work in base four. So we have a number of objects. We want to express the number as a number in base four because we're working in base four. We want to look for sets of four objects to bundle. So let's see. Here's a set. Here's another set. Here's another set of four. And another set of four. Oh, and look, I have one, two, three. I have four objects here. I can bundle these into a single set. So I'll write down the units, one big thing. Now there are two smaller units that we actually used, medium and small, so we do have to indicate those. And arithmetic is bookkeeping. We're going to write down how many of each type of unit. So I have one big, zero medium, and zero small. And I'll write in positional form by dropping the unit designations and reminding ourselves that we're working as a number in base four. And again, it is very, very, very, very, very, very, very important to read this not as 100 base four, but this is 100 base four. There's one of something, zero of something, and zero of something else.