 A useful thing to be able to do when dealing with a large number of objects is what's called bundling and trading. This is centered around what we call a place value chart, and it's a useful way of converting a concrete collection of objects to an abstract representation of the number in base n. And so, for example, let's say I have a whole bunch of these things right here, and what I'd like to do is I'd like to express the number of these things in base 3. Now, the hard way to do this is to count normally, so to speak, the number of objects we have here, invoke some complex algebraic formula, and write down what the resulting value is in base 3. And that's fine if you want to spend the time memorizing yet another formula, memorizing yet another process that really doesn't have much conceptual depth to it, and doing a lot of extra work. But let's do it the easy and more natural way of doing this. First of all, it's convenient to set down a place value chart, an empty chart where the set of objects is going to be in our rightmost place. So here I'm going to drop all the objects that we have, and I have three places here because I feel like writing three places. If I turns out that I need more, I'll just add another place over to the left as necessary, but I start off with some number of places. It's helpful but not absolutely necessary to draw a representation of what our units look like. So here in my first column, my units are the single object, and because I'm working in base 3, a set of three objects is going to form the next unit. So I'm going to take this. I'm going to take three of them, and that's going to be what my next unit looks like. And then my next unit is going to be three of these objects, so I'll take three of these, put them together, and that will form my unit in the next place, and it looks something like that. And again, because I've only drawn three places, that's as far as I'm going to go, but we could have larger sets. We could take three of these, bundle them, and put a unit in our next place, but I'm not going to bother with that at this point. All right, well, the next thing we're going to do is because I can take a set of three of these and make it one of the next things over, I'm going to start collecting sets of three. I'm going to start bundling these into sets of three. So there's a set, there's a set, there's a set, and so on. And I have these two left over, and lo and behold, these are in the wrong place. So I don't want to put them in the wrong place, that's disorganized, so I'll move them to where they belong, and now I can repeat the process. So again, three of these will form one of these. So I'll bundle sets of three, here's another set of three, I'll bundle them, and now I have new objects. Again, these are in the wrong place, so I'll move them over to where they belong. And I could take three of these, except I don't have them. So I'm not able to do any more bundling, and everything is in the place where it should be. Now I want to write this as a number expressed in base three. Well, here's what I need for it. Here's one, two of these, there's one of these, and there's one, two of these. And so I can write the numbers for the amounts present, that's going to be two, one, two. And my final expression of the number is going to be two, one, two, with a spell that indicated that I'm working in base three, and there's going to be my final expression, two, one, two, base three.